diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzhoif" "b/data_all_eng_slimpj/shuffled/split2/finalzzhoif"
new file mode 100644--- /dev/null
+++ "b/data_all_eng_slimpj/shuffled/split2/finalzzhoif"
@@ -0,0 +1,5 @@
+{"text":"\\section{The $U(3)\\times U(3)$ NJL model}\n\nThe $U(3)\\times U(3)$ NJL model with scalar-pseudoscalar and vector-axial-vector sectors is\nused in the present work. To solve the $U_A(1)$ problem, the six-quark t`Hooft interaction is\nadded to the Lagrangian of the model \\cite{Klimt:1989pm,Klevansky:1992qe}\n\\begin{eqnarray}\n \\mathcal{L}& =& {\\bar q}(i{\\hat \\partial} - m^0)q\n  + \\frac{G}{2}\\sum_{i=0}^8 [({\\bar q} {\\lambda}_i q)^2 +({\\bar q}i{\\gamma}_5{\\lambda}_i q)^2]\n  + \\frac{G_V}{2}\\sum_{i=0}^8 [({\\bar q} {\\gamma}_\\mu {\\lambda}_i q)^2 +({\\bar q}{\\gamma}_5{\\gamma}_\\mu{\\lambda}_i  q)^2]\n   \\nonumber \\\\\n &&- K \\left( {\\det}[{\\bar q}(1+\\gamma_5)q]+{\\det}[{\\bar q}(1-\\gamma_5)q] \\right),\n\\label{Ldet}\n\\end{eqnarray}\nwhere $\\lambda_i$ (i=1,...,8) are the Gell-Mann matrices and $\\lambda^0 =\n{\\sqrt{\\frac{2}{3}}}${\\bf 1}, with {\\bf 1} being the unit matrix; $m^0$ is the current quark\nmass matrix with diagonal elements $m^0_u$, $m^0_d$, $m^0_s$ $(m^0_u \\approx m^0_d)$, $G$ and\n$G_V$ are the scalar--pseudoscalar and vector--axial-vector four-quark coupling constants; $K$\nis the six-quark coupling constants. The six-quark interaction can be reduced to an effective\nfour-fermion vertex after the contraction of one of the quark pairs. The details are given in\nappendix A.\n\nLight current quarks transform to massive constituent quarks as a result of spontaneous chiral\nsymmetry breaking. Constituent quark masses can be found from the Dyson-Schwinger equation for\nthe quark propagators (gap equations)\n\\begin{eqnarray}\nm_u&=&m_u^0 + 8 m_u G I_1(m_u)+32 m_u m_s K I_1(m_u) I_1(m_s)\\nonumber\\\\\nm_s&=&m_s^0 + 8 m_s G I_1(m_s)+32 K \\left(m_uI_1(m_u)\\right)^2, \\label{gapNJL}\n\\end{eqnarray}\nwhere $I_1(m)$ is the quadratically divergent integral. The modified Pauli-Villars (PV)\nregularization with two substractions with same $\\Lambda$ is used for the regularization of\ndivergent integrals\\footnote{Any function $f(m^2)$ of mass $m^2$ is regularized by using the\nrule\n\\begin{eqnarray}\nf(m^2)\\to f(m^2)-f(m^2+\\Lambda^2)+\\Lambda^2 f^\\prime(m^2+\\Lambda^2).\\nonumber\n\\end{eqnarray}} (see\n\\cite{Bernard:1992mp,Bernard:1995hm,Bajc:1996gt,Schuren:1991sc}). In this case the\nquadratically and logarithmically divergent integrals $I_1(m)$ and $I_2(m)$ have the same form\nas in the four-momentum cut-off scheme\n\\begin{eqnarray}\nI_1(m) &=& \\frac{N_c}{4 \\pi^2}\n\\left[\\Lambda^2-m^2\\ln\\left(\\frac{\\Lambda^2}{m^2}+1\\right)\\right], \\quad\nI_2 (m) =\\frac{N_c}{4 \\pi^2}\n\\left[\\ln\\left(\\frac{\\Lambda^2}{m^2}+1\\right)-\\left(1+\\frac{m^2}{\\Lambda^2}\\right)^{-1}\\right]\n\\nonumber.\n\\end{eqnarray}\n\nMoreover, the Pauli-Villars regularization is suitable for the description of the vector\nsector because it preserves gauge invariance.\n\nMasses and vertex functions of the mesons can be found from the Bethe-Salpeter equation. The\nexpression for the quark-antiquark scattering matrix is\n\\begin{eqnarray}\n\\hat{T}=\\mathbf{G}+\\mathbf{G}\\mathbf{\\Pi}(p^2)\\hat{T}=\\frac{1}{\\mathbf{G}^{-1}-\\mathbf{\\Pi}(p^2)},\n\\end{eqnarray}\nwhere $\\mathbf{G}$ and $\\mathbf{\\Pi}(p^2)$ are the corresponding matrices of the four-quark\ncoupling constant and polarization loops. The particle mass can be found from the equation\n$\\mathrm{det}(\\mathbf{G}^{-1}-\\mathbf{\\Pi}(M^2))=0$ and near the poles the corresponding part\nof the $\\hat{T}$ matrix can be expressed in the form\n\\begin{eqnarray}\n\\hat{T}=\\frac{\\bar{V} \\otimes V}{p^2-M^2},\n\\end{eqnarray}\nwhere $V$ and $M$ are the vertex function and mass of the meson, and $\\bar{V} = \\gamma^0\nV^\\dag \\gamma^0$. Details of calculations for different channels are presented in appendices\nB, C. Here we discuss only general properties.\n\nThe most simple situation takes place for the vector and the isovector scalar meson with equal\nquark masses (say $\\rho$ and $a_0$). In this case, the coupling constant and polarization\noperator are just numbers (not matrices). For pseudoscalar mesons, additional axial-vector\ncomponents appear in the vertex function due to the pseudoscalar--axial-vector mixing (in the\nscalar case this transition loop is proportional to the difference of quark masses). An\nadditional complication takes place for $\\eta$ and $\\eta^\\prime$ due to the singlet-octet\nmixing (or mixing of strange and non-strange quarks due to the t`Hooft interaction).\nTherefore, the vertex function of this meson has four components: strange and non-strange\npseudoscalar and axial-vector.\n\n\\section{Fixing model parameters}\n\nThe model has six parameters: the coupling constants $G$, $G_V$, $K$, PV cut-off $\\Lambda$,\nand constituent quark masses $m_u$ and $m_s$. We use two parametrization schemes. In the first\none, the model parameters are defined using masses of the pion, kaon, $\\rho$ and $\\eta$ mesons\nand the weak pion decay constant $f_\\pi$. Note that the number of input parameters is greater\nthan the number of physical observables by one. This allows us, following\n\\cite{Bernard:1995hm}, to take the mass of the $u$ quark slightly larger than the half of the\n$\\rho$-meson mass.\nAs a result, we have the following set (set I) of model parameters\n\\begin{eqnarray}\n m_u = 390\\,\\mathrm{MeV},\\,\n m_s=496\\,\\mathrm{GeV},\\,\n G=6.62\\,\\mathrm{GeV}^{-2},\\,\n G_V=-11.29\\,\\mathrm{GeV}^{-2},\\,\n K = 123\\,\\mathrm{GeV}^{-5},\\,\n\\Lambda=1\\,\\mathrm{GeV}.\n\\end{eqnarray}\nThe values of the current quark masses $m^0_u,m^0_s$ are defined from the gap\nequations (\\ref{gapNJL}) $m^0_u=3.9$ MeV and $m^0_s=70$ ($m^0_u\/m^0_s=18$).\n\nFor this set of model parameters, the two-photon decay width of the $\\eta$ meson\n$\\Gamma_{\\eta\\to \\gamma\\gamma}=0.37$ KeV, is smaller than the experimental one:\n$\\Gamma^{\\mathrm{exp}}_{\\eta\\to\\gamma\\gamma}=0.510\\pm0.026$ \\cite{PDBook}.\n\nIn the set II the model parameters are fixed in order to reproduce the two-photon decay width\nof the $\\eta$ meson instead of its mass (the $\\eta$ meson mass in this case $M_\\eta=530$ MeV)\n\\begin{eqnarray}\n m_u = 390\\,\\mathrm{MeV},\\,\n m_s=506\\,\\mathrm{GeV},\\,\n G=8.04\\,\\mathrm{GeV}^{-2},\\,\n G_V=-11.29\\,\\mathrm{GeV}^{-2},\\,\n K = 77\\,\\mathrm{GeV}^{-5},\\,\n\\Lambda=1\\,\\mathrm{GeV}.\n\\end{eqnarray}\nThe current quark masses are $m^0_u=3.9$ MeV and $m^0_s=78$ MeV ($m^0_u\/m^0_s=20$).\n\n\\section{Decay $\\eta \\to \\pi^0 \\gamma \\gamma$}\n\\begin{figure}\n\\resizebox{0.8\\textwidth}{!}{\\includegraphics{EtPiGaGa}} \\caption{\\label{fig:etpigaga}\nDiagrams contributing to the amplitude of the process $\\eta \\to \\pi^0 \\gamma \\gamma$.}\n\\end{figure}\n\nThe general form of the $\\eta \\to \\pi^0 \\gamma \\gamma$ decay amplitude contains two\nindependent tensor structures \\cite{Ecker:1987hd}\n\\begin{eqnarray}\nT= T^{\\mu\\nu}\\epsilon^{1}_\\mu\\epsilon^{2}_\\nu,\\quad T^{\\mu\\nu}=A(x_{1},x_{2})(q_{1}^{\\nu}\nq_{2}^{\\mu} - q_{1} \\cdot q_{2} g^{\\mu\\nu}) + B(x_{1},x_{2}) \\left[ -M_{\\eta}^{2} x_{1}x_{2}\ng^{\\mu\\nu} - \\frac{q_{1} \\cdot q_{2}}{M_{\\eta}^{2}} p^{\\mu}p^{\\nu} + x_{1} q_{2}^{\\mu} p^{\\nu}\n+ x_{2} p^{\\mu} q_{1}^{\\nu} \\right], \\label{ampform}\n\\end{eqnarray}\nwhere $p$, $q_1$, $q_2$ are the momentum of the $\\eta$ meson and photons, $\\epsilon^{1}_\\mu$\nand $\\epsilon^{2}_\\nu$ are the polarization vectors of the photons, and $x_i= p\\cdot\nq_i\/M_\\eta^2$.\n\nThe $\\eta \\to \\pi^0 \\gamma \\gamma$ decay width has the form\n\\begin{eqnarray}\n \\Gamma & = & \\frac{M_{\\eta}^{5}}{256 \\pi^{2}}\n  \\int\\limits_{0}^{(1-y)\/2} dx_{1} \\int\\limits_{x_2^{\\mathrm{min}} }^{x_2^{\\mathrm{max}}} dx_{2}\n  \\left\\{ \\left| A(x_{1},x_{2}) + \\frac{1}{2} B(x_{1},x_{2}) \\right|^{2} \\left[ 2(x_{1}+x_{2})\n+y -1 \\right]^{2} \\right. \\nonumber \\\\\n  &  + &  \\left. \\frac{1}{4} \\left| B(x_{1},x_{2}) \\right| ^{2} \\left[ 4 x_{1} x_{2}\n- \\left[ 2(x_{1}+x_{2})+ y-1 \\right] \\right] ^{2} \\right\\}  ,\\\\\n&&x_2^{\\mathrm{min}}={(1-2x_1-y)\/2 },\\quad x_2^{\\mathrm{max}}={(1-2x_1-y)\/2(1-2x_1)},\n\\quad y =M_\\pi^2\/M_\\eta^2.\\nonumber\n\\end{eqnarray}\n\nIn the NJL model the amplitude for the $\\eta \\to \\pi^0 \\gamma \\gamma$ decay process is\ndescribed by three types of diagrams (see Fig. \\ref{fig:etpigaga}): the quark box and exchange\nof scalar($a_0$) and vector ($\\rho,\\omega$) resonances. Let us consider theses contributions\nin detail.\n\nThe scalar meson exchange has the simplest form. It gives a contribution only to $A(x_1,x_2)$.\nThis contribution consists of three parts and can be written in the form (see appendices B and\nC for the definition of polarization loops and vertex functions):\n\\begin{eqnarray}\n A(x_1,x_2) &=& \\frac{g_{a_0 \\eta \\pi}(2q_1\\cdot q_2)g_{a_0 \\gamma \\gamma}(2q_1\\cdot q_2)}{G_{a_0}^{-1}-\\Pi_{SS}^{uu}(2 q_1\\cdot q_2 )}\n, \\quad q_1\\cdot q_2 = M_{\\eta}^{2}\\left(x_1+x_2-\\frac{1}{2}\\right)+\\frac{M_\\pi^2}{2}\\nonumber\\\\\ng_{a_0\\gamma\\gamma}(p^2)&=& \\frac{1}{2 \\pi^2}  \\int \\limits_0^1 dx_1 \\int \\limits_0^{1-x_1}\ndx_2 \\frac{m_u(1-4x_1x_2)}{(p^2 x_1 x_2-m_u^2-\\Lambda^2)^2 (p^2 x_1 x_2-m_u^2)} \\\\\ng_{a_0 \\eta \\pi}(p^2)&=& -i 2 N_c N_f\\int \\frac{d^4_\\Lambda k}{(2\\pi)^4}\n  \\mathrm {Tr}_D\\left\\{V_{a_0}S_u(k+q_1)V_{\\pi}S_u(k)V_{\\eta}S_u(k-q_2)\\right\\}.\\nonumber\n\\end{eqnarray}\nhere $\\mathrm {Tr}_D$ is the trace over Dirac indices, index $\\Lambda$ in the measure of\nintegration means PV regularization of the integral and $S_j(p)=(\\hat{p}-m_j)^{-1}$.\n\nThe amplitude with the vector meson ($\\rho,\\omega$) exchanges consists of two quark triangles\nof anomalous type (see appendix D) and the vector meson propagator. It gives the following\ncontributions\n\\begin{eqnarray}\n B(x_{1},x_{2})&=&\\sum\\limits_{j=\\rho,\\omega}\\,\\sum\\limits_{i=1,2}\n \\frac{g_{\\eta j \\gamma}(M_\\eta^2,M_\\eta^2(1-2 x_i),0)g_{\\pi j \\gamma}(M_\\pi^2,M_\\eta^2(1-2 x_i),0)}\n{G_{2}^{-1}-\\Pi_{VV}^{uu}(M_\\eta^2(1-2 x_i))},\\\\\n A(x_{1},x_{2})&=&\\sum\\limits_{j=\\rho,\\omega}\\, \\sum\\limits_{i=1,2}\n \\frac{g_{\\eta j \\gamma}(M_\\eta^2,M_\\eta^2(1-2 x_i),0)g_{\\pi j \\gamma}(M_\\pi^2,M_\\eta^2(1-2 x_i),0)M_\\eta^2(1-x_i)}\n{G_{2}^{-1}-\\Pi_{VV}^{uu}(M_\\eta^2(1-2 x_i))}.\\nonumber\n\\end{eqnarray}\n\nThe box diagram is of a more complicated structure. It consists of three types of boxes (plus\nthree crossed) and contains the diagrams with pseudoscalar and axial-vector components of the\n$\\pi$ and $\\eta$ mesons\n\\begin{eqnarray}\n  T_{\\mu\\nu}=-i e^{2} \\int \\frac{d^4_\\Lambda k}{(2\\pi)^4}\n  \\mathrm {Tr}_D \\biggl(\\biggr. &&V_{\\pi} S(k) V_{\\eta}     S(k+p-q_1-q_{2}) \\gamma_{\\nu} S(k+p-q_1)   \\gamma_{\\mu} S(k+p)+\\nonumber\\\\\n&&+V_{\\pi} S(k) V_{\\eta}     S(k+q_2)         \\gamma_{\\nu} S(k+p-q_1)   \\gamma_{\\mu} S(k+p)\\\\\n&&+V_{\\pi} S(k) \\gamma_{\\nu} S(k+q_2)         \\gamma_{\\mu} S(k+q_1+q_2) V_{\\eta}     S(k+p)\n  +\\left\\{q_1 \\leftrightarrow q_2 ,\\mu \\leftrightarrow \\nu \\right\\}\\biggl.\\biggr)\\nonumber\n\\end{eqnarray}\nWe calculate these diagrams numerically. In order to check the integration procedure, we\ncalculate all coefficients of different tensor structures and verify if they have gauge\ninvariant form (\\ref{ampform}).\n\nThe obtained results for the decay width are given in the Table 1 for two sets of model\nparameters. The main contribution comes from the box diagram. The contribution from vector\nmesons has a constructive interference while the scalar $a_0$ contribution has a destructive\none. The results are in satisfactory agreement with Crystal Ball data $0.45\\pm0.12$\n\\cite{Prakhov:2005vx} and the present value $0.57\\pm0.21$ given in PDG\\cite{PDBook}.\n\n\\begin{table}\n\\caption{$\\eta\\to\\pi^0\\gamma\\gamma$ decay width.}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}\n \\hline\n Contribution        &model 1& model 2  \\\\\n \\hline\n vector mesons       & 0.17  & 0.20     \\\\\n scalar meson        & 0.03  & 0.12    \\\\\n vector+scalar mesons& 0.10  & 0.12    \\\\\n box                 & 0.28  & 0.35      \\\\\n box+vector          & 0.78  & 0.95    \\\\\n total               & 0.53  & 0.45 \\\\\n \\hline\n\\end{tabular}\n\\end{table}\n\nIt is also very instructive to consider the invariant mass distribution. In Figures\n\\ref{fig:DifEtaPiGaGaS1} and \\ref{fig:DifEtaPiGaGaS2} the invariant mass distribution of the\ntwo-photons is shown for the scalar meson contribution, vector mesons contribution, scalar +\nvector mesons and total. In Figure \\ref{fig:DifEtaPiGaGaS1S2Os}, the results of our\ncalculations of the invariant mass distribution are compared with the calculation in the\nchiral unitary approach \\cite{Oset:2002sh}.\n\n\\begin{figure}\n\\resizebox{0.5\\textwidth}{!}{\\includegraphics{DGEPGG1v2.eps}}\n\\caption{\\label{fig:DifEtaPiGaGaS1} Invariant mass distribution of the two-photons of the\nscalar meson contribution(dots), vector meson contributions(short dash), scalar + vector\nmesons(dash-dot), quark box(long dash) and total(continuous line) for the set I.}\n\\end{figure}\n\\begin{figure}\n\\resizebox{0.5\\textwidth}{!}{\\includegraphics{DGEPGG2v2.eps}}\n\\caption{\\label{fig:DifEtaPiGaGaS2} Invariant mass distribution of the two-photons of the\nscalar meson contribution(dots), vector meson contributions(short dash), scalar + vector\nmesons(dash-dot) , quark box(long dash) and total(continuous line) for the set II.}\n\\end{figure}\n\\begin{figure}\n\\resizebox{0.5\\textwidth}{!}{\\includegraphics{DGEPGG123v2.eps}}\n\\caption{\\label{fig:DifEtaPiGaGaS1S2Os} Invariant mass distribution of the two-photons of the\ntotal contributions for the set I(dashes), set II(dots) together with the results of the\nchiral unitary approach \\cite{Oset:2002sh}.}\n\\end{figure}\n\\section{Conclusions}\n\nEarlier calculations of the process $\\eta \\to \\pi^0 \\gamma\\gamma$ in the NJL model do not\ninclude the momentum dependence of quark loops and pseudoscalar--axial-vector transitions and\nare in satisfactory agreement with the GAMS experiment.\n\nRecently, the new experimental data on this decay have been obtained and the value of the\ndecay width is almost two times smaller. A number of theoretical estimates is also obtained,\nand it seems that the momentum dependence of amplitudes is important for a correct description\nof this process ( ``all-order'' estimate in ChPT).\n\nIn the present work, the contributions from quark box, scalar and vector pole diagrams are\nconsidered with the full momentum dependence. The pseudoscalar--axial-vector transitions are\nalso taken into account.\n\nThe obtained result is consistent with recent experiments, theoretical estimates of ChPT\n\\cite{Ametller:1991dp,Bellucci:1995ay} and the chiral unitary approach\\cite{Oset:2002sh}.\n\n\nIn future, we plan to consider the polarizability of pions and also decays of vector mesons\n$\\rho(\\omega)\\to\\eta(\\pi)\\pi\\gamma$.\n\\begin{acknowledgments}\nThe authors thank I. V. Anikin, A. E. Dorokhov, A. A. Osipov and V. L. Yudichev for useful\ndiscussions. The authors acknowledge the support of the Russian Foundation for Basic Research,\nunder contract 05-02-16699.\n\\end{acknowledgments}\n\n\n\\section*{Appendixes}\n\\subsection{Lagrangian}\n\nLagrangian (\\ref{Ldet}) can be rewritten in the form (see\n\\cite{Klimt:1989pm,Klevansky:1992qe})\n\\begin{eqnarray}\n&&\\mathcal{L} =\n {\\bar q}(i{\\hat \\partial} - m^0)q +\n  {\\frac{1}{2}} \\sum_{i=1}^9\n   [G_i^{(-)} ({\\bar q}{\\lambda^\\prime}_i q)^2 +G_i^{(+)}({\\bar q}i{\\gamma}_5{\\lambda^\\prime}_i q)^2] +\n     \\nonumber \\\\\n  &&\\qquad+ G^{(-)}_{us}({\\bar q}           {\\lambda}_u q)({\\bar q}            {\\lambda}_s q)\n          + G^{(+)}_{us}({\\bar q}i{\\gamma}_5{\\lambda}_u q)({\\bar q}i {\\gamma}_5{\\lambda}_s q)\n  + \\frac{G_V}{2}\\sum_{i=0}^8 [({\\bar q}{\\gamma}_\\mu {         \\lambda}_i q)^2\n                              +({\\bar q}{\\gamma}_5{\\gamma}_\\mu{\\lambda}_i q)^2],\n\\label{LGus}\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n&&{\\lambda^\\prime}_i={\\lambda}_i ~~~ (i=1,...,7),~~~\\lambda^\\prime_8 = \\lambda_u =\n({\\sqrt 2}\n\\lambda_0 + \\lambda_8)\/{\\sqrt 3},\\nonumber\\\\\n&&\\lambda^\\prime_9 = \\lambda_s = (-\\lambda_0 + {\\sqrt 2}\\lambda_8)\/{\\sqrt 3}, \\label{DefG}\\\\\n&&G_1^{(\\pm)}=G_2^{(\\pm)}=G_3^{(\\pm)}= G \\pm 4Km_sI_1 (m_s), \\nonumber \\\\\n&&G_4^{(\\pm)}=G_5^{(\\pm)}=G_6^{(\\pm)}=G_7^{(\\pm)}= G \\pm 4Km_uI_1 (m_u),\n\\nonumber \\\\\n&&G_u^{(\\pm)}= G \\mp 4Km_sI_1(m_s), ~~~ G_s^{(\\pm)}= G, ~~~ G_{us}^{(\\pm)}= \\pm 4{\\sqrt\n2}Km_uI_1 (m_u).\\nonumber\n\\end{eqnarray}\n\n\\subsection{Polarization loops}\nPolarization loops in different channels after the PV regularization\n\\begin{eqnarray}\ne^{-izm_im_j} \\to R_{ij}(z)=e^{-izm_im_j}\\left[1-(1+iz\\Lambda^2)e^{-iz\\Lambda^2}\\right]\n\\end{eqnarray}\ntake the form (see \\cite{Bernard:1995hm} for the expressions for the polarization loops with\nequal indices)\n\\begin{eqnarray}\n \\Pi_{PP}^{ij}(p^2) &=&\\frac{N_c}{4\\pi^2}\\int_{-1}^1dy\\int_0^{\\infty}\\frac{dz}{z}R_{ij}(z)e^{izA}\n                \\left[-\\frac{i}{z}+\\frac{1}{2}p^2(1-y^2)-\\frac{1}{2}\\left[(m_i-m_j)^2-y(m_i^2-m_j^2)\\right]\\right] ,\\nonumber\\\\\n \\Pi_{SS}^{ij}(p^2) &=&\\Pi_{PP}^{ij}(p^2)-2m_im_j\\frac{N_c}{4\\pi^2}\\int_{-1}^1dy\\int_0^{\\infty}\\frac{dz}{z}R_{ij}(z)e^{izA} ,\\nonumber\\\\\n \\Pi_{VV}^{ij,{\\mu\\nu}}(p^2) &=&\\left(g^{\\mu\\nu}-\\frac{p^{\\mu}p^{\\nu}}{p^2}\\right) \\Pi_{VV}^{ij}(p^2) + \\frac{p^{\\mu}p^{\\nu}}{p^2} \\Pi_{VV}^{ij,L}(p^2),\\nonumber\\\\\n \\Pi_{VV}^{ij,L}(p^2) &=&\\frac{N_c}{8\\pi^2}\\int_{-1}^1dy\\int_0^{\\infty}\\frac{dz}{z}R_{ij}(z)e^{izA}\n                \\left[(m_i-m_j)^2-y(m_i^2-m_j^2)\\right]\\nonumber,\\\\\n \\Pi_{VV}^{ij}(p^2) &=& \\Pi_{VV}^{ij,L}(p^2)- p^2 \\frac{N_c}{8\\pi^2}\\int_{-1}^1dy(1-y^2)\\int_0^{\\infty}\\frac{dz}{z}R_{ij}(z)e^{izA}\n                \\nonumber,\\\\\n \\Pi_{AA}^{ij,{\\mu\\nu}}(p^2) &=&\\left(g^{\\mu\\nu}-\\frac{p^{\\mu}p^{\\nu}}{p^2}\\right) \\Pi_{AA}^{ij,T}(p^2) + \\frac{p^{\\mu}p^{\\nu}}{p^2} \\Pi_{AA}^{ij}(p^2),\\nonumber\\\\\n \\Pi_{AA}^{ij,T}(p^2) &=&\\Pi_{VV}^{ij}(p^2)+2m_im_j\\frac{N_c}{4\\pi^2}\\int_{-1}^1dy\\int_0^{\\infty}\\frac{dz}{z}R_{ij}(z)e^{izA},\\\\\n \\Pi_{AA}^{ij}(p^2) &=&\\Pi_{VV}^{ij,L}(p^2)+2m_im_j\\frac{N_c}{4\\pi^2}\\int_{-1}^1dy\\int_0^{\\infty}\\frac{dz}{z}R_{ij}(z)e^{izA}\\nonumber,\\\\\n \\Pi_{PA}^{ij,\\mu}(p^2) &=& \\frac{p^{\\mu}}{\\sqrt{p^2}} \\Pi_{PA}^{ij}(p^2) = p^{\\mu} i(m_i+m_j) \\frac{N_c}{8\\pi^2}\\int_{-1}^1dy\\int_0^{\\infty}\\frac{dz}{z}R_{ij}(z)e^{izA}\\nonumber,\\\\\n \\Pi_{AP}^{ij,\\mu}(p^2) &=& \\frac{p^{\\mu}}{\\sqrt{p^2}} \\Pi_{AP}^{ij}(p^2) =-p^{\\mu} i(m_i+m_j) \\frac{N_c}{8\\pi^2}\\int_{-1}^1dy\\int_0^{\\infty}\\frac{dz}{z}R_{ij}(z)e^{izA}\\nonumber,\\\\\n                &&A=\\frac{p^2}{4}(1-y^2)-\\frac{1}{2}\\left[(m_i-m_j)^2-y(m_i^2-m_j^2)\\right]\\nonumber.\n\\end{eqnarray}\n\n\n\\subsection{Vertex functions}\n\nThe most simple form have the vertex functions for the vector $\\rho$ and the isovector scalar\nmeson $a_0$, namely \\footnote{We suppress flavor indices.}:\n\\begin{eqnarray}\nV_{a_0}=g_{a_0} \\mathbf{I} a_0, \\quad V_{\\rho}= g_{\\rho} \\gamma_\\mu \\rho^\\mu.\n\\end{eqnarray}\nThe matrices $\\mathbf{G}$ and $\\mathbf{\\Pi}$ for $a_0$ and $\\rho$ mesons have the form\n\\begin{eqnarray}\n\\mathbf{G}_{a_0} &=& G_1^{(-)},\\, \\mathbf{\\Pi}_{a_0}(p^2)  = \\Pi_{SS}^{uu}(p^2), \\\\\n\\mathbf{G}_{\\rho}&=&G_2  \\quad     , \\mathbf{\\Pi}_{\\rho}(p^2) = \\Pi_{VV}^{uu}(p^2)\\nonumber\n\\end{eqnarray}\n\nFor the pion and kaon,  additional axial-vector components appear in the vertex function due\nto pseudoscalar--axial-vector mixing\n\\begin{eqnarray}\n V_{\\pi}=g_{\\pi} i \\gamma_5 (1+\\Delta_\\pi \\hat{p}) \\pi,\\,\\quad\n V_{K}  =g_{K} i\\gamma_5(1+\\Delta_K \\hat{p}) K\n\\end{eqnarray}\nHere $\\mathbf{G}$ and $\\mathbf{\\Pi}$ are\n\\begin{eqnarray}\n\\mathbf{G}_{\\pi} =\n\\begin{pmatrix} G_1^{(+)}&0\\\\\n                0        &G_2\\end{pmatrix}\n, \\mathbf{\\Pi}_{\\pi}(p^2) =\n\\begin{pmatrix}\n\\Pi^{uu}_{PP}(p^2) & \\Pi^{uu}_{PA}(p^2)\\\\\n\\Pi^{uu}_{AP}(p^2) & \\Pi^{uu}_{AA}(p^2)\n\\end{pmatrix},\\\\\n\\mathbf{G}_{K} =\n\\begin{pmatrix} G_4^{(+)}&0\\\\\n                0        &G_2\\end{pmatrix}\n, \\mathbf{\\Pi}_{K}(p^2) =\n\\begin{pmatrix}\n\\Pi^{us}_{PP}(p^2) & \\Pi^{us}_{PA}(p^2)\\\\\n\\Pi^{us}_{AP}(p^2) & \\Pi^{us}_{AA}(p^2)\n\\end{pmatrix}.\\nonumber\n\\end{eqnarray}\n\nTherefore, the vertex function of the $\\eta$ meson have four components: strange and\nnon-strange pseudoscalar and axial-vector\n\\begin{eqnarray}\nV_{\\eta}&=&\n  g_{\\eta_u} i \\gamma_5 (1+\\Delta_{\\eta_u} \\hat{p}) \\eta_u\n +g_{\\eta_s} i \\gamma_5 (1+\\Delta_{\\eta_s} \\hat{p}) \\eta_s =\\\\\n &=& g_{\\eta} i \\gamma_5( \\cos\\Theta_\\eta\\eta_u-\\sin\\Theta_\\eta\\eta_s +\\Delta_{\\eta} \\hat{p}( \\cos\\widetilde{\\Theta}_\\eta \\eta_u\n -\\sin\\widetilde{\\Theta}_\\eta\\eta_s)),\\nonumber\n\\end{eqnarray}\nwhere $\\Theta_\\eta$ and $\\widetilde{\\Theta}_\\eta$ are the mixing angles for pseudoscalar and\naxial-vector components. The matrices $\\mathbf{G}$ and $\\mathbf{\\Pi}(p^2)$ are four-by-four\nmatrices\n\\begin{eqnarray}\n\\mathbf{G} =\n\\begin{pmatrix} \\mathbf{G}^{(+)}&0\\\\\n                0        &\\mathbf{G_2}\\end{pmatrix}\n, \\mathbf{G}^{(+)} =\n\\begin{pmatrix} G_u^{(+)}&G_{us}^{(+)}\\\\\n                G_{us}^{(+)}&G_s^{(+)}\\end{pmatrix},\\mathbf{G_2}=\\mathrm{diag}\\{G_2,G_2\\}\\\\\n\\mathbf{\\Pi}(p^2) =\n\\begin{pmatrix}\n\\mathbf{\\Pi}_{PP}(p^2) & \\mathbf{\\Pi}_{PA}(p^2)\\\\\n\\mathbf{\\Pi}_{AP}(p^2) & \\mathbf{\\Pi}_{AA}(p^2)\n\\end{pmatrix}\n, \\mathbf{\\Pi}_{ij}(p^2) = \\mathrm{diag}\\{\\Pi^{uu}_{ij(p^2)},\\Pi^{ss}_{ij}(p^2)\\},\ni,j=P,A\\nonumber\n\\end{eqnarray}\n\n\n\\subsection{Amplitudes $\\eta\\to\\gamma\\gamma$, $\\rho\\to\\eta(\\pi)\\gamma$}\n\nThe amplitude for the two-photon decay width of the pseudoscalar meson has the form\n\\begin{eqnarray}\nA(P\\to\\gamma\\gamma) \\ = \\ e^2\\; g_{P\\gamma\\gamma}(M_P^2,q_1^2,q_2^2)\\\n\\epsilon_{\\mu\\nu\\alpha\\beta} \\ \\epsilon_1^{\\mu} \\epsilon_2^{\\nu} \\;q_1^\\alpha\nq_2^\\beta\\ ,\n\\end{eqnarray}\nwhere $q_1$, $q_2$ are the momentum of photons and $\\epsilon^{1}_\\mu$, $\\epsilon^{2}_\\nu$ are\nthe polarization vectors of the photons,\n\\begin{eqnarray}\ng_{\\pi\\gamma\\gamma}(M_\\pi^2,q_1^2,q_2^2) & = & I_u(M_\\pi^2,q_1^2,q_2^2)g_{\\pi},  \\\\\ng_{\\eta\\gamma\\gamma}(M_\\eta^2,q_1^2,q_2^2) & = &\\frac{5}{3}\nI_u(M_\\eta^2,q_1^2,q_2^2)g_{\\eta_u}-\\frac{\\sqrt 2}{3} I_s(M_\\eta^2,q_1^2,q_2^2)g_{\\eta_s}\n.\\nonumber\n\\end{eqnarray}\nThe loop integrals $I_j(M_P^2)$ are given by\n\\begin{eqnarray}\nI_j(M_P^2,q_1^2,q_2^2) & = &\n \\frac{1}{2 \\pi^2}  \\int \\limits_0^1 dx_1 \\int \\limits_0^{1-x_1} dx_2\n \\frac{m_j}{m_j^2-x_1(1-x_1-x_2)q_1^2-x_2(1-x_1-x_2) q_2^2-x_1x_2 M_P^2}.\n\\end{eqnarray}\nThe amplitudes for the processes $\\rho(\\omega)\\to\\eta(\\pi)\\gamma$ have the form\n\\begin{eqnarray}\nA(P V \\gamma) \\ = \\ g_\\rho e\\; g_{P\\rho\\gamma}(M_P^2,q_1^2,q_2^2)\\\n\\epsilon_{\\mu\\nu\\alpha\\beta} \\ \\epsilon_1^{\\mu} \\epsilon_2^{\\nu} \\;q_1^\\alpha q_2^\\beta\\ ,\n\\end{eqnarray}\nhere $q_1$ and $\\epsilon^{1}_\\mu$ are the momentum and the polarization vector of\n$\\rho(\\omega)$ meson.\n\\begin{eqnarray}\ng_{\\pi \\rho\\gamma}(M_\\pi^2,q_1^2,q_2^2)  & = & I_u(M_\\pi^2,q_1^2,q_2^2)g_{\\pi},\\quad\ng_{\\eta\\rho\\gamma}(M_\\eta^2,q_1^2,q_2^2)  = 3I_u(M_\\eta^2,q_1^2,q_2^2)g_{\\eta_u}, \\nonumber\\\\\ng_{\\pi \\omega\\gamma}(M_\\pi^2,q_1^2,q_2^2)  & = &3I_u(M_\\pi^2,q_1^2,q_2^2)g_{\\pi} ,\\quad\ng_{\\eta\\omega\\gamma}(M_\\eta^2,q_1^2,q_2^2)  =  I_u(M_\\eta^2,q_1^2,q_2^2)g_{\\eta_u}.\n\\end{eqnarray}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Calculation Details}\\label{sec:calculation_details}\n\n\n\\begin{table}[tb]\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline\n Lattice Volume & $M_{\\pi}$ (MeV) & $N_{\\rm cfgs}$ & $N_{\\rm tsrcs}$ for $c\\bar{c}$, $c\\bar{s}$, $c\\bar{l}$ & $N_{\\rm vecs}$ \\\\\n\\hline\n $24^{3}\\times 128$ & 391 & 553 & 32, 16, 16 & 162 \\\\\n\\hline\n $32^3\\times 256$ & 236 &  484 & 1, 1, 2 & 384  \\\\\n\\hline\n\\end{tabular}\n\\caption{The lattice gauge field ensembles used.  The volume is given as\n$(L\/a_s)^3 \\times (T\/a_t)$ where $L$ and $T$ are respectively the spatial and temporal extents of the lattice.  The number of gauge field configurations used, $N_{\\rm cfgs}$, and the number\nof perambulator time-sources used per configuration, $N_{\\rm tsrcs}$, are shown along with the number of eigenvectors used in the distillation framework~\\cite{Peardon:2009}, $N_{\\rm vecs}$.}\n\\label{tab:lattice_details}\n\\end{center}\n\\end{table}\n\nIn this study we use an anisotropic lattice formulation where the temporal lattice spacing, $a_t$, is smaller than the spatial lattice spacing, $a_s \\approx 0.12$~fm, with an anisotropy $\\xi \\equiv a_s \/ a_t \\approx 3.5$.  The gauge sector is described by a tree-level Symanzik-improved anisotropic action, while the fermionic sector uses a tadpole-improved anisotropic Sheikholeslami-Wohlert (clover) action with stout-smeared gauge fields~\\cite{Morningstar:2003} and $N_{f}=2+1$ flavours of dynamical quarks.  For both ensembles the heavier dynamical quark is tuned to approximate the physical strange quark, but the ensembles differ in light quark mass giving the two different pion masses.  Table~\\ref{tab:lattice_details} summarises these lattice ensembles -- full details are given in Refs.~\\cite{Edwards:2008,Lin:2009}.\n\nWe use the same relativistic action for the charm quark as for the light and strange quarks (with tadpole-improved tree-level clover coefficients). The charm-quark mass and anisotropy parameters are tuned to reproduce the physical $\\eta_c$ mass and a relativistic dispersion relation -- this process was described for the $M_{\\pi} \\sim 400$ MeV ensemble in Ref.\\ \\cite{Liu:2012}. Throughout this work we do not correct experimental data for electromagnetic effects. For the $M_{\\pi} \\sim 240$ MeV ensemble, the momentum dependence of the $\\eta_c$ energy after tuning is shown in Figure~\\ref{fig:dispersion}.  The momentum is quantised by the periodic boundary conditions on the cubic spatial volume, $\\vec{p} = \\frac{2\\pi}{L} \\vec{n}$, where $\\vec{n}=(n_x,n_y,n_z)$ and $n_i \\in \\{0, 1, 2, \\dots, L\/a_s - 1 \\}$.  A reasonable fit to the dispersion relation,\n\\begin{equation}\n\\label{equ:dispersion}\n(a_t E)^2 = (a_t M)^2 + \\left( \\frac{2\\pi}{\\xi L\/a_s} \\right)^2 n^2 \\, ,\n\\end{equation}\nis obtained giving $\\xi_{\\eta_c} = 3.456(4)$, in agreement with the anisotropy measured from the pion dispersion relation on this ensemble, $\\xi_{\\pi} = 3.453(6)$~\\cite{Wilson:2015dqa}.\nThe fit gives $M_{\\eta_c} = 2945(17)$ MeV compared to the experimental value $2983.6(6)$~MeV~\\cite{PDG2015}, and so we estimate that the systematic uncertainty from tuning the charm-quark mass is of order 1\\%.\nFigure~\\ref{fig:dispersion} also shows the momentum dependence of the $D$ meson energy; a fit to Eq.~\\ref{equ:dispersion} gives $\\xi_{D} = 3.443(7)$, in reasonable agreement with $\\xi_{\\pi}$ and $\\xi_{\\eta_c}$.\n\n\\begin{figure}[tb]\n\\begin{minipage}{.5\\linewidth}\n\\includegraphics[width=0.95\\textwidth]{Plots\/dispersion_etac.pdf}\n\\end{minipage}\n\\begin{minipage}{.5\\linewidth}\n\\includegraphics[width=0.95\\textwidth]{Plots\/dispersion_D.pdf}\n\\end{minipage}\n\\caption{Points show the dependence of the $\\eta_c$ (left panel) and $D$ (right panel) energy on momentum; error bars show the one sigma statistical uncertainty on either side of the mean.  Lines are fits to the relativistic dispersion relation, Eq.~\\ref{equ:dispersion}, giving $\\xi_{\\eta_c} = 3.456(4)$ ($\\chi^{2}\/N_\\mathrm{d.o.f} = 1.08$) and $\\xi_D = 3.443(7)$ ($\\chi^{2}\/N_\\mathrm{d.o.f} = 0.38)$.}\n\\label{fig:dispersion}\n\\end{figure}\n\nTo give results in physical units, we set the scale via $a_t^{-1} = M_\\Omega^{\\mathrm{phys}} \/ (a_t M_\\Omega)$ using the $\\Omega$ baryon mass measured on this ensemble, $a_t M_\\Omega = 0.2789(16)$~\\cite{Wilson:2015dqa}, leading to $a_t^{-1} = 5997$ MeV. When quoting masses we reduce the already small systematic uncertainty from tuning the charm-quark mass by subtracting $M_{\\eta_c}$ ($\\tfrac{1}{2} M_{\\eta_c}$) from the mass of charmonia (open-charm mesons), rendering it negligible compared to other systematic uncertainties.\n\nThe aim of this work is to study how the spectra change as we vary the light-quark mass and only statistical uncertainties are given in the spectra we present in the following sections. While a full error budget is beyond the scope of this work, the uncertainties arising from working at a finite lattice spacing and in a finite volume were discussed in Ref.~\\cite{Liu:2012}, where they were estimated to be small and have no overall qualitative effect on the spectrum. The uncertainty arising from the ambiguity in how to set the scale can be estimated by choosing a different reference observable. For example, setting the scale on the $M_\\pi \\sim 240$ MeV ensemble using the $h_c$ -- $\\eta_c$ mass splitting gives $a_t^{-1} = 5960$ MeV, $0.6\\%$ lower than from using $M_\\Omega$. On the other hand, using the $\\eta_c(2S)$ -- $\\eta_c(1S)$ mass splitting gives $a_t^{-1} = 5787$ MeV, $4\\%$ lower than when using the $\\Omega$ baryon mass. \n\nAnother source of systematic uncertainty comes from ignoring the unstable nature of states above threshold (see Refs.~\\cite{Dudek:2010,Liu:2012,Moir:2013ub}). Although this is difficult to estimate, for a narrow resonance a conservative approach is to consider the uncertainty to be of the order of the width~\\cite{Dudek:2010}.\n\n\n\n\\subsection{Calculation of spectra}\n\nWe follow the methodology presented in Refs.~\\cite{Dudek:2010,Liu:2012,Moir:2013ub} to compute the spectra.  In brief, meson masses and other spectral information are obtained from the analysis of the time dependence of two-point Euclidean correlation functions,\n\\begin{equation}\nC_{ij}(t) = \\langle 0 | \\mathcal{O}_i(t) \\mathcal{O}^\\dagger_j(0) | 0 \\rangle~,\n\\end{equation}\nwhere $\\mathcal{O}^\\dagger(0)$ [$\\mathcal{O}(t)$] is the creation operator [annihilation operator] and $t$ is the time separation.  When computing charmonium correlators, disconnected Wick diagrams, where the charm quark and antiquark annihilate, are not included -- these are OZI suppressed and so are expected to only give a small contribution in charmonium.  There are no such disconnected contributions to the open-charm meson correlators considered here.\n\nThe hypercubic lattice has a reduced symmetry compared to an infinite volume continuum so states at rest are labelled by the irreducible representations (\\emph{irreps}), $\\Lambda$, of the octahedral group, $O_h$, rather than spin~\\cite{Johnson:1982yq}.  \nA method to ameliorate this issue and determine the continuum spin, $J$, of extracted states is given in Refs.~\\cite{Dudek:2010,Liu:2012,Moir:2013ub} which also contain demonstrations of its efficacy.  Parity, $P$, and any relevant flavour quantum numbers, e.g.\\ charge-conjugation, $C$, are still good quantum numbers in our lattice formulation.\n\nIn each quantum-number channel, the distillation technique~\\cite{Peardon:2009} is used to compute correlation functions involving a large basis of derivative-based fermion-bilinear interpolating operators~\\cite{Dudek:2010}.\\footnote{To investigate more completely the resonant nature of states above threshold we would need to supplement the basis with operators of additional structures, e.g.~multi-meson operators, as in Ref.~\\cite{Moir:2016srx}.}  The resulting matrices of correlation functions, $C_{ij}(t)$, are analysed using a variational procedure~\\cite{Michael:1985,Luscher:1990,Blossier:2009kd} as described in Ref.~\\cite{Dudek:2010}. This amounts to solving a generalised eigenvalue problem,\n$C_{ij}(t) v^{\\mathfrak{n}}_j = \\lambda^{\\mathfrak{n}}(t,t_0) C_{ij}(t_0) v^{\\mathfrak{n}}_j $, where $t_{0}$ is a carefully chosen reference time-slice.  For sufficiently large times, the eigenvalues, $\\lambda^{\\mathfrak{n}}(t,t_0)$, known as principal correlators, are proportional to $e^{-M_{\\mathfrak{n}}(t-t_{0})}$ where $M_{\\mathfrak{n}}$ is the energy of the $\\mathfrak{n}^{th}$ state.  \nEnergies are extracted from a fit to the form, $(1 - A_{\\mathfrak{n}}) e^{-M_{\\mathfrak{n}}(t-t_0)} + A_\\mathfrak{n} e^{-M'_{\\mathfrak{n}} (t-t_0)}$, \nwhere the fit parameters are $M_{\\mathfrak{n}}$, $A_{\\mathfrak{n}}$ and $M'_{\\mathfrak{n}}$.  The second exponential proves useful in stabilising the fit because it `mops up' excited state contamination.\nThe eigenvectors, $v^{\\mathfrak{n}}_{j}$, are related to the operator-state overlaps (or matrix elements), $Z_i^{(\\mathfrak{n})} \\equiv \\langle \\mathfrak{n} | \\mathcal{O}_i^\\dagger | 0 \\rangle$, and contain information on the structure of a state -- they are used in our method for determining the continuum spin.\n\n\\begin{figure}[tb]\n\\begin{center}\n\\includegraphics[width=0.9\\textwidth]{Plots\/T1mmprincorr.pdf}\n\\includegraphics[width=0.9\\textwidth]{Plots\/T1mmoverlaps.pdf}\n\\includegraphics[width=0.9\\textwidth]{Plots\/T1mmoverlaps24.pdf}\n\\end{center}\n\\caption{Top row: principal correlators for a selection of low-lying charmonium states in the $T_1^{--}$ irrep on the $M_\\pi \\sim 240$ MeV ensemble. The data (points) and fits (curves) for $t_0 = 11$ are plotted as $\\lambda^{(\\mathfrak{n})} e^{M_\\mathfrak{n}(t-t_0)}$ showing the central values and one sigma statistical uncertainties.  In each case the fit is reasonable with $\\chi^2\/N_\\mathrm{d.o.f} \\sim 1$.  Red parts of the curves show the time regions used in the fits; blue points were not included in the fits.  \nMiddle row: the operator-state overlaps, $Z$, for the state above, normalised so that the largest value for an operator across all states is equal to unity.  Colour coding is described in the text and the error bars indicate the one sigma statistical uncertainty.\nBottom row: overlaps for the corresponding state on the $M_\\pi \\sim 400$ MeV ensemble.}\n\\label{fig:prin_corrs}\n\\end{figure}\n\nFigure~\\ref{fig:prin_corrs} shows a selection of principal correlators from charmonium correlation functions in the $\\Lambda^{PC} = T_{1}^{--}$ irrep on the $M_\\pi \\sim 240$ MeV ensemble.  The leading time dependence, $e^{-M_{\\mathfrak{n}}(t-t_0)}$, has been divided out yielding a plateau when a single exponential dominates.\nBeneath each principal correlator we show the overlap, $Z$, of each operator onto that state and below that, for comparison, the overlaps for the corresponding state on the $M_\\pi \\sim 400$ MeV ensemble.  The operators were constructed to have definite $J^{PC}$ in the continuum: red bars correspond to $J=1$, blue to $J=3$ and yellow to $J=4$.  It is clear that each state is dominated by operators from a given $J$, demonstrating that the spin-identification methodology~\\cite{Dudek:2010} can be used -- this pattern is repeated for each of the spectra we determine. The darker shade of red and lighter shade of blue represent operators that are proportional to the spatial part of the field strength tensor, $F_{ij}$. We identify a state as hybrid, i.e. a meson with excited gluonic degrees of freedom~\\cite{Dudek:2010}, when overlaps from these operators onto a given state are large compared to their overlaps onto other states\\footnote{In Figure~\\ref{fig:prin_corrs} the apparently considerable overlap of the $J=3$ state with hybrid operators is an artefact of the normalisation; in absolute terms these overlaps are small and we do not identify that state as a hybrid.}.\n\n\n\\section{Comparison of the spectra at two light quark masses}\\label{sec:comparison}\n\nThe principal difference between the spectra presented in Refs.~\\cite{Liu:2012,Moir:2013ub} and this work is the light quark mass, corresponding to $M_\\pi\\sim 400$ MeV in those references and $M_\\pi\\sim 240$ MeV here.  Figures~\\ref{fig:charmonium_comparison}, \\ref{fig:Ds_comparison} and \\ref{fig:D_comparison} show comparisons of the charmonia, $D_s$ and $D$ spectra at the two light quark masses -- it can be seen that, in general, we observe only a mild light quark mass dependence throughout the entire spectra, with no change in the overall pattern of states. The systematic uncertainties were discussed in Section \\ref{sec:calculation_details}.\n\nWe note in passing that we achieve a greater statistical precision on the $M_{\\pi} \\sim 400$ MeV ensemble due to the larger number of time-sources used (see Table \\ref{tab:lattice_details}).  In the discussion that follows some notable features in each spectrum are highlighted and in Section \\ref{sec:mixing} we investigate the mixing between spin-triplet and spin-singlet open-charm mesons.\n\n\n\\subsection{Charmonium}\n\n\\begin{figure}[t]\n\\includegraphics[width=0.99\\textwidth]{Plots\/pionmasscomparison_2.pdf}\n\\caption{Charmonium spectrum, labelled by $J^{PC}$, with $M_\\pi\\sim 240$ MeV (left column for each $J^{PC}$) compared to the spectrum with $M_\\pi\\sim 400$ MeV from Ref.~\\cite{Liu:2012} (right column for each $J^{PC}$). As in earlier figures, red and blue boxes highlight states identified as constituents of, respectively, the lightest and first-excited supermultiplet of hybrid mesons. Dashed lines show some of the lower thresholds using computed masses for $M_\\pi\\sim 240$ MeV (coarse dashing) and $M_\\pi\\sim 400$ MeV (fine dashing): green is $\\eta_c \\pi \\pi$, red is $D \\bar{D}$ and blue is $D \\bar{D}^*$.}\n\\label{fig:charmonium_comparison}\n\\end{figure}\n\nIn charmonium the light quark dependence enters through the sea quark content in the dynamical gauge field ensembles. As shown in Figure~\\ref{fig:charmonium_comparison}, for the low-lying states the masses are generally consistent between the two ensembles within statistical uncertainties.  An exception is the hyperfine splitting, $M_{J\/\\psi} - M_{\\eta_c}$, where we find a small but statistically significant increase when the light quark mass is decreased.\n\nA second notable feature is that the masses of states higher up in the spectrum are generally larger on the $M_\\pi \\sim 240$ MeV ensemble.  This is particularly the case for the hybrids, implying a small increase in their mass as $M_\\pi$ is reduced; as a consequence the splitting between the hybrids and low-lying conventional mesons increases, albeit in a rather mild fashion. However, it is important to note that at higher energies the statistical uncertainties are larger and neglecting the unstable nature of states may be more important. We emphasise that the overall pattern of hybrid mesons is unaffected by decreasing the light quark mass. \n\n\n\\subsection{$D_s$ mesons}\n\n\\begin{figure}[t]\n\\includegraphics[width=0.99\\textwidth]{Plots\/Ds_840_860_comp_2.pdf}\n\\caption{As Figure~\\ref{fig:charmonium_comparison} but for the $D_{s}$ meson spectrum labelled by $J^P$.}\n\\label{fig:Ds_comparison}\n\\end{figure}\n\n\\begin{figure}[t]\n\\includegraphics[width=0.99\\textwidth]{Plots\/D_840_860_comp_2.pdf}\n\\caption{As Figure~\\ref{fig:charmonium_comparison} but for the $D$ meson spectrum labelled by $J^P$.}\n\\label{fig:D_comparison}\n\\end{figure}\n\nAs for charmonium, and shown in Fig.\\ \\ref{fig:Ds_comparison}, only mild dependence on the light quark mass is observed throughout the $D_s$ meson spectrum. The largest change in the low-lying states is for the lightest $0^{+}$ (our candidate for the $D^{*}_{s0}(2317)$). However, this state is expected to be heavily influenced by the nearby $DK$ threshold to which it can couple in $S$ wave, and interestingly, it has decreased just enough to remain below the threshold, in agreement with the experimental situation.\n\nOnce again we observe a tendency for the hybrid states, coloured red in Fig.\\ \\ref{fig:Ds_comparison}, to increase in mass, and hence the splitting between the hybrids and the lowest conventional $D_{s}$ mesons to increase, as $M_\\pi$ is reduced.  However, there is no change to their overall pattern.\n\n\n\\subsection{$D$ mesons}\n\nFigure~\\ref{fig:D_comparison} shows that, in the $D$ meson spectrum, the light quark mass dependence is also relatively mild and there is no change to the pattern of states.  As expected, the masses generally decrease with decreasing pion mass -- a $D$ meson contains a valence light quark unlike a charmonium or $D_s$ meson.\nThe most significant differences are observed for the lightest $0^+$ and $1^+$ states and this may be because they are strongly influenced by nearby thresholds that can couple in $S$ wave, namely $D \\pi$ and $D^* \\pi$ respectively.  However, the mass of the second-lightest $1^+$, which is also in the vicinity of the $D^* \\pi$ threshold, does not change significantly.  This may be because the mass difference between the charm quark and the light quark is large enough for the expectations of the heavy-quark limit to be a reasonable guide.  In this limit, one of the $1^+$ states can decay to $D^{*}\\pi$ only in $S$ wave, whereas the other can decay to $D^{*}\\pi$ only in $D$ wave~\\cite{Isgur:1991wq}; the latter would be expected to be influenced less by the position of the $D^* \\pi$ threshold.\n\nAt higher energies in the spectrum, there are generally only small or statistically insignificant mass shifts while, as for charmonia and $D_{s}$ mesons, there is a general trend for the hybrid mesons to become heavier as the light-quark mass decreases. This change is somewhat less clear in the $D$ meson spectrum because of the opposing trend for mesons to become lighter as the light-quark mass decreases.\n\n\n\n\n\\subsection{Mixing of spin-triplet and spin-singlet open-charm mesons}\\label{sec:mixing}\n\nAs discussed in Section~\\ref{sec:spectra:opencharm}, charge conjugation is not a good quantum number for open-charm mesons and, consequently, quark model spin-singlet $(^{1}L_{J=L})$ and spin-triplet $(^{3}L_{J=L})$ states with the same $J=L$ can mix. Quantifying this mixing at different light quark masses can provide an insight into the flavour symmetry breaking. Using a two-state hypothesis and assuming energy-independent mixing we can determine the mixing angle defined in Eq.~(6.1) of~\\cite{Moir:2013yfa} from ratios of operator overlaps (interpreted non-relativistically) as described in that reference.\n\n\\begin{table}[tb]\n\\begin{center}\n\\begin{tabular}{c c c|ccc|c}\n&&&&$|\\theta|\/^{\\circ}$&&\\\\\n & $J^P$ & $M_\\pi \\, \/ \\, \\mathrm{MeV}$ & $\\sim(\\rho - \\rho_2)$ &  $\\sim \\pi$ & $ \\sim \\pi_2$ & Heavy-quark limit\\\\\n\\hline \\hline\nc-s &$1^+$         &240 &60.2(0.4) &63.1(0.7) &65.4(0.7) &\\multirow{2}{*}{54.7 or 35.3} \\\\\n    &              &400 &60.9(0.6) &64.9(0.2) &66.4(0.4) & \\\\ \\cline{2-7}\n    &$2^-$         &240 &56.3(0.9) &60.7(0.8) &63.5(0.9) &\\multirow{2}{*}{50.8 or 39.2} \\\\\n    &              &400 &64.9(1.9) &68.7(2.0) &70.9(1.8) & \\\\ \\cline{2-7}\n   &$1^-$ (hybrid) &240 &58.9(1.0) &66.2(1.9) &65(2.0) & \\\\\n    &              &400 &59.9(1.7) &67.9(0.9) &67.3(0.9) & \\\\\n\\hline \\hline\nc-l &$1^+$         &240 &52.7(0.9) &61.4(0.4) &67.1(1.0)   &\\multirow{2}{*}{54.7 or 35.3} \\\\\n    &              &400 &60.1(0.4) &62.6(0.2) &65.4(0.2) & \\\\ \\cline{2-7}\n    &$2^-$         &240 &50.4(0.7) &57.5(0.8) &61.4(0.9) &\\multirow{2}{*}{50.8 or 39.2} \\\\\n    &              &400$^{\\star}$  &63.3(2.2) &67.8(3.7) &71.1(3.9) & \\\\ \\cline{2-7} \n   &$1^-$ (hybrid) &240 &57.8(1.1) &71.4(2.2) &69.9(2.5) & \\\\\n    &              &400 &59.7(1.1) &68.4(0.8) &67.4(0.9) & \\\\\n\\hline \\hline\n\\end{tabular}\n\\caption{Absolute value of the mixing angles for the lightest pairs of $1^+$, $2^-$ and hybrid $1^-$ states in the charm-strange (c-s) and charm-light (c-l) sectors on the two ensembles.  The mixing angles expected in the heavy-quark limit are also shown.  In the $M_\\pi \\sim 400$ MeV case highlighted by the $^{\\star}$, we have subtracted the angle given in Ref.~\\cite{Moir:2013yfa} from $90^{\\circ}$ so that the mass ordering of the states is consistent between the two ensembles.}\n\\label{tab:mixing}\n\\end{center}\n\\end{table}\n\nIn Table \\ref{tab:mixing}, we show the mixing angles for the lightest pairs of $P$-wave ($J^P=1^+$), $D$-wave ($J^{P} = 2^-$) and $ J^{P} = 1^-$ hybrid states extracted using three different operators for the two different ensembles.  The variation between mixing angles determined using the three different operators gives an estimate of the size of the systematic uncertainties as discussed in Ref.~\\cite{Moir:2013yfa}.  The $1^+$ mixing angle from the $\\rho - \\rho_2$ operator in the charm-light sector is closer to the heavy-quark limit value on the $M_\\pi \\sim 240$ MeV ensemble, but the analogous angle in the charm-strange sector does not differ significantly between the ensembles.  For both charm-light and charm-strange mesons, the $2^{-}$ mixing angle is closer to the heavy-quark limit value for the lighter pion mass whereas the $1^-$ hybrid mixing angle shows no significant difference between the two ensembles.\n\nIn all cases on both ensembles, the determined mixing angles lie between the flavour-symmetry limit ($0^{\\circ}$ or $90^{\\circ}$) and the heavy-quark limit values.  This is expected since the charm quark, although much heavier than the light and strange quarks, is not heavy enough for the heavy-quark limit to apply strictly.\n\n\\section{Summary and Outlook}\\label{sec:conclusions}\n\nWe have presented spectra of excited hidden and open-charm mesons obtained from dynamical lattice QCD calculations with a pion mass of approximately $240$ MeV.  The use of distillation in combination with large variational bases of interpolating operators allows us to extract highly excited mesons, while the spin identification scheme has allowed a robust identification of the $J^{P(C)}$ of states as high as spin four, including states with exotic quantum numbers.  \nThe majority of mesons we extract can be interpreted in terms of the $n^{2S+1}L_{J}$ pattern expected from quark potential models.  However, excess states, with both exotic and non-exotic quantum numbers, that do not fit this pattern are also determined.  By examining the operator overlaps we identify these as hybrid mesons, i.e. having excited gluonic degrees of freedom.  The supermultiplets of hybrid mesons follow a pattern consistent with a quark-antiquark combination in $S$ or $P$-wave coupled to a $1^{+-}$ gluonic excitation.\nThe pattern and energy scale of hybrids are the same as that found in the light meson and baryon sectors~\\cite{Dudek:2010,Dudek:2011b,Dudek:2012,Edwards:2012fx,Dudek:2013yja}, studies of charmed baryons~\\cite{Padmanath:2013zfa,Padmanath:2015jea} and in our earlier work on charmonia and open-charm mesons~\\cite{Liu:2012,Moir:2013ub}.\n\nComparing the spectra to those from a similar lattice calculation with a pion mass of approximately $400$ MeV, we find that the overall qualitative features \nare the same and, even in the case of charm mesons with a valence light quark, we find only small quantitative differences. \nThe hybrid mesons appear to show a mild increase in mass as the pion mass is decreased but the pattern of states and supermultiplet structure is unchanged.\n\nWe also compared the spin-singlet -- spin-triplet mixing angles for the lightest pairs of charm-strange and charm-light $P$-wave $(J^{P} = 1^{+})$, $D$-wave $(J^{P} = 2^{-})$ and hybrid $(J^{P} = 1^{-})$ states between the two lattice ensembles.  Using a non-relativistic interpretation of operator overlaps, our results suggest that the mixing angles for the charm-light $1^{+}$ and the charm-light and charm-strange $2^{-}$ states become closer to those expected in the heavy-quark limit as the pion mass is reduced.  Conversely, we find no significant difference in the hybrid $1^{-}$ mixing angles between the two ensembles.\n\nAs discussed earlier, a limitation of these calculations is that we have not accounted for the unstable nature of states above threshold.  \nThis issue has already been addressed for a variety of mesons appearing as bound states and resonances in coupled-channel $D\\pi$, $D\\eta$ and $D_{s}\\bar{K}$ scattering~\\cite{Moir:2016srx}.  The work presented here lays the foundation for extending these scattering calculations to pion masses \ncloser to the physical value, as well as to other scattering channels involving hidden and open-charm mesons. \n\n\\section{Introduction}\\label{sec:introduction}\n\nThe experimental status of the charm sector of Quantum Chromodynamics (QCD) has changed dramatically over the last decade \\cite{PDG2015}.  The discovery of a plethora of unexpected charmonium-like states, commonly known as ``$X, Y, Z$'s'', has highlighted the need for a more complete theoretical understanding of the spectrum.  Many different interpretations have been put forward: some are suggested to be hybrid mesons (a quark-antiquark pair with excited gluonic degrees of freedom) and others two quarks and two antiquarks in a tightly-bound configuration (tetra-quark), a molecular-like combination of two mesons, or a charmonium-like core surrounded by light degrees of freedom (hadro-quarkonium).  There are similar puzzles in the open-charm sector ($D$ and $D_s$ mesons) where the measured masses and widths of the low-lying $D^{*}_{s0}(2317)^{\\pm}$ and $D_{s1}(2460)^{\\pm}$ states are significantly smaller and narrower than expected from quark models.  For some recent reviews see Refs.~\\cite{Brambilla:2010cs,Brambilla:2014jmp,Olsen:2015zcy,Swanson:2015wgq,Prencipe:2015kva}.\n\nIn principle these states can be understood within Quantum Chromodynamics (QCD) using lattice QCD, a non-perturbative, \\textit{ab initio} formulation of the theory.  Spurred on by the experimental situation, there have been many lattice QCD calculations of hidden and open-charm mesons. The majority have focused on lowest-lying states below threshold, achieving unprecedented precision with the various systematic effects under control (some recent examples can be found in Refs.~\\cite{Namekawa:2011wt, McNeile:2012qf, Dowdall:2012ab, Donald:2012ga, Galloway:2014tta}). On the other hand, there have been a number of investigations of excited charmonia and open-charm mesons~\\cite{Dudek:2007,Bali:2011rd, Bali:2011dc, Mohler:2011ke, Bali:2015lka, Kalinowski:2015bwa, Cichy:2016bci}, all of which have some systematic uncertainties not fully accounted for and extract a more limited set of states than we consider here.\n\nIn a previous lattice QCD study, the Hadron Spectrum Collaboration used large bases of interpolating operators with various structures to robustly extract many excited and high-spin states and, crucially, to identify their continuum quantum numbers.  Highlights included the presence of states with exotic quantum numbers (i.e.\\ those forbidden with solely a quark-antiquark pair) and the identification of ``supermultiplets'' of hybrid mesons.  However, these calculations were performed with unphysically-heavy light quarks corresponding to $M_{\\pi} \\sim 400$ MeV.  The results provided useful benchmarks for other approaches such as nonrelativistic effective field theories, for example see Ref.~\\cite{Berwein:2015vca}.\n\nThe current work extends these earlier investigations by performing similar calculations with light-quark masses significantly closer to their physical values, corresponding to $M_{\\pi} \\sim 240$ MeV.  The spectra at the two light quark masses are compared, focusing on the overall qualitative picture and, in particular, whether changes in the pattern of states with exotic quantum numbers or other hybrid mesons are observed.  This allows us to explore the light-quark mass dependence of excited heavy quarkonia which has been suggested to be significant~\\cite{Guo:2012tg}.\n\nIn this study the unstable nature of states above threshold is not considered -- a point discussed in~\\cite{Dudek:2010,Liu:2012,Moir:2013ub} -- and so the spectra should only be considered a guide to the pattern of resonances.  In the charm sector, we have already addressed this limitation for a variety of states appearing as bound-states and resonances in coupled-channel $D\\pi$, $D\\eta$ and $D_{s}\\bar{K}$ scattering~\\cite{Moir:2016srx} for $M_{\\pi} \\sim 400$ MeV and investigations of various other channels involving charm quarks are underway.  This paper lays the foundation for extending those studies to $M_{\\pi} \\sim 240$ MeV, where the additional light-quark mass, closer to the physical value, will enable us to study the evolution with light-quark mass of hidden and open-charm bound-states and resonances.\n\nA number of other investigations of near-threshold bound states, scattering and resonances in the charm sector have appeared over the last few years~\\cite{Ozaki:2012ce, Mohler:2012na, Liu:2012zya, Prelovsek:2013cra, Prelovsek:2013xba, Mohler:2013rwa, Chen:2014afa, Lang:2014yfa, Prelovsek:2014swa, Padmanath:2015era, Lang:2015sba, Chen:2015jwa, Chen:2016lkl,Ikeda:2016zwx}.  There have also been studies addressing the existence of four-quark configurations (mostly considering static heavy quarks)~\\cite{Bicudo:2012qt, Brown:2012tm, Ikeda:2013vwa, Bicudo:2015vta, Bicudo:2015kna, Francis:2016hui, Alberti:2016dru, Peters:2016isf,Bicudo:2016jwl}. However, these are mainly exploratory and more comprehensive calculations as described in Ref.~\\cite{Moir:2016srx} are called for.\n\nThe remainder of the manuscript is organised as follows.  In Section~\\ref{sec:calculation_details} we describe the lattice ensembles used in this study, provide some details on the tuning of the anisotropy and charm-quark mass, and give a brief overview of the analysis of two-point correlation functions.  In Section~\\ref{sec:spectra} we present and interpret the charmonium, $D_s$ and $D$ meson spectra from the calculations with $M_{\\pi} \\sim 240$ MeV.  In Section~\\ref{sec:comparison} we compare these spectra to those from earlier computations with $M_{\\pi} \\sim 400$ MeV and we present a summary in Section~\\ref{sec:conclusions}.\n\n\\section{Charmonium and Open-Charm Spectra}\n\\label{sec:spectra}\n\nIn this section we present the spectra, labelled by $J^{P(C)}$, computed on the $M_{\\pi} \\sim 240$ MeV ensemble.  \nResults for charmonium are described first followed by those for $D_s$ and $D$ mesons.\n\n\n\\subsection{Charmonium}\n\\label{sec:spectra:charmonium}\n\n\n\\begin{figure}[tb]\n\\begin{center}\n     \\includegraphics[width=0.99\\textwidth]{Plots\/charmonium_spinplot.pdf}\n\\end{center}\n    \\caption{Charmonium spectrum up to around $4.5$ GeV labelled by $J^{PC}$; the left (right) panel shows the negative (positive) parity states.  Green, red and blue boxes are the masses computed on our $M_\\pi\\sim 240$ MeV ensemble while black boxes are experimental values from the PDG summary tables~\\cite{PDG2015}. As discussed in the text, we show the calculated (experimental) masses with the calculated (experimental) $\\eta_c$ mass subtracted.  The vertical size of the boxes represents the one-sigma statistical (or experimental) uncertainty on either side of the mean.  Red and blue boxes correspond to states identified as hybrid mesons grouped into, respectively, the lightest and first-excited supermultiplet, as described in the text.  Dashed lines show the location of some of the lower thresholds for strong decay using computed (coarse green dashing) and experimental (fine grey dashing) masses.}   \n  \\label{fig:charmonium_spectrum}\n\\end{figure}\n\nThe charmonium spectrum computed on the $M_\\pi\\sim 240$ MeV ensemble is shown in Figure~\\ref{fig:charmonium_spectrum} and the results are tabulated in \nAppendix~\\ref{app:tables}.\nFor flavour singlets such as charmonium, charge-conjugation, $C$, and parity, $P$, are both good quantum numbers and so states are labelled by $J^{PC}$.  As discussed above, masses are presented after subtracting the $\\eta_c$ mass to reduce the systematic uncertainty arising from tuning the charm quark mass. \nDashed lines indicate the location of some thresholds for strong decay: $\\eta_c \\pi \\pi$ (the lowest threshold if the charm quark and antiquark do not annihilate), $D\\bar{D}$ and $D\\bar{D}^*$.  Since the resonant nature of states above threshold is not investigated in this work, a conservative approach is to only consider the mass values accurate up to the order of the hadronic width~\\cite{Dudek:2010}.\n\nAs found in Ref.~\\cite{Liu:2012}, many of the states with non-exotic $J^{PC}$ follow the $n^{2S+1}L_{J}$ pattern predicted by quark potential models, where $J$ is the total spin of the meson with relative orbital angular momentum $L$, quark-antiquark spin $S$ and radial quantum number $n$.  \nWe find all states up to $J=4$ expected by such models.\n\nFigure~\\ref{fig:charmonium_spectrum} also shows the states (coloured red and blue) that do not fit the $n^{2S+1}L_{J}$ pattern.  Four of these have exotic $J^{PC}$ quantum numbers, $0^{+-}, 1^{-+}, 2^{+-}$, and we find that they, as well as the excess states with non-exotic quantum numbers, have relatively large overlaps onto operators that are proportional to the spatial components of the field strength tensor, $F_{ij}$ (i.e. operators that have a non-trivial gluonic structure), something not seen for the other states in the spectrum. Furthermore, on removing operators proportional to $F_{ij}$ from the variational basis we generally observe a reduction in the quality of the signal for these states. We therefore follow Refs.~\\cite{Dudek:2010,Liu:2012} and interpret these excess states as hybrid mesons.\n\nAs discussed in detail in Ref.~\\cite{Liu:2012}, the hybrid states can be grouped into supermultiplets.  We find that the set $\\left[(0^{-+}, 1^{-+}, 2^{-+}), 1^{--}\\right]$, highlighted in red in Figure~\\ref{fig:charmonium_spectrum}, forms the lightest charmonium hybrid supermultiplet, while the\nstates highlighted in blue, $(0^{++}, 1^{++}, 2^{++})$, $(0^{+-}, 1^{+-}, 1^{+-}, 1^{+-}, 2^{+-}, 2^{+-}, 3^{+-})$, form the first excited hybrid supermultiplet. These patterns are consistent with a quark-antiquark pair coupled to a $1^{+-}$ gluonic excitation; the lightest hybrid supermutiplet has the quark-antiquark pair in $S$-wave and the first excited hybrid supermultiplet has it in $P$-wave.\nThe lightest hybrids appear $\\sim 1.2$ - $1.3$ GeV above the lightest $S$-wave meson multiplet.\nThis pattern of hybrids and their energy scale are consistent with what was found in the light meson and baryon sectors~\\cite{Dudek:2010,Dudek:2011b,Dudek:2012,Edwards:2012fx,Dudek:2013yja}, studies of charmed baryons~\\cite{Padmanath:2013zfa,Padmanath:2015jea} and in our previous work on charmonia and open-charm mesons~\\cite{Liu:2012,Moir:2013ub}.\n\nAs noted in Section~\\ref{sec:calculation_details}, these calculations are performed at a single spatial lattice spacing. On the $400$ MeV ensemble we estimated a scale of $40$ MeV for the discretisation uncertainty arising from $\\mathcal{O}(a_{s})$ corrections to charmonia~\\cite{Liu:2012}. Since the $240$ MeV ensemble has the same spatial lattice spacing, we expect the 40 MeV scale to also be a reasonable estimate for the discretisation uncertainty here.\n\n\n\\subsection{$D_s$ and $D$ mesons}\n\\label{sec:spectra:opencharm}\n\nFor flavoured mesons, such as $D_{s}$ and $D$, charge conjugation is no longer a good quantum number and states are labelled only by $J^{P}$. Figures~\\ref{fig:Ds_spectrum} and \\ref{fig:D_spectrum} show the $D_s$ and $D$ meson spectra respectively; these results are tabulated in Appendix~\\ref{app:tables}.\nMasses are presented with half the mass of the $\\eta_c$ subtracted in order to reduce the systematic uncertainty arising from tuning the charm quark mass.\nDashed lines indicate some of the lower strong-decay thresholds ($DK$ for the $D_s$ spectrum and $D\\pi$ and $D^{*}\\pi$ for the $D$ meson spectrum).\n\nAs for charmonium, the $D_s$ and $D$ spectra can be interpreted in terms of a $n^{2S+1}L_{J}$ pattern and we identify complete $S,P,D$ and $F$-wave multiplets.\nWithin the negative parity sector of both spectra, there are four states, highlighted in red, that do not appear to fit this pattern.  Due to their relatively large overlap with operators featuring a non-trivial gluonic structure, these are identified as the members of the lightest hybrid meson supermultiplet in each flavour sector.  The pattern is again consistent with a $1^{+-}$ gluonic excitation coupled to an $S$-wave quark-antiquark pair and they appear at an energy $\\sim 1.2$ - $1.3$ GeV above the lightest conventional multiplet.  However, unlike in charmonium, the first excited hybrid supermultiplet is not \nrobustly determined for open-charm mesons and is not shown here. \n\n\\begin{figure}[tb]\n\\begin{center}\n\\includegraphics[width=0.99\\textwidth]{Plots\/Ds_spin_spectrum.pdf}\n\\caption{$D_s$ meson spectrum labelled by $J^P$; the left (right) panel shows the negative (positive) parity states.  Green and red boxes are the masses computed on the $M_\\pi\\sim 240$ MeV ensemble while black boxes are experimental masses of the neutral $D$ mesons from the PDG summary tables~\\cite{PDG2015}.  As discussed in the text, the calculated (experimental) masses are shown with with half the calculated (experimental) $\\eta_c$ mass subtracted.  The vertical size of the boxes indicates the one-sigma statistical (or experimental) uncertainty on either side of the mean.  Red boxes show states identified as constituting the lightest hybrid supermultiplet, as described in the text.  \nDashed lines indicate the $DK$ threshold using computed (coarse green dashing) and experimental (fine grey dashing) masses.}\n\\label{fig:Ds_spectrum}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}[tb]\n\\begin{center}\n\\includegraphics[width=0.99\\textwidth]{Plots\/D_spin_spectrum.pdf}\n\\caption{As Figure~\\ref{fig:Ds_spectrum} but for the $D$ meson spectrum.\nDashed lines show the $D\\pi$ and $D^\\ast\\pi$ thresholds using computed (coarse green dashing) and experimental (fine grey dashing) masses.}\n\\label{fig:D_spectrum}\n\\end{center}\n\\end{figure}\n\n\n\\section{Tables of Results}\\label{app:tables}\n\nIn Tables \\ref{tab:charmonium_summary}, \\ref{tab:Ds_summary} and \\ref{tab:D_summary} we present numerical values for, respectively, the charmonium, $D_{s}$ and $D$ meson masses obtained for $M_\\pi \\sim 240$~MeV. \nMasses are given in MeV with either the mass of the $\\eta_{c}$ subtracted (charmonium) or half the mass of the $\\eta_{c}$ subtracted (open-charm mesons)\nin order to minimise the systematic uncertainty in tuning the charm quark mass. \nIn all cases the quoted error corresponds to the (one-sigma) statistical uncertainty.  As discussed earlier, above the lowest multi-hadron threshold in each channel states can decay strongly into lighter hadrons and, aside from any other systematic uncertainties, we only expect the masses to be correct up to around the width of the state~\\cite{Dudek:2010}.\n\n\n\\begin{table}[h!]\n\\begin{center}\n\\begin{tabular}{|c|cccccc|}\n\\hline\n$J^{PC}$ &  &  & $M-M_{\\eta_c}$ &\\hspace{-0.7cm} $(MeV)$  &  & \\\\\n\\hline\n$0^{-+}$ & 0 & 679(6) & 1197(7) & 1295(18)&  &  \\\\\n$1^{--}$ & 88(1) & 728(7) & 865(7) & 1316(17)& 1345(27) & 1427(17) \\\\\n$2^{--}$ & 879(7) & 1352(21) &  & &  &  \\\\\n$2^{-+}$ & 888(7) & 1414(24) & 1472(21) &  &  &  \\\\\n$3^{--}$ & 902(6) & 1442(18) & 1484(40) & &  &  \\\\\n$4^{-+}$ & 1474(19)  &  &  & &  &  \\\\\n$4^{--}$ & 1450(18) &  &  & &  &  \\\\\n\\hline\n$0^{++}$ & 466(3)  & 989(10) & 1485(25) & 1607(46)&  &  \\\\\n$1^{++}$ & 531(4) & 1038(12) & 1486(25) & 1534(35)&  &  \\\\\n$1^{+-}$ & 545(4) & 1041(12) & 1454(23) & 1587(27) & 1643(47) & 1681(53)   \\\\\n$2^{++}$ & 571(4) & 1065(13) & 1154(11) & 1173(11) & 1639(32) &  \\\\\n$3^{++}$ & 1166(11) &  &  & &  &  \\\\\n$3^{+-}$ & 1173(11) & 1660(34) &  & &  &  \\\\\n$4^{++}$ & 1181(12) &  &  & &  &  \\\\\n\\hline\n$1^{-+}$ & 1326(23) &  &  & &  &  \\\\\n$0^{+-}$ & 1453(27) &  &  & &  &  \\\\\n$2^{+-}$ & 1518(18) & 1647(26) &  & &  &  \\\\\n\\hline\n\\end{tabular}\n\\caption{Summary of the charmonium spectrum presented in Figure \\ref{fig:charmonium_spectrum}.  Masses are shown with $M_{\\eta_{c}}$ subtracted.  Quoted uncertainties are statistical only.}\n\\label{tab:charmonium_summary}\n\\end{center}\n\\end{table}\n\n\n\\begin{table}[h!]\n\\begin{center}\n\\begin{tabular}{|c|cccccc|}\n\\hline\n$J^P$ &  &  & $M-M_{\\eta_c}\/2$& \\hspace{-0.7cm}$(MeV)$  &  &   \\\\\n\\hline\n$0^-$ & 467(11) & 1225(17) & 1679(27) & 1873(31) & & \\\\\n$1^-$ & 593(12) & 1286(12) & 1399(21) & 1740(30) & 1891(33) & 1898(38)\\\\\n$2^-$ & 1424(19) & 1440(20) & 1952(35) & 1993(36) & 2002(32) & \\\\\n$3^-$ & 1481(19) & 2029(28) &  &  &  & \\\\\n$4^-$ & 2075(29) & 2109(31) &  &  &  & \\\\\n\\hline\n$0^+$ & 886(14)  & 1567(35) & 1934(51) & &  & \\\\\n$1^+$ & 1022(15) & 1064(16) & 1612(25) & 1670(26) & 1929(44) & 2030(35) \\\\\n$2^+$ & 1100(15) & 1675(24) & 1773(23) & 2000(37) & & \\\\\n$3^+$ & 1766(22) & 1779(22) &  &  &  & \\\\\n$4^+$ & 1811(24) &  &  &  &  & \\\\\n\\hline\n\\end{tabular}\n\\caption{Summary of the $D_{s}$ meson spectrum presented in Figure \\ref{fig:Ds_spectrum}.  Masses are shown with $M_{\\eta_{c}} \/ 2$ subtracted.  Quoted uncertainties are statistical only.}\n\\label{tab:Ds_summary}\n\\end{center}\n\\end{table}\n\n\n\\clearpage\n\n\n\\begin{table}[h!]\n\\begin{center}\n\\begin{tabular}{|c|cccccc|}\n\\hline\n$J^P$ &  &  & $M-M_{\\eta_c}\/2$ &\\hspace{-0.7cm}$(MeV)$  &  &   \\\\\n\\hline\n$0^-$ & 382(10) & 1138(17) & 1569(26) & 1783(29) & 2176(37)  &\\\\\n$1^-$ & 509(11) & 1233(22) & 1315(21) & 1610(33) & 1801(34) & 1838(36)\\\\\n$2^-$ & 1352(19) & 1429(20) & 1912(34) & 1935(34) & & \\\\\n$3^-$ & 1441(19) & 2032(26) &  &  &  & \\\\\n$4^-$ & 2037(29) & &  &  &  & \\\\\n\\hline\n$0^+$ & 770(15) & 1494(25) & 2201(45) & 1874(26) &  & \\\\\n$1^+$ & 881(17) & 984(14) & 1559(27) & 1603(26) & &\\\\\n$2^+$ & 1020(16) & 1623(26) & 1665(29) & 1925(36) & & \\\\\n$3^+$ & 1724(21) & 1743(21) &  &  &  & \\\\\n$4^+$ & 1804(22) &  &  &  &  & \\\\\n\\hline\n\\end{tabular}\n\\caption{Summary of the $D$ meson spectrum presented in Figure \\ref{fig:D_spectrum}.  Masses are shown with $M_{\\eta_{c}} \/ 2$ subtracted.  Quoted uncertainties are statistical only.}\n\\label{tab:D_summary}\n\\end{center}\n\\end{table}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\n\nThis paper studies the dynamics of fluid drops or bubbles immersed in another fluid filling a porous media under the action of gravity. This process is governed by the classical Darcy's law \n\\begin{equation}\\label{Darcy}\n\\frac{\\mu(x,t)}{\\kappa(x,t)} u(x,t)=-\\nabla p(x,t)-g(0,\\rho(x,t)), \n\\end{equation} \nwhere  $u$ is the velocity of the fluid, $p$ is the pressure, $\\rho$ is the density and $\\mu$ is the viscosity of the fluid. Above $x\\in\\mathbb{R}^2$ and $t\\geq 0$.  Here the medium is assumed to be homogeneous so  that the permeability $\\kappa(x,t)=\\kappa \\ge0$ is constant, as is the gravitational acceleration $g>0$. While Darcy's law was first derived experimentally \\cite{Darcy56}, it can be rigorously obtained through homogenization \\cite{Tartar89,Hornung97}. This physical scenario is mathematically analogous to the evolution of an incompressible flow in a Hele-Shaw cell \\cite{Gancedo2017} where the fluid is set inside two parallel plates that are close enough together so that the resulting dynamics are two dimensional. In particular, the results in this paper can be applied to the Hele-Shaw problem. \n\n\n\n\nThe presence of two immiscible fluids is modeled by taking the viscosity $\\mu$ and the density $\\rho$ as piece-wise constant functions:\n\\begin{equation}\\label{patchsolution}\n\\mu(x,t)=\\left\\{\\begin{array}{rl}\n\\mu^1,& x\\in D(t),\\\\\n\\mu^2,& x\\in \\mathbb{R}^2\\smallsetminus D(t),\n\\end{array}\\right.\n\\quad \n\\rho(x,t)=\\left\\{\\begin{array}{rl}\n\\rho^1,& x\\in D(t),\\\\\n\\rho^2,& x\\in \\mathbb{R}^2\\smallsetminus D(t),\n\\end{array}\\right.\n\\end{equation}\nwhere $D(t)$ is a simply connected bounded domain, namely, the \\textit{bubble}. Thus, there is a sharp interface between the fluids, moving with the flow, which we assume to be incompressible:\n\\begin{equation}\\label{incom}\n\\nabla \\cdot u(x,t)=0.\n\\end{equation}\nWe consider the physically relevant case where surface tension at the free boundary is taken into consideration. The Laplace-Young's formula then states that \\cite{Hou94}:\n\\begin{equation}\\label{surfacetension}\np^1(x,t)-p^2(x,t)=\\sigma K(x,t),\\hspace{1cm}x\\in\\partial D(t),\n\\end{equation}\nwhere $K(x,t)$ denotes the curvature of the curve $\\partial D(t)$, $\\sigma>0$ is the constant surface tension coefficient and $p^1(x,t)$, $p^2(x,t)$ are the limits of the pressure at $x$ from inside and outside, respectively.\nWe are then dealing with the Muskat problem, where the main mathematical interest is to study the dynamics of the free boundary $\\partial D(t)$, especially between water and oil \\cite{Muskat34}. \nIt is remarkable that the evolution equation for the free boundary is well-defined even though the velocity is not continuous. The discontinuity in the velocity is due to the density, viscosity and pressure jumps. But the interface evolution is dictated only by the normal velocity, which is continuous by the incompressibility condition. \n\nIn this sense it is indeed possible to obtain a self-evolution equation for the interface $\\partial D(t)$ which is called the contour evolution system.   This system is equivalent to the Eulerian-Lagrangian formulation \\eqref{Darcy}, \\eqref{patchsolution}, \\eqref{incom}, \\eqref{surfacetension} understood in a weak sense.\nDue to the irrotationality of the velocity in each domain $D(t)$, the vorticity is concentrated on the interface $\\partial D(t)$. That is, the vorticity is given by a delta distribution as follows\n$$\n\\nabla^\\perp\\cdot u(x,t)=\\omega(\\alpha,t)\\delta(x=z(\\alpha,t)),\n$$\nwhere $\\omega(\\alpha,t)$ is the amplitude of the vorticity and $z(\\alpha,t)$ is a parameterization of $\\partial D(t)$ with\n$$\n\\partial D(t)=\\{z(\\alpha,t)=(z_1(\\alpha,t), z_2(\\alpha,t)):\\, \\alpha\\in[-\\pi,\\pi]\\}.\n$$\nThe Biot-Savart law then yields that\n\\begin{equation*}\nu(x,t)=\\frac1{2\\pi} \\text{pv}\\int_{-\\pi}^\\pi \\frac{(x-z(\\beta,t))^\\perp}{|x-z(\\beta,t)|^2}\\omega(\\beta,t)d\\beta,\\hspace{1cm}x\\neq z(\\beta,t),\n\\end{equation*}\nand taking limits in the normal direction to $z(\\alpha,t)$ one finds\n\\begin{equation*}\n\\begin{aligned}\nu_1(z(\\alpha,t),t)&=BR(z,\\omega)(\\alpha,t)-\\frac12\\frac{\\omega(\\alpha,t)}{|\\partial_{\\alpha}z(\\alpha,t)|^2}\\partial_{\\alpha}z(\\alpha,t),\\\\\nu_2(z(\\alpha,t),t)&=BR(z,\\omega)(\\alpha,t)+\\frac12\\frac{\\omega(\\alpha,t)}{|\\partial_{\\alpha}z(\\alpha,t)|^2}\\partial_{\\alpha}z(\\alpha,t),\n\\end{aligned}\n\\end{equation*}\nwhere $BR$ is the Birkhoff-Rott integral that is given by\n\\begin{equation}\\label{eqBR}\nBR(z,\\omega)(\\alpha,t)=\\frac1{2\\pi} \\text{pv}\\int_{-\\pi}^\\pi \\frac{(z(\\alpha,t)-z(\\beta,t))^\\perp}{|z(\\alpha,t)-z(\\beta,t)|^2}\\omega(\\beta,t)d\\beta.\n\\end{equation}\nTaking the dot product with $\\partial_\\alpha z$ in the above equations for $u_1$ and $u_2$ and subtracting one from the other, one then finds that the vorticity strength is given by the jump in the tangential velocity\n\\begin{equation*}\n\\omega(\\alpha,t)=\\left(u_2(z(\\alpha,t),t)-u_1(z(\\alpha,t),t)\\right)\\cdot\\partial_{\\alpha}z(\\alpha,t).\n\\end{equation*}\nThen using Darcy's law \\eqref{Darcy} yields the non-local implicit identity\n\\begin{multline}\\label{eqOmega}\n\\omega(\\alpha,t)=2 A_\\mu |\\partial_{\\alpha}z(\\alpha,t)| \\mathcal{D}(z,\\omega)(\\alpha,t) +2A_\\sigma\\partial_{\\alpha}K(z(\\alpha,t))-2A_\\rho\\partial_{\\alpha}z_2(\\alpha,t),\n\\end{multline}\nwhere\n\\begin{equation}\\label{Dz}\n\\mathcal{D}(z,\\omega)(\\alpha,t)=-BR(z,\\omega)(\\alpha,t)\\cdot\\frac{\\partial_{\\alpha}z(\\alpha,t)}{|\\partial_{\\alpha}z(\\alpha,t)|},\n\\end{equation}\nand\n\\begin{equation}\\label{AmuAsigmaArho}\nA_\\mu=\\frac{\\mu_2-\\mu_1}{\\mu_2+\\mu_1}, \\quad \nA_{\\sigma}=\\frac{\\kappa \\sigma}{\\mu_2+\\mu_1}, \\quad \nA_\\rho = g\\kappa \\frac{\\rho_2-\\rho_1}{\\mu_2+\\mu_1}.\n\\end{equation}\nFurther, in \\eqref{eqOmega} the curvature is given by\n\\begin{equation}\\label{curvature}\n\tK(\\alpha,t)=\\frac{\\partial_{\\alpha}z(\\alpha,t)^\\perp\\cdot\\partial_{\\alpha}^2z(\\alpha,t)}{|\\partial_{\\alpha}z(\\alpha,t)|^3}.\n\\end{equation}\nSince the fluids are immiscible, the interface is just advected by the normal velocity of the fluid flow:\n\\begin{equation*}\nz_t(\\alpha,t)\\cdot \\partial_\\alpha z(\\alpha,t)^\\perp=BR(z,\\omega)(\\alpha,t)\\cdot \\partial_\\alpha z(\\alpha,t)^\\perp.\n\\end{equation*}\nTherefore a tangential velocity $T(z(\\alpha,t))$ can be introduced to change the parametrization of the interface, without altering its shape.  Let $\\partial_\\alpha z(\\alpha,t)=z_\\alpha(\\alpha,t)$. Then we denote the unit tangent and normal vectors by\n\t\\begin{equation}\\label{vectors}\n\t\\tau(\\alpha,t)=\\frac{z_\\alpha(\\alpha,t)}{|z_\\alpha(\\alpha,t)|},\\quad n(\\alpha,t)=\\frac{z_\\alpha(\\alpha,t)^\\perp}{|z_\\alpha(\\alpha,t)|}.\n\t\\end{equation}\nWithout changing the shape of the interface we can replace the above equation by \t\n\\begin{equation}\\label{eqcurve}\nz_t(\\alpha,t)=\\big(BR(z,\\omega)(\\alpha,t)\\cdot n(\\alpha,t)\\big)n(\\alpha,t)+T(z(\\alpha,t))\\tau(\\alpha,t).\n\\end{equation}\nTherefore we have a closed system of equations for the the contour evolution system with (\\ref{eqcurve}), (\\ref{eqBR}), (\\ref{eqOmega}), and  (\\ref{Dz}).\n\nGiven its origins in petrochemical engineering and its mathematical equivalence with Hele-Shaw flows \\cite{SaffmanTaylor58}, the Muskat problem has long attracted a lot of attention from physics \\cite{Bear72,Saffman86}. Mathematically, the Muskat problem poses many challenges, since even the well-posedness of the problem is not always guaranteed. Indeed, when one neglects surface tension, the well-posedness depends on the  Rayleigh-Taylor condition (which is also called the Saffman-Taylor condition for the Muskat problem). If the fluids have different densities, this condition requires the denser fluid to be below the less dense fluid. When this condition is satisfied, i.e., in the stable setting \\cite{Ambrose04}, local-in-time existence for large initial data is known for both density and viscosity jump cases in 2d and 3d \\cite{CG07,CCG11, CCG13,CGS16}\nfor subcritical spaces\n\\cite{CGSV17,Mat16,AlazardOneFluid2019,AlazardLazar2019,NguyenPausader2019}.\nHowever, finite time singularities can arise even from these stable configurations. As a matter of fact, the Muskat problem was the first incompressible model where blow-up was proved starting with well-posed initial data \\cite{CCFGL12,CCFG13,GG14,CCFG16}.\n\nFrom the previous considerations, it is an important question to determine under which conditions the solution exists and remains regular globally in time.  For the non-surface tension case, the global existence in the stable setting was first obtained  for small enough initial data in subcritical norms, allowing both density and viscosity jumps \\cite{SCH04,CG07,EM11, CGS16} and later for some critical norms \\cite{BSW14,CGSV17}. Very recently, global well-posedness results appeared that allow initial data of \\textit{medium} size in critical spaces, meaning initial data explicitly bounded independent of any parameter: first only for the density jump case \\cite{CCGS13,CCGRS16,Cam17,Cam20}, and later extended to the density-viscosity jump case \\cite{GGPS19}.  In particular in \\cite{GGPS19} there is a medium-size bound for the initial data that is independent of any parameter of the system when $|A_\\mu|=1$, and that value of the bound for the initial data is improved when $|A_\\mu|<1$. In all these results, the magnitude of the slope of the first derivative appears as a crucial quantity. However, this restriction is removed in \\cite{CordobaLazar2018,GancedoLazar2020} by assuming smallness in the critical $L^2$ based Sobolev norm.\n\n\n\n On the other hand, in the unstable scenario, the problem is ill-posed in all Sobolev spaces $H^s$, $s>0$ \\cite{GGPS19}, unless surface tension is taken into account. In that case, surface tension controls the instabilities at large scales, giving well-posedness. Classical results for this scenario can be found in \\cite{DuchonRobert1984,Chen93,ES97}. See the recent work \\cite{NguyenST2019} for low regularity initial data. Unstable scenarios are known \\cite{Ott97} which exhibit exponential growth locally in time of low order norms \\cite{GHS07}, and finger shaped unstable stationary solutions were also studied \\cite{EM11}. In particular, Rayleigh-Taylor stable solutions with surface tension converge to the solution without surface tension \\cite{Ambrose2014} with optimal decay rate or low regularity \\cite{FlynnNguyen2020}. Here, this is not the case as the scenario we deal with is Rayleigh-Taylor unstable. \nRecently, while writing this paper, unstable fluid layers have been proved to exist globally in time for initial near flat configurations \\cite{GG-BS2019}.\n \nIn this paper, we aim to improve the understanding of the effects produced by the surface tension for bubble-shaped interfaces. In particular, we consider the movement of fluid bubbles under the effect of gravity in another fluid with both different densities and viscosities. This is a highly unstable situation, as the Rayleigh-Taylor condition cannot hold for a closed curve. The function that provides the Rayleigh-Taylor condition has mean zero in this scenario. Moreover, as one expects, we will show that a less dense bubble moves upwards. But this means that on the top part of the interface, the less viscous fluid may push the more viscous one and the denser one is on top of the lighter one: both classic conditions in the linear Rayleigh-Taylor analysis are violated here. So that in our scenario, gravity effects make hard to find global-in-time control.\nPrevious results dealing with this setting \\cite{CP93,YT11,YT12} assumed no gravity force (i.e., $g=0$ or no density jump) and required small initial data in high regularity spaces (such as $H^r$ for $r\\geq 4$).\n\n We show here that even without surface tension, circle shaped curves are steady state solutions evolving vertically due to gravity. Furthermore, this surprising state in this unstable configuration allows to find global-in-time existence for capillarity bubbles. We will show that if the initial interface of a bubble is close to a circle with respect to a  constant depending on the dimensionless constants\n$$|A_\\mu|\\quad\\mbox{and}\\quad  \\frac{R^2|A_\\rho|}{A_\\sigma},\\quad\\mbox{with}\\quad \\pi R^2=|D(0)|,$$\n then the solution exists globally in time and, moreover, it becomes instantly analytic. In particular, in our proof it is possible to compute the explicit numerical condition that the initial data must satisfy. It is interesting to notice that only two quantities are involved, where the second represents the ratio between gravity force per length and surface tension, \n$$\\frac{|A_{\\rho}|R^2}{A_\\sigma}=\\frac{gR^2|\\rho_2-\\rho_1|}{\\sigma}.$$\nWe will also show that these bubbles converge exponentially fast in time to a circle that moves vertically with constant velocity equal to $A_\\rho$ (upwards if $A_\\rho>0$). Due to the incompressibility condition, the area of the bubble is preserved during the process.\nWe give precise statements of these results in Section \\ref{MainResults}. In next section, we provide the contour equations we use throughout the paper.\n\nNote that the parameterization that is used in for instance \\cite{CG07,CCGS13,CCGRS16,Cam17,GGPS19} is difficult to use in our scenario (those results are close to a horizontal line while in contrast the results in this paper are close to a circle) because our system in general sends the solution to a nearby circular steady state, and not to the one that we linearize around. The steady state that the solution converges to is determined by the dynamics and, without another conservation law that may not exist, then the limit can not be predicted by the initial data alone. In particular, the standard parametrization for star-shape bubbles given by\n\\begin{equation*}\n    z(\\alpha,t)=R(1+f(\\alpha,t))(\\cos{(\\alpha)},\\sin{(\\alpha)})\n\\end{equation*}\ndoes not do a good job of describing the nearby circular steady states, and therefore it is hard to use in this context. In particular, unless the center is the origin, which one cannot know \\textit{a priori}, circles parametrized in this form do not have a simple expression.\nFrom the analytical point of view, \nlooking at the decay on the Fourier side, it is easy to find that there is no dissipation at the linear level for the $\\pm 1$ Fourier coefficients, which corresponds to the fact that the center of a circle parametrized in this way is given by the $\\pm 1$ Fourier coefficients of $f$. At the nonlinear level, these Fourier coefficients are present and mixed in the evolution together with the rest of them, making it difficult to control globally in time. In order to handle this issue we reparametrize the interface getting a tangent vector to the curve with length independent of the parameter $\\alpha$  so that $\\partial_\\alpha |z_\\alpha|=0$  \\cite{Hou94}. Therefore, the system can be reformulated in terms of the angle formed between the tangent and the horizontal, $\\alpha+\\vartheta(\\alpha,t)$, and the length of the curve as follows\n$$\nz_\\alpha(\\alpha,t)=\\frac{L(t)}{2\\pi}(\\cos(\\alpha+\\vartheta(\\alpha,t)),\\sin(\\alpha+\\vartheta(\\alpha,t))).\n$$\nThe main unknown to control in this setting is $\\vartheta$. \n\nIn this parametrization circles correspond to a constant value of $\\vartheta$. \nThe evolution  of the zero frequency of $\\vartheta$ is decoupled from the rest.\nWhile the $\\pm 1$ are also neutral in this formulation, the simple compatibility condition\n$$\n\\int_{-\\pi}^\\pi z_\\alpha(\\alpha,t)d\\alpha=0\n$$\nused in \\cite{YT11,YT12}, allows us to control the $\\pm 1$ Fourier coefficients of $\\vartheta$ in terms of the higher modes.   For the higher Fourier modes we can use the dissipation due to surface tension. All the frequencies, together with the initial condition, determine the evolution of the center of the bubble.\n\nOn the other hand, in the analysis done around circles, it is possible to check that the Fourier coefficients of different frequencies interact together in the evolution, even at the linear level. If the ratio $|A_\\rho|R^2\/A_\\sigma$  between gravity and surface tension forces is large, it is not straightforward how to take advantage of the dissipation. Thus it is not clear how to obtain the global-in-time result in terms only of the size of the initial data and not upon the size of the parameters. In order to obtain a global result that does not rely on the size of the physical parameters of the problem, we preform a transformation in Fourier space of the infinite-dimensional nonlinear system and we prove that this transformation \\textit{diagonalizes} the linear system so that our result holds for any size of the physical parameters.  In particular, we show that it is possible to obtain explicitly the size of the smallness constant. Finally, the analysis we perform is for low regularity initial data, ($z_0(\\alpha)\\in C^{1,\\frac12}(\\mathbb T)$), allowing unbounded  initial curvature and providing instant (analytic) smoothing. \n\n\n\n\\subsection{Outline} The rest of the paper is structured as follows.  In Section \\ref{formulation}, we explain the contour dynamics formulation of the system of equations for the interface and we derive the full linearization.  Section \\ref{MainResults} records the notation that will be used in the rest of the paper and explains the main theorem proving global existence, uniqueness and exponential large time decay.  Then Section \\ref{IFTSection} gives the proof of the implicit function theorem to obtain the implicit relation between $\\hat{\\theta}(\\pm 1)$ and the higher Fourier modes.\nIn Section \\ref{sec:FourierRandS} we prove the Fourier multiplier estimates for the operators $\\mathcal{R}$ and $\\mathcal{S}$.\nSection \\ref{secw} proves the a priori estimates on the vorticity strength $\\omega$.\nIn Section \\ref{secanalytic} we use all the previous estimates to prove the global existence and instant analyticity of solutions.\nLastly in Section \\ref{sec:uniqueness} we explain the proof of uniqueness.\n\n\n\n\n\n\\section{Contour dynamics formulation}\\label{formulation}\n\n\nIn this section we introduce the contour evolution equations that will be used throughout the paper. We suppress the dependence in $t$ for clarity of notation. We note that in the introduction it was convenient to introduce the system using vector notation.  However in the rest of the paper, we will study the equation using complex notation.    In Section \\ref{sec:ComplexVector} we explain some complex notation used in the rest of this paper.  In Section \\ref{subsec:parametrization}, we rewrite the equations (\\ref{eqcurve}), (\\ref{eqBR}), (\\ref{eqOmega}), (\\ref{Dz}) in terms of the length of the curve and the angle of the tangent vector \\cite{Hou94}.  Then in Section \\ref{subsec:evolutionSystem} we derive an equivalent expression for the evolution of the length of the curve.  \nIn Section \\ref{sec:FourierTransCalc}, we explain the calculations that we will use involving the Fourier transform in our further decompositions.\nLastly in Section \\ref{sec:linearization}  we decompose the equations into linear and nonlinear parts.  In particular, the calculation of the expression for the linearized operator is given in Proposition \\ref{linearfourier}.\n\n\n\\subsection{Complex notation and vector notation}\\label{sec:ComplexVector}\nIn particular given $x=(x_1, x_2)$ and $y=(y_1, y_2)$ in vector notation and given $z=x_1+ix_2$ and $w=y_1+iy_2$ in complex notation, then the inner product is expressed as\n\\begin{equation}\\notag\n    x\\cdot y = x_1 y_1 + x_2 y_2 = \\text{Re}\\hspace{0.05cm}(\\overline{z} w).    \n\\end{equation}\nHere $\\overline{z} = x_1 - i x_2$ is the complex conjugate.  Similarly in two dimensions in vector notation we can write  $x \\wedge y \\overset{\\mbox{\\tiny{def}}}{=} x_1 y_2 - x_2 y_1$ in vector notation and this is equal to $\\text{Im}\\hspace{0.05cm}{(\\overline{x}y)}$ in complex notation.  Then for a vector the perpendicular is $x^\\perp = (-x_2, x_1)$, and in complex notation the perpendicular is $iz= -x_2 + i x_1$.  We will use the complex notation in most of the rest of the paper.\n\n\n\n\n\\subsection{Parametrization}\\label{subsec:parametrization}\n\nNow we define $\\vartheta(\\alpha)$ so\nthat $\\alpha+\\vartheta(\\alpha)$ is the angle formed between the tangent to the curve and the horizontal. In complex notation, this means that\n\\begin{equation}\\label{param}\n    z_\\alpha(\\alpha)=|z_\\alpha(\\alpha)|e^{i(\\alpha+\\vartheta(\\alpha))}.\n\\end{equation}\nIn this formulation the normal and tangential vectors from \\eqref{vectors} are \n\\begin{equation*}\n    n(\\alpha)=ie^{i(\\alpha+\\vartheta(\\alpha))}, \\quad \\tau(\\alpha) = e^{i(\\alpha+\\vartheta(\\alpha))}.\n\\end{equation*}\nWe will then denote the normal velocity by $U(\\alpha)$ with\n\\begin{equation}\\label{U}\n    U(\\alpha)=\\text{Re}\\hspace{0.05cm}(\\overline{BR}(z,\\omega)(\\alpha) n(\\alpha))=\\text{Re}\\hspace{0.05cm}(\\overline{BR}(\\omega)(\\alpha) ie^{i(\\alpha+\\vartheta(\\alpha))}),  \n\\end{equation}\nwith the Birkhoff-Rott integral \\eqref{eqBR} given by\n\\begin{equation*}\n    \\overline{BR}(z,\\omega)(\\alpha)=\\frac{1}{2\\pi i}\\text{pv}\\int_{-\\pi}^{\\pi}\\frac{\\omega(\\beta)}{z(\\alpha)-z(\\beta)}d\\beta.\n\\end{equation*}\nNote that $\\overline{BR}(z,\\omega)$ is the complex conjugate of \\eqref{eqBR} written in complex notation, this holds in particular since $\\omega(\\beta)$ is seen to be real as in \\eqref{omega} below.\nThese expressions can be written in terms of $z_\\alpha(\\alpha)$ by noticing that\n\\begin{equation*}\n    z(\\alpha)-z(\\alpha-\\beta)=\\int_0^\\alpha z_\\alpha(\\eta)d\\eta-\\int_0^{\\alpha-\\beta} z_\\alpha(\\eta)d\\eta =\\beta\\int_0^1 z_\\alpha(\\alpha+(s-1)\\beta)ds.\n\\end{equation*}\nThe equation \\eqref{eqcurve} then reads as follows\n\\begin{equation}\\label{eqcurve.theta}\n    z_t(\\alpha)=U(\\alpha)n(\\alpha)+T(\\alpha)\\tau(\\alpha).\n\\end{equation}\nTaking a derivative in $\\alpha$ and projecting into normal and tangential components, we obtain the evolution equations for $\\vartheta(\\alpha)$ and $|z_\\alpha(\\alpha)|$:\n\\begin{equation}\\label{thetaeqaux}\n\\begin{aligned}\n    \\vartheta_t(\\alpha)&=\\frac{1}{|z_\\alpha(\\alpha)|}\\Big(U_\\alpha(\\alpha)+T(\\alpha)(1+\\vartheta_\\alpha(\\alpha))\\Big),\\\\\n    |z_\\alpha(\\alpha)|_t&=T_\\alpha(\\alpha)-(1+\\vartheta_\\alpha(\\alpha))U(\\alpha).\n\\end{aligned}\n\\end{equation}\nNow, we can choose a tangential velocity $T(\\alpha)$ so that the parametrization of $z(\\alpha)$ has a tangent vector whose modulus does not depend on $\\alpha$. Indeed, we impose\n\\begin{equation}\\label{modul}\n    |z_\\alpha(\\alpha)|=\\frac{1}{2\\pi}\\int_{-\\pi}^{\\pi}|z_\\alpha(\\eta)|d\\eta=\\frac{L(t)}{2\\pi},\n\\end{equation}\nwhere $L(t)$ is the length of the curve at time $t$. We then differentiate in time the equation above and use equation \\eqref{thetaeqaux}$_2$ twice to obtain that\n\\begin{equation}\\label{T}\n    T(\\alpha)=\\int_0^{\\alpha}(1+\\vartheta_\\alpha(\\eta))U(\\eta)d\\eta-\\frac{\\alpha}{2\\pi}\\int_{-\\pi}^{\\pi}(1+\\vartheta_\\alpha(\\eta))U(\\eta)d\\eta+T(0),\n\\end{equation}\nwhere $T(0)$ simply provides a change of frame in the parametrization. \nTherefore, after substitution of this expression of $T(\\alpha)$ into \\eqref{thetaeqaux} and using the relation $|z_\\alpha(\\alpha)|=\\frac{L(t)}{2\\pi}$, the evolution system in terms of $\\vartheta(\\alpha)$ and $L(t)$ is the following\n\\begin{equation}\\label{eqcurve3}\n\\begin{aligned}\n    \\vartheta_t(\\alpha)&=\\frac{2\\pi}{L(t)} U_\\alpha(\\alpha)+\\frac{2\\pi}{L(t)}T(\\alpha)(1+\\vartheta_\\alpha(\\alpha)),\\\\\n    L_t(t)&=-\\int_{-\\pi}^{\\pi} (1+\\vartheta_\\alpha(\\alpha))U(\\alpha)d\\alpha,\n\\end{aligned}\n\\end{equation}\nwhere $T(\\alpha)$ is defined in \\eqref{T}, with $T(0)$ free to choose, \nand $U(\\alpha)$ is given by \\eqref{U} with\n\\begin{equation}\\label{BR}\n    \\overline{BR}(\\omega)(\\alpha)=\\frac{1}{ i L(t) }\\text{pv}\\int_{-\\pi}^{\\pi}\\frac{\\omega(\\alpha-\\beta)}{\\int_0^1 e^{i(\\alpha+(s-1)\\beta)}e^{i\\vartheta(\\alpha+(s-1)\\beta)}ds}\\frac{d\\beta}{\\beta}.\n\\end{equation}\nRecalling the expression of the curvature in terms of the angle using \\eqref{curvature}, \n\\begin{equation*}\n    K(z)(\\alpha)=\\frac{2\\pi}{L(t)}(1+\\vartheta_\\alpha(\\alpha)),\n\\end{equation*}\nthe equation for the vorticity strength $\\omega(\\alpha)$ in \\eqref{eqOmega} reads as follows\n \\begin{equation}\\label{omega}\n    \\omega(\\alpha)=2A_\\mu \\frac{L(t)}{2\\pi} \\mathcal{D}(\\omega)(\\alpha)+2A_\\sigma\\frac{2\\pi}{L(t)} \\vartheta_{\\alpha\\alpha}(\\alpha)-2A_\\rho\\frac{L(t)}{2\\pi}\\sin{(\\alpha+\\vartheta(\\alpha))},\n\\end{equation}\n with $\\mathcal{D}(z,\\omega)(\\alpha)$ in \\eqref{Dz} given by\n\\begin{equation}\\label{md}\n    \\mathcal{D}(\\omega)(\\alpha)=-\\text{Re}\\hspace{0.05cm}(\\overline{BR}(\\omega)(\\alpha) e^{i(\\alpha+\\vartheta(\\alpha))}).\n\\end{equation}\nIn addition, \nwe notice that because $|z_\\alpha(\\alpha)|$ is constant in $\\alpha$ and $z(\\alpha)$ is a closed curve, then the following constraint must hold\n\\begin{equation}\\label{constraint}\n    0=\\int_{-\\pi}^{\\pi} \\frac{z_\\alpha(\\alpha)}{|z_\\alpha(\\alpha)|} d\\alpha=\\int_{-\\pi}^{\\pi} e^{i(\\alpha+\\vartheta(\\alpha))}d\\alpha.\n\\end{equation}\nFinally, once the system \\eqref{T}-\\eqref{md} is solved, one can track the evolution of a single point, say that with $\\alpha=0$, by integrating in time \\eqref{eqcurve} (notice that the right hand side of \\eqref{eqcurve} has been shown to depend only on $z_\\alpha$, given by \\eqref{param}, \\eqref{modul}, and \\eqref{eqcurve3}).\n\n\\subsection{Evolution System}\\label{subsec:evolutionSystem}\nFor our purposes, equation \\eqref{eqcurve3}$_2$ is not  convenient  to study $L(t)$. Instead, we will make use of the fact that the fluid is incompressible, and thus the volume is preserved.\nThe volume is given in terms of the curve $z(\\alpha)$ by\n\\begin{equation*}\n    V=\\frac12\\int_{-\\pi}^\\pi z(\\alpha)\\wedge  z_\\alpha(\\alpha)d\\alpha,\n\\end{equation*}\nwhich in complex notation reads as\n\\begin{equation*}\n    V=\\frac12\\text{Im}\\hspace{0.05cm}\\int_{-\\pi}^\\pi  \\overline{z(\\alpha)} z_\\alpha(\\alpha) d\\alpha.\n\\end{equation*}\nSince $U(\\alpha)$ in \\eqref{U} is a total derivative in $\\alpha$, the conservation of volume is obtained by simply taking a time derivative in the equation above.\nNow, from \\eqref{param} and \\eqref{modul} we have that\n$$\nz_\\alpha(\\alpha)=\\frac{L(t)}{2\\pi}e^{i(\\alpha+\\vartheta(\\alpha))},\n$$\nand we can write\n\\begin{equation*}\n    z(\\alpha)=z(0)+\\int_0^\\alpha z_\\alpha(\\eta)d\\eta.\n\\end{equation*}\nThen the conservation of volume  writes as follows\n\\begin{equation}\\label{volumelength}\n\\begin{aligned}\n    V_0&=\\pi R^2\n    \\\\\n    &=\\frac{1}{2}\\Big(\\frac{L(t)}{2\\pi}\\Big)^2\\text{Im}\\hspace{0.05cm} \\Big(\\int_{-\\pi}^\\pi\\int_0^\\alpha e^{i(\\alpha-\\eta)}e^{i(\\vartheta(\\alpha)-\\vartheta(\\eta))}d\\eta d\\alpha\\Big)\\\\\n    &=\\frac{1}{2}\\Big(\\frac{L(t)}{2\\pi}\\Big)^2\\text{Im}\\hspace{0.05cm} \\Big(2\\pi i\n    +\\int_{-\\pi}^\\pi\\int_0^\\alpha e^{i(\\alpha-\\eta)} \\sum_{n\\geq1}\\frac{i^n}{n!}(\\vartheta(\\alpha)-\\vartheta(\\eta))^n d\\eta d\\alpha\\Big).\n\\end{aligned}\n\\end{equation}\nThis yields the following equation for $L(t)$:\n\\begin{equation}\\label{Lequation}\n\\Big(\\frac{L(t)}{2\\pi}\\Big)^2=R^2\\Big(1+\\frac{1}{2\\pi}\\text{Im}\\hspace{0.05cm} \\int_{-\\pi}^\\pi\\int_0^\\alpha e^{i(\\alpha-\\eta)} \\sum_{n\\geq1}\\frac{i^n}{n!}(\\vartheta(\\alpha)-\\vartheta(\\eta))^n d\\eta d\\alpha\\Big)^{-1}.\n\\end{equation}\nThis is the equation for $L(t)$ that will be use later in the paper,\n\nConversely, it is not hard to check that \\eqref{Lequation} implies \\eqref{eqcurve3}$_2$. In fact, taking a time derivative of \\eqref{volumelength}, and assuming $L(t)\\neq0$, gives that\n\\begin{equation*}\n    \\begin{aligned}\n    L'(t)&=\\frac{1}{4\\pi R^2}\\Big(\\frac{L(t)}{2\\pi}\\Big)^3\\text{Im}\\hspace{0.05cm} \\int_{-\\pi}^\\pi\\int_0^\\alpha i e^{i(\\alpha-\\eta)} e^{i(\\vartheta(\\alpha)\\!-\\!\\vartheta(\\eta))}(\\vartheta_t(\\alpha)\\!-\\!\\vartheta_t(\\eta)) d\\eta d\\alpha\\\\\n    &=\\frac{1}{4\\pi R^2}\\Big(\\frac{L(t)}{2\\pi}\\Big)^3\\text{Im}\\hspace{0.05cm} \\int_{-\\pi}^\\pi i e^{i\\alpha} e^{i\\vartheta(\\alpha)}\\vartheta_t(\\alpha)\\int_0^\\alpha  e^{-i\\eta} e^{-i\\vartheta(\\eta)}d\\eta d\\alpha\\\\\n    &\\quad-\\frac{1}{4\\pi R^2}\\Big(\\frac{L(t)}{2\\pi}\\Big)^3\\text{Im}\\hspace{0.05cm} \\int_{-\\pi}^\\pi i e^{i\\alpha} e^{i\\vartheta(\\alpha)}\\int_0^\\alpha  e^{-i\\eta} e^{-i\\vartheta(\\eta)}\\vartheta_t(\\eta)d\\eta d\\alpha.\n    \\end{aligned}\n\\end{equation*}\nThus writing in the first term $e^{i\\alpha} e^{i\\vartheta(\\alpha)}\\vartheta_t(\\alpha)=\\partial_\\alpha\\int_0^\\alpha e^{i\\eta} e^{i\\vartheta(\\eta)}\\vartheta_t(\\eta)$ and integrating by parts we obtain that\n\\begin{equation}\\label{auxL}\n    \\begin{aligned}\n    L'(t)=\\frac{-1}{2\\pi R^2}\\Big(\\frac{L(t)}{2\\pi}\\Big)^3\\text{Im}\\hspace{0.05cm} \\int_{-\\pi}^\\pi i e^{i\\alpha} e^{i\\vartheta(\\alpha)}\\int_0^\\alpha  e^{-i\\eta} e^{-i\\vartheta(\\eta)}\\vartheta_t(\\eta)d\\eta d\\alpha.\n    \\end{aligned}\n\\end{equation}\nUsing the equation for $\\vartheta$ in \\eqref{eqcurve3}$_1$, we have\n\\begin{equation*}\n    \\begin{aligned}\n     \\frac{L(t)}{2\\pi}&\\int_0^\\alpha  e^{-i\\eta} e^{-i\\vartheta(\\eta)}\\vartheta_t(\\eta)d\\eta=\n    \\int_0^\\alpha  e^{-i\\eta-i\\vartheta(\\eta)}\\big(U_\\alpha(\\eta)+ T(\\eta)(1+\\vartheta_\\alpha(\\eta))\\big)\n     d\\eta\\\\\n     &=e^{-i\\alpha-i\\vartheta(\\alpha)}U(\\alpha)-e^{-i\\vartheta(0)}U(0)+i\\int_0^\\alpha  e^{-i\\eta-i\\vartheta(\\eta)}(1+\\vartheta_\\alpha(\\eta))U(\\eta)d\\eta\\\\\n     &\\quad+i\\int_0^\\alpha  \\partial_\\eta\\big(e^{-i\\eta-i\\vartheta(\\eta)}\\big)T(\\eta) d\\eta.\n    \\end{aligned}\n\\end{equation*}\nWe then integrate by parts once more in the last term, also using \\eqref{T}, to obtain\n\\begin{equation*}\n    \\begin{aligned}\n    i&\\int_0^\\alpha  \\partial_\\eta\\big(e^{-i\\eta-i\\vartheta(\\eta)}\\big)T(\\eta) d\\eta=i e^{-i\\alpha-i\\vartheta(\\alpha)}T(\\alpha)-i e^{-i\\vartheta(0)}T(0)\\\\\n    &-i\\int_0^\\alpha e^{-i\\eta-i\\vartheta(\\eta)}(1+\\vartheta_\\alpha(\\eta))U(\\eta)d\\eta +\\frac{i}{2\\pi}\\int_{-\\pi}^\\pi(1+\\theta_\\alpha(\\eta))U(\\eta)d\\eta \\int_0^\\alpha e^{-i\\eta-i\\vartheta(\\eta)}d\\eta.\n    \\end{aligned}\n\\end{equation*}\nSubstituting into the previous equation we find that\n\\begin{equation*}\n    \\begin{aligned}\n     \\frac{L(t)}{2\\pi}&\\int_0^\\alpha  e^{-i\\eta} e^{-i\\vartheta(\\eta)}\\vartheta_t(\\eta)d\\eta=\n    e^{-i\\alpha-i\\vartheta(\\alpha)}U(\\alpha)-e^{-i\\vartheta(0)}U(0)+i e^{-i\\alpha-i\\vartheta(\\alpha)}T(\\alpha)\\\\\n    &-i e^{-i\\vartheta(0)}T(0)\n     +\\frac{i}{2\\pi}\\int_{-\\pi}^\\pi(1+\\theta_\\alpha(\\eta))U(\\eta)d\\eta \\int_0^\\alpha e^{-i\\eta-i\\vartheta(\\eta)}d\\eta.\n    \\end{aligned}\n\\end{equation*}\nThus, plugging this back into \\eqref{auxL} and using relation \\eqref{constraint} gives\n\\begin{equation*}\n    \\begin{aligned}\n    L'(t)=\\frac{-1}{2\\pi R^2}\\Big(\\frac{L(t)}{2\\pi}\\Big)^2 \\Big(\\text{Im}\\hspace{0.05cm}\\! \\int_{-\\pi}^\\pi \\! \\!\\!e^{i\\alpha+i\\vartheta(\\alpha)}\\! \\int_0^\\alpha\\! \\! e^{-i\\eta-i\\vartheta(\\eta)} d\\eta d\\alpha\\Big)\\!\\int_{-\\pi}^\\pi\\!\\! (1\\!+\\!\\vartheta_\\alpha(\\eta))U(\\eta)d\\eta,\n    \\end{aligned}\n\\end{equation*}\nwhich recalling \\eqref{volumelength} implies \\eqref{eqcurve3}$_2$. \n\nWe will later show (see Subsection \\ref{sec:Lestimate}) that the condition on the initial data will guarantee that $L(t)>0$ for all time, and thus the formulations using \\eqref{eqcurve3}$_2$ and \\eqref{Lequation} are equivalent.\n\nIn summary, the closed system of equations that define the evolution of the Muskat bubble can be expressed by \\eqref{eqcurve3}$_1$ and \\eqref{Lequation}, together with \\eqref{constraint}.  We will study the evolution of this system to prove our main results.\n\n\n\n\n\n\\subsection{Calculations Involving the Fourier Transform}\\label{sec:FourierTransCalc}\n\nIn this section we recall basic calculations for the periodic Fourier transform that will be used in the next section.  In particular we define the Fourier transform of a periodic function $g$ with domain $\\mathbb{T}=[-\\pi, \\pi]$ as: \n\\begin{equation}\\notag\n    \\mathcal{F}(g)(k)\\overset{\\mbox{\\tiny{def}}}{=}\\widehat{g}(k)=\\frac{1}{2\\pi}\\int_{-\\pi}^\\pi g(\\alpha)e^{-i k\\alpha}d\\alpha,\n\\end{equation}\nand the corresponding Fourier series\n\\begin{equation}\\notag\n    g(\\alpha)=\\sum_{k\\in\\mathbb{Z}}\\widehat{g}(k)e^{i k\\alpha}. \n\\end{equation}\nFor later use, we also define the periodic Hilbert transform as \n\\begin{equation}\\label{defHilbertTransform}\n    \\mathcal{H} (g)(\\alpha)\\overset{\\mbox{\\tiny{def}}}{=}\\frac{1}{2\\pi}\\text{pv}\\int_{-\\pi}^{\\pi}\\frac{g(\\alpha-\\beta)}{\\tan{(\\beta\/2)}}.\n\\end{equation}\nWe notice that $\\mathcal{H} (g)(\\alpha)=-\\frac{1}{2\\pi}\\text{pv}\\int_{-\\pi}^{\\pi}\\frac{g(\\alpha+\\beta)}{\\tan{(\\beta\/2)}}$.  Adding half of these together one can calculate that $\\mathcal{H} (c)=0$ if $c\\in \\mathbb{C}$ is a constant and \n$\n\\mathcal{H} (g)(\\alpha)=-i\\sum_{k \\ne 0} \\text{sgn}(k) \\widehat{g}(k)e^{i k\\alpha}.\n$\nAnd further $\\mathcal{F}(\\mathcal{H} (g))(k) = -i \\text{sgn}(k) \\widehat{g}(k)$.  Then we define the operator $\\Lambda$ using the Fourier transform as \n$\n\\mathcal{F}(\\Lambda g)(k)\\overset{\\mbox{\\tiny{def}}}{=} |k| \\widehat{g}(k).\n$\nAnd we observe that $\\mathcal{H} (g_\\alpha )(\\alpha)=\\sum_{k\\in \\mathbb{Z}} |k| \\widehat{g}(k)e^{i k\\alpha} = \\Lambda g$.    And furthermore $$\\partial_\\alpha \\mathcal{H} (g_{\\alpha\\alpha} )(\\alpha)=-\\sum_{k\\in \\mathbb{Z}} |k|^3 \\widehat{g}(k)e^{i k\\alpha} = -\\Lambda^3 g.$$  \nAlso one can compute by plugging in the Fourier series that\n\\begin{equation}\\label{fourierCalcAlpha}\n        \\mathcal{F}\\Big(\\int_0^\\alpha g(\\eta)d\\eta-\\frac{\\alpha}{2\\pi}\\int_{-\\pi}^\\pi g(\\eta)d\\eta \\Big)(k)=\\left\\{\n    \\begin{aligned}-\\frac{i}{k} \\widehat{g}(k),\\quad k\\neq0,\\\\\n        \\sum_{j\\neq0}\\frac{i}{j}\\widehat{g}(j),\\quad k=0,\n    \\end{aligned}\\right.\n\\end{equation}\nThese calculations will be used in the next section when we take the Fourier transform of the linearization.\n\n\n\\subsection{Linearization and Nonlinear Expansion}\\label{sec:linearization}\n\nWe proceed next to decompose the equation for $\\vartheta$ in the system \\eqref{T}-\\eqref{md} into linear and nonlinear parts. We will Taylor expand the nonlinear terms around the zero frequency of $\\vartheta(\\alpha)$. Define\n\\begin{equation}\\label{theta}\n    \\theta(\\alpha)=\\vartheta(\\alpha)-\\hat{\\vartheta}(0).\n\\end{equation}\nTaking into account that\n\\begin{equation*}\n    \\int_0^1 e^{i(\\alpha+(s-1)\\beta)}ds=e^{i\\alpha}\\frac{1-e^{-i\\beta}}{i\\beta},\n\\end{equation*}\nwe write the denominator of \\eqref{BR} as follows\n\\begin{multline*}\n    \\int_0^1 e^{i(\\alpha+(s-1)\\beta)}e^{i\\hat{\\vartheta}(0)}e^{i\\theta(\\alpha+(s-1)\\beta)}ds=\\\\\n    e^{i\\hat{\\vartheta}(0)}e^{i\\alpha}\\frac{1-e^{-i\\beta}}{i\\beta}\\left(e^{-i\\alpha}\\frac{i\\beta}{1-e^{-i\\beta}}\\int_0^1 e^{i(\\alpha+(s-1)\\beta)}e^{i\\theta(\\alpha+(s-1)\\beta)}ds-1+1\\right).\n\\end{multline*}\nThen after performing a Taylor expansion, \\eqref{BR} is given by \n\\begin{multline*}\n    \\overline{BR}(\\omega)(\\alpha)=\\frac{e^{-i\\hat{\\vartheta}(0)}}{iL(t)} \\text{pv}\\int_{-\\pi}^{\\pi}\\frac{\\omega(\\alpha-\\beta)}{\\beta e^{i\\alpha}\\frac{1-e^{-i\\beta}}{i\\beta}}\n    \\\\\\cdot\\sum_{n\\geq 0}\\left(1-\\frac{i\\beta}{1-e^{-i\\beta}}\\int_0^1 e^{i(s-1)\\beta}e^{i\\theta(\\alpha+(s-1)\\beta)}ds\\right)^nd\\beta.\n\\end{multline*}\nBy further Taylor expanding the exponential term, we find that\n\\begin{multline*}\n    \\overline{BR}(\\omega)(\\alpha)=\\frac{e^{-i\\hat{\\vartheta}(0)}e^{-i\\alpha}}{iL(t)}\\sum_{n\\geq 0}\\text{pv}\\int_{-\\pi}^{\\pi}\\frac{\\omega(\\alpha-\\beta)}{\\beta }\\frac{(-1)^n(i\\beta)^{n+1}}{(1-e^{-i\\beta})^{n+1}}\\\\\n    \\cdot\\Big(\\sum_{m\\geq 1}\\frac{i^m}{m!}\\int_0^1 e^{i(s-1)\\beta}(\\theta(\\alpha+(s-1)\\beta))^m ds\\Big)^nd\\beta.\n\\end{multline*}\nWe further Taylor expand $e^{i\\theta(\\alpha)}$.  Then plugging these expansions into \\eqref{U} provides the series for $U(\\alpha)$\n\\begin{multline*}\n    U(\\alpha)=\\text{Re}\\hspace{0.05cm}\\left(\\frac{1}{L(t)}\\sum_{n,l\\geq0}\\frac{(i\\theta(\\alpha))^l}{l!}\\text{pv}\\int_{-\\pi}^{\\pi}\\frac{\\omega(\\alpha-\\beta)(-1)^n(i\\beta)^{n+1}}{\\beta(1-e^{-i\\beta})^{n+1}}\n    \\right.\\\\\n    \\left.\\cdot\\left(\\sum_{m\\geq 1}\\frac{i^m}{m!}\\int_0^1 e^{i(s-1)\\beta}(\\theta(\\alpha+(s-1)\\beta))^m ds\\right)^nd\\beta\\right).\n\\end{multline*}\nFor convenience, we introduce the following notation for the operators $\\mathcal{R}$ and $\\mathcal{S}$.  We first define $\\mathcal{R}$:  \n\\begin{equation}\\label{R}\n    \\mathcal{R}(f)(\\alpha)\\!=\\!\\frac{i}{\\pi}\\text{pv}\\!\\!\\int_{-\\pi}^{\\pi}\\!\\!\\!\\frac{f(\\alpha\\!-\\!\\beta)}{\\beta}\\frac{\\beta^2}{(1\\!-\\!e^{-i\\beta})^2}\\!\\!\\int_0^1\\!\\!e^{i(s-1)\\beta}\\theta(\\alpha\\!+\\!(s-1)\\beta) ds d\\beta.\n\\end{equation}\nAbove $\\mathcal{R}$ is chosen to be a linear function in $\\theta$, it corresponds to $l=0$, $n=1$ and $m=1$ in $U(\\alpha)$ above. Then, we further define the operator $\\mathcal{S}$ to be the nonlinear in $\\theta$ terms inside $U(\\alpha)$ above:\n\\begin{multline}\\label{S}\n    \\mathcal{S}(f)(\\alpha)=\n    \\frac{1}{\\pi}\\sum_{\\substack{n,l\\geq0 \\\\ n+l\\geq 2}}\\frac{(-1)^n i^{l+n+1}(\\theta(\\alpha))^l}{l!}\\text{pv}\\int_{-\\pi}^{\\pi}\\frac{f(\\alpha-\\beta)\\beta^{n+1}}{\\beta(1-e^{-i\\beta})^{n+1}}\\\\\n    \\cdot\n    \\left(\\sum_{m\\geq 1}\\frac{i^m}{m!}\\int_0^1 e^{i(s-1)\\beta}(\\theta(\\alpha+(s-1)\\beta))^m ds\\right)^n d\\beta\\\\\n    +\\frac{1}{\\pi}\\text{pv}\\int_{-\\pi}^{\\pi}\\frac{f(\\alpha-\\beta)\\beta^{2}}{\\beta(1-e^{-i\\beta})^{2}}\\sum_{m\\geq 2}\\frac{i^m}{m!}\\int_0^1 e^{i(s-1)\\beta}(\\theta(\\alpha+(s-1)\\beta))^m ds d\\beta.\n\\end{multline}\nThe $\\mathcal{S}$ operator corresponds to the terms in $U(\\alpha)$ above where $n,l \\ge 0$ and $n+l\\ge 2$ plus the case where $l=0$, $n=1$ and $m\\ge 2$. \n\nFor the cases in $U(\\alpha)$ where $n=l=0$ and $n=0$, $l=1$ we further notice that\n\\begin{equation}\\label{hilbert}\n    \\frac{1}{\\pi}\\text{pv}\\int_{-\\pi}^{\\pi}\\frac{f(\\alpha-\\beta)}{1-e^{-i\\beta}}d\\beta=-i\\mathcal{H}  f(\\alpha)+\\hat{f}(0),\n\\end{equation}\nwhere $\\mathcal{H}  f$ denotes the periodic Hilbert transform of $f$ as given in \\eqref{defHilbertTransform}.  The previous identity \\eqref{hilbert} is obtained multiplying above and below by $1-e^{i\\beta}$ and using the trigonometric identities\n$1-\\cos{(\\beta)}=2\\sin^2{(\\beta\/2)}$ and  $\\sin{(\\beta)}=2\\sin{(\\beta\/2)}\\cos{(\\beta\/2)}.\n$\nFurther $\\omega(\\alpha)$ in \\eqref{omega} can be written as an exact derivative, its mean value is zero and therefore $\\hat{\\omega}(0)=0$. \nThese calculations show that we can write the expression inside \\eqref{U} as\n\\begin{equation}\\notag\n    i\\overline{BR}(\\omega)(\\alpha) e^{i(\\alpha+\\vartheta(\\alpha))}\n    =\n    \\frac{\\pi}{L(t)} \\left(i \\theta(\\alpha) \\mathcal{H}(\\omega) + \\mathcal{H}(\\omega) +\\mathcal{R}(\\omega)(\\alpha)+\\mathcal{S}(\\omega)(\\alpha)\\right).\n\\end{equation}\nThus, noticing that the term with $n=0$, $l=1$ vanishes in $U(\\alpha)$ in \\eqref{U} because it is purely imaginary, using the notation above we can write $U(\\alpha)$ in the following manner\n\\begin{equation}\\label{Udecomp}\n    U(\\alpha)=\\frac{\\pi}{L(t)}\\Big(\\mathcal{H}  \\omega(\\alpha)+ \\text{Re}\\hspace{0.05cm}\\hspace{0.05cm}\\mathcal{R}(\\omega)(\\alpha)+ \\text{Re}\\hspace{0.05cm}\\hspace{0.05cm}\\mathcal{S}(\\omega)(\\alpha)\\Big).\n\\end{equation}\nProceeding similarly, in \\eqref{md}, one finds that\n\\begin{equation}\\label{Ddecomp}\n    \\mathcal{D}(\\omega)(\\alpha)=\\frac{-\\pi}{L(t)}\\Big(\\theta(\\alpha)\\mathcal{H}  \\omega(\\alpha)+ \\text{Im}\\hspace{0.05cm}\\hspace{0.05cm}\\mathcal{R}(\\omega)(\\alpha)+ \\text{Im}\\hspace{0.05cm}\\hspace{0.05cm}\\mathcal{S}(\\omega)(\\alpha)\\Big).\n\\end{equation}\nWe shall now split all the terms into zero, first or higher order polynomials of $\\theta(\\alpha)$. First, the vorticity strength \\eqref{omega} is split as follows\n\\begin{equation*}\n    \\omega(\\alpha)=\\omega_0(\\alpha)+\\omega_1(\\alpha)+\\omega_{\\geq2}(\\alpha),\n\\end{equation*}\nwhere\n\\begin{equation}\\label{omegasplit}\n\\left\\{\\begin{aligned}\n    \\omega_0(\\alpha)&\\!=\\!-A_\\rho\\frac{L(t)}{\\pi}\\sin{(\\alpha+\\hat{\\vartheta}(0))},\\\\\n    \\omega_1(\\alpha)&\\!=\\!A_\\mu\\frac{L(t)}{\\pi}\\mathcal{D}_1(\\omega_0)(\\alpha)\\!+\\!2A_\\sigma\\frac{2\\pi}{L(t)}\\theta_{\\alpha\\alpha}\\!\n    \\\\\n    &\\hspace{1.6cm}-\\!A_\\rho\\frac{L(t)}{\\pi}\\cos{(\\alpha\\!+\\!\\hat{\\vartheta}(0))}\\theta(\\alpha),\\\\\n    \\omega_{\\geq2}(\\alpha)&\\!=\\!A_\\mu\\frac{L(t)}{\\pi}\\mathcal{D}_{\\geq2}(\\omega)(\\alpha)\\!\n    \\\\ &\\hspace{1.6cm} -\\!A_\\rho\\frac{L(t)}{\\pi}\\sin{(\\alpha\\!+\\!\\hat{\\vartheta}(0))}\\sum_{j\\geq1}\\!\\!\\frac{(-1)^j(\\theta(\\alpha))^{2j}}{(2j)!}\\\\\n    &\\hspace{1.6cm}-A_\\rho\\frac{L(t)}{\\pi}\\cos{(\\alpha+\\hat{\\vartheta}(0))}\\sum_{j\\geq1}\\frac{(-1)^j(\\theta(\\alpha))^{1+2j}}{(1+2j)!},\\\\\n    \\omega_{\\geq1} &\\!=\\! \\omega_{1}+\\omega_{\\geq2}.\n\\end{aligned}\\right.\n\\end{equation}\nAbove we used the trigonometric identity\n$\\sin(a+b) = \\sin(a)\\cos(b)+\\cos(a)\\sin(b)$,\nas well as the Taylor expansions for sine and cosine. \nThen $\\mathcal{D}_1(\\omega_0)(\\alpha)$ and $\\mathcal{D}_{\\geq2}(\\omega)(\\alpha)$ are obtained, in turn, by introducing \\eqref{omegasplit} into \\eqref{Ddecomp} as follows\n\\begin{equation*}\n    \\mathcal{D}(\\omega)(\\alpha)=\\mathcal{D}_1(\\omega_0)(\\alpha)+\\mathcal{D}_{\\geq2}(\\omega)(\\alpha),\n\\end{equation*}\nwhere\n\\begin{equation}\\label{mdsplit}\n\\left\\{\\begin{aligned}\n    \\mathcal{D}_1(\\omega_0)(\\alpha)&=\\frac{-\\pi}{L(t)}\\Big(\\theta(\\alpha)\\mathcal{H}  \\omega_0(\\alpha)+\\text{Im}\\hspace{0.05cm}\\hspace{0.05cm}\\mathcal{R}(\\omega_0)(\\alpha)\\Big),\\\\\n    \\mathcal{D}_{\\geq2}(\\omega)(\\alpha)&=\\frac{-\\pi}{L(t)}\\Big(\\theta(\\alpha)\\mathcal{H}  \\omega_{\\geq1}(\\alpha)+\\text{Im}\\hspace{0.05cm}\\hspace{0.05cm} \\mathcal{R}(\\omega_{\\geq1})(\\alpha)+\\text{Im}\\hspace{0.05cm}\\hspace{0.05cm}\\mathcal{S}(\\omega)(\\alpha)\\Big).\n\\end{aligned}\\right.\n\\end{equation}\nAnalogously, the splitting for $U(\\alpha)$ from \\eqref{Udecomp} is\n\\begin{equation*}\n    U(\\alpha)=U_0(\\alpha)+U_1(\\alpha)+U_{\\geq2}(\\alpha),\n\\end{equation*}\\vspace{-0.5cm}\nwith\n\\begin{equation}\\label{Usplit}\n\\left\\{\\begin{aligned}\n    U_0(\\alpha)&=\\frac{\\pi}{L(t)}\\mathcal{H}  \\omega_0(\\alpha),\\\\\n    U_1(\\alpha)&=\\frac{\\pi}{L(t)}\\Big(\\mathcal{H}  \\omega_1(\\alpha)+\\text{Re}\\hspace{0.05cm}\\hspace{0.05cm}\\mathcal{R}(\\omega_0)(\\alpha)\\Big),\\\\\n    U_{\\geq2}(\\alpha)&=\\frac{\\pi}{L(t)}\\Big(\\mathcal{H}  \\omega_{\\geq2}(\\alpha)+\\text{Re}\\hspace{0.05cm}\\hspace{0.05cm} \\mathcal{R}(\\omega_{\\geq1})(\\alpha)+\\text{Re}\\hspace{0.05cm}\\hspace{0.05cm}\\mathcal{S}(\\omega)(\\alpha)\\Big).\n\\end{aligned}\\right.\n\\end{equation}\nRecalling the expression for $T(\\alpha)$ in \\eqref{T}, we find that\n\\begin{equation*}\n    T(\\alpha)=T_0(\\alpha)+T_1(\\alpha)+T_{\\geq2}(\\alpha),\n\\end{equation*}\nwhere, using $\\int_{-\\pi}^{\\pi} U_0(\\eta) d\\eta =0$, we have\n\\begin{equation}\\label{Tsplit}\n\\left\\{\\begin{aligned}\n    T_0(\\alpha)=& T(0)+\\int_0^\\alpha U_0(\\eta)d\\eta,\\\\\n    T_1(\\alpha)=& \\int_0^\\alpha  U_1(\\eta)d\\eta-\\frac{\\alpha}{2\\pi}\\int_{-\\pi}^{\\pi}U_1(\\eta)d\\eta \\\\\n    &+\\int_0^\\alpha\\theta_\\alpha(\\eta)U_0(\\eta)d\\eta-\\frac{\\alpha}{2\\pi}\\int_{-\\pi}^{\\pi}\\theta_\\alpha(\\eta)U_0(\\eta)d\\eta,\\\\\n    T_{\\geq2}(\\alpha)=&\\int_0^\\alpha (1+\\theta_\\alpha(\\eta))U_{\\geq2}(\\eta)d\\eta-\\frac{\\alpha}{2\\pi}\\int_{-\\pi}^{\\pi}(1+\\theta_\\alpha(\\eta))U_{\\geq2}(\\eta)d\\eta\\\\\n    &+\\int_0^\\alpha\\theta_\\alpha(\\eta)U_1(\\eta)d\\eta-\\frac{\\alpha}{2\\pi}\\int_{-\\pi}^{\\pi}\\theta_\\alpha(\\eta)U_1(\\eta)d\\eta.\n\\end{aligned}\\right.\n\\end{equation}\nThese are all the splittings that we will use in the following.\n\nWe first examine the zero order terms from \\eqref{thetaeqaux}$_1$.   The zero order terms on the right side of the equality \\eqref{thetaeqaux}$_1$ would be\n\\begin{equation}\\notag\n        \\Theta(\\alpha)=(U_0)_\\alpha(\\alpha)+T_0(\\alpha).\n\\end{equation}\nNow a direct calculation from \\eqref{omegasplit} shows that \n\\begin{equation}\\label{hilbertTcalc}\n    \\mathcal{H}(\\omega_0)(\\alpha) \n    = A_\\rho\\frac{L(t)}{\\pi}\\cos{(\\alpha+\\hat{\\vartheta}(0))}.\n\\end{equation}\nThen we plug this into \\eqref{Usplit}$_1$ and \\eqref{Tsplit}$_1$ to obtain\n\\begin{equation}\\label{u0t0}\n\\left\\{\n\\begin{aligned}\n    U_0(\\alpha)&=A_\\rho \\cos{(\\alpha+\\hat{\\vartheta}(0))},\\\\\n     T_0(\\alpha)&=A_\\rho\\sin{(\\alpha+\\hat{\\vartheta}(0))}-A_\\rho\\sin{\\hat{\\vartheta}(0)}+T(0).\n\\end{aligned}\\right.\n\\end{equation}\nIn particular then the zero order term $\\Theta(\\alpha)$ does not depend on $\\alpha$,\n\\begin{equation*}\n    \\Theta(\\alpha)=-A_\\rho \\sin{\\hat{\\vartheta}(0)}+T(0).\n\\end{equation*}\nNow we choose \n\\begin{equation}\\label{T0}\n    T(0)=A_\\rho \\sin{\\hat{\\vartheta}(0)}.\n\\end{equation}\nThus the parametrization of the circle solution is independent of time (see Proposition \\ref{circles} below).  Further $\\Theta(\\alpha)=0.$\n\n\nNow we introducing the splittings \\eqref{Usplit} and \\eqref{Tsplit} into the equation for $\\vartheta$ in \\eqref{eqcurve3}, we find that\n\\begin{equation}\\label{system}\n\\left\\{\n\\begin{aligned}\n    \\vartheta_t(\\alpha)&=\\frac{2\\pi}{L(t)}\\Big(\\mathcal{L}(\\alpha)+N(\\alpha)\\Big),\\\\\n    \\mathcal{L}(\\alpha)&=(U_1)_\\alpha(\\alpha)+T_1(\\alpha)+T_0(\\alpha)\\theta_\\alpha(\\alpha),\\\\\n    N(\\alpha)&=(U_{\\geq 2})_\\alpha(\\alpha)+T_{\\geq 2}(\\alpha)(1+\\theta_\\alpha(\\alpha))+T_1(\\alpha)\\theta_\\alpha(\\alpha).\n\\end{aligned}\\right.\n\\end{equation}\nNow we will expand the linear terms in $\\mathcal{L}(\\alpha)$ in \\eqref{system}.  To do this we first split $U_1(\\alpha)$ in \\eqref{Usplit} into parts corresponding to the parameters $A_\\rho$, $A_\\sigma$ and $A_\\mu$ respectively as  \n\\begin{equation*}\n    U_1(\\alpha)=A_\\rho U_{1\\rho}(\\alpha)+4A_\\sigma\\frac{\\pi^2}{(L(t))^2} U_{1\\sigma}(\\alpha)+A_\\mu A_\\rho U_{1\\mu}(\\alpha),\n\\end{equation*}\nTo calculate these terms we plug $\\omega_0(\\alpha)$ and $\\omega_1(\\alpha)$ from \\eqref{omegasplit} into $U_1(\\alpha)$ in \\eqref{Usplit} using also $\\mathcal{D}_1(\\omega_0)(\\alpha)$ from \\eqref{mdsplit} and \\eqref{hilbertTcalc}.  \nWe obtain\n\\begin{equation}\\label{uterms}\n\\left\\{\n\\begin{aligned}\n    U_{1\\rho}(\\alpha)&=-\\mathcal{H} \\big(\\theta(\\alpha)\\cos{(\\alpha+\\hat{\\vartheta}(0))}\\big)-\\text{Re}\\hspace{0.05cm}\\hspace{0.05cm}\\mathcal{R}(\\sin{(\\alpha+\\hat{\\vartheta}(0))}),\\\\\n    U_{1\\sigma}(\\alpha)&=\\mathcal{H}  \\theta_{\\alpha\\alpha}(\\alpha),\\\\\n    U_{1\\mu}(\\alpha)&= - \\mathcal{H}  \\big(\\theta(\\alpha)\\cos{(\\alpha+\\hat{\\vartheta}(0))}\\big)+\\mathcal{H} \\hspace{0.03cm}\\text{Im}\\hspace{0.05cm}\\hspace{0.05cm}\\mathcal{R}(\\sin{(\\alpha+\\hat{\\vartheta}(0))}).\n\\end{aligned}\\right.\n\\end{equation}\nWe analogously write the linear part, $\\mathcal{L}(\\alpha)$ in \\eqref{system}, \nas follows\n\\begin{equation*}\n    \\mathcal{L}(\\alpha)=A_\\rho \\mathcal{L}_\\rho(\\alpha)+4A_\\sigma\\frac{\\pi^2}{(L(t))^2}\\mathcal{L}_\\sigma(\\alpha)+A_\\mu A_\\rho\\mathcal{L}_\\mu(\\alpha),\n\\end{equation*}\nwhere\n\\begin{equation}\\label{linearterms}\n\\left\\{\n\\begin{aligned}\n    \\mathcal{L}_{\\rho}(\\alpha)&=(U_{1\\rho})_\\alpha(\\alpha)+\\int_0^\\alpha U_{1\\rho}(\\eta)d\\eta-\\frac{\\alpha}{2\\pi}\\int_{-\\pi}^{\\pi}U_{1\\rho}(\\eta)d\\eta\n    \\\\\n    &\\quad+\\int_0^{\\alpha}\\theta_\\alpha(\\eta)\\cos{(\\eta+\\hat{\\vartheta}(0))}d\\eta-\\frac{\\alpha}{2\\pi}\\int_{-\\pi}^{\\pi}\\theta_\\alpha(\\eta)\\cos{(\\eta+\\hat{\\vartheta}(0))}d\\eta\\\\\n    &\\quad+\\theta_\\alpha(\\alpha)\\sin{(\\alpha+\\hat{\\vartheta}(0))},\\\\\n    \\mathcal{L}_\\sigma(\\alpha)&=-\\Lambda^3 \\theta(\\alpha)+\\int_0^\\alpha \\mathcal{H}  \\theta_{\\alpha\\alpha}(\\eta)d\\eta,\\\\\n    \\mathcal{L}_\\mu(\\alpha)&=(U_{1\\mu})_\\alpha(\\alpha)+\\int_0^\\alpha U_{1\\mu}(\\eta)d\\eta.\n\\end{aligned}\\right.\n\\end{equation}\nHere we used that $\\partial_\\alpha \\mathcal{H}\\theta_{\\alpha\\alpha} = -\\Lambda^3 \\theta(\\alpha)$.  We also used that \n$$\\int_{-\\pi}^{\\pi}\\mathcal{H}  \\theta_{\\alpha\\alpha}(\\eta)d\\eta = \\int_{-\\pi}^{\\pi}U_{1\\mu}(\\eta)d\\eta =0$$ since both integrals are of a Hilbert transform and thus have zero value for the zero Fourier frequency.\nThis completes our decomposition of the equation \\eqref{eqcurve} into \\eqref{system}.  \n\nIn the following, we explain the steady states circles for equation \\eqref{eqcurve} using the reformulation of the equations given above.  \n\n\\begin{prop}\\label{circles}\n\tA circle of radius $R$, defined by \\eqref{circles} and \\eqref{modul} with $\\vartheta(\\alpha)=\\widehat{\\vartheta}(0)$ constant in time and $L(t)=2\\pi R$, is a time-independent solution of \\eqref{T}-\\eqref{md} with $T(0)$ given by \\eqref{T0}.\n\tIt corresponds to the solution of \\eqref{eqcurve} given by a circle of radius $R$ moving vertically with velocity $A_\\rho$.\t\n\\end{prop}\n\n\\begin{proof}\nFor $\\vartheta(\\alpha)=\\widehat{\\vartheta}(0)$, all the linear and nonlinear terms in the decompositions \\eqref{omegasplit}-\\eqref{Tsplit} are zero. Thus, with $L(t)=2\\pi R$,  as in \\eqref{u0t0} with \\eqref{T0} we have\n\\begin{align*}\n    U(\\alpha)&=U_0(\\alpha)=A_\\rho\\cos{(\\alpha+\\widehat{\\vartheta}(0))},\\\\\n    T(\\alpha)&=T_0(\\alpha) = A_\\rho\\sin{(\\alpha+\\widehat{\\vartheta}(0))}.\n\\end{align*} \nBoth equations in \\eqref{eqcurve3} are then trivially satisfied; equation \\eqref{eqcurve3}$_1$ is decomposed as  \\eqref{system} with $\\mathcal{L}(\\alpha)=N(\\alpha)=0$.\t\n\t\nThen we integrate \\eqref{param} to obtain\n\\begin{equation*}\n\tz(\\alpha,t)=z(0,t)+R\\int_0^\\alpha e^{i(\\eta+\\widehat{\\vartheta}(0))} d\\eta.\n\\end{equation*}\nWe differentiate the above in time, and then use \\eqref{eqcurve.theta} to obtain\n\\begin{equation*}\n\\begin{aligned}\n\tz_t(\\alpha,t)&=z_t(0,t)=U(0,t)n(0,t)+T(0,t)\\tau(0,t)\\\\\n\t&=A_\\rho\\cos{(\\widehat{\\vartheta}(0))}ie^{i\\widehat{\\vartheta}(0)}+A_\\rho\\sin{(\\widehat{\\vartheta}(0))}e^{i\\widehat{\\vartheta}(0)}=0+i A_\\rho.\n\\end{aligned}\t\n\\end{equation*}\t\nThis completes the proof.\n\\end{proof}\n\nNext, we compute the Fourier transform of the linearized system. Because the function $\\theta(\\alpha)$ is real and has zero average, we only need to compute the positive frequencies.\n\n\n\n\\begin{prop}\\label{linearfourier}\n\t(Linear system in Fourier variables.)\n\tFor $k\\geq 1$, $k\\neq2$, the Fourier transform of the linear terms \\eqref{linearterms} are given by  \n\t\\begin{equation*}\n\t\\begin{aligned}\n\t    \\widehat{\\mathcal{L}}(k)&=-A_\\sigma \\frac{4\\pi^2}{L(t)^2}k(k^2\\!-\\!1)\\hat{\\theta}(k)\\!-\\!(1\\!+\\!A_\\mu)A_\\rho\\frac{(k^2\\!-\\!1)(k\\!+\\!1)}{k(k\\!+\\!2)}e^{-i\\hat{\\vartheta}(0)}\\hat{\\theta}(k+1),\t\n\t\\end{aligned}\n\t\\end{equation*}\n\tand for $k=2$,\n\t\\begin{equation*}\n\t\\begin{aligned}\n\t    \\widehat{\\mathcal{L}}(2)&=-A_\\sigma \\frac{4\\pi^2}{L(t)^2}6\\hat{\\theta}(2)+(1-A_\\mu)A_\\rho\\frac{3}{2}\\left(\\frac34-\\log{2}\\right)e^{i\\hat{\\vartheta}(0)}\\hat{\\theta}(1)\\\\\n\t    &\\quad-(1+A_\\mu)A_\\rho\\frac{9}{8}e^{-i\\hat{\\vartheta}(0)}\\hat{\\theta}(3).\n\t\\end{aligned}\n\t\\end{equation*}\n\\end{prop}\n\n\\begin{proof}\nFirst, we note that, for a general function $f(\\alpha)$ and $k\\neq0$ we have \\eqref{fourierCalcAlpha}.\nTherefore, for $k\\geq 1$, the Fourier coefficients of\n\\eqref{linearterms} are given by\n\\begin{equation}\\label{Lfourier}\n\\begin{aligned}\n    \\widehat{\\mathcal{L}}_\\sigma(k)&=-k(k^2-1)\\hat{\\theta}(k),\\\\\n    \\widehat{\\mathcal{L}}_\\mu(k)&=i\\left(k-\\frac1{k}\\right)\\widehat{U}_{1\\mu}(k),\\\\\n    \\widehat{\\mathcal{L}}_\\rho(k)\n    &=\n    \\left(k-\\frac1{k}\\right)\n    \\left(i\\widehat{U}_{1\\rho}(k)+\\frac{e^{i\\hat{\\vartheta}(0)}}{2}\\hat{\\theta}(k-1)-\\frac{e^{-i\\hat{\\vartheta}(0)}}{2}\\hat{\\theta}(k+1)\\right),\n\\end{aligned}\n\\end{equation}\nso it remains to compute $\\widehat{U}_{1\\mu}(k)$ and $\\widehat{U}_{1\\rho}(k)$.\nFrom \\eqref{uterms}, we can write \n\\begin{align*}\n    \\widehat{U}_{1\\mu}(k)\n    =&\n    \\frac{i}{2}\\Big(e^{i\\hat{\\vartheta}(0)}\\hat{\\theta}(k\\!-\\!1)\\!\n    +\\!e^{-i\\hat{\\vartheta}(0)}\\hat{\\theta}(k\\!+\\!1)\\Big)\\!\n    \\\\\n    &-\\!i\\mathcal{F}\\Big(\\text{Im}\\hspace{0.05cm}\\hspace{0.05cm} \\mathcal{R}(\\sin{(\\alpha\\!+\\!\\hat{\\vartheta}(0))})\\Big)(k),\n\\end{align*}\nand\n\\begin{align*}\n    \\widehat{U}_{1\\rho}(k)\n    =&\\frac{i}2 \\Big(e^{i\\hat{\\vartheta}(0)}\\hat{\\theta}(k\\!-\\!1)\\!+\\!e^{-i\\hat{\\vartheta}(0)}\\hat{\\theta}(k\\!+\\!1)\\Big)\\!\n    \\\\\n    &-\\!\\mathcal{F}\\Big(\\text{Re}\\hspace{0.05cm}\\hspace{0.05cm} \\mathcal{R}(\\sin{(\\alpha\\!+\\!\\hat{\\vartheta}(0))})\\Big)(k).\n\\end{align*}\nRecalling the expression of $\\mathcal{R}$ in \\eqref{R}, we have that\n\\begin{multline*}\n    \\mathcal{F}\\Big(\\text{Im}\\hspace{0.05cm}\\hspace{0.05cm} \\mathcal{R}(\\sin{(\\alpha+\\hat{\\vartheta}(0))})\\Big)(k)=\\\\\n    \\frac1{\\pi}\\text{pv}\\!\\int_{-\\pi}^{\\pi}\\int_0^1\\!\\! \\text{Im}\\hspace{0.05cm}\\left(\\!\\frac{i\\beta e^{i(s-1)\\beta}}{(1-e^{-i\\beta})^2}\\!\\right)\\!\\mathcal{F}\\Big(\\theta(\\alpha+(s-1)\\beta)\\sin{(\\alpha\\!-\\!\\beta\\!+\\!\\hat{\\vartheta}(0))}\\Big)(k)ds d\\beta.\n\\end{multline*}\nUsing that\n\\begin{equation*}\n    \\text{Im}\\hspace{0.05cm}\\left(\\frac{i\\beta e^{i(s-1)\\beta}}{(1-e^{-i\\beta})^2}\\right)=-\\frac{\\beta\\cos{(\\beta s)}}{4\\sin^2{(\\beta\/2)}},\n\\end{equation*}\nand computing the Fourier transform inside the integral, \nwe obtain that\n\\begin{multline}\\nota\n    \\mathcal{F}\\Big(\\text{Im}\\hspace{0.05cm}\\hspace{0.05cm} \\mathcal{R}(\\sin\\!{(\\alpha\\!+\\!\\hat{\\vartheta}(0))})\\!\\Big)\\!(k)=\\\\\n    -\\frac{ e^{i\\hat{\\vartheta}(0)}}{2\\pi}\\hat{\\theta}(k\\!-\\!1)\\text{pv}\\!\\!\\int_{-\\pi}^{\\pi}\\!\\int_0^1 \\!\\!\\!\\frac{\\beta\\cos{(\\beta s)}}{4\\sin^2{(\\beta\/2)}}\\sin{((k\\!-\\!1)(s\\!-\\!1)\\beta\\!-\\!\\beta)}ds d\\beta\\\\\n    +\\frac{ e^{-i\\hat{\\vartheta}(0)}}{2\\pi}\\hat{\\theta}(k\\!+\\!1)\\text{pv}\\!\\!\\int_{-\\pi}^{\\pi}\\!\\int_0^1 \\!\\!\\!\\frac{\\beta\\cos{(\\beta s)}}{4\\sin^2{(\\beta\/2)}}\\sin{((k\\!+\\!1)(s\\!-\\!1)\\beta\\!+\\!\\beta)}ds d\\beta.\t\n\\end{multline}\nTaking into account that\n\\begin{equation*}\n    \\text{Re}\\hspace{0.05cm}\\left(\\frac{i\\beta e^{i(s-1)\\beta}}{(1-e^{-i\\beta})^2}\\right)=\\frac{\\beta\\sin{(\\beta s)}}{4\\sin^2{(\\beta\/2)}},\n\\end{equation*}\nand proceeding analogously, the following expression is found for the real part: \n\\begin{multline}\\notag\n    \\mathcal{F}\\Big(\\text{Re}\\hspace{0.05cm}\\hspace{0.05cm} \\mathcal{R}(\\sin\\!{(\\alpha\\!+\\!\\hat{\\vartheta}(0))})\\!\\Big)\\!(k)=\\\\\n    -\\frac{i e^{i\\hat{\\vartheta}(0)}}{2\\pi}\\hat{\\theta}(k\\!-\\!1)\n    \\text{pv}\\!\\!\\int_{-\\pi}^{\\pi}\\!\\int_0^1 \\!\\!\\!\\frac{\\beta\\sin{(\\beta s)}}{4\\sin^2{(\\beta\/2)}}\\cos{((k\\!-\\!1)(s\\!-\\!1)\\beta\\!-\\!\\beta)}ds d\\beta\n    \\\\\n    +\\frac{i e^{-i\\hat{\\vartheta}(0)}}{2\\pi}\\hat{\\theta}(k\\!+\\!1)\\text{pv}\\!\\!\\int_{-\\pi}^{\\pi}\\!\\int_0^1 \\!\\!\\!\\frac{\\beta\\sin{(\\beta s)}}{4\\sin^2{(\\beta\/2)}}\\cos{((k\\!+\\!1)(s\\!-\\!1)\\beta\\!+\\!\\beta)}ds d\\beta.\t\n\\end{multline}\nThe above integrals are calculated in Lemma \\ref{lemmaI1I2} below. Plugging in their values, we have \n\\begin{multline}\\label{fourierimag}\n    \\mathcal{F}\\Big(\\text{Im}\\hspace{0.05cm}\\hspace{0.05cm} \\mathcal{R}(\\sin\\!{(\\alpha\\!+\\!\\hat{\\vartheta}(0))})\\!\\Big)\\!(k)=\n    \\frac{ e^{-i\\hat{\\vartheta}(0)}}{2\\pi}\\hat{\\theta}(k\\!+\\!1)\\frac{-k\\pi}{2+k} 1_{k \\ge 1}\\\\\n    -\\frac{ e^{i\\hat{\\vartheta}(0)}}{2\\pi}\\hat{\\theta}(k\\!-\\!1)\n        \\left(-\\pi 1_{k \\ge 1, k\\ne 2}+ \\pi \\left(\\frac{1}{2}-\\log 4 \\right)1_{k =2} \\right),\t\n\\end{multline}\nand\n\\begin{multline}\\label{fourierreal}\n    \\mathcal{F}\\Big(\\text{Re}\\hspace{0.05cm}\\hspace{0.05cm} \\mathcal{R}(\\sin\\!{(\\alpha\\!+\\!\\hat{\\vartheta}(0))})\\!\\Big)\\!(k)=\n    -\\frac{i e^{i\\hat{\\vartheta}(0)}}{2\\pi}\\hat{\\theta}(k\\!-\\!1)\\pi \\left(\\log 4-\\frac{3}{2} \\right)1_{k =2} \n    \\\\\n    +\\frac{i e^{-i\\hat{\\vartheta}(0)}}{2\\pi}\\hat{\\theta}(k\\!+\\!1)\n    \\frac{2\\pi}{2+k} 1_{k \\ge 1}.\t\n\\end{multline}\nWe conclude the Fourier transform of $U_{1\\mu}$ and $U_{1\\rho}$ are given by\n\\begin{equation*}\n    \\widehat{U}_{1\\mu}(k)=\n    \\left\\{\n    \\begin{aligned}\n        &\\frac{i}2e^{-i\\hat{\\vartheta}(0)}\\Big(1+\\frac{k}{2+k}\\Big)\\hat{\\theta}(k+1),\\qquad\\hspace{1.8cm} k=1,3,4,\\dots,\\\\\n        &\\frac{i}2e^{i\\hat{\\vartheta}(0)}\\Big(\\frac32-\\log{(4)}\\Big)\\hat{\\theta}(1)+\\frac{3i}4e^{-i\\hat{\\vartheta}(0)}\\hat{\\theta}(3),\\qquad k=2,\n    \\end{aligned}\\right.\n\\end{equation*}\t\nand \n\\begin{equation*}\n    \\widehat{U}_{1\\rho}(k)=\n    \\left\\{\n    \\begin{aligned}\n        &\\frac{i}2e^{i\\hat{\\vartheta}(0)}\\hat{\\theta}(k\\!-\\!1)\\!+\\!\\frac{i}2e^{-i\\hat{\\vartheta}(0)}\\Big(1\\!-\\!\\frac{2}{2\\!+\\!k}\\Big)\\hat{\\theta}(k\\!+\\!1),\\hspace{0.3cm} k=1,3,4,\\dots,\\\\\n        &\\frac{i}2e^{i\\hat{\\vartheta}(0)}\\Big(-\\frac12+\\log{4}\\Big)\\hat{\\theta}(1)+\\frac{i}4e^{-i\\hat{\\vartheta}(0)}\\hat{\\theta}(3),\\qquad\\hspace{0.4cm} k=2.\n    \\end{aligned}\\right.\n\\end{equation*}\t\nSubstituting these expressions into \\eqref{Lfourier} gives that\n\\begin{equation*}\n    \\widehat{\\mathcal{L}}_{\\mu}(k)=\n    \\left\\{\n    \\begin{aligned}\n        &-e^{-i\\widehat{\\vartheta}(0)}\\frac{(k^2-1)(k+1)}{k(k+2)}\\widehat{\\theta}(k+1),\\hspace{0.4cm} k=1,3,4,\\dots,\\\\\n        &-\\frac34\\Big(\\frac32-\\log{4}\\Big)e^{i\\widehat{\\vartheta}(0)}\\widehat{\\theta}(1)-\\frac98e^{-i\\widehat{\\vartheta}(0)}\\widehat{\\theta}(3),\\qquad\\hspace{0.4cm} k=2,\n    \\end{aligned}\\right.\n\\end{equation*}\nand\n\\begin{equation*}\n    \\widehat{\\mathcal{L}}_{\\rho}(k)=\n    \\left\\{\n    \\begin{aligned}\n        &-e^{i\\widehat{\\vartheta}(0)}\\frac{(k^2-1)(k+1)}{k(k+2)}\\widehat{\\theta}(k-1),\\hspace{0.4cm} k=1,3,4,\\dots,\\\\\n        &\\frac34\\Big(\\frac32-\\log{4}\\Big)e^{i\\widehat{\\vartheta}(0)}\\widehat{\\theta}(1)-\\frac98e^{-i\\widehat{\\vartheta}(0)}\\widehat{\\theta}(3),\\qquad\\hspace{0.4cm} k=2,\n    \\end{aligned}\\right.\n\\end{equation*}\nFinally, adding them according to \\eqref{linearterms}, the result follows.\n\\end{proof}\n\n\\begin{lemma}\\label{lemmaI1I2}\nFor $k\\in\\mathbb{Z}\\setminus\\{0\\}$, define the integrals \n\\begin{equation*} \n\tI_1(k)=\\int_{-\\pi}^{\\pi}\\!\\int_0^1\\!\\frac{\\beta\\cos{(\\beta s)}}{4\\sin^2{(\\beta\/2)}}\\sin{((k\\!-\\!1)(s\\!-\\!1)\\beta\\!-\\!\\beta)}dsd\\beta,\n\\end{equation*}\n\\begin{equation*}\n\tI_2(k)=\\int_{-\\pi}^{\\pi}\\!\\int_0^1\\!\\frac{\\beta\\sin{(\\beta s)}}{4\\sin^2{(\\beta\/2)}}\\cos{((k\\!-\\!1)(s\\!-\\!1)\\beta\\!-\\!\\beta)}dsd\\beta.\n\\end{equation*}\nFor $k\\geq 1$ and $k\\neq 2$,\n\\begin{equation*}\n\tI_1(k)=-\\pi,\\qquad I_2(k)=0,\n\\end{equation*}\nwhile for $k\\leq -1$. \n\\begin{equation*}\n\tI_1(k)=\\frac{-k\\pi}{2-k},\\qquad I_2(k)=\\frac{2\\pi}{2-k}.\n\\end{equation*}\t\nThe value $k=2$ is given by\n\\begin{equation*}\n\tI_1(2)=\\pi\\Big(\\frac12-\\log{4}\\Big),\\qquad I_2(2)=\\pi\\Big(\\log{4}-\\frac32\\Big).\n\\end{equation*}\n\\end{lemma}\n\n\n\\begin{remark}\nNotice that\n\\begin{equation*} \n\tI_1(-k)=-\\int_{-\\pi}^{\\pi}\\!\\int_0^1\\!\\frac{\\beta\\cos{(\\beta s)}}{4\\sin^2{(\\beta\/2)}}\\sin{((k\\!+\\!1)(s\\!-\\!1)\\beta\\!+\\!\\beta)}dsd\\beta,\n\\end{equation*}\nand \n\\begin{equation*}\n\tI_2(-k)=\\int_{-\\pi}^{\\pi}\\!\\int_0^1\\!\\frac{\\beta\\sin{(\\beta s)}}{4\\sin^2{(\\beta\/2)}}\\cos{((k\\!+\\!1)(s\\!-\\!1)\\beta\\!+\\!\\beta)}dsd\\beta.\n\\end{equation*}\nThus Lemma \\ref{lemmaI1I2} covers all the integrals in \\eqref{fourierimag} and \\eqref{fourierreal}.\n\\end{remark}\n\n\n\n\\begin{proof}\n\tBoth integrals are computed similarly. We only show the details for $I_1(k)$. \n\tFirst, consider the case $k\\neq2$.\n    Using complex exponentials, we write the numerator as follows\n    \\begin{equation*}\n        \\begin{aligned}\n            \\cos{(\\beta s)} \\sin{((k\\!-\\!1)(s\\!-\\!1)\\beta\\!-\\!\\beta)}&=   \\frac{1}{4i}\\Big(e^{ik(s-1)\\beta}-e^{-i(k(s-1)\\beta-2s\\beta)}\\\\\n            &\\quad+e^{i(k(s-1)\\beta-2s\\beta)}-e^{-i(k(s-1)\\beta)}\\Big).   \n        \\end{aligned}\n    \\end{equation*}\n    Thus integration in $s$ gives that\n    \\begin{equation*}\n        \\begin{aligned}\n            I_1(k)&\\!=\\!\\frac{-1}{16}\\!\\int_{-\\pi}^{\\pi}\\!\\Big(\\frac{1\\!-\\!e^{-i k\\beta}}{k}\\!+\\!\\frac{e^{2i\\beta}\\!-\\!e^{i k\\beta}}{k-2}\\!+\\!\\frac{e^{-2i\\beta}\\!-\\!e^{-i k\\beta}}{k-2}\\!+\\!\\frac{1\\!-\\!e^{i k\\beta}}{k}\\Big)\\frac{d\\beta}{\\sin^2{(\\beta\/2)}}\\\\\n            &\\!= \\!\\frac{-1}{16}\\!\\int_{-\\pi}^{\\pi}\\!\\Big(\\frac{2\\!-\\!e^{i k\\beta}\\!-\\!e^{-i k\\beta}}{k}\\!+\\!\\frac{e^{2i\\beta}\\!+\\!e^{-2i \\beta}}{k-2}\\!-\\!\\frac{e^{i k\\beta}\\!+\\!e^{-i k\\beta}}{k-2}\\Big)\\frac{d\\beta}{\\sin^2{(\\beta\/2)}}.\n        \\end{aligned}\n    \\end{equation*}\nWe then write the denominator in complex form too\n\\begin{equation*}\n    \\begin{aligned}\n        \\sin^2{(\\beta\/2)}=\\frac{-1}4\\big(e^{i\\beta\/2}-e^{-i\\beta\/2}\\big)^2,\n    \\end{aligned}\n\\end{equation*}\nand formally expand it\n\\begin{equation*}\n    \\begin{aligned}\n        \\big(\\sin{\\beta\/2)}\\big)^{-2}\\!=-4e^{-i\\beta}\\big(1\\!-\\!e^{-i\\beta}\\big)^{-2}\\!=-4e^{-i\\beta}\\sum_{l\\geq1} l e^{-i(l-1)\\beta}\\!=\\!-4\\sum_{l\\geq1}l e^{-i l\\beta},\n    \\end{aligned}\n\\end{equation*}\nwhere we have used that\\begin{equation*}\n\\frac{1}{(1-x)^2}=\\sum_{l\\geq1} l x^{l-1}.    \n\\end{equation*}\nTherefore, for $k\\neq0,2$, we have\n \\begin{equation*}\n        \\begin{aligned}\n            I_1(k)\\!= \\!\\frac{1}{4}\\sum_{l\\geq1}l\\!\\int_{-\\pi}^{\\pi}\\!\\Big(&\\frac{2e^{-i l\\beta}\\!-\\!e^{i (k-l)\\beta}\\!-\\!e^{-i (k+l)\\beta}}{k}\\!+\\!\\frac{e^{i(2-l)\\beta}\\!+\\!e^{-i(2+l) \\beta}}{k-2}\\\\\n            &\\quad-\\frac{e^{i (k-l)\\beta}\\!+\\!e^{-i( k+l)\\beta}}{k-2}\\Big)d\\beta.\n        \\end{aligned}\n    \\end{equation*}\nThus performing the integral in $\\beta$ we obtain that\n\\begin{equation*}\n        \\begin{aligned}\n            I_1(k)&\\!= \\!\\frac{1}{4}\\sum_{l\\geq1}l\\Big(\\frac{2\\pi}{k}\\big(-\\delta(k-l)-\\delta(k+l)\\big)\\\\\n            &\\hspace{1.7cm}+\\!\\frac{2\\pi}{k\\!-\\!2}\\big(\\delta(2\\!-\\!l)\\!+\\!\\delta(2\\!+\\!l)\\!-\\!\\delta(k\\!-\\!l)\\!-\\!\\delta(k\\!+\\!l)\\big)\\Big)\\\\\n            &=\\frac{1}{4}\\Big(-2\\pi \\text{sgn}(k)+\\frac{2\\pi}{k-2}2-\\frac{2\\pi}{k-2}|k|\\Big),\n        \\end{aligned}\n    \\end{equation*}\n\twhich gives\n\t\\begin{equation*}\n        \\begin{aligned}\n            I_1(k)&\\!=\\frac{\\pi}{2}\\Big(\\!- \\text{sgn}(k)\\!+\\!\\frac{2\\!-\\!|k|}{k\\!-\\!2}\\Big)=\\frac{\\pi}{2}\\frac{-2|k|\\!+\\!2(\\text{sgn}(k)\\!+\\!1)}{k\\!-\\!2}.\n        \\end{aligned}\n    \\end{equation*}\n\tIt follows then that for $k\\geq1$ ($k\\neq2$),\n\t\\begin{equation*}\n\t    I_1(k)=-\\pi,\n\t\\end{equation*}\n    and for $k\\leq-1$,\n    \\begin{equation*}\n        I_1(k)=\\frac{\\pi}2\\frac{2k}{k-2}=\\pi\\frac{|k|}{2+|k|}.\n    \\end{equation*}\n    The above computations can be justified by writing $x=\\lambda e^{-i\\beta}$ with $0<\\lambda<1$. Then, $\\big(1-\\lambda e^{-i\\beta}\\big)^{-2}=\\sum_{l\\geq1} l\\lambda^{l-1}e^{-i(l-1)\\beta}$ converges uniformly and one can repeat the steps above and take the limit $\\lambda\\to1$.\n    \n    Lastly, we deal with the case $k=2$. We first rewrite it as follows\n\t\\begin{equation*}\n\t\\begin{aligned}\n\t    I_1(2)=\\int_{-\\pi}^{\\pi}\\int_0^1\\frac{\\beta}{8\\sin^2{(\\beta\/2)}}\\left(\\sin{(2\\beta s-2\\beta)}-\\sin{(2\\beta)}\\right)ds d\\beta,\n\t\\end{aligned}\n\t\\end{equation*}\n\tso after integration in $s$ we obtain\n\t\\begin{equation*}\n\t\\begin{aligned}\n\t    I_1(2)=\\int_{-\\pi}^{\\pi}\\frac{\\beta}{8\\sin^2{(\\beta\/2)}}\\left(\\frac{-1+\\cos{(2\\beta)}}{2\\beta}-\\sin{(2\\beta)}\\right)ds d\\beta.\n\t\\end{aligned}\n\t\\end{equation*}\n\tUsing repeatedly the double angle formula, we find that\n\t\\begin{equation*}\n\t\\begin{aligned}\n\t    I_1(2)&=-\\frac12\\int_{-\\pi}^{\\pi}\\cos^2{(\\beta\/2)}d\\beta-\\frac12\\int_{-\\pi}^{\\pi}\\frac{\\beta\\cos{(\\beta)}\\cos{(\\beta\/2)}}{\\sin{(\\beta\/2)}}d\\beta\t\\\\\n\t    &=-\\frac{\\pi}2-\\frac12\\int_{-\\pi}^{\\pi}\\frac{\\beta\\cos{(\\beta)}\\cos{(\\beta\/2)}}{\\sin{(\\beta\/2)}}d\\beta,\n\t\\end{aligned}\n\t\\end{equation*}\n\twhich can be further simplified\n\t\\begin{equation*}\n\t\\begin{aligned}\n\t    I_1(2)&=-\\frac{\\pi}2+\\int_{-\\pi}^{\\pi}\\beta\\sin{(\\beta\/2)}\\cos{(\\beta\/2)}d\\beta -\\frac12\\int_{-\\pi}^{\\pi}\\frac{\\beta\\cos{(\\beta\/2)}}{\\sin{(\\beta\/2)}}d\\beta\\\\\n\t    &=\\frac{\\pi}2-\\frac12\\int_{-\\pi}^{\\pi}\\frac{\\beta\\cos{(\\beta\/2)}}{\\sin{(\\beta\/2)}}d\\beta.\n\t\\end{aligned}\n\t\\end{equation*}\n\tIntegration by parts gives that\n\t\\begin{equation*}\n\t    I_1(2)=\\frac{\\pi}2+2\\int_0^\\pi \\log{|\\sin{(\\beta\/2)}|}d\\beta.\n\t\\end{equation*}\n\tThis integral above is related to the known \\text{Clausen function} \\cite{Cl32} of order two (whose value at $\\pi$ is zero):\n\t\t\\begin{equation*}\n\t\t0=Cl_2(\\pi) =\n\t    \\int_0^\\pi \\log{|2\\sin{(\\beta\/2)}|}d\\beta=\n\t    \\int_0^\\pi \\log{|\\sin{(\\beta\/2)}|}d\\beta+\\pi\\log{2},\n\t\\end{equation*}\n\tWe thus conclude \n\t\t\\begin{equation*}\n\t    I_1(2)=\\frac{\\pi}2+2\\int_0^\\pi \\log{|\\sin{(\\beta\/2)}|}d\\beta=\\frac{\\pi}2-2\\pi\\log{2}=\\pi\\Big(\\frac12-\\log{4}\\Big).\n\t\\end{equation*}\nThis completes the proof.\n\\end{proof}\n\n\nWe notice that in Proposition \\ref{linearfourier} the first frequency mode is neutral at the linear level. However, the restriction \\eqref{constraint} is an equation that relates this frequency with all the higher ones. Thus, the rough idea is to proceed as follows: apply an implicit function theorem to \\eqref{constraint} to solve $\\widehat{\\theta}(-1)$ and $\\widehat{\\theta}(1)$ in terms of $\\widehat{\\theta}(k)$ for $|k|\\geq2$; use \\eqref{system} to compute $\\widehat{\\theta}(k)$ for $|k|\\geq2$, for which the linear operator provides dissipation when we are able to control the nonlinear terms.  Then we will use \\eqref{Lequation} to control $L(t)$ in terms of $\\theta(t)$.  Finally we can compute the evolution of the zero frequency $\\widehat{\\vartheta}(0)$ from \\eqref{system}. \n\n\n\n\n\n\n\\section{Notation and main results}\\label{MainResults}\n\nWe introduce the notation that will be used in the rest of the paper in Section \\ref{sec:Notation}.   We will then state the main results in Section \\ref{sec:globalTHM}.\n\n\\subsection{Notation}\\label{sec:Notation}\nWe recall the complex vector notation introduced in Section \\ref{sec:ComplexVector} and the Fourier transform notation introduced in Section \\ref{sec:FourierTransCalc}.  We use $\\mathbb{T}=[-\\pi,\\pi]$ as our domain with periodic boundary conditions. \n\nWe introduce the space $\\mathcal{F}^{0,1}$ to denote the Wiener algebra, i.e., the space of absolutely convergent Fourier series. The norm in this space is\n\\begin{equation}\\notag \\label{fzerone.def}\n\\|f\\|_{\\mathcal{F}^{0,1}}\\overset{\\mbox{\\tiny{def}}}{=}\\sum_{k\\in\\mathbb{Z}} |\\hat{f}(k)|.\n\\end{equation}\nWe analogously introduce the homogeneous spaces $\\dot{\\mathcal{F}}^{s,1}$ with norm\n\\begin{equation}\\notag \\label{fsone.def}\n\\|f\\|_{\\dot{\\mathcal{F}}^{s,1}}\\overset{\\mbox{\\tiny{def}}}{=}\\sum_{k\\ne 0} |k|^s|\\hat{f}(k)|, \\quad s\\ge 0.\n\\end{equation}\nFurther in Section \\ref{sec:uniqueness} we will use the notation\n\\begin{equation}\\label{max.fcns}\n    \\|f_{1}, f_{2}, \\ldots , f_{k}\\|_{\\dot{\\mathcal{F}}^{s,1}} \\overset{\\mbox{\\tiny{def}}}{=} \n    \\sum_{j=1}^{k}\\|f_{j}\\|_{\\dot{\\mathcal{F}}^{s,1}}.\n\\end{equation}\nMoreover, we will use the spaces of analytic functions $\\dot{\\mathcal{F}}^{s,1}_\\nu$ where these norms include exponential weights as follows:\n\\begin{equation}\\label{fzeroonenunorm}\n\\|f\\|_{\\mathcal{F}^{0,1}_\\nu}\\overset{\\mbox{\\tiny{def}}}{=}\\sum_{k\\in\\mathbb{Z}}e^{\\nu(t)|k|}|\\hat{f}(k)|,\n\\end{equation}\nand\n\\begin{equation}\\label{fsonenorm}\n\\|f\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\\overset{\\mbox{\\tiny{def}}}{=}\\sum_{k\\ne 0}e^{\\nu(t)|k|}|k|^s|\\hat{f}(k)|, \\quad s\\ge 0,\n\\end{equation}\nwith  \n\\begin{equation}\\label{nu}\n    \\nu(t)\\overset{\\mbox{\\tiny{def}}}{=}\\nu_0 \\frac{t}{1+t}>0,\n\\end{equation}\nwhere $0<\\nu'(t)\\leq \\nu_0$ is bounded and small enough for all time when $\\nu_0>0$ is chosen small enough. \n\n\nWe recall the embeddings $\\dot{\\mathcal{F}}^{s_2,1}_\\nu\\hookrightarrow \\dot{\\mathcal{F}}^{s_1,1}_\\nu$ for $0<s_1\\leq s_2$, with norm inequality\n\\begin{equation}\\label{embed}\n\\|f\\|_{\\dot{\\mathcal{F}}^{s_1,1}_\\nu}\\leq \\|f\\|_{\\dot{\\mathcal{F}}^{s_2,1}_\\nu}, \\qquad  0<s_1\\leq s_2.\n\\end{equation}\nThis inequality also holds if $s_1=0$ with $\\mathcal{F}^{0,1}$ in the lower bound provided that $\\hat{f}(0)=0$.\nWe also recall the interpolation inequality \n\\begin{equation}\\label{interpolation}\n\\|f\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\\leq \\|f\\|_{\\dot{\\mathcal{F}}^{s_1,1}_{\\nu}}^{1-\\sigma}\\|f\\|_{\\dot{\\mathcal{F}}^{s_2,1}_{\\nu}}^\\sigma,\\qquad s=(1-\\sigma)s_1+\\sigma s_2,\n\\end{equation}\nfor $0 \\le \\sigma \\le 1$, $s, s_1, s_2 \\ge 0$.\n\nWe further will use the discrete delta function $\\delta(k)$ for $k\\in \\mathbb{Z}$ which is defined  as $\\delta(0)=1$ and $\\delta(k)=0$ for $k\\ne 0$.  We also define $1_A$ to be the standard indicator function of the set $A$ so that $1_A(x)=1$ if $x\\in A$ and $1_A(x)=0$ if $x\\notin A$.  \n\nWe define the $\\ell^p=\\ell^p(\\mathbb{Z})$ norms for $1\\le p < \\infty$ of a sequence $a=\\{a_k\\}_{k\\in \\mathbb{Z}}$ as\n\\begin{equation}\\notag\n    \\| a\\|_{\\ell^p} \\overset{\\mbox{\\tiny{def}}}{=} \\left( \\sum_{k\\in\\mathbb{Z}} |a_k|^p \\right)^{1\/p},\n\\end{equation}\nand for $p=\\infty$ we use\n\\begin{equation}\\notag\n    \\| a\\|_{\\ell^\\infty} \\overset{\\mbox{\\tiny{def}}}{=} \\sup_{k\\in\\mathbb{Z}} |a_k|.\n\\end{equation}\nWe also define the following notation for $s\\ge 0$:\n\\begin{equation}\\label{bfcn.def}\n    b(n,s) \n    \\overset{\\mbox{\\tiny{def}}}{=} \\left\\{\n    \\begin{aligned} \n    1,\\quad 0 \\le s \\le 1,\\\\\n        n^{s-1}, \\quad s >1,\n    \\end{aligned}\\right.\n\\end{equation}\nFurther define the high frequency cut-off operator $\\mathcal{J}_N$ for $N\\ge 0$ by\n\\begin{equation}\\label{CutOffHigh}\n    \\widehat{\\mathcal{J}_N f}(k) \\overset{\\mbox{\\tiny{def}}}{=} 1_{|k|\\leq N}\\widehat{f}(k).\n\\end{equation}\nWe will use these norms and notations in the rest of the paper.\n\n\nThroughout the paper, we will denote \n\\begin{equation*}\n    C_i=C_i(x)=C_i\\left(x; A_\\mu,\\frac{|A_\\rho|R^2}{A_\\sigma}\\right)>0\n\\end{equation*}\nas functions that are increasing in $x\\geq0$ and might depend on the physical parameters such as $A_\\mu,\\frac{|A_\\rho|R^2}{A_\\sigma}$,  with the property that $C_i(x)\\approx 1$  for all $-1\\leq A_\\mu\\leq1$, $\\frac{|A_\\rho|R^2}{A_\\sigma}\\geq0$. Typically, $x$ will be the norm $\\|\\theta\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}$ with $s=0$ or $s=1\/2$.\n\n\nWe denote $f*g$ as the standard convolution of $f$ and $g$.\nWe use the iterated convolution notation\n\\begin{equation}\\label{iteratedConvlution}\n*^kf = \\underbrace{f *  \\cdots *f}_\\text{$k-1$ convolutions of $k$ copies of $f$}\n\\end{equation}\nThus for instance $*^2f = f*f$.  We then sometimes also use the notation $g*^kf = g * \\underbrace{f *  \\cdots *f}_\\text{$k-1$ convolutions}$ to avoid an additional $*$ in the notation.\n\n\n\n\n\n\\subsection{Main Results}\\label{sec:globalTHM} \n\nThe main result of this paper states that, for any value of the physical parameters $A_\\rho$, $A_\\mu$, $A_\\sigma$, a bubble in a porous medium, with arbitrary volume $\\pi R^2$ and shape that is not too far from a circle, converges exponentially fast to a circle that rises (or falls) with constant velocity proportional to the difference between the inner and outer density. The initial interface needs to have barely more than a  continuous tangent vector, allowing in particular for unbounded curvature.  In particular since we suppose $\\theta_0\\in \\dot{\\mathcal{F}}^{\\frac12,1}$ then the initial interface has regularity $W^{\\frac32,\\infty}$,  in terms of the tangent vector the initial regularity is $W^{\\frac12,\\infty}$.  In particular the initial curvature doesn't need to be bounded.\nMoreover, the interface becomes instantaneously analytic.\n\nWe summarize here the system that models our problem. For clarity, we write the zero frequency of $\\vartheta$ apart because it is decoupled from the rest, and the equation for $\\theta$ with the linear and nonlinear terms separated:\n\\begin{equation}\\label{finalsystem}\n\\left\\{\\begin{aligned}\n \\widehat{\\vartheta}_t(0)&=\\frac{2\\pi}{L(t)}\\widehat{T}*\\widehat{(1+\\theta_\\alpha)}(0),\\\\\n     \\theta_t(\\alpha)&=\\frac{2\\pi}{L(t)}\\Big(\\mathcal{L}(\\alpha)+N(\\alpha)\\Big),\\\\\n    L(t)&=2\\pi R\\Big(1+\\frac{1}{2\\pi}\\text{Im}\\hspace{0.05cm} \\int_{-\\pi}^\\pi\\int_0^\\alpha e^{i(\\alpha-\\eta)} \\sum_{n\\geq1}\\frac{i^n}{n!}(\\theta(\\alpha)-\\theta(\\eta))^n d\\eta d\\alpha\\Big)^{-\\frac12},\\\\\n    0&=\\int_{-\\pi}^{\\pi} e^{i(\\alpha+\\theta(\\alpha))}d\\alpha,\n    \\end{aligned}\\right.\n\\end{equation}\nwhere the linear and nonlinear terms $\\mathcal{L}(\\alpha)$, $N(\\alpha)$ are given by \\eqref{system}\nwith $T(\\alpha)$ defined in \\eqref{T}, \\eqref{Tsplit}, and \\eqref{T0}, $U(\\alpha)$ in \\eqref{U} and \\eqref{Usplit}.\n\n\n\\begin{thm}\\label{thm:global}\n\tFix $A_\\mu\\in[-1,1]$, $A_\\rho\\in\\mathbb{R}$, $A_\\sigma>0$, and $R>0$.\n\tAssume that the initial data  $\\vartheta_0(\\alpha)=\\widehat{\\vartheta}_0(0)+\\theta_0(\\alpha)\\in \\dot{\\mathcal{F}}^{\\frac12,1}$ satisfies the medium-size condition \n    \\begin{equation}\\label{condition}\n     \\|\\theta_0\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}}<K\\Big(\\frac{|A_\\rho|R^2}{A_\\sigma}, A_\\mu\\Big),\n    \\end{equation}\n    where $K$ is defined in \\eqref{Kdef}.\n\tThen, for any $T>0$, there exists a unique global strong solution   $\\vartheta(\\alpha, t)=\\widehat{\\vartheta}(0,t)+\\theta(\\alpha,t)$ to the system  \\eqref{finalsystem}, which lies in the space\n\t\\begin{equation*}\t    \\vartheta\\in C([0,T];\\dot{\\mathcal{F}}^{\\frac12,1}_\\nu)\\cap L^1([0,T];\\dot{\\mathcal{F}}^{\\frac72,1}_\\nu),\\quad 0<T<\\infty,\n\t\\end{equation*}\n\twith $\\nu(t)$ given by \\eqref{nu}.\n\tIn particular the solution becomes instantaneously analytic. Moreover, the following energy inequality is satisfied for $0\\leq t\\leq T$:\n\t\\begin{equation}\\label{estimatef12}\n    \t\\|\\theta\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}_\\nu}(t)+ \\frac{A_\\sigma}{R^3}  \\mathcal{D} \\int_0^t \\|\\theta\\|_{\\dot{\\mathcal{F}}^{\\frac72,1}_\\nu}(\\tau) d\\tau \\leq C_S^2\\|\\theta_0\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}},\n\t\\end{equation}\n\twith $$\\mathcal{D}=\\mathcal{D}\\Big(C_S\\|\\theta_0\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}_\\nu},\\frac{|A_\\rho|R^2}{A_\\sigma},\n       A_\\mu,\\frac{A_\\sigma}{R^3},\\nu\\Big)>0$$\n       defined in \\eqref{dissipation}, and $C_S=C_S\\big(A_\\mu,\\frac{|A_\\rho| R^2}{A_\\sigma}\\big)$ defined in \\eqref{CSbounds}. \n       In addition,  we have the uniform in time estimate\n\t\\begin{equation}\\label{decay}\n\t    \\begin{aligned}\n\t        \\|\\theta\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}_\\nu}(t)\\leq C_S^2 \\|\\theta_0\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}}e^{-\\frac{A_\\sigma}{R^3}\\mathcal{D} t}.\n\t    \\end{aligned}\n\t\\end{equation}\n\tFurthermore, the zero frequency $\\widehat{\\vartheta}(0)$ remains bounded for all times\n\t\\begin{equation}\\label{zeroBoundUniform}\n\t     |\\widehat{\\vartheta}(0,t)|\\leq |\\widehat{\\vartheta}_0(0)|+C_{42}\\|\\theta_0\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}},\n\t\\end{equation}\n\twhere $C_{42}$ is defined in \\eqref{C42}.\n\\end{thm}\n\nWe remark that none of the uniform constants in Theorem \\ref{thm:global} depend upon our choice of $T>0$, and $T$ can be taken arbitrarily large.\n\n\\begin{remark}\nFrom Proposition \\ref{circles}, the large time decay in \\eqref{decay} implies the exponential convergence to rising or falling circles. Moreover, as part of the proof, it will be proven in \\eqref{Lboundaux} that the length satisfies for all times $t\\geq0$ that\n\\begin{equation}\\label{Lfinalbound}\n     \\begin{aligned}\n        \\frac{R}{\\sqrt{1+\\frac{\\pi}{2} \\big(e^{2\\|\\theta\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}}(t)}-1\\big)}}\\leq\\frac{L(t)}{2\\pi}\\leq \\frac{R}{\\sqrt{1-\\frac{\\pi}{2}\\big(e^{2\\|\\theta\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}}(t)}-1\\big)}},\n    \\end{aligned}\n    \\end{equation}\n    which also shows that $L(t)\\to 2\\pi R$ as $t\\to \\infty$.\n\\end{remark}\n\n\nThe size condition \\eqref{condition} of the theorem above is explicit: for any value of the physical parameters, it gives a bound for the norm of the initial data that can be computed. We also notice that, thanks to the diagonalization performed in Section \\ref{secanalytic}, the dissipative character of the equation is shown in \\eqref{estimatef12} for any $A_\\sigma>0$, no matter how large the gravity effects are.\n\n\n\n\n\\section{Implicit function theorem}\\label{IFTSection}\n\n\nIn this section, we will prove an explicit uniform upper bound for the $\\pm 1$ frequencies of $\\theta$ in terms of the higher Fourier frequencies of $\\theta$.  The main result of this section is the implicit function theorem in Proposition \\ref{IFTprop}.\n\n\n\n\n\nFor ease of notation, only in this section we will use the following space.  For $s \\ge 0$, we define the normed space\n\\begin{equation}\\label{tildefsone}\n    \\tilde{\\mathcal{F}}^{s,1} \\overset{\\mbox{\\tiny{def}}}{=} \\left\\{ u: \\mathbb{T} \\rightarrow \\mathbb{T} \\ | \\ \\hat{u}(0) = \\hat{u}(\\pm 1) = 0 \\text{ and } \\|u\\|_{\\tilde{\\mathcal{F}}^{s,1}}  < \\infty \\right\\}.\n\\end{equation}\nHere we use the norm\n\\begin{equation*}\n    \\|u\\|_{\\tilde{\\mathcal{F}}^{s,1}} \\overset{\\mbox{\\tiny{def}}}{=} \\sum_{|k|\\geq 2} |k|^{s}|\\hat{u}(k)|.\n\\end{equation*}\nIn view of \\eqref{tildefsone}, recalling \\eqref{CutOffHigh}, we consider \n$\n\\tilde{\\theta} \\overset{\\mbox{\\tiny{def}}}{=} \\left(I-\\mathcal{J}_2 \\right)\\theta.\n$\nWe then remark that \n$$\n\\tilde{\\theta}(\\alpha) = \n\\left(I-\\mathcal{J}_2 \\right)\\theta(\\alpha)=\\sum_{|k|\\ge 2} \\hat{\\theta}(k) e^{ik\\alpha}.\n$$\nAn implicit relation between the frequencies $\\hat{\\theta}(\\pm 1)$ and the function $\\tilde{\\theta}$ can be derived from \\eqref{constraint} and is given by\n\\begin{equation}\\label{implicitrelation}\n\t\\int_{-\\pi}^{\\pi} e^{i\\alpha + i\\hat{\\theta}(-1)e^{-i\\alpha} + i\\hat{\\theta}(1)e^{i\\alpha} + i\\tilde{\\theta}(\\alpha)} d\\alpha = 0.\n\\end{equation}\nNote that $\\hat{\\theta}(-1) = \\overline{\\hat{\\theta}(1)}=\\text{Re}\\hspace{0.05cm} \\hat{\\theta}(1)- i \\text{Im}\\hspace{0.05cm} \\hat{\\theta}(1)$ since $\\theta$ is real.  Further\n\\begin{equation}\\notag\n\\hat{\\theta}(-1)e^{-i\\alpha} + \\hat{\\theta}(1)e^{i\\alpha}\n=\n2\\text{Re}\\hspace{0.05cm} \\hat{\\theta}(1) \\cos(\\alpha) - \\text{Im}\\hspace{0.05cm} \\hat{\\theta}(1)\\sin(\\alpha).\n\\end{equation}\nWe will also use the following notation\n\\begin{equation}\\label{psi}\n    \\psi(\\alpha, x, u) = \\alpha + 2(x_{1}\\cos(\\alpha) - x_{2}\\sin(\\alpha)) + u(\\alpha),\n    \\quad x\\in\\mathbb{R}^{2}.\n\\end{equation}\nThen we express the integral in \\eqref{implicitrelation} as a vector as follows\n\\begin{equation}\\label{g}\n\tg(u, x) = \n\t\\begin{bmatrix}\n\t\t\\int_{-\\pi}^{\\pi} \\cos(\\psi(\\alpha, x, u))d\\alpha\n\t\t\\\\\n\t\t\\int_{-\\pi}^{\\pi} \\sin(\\psi(\\alpha, x, u))d\\alpha\n\t\\end{bmatrix}\n\t=\t\\begin{bmatrix}\n\t\tg_1(u, x)\n\t\t\\\\\n\t\tg_2(u, x)\n\t\\end{bmatrix}.\n\\end{equation}\nHere $g: \\tilde{\\mathcal{F}}^{0,1} \\times \\mathbb{R}^{2} \\rightarrow \\mathbb{R}^{2}$.\nNow we can rewrite the relation \\eqref{implicitrelation} as\n$$\ng(\\tilde{\\theta}, (\\text{Re}\\hspace{0.05cm}\\hat{\\theta}(1), \\text{Im}\\hspace{0.05cm} \\hat{\\theta}(1))) = 0.\n$$\nThe main result in this section is the following proposition.\n\n\n\n\\begin{prop}\\label{IFTprop}\nFix $0<r<\\frac{1}{2}\\log(\\frac{5}{4})$.\nConsider $\\|\\theta\\|_{\\mathcal{F}^{0,1}} < r$.  Let\n$$\nB_{\\tilde{\\mathcal{F}}^{0,1}}(0,r) \\overset{\\mbox{\\tiny{def}}}{=} \\{ u \\in \\tilde{\\mathcal{F}}^{0,1}: \\|u\\|_{\\tilde{\\mathcal{F}}^{0,1}} < r\\}.\n$$\nThen there exists a unique function $F:B_{\\tilde{\\mathcal{F}}^{0,1}}(0,r)\\to \\mathbb R^2$ such that \n$F(\\tilde{\\theta})=(\\text{Re}\\hspace{0.05cm}\\hat{\\theta}(1), \\text{Im}\\hspace{0.05cm} \\hat{\\theta}(1))$ and \\eqref{implicitrelation} is satisfied.\nWe further have\n\\begin{equation}\\label{frequencyrelation}\n\t\t\t|\\hat{\\theta}(+1)|+|\\hat{\\theta}(-1)|  \\leq   C_{I}(r)r \\sum_{|k|\\ge 2} |\\hat{\\theta}(k)|.\n\\end{equation}\nHere the constant $C_I(r)>0$ is given by \n\\begin{equation}\\label{c1.implicit}\nC_I(r) \\overset{\\mbox{\\tiny{def}}}{=} \\frac{1}{r}\\frac{2\\exp(r)(\\exp(r)-1)}{1- 4(\\exp(2r)-1)}.\n\\end{equation}\nWe note that $C_I(r)$ is an increasing function of $r$ and that $C_I(r)\\to 2$ as $r\\to 0$ and $C_I(r)\\to \\infty$ as $r\\to \\frac{1}{2}\\log(\\frac{5}{4})$. \n\\end{prop}\n\nThis proposition is shown by an implicit function theorem argument on the function described in \\eqref{g} around the value $g(0,0) = 0$. The remainder of this section is dedicated to proving Proposition \\ref{IFTprop}.\n\nFirst, we compute the Fr{\\'e}chet derivatives with respect to $u \\in \\tilde{\\mathcal{F}}^{0,1}$ and $x\\in \\mathbb{R}^{2}$.  Below we use the notation $D_u$ to denote the one component Fr{\\'e}chet derivative of $g(u,x)$ so that $D_u g(u,x)$ is a two dimensional vector, and we denote $D_x$ to denote the two component Fr{\\'e}chet derivative of $g(u,x)$ so that $D_xg(u,x)$ is a $2\\times 2$ matrix.  We will also use the notation $D=(D_u, D_x)$ to denote the derivative in the variables $(u,x)$ and then $Dg(u,x)$ can be represented as a $3\\times 3$ matrix.\n\n\n\\begin{lemma}\\label{Frechet:Lemma}\n\tThe Fr\\'echet derivatives of $g$, recalling \\eqref{psi}, are given by\n\t\\begin{equation}\\label{Frechetu}\n\t\tD_{u}g(u,x)h\n\t\t=\n\t\t\\int_{-\\pi}^{\\pi}d\\alpha ~ h(\\alpha)\t\\begin{bmatrix}\n\t\t -\\sin(\\psi(\\alpha, x, u))\n\t\t\\\\\n\t\t\\cos(\\psi(\\alpha, x, u))\n\t\\end{bmatrix},\n\t\\end{equation}\n\tfor $h\\in \\tilde{\\mathcal{F}}^{0,1}$ and\n\t\\begin{equation}\\label{Frechetx}\n\t\tD_{x}g(u,x)y \n\t\t\t\t=\n\t\t\\begin{bmatrix}\n\t\t -2\\int_{-\\pi}^{\\pi}d\\alpha \\sin(\\psi) \\cos\\alpha & 2\\int_{-\\pi}^{\\pi}d\\alpha \\sin(\\psi) \\sin\\alpha\n\t\t\\\\\n\t\t 2\\int_{-\\pi}^{\\pi}d\\alpha \\cos(\\psi) \\cos\\alpha \n\t\t & -2\\int_{-\\pi}^{\\pi}d\\alpha \\cos(\\psi) \\sin\\alpha\n\t\\end{bmatrix}\n\t\\begin{bmatrix}\n\t\t y_{1}\n\t\t\\\\\n\t\ty_{2}\n\t\\end{bmatrix},\n\t\\end{equation}\n\tfor $y = \\begin{bmatrix}\n\t\t y_{1}\n\t\t\\\\\n\t\ty_{2}\n\t\\end{bmatrix} \\in \\mathbb{R}^{2}$.\n\\end{lemma}\n\nFor simplicity, we write \\eqref{g} in complex notation as\n\\begin{equation}\\notag\n\tg(u, x) = \\int_{-\\pi}^{\\pi} e^{i\\alpha + 2i(x_{1}\\cos(\\alpha) -x_{2}\\sin(\\alpha)) + iu(\\alpha)}d\\alpha.\n\\end{equation}\nIn particular then in complex notation \\eqref{Frechetu} takes the form\n\t\\begin{equation}\\notag\n\t\tD_{u}g(u,x)h = \\int_{-\\pi}^{\\pi} i h(\\alpha) e^{i\\psi(\\alpha, x, u)} d\\alpha.\n\t\\end{equation}\nWe will prove Lemma \\ref{Frechet:Lemma} using this expression for $g(u,x)$.\n\n\n\\begin{proof}[Proof of Lemma \\ref{Frechet:Lemma}]\n\tComputing the derivative with respect to $u$, we have\n\t\\begin{multline*}\n\t\t|g(u+h, x) - g(u,x) - D_{u}g(u,x)h| \\\\= \\Big|\\int_{-\\pi}^{\\pi}  e^{i\\psi(\\alpha, x, u)}(e^{i h(\\alpha)} - 1 - ih(\\alpha)) d\\alpha \\Big|\n\t\t\\\\\n\t\t\\leq 2\\pi \\|e^{ih(\\alpha)} - 1 - ih(\\alpha)\\|_{L^{\\infty}} \\leq 2\\pi \\sum_{n=2}^{\\infty} \\frac{\\|h\\|_{L^{\\infty}}^{n}}{n!} \n\t\t\\\\\n\t\t\\leq   \\Big(2\\pi \\sum_{n=1}^{\\infty} \\frac{\\|h\\|_{\\tilde{\\mathcal{F}}^{0,1}}^{n}}{(n+1)!} \\Big) \\|h\\|_{\\tilde{\\mathcal{F}}^{0,1}}\n\t\t\t\t\\leq   \\Big(2\\pi e^{\\|h\\|_{\\tilde{\\mathcal{F}}^{0,1}}} \\Big) \\|h\\|_{\\tilde{\\mathcal{F}}^{0,1}},\n\t\\end{multline*}\n\tsince for $h\\in\\tilde{\\mathcal{F}}^{0,1}$ we have that $\\hat{h}(0) = \\hat{h}(\\pm 1) = 0$ and so\n\t$$ \\|h\\|_{L^{\\infty}} \\leq \\sum_{k\\in\\mathbb{Z}} |\\hat{h}(k)| = \\sum_{|k| \\geq 2}  |\\hat{h}(k)|.$$\n\tThus, as $h \\rightarrow 0$, we obtain validation of \\eqref{Frechetu}. Then \\eqref{Frechetx} is proven in a similar way.  \\end{proof}\n\nWe now prove Proposition \\ref{IFTprop}.\n\n\\begin{proof}[Proof of Proposition \\ref{IFTprop}]\nFirst notice that from \\eqref{Frechetx} we have \n\\begin{equation}\\label{dxg00}\n    \\begin{aligned}\n\tD_{x} g(0,0)y\n\t& = \t\t\\begin{bmatrix}\n\t\t -2\\int_{-\\pi}^{\\pi}d\\alpha \\sin(\\alpha) \\cos\\alpha & 2\\int_{-\\pi}^{\\pi}d\\alpha \\sin(\\alpha) \\sin\\alpha\n\t\t\\\\\n\t\t 2\\int_{-\\pi}^{\\pi}d\\alpha \\cos(\\alpha) \\cos\\alpha \n\t\t & -2\\int_{-\\pi}^{\\pi}d\\alpha \\cos(\\alpha) \\sin\\alpha\n\t\\end{bmatrix}\n\t\\begin{bmatrix}\n\t\ty_{1}\\\\\n\t\ty_{2}\n\t\\end{bmatrix} \n\t\\\\\n\n\t&= 2\\pi \\begin{bmatrix}\n\t\t0 &  1\\\\\n\t\t1 & 0\n\t\\end{bmatrix} \\begin{bmatrix}\n\t\ty_{1}\\\\\n\t\ty_{2}\n\t\\end{bmatrix}\n\t=2\\pi\\begin{bmatrix}\n\t\ty_{2}\\\\\n\t\ty_{1}\n\t\\end{bmatrix},\n\t\\end{aligned}\n\\end{equation}\nTherefore $D_{x} g(0,0)^{-1} = \\frac{1}{2\\pi}\\begin{bmatrix}\n\t\t0 &  1\\\\\n\t\t1 & 0\n\t\\end{bmatrix}$.  \nFor simplicity, we normalize the function $g$ around $(0,0)$ by defining\n\\begin{equation}\\label{tildeg}\n\t\\tilde{g}(u,x) = \\begin{bmatrix}\n\t\t\\tilde{g}_{1}(u,x)\\\\\n\t\t\\tilde{g}_{2}(u,x)\n\t\\end{bmatrix} \n\t=  D_{x}g(0,0)^{-1}\\begin{bmatrix}\n\t\tg_{1}(u,x)\\\\\n\t\tg_{2}(u,x)\n\t\\end{bmatrix}\n\t=\\frac{1}{2\\pi}\\begin{bmatrix}\n\t\tg_{2}(u,x)\\\\\n\t\tg_{1}(u,x)\n\t\\end{bmatrix},\n\\end{equation}\nand thus, $D_{x}\\tilde{g}(0,0) = \\mathbb{I}_{\\mathbb{R}^{2}}$ which is the identity matrix on $\\mathbb{R}^{2}$. Next, define the function $\\phi: \\tilde{\\mathcal{F}}^{0,1} \\times \\mathbb{R}^{2} \\rightarrow \\tilde{\\mathcal{F}}^{0,1} \\times \\mathbb{R}^{2}$ by\n\\begin{equation}\\label{phi.def}\n\\phi(u,x) = [D\\tilde{\\phi}(0,0)]^{-1}\\tilde{\\phi}(u,x), \\quad\\text{  where  } \\quad \\tilde{\\phi}(u,x) \n=\n\\begin{bmatrix}\nu   \\\\\n\t\\tilde{g}_{1}(u,x)\\\\\n\t\t\\tilde{g}_{2}(u,x)\n\\end{bmatrix}.\n\\end{equation}\nThen also $D\\tilde{\\phi}(u,x)$ is a $3 \\times 3$ matrix.  We will obtain the implicit function $F$ given in Proposition \\ref{IFTprop} by inverting $\\phi$ in a neighborhood of the point $(0, 0) \\in \\tilde{\\mathcal{F}}^{0,1} \\times \\mathbb{R}^{2}$. We will also calculate $[D\\tilde{\\phi}(0,0)]^{-1}$,  notice that \n\\begin{equation}\\notag\nD\\tilde{\\phi}(0,0) \n=\n    \\begin{bmatrix}\n\t\t\\mathbb{I}_{\\tilde{\\mathcal{F}}^{0,1}} & 0\\\\\n\t\tD_{u}\\tilde{g}(0,0) & D_{x}\\tilde{g}(0,0)\n\t\\end{bmatrix}\n\t=\n\t\t\\begin{bmatrix}\n\t\t\\mathbb{I}_{\\tilde{\\mathcal{F}}^{0,1}} &  0\\\\\n\t\t  0  & \\mathbb{I}_{\\mathbb{R}^{2}}\n\t\\end{bmatrix}.\n\\end{equation}\nHere $\\mathbb{I}_{\\tilde{\\mathcal{F}}^{0,1}}$ is the identity map on $\\tilde{\\mathcal{F}}^{0,1}$.\nThe last equality holds since\n\\begin{equation*}\n\\begin{aligned}\nD_{u}\\tilde{g}(0,0)h \n&= D_{x}g(0,0)^{-1}D_{u}g(0,0)h \n\\\\\n&=\\frac{1}{2\\pi}\n\t\t\\int_{-\\pi}^{\\pi}d\\alpha ~ h(\\alpha)\t\\begin{bmatrix}\n\t\t0 &  1\\\\\n\t\t1 & 0\n\t\\end{bmatrix}\t\\begin{bmatrix}\n\t\t -\\sin(\\psi(\\alpha, x, u))\n\t\t\\\\\n\t\t\\cos(\\psi(\\alpha, x, u))\n\t\\end{bmatrix}=  0,\n\\end{aligned}\n\\end{equation*}\nsince $h\\in\\tilde{\\mathcal{F}}^{0,1}$ so that $\\hat{h}(\\pm 1) =0$.  Therefore $[D\\tilde{\\phi}(0,0)]^{-1} = \t\t\\begin{bmatrix}\n\t\t\\mathbb{I}_{\\tilde{\\mathcal{F}}^{0,1}} &  0\\\\\n\t\t  0  & \\mathbb{I}_{\\mathbb{R}^{2}}\n\t\\end{bmatrix}$.\n\n\nFor two norms $\\| \\cdot \\|$ and $| \\cdot |$, we will use the notation that \n\\begin{equation}\\label{littleo}\n   f=o(|h|), \\quad \\text{if} \\quad \n   \\|f\\|\\to 0 \\quad \\text{and} \\quad \\frac{\\|f\\|}{|h|}\\to 0 \\quad \\text{as}\\quad  |h|\\to 0.\n\\end{equation}\nNow, to invert $\\phi$, we first define the function \n\\begin{equation}\\label{tau.def}\n    \\tau(u,x) \\overset{\\mbox{\\tiny{def}}}{=} (u,x) - \\phi(u,x).\n\\end{equation}\nWe will show that $\\tau(u,x)$ is a contraction map by computing $D\\tau$. We will calculate that\n$$\nD\\tau(u,x) = \\mathbb{I} - [D\\tilde{\\phi}(0,0)]^{-1} D\\tilde{\\phi}(u,x),\n$$\nwhere $\\mathbb{I}$ is the identity map on $\\tilde{\\mathcal{F}}^{0,1} \\times \\mathbb{R}^{2}$. To this end, we compute \n\\begin{multline}\\notag\n\t\\tilde{\\phi}(u+h,x+y) - \\tilde{\\phi}(u,x)\n\t\\\\ = \\tilde{\\phi}(u+h,x+y) - \\tilde{\\phi}(u,x+y) + \\tilde{\\phi}(u,x+y) - \\tilde{\\phi}(u,x)\n\t\\\\\n\t= \\begin{bmatrix}\n\t\t\\mathbb{I}_{\\tilde{\\mathcal{F}}^{0,1}} & 0\\\\\n\t\tD_{u}\\tilde{g}(u,x) & D_{x}\\tilde{g}(u,x)\n\t\\end{bmatrix}\n\t\\begin{bmatrix}\n\t\th\\\\\n\t\ty\n\t\\end{bmatrix} + o(\\|h\\|_{\\tilde{\\mathcal{F}}^{0,1}} + |y|).\n\\end{multline}\nThen using, \n\\eqref{littleo},  the above holds as $\\|h\\|_{\\tilde{\\mathcal{F}}^{0,1}} + |y| \\to 0$.\n Hence,\n\\begin{equation*}\n\\begin{aligned}\n\tD\\tau(u,x) &=  \\mathbb{I} - \n\t\\begin{bmatrix}\n\t\t\\mathbb{I}_{\\tilde{\\mathcal{F}}^{0,1}} &  0\\\\\n\t\t  0  & \\mathbb{I}_{\\mathbb{R}^{2}}\n\t\\end{bmatrix} \\!\\!\n\t\\begin{bmatrix}\n\t\t\\mathbb{I}_{\\tilde{\\mathcal{F}}^{0,1}} & 0\\\\\n\t\tD_{u}\\tilde{g}(u,x) & D_{x}\\tilde{g}(u,x)\n\t\\end{bmatrix} \\\\\n\t&=\n\t\\begin{bmatrix}\n\t\t0  & 0\\\\\n\t\t-D_{u}\\tilde{g}(u,x) & \\mathbb{I}_{\\mathbb{R}^{2}}- D_{x}\\tilde{g}(u,x)\n\t\\end{bmatrix}.\n\t\\end{aligned}\n\\end{equation*}\nTo obtain $\\tau$ as a contraction on some ball $B_{R}(0) \\subset \\tilde{\\mathcal{F}}^{0,1} \\times \\mathbb{R}^{2}$, it is sufficient to show that the following holds\n\\begin{equation}\\label{contraction}\n\t\\Big\\|D\\tau(u,x) \\begin{bmatrix}\n\t\th\\\\\n\t\ty\n\t\\end{bmatrix}\\Big\\|_{\\tilde{\\mathcal{F}}^{0,1}\\times\\mathbb{R}^{2}} < r \\|(h,y)\\|_{\\tilde{\\mathcal{F}}^{0,1}\\times\\mathbb{R}^{2}},\n\\end{equation}\nfor some $0< r< 1$ and\nfor any $(h,y)\\in B_{R}(0) \\subset \\tilde{\\mathcal{F}}^{0,1} \\times \\mathbb{R}^{2}$  where \n\\begin{equation}\\notag\n    B_{R}(0) \\overset{\\mbox{\\tiny{def}}}{=} \\{(u,x) \\ | \\ |x|+\\|u\\|_{\\tilde{\\mathcal{F}}^{0,1}} < R \\}, \\quad R>0.\n\\end{equation}\nHere we also denote $\\|(h,y)\\|_{\\tilde{\\mathcal{F}}^{0,1}\\times\\mathbb{R}^{2}} = \\|h\\|_{\\tilde{\\mathcal{F}}^{0,1}}+ |y|$.\n\n\n\n\n\n\nSince we have \\eqref{dxg00} then, using \\eqref{Frechetu} and \\eqref{Frechetx}, condition \\eqref{contraction} becomes the condition that the inequality\n\\begin{multline}\\label{contractioncondition}\n\\Big\\|D\\tau(u,x) \\begin{bmatrix}\n\t\th\\\\\n\t\ty\n\t\\end{bmatrix}\\!\\Big\\|_{\\tilde{\\mathcal{F}}^{0,1}\\times\\mathbb{R}^{2}}\\!\\!\n\t=\n\t\\\\\n\t\\Bigg|\\frac{1}{2\\pi} \t\t\\int_{-\\pi}^{\\pi}d\\alpha ~ h(\\alpha)\t\\begin{bmatrix}\n\t\t -\\cos(\\psi(\\alpha, x, u))\n\t\t\\\\\n\t\t\\sin(\\psi(\\alpha, x, u))\n\t\\end{bmatrix}\n\t\\\\\n\t+\\! \\frac{1}{2\\pi} \n\t\t\t\\begin{bmatrix}\n\t\t\t\t\t 2\\int_{-\\pi}^{\\pi}d\\alpha \\cos(\\psi) \\left(y_1 \\cos\\alpha -y_2\\sin\\alpha\\right) - 2\\pi y_1\n\t\t \\\\\n\t\t -2\\int_{-\\pi}^{\\pi}d\\alpha \\sin(\\psi) \\left(y_1 \\cos\\alpha -y_2\\sin\\alpha\\right)- 2\\pi y_2\n\t\\end{bmatrix}\n\t\\Bigg|\n\t\\\\\n\t< r\\|h\\|_{\\tilde{\\mathcal{F}}^{0,1}} + r|y|,\n\\end{multline}\nholds for a fixed $0< r< 1$, for all $(u,x)\\in B_{R}(0)$, and for any $(h,y)\\in \\tilde{\\mathcal{F}}^{0,1}\\times \\mathbb{R}^{2}$.  We also use the definition \\eqref{psi} in the integral above.\n\n\nWe further {\\it claim} that condition \\eqref{contractioncondition} is satisfied for any ball \n$B_{R}(0)$ such that $2|x|+\\|u\\|_{\\tilde{\\mathcal{F}}^{0,1}} < \\log(2)$ holds for all $(u,x)\\in B_{R}(0)$. \n\n\n\n {\\it Proof of the claim:}\tLet\n  \t\\begin{equation}\\label{A1}\n \tA_{1} = \n \t\\begin{bmatrix}\n\t\tA_{11}\n\t\t\\\\\n\t\tA_{12}\n\t\\end{bmatrix}\n \t=\n \t \\frac{1}{2\\pi} \t\t\\int_{-\\pi}^{\\pi}d\\alpha ~ h(\\alpha)\t\\begin{bmatrix}\n\t\t -\\cos(\\psi(\\alpha, x, u))\n\t\t\\\\\n\t\t\\sin(\\psi(\\alpha, x, u))\n\t\\end{bmatrix}.\\end{equation}\nWe will use the complex exponential representation of $\\cos(\\psi(\\alpha, x, u))$ and $\\sin(\\psi(\\alpha, x, u))$ with \\eqref{psi}.  We will further Taylor expand $\\exp\\left(i 2(x_{1}\\cos(\\alpha)-x_{2}\\sin(\\alpha))+i u(\\alpha)) \\right)$ \n\tand use that\n\t$\\frac{1}{2\\pi}\\int_{-\\pi}^{\\pi} h(\\alpha) e^{\\pm i\\alpha} d\\alpha = \\hat{h}(\\mp 1) = 0$ since $h\\in \\tilde{\\mathcal{F}}^{0,1}$.   We obtain\n\t\t\\begin{multline*}\n\t\t|A_{11}| \n\t\t\\le \\frac{1}{4\\pi}\\Big| \\int_{-\\pi}^{\\pi} h(\\alpha) e^{i\\alpha}\\sum_{n=1}^{\\infty} \\frac{i^{n}(2(x_{1}\\cos(\\alpha)-x_{2}\\sin(\\alpha))+u(\\alpha))^{n}}{n!} d\\alpha \\Big|\\\\\n+ \\frac{1}{4\\pi}\\Big| \\int_{-\\pi}^{\\pi} h(\\alpha) e^{-i\\alpha}\\sum_{n=1}^{\\infty} \\frac{i^{n}(2(x_{1}\\cos(\\alpha)-x_{2}\\sin(\\alpha))+u(\\alpha))^{n}}{n!} d\\alpha \\Big|.\n\t\\end{multline*}\n\tFor simplicity we estimate the term with $e^{i\\alpha}$ below:\n\\begin{multline}\\notag\n\\frac{1}{2\\pi}\\Big| \\int_{-\\pi}^{\\pi} h(\\alpha) e^{i\\alpha}\\sum_{n=1}^{\\infty} \\frac{i^{n}(2(x_{1}\\cos(\\alpha)-x_{2}\\sin(\\alpha))+u(\\alpha))^{n}}{n!} d\\alpha \\Big|\n\\\\\n\t\t= \\frac{1}{2\\pi}\\Big| \\int_{-\\pi}^{\\pi} h(\\alpha) e^{i\\alpha}\\sum_{n=1}^{\\infty}\\sum_{k=0}^{n} {n \\choose k} \\frac{2^{k}(x_{1}\\cos(\\alpha)-x_{2}\\sin(\\alpha))^{k}u(\\alpha)^{n-k}}{n!} d\\alpha   \\Big|\n\\\\\n\t\t\\leq \\sum_{n=1}^{\\infty}\\sum_{k=0}^{n} {n \\choose k}\\frac{1}{n!}\\Big|\\Big(\\hat{h} \\ast^{k} \\mathcal{F}\\{2(x_{1}\\cos(\\alpha)-x_{2}\\sin(\\alpha))\\}\\ast^{n-k}\\hat{u}\\Big)(-1)\\Big|\n\\\\\n\\leq \\sum_{n=1}^{\\infty}\\sum_{k=0}^{n} {n \\choose k}\\frac{1}{n!} \\|\\hat{h}\\|_{\\ell^{1}}\\|\\hat{u}\\|_{\\ell^{1}}^{n-k}2^{k-1}|x|^{k}.\n\\end{multline}\nThe last line is obtained using Young's inequality for convolutions and \n\t$$\\widehat{\\cos}(k) = \\frac{1}{2}(\\delta(1-k) + \\delta(-1-k)), \\quad   \\widehat{\\sin}(k) = \\frac{-i}{2}(\\delta(1-k) - \\delta(-1-k)),$$\n and thus the $\\ell^{1}$ norm of the Fourier transform is\n \\begin{equation}\\label{ell1ft.def}\n    \\|(2(x_{1}\\cos(\\alpha)-x_{2}\\sin(\\alpha)))^{\\wedge}\\|_{\\ell^{1}} = 2|x|,\n\\end{equation}\n\tand the $\\ell^{\\infty}$ norm of the Fourier transform is\n\t$$\\|(2(x_{1}\\cos(\\alpha)-x_{2}\\sin(\\alpha)))^{\\wedge}\\|_{\\ell^{\\infty}} = |x|.$$\nThen, rewriting the series in function form using Taylor's theorem, we get\n\\begin{equation}\\notag\n    \t|A_{11}| \\leq \\frac{1}{2}\\|h\\|_{\\tilde{\\mathcal{F}}^{0,1}}\\Big( e^{2|x| + \\|u\\|_{\\tilde{\\mathcal{F}}^{0,1}}} - 1\\Big).\n\\end{equation}\nDoing the same for the second term in \\eqref{A1} we obtain\n\\begin{equation}\\label{A1estimate}\n    \t|A_{1}| \\leq \\|h\\|_{\\tilde{\\mathcal{F}}^{0,1}}\\Big( e^{2|x| + \\|u\\|_{\\tilde{\\mathcal{F}}^{0,1}}} - 1\\Big).\n\\end{equation}\n\tSince the latter term in \\eqref{contractioncondition} does not involve $h$, for this term we need \n$$\n e^{2|x| + \\|u\\|_{\\tilde{\\mathcal{F}}^{0,1}}} - 1 < r < 1.\n$$\nThese estimates give the upper bound in terms of $\\|h\\|_{\\tilde{\\mathcal{F}}^{0,1}}$ in \\eqref{contractioncondition}.\n\t\n\t\n\t\n\n\n\n\t\n\tThe other term in \\eqref{contractioncondition} will give us the upper bound in terms of $|y|$. Now let\n\t\\begin{equation*}\n\t    A_{2} = \\begin{bmatrix} A_{21} \\\\ A_{22} \\end{bmatrix}=  \\frac{1}{2\\pi} \n\t\t\t\\begin{bmatrix}\n\t\t\t\t\t 2\\int_{-\\pi}^{\\pi}d\\alpha \\cos(\\psi) \\left(y_1 \\cos\\alpha -y_2\\sin\\alpha\\right) - 2\\pi y_1\n\t\t \\\\\n\t\t -2\\int_{-\\pi}^{\\pi}d\\alpha \\sin(\\psi) \\left(y_1 \\cos\\alpha -y_2\\sin\\alpha\\right)- 2\\pi y_2\n\t\\end{bmatrix}.\n\t\\end{equation*}\n\tThen using \\eqref{psi}, again we use the complex exponential form of $\\cos(\\psi)$ and by Taylor expansion we have\n\t\\begin{multline}\\notag\n\t\tA_{21} = \n\t\t\\frac{1}{2\\pi} \\int_{-\\pi}^{\\pi} d\\alpha (y_{1}\\cos(\\alpha)-y_{2}\\sin(\\alpha))e^{i\\alpha}\\sum_{n=1}^{\\infty}\\frac{i^{n}(\\psi(\\alpha,x,u)-\\alpha)^{n}}{n!} \n\t\t\\\\\n+\t\\frac{1}{2\\pi} \\int_{-\\pi}^{\\pi} d\\alpha (y_{1}\\cos(\\alpha)-y_{2}\\sin(\\alpha))e^{-i\\alpha}\\sum_{n=1}^{\\infty}\\frac{i^{n}(\\psi(\\alpha,x,u)-\\alpha)^{n}}{n!}.\n\t\\end{multline}\nWe estimate the term with $e^{i\\alpha}$ as \n\t\\begin{multline}\\notag\n\t\t\\frac{1}{2\\pi}\\Big| \\int_{-\\pi}^{\\pi} d\\alpha (y_{1}\\cos\\alpha-y_{2}\\sin\\alpha)e^{i\\alpha}\\sum_{n=1}^{\\infty}\\frac{i^{n}(\\psi(\\alpha,x,u)-\\alpha)^{n}}{n!} \\Big|\n\t\t\\\\\n =\\Big| \\sum_{n=1}^{\\infty}\\frac{1}{2\\pi}\\int_{-\\pi}^{\\pi} (y_{1}\\cos\\alpha-y_{2}\\sin\\alpha)e^{i\\alpha}\\frac{(2(x_{1}\\cos\\alpha-x_{2}\\sin\\alpha)+u(\\alpha))^{n}}{n!} d\\alpha \\Big|\n\t\t\\\\\n\t\t\\leq \\frac{1}{2}\\sum_{n=1}^{\\infty} \\frac{1}{n!}\\|2(y_{1}\\cos\\alpha-y_{2}\\sin\\alpha)^{\\wedge}\\|_{\\ell^{\\infty}}\\|(2(x_{1}\\cos\\alpha-x_{2}\\sin\\alpha)+u(\\alpha))^{\\wedge}\\|_{\\ell^{1}}^{n}\n\t\t\\\\\n\t\t\\leq \\frac{1}{2}\\Big( e^{2|x| + \\|u\\|_{\\tilde{\\mathcal{F}}^{0,1}}} - 1\\Big) |y|.\n\t\\end{multline}\n\tAll the other terms are estimated in the same way.\n\tAdding all the estimates together we obtain the condition \n\\begin{equation}\\notag\n    |A_2| \\le 2\\Big( e^{2|x| + \\|u\\|_{\\tilde{\\mathcal{F}}^{0,1}}} - 1\\Big) |y|.\n\\end{equation}\nThis is a bigger coefficient in front of $|y|$ than the coefficient in the bound for $A_{1}$. Thus, the following condition\n$$\n2\\Big( e^{2|x| + \\|u\\|_{\\tilde{\\mathcal{F}}^{0,1}}} - 1\\Big) < 1,\n$$ \nis sufficient. This yields the claim that condition \\eqref{contractioncondition} is satisfied on any ball contained in the set of $(x,u)$ such that $$2|x|+\\|u\\|_{\\tilde{\\mathcal{F}}^{0,1}} < \\log(3\/2).$$  This completes the proof of the {\\it claim}.\n\t\n\n\nNow  $\\tau$ satisfies the contraction \\eqref{contraction} for $(u,x)\\in B_{R}(0)$ for $R$ chosen to satisfy \\eqref{contractioncondition}.  Recalling \\eqref{phi.def} and \\eqref{tau.def}, for $v,w\\in\\tilde{\\mathcal{F}}^{0,1}\\times\\mathbb{R}^{2}$ and for $0<r<1$ fixed in \\eqref{contractioncondition}, we obtain\n\\begin{align*}\n\t\\|v-w\\|_{\\tilde{\\mathcal{F}}^{0,1}\\times\\mathbb{R}^{2}} &\\leq \\|\\phi(v)-\\phi(w)\\|_{\\tilde{\\mathcal{F}}^{0,1}\\times\\mathbb{R}^{2}} + \\|\\tau(v)-\\tau(w)\\|_{\\tilde{\\mathcal{F}}^{0,1}\\times\\mathbb{R}^{2}}\\\\ &\\leq \\|\\phi(v)-\\phi(w)\\|_{\\tilde{\\mathcal{F}}^{0,1}\\times\\mathbb{R}^{2}} + r\\|v-w\\|_{\\tilde{\\mathcal{F}}^{0,1}\\times\\mathbb{R}^{2}}.\n\\end{align*} \nWe conclude that\n\\begin{equation*}\n\t\\|v-w\\|_{\\tilde{\\mathcal{F}}^{0,1}\\times\\mathbb{R}^{2}} \n\t\\leq (1-r)^{-1}\\|\\phi(v)-\\phi(w)\\|_{\\tilde{\\mathcal{F}}^{0,1}\\times\\mathbb{R}^{2}}.\n\\end{equation*}\nHence, $\\phi$ is an injection on $B=B_R(0)$ and so there exists an inverse $\\phi^{-1} : \\phi(B)\\rightarrow B$. Now we define the map $F(u)$ by\n$$\nF(u) \\overset{\\mbox{\\tiny{def}}}{=} \\pi_{2}\\circ \\phi^{-1}(u,(0,0)) \\text{ \\ \\ where \\ \\ } \\pi_{2}(a,(b_{1},b_{2})) = (b_{1},b_{2}).\n$$\nFurther define the map $\\pi_{1}$ by $\\pi_{1}(a,(b_{1},b_{2})) = a$.\n\nGiven $u\\in \\pi_{1}(B)$, by the definition of $F(u)$ there exists $u'\\in \\pi_{1}(B)$ such that $(u,(0,0)) = \\phi(u',F(u))$.  Then $u=u'$ by \\eqref{phi.def}. Further $\\phi(u,F(u)) = (u,\\tilde{g}(u,F(u)))$ and therefore $\\tilde{g}(u,F(u)) = 0$ using \\eqref{phi.def} and \\eqref{tildeg}.  We have thus shown that $g(u,F(u)) = 0$ from \\eqref{g}.  We conclude that $(u,(0,0)) \\in \\phi(B)$  for all $u\\in \\pi_{1}(B)$ and so $(u,F(u)) = \\phi^{-1}(u,(0,0)) \\in B$.  Therefore $g(u,F(u)) = 0$ for any $u\\in \\pi_{1}(B)$. This concludes the proof of the existence of the function $F$ described in Proposition \\ref{IFTprop}. \n\n\n\n\n\n\n\n\nTo obtain the estimate \\eqref{frequencyrelation} on $B$, first note that for $x = (x_{1},x_{2})$ we have\n\\begin{equation*}\n\\begin{aligned}\n|x| &= |\\pi_{2}\\circ \\phi^{-1}(u,(0,0))| = |\\pi_{2}\\circ \\phi^{-1}(u,(0,0)) - \\pi_{2}\\circ \\phi^{-1}(0,(0,0))|\\\\\n&\\leq \\|u\\|_{\\tilde{\\mathcal{F}}^{0,1}}\\|D_{u}(\\pi_{2}\\circ\\phi^{-1})\\|.\n\\end{aligned}\n\\end{equation*}\nAbove note that $g(0,F(0))=0$ and so $F(0)=0$.  \nAlso the norm on $D_{u}(\\pi_{2}\\circ\\phi^{-1})$ on is the operator norm.   We calculate $D_{u}(\\pi_{2}\\circ\\phi^{-1})$  as follows:\n$$\\pi_{2}\\circ \\phi^{-1}(u+h,y)-\\pi_{2}\\circ \\phi^{-1}(u,y) = \\pi_{2}(u+h,x_{h}) - \\pi_{2}(u,x) = x_{h}-x,$$\nwhere we suppose $(u,y) \\in \\phi(B)$ and $(u+h,y) \\in \\phi(B)$ for small $h$.  Therefore\n\\begin{equation*}\n\\tilde{g}(u+h,x_{h}) = y \\text{ \\ \\ and \\ \\ }  \\tilde{g}(u,x) = y.    \n\\end{equation*}\nWe conclude that $[\\tilde{g}(u+h,x_{h})- \\tilde{g}(u,x)] = 0$.\nThus\n$$0 = \\tilde{g}(u+h,x_{h}) - \\tilde{g}(u,x_{h}) +\\tilde{g}(u,x_{h})- \\tilde{g}(u,x).$$\nThen, using \\eqref{littleo}, we have\n$$\n0 = D_{u}\\tilde{g}(u,x_{h})h +o(\\|h\\|_{\\tilde{\\mathcal{F}}^{0,1}}) + D_{x}\\tilde{g}(u,x)(x_{h}-x) +o(|x_{h}-x|).\n$$\nWe can now conclude that\n\\begin{multline}\\notag\n    x_{h}-x + D_{x}\\tilde{g}(u,x)^{-1}o(|x_{h}-x|) \\\\ = -D_{x}\\tilde{g}(u,x)^{-1}D_{u}\\tilde{g}(u,x)h -D_{x}\\tilde{g}(u,x)^{-1}o(\\|h\\|_{\\tilde{\\mathcal{F}}^{0,1}}).\n\\end{multline}\nWe will show that $D_{x}\\tilde{g}(u,x)^{-1}$ is a bounded operator with bound \\eqref{detandTbound} for $(x,u)$ satisfying \\eqref{xucriteria}. Thus, for $(x,u)$ satisfying \\eqref{xucriteria}, we have\n$$D_{u}(\\pi_{2}\\circ \\phi^{-1}) = -D_{x}\\tilde{g}(u,x)^{-1}D_{u}\\tilde{g}(u,x).$$\nThen we conclude that\n\\begin{equation}\\label{xinitialbound}\n|x| \\leq \\|D_{x}\\tilde{g}(u,x)^{-1}\\| \\|D_{u}\\tilde{g}(u,x)\\| \\|u\\|_{\\tilde{\\mathcal{F}}^{0,1}}.\n\\end{equation}\nThe term $D_{u}\\tilde{g}(u,x)$, using \\eqref{Frechetu} with \\eqref{dxg00}, is given by\n$$D_{u}\\tilde{g}(u,x) = D_{x}g(0,0)^{-1}D_{u}g(u,x) = -A_{1},$$\nwhere  $A_{1}$ is given by \\eqref{A1} with \\eqref{psi}. Hence, using \\eqref{A1estimate}, we have \n\\begin{equation}\\label{Dugbound}\n\\|D_{u}\\tilde{g}(u,x)\\| \\leq \t\\Big(e^{2|x| + \\|u\\|_{\\tilde{\\mathcal{F}}^{0,1}}} - 1\\Big).\n\\end{equation}\nConsider the operator $D_{x}\\tilde{g}(u,x)^{-1}$.  \nUsing \\eqref{Frechetx} and \\eqref{dxg00} with \\eqref{psi}, we calculate that\n\\begin{equation*}\n    D_{x}\\tilde{g}(u,x) =\\! \\frac{1}{\\pi}\\! \\begin{bmatrix}\n    \\!\\int_{-\\pi}^{\\pi}\\cos(\\alpha)\\cos(\\psi(\\alpha)) d\\alpha & -\\int_{-\\pi}^{\\pi}\\sin(\\alpha)\\cos(\\psi(\\alpha)) d\\alpha\\\\\\\\\n    -\\int_{-\\pi}^{\\pi}\\cos(\\alpha)\\sin(\\psi(\\alpha)) d\\alpha &  \\int_{-\\pi}^{\\pi}\\sin(\\alpha)\\sin(\\psi(\\alpha)) d\\alpha\n    \\end{bmatrix}\\!.    \n\\end{equation*}\nInverting this matrix, we obtain\n\\begin{equation}\\label{aux1}\n    \\begin{aligned}\n    D_{x}\\tilde{g}(u,x)^{-1} = \\frac{1}{\\det(D_{x}\\tilde{g}(u,x))} T(\\psi),\n    \\end{aligned}\n\\end{equation}\nwhere\n\\begin{equation}\\label{T.psi.def}\n    T(\\psi)=\\frac{1}{\\pi} \\begin{bmatrix}\n\\int_{-\\pi}^{\\pi}\\sin(\\alpha)\\sin(\\psi(\\alpha)) d\\alpha\\ \\ \\ \\ \\int_{-\\pi}^{\\pi}\\sin(\\alpha)\\cos(\\psi(\\alpha)) d\\alpha\n\\\\\n\\\\\n\\int_{-\\pi}^{\\pi}\\cos(\\alpha)\\sin(\\psi(\\alpha)) d\\alpha \\ \\ \\ \\  \\int_{-\\pi}^{\\pi}\\cos(\\alpha)\\cos(\\psi(\\alpha)) d\\alpha\n\\end{bmatrix}.\n\\end{equation}\nWe will calculate a lower bound for $d\\overset{\\mbox{\\tiny{def}}}{=}\\det(D_{x}\\tilde{g}(u,x))$:  \n\\begin{align}\nd\n&\\!=\\! \\frac1{\\pi^{2}}\\!\\int_{-\\pi}^{\\pi}\\!\\!\\cos(\\alpha)\\cos(\\psi(\\alpha)) d\\alpha\\! \\int_{-\\pi}^{\\pi}\\!\\!\\sin(\\beta)\\sin(\\psi(\\beta)) d\\beta \\nonumber\\\\ \\label{det}\n&\\quad- \\frac1{\\pi^{2}}\\! \\int_{-\\pi}^{\\pi}\\!\\!\\!\\!\\cos(\\alpha)\\sin(\\psi(\\alpha)) d\\alpha \\!\\int_{-\\pi}^{\\pi}\\!\\!\\!\\sin(\\beta)\\cos(\\psi(\\beta)) d\\beta\\\\\n&=\\frac1{\\pi^{2}}\\int_{-\\pi}^{\\pi}\\!\\!\\int_{-\\pi}^{\\pi}\\!\\!\\cos(\\alpha)\\sin(\\beta)\\sin(\\psi(\\beta)-\\psi(\\alpha))d\\alpha d\\beta.\\nonumber\n\\end{align}\nNext, recalling \\eqref{psi}, define \n$$\nR(\\alpha,\\beta) \\overset{\\mbox{\\tiny{def}}}{=} 2x_{1}(\\cos\\beta-\\cos\\alpha)-2x_{2}(\\sin\\beta-\\sin\\alpha) + u(\\beta)-u(\\alpha).\n$$\nThen we have\n\\begin{align}\n|\\sin(\\psi(\\beta)-\\psi(\\alpha))& -\\sin(\\beta - \\alpha)|  = |\\sin(\\beta-\\alpha + R(\\alpha,\\beta))-\\sin(\\beta - \\alpha)| \\nonumber\\\\\\label{sinAalphbeta}\n&=  |\\sin(\\beta-\\alpha)(\\cos(R(\\alpha,\\beta)) - 1)+\\cos(\\beta-\\alpha)\\sin(R(\\alpha,\\beta))| \\\\\\nonumber\n&\\leq |\\cos(R(\\alpha,\\beta)) - 1| + |\\sin(R(\\alpha,\\beta))| .\n\\end{align}\nSince $|R(\\alpha,\\beta)| \\leq 4|x| + 2\\|u\\|_{\\mathcal{F}^{0,1}}$, we have that\n\\begin{multline}\\label{sinAalphbetabound}\n|\\cos(R(\\alpha,\\beta)) - 1| + |\\sin(R(\\alpha,\\beta))| \n\\\\\n\\leq \\sum_{n=1}^{\\infty} \\frac{(4|x|\\! +\\! 2\\|u\\|_{\\mathcal{F}^{0,1}})^{2n}}{(2n)!}\\! +\\! \\sum_{n=0}^{\\infty}\\frac{(4|x| \\!+\\! 2\\|u\\|_{\\mathcal{F}^{0,1}})^{2n+1}}{(2n+1)!} = e^{4|x| + 2\\|u\\|_{\\mathcal{F}^{0,1}}} \\!-\\! 1,\n\\end{multline}\nwhich tends to zero as $(u,x)$ go to zero. Hence, by \\eqref{sinAalphbeta} and \\eqref{sinAalphbetabound} we have\n\\begin{multline*}\n\\int_{-\\pi}^{\\pi}\\int_{-\\pi}^{\\pi}\\cos(\\alpha)\\sin(\\beta)(\\sin(\\psi(\\beta)-\\psi(\\alpha)) -\\sin(\\beta - \\alpha))d\\alpha d\\beta \\\\ \\geq -4\\pi^{2}\\|\\sin(\\psi(\\beta)-\\psi(\\alpha)) -\\sin(\\beta - \\alpha)\\|_{L^{\\infty}} \\\\\n\\geq -4\\pi^{2} (e^{4|x| + 2\\|u\\|_{\\mathcal{F}^{0,1}}} - 1)\n\\end{multline*}\nand\n$$\\int_{-\\pi}^{\\pi}\\int_{-\\pi}^{\\pi}\\cos(\\alpha)\\sin(\\beta) \\sin(\\beta - \\alpha)d\\alpha d\\beta = \\pi^{2}.$$\nWe thus conclude \n\\begin{multline*}\n    \\Big|\\int_{-\\pi}^{\\pi}\\int_{-\\pi}^{\\pi}\\cos(\\alpha)\\sin(\\beta)\\sin(\\psi(\\alpha)-\\psi(\\beta))d\\alpha d\\beta\\Big| \\\\\n    = \\Big|\\int_{-\\pi}^{\\pi}\\int_{-\\pi}^{\\pi}\\cos(\\alpha)\\sin(\\beta)(\\sin(\\psi(\\alpha)-\\psi(\\beta)) -\\sin(\\alpha - \\beta))d\\alpha d\\beta\\\\ \n    + \\int_{-\\pi}^{\\pi}\\int_{-\\pi}^{\\pi}\\cos(\\alpha)\\sin(\\beta) \\sin(\\alpha - \\beta)d\\alpha d\\beta\\Big|\\\\\n    \\geq -4\\pi^{2} (e^{4|x| + 2\\|u\\|_{\\mathcal{F}^{0,1}}} - 1) + \\pi^{2}.\n\\end{multline*}\nThus, the determinant given by \\eqref{det} is bounded from below by\n\\begin{equation}\\label{detbound}\n|\\det(D_{x}\\tilde{g}(u,x))| \\geq 1+4\\big(1- e^{4|x| + 2\\|u\\|_{\\mathcal{F}^{0,1}}}\\big).\n\\end{equation}\nTherefore, it only remains to estimate the norm $\\|T(\\psi)\\|$ to obtain a bound for the norm of $D_{x}\\tilde{g}(u,x)^{-1}$ in \\eqref{aux1}.\n\n\n\n\n\nRecalling \\eqref{T.psi.def}, we calculate that \n\\begin{equation}\\notag\n    T(\\psi)\\begin{bmatrix}\ny_{1}\\\\\ny_{2}\n\\end{bmatrix}\n=\n    \\frac{1}{\\pi} \\begin{bmatrix}\n\\int_{-\\pi}^{\\pi}\\sin(\\alpha)\\left( y_1 \\sin(\\psi(\\alpha))+y_2\\cos(\\psi(\\alpha)) \\right)d\\alpha\n\\\\\n\\int_{-\\pi}^{\\pi}\\cos(\\alpha)\\left( y_1 \\sin(\\psi(\\alpha))+y_2\\cos(\\psi(\\alpha))\\right) d\\alpha\n\\end{bmatrix}.\n\\end{equation}\nWe have for $\\phi(\\alpha) = \\psi(\\alpha) - \\alpha = 2(x_{1}\\cos(\\alpha) - x_{2}\\sin(\\alpha)) + u(\\alpha)$ that\n\\begin{multline}\\notag\n\\sin\\alpha\\left( y_1 \\sin(\\psi(\\alpha))+y_2\\cos(\\psi(\\alpha)) \\right)\n\\\\\n    =\n    -\\frac{1}{4}y_1\n    \\left(\n    e^{i\\phi}e^{2i\\alpha}-e^{-i\\phi}-e^{i\\phi}\n    +e^{-i\\phi}e^{-2i\\alpha}\n    \\right)\n    \\\\\n        +\\frac{1}{4i}y_2\n    \\left(\n    e^{i\\phi}e^{2i\\alpha}+e^{-i\\phi}-e^{i\\phi}\n    -e^{-i\\phi}e^{-2i\\alpha}\n    \\right).\n\\end{multline}\nAnd similarly\n\\begin{multline}\\notag\n\\cos\\alpha\\left( y_1 \\sin(\\psi(\\alpha))+y_2\\cos(\\psi(\\alpha)) \\right)\n\\\\\n    =\n    \\frac{1}{4i}y_1\n    \\left(\n    e^{i\\phi}e^{2i\\alpha}-e^{-i\\phi}+e^{i\\phi}\n    -e^{-i\\phi}e^{-2i\\alpha}\n    \\right)\n    \\\\\n        +\\frac{1}{4}y_2\n    \\left(\n    e^{i\\phi}e^{2i\\alpha}+e^{-i\\phi}+e^{i\\phi}\n    +e^{-i\\phi}e^{-2i\\alpha}\n    \\right).\n\\end{multline}\nWe conclude that\n\\begin{multline}\\notag\n    T(\\psi)\n=\n\\\\\n\\frac{1}{2}\n\\begin{bmatrix}\n\\left(\\text{Re}\\hspace{0.05cm}{\\mathcal{F}\\{e^{i\\phi}\\}(0)}-\\text{Re}\\hspace{0.05cm}{\\mathcal{F}\\{e^{i\\phi}\\}(-2)} \\right)\n&\n\\left(\\text{Im}\\hspace{0.05cm}{\\mathcal{F}\\{e^{i\\phi}\\}(-2)}-\\text{Im}\\hspace{0.05cm}{\\mathcal{F}\\{e^{i\\phi}\\}(0)} \\right)\n\\\\\n\\left(\\text{Im}\\hspace{0.05cm}{\\mathcal{F}\\{e^{i\\phi}\\}(-2)}-\\text{Im}\\hspace{0.05cm}{\\mathcal{F}\\{e^{i\\phi}\\}(0)} \\right)\n& \\left(\\text{Re}\\hspace{0.05cm}{\\mathcal{F}\\{e^{i\\phi}\\}(-2)}+\\text{Re}\\hspace{0.05cm}{\\mathcal{F}\\{e^{i\\phi}\\}(0)} \\right)\n\\end{bmatrix}.\n\\end{multline}\nHence \n\\begin{equation*}\n\\Big|T(\\psi) \\begin{bmatrix}\ny_{1}\\\\\ny_{2}\n\\end{bmatrix} \\Big| \\leq 2 |y|\\|\\widehat{e^{i\\phi(\\alpha)}}\\|_{\\ell^{1}}.\n\\end{equation*}\nHence, using \\eqref{ell1ft.def}, we obtain the bound\n\\begin{align*}\n\\|T(\\psi)\\| &\\leq \\sum_{n=0}^{\\infty} \\frac{\\|\\widehat{\\phi^{n}}\\|_{\\ell^{1}}}{n!}\\\\ \n&\\leq  \\sum_{n=1}^{\\infty}\\sum_{k=0}^{n} {n \\choose k}\\frac{1}{n!}\\Big\\| \\ast^{k} |\\mathcal{F}\\{2(x_{1}\\cos\\alpha-x_{2}\\sin\\alpha)\\}|\\ast^{n-k}|\\hat{u}|\\Big\\|_{\\ell^{1}}\\\\\n&\\leq  \\sum_{n=1}^{\\infty}\\sum_{k=0}^{n} {n \\choose k}\\frac{1}{n!} 2^{k}|x|^{k}\\|u\\|_{\\tilde{\\mathcal{F}}^{0,1}}^{n-k}.\n\\end{align*}\nThis finally yields the result \n\\begin{equation}\\label{Tbound}\n\\|T(\\psi)\\|\\leq 2e^{2|x|+\\|u\\|_{\\tilde{\\mathcal{F}}^{0,1}}}.\n\\end{equation}\nFrom \\eqref{aux1}, using \\eqref{detbound} and \\eqref{Tbound}, we obtain \n\\begin{equation}\\label{detandTbound}\n\\|D_{x}\\tilde{g}(u,x)^{-1}\\| \\leq \\frac{ 2e^{2|x|+\\|u\\|_{\\tilde{\\mathcal{F}}^{0,1}}}}{1+ 4(1- e^{4|x|+2\\|u\\|_{\\tilde{\\mathcal{F}}^{0,1}}})}.\n\\end{equation}\nIn conclusion, we have from \\eqref{xinitialbound}, \\eqref{Dugbound} and \\eqref{detandTbound} that\n\\begin{equation}\\label{xbound}\n2|x| \\leq  \\frac{ 2e^{2|x|+\\|u\\|_{\\tilde{\\mathcal{F}}^{0,1}}}(e^{2|x| + \\|u\\|_{\\tilde{\\mathcal{F}}^{0,1}}} - 1)}{1- 4( e^{4|x|+2\\|u\\|_{\\tilde{\\mathcal{F}}^{0,1}}}-1)} \\|u\\|_{\\tilde{\\mathcal{F}}^{0,1}}.\n\\end{equation}\nFrom the denominator term, we need $$ e^{4|x|+2\\|u\\|_{\\tilde{\\mathcal{F}}^{0,1}}} - 1 < \\frac{1}{4}.$$\nThis yields the condition\n\\begin{equation}\\label{xucriteria}\n2|x| + \\|u\\|_{\\tilde{\\mathcal{F}}^{0,1}} < \\frac{1}{2}\\log\\big(5\/4\\big),\n\\end{equation}\nwhich ensures that the condition \n$2|x| + \\|u\\|_{\\tilde{\\mathcal{F}}^{0,1}} < \\log(3\/2)$ is also satisfied.   Using $|x| = |\\hat{\\theta}(\\pm 1)|$ and $u = \\tilde{\\theta}$, we obtain \n$$2|x| + \\|u\\|_{\\tilde{\\mathcal{F}}^{0,1}} = |\\hat{\\theta}(1)| +  |\\hat{\\theta}(-1)| + \\|\\tilde{\\theta}\\|_{\\tilde{\\mathcal{F}}^{0,1}} =  \\|\\theta\\|_{\\mathcal{F}^{0,1}}.$$\nThus, for $\\|\\theta\\|_{\\mathcal{F}^{0,1}} <\\frac{1}{2}\\log\\big(\\frac{5}{4}\\big)$ and using \\eqref{xbound}, we obtain Proposition \\ref{IFTprop}.\n\\end{proof}\n\n\nThe result in Proposition \\ref{IFTprop} will be needed to close the \\textit{a priori} estimates in Subsection \\ref{subsecGlobal} . In particular, Proposition \\ref{linearfourier} show that the modes $\\pm 1$ are neutral at the linear level (see also \\eqref{auxx2}). \n\n\n\n\\section{Fourier multiplier estimates}\\label{sec:FourierRandS}\n\n\n\n\nIn this section, we will prove the crucial estimates for the operators $\\mathcal{R}$ from \\eqref{R}  (in Proposition \\ref{Restimates.prop}) and $\\mathcal{S}$ from \\eqref{S} (in Proposition \\ref{Smultiplierprop}) in the norms ${\\mathcal{F}^{0,1}_\\nu}$ and  ${\\dot{\\mathcal{F}}^{s,1}_\\nu}$ for $s> 0$ from \\eqref{fzeroonenunorm} and \\eqref{fsonenorm} respectively.  \n\nWe will use the following facts throughout the section:\n\n\\begin{lemma}\\label{triangleprop}\nWe have the estimate\n\t\\begin{equation}\\label{zeroproduct}\n\t\t\\|g_{1}g_{2}\\cdots g_{n}\\|_{\\mathcal{F}^{0,1}_\\nu} \\leq \\prod_{k=1}^{n} \\|g_{k}\\|_{\\mathcal{F}^{0,1}_\\nu}.\n\t\\end{equation}\n\tFor $s>0$, recalling \\eqref{bfcn.def}, we have\n\t\\begin{equation}\\label{sproduct}\n\t\t\\|g_{1}g_{2}\\cdots g_{n}\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu} \\leq b(n,s)\\sum_{j=1}^{n} \\|g_{j}\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\n\t\t\\prod_{\\substack{k=1 \\\\ k\\ne j}}^{n}  \\|g_{k}\\|_{\\mathcal{F}^{0,1}_\\nu}.\n\t\\end{equation}\n\\end{lemma}\n\n\n\\begin{remark}\\label{nuremark}\nWe note that these results and all the apriori estimates in this paper further hold with ${\\mathcal{F}^{0,1}_\\nu}$ and ${\\dot{\\mathcal{F}}^{s,1}_\\nu}$ replaced by ${\\mathcal{F}^{0,1}}$ and ${\\dot{\\mathcal{F}}^{s,1}}$ respectively since we can simply take $\\nu=0$.\n\\end{remark}\n\n\n\n\n\\begin{proof}\nFirst consider the case $0< s \\leq 1$ and $n=2$ in \\eqref{sproduct}. Then we use the inequalities\n$$|k|^{s} \\leq |k-j|^{s} + |j|^{s}, $$\nand\n$$e^{\\nu(t)|k|} \\leq e^{\\nu(t)|k-j|}e^{\\nu(t)|j|}, $$\nto see that\n\\begin{align*}\n\t\\|g_1 g_2\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu} &= \\sum_{k\\in\\mathbb{Z}} e^{\\nu(t)|k|} e^{\\nu(t)|k|} |k|^{s} \n\t\\left| \\widehat{g_1 g_2}(k) \\right|\n\t\\\\\n \t&\\leq \\sum_{j,k\\in\\mathbb{Z}} e^{\\nu(t)|k-j|}e^{\\nu(t)|j|}(|k-j|^{s} + |j|^{s})  \\left| \\hat{g_1}(k-j)\\hat{g_2}(j) \\right| \\\\\n\t&\\leq \\|g_1\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\\|g_2\\|_{\\mathcal{F}^{0,1}_\\nu} + \\|g_1\\|_{\\mathcal{F}^{0,1}_\\nu}\\|g_2\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}.\n\\end{align*}\nFor general $n$ and $s>0$, recalling \\eqref{bfcn.def}, we use the following inequality\n\\begin{equation}\\label{s.inequality}\n    |k|^{s} \\leq b(n,s) (|k-k_{1}|^{s} + |k_{1}-k_{2}|^{s} + \\ldots + |k_{n-2} - k_{n-1}|^{s} + |k_{n-1}|^{s}).\n\\end{equation}\nThen the proofs of all the other cases in \\eqref{zeroproduct} and \\eqref{sproduct} follow similarly.  \n\\end{proof}\n\n\n\nWe will also use the following repeatedly in this section:\n\\begin{prop}\nWe have the useful bound for $\\beta \\in \\mathbb{T}=[-\\pi, \\pi]$ and $l \\ge 1$:\n\\begin{equation}\\label{usefuloperatorbound}\n\\Big|\\Big( \\frac{i\\beta}{(1-e^{-i\\beta})}\\Big)^{l}-1 \\Big| \\leq |\\beta|\\frac{l}{2}\\Big(\\frac{\\pi}{2}\\big)^{l-1}\\sqrt{1 + \\frac{\\pi^{2}}{4}}.\n\\end{equation}\n\\end{prop}\n\n\\begin{proof}\nBy the mean value theorem we have\n\\begin{equation*}\n\t\t\\Big|\\Big( \\frac{i\\beta}{(1-e^{-i\\beta})}\\Big)^{l}-1 \\Big| \\leq |\\beta| \\Big\\|\\frac{d}{d\\beta}\\Big( \\frac{i\\beta}{(1-e^{-i\\beta})}\\Big)^{l}  \\Big\\|_{L^{\\infty}({\\mathbb{T}})}.\n\t\\end{equation*}\n\tWe compute the above derivative as:\n\t\\begin{equation*}\n\t\t\\frac{d}{d\\beta}\\Big( \\frac{i\\beta}{1-e^{-i\\beta}}\\Big)^{l}  = l\\Big(\\frac{i\\beta}{1-e^{-i\\beta}}\\Big)^{l-1}\\frac{d}{d\\beta}\\Big(\\frac{i\\beta}{1-e^{-i\\beta}}\\Big).\n\t\\end{equation*}\n\tFirst, for $\\beta\\in\\mathbb{T}$, we have \n\t\\begin{align*}\n\t\t\\Big|\\frac{\\beta}{(1-e^{-i\\beta})}\\Big|&= \\frac{|\\beta|}{\\sqrt{(1-\\cos(\\beta))^{2}+\\sin^{2}(\\beta)}}\n\t\t= \\frac{|\\beta|}{\\sqrt{2-2\\cos(\\beta)}}\\\\\n\t\t&= \\frac{|\\beta|}{2\\sin(\\beta\/2)}\n\t\t\\leq \\frac{\\pi}{2}.\n\t\\end{align*}\n\t\tNext, we have\n\t\\begin{align*}\n\t\t\\Big|\\frac{d}{d\\beta}\\Big(\\frac{\\beta}{1-e^{-i\\beta}}\\Big)\\Big| & = \\frac{|1-e^{-i\\beta}-i\\beta e^{-i\\beta}|}{|(1-e^{-i\\beta})^{2}|}\\\\\n\t\t&=\\frac{|(1-\\cos(\\beta)-\\beta\\sin(\\beta)) + i (\\sin(\\beta)-\\beta\\cos(\\beta))|}{4\\sin^{2}(\\beta\/2)}.\n\t\\end{align*}\n\tIn absolute value, the real term in the numerator is\n\t$$|1-\\cos(\\beta)-\\beta\\sin(\\beta)| = |2\\sin^{2}(\\beta\/2) -2\\beta\\sin(\\beta\/2)\\cos(\\beta\/2)|.$$\n\tHence for $\\beta\\in\\mathbb{T}$ we have  \n\t$$\\frac{|1-\\cos(\\beta)-\\beta\\sin(\\beta)| }{4\\sin^{2}(\\beta\/2)} \\leq \\Big|\\frac{1}{2} - \\frac{\\beta}{2\\tan(\\beta\/2)}\\Big|\\leq \\frac{1}{2}.$$\n\tNext, the imaginary part can be checked to have a maximum of \n\t$$ \\frac{|\\sin(\\beta)-\\beta\\cos(\\beta)|}{4\\sin^{2}(\\beta\/2)} \\leq \\frac{\\pi}{4}.$$\n\tThus we have \n\t\\begin{equation}\\notag\n\t\\Big|\\frac{d}{d\\beta}\\Big(\\frac{\\beta}{1-e^{-i\\beta}}\\Big)\\Big|\\leq \\frac{1}{2}\\sqrt{1 + \\frac{\\pi^{2}}{4}}.\n\t\\end{equation}\n\tWe conclude that\n\t$$\\Big|\\frac{d}{d\\beta}\\Big[\\Big( \\frac{i\\beta}{1-e^{-i\\beta}}\\Big)^{l}-1\\Big] \\Big| \\leq \\frac{1}{2}l\\Big(\\frac{\\pi}{2}\\Big)^{l-1}\\sqrt{1 + \\frac{\\pi^{2}}{4}}.$$\n\tThis yields \\eqref{usefuloperatorbound}.\n\\end{proof}\n\nWe will now estimate the operator $\\mathcal{R}$ from \\eqref{R} as follows. \n\n\\begin{prop}\\label{Restimates.prop}\n\tThe operator $\\mathcal{R}$ from \\eqref{R} satisfies the estimates\n\t\\begin{equation}\\label{Restimates}\n\\begin{aligned}\n\t\t\\|\\mathcal{R}(f)\\|_{\\mathcal{F}^{0,1}_\\nu} &\\leq {C_{\\mR}}\\|f\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu},\n\t\n\t\t\\\\\n\t\t\\|\\mathcal{R}(f)\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu} &\\leq {b(2,s)} {C_{\\mR}}(\\|f\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu} + \\|f\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}),\\qquad s>0.\n\\end{aligned}\n\\end{equation}\nwhere $b(2,s)$ is from \\eqref{bfcn.def} and the constant\n\\begin{equation}\\label{CR}\n{C_{\\mR}}\t\\overset{\\mbox{\\tiny{def}}}{=} 1+\\tilde{\\mathcal{C}}>0,    \n\\end{equation}\nand $\\tilde{\\mathcal{C}}>0$ is defined in \\eqref{catalanconstant}.\n\\end{prop}\n\nIn the rest of this section we will adopt the convention that \n\t\\begin{equation}\\label{convention}\n\t\\frac{1-e^{-i\\beta(1+k_1)}}{1+k_1}  = i\\beta \\quad \\text{for} ~k_1=-1. \n\t\\end{equation}\nThis convention will allow us to write many formula's succinctly.\n\n\\begin{proof}\nTaking the Fourier transform of the operator $\\mathcal{R}$ from \\eqref{R} and using the convention \\eqref{convention} we obtain \n\t\\begin{equation*}\n\t   \\begin{aligned}\n\t   \t\\widehat{\\mathcal{R}(f)}(k) &=\\!\\frac{i}{\\pi}\\text{pv}\\!\\!\\int_{-\\pi}^{\\pi}\\!\\frac{\\hat{f}(k)e^{-ik\\beta}\\beta}{(1\\!-\\!e^{-i\\beta})^2}\\ast \\!\\!\\int_0^1\\!\\!e^{i(s-1)\\beta(1+k)}\\hat{\\theta}(k) dsd\\beta\\\\\n\t\t&=\\!\\frac{i}{\\pi}\\sum_{k_1\\in\\mathbb{Z}}\\text{pv}\\!\\!\\int_{-\\pi}^{\\pi}\\!\\frac{\\hat{f}(k-k_1)e^{-i(k-k_{1})\\beta}\\beta}{(1\\!-\\!e^{-i\\beta})^2} \\!\\!\\int_0^1\\!\\!e^{i(s-1)\\beta(1+k_1)}\\hat{\\theta}(k_1) dsd\\beta\\\\\n\t\t&= \\!\\frac{1}{\\pi}\\sum_{k_1\\in\\mathbb{Z}}\\text{pv}\\!\\!\\int_{-\\pi}^{\\pi}\\!\\frac{\\hat{f}(k-k_1)e^{-i(k-k_{1})\\beta}}{(1\\!-\\!e^{-i\\beta})^2} \\frac{1-e^{-i\\beta(1+k_1)}\\hat{\\theta}(k_1)}{1+k_1} d\\beta.\n\t    \\end{aligned}\n\t\\end{equation*}\n\t\tUsing the convention \\eqref{convention}, we have\n\t\t\\begin{equation*}\n\t\\widehat{\\mathcal{R}(f)}(k)=  \\sum_{k_1\\in\\mathbb{Z}} \\hat{f}(k-k_1)\\hat{\\theta}(k_1)I(k,k_1),\n\t\\end{equation*}\n   where\n\t\\begin{equation}\\label{I.function.def}\n\t I(k,k_1)\\overset{\\mbox{\\tiny{def}}}{=} \\frac{1}{\\pi}\\text{pv}\\!\\int_{-\\pi}^{\\pi}\\!\\frac{e^{-i(k-k_{1})\\beta}}{(1\\!-\\!e^{-i\\beta})^2} \\frac{1-e^{-i\\beta(1+k_1)}}{1+k_1} d\\beta.\n\t\\end{equation}\n\tFor $k_{1} = -1$, using \\eqref{convention}, we split \\eqref{I.function.def} as \n\t\\begin{equation*}\n\t\tI(k,-1) = \\frac{1}{\\pi} \\text{pv}\\int_{-\\pi}^{\\pi}\\frac{e^{-i(k+1)\\beta}}{1-e^{-i\\beta}} d\\beta \n\t\t+ \\frac{1}{\\pi}\\text{pv}\\int_{-\\pi}^{\\pi}\\frac{e^{-i(k+1)\\beta}}{1-e^{-i\\beta}} \\Big(\\frac{i\\beta}{1-e^{-i\\beta}} - 1 \\Big).\n\t\\end{equation*}\nWe will first calculate each of these two integrals separately below.\t\n\t\n\n\t\nWe can calculate the general integral formula:\n\\begin{equation}\\label{j1}\n \\frac{1}{\\pi} \\text{pv} \\int_{-\\pi}^{\\pi} \\frac{e^{-i \\ell \\beta}}{1-e^{-i\\beta}} d\\beta\n =\n 1_{\\ell \\le 0}(\\ell) -  1_{\\ell \\ge 1}(\\ell).\n\\end{equation}\nThis will be used several times below.  It is proven using  \\eqref{hilbert} and noticing that $\\mathcal{F}(e^{-i \\ell \\beta})(0)=\\delta(\\ell)$.  Further from \\eqref{defHilbertTransform} we have\n$$\n-i\\mathcal{H} (e^{-i \\ell \\beta})(0) = \n-\\frac{1}{2\\pi}\\text{pv}\\int_{-\\pi}^{\\pi}\\frac{\\sin(\\ell\\beta)}{\\tan{(\\beta\/2)}} d\\beta\n= 1_{\\ell \\le -1}(\\ell) -  1_{\\ell \\ge 1}(\\ell).\n$$  \nCombining these calculations gives \\eqref{j1}.\n\t\n\tFrom \\eqref{j1}, the first term in $I(k,-1)$ is $\\pm 1$\n\tdepending on the sign of $k+1$. For the second integral, we use \\eqref{usefuloperatorbound} for $l=1$ to obtain\n\t\\begin{align*}\n\t\\Big|\\frac{1}{\\pi}\\text{pv}\\int_{-\\pi}^{\\pi}\\frac{e^{-i(k+1)\\beta}}{1-e^{-i\\beta}} \\Big(\\frac{i\\beta}{1-e^{-i\\beta}} - 1 \\Big)\\Big| &\\leq \\frac{1}{\\pi}\\sqrt{\\frac{1}{4} + \\frac{\\pi^{2}}{16}} \\int_{-\\pi}^{\\pi} \\frac{|\\beta|}{2|\\sin(\\beta\/2)|} d\\beta\\\\\n\t&\\leq \\tilde{\\mathcal{C}},\n\t\\end{align*}\n\twhere \n\t\\begin{equation}\\label{catalanconstant}\n\t\t\\tilde{\\mathcal{C}} \\overset{\\mbox{\\tiny{def}}}{=} \\frac{4}{\\pi}\\mathcal{V}\\sqrt{1 + \\frac{\\pi^{2}}{4}}.\n\t\\end{equation}\nIn this calculation we used that $|1-e^{-i\\beta}| = 2|\\sin(\\beta\/2)|$.  Here $\\mathcal{V} \\approx 0.916$ is Catalan's constant:\n\t\\begin{equation}\\notag\n\t    \\mathcal{V} = \\frac{1}{4}\\int_{-\\pi\/2}^{\\pi\/2} \\frac{\\beta}{\\sin\\beta} d\\beta\n\t    =\\frac{1}{16}\\int_{-\\pi}^{\\pi} \\frac{\\beta}{\\sin(\\beta\/2)} d\\beta.\n\t\\end{equation}\n\tHence, for $k_{1}= -1$, using \\eqref{CR}, we have the bound\n\t\\begin{equation}\\label{k1negativeone}\n\t|I(k,-1)| \\leq 1+ \\tilde{\\mathcal{C}} = {C_{\\mR}}.\n\t\\end{equation}\nThis will be our main estimate for the case for $k_{1}= -1$.\t\n\t\n\n\t\nNow generally in \\eqref{I.function.def} for $k_1>-1$ we have\n\t\\begin{equation}\\label{big.one.def}\n    \t\\frac{1-e^{-i\\beta (1+k_{1})}}{1-e^{-i\\beta}} = \\sum_{r=0}^{k_{1}} e^{-ir\\beta}\n\\end{equation}\n\tand\n    if $k_1 \\leq -2$ then we have\n    \\begin{equation}\\label{small.one.def}\n\\frac{1-e^{-i\\beta (1+k_{1})}}{1-e^{-i\\beta}} = -\\sum_{r=1}^{-1-k_{1}} e^{i\\beta r}.\n\\end{equation}\nThus for $k_1>-1$, using \\eqref{I.function.def},  \\eqref{j1} and \\eqref{big.one.def} we have\n\\begin{multline*}\n\tI(k,k_1) \n\t= \\frac{1}{1+k_1}\\sum_{r=0}^{k_1} \\frac{1}{\\pi}\\text{pv}\\!\\int_{-\\pi}^{\\pi}\\!\\frac{e^{-i(k-k_{1}+r)\\beta}}{1\\!-\\!e^{-i\\beta}}  d\\beta\n\t\\\\\n\t=\\frac{1}{1+k_1}\\sum_{r=0}^{k_1} \\left( 1_{k-k_{1}+r \\le 0} -  1_{k-k_{1}+r \\ge 1} \\right),\n\t\\end{multline*}\n\twhile for $k_1 \\leq -2$ we similarly, using \\eqref{small.one.def}, have\n\t\\begin{multline*}\n\tI(k,k_1) = - \\frac{1}{1+k_1}\\sum_{r=1}^{-1-k_1} \\frac{1}{\\pi}\\text{pv}\\!\\int_{-\\pi}^{\\pi}\\!\\frac{e^{-i(k-k_{1}-r)\\beta}}{1\\!-\\!e^{-i\\beta}}  d\\beta\n\t\\\\\n\t=\\frac{-1}{1+k_1}\\sum_{r=1}^{-1-k_1} \\left( 1_{k-k_{1}-r \\le 0} -  1_{k-k_{1}-r \\ge 1} \\right),\n\t\\end{multline*}\n\tIn both cases, $k_1>-1$ and $k_1 \\leq -2$, we conclude that \n\t    \\begin{equation}\\label{I.est.rest}\n    |I(k,k_1)| \\leq 1.\n    \\end{equation}\nHence we conclude that\n\t\\begin{equation*}\n\t|\\widehat{\\mathcal{R}(f)}(k)| \\leq  \t{C_{\\mR}}|(\\hat{f}\\ast \\hat{\\theta})(k)|.   \n\t\\end{equation*}\n\tApplying ${\\mathcal{F}^{0,1}_\\nu}$ and ${\\dot{\\mathcal{F}}^{s,1}_\\nu}$ norms to both sides, using Lemma \\ref{triangleprop}, gives the result.\n\\end{proof}\n\n\n\n\t\t\nNext, we proceed with the estimates for $\\mathcal{S}$ from \\eqref{S}.\n\\begin{prop}\\label{Smultiplierprop}\n\tThe operator $\\mathcal{S}$ in \\eqref{S} satisfies the estimates\n\t\\begin{equation}\\label{Sestimates}\n    \\begin{aligned}\n    \t\\|\\mathcal{S}(f)\\|_{\\mathcal{F}^{0,1}_\\nu} &\\leq C_1 \\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^2\\|f\\|_{\\mathcal{F}^{0,1}_\\nu},\\\\\n\t\t\t\\|\\mathcal{S}(f)\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}& \\leq C_3\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^{2}\\|f\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}+C_4\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|f\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu},     \\quad s>0,\n    \\end{aligned}\n\t\\end{equation}\n\twhere the positive constant $C_1$ is given by \\eqref{C1}.  Further the positive constants $C_{3}$ and $C_{4}$ for $s>0$ are given in \\eqref{C3}.\n\\end{prop}\n\n\\begin{proof}\nFrom \\eqref{S}, we split  the operator $\\mathcal{S}$ as: \n\t\\begin{equation}\\label{Ssum}\n\t\\mathcal{S}(f)(\\alpha) = \n\t{\\mathcal{A}}(f)(\\alpha) + {\\mathcal{B}}(f)(\\alpha),\n\t\\end{equation}\n\twhere\n\t\\begin{multline}\\label{breveS}\n\t{\\mathcal{A}}(f)(\\alpha) \\overset{\\mbox{\\tiny{def}}}{=} \\\\ \\frac{1}{\\pi}\\text{pv}\\int_{-\\pi}^{\\pi}\\frac{f(\\alpha-\\beta)\\beta^{2}}{\\beta(1-e^{-i\\beta})^{2}}\n\t\\sum_{m\\geq 2}\\frac{i^m}{m!}\\int_0^1 e^{i(s-1)\\beta}(\\theta(\\alpha+(s-1)\\beta))^m ds d\\beta,\n\t\\end{multline}\n\tand\n\t\\begin{equation}\\label{tildeS}\n\t\t{\\mathcal{B}}(f)(\\alpha)\\overset{\\mbox{\\tiny{def}}}{=} \\sum_{\\substack{n,l\\geq0 \\\\ n+l\\geq 2}}\\frac{(-1)^n i^{l+n+1}(\\theta(\\alpha))^l}{l!} \\mathcal{S}_{n}(f)(\\alpha).\n\t\\end{equation}\nHere for $n\\ge0$ we define\n\t\\begin{equation}\\label{Snl}\n\t\t\\mathcal{S}_{n}(f)(\\alpha) \\overset{\\mbox{\\tiny{def}}}{=} \\frac{1}{\\pi} \\text{pv}\\int_{-\\pi}^{\\pi}\\frac{f(\\alpha-\\beta)\\beta^{n+1}}{\\beta(1-e^{-i\\beta})^{n+1}}\n\t\tM(\\alpha,\\beta)^{n} d\\beta,\n\t\\end{equation}\n\twith\n\t\\begin{equation}\\label{M}\n\t\tM(\\alpha,\\beta) \\overset{\\mbox{\\tiny{def}}}{=} \\sum_{m\\geq 1}\\frac{i^m}{m!}\\int_0^1 e^{i(s-1)\\beta}(\\theta(\\alpha+(s-1)\\beta))^m ds.\n\t\\end{equation}\nWe take the Fourier transform to obtain\n\t\\begin{equation}\\label{fourierS}\n\t\t\\widehat{{\\mathcal{B}}}(f)(k) = \\sum_{\\substack{n,l\\geq0 \\\\ n+l\\geq 2}}\\frac{(-1)^ni^{l+n+1}(\\ast^{l}\\hat{\\theta}(k))}{l!} \\ast \\widehat{\\mathcal{S}_{n}(f)}(k),\n\t\\end{equation}\n\twhere \n\t\\begin{equation}\\label{fourierSnl}\n\t\t\\widehat{\\mathcal{S}_{n}(f)}(k) \\overset{\\mbox{\\tiny{def}}}{=} \\frac{1}{\\pi} \\text{pv}\\int_{-\\pi}^{\\pi}\\frac{\\hat{f}(k)e^{-ik\\beta}\\beta^{n+1}}{\\beta(1-e^{-i\\beta})^{n+1}}\n\t\t\\ast^{n}\\widehat{M}(k,\\beta) d\\beta.\n\t\\end{equation}\nFor $k_{1} \\neq -1$, from \\eqref{M} we have\n\t\\begin{align*}\n\t\t\\widehat{M}(k_{1},\\beta) &= \\sum_{m\\geq 1}\\frac{i^m}{m!}\\int_0^1 e^{i(s-1)\\beta}\\Big(\\ast^{m}(\\hat{\\theta}(k_{1}) e^{ik_{1}(s-1)\\beta})\\Big) ds\\\\\n\t\t&= \\sum_{m\\geq 1}\\sum_{k_{2},\\ldots,k_{m}\\in\\mathbb{Z}}\\frac{i^m}{m!} \\int_0^1 e^{i(s-1)\\beta}\\left(\\prod_{j=1}^{m-1}\\hat{\\theta}(k_{j}-k_{j+1}) e^{i(k_{j}-k_{j+1})(s-1)\\beta}\\right)\\\\\n\t\t&\\hspace{3in} \\cdot\\hat{\\theta}(k_{m}) e^{ik_{m}(s-1)\\beta}  ds\\\\\n\t\t&= \\sum_{m\\geq 1}\\sum_{k_{2},\\ldots,k_{m}\\in\\mathbb{Z}}\\frac{i^m}{m!}\\int_0^1 e^{i(s-1)\\beta(1+k_{1})}\\left(\\prod_{j=1}^{m-1}\\hat{\\theta}(k_{j}-k_{j+1}) \\right) \\hat{\\theta}(k_{m}) ds\\\\\n\t\t&= \\sum_{m\\geq 1}\\sum_{k_{2},\\ldots,k_{m}\\in\\mathbb{Z}}\\frac{i^m}{m!} \\left(\\prod_{j=1}^{m-1}\\hat{\\theta}(k_{j}-k_{j+1}) \\right) \\hat{\\theta}(k_{m}) \\frac{1-e^{-i\\beta (1+k_{1})}}{i\\beta (1+k_{1})}\\\\\n\t\t&=  \\sum_{m\\geq 1}\\frac{i^m}{m!} \\left(\\ast^{m}\\widehat{\\theta}(k_{1})\\right) \\frac{1-e^{-i\\beta (1+k_{1})}}{i\\beta (1+k_{1})}.\n\t\\end{align*}\nAnd when $k_{1} = -1$,  analogously we have\n\t$$\\widehat{M}(k_{1},\\beta) =  \\sum_{m\\geq 1}\\frac{i^m}{m!} \\left(\\ast^{m}\\widehat{\\theta}(k_{1})\\right).$$\nThus we define\n\t\\begin{equation}\\label{P}\n\t\tP(k) \\overset{\\mbox{\\tiny{def}}}{=} \\sum_{m\\geq 1}\\frac{i^m}{m!} (\\ast^{m}\\widehat{\\theta}(k)).\n\t\\end{equation}\nNow we use the convention \\eqref{convention} and plug the above into \\eqref{fourierSnl} to obtain\n\t\\begin{align}\n\t\t\\widehat{\\mathcal{S}_{n}}(k_{1}) &= \\frac{1}{\\pi} \\text{pv}\\int_{-\\pi}^{\\pi}\\frac{\\hat{f}(k_{1})e^{-ik_{1}\\beta}\\beta^{n+1}}{\\beta(1-e^{-i\\beta})^{n+1}}\\ast\n\t\t\\left(\\ast^{n}P(k_{1})\\frac{1-e^{-i\\beta (1+k_{1})}}{i\\beta (1+k_{1})}\\right) d\\beta \\nonumber\\\\\n\t\t&= \\frac{1}{i^{n}}\\sum_{k_{2},\\ldots, k_{n+1}\\in \\mathbb{Z}}I(k_{1},\\ldots,k_{n+1})\\hat{f}(k_{n+1})\\prod_{j=1}^{n}P(k_{j}-k_{j+1}) \\hspace{0.05cm}, \\label{Snfourier}\n\t\\end{align}\n\twhere $I = I(k_{1},\\ldots,k_{n+1})$ is defined by \n\t\\begin{equation}\\label{I}\n\t\tI \\overset{\\mbox{\\tiny{def}}}{=} \\frac{1}{\\pi}\\text{pv} \\int_{-\\pi}^{\\pi} \\frac{e^{-i k_{n+1}\\beta}}{1-e^{-i\\beta}}\\prod_{j=1}^{n}\n\t\t\\frac{1-e^{-i\\beta (1+k_{j}-k_{j+1})}}{(1+k_{j}-k_{j+1})(1-e^{-i\\beta})}d\\beta.\n\t\\end{equation}\n\tWe suppose that $l$ elements of $\\{k_{j}-k_{j+1}\\}_{j=1}^n$ satisfy $k_{j} - k_{j+1} =  -1$ for $0\\le l \\le n$.  Then ordering the subscripts such that $k_{j} - k_{j+1} \\neq  -1$ for $j= 1,\\ldots, n-l$, we obtain with the convention \\eqref{convention} that the integral $I = I(k_{1},\\ldots, k_{n+1})$ given by \\eqref{I} becomes \n\t\\begin{equation*}\n\t\tI = \\frac{1}{\\pi}\\text{pv} \\int_{-\\pi}^{\\pi} \\frac{e^{-i k_{n+1}\\beta}}{1-e^{-i\\beta}}\\frac{(i\\beta)^{l}}{(1-e^{-i\\beta})^l}\n\t\t\\prod_{j=1}^{n-l}\n\t\t\\frac{1-e^{-i\\beta (1+k_{j}-k_{j+1})}}{(1+k_{j}-k_{j+1})(1-e^{-i\\beta})}d\\beta.\n\t\\end{equation*}\n\tNow, if $k_{j}-k_{j+1} > -1$ then similar to \\eqref{big.one.def} we have\n\\begin{equation}\\notag \n    \t\\frac{1-e^{-i\\beta (1+k_{j}-k_{j+1})}}{1-e^{-i\\beta}} = \\sum_{r_{j}=0}^{k_{j}-k_{j+1}} e^{-ir_{j}\\beta}\n\\end{equation}\n\tand\n    if $k_{j}-k_{j+1} \\leq -2$ then similar to \\eqref{small.one.def} we have\n    \\begin{equation}\\notag \n\\frac{1-e^{-i\\beta (1+k_{j}-k_{j+1})}}{1-e^{-i\\beta}} = \\sum_{r_{j}=1}^{-1-(k_{j}-k_{j+1})} -e^{i\\beta r_j}.\n\\end{equation}\n    Hence, if $k_{j}-k_{j+1} \\leq -2$ only for $j=m,\\ldots,n-l$ then\n    \\begin{multline*}\n    \\prod_{j=1}^{n-l}\n\t\t\\frac{1-e^{-i\\beta (1+k_{j}-k_{j+1})}}{1-e^{-i\\beta}}\\\\ = \\sum_{r_{1}=0}^{k_{1}-k_{2}} e^{-ir_{1}\\beta} \\cdots \\sum_{r_{m-1}=0}^{k_{m-1}-k_{m}} e^{-ir_{m-1}\\beta} \\sum_{r_{m}=1}^{k_{m+1}-1-k_{m}} -e^{ir_{m}\\beta}\\\\ \\cdots \\sum_{r_{n-l}=1}^{k_{n-l+1}-1-k_{n-l}} -e^{ir_{n-l}\\beta}\\\\\n\t\t=\\sum_{r_{1}=0}^{k_{1}-k_{2}}\\cdots \\sum_{r_{m-1}=0}^{k_{m-1}-k_{m}}  \\sum_{r_{m}=1}^{k_{m+1}-1-k_{m}} \\cdots \\sum_{r_{n-l}=1}^{k_{n-l+1}-1-k_{n-l}}(-1)^{n-l-m+1} e^{-i\\tilde{A}\\beta}\n\t\t\\end{multline*}\n\t\twhere\n\t\t$$\\tilde{A} = r_{1} + \\ldots + r_{m-1} -r_{m} -\\ldots -r_{n-l}.$$\n\t\tHence, in this case\n\t\\begin{multline}\\label{IJ}\n\t    I =  \\prod_{j=1}^{n-l}\n\t\t\\frac{1}{1+k_{j}-k_{j+1}}\\sum_{r_{1}=0}^{k_{1}-k_{2}}\\cdots \\sum_{r_{m-1}=0}^{k_{m-1}-k_{m}}  \\sum_{r_{m}=1}^{k_{m+1}-1-k_{m}} \\\\ \\cdots \\sum_{r_{n-l}=1}^{k_{n-l+1}-1-k_{n-l}}(-1)^{n-l-m+1}\n\t\t\\frac{1}{\\pi}\\text{pv} \\int_{-\\pi}^{\\pi} \\frac{e^{-i A\\beta}(i\\beta)^{l}}{(1-e^{-i\\beta})^{l+1}} d\\beta\n\t\\end{multline}\n\twhere\n\t$$A = k_{n+1}+\\tilde{A}.$$\n\tFor the inner integral, which we define as $J$, we have\n\t\\begin{equation}\\label{j1j2}\n\t\tJ \\overset{\\mbox{\\tiny{def}}}{=} \\frac{1}{i^{l}\\pi}\\text{pv} \\int_{-\\pi}^{\\pi} \\frac{e^{-i A\\beta}(i\\beta)^{l}}{(1-e^{-i\\beta})^{l+1}} d\\beta = \\frac{1}{i^{l}}( J_{1} + J_{2}), \n\t\\end{equation}\n\twhere we calculate $J_1$ as in \\eqref{j1}\n\tand\n\t\\begin{equation}\\label{j2}\n\t\tJ_{2} \\overset{\\mbox{\\tiny{def}}}{=}  \\frac{1}{\\pi} \\text{pv} \\int_{-\\pi}^{\\pi} \\frac{e^{-i A\\beta}}{1-e^{-i\\beta}}\\Big[\\Big( \\frac{i\\beta}{(1-e^{-i\\beta})}\\Big)^{l}-1\\Big] d\\beta.\n\t\\end{equation}\n\tWe can bound $J_{2}$ as well. Using the bound \\eqref{usefuloperatorbound} in \\eqref{j2}, we have \n\t\\begin{align*}\n\t\t|J_{2}| &\\leq \\frac{1}{2\\pi}l\\Big(\\frac{\\pi}{2}\\Big)^{l-1}\\sqrt{1 + \\frac{\\pi^{2}}{4}} \\int_{-\\pi}^{\\pi} \\frac{|\\beta|}{2|\\sin(\\beta\/2)|} d\\beta\\\\\n\t\t&=\\tilde{\\mathcal{C}} l\\Big(\\frac{\\pi}{2}\\Big)^{l-1},\n\t\\end{align*}\n\twhere the calculation above is similar to the calculation above \\eqref{catalanconstant} and the constant $\\tilde{\\mathcal{C}}$ is given by \\eqref{catalanconstant}.  \n\n\t\n\t\nThus, from \\eqref{IJ} and \\eqref{j1j2}, we have\n\t\\begin{align*}\n\t\t|I|&\\leq\\prod_{j=1}^{n-l}\n\t\t\\frac{1}{|1+k_{j}-k_{j+1}|}\\sum_{r_{1}=0}^{k_{1}-k_{2}}\\cdots \\sum_{r_{m-1}=0}^{k_{m-1}-k_{m}}  \\sum_{r_{m}=1}^{k_{m+1}-1-k_{m}} \\\\ \n\t\t&\\hspace{2in}\\cdots \\sum_{r_{n-l}=1}^{k_{n-l+1}-1-k_{n-l}}  \\left(1 + \\tilde{\\mathcal{C}} l\\big(\\frac{\\pi}{2}\\big)^{l-1}\\right)\\\\\n\t\t&\\leq\\prod_{j=1}^{n-l}\n\t\t\\frac{1}{|1+k_{j}-k_{j+1}|}\\sum_{r_{1}=0}^{k_{1}-k_{2}}\\cdots \\sum_{r_{m-1}=0}^{k_{m-1}-k_{m}}  \\sum_{r_{m}=1}^{k_{m+1}-1-k_{m}} \n\t\t\\\\ \n\t\t&\\hspace{2in}\\cdots \\sum_{r_{n-l}=1}^{k_{n-l+1}-1-k_{n-l}} \\left( 1 + \\tilde{\\mathcal{C}} n\\big(\\frac{\\pi}{2}\\big)^{n-1}\\right).\n\t\\end{align*}\nThen we denote\n\\begin{equation}\\label{an.def}\n\t    a_n\\overset{\\mbox{\\tiny{def}}}{=} 1 + \\tilde{\\mathcal{C}} n\\big(\\frac{\\pi}{2}\\big)^{n-1}.\n\\end{equation}\nTherefore we have\n\t\\begin{equation}\\label{Ibound}\n\t\t|I(k_{1},\\ldots,k_{n+1})| \\leq a_n \\ \\ \\forall  (k_{1},\\ldots, k_{n+1}) \\in \\mathbb{Z}^{n+1}.\n\t\\end{equation}\t\n\tPlugging \\eqref{Ibound} into the estimate for \\eqref{Snfourier}, we obtain\n\t\\begin{align}\n\t\t|\\widehat{\\mathcal{S}_{n}}(k_{1})|&\\leq  \\sum_{k_{2},\\ldots, k_{n+1}\\in \\mathbb{Z}} |I(k_{1},\\ldots,k_{n+1})||\\hat{f}(k_{n+1})|\n\t\t\\prod_{j=1}^n |P(k_{j}-k_{j+1})| \\nonumber\\\\\n\t\t&\\leq  \\sum_{k_{2},\\ldots, k_{n+1}\\in \\mathbb{Z}}  a_n |\\hat{f}(k_{n+1})|\\prod_{j=1}^n |P(k_{j}-k_{j+1})|\\nonumber\\\\\n\t\t& = a_n(|\\hat{f}|\\ast^{n} |P|)(k_{1}). \\label{Snfbound}\n\t\\end{align}\t\n\tWe will use this estimate below to obtain the appropriate upper bound for $\\mathcal{S}(f)$ given by \\eqref{fourierS}. \n\t\n\n\n\t\nRecalling \\eqref{nu}, in the rest of this proof for convenience of notation we define the $\\ell^1_\\nu$ norm of a sequence $a=\\{a_k\\}_{k\\in \\mathbb{Z}}$ as\n\\begin{equation}\\notag\n    \\| a\\|_{\\ell^{1}_\\nu} \\overset{\\mbox{\\tiny{def}}}{=} \\sum_{k\\in\\mathbb{Z}} e^{\\nu |k|} |a_k|^p.\n\\end{equation}\nThen for $s=0$ using \\eqref{tildeS} and \\eqref{fourierS} we have\n\t\\begin{align}\n\t\t\\|{\\mathcal{B}}(f)\\|_{\\mathcal{F}^{0,1}_\\nu} &= \\Big\\|\\sum_{\\substack{n,l\\geq 0 \\\\ n+l\\geq 2}}\\frac{(-1)^ni^{l+n+1}(\\ast^{l}\\hat{\\theta}(k))}{l!} \\ast \\widehat{\\mathcal{S}_{n}(f)}(k) \\Big\\|_{\\ell^{1}_{\\nu}} \\nonumber \\\\\n\t\t&\\leq \\sum_{\\substack{n\\geq 0,l\\geq 0 \\\\n+l \\geq 2}}\\frac{1}{l!}\\|\\hat{\\theta}(k)\\|_{\\ell^{1}_{\\nu}}^{l}\\|\\widehat{\\mathcal{S}_{n}(f)}(k)\\|_{\\ell^{1}_{\\nu}} \\label{SSn0}.\n\t\\end{align}\n\tLet us examine the bound on the $\\widehat{\\mathcal{S}_{n}(f)}$ term in \\eqref{SSn0} using \\eqref{Snfbound}. For the quantity $P$ given by \\eqref{P}, \twe have\n\t\\begin{equation}\\label{P0}\n\t\t\\|P\\|_{\\ell^{1}_{\\nu}} \\leq \\sum_{m\\geq 1}\\frac{1}{m!} \\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^{m} = \\exp(\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu})-1.\n\t\\end{equation}\n\tFrom \\eqref{Snfbound} and \\eqref{an.def}, we then have\n\t\\begin{equation}\\label{Snfzero}\n\t\t\\|\\mathcal{S}_{n}(f)\\|_{\\mathcal{F}^{0,1}_\\nu} \\leq   a_n\\|f\\|_{\\mathcal{F}^{0,1}_\\nu}(\\exp(\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu})-1)^{n}.\n\t\\end{equation}\n\tHence, from \\eqref{SSn0}, we have\n\t\\begin{equation}\\label{SSn00}\n\t\t\\|{\\mathcal{B}}(f)\\|_{\\mathcal{F}^{0,1}_\\nu} \\leq \n\t\t\\! \\! \\! \\! \\! \\sum_{\\substack{n\\geq 0,l\\geq 0 \\\\n+l \\geq 2}}\\frac{a_n}{l!}\\|\\hat{\\theta}(k)\\|_{\\ell^{1}_{\\nu}}^{l}   \\|f\\|_{\\mathcal{F}^{0,1}_\\nu}(\\exp(\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu})-1)^{n}.\n\t\\end{equation}\nWe will separately look at the above sum when $n=0$, $n=1$ and $n\\geq 2$. \n\n\n\t\t\t\nFirst, when $n=0$ then $a_0=1$ and we have\n\t\\begin{equation}\\label{SSn00n0}\n\t\t\\sum_{l\\geq 2}\\frac{1}{l!}\\|\\hat{\\theta}(k)\\|_{\\ell^{1}_{\\nu}}^{l}   \\|f\\|_{\\mathcal{F}^{0,1}_\\nu} = \\frac{\\exp(\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}) - 1 - \\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}}{\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^{2}} \\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^{2}\\|f\\|_{\\mathcal{F}^{0,1}_\\nu}.\n\t\\end{equation}\n\tWhen $n=1$ then $a_1=1+\\tilde{\\mathcal{C}}$ and we have\n\t\\begin{multline}\\label{SSn0n1}\n\t\t\\sum_{l\\geq 1}\\frac{1}{l!}\\|\\hat{\\theta}(k)\\|_{\\ell^{1}_{\\nu}}^{l}\\Big(1 + \\tilde{\\mathcal{C}} \\Big)(\\exp(\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu})-1)   \\|f\\|_{\\mathcal{F}^{0,1}_\\nu} \\\\ =\\Big(1 + \\tilde{\\mathcal{C}}\\Big)\\frac{(\\exp(\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu})-1)^{2}}{\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^{2}}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^{2}\\|f\\|_{\\mathcal{F}^{0,1}_\\nu}.\n\t\\end{multline}\n\tFinally, for $n\\geq 2$ using also \\eqref{an.def} we have\n\t\\begin{multline}\\label{SSn0n2}\n\t\t\\sum_{n\\geq 2,l\\geq 0 }\\frac{a_n}{l!}\\|\\hat{\\theta}(k)\\|_{\\ell^{1}_{\\nu}}^{l}   \\|f\\|_{\\mathcal{F}^{0,1}_\\nu}(\\exp(\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu})-1)^{n} \\\\= \\sum_{n\\geq 2,l\\geq 0 }\\frac{1}{l!}\\|\\hat{\\theta}(k)\\|_{\\ell^{1}_{\\nu}}^{l} \\|f\\|_{\\mathcal{F}^{0,1}_\\nu}(\\exp(\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu})-1)^{n}\\\\ \n\t\t+ \\sum_{n\\geq 2,l\\geq 0 }\\frac{1}{l!}\n\t\t\\tilde{\\mathcal{C}} n\\big(\\frac{\\pi}{2}\\big)^{n-1}\n\t\t\\|\\hat{\\theta}(k)\\|_{\\ell^{1}_{\\nu}}^{l}   \n\t\t\\|f\\|_{\\mathcal{F}^{0,1}_\\nu}(\\exp(\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu})-1)^{n}.\n\t\\end{multline}\nNow considering the first sum on the right hand side of \\eqref{SSn0n2}, we have\n\t\\begin{multline}\\label{SSn0n21}\n\t\t\\sum_{n\\geq 2,l\\geq 0 }\\frac{1}{l!}\\|\\hat{\\theta}(k)\\|_{\\ell^{1}_{\\nu}}^{l} \\|f\\|_{\\mathcal{F}^{0,1}_\\nu}(\\exp(\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu})-1)^{n}\\\\ = \\frac{\\exp(\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu})(\\exp(\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu})-1)^{2}}{\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^{2}}\\sum_{n\\geq 2} \\|f\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^{2}(\\exp(\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu})-1)^{n-2}\\\\\n\t\t=\\frac{\\exp(\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu})(\\exp(\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu})-1)^{2}}{\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^{2}(2-\\exp(\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}))} \\|f\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^{2}.\n\t\\end{multline}\nThen the second sum on the right hand side of \\eqref{SSn0n2} is\n\\begin{multline*}\n\\sum_{n\\geq 2,l\\geq 0 }\\frac{1}{l!}\\tilde{\\mathcal{C}} n\\big(\\frac{\\pi}{2}\\big)^{n-1}\\|\\hat{\\theta}(k)\\|_{\\ell^{1}_{\\nu}}^{l}   \\|f\\|_{\\mathcal{F}^{0,1}_\\nu}(\\exp(\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu})-1)^{n} \n\\\\\n=\\frac{\\exp(\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu})(\\exp(\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu})-1)}{\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}}\\\\ \\cdot \\sum_{n\\geq 2}\\tilde{\\mathcal{C}} n\\big(\\frac{\\pi}{2}(\\exp(\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu})-1)\\big)^{n-1} \\|f\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}.\n\\end{multline*}\nNow using that $$\\sum_{ n\\geq 2} nx^{n-1} = -1 + \\sum_{n\\geq 1} nx^{n-1} = -1 + \\frac{1}{(1-x)^{2}} = \\frac{x(2-x)}{(1-x)^{2}}, $$\n\twe obtain\n\\begin{multline}\\label{SSn00n22}\n\t\t\\sum_{n\\geq 2,l\\geq 0 }\\frac{1}{l!}\\|\\hat{\\theta}(k)\\|_{\\ell^{1}_{\\nu}}^{l}   \\tilde{\\mathcal{C}} n\\big(\\frac{\\pi}{2}\\big)^{n-1}\\|f\\|_{\\mathcal{F}^{0,1}_\\nu}(\\exp(\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu})-1)^{n} \\\\\n=\\frac{\\pi\\tilde{\\mathcal{C}}}{2}\n\t\t\\frac{\\exp(\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu})(\\exp(\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu})-1)^{2}}{\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^{2}}\\|f\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^{2}\\\\\\cdot\\Big( \\frac{2-\\frac{\\pi}{2}(\\exp(\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu})-1)}{\\big(1-\\frac{\\pi}{2}(\\exp(\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu})-1)\\big)^{2}}\\Big).\n\\end{multline}\t\n\tCombining \\eqref{SSn00n0}, \\eqref{SSn0n1}, \\eqref{SSn0n2}, \\eqref{SSn0n21} and \\eqref{SSn00n22} into \\eqref{SSn00}, we obtain in the space $\\mathcal{F}^{0,1}_\\nu$ with $\\nu=0$ that\n\t\\begin{equation*}\n\t\t\\|{\\mathcal{B}}(f)\\|_{\\mathcal{F}^{0,1}_\\nu} \\leq \\tilde{C}_1\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^{2}\\|f\\|_{\\mathcal{F}^{0,1}_\\nu},\n\t\\end{equation*} \n\twhere $\\tilde{C}_1\\overset{\\mbox{\\tiny{def}}}{=} \\tilde{C}_1(\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu})$ is an increasing function of $\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}$ given by\n\t\\begin{multline}\\label{tildeC1}\n\t\t\\tilde{C}_1=\n\t\t\\\\\n\t\t\\frac{\\pi\\tilde{\\mathcal{C}}}{2}\\frac{\\exp(\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu})(\\exp(\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu})-1)^{2}}{\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^{2}} \\frac{2-\\frac{\\pi}{2}(\\exp(\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu})-1)}{\\big(1-\\frac{\\pi}{2}(\\exp(\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu})-1)\\big)^{2}} \\\\+ \\frac{\\exp(\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}) - 1 - \\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}}{\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^{2}} + \\Big(1 + \\tilde{\\mathcal{C}}\\Big)\\frac{(\\exp(\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu})-1)^{2}}{\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^{2}} \\\\+\\frac{\\exp(\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu})(\\exp(\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu})-1)^{2}}{\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^{2}(2-\\exp(\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}))}.\n\t\\end{multline}\n This completes the estimate \\eqref{Sestimates} for the operator \\eqref{tildeS} and for $s=0$.\t\n\t\n\n\n\n\t\t\t\t\nFor the operator \\eqref{tildeS} and $s>0$, from \\eqref{fourierS}, \\eqref{fourierSnl}, \\eqref{P}, \\eqref{Snfourier} with \\eqref{I} we have\n\t\\begin{multline}\\label{S1S2S3}\n\t\\|{\\mathcal{B}}(f)\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\n\t= \\Big\\||k_{1}|^{s}\\sum_{\\substack{n,l\\geq 0 \\\\ n+l\\geq 2}}\\frac{(-1)^n i^{l+n+1}(\\ast^{l}\\hat{\\theta}(k_{1}))}{l!} \\ast \\widehat{\\mathcal{S}_{n}(f)}(k_{1}) \\Big\\|_{\\ell^{1}_{\\nu}} \n\t\\\\\n\t= \\Big\\||k_{1}|^{s}\\sum_{\\substack{n,l\\geq 0 \\\\ n+l\\geq 2}}\\frac{(\\ast^{l}\\hat{\\theta}(k_{1}))}{l!} \\ast \\sum_{k_{2},\\ldots, k_{n+1}\\in \\mathbb{Z}}I(k_{1},\\ldots,k_{n+1})\\hat{f}(k_{n+1})\\prod_{j=1}^{n}P(k_{j}-k_{j+1})  \\Big\\|_{\\ell^{1}_{\\nu}}\n\t\\\\\n\t\\leq S_{1} + S_{2} + S_{3},\n\t\\end{multline}\n\twhere we use $a_n$ from \\eqref{an.def} and \\eqref{bfcn.def} so that,\n\tusing \\eqref{s.inequality} and \\eqref{Ibound} with \\eqref{P}, the terms $S_i$ are given by\n\t\\begin{equation*}\n\t    \\begin{aligned}\n\tS_{1}  &=  \\sum_{\\substack{n,l\\geq 0 \\\\ n+l\\geq 2}} \\frac{{b}(l+n+1,s)}{l!}a_n l\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^{l-1}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\\|f\\|_{\\mathcal{F}^{0,1}_\\nu}\\|P\\|_{\\ell^{1}_{\\nu}}^{n},\\\\\n\tS_{2} &= \\sum_{\\substack{n,l\\geq 0 \\\\ n+l\\geq 2}} \\frac{{b}(l+n+1,s)}{l!}a_n \\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^{l}\\|f\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\\|P\\|_{\\ell^{1}_{\\nu}}^{n},\\\\\n\tS_{3} &= \\sum_{\\substack{n,l\\geq 0 \\\\ n+l\\geq 2}} \\frac{{b}(l+n+1,s)}{l!}a_n n \\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^{l}\\|f\\|_{\\mathcal{F}^{0,1}_\\nu}\\|P\\|_{_{\\ell^{1}_{\\nu}}}^{n-1}\\|\\mathcal{F}^{-1}(P)\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu} .\n\t    \\end{aligned}\n\t\\end{equation*}\n\tWe recall from \\eqref{P} and \\eqref{P0} that\n\t$\\|P\\|_{\\ell^{1}_{\\nu}}\\le \\exp{(\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu})}-1$  and notice with \\eqref{s.inequality} that for $s>0$ we have\n\t\\begin{equation}\\label{Ps.estimate}\n\t\\|\\mathcal{F}^{-1}(P)\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu} \\leq \\sum_{m\\geq 1}\\frac{{b}(m,s)}{(m-1)!} \\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^{m-1}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}.\n\t\t\\end{equation}\n\tThen, we have that\n\t\\begin{equation*}\n\tS_{2}\\le \\tilde{C}_3\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^2\\|f\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu},\n\t\\end{equation*}\n\twhere\n\t\\begin{equation}\\label{tildeC3}\n\t\\begin{aligned}\n\t \\tilde{C}_3&\\overset{\\mbox{\\tiny{def}}}{=} \\sum_{\\substack{n\\geq 0,l\\geq 0 \\\\ n+l\\geq 2}} \\frac{{b}(l+n+1,s)}{l!} a_n\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^{n+l-2}\\Big(\\frac{e^{\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}}-1}{\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}}\\Big)^n,\n\t\\end{aligned}\n\t\\end{equation}\n\tand\n\\begin{equation*}\nS_{1}  +S_{3}\\le  \\tilde{C}_4\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\\|f\\|_{\\mathcal{F}^{0,1}_\\nu},\n\\end{equation*}\nwhere \n\\begin{equation}\\label{tildeC4}\n\\begin{aligned}\n\\tilde{C}_4&\\overset{\\mbox{\\tiny{def}}}{=} \\!\\!\\!\\sum_{\\substack{n\\geq 0,l\\geq 0 \\\\ n+l\\geq 2}}\\!\\!\\!\\! \\frac{{b}(l\\!+\\!n\\!+\\!1,s)}{l!} a_n\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^{n+l-2}\\Big(l\\Big(\\frac{e^{\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}}\\!-\\!1}{\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}}\\Big)^n+\\! \\tilde{C}_5\\Big),\n\\end{aligned}\n\\end{equation}\nwith \n\\begin{equation}\\label{C5}\n\\tilde{C}_5\\overset{\\mbox{\\tiny{def}}}{=} n\\Big(\\frac{e^{\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}}\\!-\\!1}{\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}}\\Big)^{n-1}\\sum_{m\\geq 1}\\frac{{b}(m,{s})}{(m-1)!} \\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^{m-1}.    \n\\end{equation}\nFinally, going back to \\eqref{S1S2S3}, we obtain the result for $s>0$ that\n\\begin{equation*}\n    \\|{\\mathcal{B}}(f)\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\\leq \\tilde{C}_3\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^2\\|f\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}+\\tilde{C}_4\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\\|f\\|_{\\mathcal{F}^{0,1}_\\nu}.\n\\end{equation*}\nThis completes the desired estimate for ${\\mathcal{B}}$ in \\eqref{Sestimates} for $s>0$.\n\n\n\n\n\t\n\t\t\t\t\nNow it only remains to bound ${\\mathcal{A}}(f)(\\alpha)$ as defined by \\eqref{breveS} in \\eqref{Sestimates}.  Analogously to \\eqref{Snfourier} and \\eqref{I}, one can obtain that\n\\begin{equation*}\n\\widehat{{\\mathcal{A}}}(k_{1}) = \\sum_{k_{2}\\in\\mathbb{Z}} I(k_{1},k_{1}-k_{2})\\hat{f}(k_{2})P_2(k_{1}-k_{2})\n\\end{equation*}\nwhere\n$$ P_2(k) = \\sum_{m\\geq 2} \\frac{i^{m}}{m!} (\\ast^{m}\\hat{\\theta}(k))$$\nand $I(k_{1},k_{1}-k_{2})$ is given by \\eqref{I.function.def} with $k=k_1$ and $k_1$ in \\eqref{I.function.def} replaced by $k_{1}-k_{2}$ using also \\eqref{convention}, so that in particular\n$$ \nI(k_{1},k_{1}-k_{2}) =  \\frac{1}{\\pi}\\text{pv} \\int_{-\\pi}^{\\pi} \\frac{e^{-i k_{2}\\beta}}{1-e^{-i\\beta}}\n\t\t\\frac{1-e^{-i\\beta (1+k_{1}-k_{2})}}{i(1+k_{1}-k_{2})(1-e^{-i\\beta})}d\\beta.\n$$\nThen analogously to \\eqref{k1negativeone} and \\eqref{I.est.rest} we have $$| I(k_{1},k_{1}-k_{2})| \\le C_\\mathcal{R},$$ with $C_\\mathcal{R}$ from \\eqref{CR}.  Note that $\\widehat{{\\mathcal{A}}}$ is the operator $\\widehat{\\mathcal{S}_{1}}$ in \\eqref{Snfourier} with $n=1$ if you replace $P$ from \\eqref{P} with $P_2$ above.  Then we have the same estimate as \\eqref{Snfbound} with $P_2$ replacing $P$ and $n=1$ recalling also \\eqref{an.def}.   We estimate $P_2$ similarly to \\eqref{P0} (except that $m\\ge 2$).  We conclude that\n\\begin{equation}\\label{breveS0bound}\n\\|{\\mathcal{A}}\\|_{\\mathcal{F}^{0,1}_\\nu} \\leq C_\\mathcal{R} \\|f\\|_{\\mathcal{F}^{0,1}_\\nu} \\|P_2\\|_{\\mathcal{F}^{0,1}_\\nu} \n\\leq C_\\mathcal{R} \\breve{C}_{1}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^{2}\\|f\\|_{\\mathcal{F}^{0,1}_\\nu}\n\\end{equation}\nwhere\n\\begin{equation}\\label{breveS1constant1}\n\\breve{C}_{1} = \\frac{\\exp(\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}) - 1 - \\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}}{\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^{2}}.\n\\end{equation}\nThus, recalling \\eqref{CR}, \\eqref{catalanconstant}, \\eqref{tildeC1} and \\eqref{breveS1constant1}, then we have\n\\begin{equation}\\label{C1}\nC_{1} = \\tilde{C}_{1} + C_\\mathcal{R} \\breve{C}_{1}.\n\\end{equation}\nWe obtain \\eqref{Sestimates} for $s=0$ by combining \\eqref{breveS0bound} with the bound above \\eqref{tildeC1}. \n\n\n\n\n\n\nWe turn to the estimate for ${\\mathcal{A}}$ in \\eqref{breveS} for $s>0$.    We will use \\eqref{s.inequality} since $s>0$ and \\eqref{bfcn.def}.  We will in this case estimate $P_2$ similarly to \\eqref{Ps.estimate} (except with $m\\ge 2$).\nWe then have the estimate\n\\begin{equation*}\n \\|{\\mathcal{A}}\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu} \\leq C_\\mathcal{R} \\sum_{m\\geq 2} \\frac{{b}(m+1,s)}{m!} \\big(\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^{m}\\|f\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu} + m \\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^{m-1}\\|f\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu} \\big).\n\\end{equation*}\nNow define\n\\begin{equation}\\label{breveCs}\n\\breve{C}_{3} \\overset{\\mbox{\\tiny{def}}}{=} \\sum_{m\\geq 0} \\frac{{b}(m+3,s)}{(m+2)!}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^{m}, \\quad\n\\breve{C}_{4} \\overset{\\mbox{\\tiny{def}}}{=} \\sum_{m\\geq 0} \\frac{{b}(m+3,s)}{(m+1)!} \\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^{m},\n\\end{equation}\nthen we have\n\\begin{equation}\\label{breveSsbound}\n\\|{\\mathcal{A}}\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu} \\leq C_\\mathcal{R} \\breve{C}_{3}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^{2}\\|f\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu} + C_\\mathcal{R} \\breve{C}_{4} \\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|f\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}.\n\\end{equation}\nHence further define\n\\begin{equation}\\label{C3}\nC_{3} \\overset{\\mbox{\\tiny{def}}}{=} \\tilde{C}_{3} + C_\\mathcal{R} \\breve{C}_{3}, \\quad C_{4} \\overset{\\mbox{\\tiny{def}}}{=} \\tilde{C}_{4} + C_\\mathcal{R} \\breve{C}_{4}.\n\\end{equation}\nThen using \\eqref{tildeC3}, \\eqref{tildeC4} (and the estimate below them) and \\eqref{breveCs} (and the estimate above it) we obtain \\eqref{Sestimates} for $s>0$.\n\\end{proof}\n\nIn the next section we prove the \\textit{a priori} estimates on the vorticity strength $\\omega$.\n\n\\section{\\textit{A priori} estimates on the vorticity strength}\\label{secw}\n\nThis section includes the \\textit{a priori} estimates for the vorticity strength $\\omega$ from \\eqref{omega} that will be used in particular in Subsection \\ref{subsecGlobal}.  The main result of the section is Proposition \\ref{vorticityestimatess}.\n\n\n\\begin{prop}\\label{vorticityestimatess}\n\tThe linear part of the vorticity, $\\omega_1$  in \\eqref{omegasplit}, satisfies the following estimates:\n\t\\begin{equation}\\label{omega1f0}\n    \\|\\omega_1\\|_{\\mathcal{F}^{0,1}_\\nu}\\leq A_\\sigma\\frac{4\\pi}{L(t)}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{2,1}_\\nu}+(1+2|A_\\mu|)|A_\\rho|\\frac{L(t)}{\\pi}e^{\\nu(t)}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu},\n    \\end{equation}\n    for $s>0$, recalling \\eqref{bfcn.def}, we have\n    \\begin{equation}\\label{omega1fss}\n    \\|\\omega_1\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\n    \\leq A_\\sigma\\frac{4\\pi}{L(t)}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{2+s,1}_\\nu}+(1+2|A_\\mu|)|A_\\rho|\\frac{L(t)}{\\pi}2b(2,s)e^{\\nu(t)}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}.\n    \\end{equation}\nThe nonlinear part  $\\omega_{\\geq2}$ from \\eqref{omegasplit} satisfies\n\\begin{equation}\\label{omega2f0}\n\t\\|\\omega_{\\geq2}\\|_{\\mathcal{F}^{0,1}_{\\nu}}\n\t\\leq |A_\\mu|A_\\sigma\\frac{4\\pi}{L(t)}C_9\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{2,1}_\\nu}\n\t\n\t+|A_\\rho| \\frac{L(t)}{\\pi}e^{\\nu(t)}C_{10}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^2,\n\\end{equation}\nwhile $s>0$ we have\n\\begin{multline}\\label{omega2fss}\n\t\\|\\omega_{\\geq2}\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\n\t\\!\\leq\\!\n\t|A_\\mu|A_\\sigma\\frac{4\\pi}{L(t)}C_{13}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{2+s,1}_\\nu}\\!\n\t\\\\\n+\n\t\\!|A_\\rho| \\frac{L(t)}{\\pi}e^{\\nu(t)}C_{14}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu},\n\\end{multline}\nwhere $C_9$ and $C_{10}$ are defined in \\eqref{C9C10} and $C_{13}$ and $C_{14}$ are in \\eqref{C13C14}.\n\\end{prop}\n\n\\begin{proof}\nFirst, recalling \\eqref{bfcn.def}, we note that \n\\begin{equation}\\label{cosine.calc.FT}\n\\begin{aligned}\n    \\|\\cos{(\\alpha\\!+\\!\\hat{\\vartheta}(0))}\\theta(\\alpha)\\|_{\\mathcal{F}^{0,1}_\\nu}&\\leq e^{\\nu(t)}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu},\\\\\n    \\|\\cos{(\\alpha\\!+\\!\\hat{\\vartheta}(0))}\\theta(\\alpha)\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}&\\leq 2 b(2,s) e^{\\nu(t)}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu},\\qquad s>0,\n\\end{aligned}    \n\\end{equation}\nas similar to the calculations in the proof of Lemma \\ref{triangleprop}.  We point out that the same calculations also hold with $\\cos{(\\alpha\\!+\\!\\hat{\\vartheta}(0))}$ replaced by $\\sin{(\\alpha\\!+\\!\\hat{\\vartheta}(0))}$.  \nFor example, to see the above in the case $0 < s \\leq 1$, \nwe have \n\\begin{multline}\\label{proof.angle.nu.ineq}\n\\|\\cos{(\\alpha\\!+\\!\\widehat{\\vartheta}(0))}\\theta(\\alpha)\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu} \n= \\sum_{k\\in\\mathbb{Z}} e^{\\nu(t)|k|}|k|^{s}|((\\cos{(\\alpha\\!+\\!\\widehat{\\vartheta}(0))})^{\\wedge}\\ast \\widehat{\\theta}) (k)|\n\\\\\n= \\sum_{k, k_{1}\\in\\mathbb{Z}} e^{\\nu(t)|k|} |k|^{s} |\\frac{1}{2}(\\delta(1-k_{1})\ne^{i\\widehat{\\vartheta}(0) }\n+ \\delta(1+k_{1})\ne^{-i\\widehat{\\vartheta}(0)}\n) \\widehat{\\theta} (k-k_{1})|\\\\\n\\leq  \\sum_{k\\in\\mathbb{Z}} e^{\\nu(t)|k|}|k|^{s}  \\frac{1}{2} (|\\widehat{\\theta} (k-1)|+|\\widehat{\\theta} (k+1)|)\\\\\n\\leq  \\frac{1}{2}\\sum_{k\\in\\mathbb{Z}} e^{\\nu(t)|k-1|}e^{\\nu(t)}(|k-1|^{s}+1)   |\\widehat{\\theta} (k-1)|\\\\ \\hspace{0.5in}+e^{\\nu(t)|k+1|}e^{\\nu(t)}(|k+1|^{s}+1) |\\widehat{\\theta} (k+1)|\\\\\n= e^{\\nu(t)}(\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu} + \\|\\theta\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}) \\leq 2e^{\\nu(t)}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}.\n\\end{multline}\nThis explains the difference between \\eqref{omega1f0} and \\eqref{omega1fss}.\nWe will explain the proof of \\eqref{omega1fss} for $s>0$.  The proof of the other case, \\eqref{omega1f0} when $s=0$, is analogous.  It follows from \\eqref{omegasplit} and \\eqref{cosine.calc.FT} that\n\\begin{equation*}\n\\|\\omega_1\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\n\\leq \n|A_\\mu|\\frac{L(t)}{\\pi}\\|\\mathcal{D}_1(\\omega_0)\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\n\\!+\\!\nA_\\sigma\\frac{4\\pi}{L(t)}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{2+s,1}_\\nu}\n\\!+\\!\n|A_\\rho| \\frac{L(t)}{\\pi}2b(2,s) e^{\\nu(t)}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu},\n\\end{equation*}\nand from \\eqref{mdsplit} with \\eqref{cosine.calc.FT} and \\eqref{hilbertTcalc} we have\n\\begin{equation*}\n\\begin{aligned}\n\\|\\mathcal{D}_1(\\omega_0)\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\\leq\\frac{\\pi}{L(t)}\\Big(|A_\\rho|\\frac{L(t)}{\\pi}2 b(2,s) e^{\\nu(t)}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}+\\|\\text{Im}\\hspace{0.05cm}\\hspace{0.05cm} \\mathcal{R}(\\omega_0)\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\\Big). \n\\end{aligned}\n\\end{equation*}\nRecalling  \\eqref{fourierimag} together with Lemma  \\eqref{cosine.calc.FT} gives the following estimate \n\\begin{equation*}\n    \\|\\text{Im}\\hspace{0.05cm}\\hspace{0.05cm} \\mathcal{R}(\\omega_0)\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\\leq |A_\\rho|\\frac{L(t)}{\\pi}2 b(2,s) e^{\\nu(t)}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}.\n\\end{equation*}\nTherefore,  \n\\begin{equation*}\n\\begin{aligned}\n\\|\\mathcal{D}_1(\\omega_0)\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\\leq 4|A_\\rho|e^{\\nu(t)}b(2,s) \\|\\theta\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}, \n\\end{aligned}\n\\end{equation*}\nand thus the estimate \\eqref{omega1fss} for $\\omega_1$ is complete.  The estimate for \\eqref{omega1f0} is proven in the same way.\n\n\n\nNow we proceed to bound $\\omega_{\\geq 2}$ from \\eqref{omegasplit} in $\\mathcal{F}^{0,1}_\\nu$.  In this case we have\n\\begin{equation}\\label{omega2aux}\n\\begin{aligned}\n\\|\\omega_{\\geq2}\\|_{\\mathcal{F}^{0,1}_\\nu}\n&\\leq \n|A_\\mu|\\frac{L(t)}{\\pi}\\|\\mathcal{D}_{\\geq2}(\\omega)\\|_{\\mathcal{F}^{0,1}_\\nu}\n+|A_\\rho| \\frac{L(t)}{\\pi}e^{\\nu(t)}\\sum_{j\\geq2}\\frac{\\|\\theta\\|_{\\mathcal{F}^{0,1}_{\\nu}}^j}{j!}\n\\\\\n&=\n|A_\\mu|\\frac{L(t)}{\\pi}\\|\\mathcal{D}_{\\geq2}(\\omega)\\|_{\\mathcal{F}^{0,1}_\\nu}\n+\n|A_\\rho| \\frac{L(t)}{\\pi}e^{\\nu(t)}C_6\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^2,\n\\end{aligned}\n\\end{equation}\nwhere\n\\begin{equation}\\label{C6}\nC_6=\\frac{e^{\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}}-1-\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}}{\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^2}.\n\\end{equation}\nFrom \\eqref{mdsplit}, one has that\n\\begin{equation*}\n\\begin{aligned}\n\\|\\mathcal{D}_{\\geq2}\\|_{\\mathcal{F}^{0,1}_\\nu}&\\leq \\frac{\\pi}{L(t)}\\Big(\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\omega_{\\geq1}\\|_{\\mathcal{F}^{0,1}_\\nu}+\\|\\mathcal{R}(\\omega_{\\geq1})\\|_{\\mathcal{F}^{0,1}_\\nu}+\\|\\mathcal{S}(\\omega)\\|_{\\mathcal{F}^{0,1}_\\nu}\\Big).\n\\end{aligned}\n\\end{equation*}\nUsing the estimates \\eqref{Restimates} and \\eqref{Sestimates} and splitting the vorticity terms as $\\omega_{\\geq1} = \\omega_{1}+\\omega_{\\geq2}$, we obtain the estimate\n\\begin{equation*}\n\\begin{aligned}\n\\|\\mathcal{D}_{\\geq2}&\\|_{\\mathcal{F}^{0,1}_\\nu}\\leq \\frac{\\pi}{L(t)}\\Big(\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\omega_{1}\\|_{\\mathcal{F}^{0,1}_\\nu}+\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\omega_{\\geq2}\\|_{\\mathcal{F}^{0,1}_\\nu}+\n{C_{\\mR}}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\omega_{1}\\|_{\\mathcal{F}^{0,1}_\\nu}\\\\\n&+{C_{\\mR}}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\omega_{\\geq2}\\|_{\\mathcal{F}^{0,1}_\\nu}+|A_\\rho|\\frac{L(t)}{\\pi}e^{\\nu(t)}C_1\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^2+C_1\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^2\\|\\omega_{1}\\|_{\\mathcal{F}^{0,1}_\\nu}\\\\\n&+C_1\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^2\\|\\omega_{\\geq2}\\|_{\\mathcal{F}^{0,1}_\\nu}\n\\Big),\n\\end{aligned}\n\\end{equation*}\nso, substituting back in \\eqref{omega2aux} and solving for $\\|\\omega_{\\geq2}\\|_{\\mathcal{F}^{0,1}_\\nu}$, we obtain that\n\\begin{equation*}\n    \\begin{aligned}\n\\|\\omega_{\\geq2}\\|_{\\mathcal{F}^{0,1}_\\nu}&\\leq C_8\\Big( |A_\\mu|C_7\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\omega_{1}\\|_{\\mathcal{F}^{0,1}_\\nu}+|A_\\mu||A_\\rho|\\frac{L(t)}{\\pi}e^{\\nu(t)}C_1\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^2\\\\\n&\\quad\\quad+|A_\\rho|\\frac{L(t)}{\\pi}e^{\\nu(t)}C_6\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^2\\Big),\n    \\end{aligned}\n\\end{equation*}\nwhere we defined\n\\begin{equation}\\label{C7C8}\n    C_7=1+{C_{\\mR}}+C_1\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu},\\qquad C_8=\\frac{1}{1-|A_\\mu|C_7\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}},\n\\end{equation}\nwith $C_1$ given in \\eqref{C1}.\nSubstituting in \\eqref{omega1f0}  we conclude that\n\\begin{equation*}\n    \\begin{aligned}\n    \\|\\omega_{\\geq2}\\|_{\\mathcal{F}^{0,1}_\\nu}&\\leq 2|A_\\mu|A_\\sigma \\frac{2\\pi}{L(t)}C_7 C_8\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{2,1}_\\nu}\\\\\n    &+|A_\\rho|\\frac{L(t)}{\\pi}e^{\\nu(t)}C_8\\Big(|A_\\mu|(1+2|A_\\mu|)C_7+|A_\\mu|C_1+C_6\\Big)\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^2,\n    \\end{aligned}\n\\end{equation*}\nwhich gives the  estimate  \\eqref{omega2f0} by defining\n\\begin{equation}\\label{C9C10}\n\\begin{aligned}\nC_9&=C_7C_8,\\\\\nC_{10}&=|A_\\mu|(1+2|A_\\mu|)C_7C_8+|A_\\mu|C_1C_8+C_6C_8,\n\\end{aligned}\n\\end{equation}\nwhere $C_1$, $C_6$, $C_7$, and $C_8$ were previously defined in \\eqref{C1}, \\eqref{C6}, and \\eqref{C7C8}.\n\n\n\n\nWe consider now the case $s>0$ in \\eqref{omega2fss}. From \\eqref{omegasplit}, also using \\eqref{proof.angle.nu.ineq}, we have\n\\begin{multline}\\label{omega2aux2}\n\\|\\omega_{\\geq2}\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\n\\leq |A_\\mu|\\frac{L(t)}{\\pi}\\|\\mathcal{D}_{\\geq2}(\\omega)\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\n+\n|A_\\rho| \\frac{L(t)}{\\pi}e^{\\nu(t)}\\sum_{j\\geq2}\\frac{b(j,s) \\|\\theta\\|_{\\mathcal{F}^{0,1}_{\\nu}}^{j}}{j!}\n\\\\\n\\quad\n+|A_\\rho| \\frac{L(t)}{\\pi}e^{\\nu(t)}\\sum_{j\\geq1}\\frac{b(j,s) \\|\\theta\\|_{\\mathcal{F}^{0,1}_{\\nu}}^{j}}{j!}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\n\\\\\n\\leq\n|A_\\mu|\\frac{L(t)}{\\pi}\\|\\mathcal{D}_{\\geq2}(\\omega)\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\n+\n|A_\\rho| \\frac{L(t)}{\\pi}e^{\\nu(t)}C_{11}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu},\n\\end{multline}\nwhere \n\\begin{equation}\\label{C11}\nC_{11}=\\sum_{j\\geq 2}\\frac{b(j,s) \\|\\theta\\|_{\\mathcal{F}^{0,1}_{\\nu}}^{j-2}}{j!}+\\sum_{j\\geq1}\\frac{b(j,s) \\|\\theta\\|_{\\mathcal{F}^{0,1}_{\\nu}}^{j-1}}{j!}.\n\\end{equation}\nRecalling \\eqref{mdsplit}, and \\eqref{s.inequality}, one has that\n\\begin{equation*}\n\\begin{aligned}\n\\|\\mathcal{D}_{\\geq2}\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}&\\leq \\frac{\\pi}{L(t)}\\Big(b(2,s) \\|\\theta\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\\|\\omega_{\\geq1}\\|_{\\mathcal{F}^{0,1}_\\nu}+ b(2,s)\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\omega_{\\geq1}\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\n\\\\\n&\\qquad\\qquad+\\|\\mathcal{R}(\\omega_{\\geq1})\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}+\\|\\mathcal{S}(\\omega)\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\\Big).\n\\end{aligned}\n\\end{equation*}\nUsing the estimates \\eqref{Restimates} and \\eqref{Sestimates} with $s>0$ and splitting the vorticity terms again, we obtain\n\\begin{multline*}\n\\|\\mathcal{D}_{\\geq2}\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\\leq \\frac{\\pi}{L(t)}\\bigg((1+b(2,s){C_{\\mR}})\\|\\theta\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\\|\\omega_{1}\\|_{\\mathcal{F}^{0,1}_\\nu}+(1+b(2,s){C_{\\mR}})\\|\\theta\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\\|\\omega_{\\geq2}\\|_{\\mathcal{F}^{0,1}_\\nu}\n\\\\\n+\n(1+b(2,s){C_{\\mR}})\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\omega_{1}\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}+(1+b(2,s){C_{\\mR}})\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\omega_{\\geq2}\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\n\\\\\n+\n|A_\\rho|\\frac{L(t)}{\\pi}e^{\\nu(t)}(C_3+C_4)\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}+C_3\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^2\\|\\omega_{1}\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\n\\\\\n+\\!C_3\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^2\\|\\omega_{\\geq2}\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\\!+\\!C_4\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\\|\\omega_{1}\\|_{\\mathcal{F}^{0,1}_\\nu}\\!+\\!C_4\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\\|\\omega_{\\geq2}\\|_{\\mathcal{F}^{0,1}_\\nu}\\!\\!\\bigg),\n\\end{multline*}\nwhich becomes \n\\begin{equation*}\n\\begin{aligned}\n\\|\\mathcal{D}_{\\geq2}\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}&\\leq \\frac{\\pi}{L(t)}\\Big(|A_\\rho|\\frac{L(t)}{\\pi}e^{\\nu(t)}(C_3\\!+\\!C_4)\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\\!+\\!C_{12}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\\|\\omega_{1}\\|_{\\mathcal{F}^{0,1}_\\nu}\\\\\n&+\\!C_2\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\omega_{1}\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\\!+\\!C_{12}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\\|\\omega_{\\geq2}\\|_{\\mathcal{F}^{0,1}_\\nu}\\!+\\!C_2\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\omega_{\\geq2}\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\\Big),\n\\end{aligned}\n\\end{equation*}\nwhere \n\\begin{equation}\\label{C12}\n\\begin{aligned}\nC_2&=1+b(2,s){C_{\\mR}}+C_3\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\\\\nC_{12}&=1+b(2,s){C_{\\mR}}+C_4\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu},\n\\end{aligned}\n\\end{equation}\nwith $C_\\mathcal{R}$ and $C_4$ given in \\eqref{CR} and \\eqref{C3}, respectively.\nSubstituting back in \\eqref{omega2aux2}, and solving for $\\|\\omega_{\\geq2}\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}$, we have\n\\begin{equation*}\n    \\begin{aligned}\n\\|\\omega_{\\geq2}\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}&\\leq |A_\\mu|\\tilde{C}_8 \\Big( \n|A_\\rho|\\frac{L(t)}{\\pi}e^{\\nu(t)}(C_3+C_4)\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\\\\\n&+\nC_{12}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\\|\\omega_{1}\\|_{\\mathcal{F}^{0,1}_\\nu}+C_2\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\omega_{1}\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}+C_{12}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\\|\\omega_{\\geq2}\\|_{\\mathcal{F}^{0,1}_\\nu}\\Big)\\\\\n&+|A_\\rho|\\frac{L(t)}{\\pi}e^{\\nu(t)}C_{11}\\tilde{C}_8\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu},\n    \\end{aligned}\n\\end{equation*}\nwhere we defined\n\\begin{equation}\\label{tildeC8}\n\\tilde{C}_8=\\frac{1}{1-|A_\\mu|C_2\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}}.\n\\end{equation}\nWe introduce the estimates \\eqref{omega2f0}, \\eqref{omega1f0}, and \\eqref{omega1fss} to obtain that\n\\begin{equation*}\n    \\begin{aligned}\n\\|\\omega_{\\geq2}&\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\\leq |A_\\mu|\\tilde{C}_8\\bigg( \n|A_\\rho|\\frac{L(t)}{\\pi}e^{\\nu(t)}(C_3+C_4)\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\\\\\n&\\hspace{-0.7cm}+\nA_\\sigma \\frac{4\\pi}{L(t)}C_{12}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{2,1}_\\nu}+(1+2|A_\\mu|)|A_\\rho|\\frac{L(t)}{\\pi}e^{\\nu(t)}C_{12}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\\\\n&\\hspace{-0.7cm}+A_\\sigma\\frac{4\\pi}{L(t)} C_2\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{2+s,1}}+(1\\!+\\!2|A_\\mu|)|A_\\rho|\\frac{L(t)}{\\pi}2b(2,s) e^{\\nu(t)}C_2\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\\\\\n&\\hspace{-0.7cm}+|A_\\mu|A_\\sigma\\frac{4\\pi}{L(t)}C_9C_{12}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{2,1}_\\nu}\\\\\n&\\hspace{-0.7cm}+|A_\\rho|\\frac{L(t)}{4\\pi}e^{\\nu(t)}C_{10}C_{12}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^2\\!\n\\bigg)\\!\\!+\\!|A_\\rho|\\frac{L(t)}{\\pi}e^{\\nu(t)}C_{11}\\tilde{C}_8\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}.\n    \\end{aligned}\n\\end{equation*}\nNext, we use the interpolation inequality \\eqref{interpolation} to find that\n\\begin{equation}\\label{interp}\n    \\|\\theta\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{2,1}_\\nu}\\leq \\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{2+s,1}_\\nu},\n\\end{equation}\nand therefore\n\\begin{equation*}\n\\begin{aligned}\n    \\|\\omega_{\\geq2}\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}&\\leq |A_\\mu|A_\\sigma \\frac{4\\pi}{L(t)}\\tilde{C}_8\\Big(C_2+C_{12}+|A_\\mu|C_9C_{12}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\Big)\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{2+s,1}_\\nu}\\\\\n    &+|A_\\rho|\\frac{L(t)}{\\pi}e^{\\nu(t)}\\tilde{C}_8\\Big(|A_\\mu|(C_3+C_4)+|A_\\mu|(1+2|A_\\mu|)(2C_2+C_{12})\\\\\n    &+|A_\\mu|C_{10}C_{12}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}+C_{11}\\Big)\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu},\n\\end{aligned}\n\\end{equation*}\nwhich gives the inequality \\eqref{omega2fss} by defining\n\\begin{equation}\\label{C13C14}\n\\begin{aligned}\nC_{13}&=\\tilde{C}_8\\Big(C_2+C_{12}+|A_\\mu|C_9C_{12}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\Big),\\\\\nC_{14}&=|A_\\mu|\\tilde{C}_8\\Big(C_3\\!+\\!C_4\\!+\\!(1\\!+\\!2|A_\\mu|)(2C_2\\!+\\!C_{12})\\!+\\!C_{10}C_{12}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\Big)\\!\n\\\\\n& \\hspace{1cm}+\\!C_{11}\\tilde{C}_8,\n\\end{aligned}\n\\end{equation}\nwhere $C_3$ and $C_4$ in \\eqref{C3}, $C_2$ in \\eqref{C12}, $\\tilde{C}_8$ in \\eqref{tildeC8}, $C_9$ and $C_{10}$ in\\eqref{C9C10}, $C_{11}$ in \\eqref{C11}, and $C_{12}$ in \\eqref{C12} were previously defined.\n\\end{proof}\n\n\nIn the next section we put together all the previous results in the paper to prove the global existence and instant analyticity as in Theorem \\ref{thm:global}\n\n\n\n\\section{Global existence and instant analyticity}\\label{secanalytic}\n\nThis section is dedicated to the proof of Theorem \\ref{thm:global}. In Subsection \\ref{sec:Lestimate} we prove the claimed bound for the length $L(t)$ from \\eqref{Lfinalbound}.  Then Subsection \\ref{subsecGlobal} proves the main \\textit{a priori}  estimates for a solution $\\theta$.  We prove the global energy inequality from \\eqref{estimatef12}.  In particular we diagonalize the linearized operator in Proposition \\ref{diagonalization}.  We prove operator bounds for the corresponding linear transformations in Lemma \\ref{lemmaS}.   In Subsection \\ref{sec:NonLinearEst} we prove the corresponding non-linear estimates that were used in the a priori estimates from the previous subsection.  In particular we prove Theorem \\ref{thm:Nbound}.  Lastly, in Subsection \\ref{subregular} we describe the scheme of the proof of our main theorem via a regularization argument.\n\n\n\n\\subsection{Estimate for $L(t)$}\\label{sec:Lestimate}\n\nIn this section, we prove \nthe bound for $L(t)$ from \\eqref{Lfinalbound}.  Equation \\eqref{Lequation}, together with the bound\n\\begin{equation}\\label{lengthauxbound}\n\\begin{aligned}\n    \\text{Im}\\hspace{0.05cm}\\Big(\\int_{-\\pi}^\\pi\\int_0^\\alpha e^{i(\\alpha-\\eta)} \\sum_{n\\geq1}\\frac{i^n}{n!}(\\theta(\\alpha)-\\theta(\\eta))^n d\\eta d\\alpha\\Big)&\\leq \\pi^2\\sum_{n\\geq1}\\frac{2^n\\|\\theta\\|_{\\mathcal{F}^{0,1}}^n}{n!}\\\\\n    &\\leq \\pi^2\\big(e^{2\\|\\theta\\|_{\\mathcal{F}^{0,1}}}-1\\big),\n\\end{aligned}\n\\end{equation}\nimplies that\n\\begin{equation}\\label{Lboundaux}\n    \\begin{aligned}\n        \\frac{R^2}{1+\\frac{\\pi}{2} \\big(e^{2\\|\\theta\\|_{\\mathcal{F}^{0,1}}}-1\\big)}\\leq\\Big(\\frac{L(t)}{2\\pi}\\Big)^2\\leq \\frac{R^2}{1-\\frac{\\pi}{2}\\big(e^{2\\|\\theta\\|_{\\mathcal{F}^{0,1}}}-1\\big)},\n    \\end{aligned}\n\\end{equation}\nand therefore\n\\begin{equation}\\label{Lbound}\n    \\begin{aligned}\n        2\\pi R C_{37}\\leq L(t)\\leq 2\\pi RC_{38},\n    \\end{aligned}\n\\end{equation}\nwhere\n\\begin{equation}\\label{C37C38}\n    C_{37}=\\frac{1}{\\sqrt{1+\\frac{\\pi}{2} \\big(e^{2\\|\\theta\\|_{\\mathcal{F}^{0,1}}}-1\\big)}},\\quad C_{38}=\\frac{1}{\\sqrt{1-\\frac{\\pi}{2}\\big(e^{2\\|\\theta\\|_{\\mathcal{F}^{0,1}}}-1\\big)}}.\n\\end{equation}\nWe have thus shown that the length of the curve is controlled  for all times. In particular, we also have the following estimates\n\\begin{equation}\\label{Lestimates}\n\\begin{aligned}\n    \\Big|R^2\\Big(\\frac{2\\pi}{L(t)}\\Big)^2-1\\Big|&\\leq  \\frac{\\pi}{2}\\big(e^{2\\|\\theta\\|_{\\mathcal{F}^{0,1}}}-1\\big),\n    \\\\\n    \\Big|R\\Big(\\frac{2\\pi}{L(t)}\\Big)-1\\Big|&\\leq  \\sqrt{1+\\frac{\\pi}{2}\\big(e^{2\\|\\theta\\|_{\\mathcal{F}^{0,1}}}-1\\big)}-1=C_{39}\\|\\theta\\|_{\\mathcal{F}^{0,1}},\n    \\\\\n    \\Big|R^3\\Big(\\frac{2\\pi}{L(t)}\\Big)^3-1\\Big|&\\leq  \\Big(1+\\frac{\\pi}{2}\\big(e^{2\\|\\theta\\|_{\\mathcal{F}^{0,1}}}-1\\big)\\Big)^{3\/2}-1=C_{40}\\|\\theta\\|_{\\mathcal{F}^{0,1}},\n    \\end{aligned}\n\\end{equation}\nwith\n\\begin{equation}\\label{C39C40}\n    C_{39}=\\frac{\\sqrt{1+\\frac{\\pi}{2}\\big(e^{2\\|\\theta\\|_{\\mathcal{F}^{0,1}}}-1\\big)}-1}{\\|\\theta\\|_{\\mathcal{F}^{0,1}}},\\quad C_{40}=\\frac{\\Big(1+\\frac{\\pi}{2}\\big(e^{2\\|\\theta\\|_{\\mathcal{F}^{0,1}}}-1\\big)\\Big)^{3\/2}-1}{\\|\\theta\\|_{\\mathcal{F}^{0,1}}}.\n\\end{equation}\nThese are the main estimates that we will use for $L(t)$.  We remark that all the estimates in this section hold with the same proof in the norms $\\mathcal{F}^{0,1}_\\nu$ and $\\dot{\\mathcal{F}}^{\\frac12,1}_\\nu$.   In the next section we prove the main a priori estimates for $\\theta$.\n\n\n\n\\subsection{\\textit{A priori} Estimates for $\\theta$}\\label{subsecGlobal}\n\nIn this section we obtain the \\textit{a priori} estimates that guarantee the global existence of the solutions, the instant in time analyticity, and the exponentially fast convergence to a circle. In particular, the main goal is to show the energy inequality \\eqref{estimatef12}.\n\n\nThe results of this section are ordered as follows: First, we write the system \\eqref{system} using the linearization \\eqref{linearfourier} of Subsection \\ref{sec:linearization}; then, we diagonalize the system according to Proposition \\ref{diagonalization} to show its dissipative character; the bounds for this change of frame are proven in Lemma \\ref{lemmaS}; the estimates for the nonlinear terms are proven in Subsection \\ref{sec:NonLinearEst}, together with the control of the length in Subsection \\ref{sec:Lestimate}.\n\n\n\\begin{proof}[Proof of the global energy inequality from  \\eqref{estimatef12}]\nEquation \\eqref{system} and  Proposition \\ref{linearfourier} show that the equation for $\\widehat{\\theta}(k)$, $1\\leq k\\neq2$, is given by\n\\begin{equation*}\n    \\begin{aligned}\n   \\widehat{\\theta}_t(k)&=-A_\\sigma \\Big(\\frac{2\\pi}{L(t)}\\Big)^3k(k^2\\!-\\!1)\\widehat{\\theta}(k)\\!-\\!\\big(1\\!+\\!A_\\mu\\big)A_\\rho\\frac{2\\pi}{L(t)}\\frac{(k^2\\!-\\!1)(k\\!+\\!1)}{k(k\\!+\\!2)}e^{-i\\widehat{\\vartheta}(0)}\\widehat{\\theta}(k\\!+\\!1)\\\\\n   &\\quad+\\frac{2\\pi}{L(t)}\\widehat{N}(k),\n    \\end{aligned}\n\\end{equation*}\nand for $k=2$, we have\n\\begin{equation*}\n    \\begin{aligned}\n   \\widehat{\\theta}_t(2)&=-A_\\sigma \\Big(\\frac{2\\pi}{L(t)}\\Big)^36\\widehat{\\theta}(2)\\!-(1+A_\\mu)A_\\rho\\frac{2\\pi}{L(t)}\\frac{9}{8}e^{-i\\widehat{\\vartheta}(0)}\\widehat{\\theta}(3)\\\\\n   &\\quad +(1-A_\\mu)A_\\rho\\frac{2\\pi}{L(t)}\\frac{3}{2}\\left(\\frac34-\\log{2}\\right)e^{i\\widehat{\\vartheta}(0)}\\widehat{\\theta}(1) +\\frac{2\\pi}{L(t)}\\widehat{N}(k).\n    \\end{aligned}\n\\end{equation*}\nWe notice in the equation above that the terms that are linear in $\\widehat{\\theta}(k)$ have time-dependent coefficients. However, this dependency happens only through $L(t)$. We will show that $L(t)$ is bounded from below and above (see Subsection \\ref{sec:Lestimate}). In fact, it is not hard to see from \\eqref{Lequation} that, to leading order, $L(t)$ equals $2\\pi R$. Thus we rewrite the equation above as follows\n\\begin{equation}\\label{thetaequation}\n    \\begin{aligned}\n   \\widehat{\\theta}_t(k)&=-\\frac{A_\\sigma}{R^3} k(k^2-1)\\widehat{\\theta}(k)-\\big(1+A_\\mu\\big)\\frac{A_\\rho}{R}\\frac{(k^2-1)(k+1)}{k(k+2)}e^{-i\\widehat{\\vartheta}(0)}\\widehat{\\theta}(k+1)\\\\\n   &\\quad+\\frac{2\\pi}{L(t)}\\widehat{N}(k)-\\frac{A_\\sigma}{R^3} k(k^2-1)\\widehat{\\theta}(k)\\Big(R^3\\Big(\\frac{2\\pi}{L(t)}\\Big)^3-1\\Big)\\\\\n   &\\quad-\\big(1+A_\\mu\\big)\\frac{A_\\rho}{R}\\frac{(k^2-1)(k+1)}{k(k+2)}e^{-i\\widehat{\\vartheta}(0)}\\widehat{\\theta}(k+1)\\Big(R\\frac{2\\pi}{L(t)}-1\\Big),\n    \\end{aligned}\n\\end{equation}\nfor $1\\leq k\\neq2$, and we decompose the equation analogously for $k=2$.\n\nNext, we write the corresponding linear system as follows\n\\begin{equation}\\label{linearZ}\n    \\begin{aligned}\n    z_t(k)=\\sum_{j\\geq1}M_{k,j}z(j),\\quad k\\geq1,\n    \\end{aligned}\n\\end{equation}\nwhere we denote\n\\begin{equation*}\n    M_{k,j}=\\left\\{\n    \\begin{aligned}\n    &-a(k),\\quad j=k,\\\\\n    &b(k),\\hspace{0.8cm} j=k+1,\\\\\n    &c(1),\\hspace{0.8cm}j=1,k=2,\\\\\n    &0,\\hspace{1.2cm} \\text{otherwise},\n    \\end{aligned}\\right.\n\\end{equation*}\nwith \n\\begin{equation}\\label{coeffs}\n    a(k)=\\frac{A_\\sigma}{R^3} k(k^2-1),\\quad b(k)=-\\big(1+A_\\mu\\big)\\frac{A_\\rho}{R}\\frac{(k^2-1)(k+1)}{k(k+2)}e^{-i\\widehat{\\vartheta}(0)},\n\\end{equation}\n\\begin{equation*}\n    c(1)=(1-A_\\mu)\\frac{A_\\rho}{R}\\frac{3}{2}\\left(\\frac34-\\log{2}\\right)e^{i\\widehat{\\vartheta}(0)}.\n\\end{equation*}\nNotice that this is an upper triangular system except for the entry $k=2$,$j=1$.  This entry, $k=2$ with $j=1$, will require special attention.  The eigenvalues of this system are $-a(k)$.  This is given in the following proposition.\n\n\n\n\n\\begin{prop}[Diagonalization]\\label{diagonalization}\nConsider $z=(z(k))_{k\\geq1},y=(y(k))_{k\\geq1}\\in \\ell^1$ and the linear operator \\begin{equation*}\n    \\begin{aligned}\n    S^{-1}:\\ell^1&\\rightarrow \\ell^1\\\\\n    z&\\mapsto y=S^{-1}z,\n    \\end{aligned}\n\\end{equation*}\ndefined by\n    \\begin{equation*}\n        y(k)=\\sum_{j\\geq1}S^{-1}_{k,j}z(j),\n    \\end{equation*}\n    with\n    \\begin{equation}\\label{Sm1}\n        S^{-1}_{k,j}=\\left\\{\n        \\begin{aligned}\n        (&-1)^{j-k}\\prod_{l=1}^{j-k} \\frac{b(k-1+l)}{a(k)-a(k+l)},\\hspace{0.4cm}j\\geq k\\geq2,\\\\\n        &1,\\hspace{1.2cm}j=k=1,\\\\\n        -&\\frac{c(1)}{a(2)},\\hspace{1.2cm}k=2,j=1,\\\\\n        &0,\\hspace{1.72cm} \\text{otherwise}.\n        \\end{aligned}\\right.\n    \\end{equation}\nThen, the inverse operator $S$ is given by\n    \\begin{equation*}\n        S_{k,j}=\\left\\{\n        \\begin{aligned}\n        &\\prod_{l=1}^{j-k} \\frac{b(k-1+l)}{a(k-1+l)-a(j)},\\hspace{0.4cm}j\\geq k\\geq2,\\\\\n        &1,\\hspace{1.2cm}j=k=1,\\\\\n        &\\frac{c(1)}{a(2)},\\hspace{1.72cm}k=2,j=1,\\\\\n        &0,\\hspace{1.72cm} \\text{otherwise}.\n        \\end{aligned}\\right.\n    \\end{equation*}\nMoreover, the linear operator $S^{-1}$ diagonalizes the system \\eqref{linearZ} as follows\n\\begin{equation*}\n    y_t(k)=-a(k)y(k),\\quad k\\geq1.\n\\end{equation*}\n\\end{prop}\n\nWe remark that since $\\prod_{l=1}^{0}=1$ by definition then $S_{k,k} = S^{-1}_{k,k} =1$ when $j=k$ for all $k\\ge 1$.   We also have the following lemma which gives uniform bounds for the operators $S$ and $S^{-1}$.\n\n\\begin{lemma}\\label{lemmaS}\n    The operator norms in $\\ell^1$ of the linear operators $S$ and $S^{-1}$ satisfy the following bounds\n    \\begin{equation}\\label{CSbounds}\n        \\begin{aligned}\n        \\|S^{-1}\\|_{\\ell^1\\rightarrow \\ell^1}\\leq C_{S},\\qquad  \\|S\\|_{\\ell^1\\rightarrow \\ell^1}\\leq C_{S},\n        \\end{aligned}\n    \\end{equation}\nwith $C_S = C_S(A_\\mu,\\frac{|A_\\rho| R^2}{A_\\sigma})$ where\n\\begin{equation*}\n    C_S\n    \\overset{\\mbox{\\tiny{def}}}{=} \\max\\Big\\{1+\\frac{1}{4}\\big(1\\!-\\!A_\\mu\\big)\\frac{|A_\\rho| R^2}{A_\\sigma}\\Big(\\frac34-\\log{2}\\Big),6\\frac{I_3\\Big(2\\sqrt{\\big(1\\!+\\!A_\\mu\\big)\\frac{|A_\\rho| R^2}{A_\\sigma}}\\Big)}{\\Big(\\big(1\\!+\\!A_\\mu\\big)\\frac{|A_\\rho| R^2}{A_\\sigma}\\Big)^{3\/2}}\\Big\\}.\n\\end{equation*}\n\\end{lemma}\n\nIn the constant $C_S$ above we used the modified Bessel function of the first kind of order three.  In general, for an integer $n\\ge 0$, we define\n\\begin{equation}\\label{bessel.In.def}\nI_n(z) \\overset{\\mbox{\\tiny{def}}}{=} \\left(\\frac{z}{2} \\right)^n\n \\sum_{j=0}^{\\infty}\n \\frac{(z\/2)^{2j}}{j!(j+n)!}.\n\\end{equation}\nWe will prove Proposition \\ref{diagonalization} and Lemma \\ref{lemmaS} after we finish the proof of the main global energy inequality. \n\n\n\\begin{remark}\\label{analytic.norm.remark.S}\nThe results in Proposition \\ref{diagonalization} and Lemma \\ref{lemmaS} also hold in the space $\\ell^1$ with weight $e^{\\nu(t)|k|}|k|^s$ for any $s\\ge 0$ without any change to the proof; i.e.\n\\begin{equation*}\n    \\|z\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\n    \\leq \n     C_S\\|y\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu},\n     \\quad \n         \\|y\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\n    \\leq \n     C_S\\|z\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu},\n\\end{equation*}\nwhere $y(k)$ and $z(k)$ are defined in Proposition \\ref{diagonalization}.\n\\end{remark}\n\nAccording to Proposition \\ref{diagonalization}, we apply the linear transformation $S^{-1}$ to \\eqref{thetaequation} to rewrite the system with the linear part in diagonal form\n\\begin{equation}\\label{thetaeqdiag}\n    \\begin{aligned}\n   (S^{-1}\\widehat{\\theta})_t(k)&=-\\frac{A_\\sigma}{R^3} k(k^2-1)(S^{-1}\\widehat{\\theta})(k)+\\frac{2\\pi}{L(t)}(S^{-1}\\widehat{N})(k)\\\\\n   &\\quad-\\frac{A_\\sigma}{R^3} (S^{-1}u)(k)\\Big(R^3\\Big(\\frac{2\\pi}{L(t)}\\Big)^3-1\\Big)\\\\\n   &\\quad-\\big(1+A_\\mu\\big)\\frac{A_\\rho}{R}e^{-i\\widehat{\\vartheta}(0)}(S^{-1}v)(k)\\Big(R\\frac{2\\pi}{L(t)}-1\\Big)\\\\\n   &\\quad+(1-A_\\mu)\\frac{A_\\rho}{R}\\frac{3}{2}\\left(\\frac34-\\log{2}\\right)e^{i\\widehat{\\vartheta}(0)}\\Big(R\\frac{2\\pi}{L(t)}-1\\Big)(S^{-1}w)(k),\n    \\end{aligned}\n\\end{equation}\nwhere we introduced the notation\n\\begin{equation*}\n    \\begin{aligned}\n    u(k)=k(k^2-1)\\widehat{\\theta}(k),\\quad v(k)=\\frac{(k^2-1)(k+1)}{k(k+2)}\\widehat{\\theta}(k+1),\\quad w(k)=1_{k=2}\\widehat{\\theta}(1).\n    \\end{aligned}\n\\end{equation*}\nTaking into account that $\\theta(\\alpha)$ is a real-valued function, we can write the norm \\eqref{fsonenorm} in terms of the positive frequencies alone\n\\begin{equation*}\n\\begin{aligned}\n    \\|\\theta\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}_\\nu}=\\sum_{k\\in\\mathbb{Z}}e^{\\nu(t)|k|}|k|^{1\/2}|\\widehat{\\theta}(k)|=2\\sum_{k\\geq1}e^{\\nu(t)k}k^{1\/2}|\\widehat{\\theta}(k)|.\n\\end{aligned}\n\\end{equation*}\nDefine\n\\begin{equation*}\n    \\begin{aligned}\n    \\widehat{y}(k)=(S^{-1}\\widehat{\\theta})(k),\\quad \\widehat{y}(-k)=\\overline{\\widehat{y}(k)},\\quad k\\geq1, \\quad \\widehat{y}(0)=0,\\quad y=\\mathcal{F}^{-1}\\widehat{y},\n    \\end{aligned}\n\\end{equation*}\nand consider the evolution of the quantity\n\\begin{equation*}\n    \\|y\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}_\\nu}=2\\sum_{k\\geq1} e^{\\nu(t)k}k^{1\/2}|\\widehat{y}(k)|.\n\\end{equation*}\nTaking the derivative in time we obtain that\n\\begin{equation*}\n\\begin{aligned}\n    \\frac{d}{dt}\\|y\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}_\\nu}\\!=\\!2\\!\\sum_{k\\geq1}\\!\\nu'(t) k^{3\/2}e^{\\nu(t)k}|\\widehat{y}(k)|\\!+\\!2\\sum_{k\\geq1}e^{\\nu(t)k}k^{1\/2}\\frac12\\frac{\\widehat{y}_t(k)\\overline{\\widehat{y}(k)}\\!+\\!\\widehat{y}(k)\\overline{\\widehat{y}_t(k)}}{|\\widehat{y}(k)|}.\n\\end{aligned}\n\\end{equation*}\nTherefore, substituting \\eqref{thetaeqdiag}, one finds the following equation\n\\begin{equation}\\label{auxx}\n\\begin{aligned}\n    \\frac{d}{dt}\\|y\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}_\\nu}\\!&=\\!2\\!\\sum_{k\\geq1}\\!\\nu'(t) k^{3\/2}e^{\\nu(t)k}|\\widehat{y}(k)|-2\\frac{A_\\sigma}{R^3}\\!\\!\\sum_{k\\geq1}\\!e^{\\nu(t)k}k^{3\/2}(k^2-1)|\\widehat{y}(k)|\n    \\\\\n    &\\quad+2\\frac{2\\pi}{L(t)}\\sum_{k\\geq1}e^{\\nu(t)k}k^{1\/2}\\frac12\\frac{(S^{-1}\\widehat{N})(k)\\overline{\\widehat{y}(k)}+\\overline{(S^{-1}\\widehat{N})(k)}\\widehat{y}(k)}{|\\widehat{y}(k)|}\\\\\n    &\\quad+Y_u+Y_v+Y_w,\n\\end{aligned}\n\\end{equation}\nwhere\n\\begin{equation*}\n    Y_u=-2\\frac{A_\\sigma}{R^3}\\Big(R^3\\Big(\\frac{2\\pi}{L(t)}\\Big)^3\\!-\\!1\\Big)\\sum_{k\\geq1}\\!\\!e^{\\nu(t)k}k^{1\/2}\\frac{(S^{-1}u)(k)\\overline{\\widehat{y}(k)}+\\overline{(S^{-1}u)(k)}\\widehat{y}(k)}{2|\\widehat{y}(k)|},\n\\end{equation*}\n\\begin{equation*}\n\\begin{aligned}\n    Y_v&=-2(1\\!+\\!A_\\mu)\\frac{A_\\rho}{R}\\Big(R\\frac{2\\pi}{L(t)}\\!-\\!1\\Big)\n    \\\\\n    &\\quad\\times\\sum_{k\\geq1}\\!\\!e^{\\nu(t)k}k^{1\/2}\\frac{e^{-i\\widehat{\\vartheta}(0)}(S^{-1}v)(k)\\overline{\\widehat{y}(k)}\\!+\\!e^{i\\widehat{\\vartheta}(0)}\\overline{(S^{-1}v)(k)}\\widehat{y}(k)}{2|\\widehat{y}(k)|},\n\\end{aligned}\n\\end{equation*}\n\\begin{equation*}\n\\begin{aligned}\n    Y_w&=2(1\\!-\\!A_\\mu)\\frac{A_\\rho}{R}\\Big(R\\frac{2\\pi}{L(t)}\\!-\\!1\\Big) \\frac32\\Big(\\frac34\\!-\\!\\log{2}\\Big)\n    \\\\\n    &\\quad\\times\\sum_{k\\geq1}e^{\\nu(t)k}k^{1\/2}\\frac12\\frac{e^{+i\\widehat{\\vartheta}(0)}(S^{-1}w)(k)\\overline{\\widehat{y}(k)}\\!+\\!e^{-i\\widehat{\\vartheta}(0)}\\overline{(S^{-1}w)(k)}\\widehat{y}(k)}{|\\widehat{y}(k)|}.\n    \\end{aligned}\n\\end{equation*}\nWe have the bounds\n\\begin{equation*}\n    \\begin{aligned}\n    |Y_u|&\\leq 2\\frac{A_\\sigma}{R^3}\\Big|R^3\\Big(\\frac{2\\pi}{L(t)}\\Big)^3-1\\Big|\\sum_{k\\geq1}\\!\\!e^{\\nu(t)k}k^{1\/2}|(S^{-1}u)(k)|,\n    \\\\\n    |Y_v|&\\leq 2(1+A_\\mu)\\frac{|A_\\rho|}{R}\\Big|R\\frac{2\\pi}{L(t)}-1\\Big|\\!\\!\\sum_{k\\geq1}\\!\\!e^{\\nu(t)k}k^{1\/2}|(S^{-1}v)(k)|,\n    \\\\\n    |Y_w|&\\leq 2(1-A_\\mu)\\frac{|A_\\rho|}{R}\\Big|R\\frac{2\\pi}{L(t)}-1\\Big| \\frac32\\Big(\\frac34-\\log{2}\\Big)\\sum_{k\\geq1}e^{\\nu(t)k}k^{1\/2}|(S^{-1}w)(k)|.\n    \\end{aligned}\n\\end{equation*}\nNoticing that $(S^{-1}w)(k)=w(k)=1_{k=2}\\widehat{\\theta}(1)$ and using the inequalities given by Lemma \\ref{lemmaS}, we have\n\\begin{equation*}\n    \\begin{aligned}\n    |Y_u|&\\leq2\\frac{A_\\sigma}{R^3}\\Big|R^3\\Big(\\frac{2\\pi}{L(t)}\\Big)^3-1\\Big|C_{S}(A_\\mu,\\frac{|A_\\rho| R^2}{A_\\sigma})\\sum_{k\\geq2}e^{\\nu(t)k}k^{1\/2}|u(k)|,\n    \\\\\n    |Y_v|&\\leq2(1+A_\\mu)\\frac{|A_\\rho|}{R}\\Big|R\\frac{2\\pi}{L(t)}-1\\Big|C_{S}(A_\\mu,\\frac{|A_\\rho| R^2}{A_\\sigma})\\sum_{k\\geq2}e^{\\nu(t)k}k^{1\/2}|v(k)|,\n    \\\\\n    |Y_w|&\\leq 2^{1\/2}3(1-A_\\mu)\\frac{|A_\\rho|}{R}\\Big|R\\frac{2\\pi}{L(t)}-1\\Big| \\Big(\\frac34-\\log{2}\\Big)e^{2\\nu(t)}|\\widehat{\\theta}(1)|.\n    \\end{aligned}\n\\end{equation*}\nSince \n\\begin{equation*}\n    k(k^2-1)\\leq k^3,\\quad \\text{for }k\\geq2,\\qquad \\frac{(k^2-1)(k+1)}{k^{1\/2}(k+2)}\\leq (k+1)^{3\/2}, \\quad \\text{for }k\\geq 1,\n\\end{equation*}\nit holds that\n\\begin{equation*}\n    |u(k)|\\leq  k^3 \\left| \\widehat{\\theta}(k)\\right|, \\qquad |v(k)|\\leq (k+1)\\widehat{\\theta}(k+1).\n\\end{equation*}\nNow we use the inequalities for $L(t)$ from  \\eqref{Lestimates} (see Subsection \\ref{sec:Lestimate}) \nto obtain that\n\\begin{equation}\\label{Ybounds}\n    \\begin{aligned}\n    |Y_u|&\\leq 2\\frac{A_\\sigma}{R^3}C_{S}C_{40}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu} \\sum_{k\\geq2}e^{\\nu(t)k}k^{\\frac72}|\\widehat{\\theta}(k)|,\\\\\n    |Y_v|&\\leq2(1+A_\\mu)\\frac{|A_\\rho|}{R}\n    C_{S}\n    C_{39}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}  \n    \\sum_{k\\geq2}e^{\\nu(t)k}(k+1)^{3\/2}|\\widehat{\\theta}(k+1)|,\\\\\n    |Y_w|&\\leq 2^{1\/2}3(1-A_\\mu)\\frac{|A_\\rho|}{R}C_{39}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu} \\Big(\\frac34-\\log{2}\\Big)e^{2\\nu(t)}|\\widehat{\\theta}(1)|.\n    \\end{aligned}\n\\end{equation}\nAbove we wrote $C_{S}=C_{S}(A_\\mu,\\frac{|A_\\rho| R^2}{A_\\sigma})$.\nGoing back to \\eqref{auxx} with the bound \\eqref{Lbound} and the inequality\n\\begin{equation*}\n    k^{3\/2}(k^2-1)\\geq \\frac34 k^{7\/2},\\quad \\text{for }k\\geq 2,\n\\end{equation*}\nwe find using also \\eqref{CSbounds} that\n\\begin{equation}\\label{auxx2}\n    \\begin{aligned}\n    \\frac{d}{dt}\\|y\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}_\\nu}&\\leq \\nu'(t)\\|y\\|_{\\dot{\\mathcal{F}}^{\\frac32,1}_\\nu} -\\frac32\\frac{A_\\sigma}{R^3}\\sum_{k\\geq2}e^{\\nu(t)k}k^{7\/2}|\\widehat{y}(k)|\n    \\\\\n    &\\quad+C_{S}(A_\\mu,\\frac{|A_\\rho| R^2}{A_\\sigma})C_{37}^{-1}\\frac{1}{R}\\|N\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}_\\nu}+|Y_u|+|Y_v|+|Y_w|\n    .\n    \\end{aligned}\n\\end{equation}\nWe will next control the $k=1$ frequency in the dissipation part above.\nTo that end recall from \\eqref{frequencyrelation} that, if $\\|\\theta\\|_{\\mathcal{F}^{0,1}}\\leq \\frac12\\log{\\big(\\frac54\\big)}$, then we have \n\\begin{equation*}\n    2|\\widehat{\\theta}(1)|\\leq 2C_I(\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu})\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\sum_{k\\geq2}|\\widehat{\\theta}(k)|,\n\\end{equation*}\nwhich implies\n\\begin{equation*}\n\\begin{aligned}\n    \\|\\theta\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu} &=2  e^{\\nu(t)} |\\widehat{\\theta}(1)|+2\\sum_{k\\geq2}e^{\\nu(t)k}k^s|\\widehat{\\theta}(k)|\\\\\n    &\\leq 2\\left(C_I(\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu})\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}+1\\right)\\sum_{k\\geq2}e^{\\nu(t)k}k^s|\\widehat{\\theta}(k)|,\\qquad s\\geq0.\n\\end{aligned}\n\\end{equation*}  \nWe will find an analogous inequality in terms of $y$.\nSince $\\widehat{y}(1)=\\widehat{\\theta}(1)$, for $s\\geq0$ we have that\n\\begin{equation*}\n\\begin{aligned}\n    \\|y\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}&=2  e^{\\nu(t)} |\\widehat{\\theta}(1)|+2\\sum_{k\\geq2}e^{\\nu(t)k}k^s|\\widehat{y}(k)|\\\\\n    &\\leq 2C_I(\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu})\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\sum_{k\\geq2}e^{\\nu(t)k}k^s|\\widehat{\\theta}(k)|+2\\sum_{k\\geq2}e^{\\nu(t)k}k^s|\\widehat{y}(k)|,\n\\end{aligned}\n\\end{equation*}\nthus substituting $\\widehat{\\theta}(k)=(S\\widehat{y})(k)$ and using \\eqref{equivaux2} with \\eqref{sigmabound} and \\eqref{bessel.In.def}, we obtain\n\\begin{equation*}\n    \\begin{aligned}\n    \\|y\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}&\\leq 2C_I(\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu})\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu} e^{2\\nu(t)}2^s\\Big|\\frac{c(1)}{a(2)}\\Big||\\widehat{y}(1)|\\\\\n    &\\quad+2\\bigg(C_I(\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu})6\\frac{I_3\\Big(2\\sqrt{\\big|1\\!+\\!A_\\mu\\big|\\frac{|A_\\rho| R^2}{A_\\sigma}}\\Big)}{\\Big(\\big|1\\!+\\!A_\\mu\\big|\\frac{|A_\\rho| R^2}{A_\\sigma}\\Big)^{3\/2}}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}+1\\bigg)\\sum_{k\\geq2}e^{\\nu(t)k}k^s|\\widehat{y}(k)|.\n    \\end{aligned}\n\\end{equation*}\nSubtracting the $\\widehat{y}(1)$ term and using that  $e^{\\nu(t)}\\leq e^{\\nu_0}$ for all $t\\geq0$, we find\n\\begin{equation}\\label{ybound.imlicit}\n    \\begin{aligned}\n    \\|y\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}&\\leq 2C_y\\sum_{k\\geq2}e^{\\nu(t)k}k^s|\\widehat{y}(k)|, \\qquad s\\geq0,\n    \\end{aligned}\n\\end{equation}\nwhere using also \\eqref{coeffs} we define\n\\begin{equation}\\label{Cy}\n    \\begin{aligned}\n    C_y&=\\frac{C_I(\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu})6\\frac{I_3\\Big(2\\sqrt{\\big|1\\!+\\!A_\\mu\\big|\\frac{|A_\\rho| R^2}{A_\\sigma}}\\Big)}{\\Big(\\big|1\\!+\\!A_\\mu\\big|\\frac{|A_\\rho| R^2}{A_\\sigma}\\Big)^{3\/2}}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}+1}{1-C_I(\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}) 2^s e^{\\nu_0}\\frac{1}{4}\\big|1\\!-\\!A_\\mu\\big|\\frac{|A_\\rho| R^2}{A_\\sigma}\\Big(\\frac34-\\log{2}\\Big)\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}}.\n    \\end{aligned}\n\\end{equation}\nThis is the bound for $\\|y\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}$ that we will use in \\eqref{auxx2}.\n\n\n\nNotice that in \\eqref{Ybounds}, regarding $Y_w$, that $3 \\sqrt{2} \\left(\\frac{3}{4}-\\log(2)\\right) \\le 1.91\\le 2$.  Also in Lemma \\ref{lemmaS} we have $C_S \\ge 1$.  Thus, from \\eqref{Ybounds}, we have \n$$\n|Y_v|+|Y_w|\n\\leq\n    2\\left(1+|A_\\mu|\\right)\\frac{|A_\\rho|}{R}\n    C_{S}\n    C_{39}\n    e^{\\nu(t)}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}  \n    \\sum_{k\\geq1}e^{\\nu(t)k}k^{3\/2}|\\widehat{\\theta}(k)|,\n$$\nNow we go back to \\eqref{auxx2}, use \\eqref{ybound.imlicit}, and substitute in the bounds for $Y_u, Y_v, Y_w$ \\eqref{Ybounds} to obtain\n\\begin{equation*}\n\\begin{aligned}\n    \\frac{d}{dt}\\|y\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}_\\nu}&\\leq \\nu'(t)\\|y\\|_{\\dot{\\mathcal{F}}^{\\frac32,1}_\\nu} -\\frac34\\frac{A_\\sigma}{R^3}C_y^{-1}\\|y\\|_{\\dot{\\mathcal{F}}^{\\frac72,1}_\\nu}\n    \\\\\n    &\\quad+C_{S}\n    C_{37}^{-1}\\frac{1}{R}\\|N\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}_\\nu}\\\\\n    &\\quad\n    +2\\frac{A_\\sigma}{R^3}C_{S}\n    C_{40}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu} \\sum_{k\\geq2}e^{\\nu(t)k}k^{\\frac72}|\\widehat{\\theta}(k)|\n    \\\\\n    &\\quad \n    +\n    2\\left(1+|A_\\mu|\\right)\\frac{|A_\\rho|}{R}\n    C_{S}\n    C_{39}\n    e^{\\nu(t)}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}  \n    \\sum_{k\\geq1}e^{\\nu(t)k}k^{3\/2}|\\widehat{\\theta}(k)|,\n\\end{aligned}\n\\end{equation*}    \nwhere we have combined the bounds for $Y_v$ and $Y_w$ as above.\nThe reverse inequalities \\eqref{CSbounds} and the embeddings \\eqref{embed} give that\n\\begin{equation}\\label{balance}\n\\begin{aligned}\n    \\frac{d}{dt}\\|y\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}_\\nu}&\\leq  \\frac{A_\\sigma}{R^3}\\bigg(-\\frac34C_y^{-1}+\\nu'(t)\\frac{R^3}{A_\\sigma}+2\\big(C_{S}\\big)^2C_{40}\\|y\\|_{\\dot{\\mathcal{F}}^{0,1}_\\nu} \\\\\n    &\\quad+2(1+|A_\\mu|)\\frac{|A_\\rho|R^2}{A_\\sigma} \\big(C_{S}\\big)^2C_{39}e^{\\nu(t)}\\|y\\|_{\\dot{\\mathcal{F}}^{0,1}_\\nu}\\bigg)\\|y\\|_{\\dot{\\mathcal{F}}^{\\frac72,1}_\\nu}\\\\\n    &+ \n    C_{S} C_{37}^{-1}\\frac{1}{R}\\|N\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}_\\nu}.\n\\end{aligned}\n\\end{equation}    \nNext, we will use \\eqref{Nbound} to control the nonlinear term $\\|N\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}_\\nu}$. Together with \\eqref{Lbound}, we obtain\n\\begin{equation*}\n    \\begin{aligned}\n        \\|N\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}_\\nu}&\\leq \\frac{A_\\sigma}{R^2}C_{35}C_{37}^{-2}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{\\frac72,1}_\\nu}\n        +|A_\\rho|e^{\\nu(t)}C_{36}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{\\frac72,1}_\\nu}.\n    \\end{aligned}\n\\end{equation*}\nThus using also \\eqref{CSbounds} we have\n\\begin{equation}\\label{nonlinearaux}\n    \\begin{aligned}\n        \\|N\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}_\\nu}&\\leq \\frac{A_\\sigma}{R^2}\\big(C_{S}\\big)^2\\Big(C_{35}C_{37}^{-2}\n        +\\frac{|A_\\rho|R^2}{A_\\sigma}e^{\\nu(t)}C_{36}\\Big)\\|y\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}_\\nu}\\|y\\|_{\\dot{\\mathcal{F}}^{\\frac72,1}_\\nu}.\n    \\end{aligned}\n\\end{equation}\nSubstitution of \\eqref{nonlinearaux}  into \\eqref{balance}, and using once more \\eqref{embed}, provides that\n\\begin{equation*}\n\\begin{aligned}\n    \\frac{d}{dt}\\|y\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}_\\nu}&\\leq - \\frac{A_\\sigma}{R^3}\\bigg(\\frac34C_y^{-1}-\\nu'(t)\\frac{R^3}{A_\\sigma}-2\\big(C_{S}\\big)^2C_{40}\\|y\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}_\\nu} \\\\\n    &-2(1+|A_\\mu|)\\frac{|A_\\rho|R^2}{A_\\sigma} \\big(C_{S}\\big)^2C_{39}e^{\\nu(t)}\\|y\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}_\\nu} \\\\   &-\\big(C_{S}\\big)^3C_{37}^{-1}\\Big(C_{35}C_{37}^{-2}\\!+\\!\\frac{|A_\\rho|R^2}{A_\\sigma}e^{\\nu(t)}C_{36}\\Big)\\|y\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}_\\nu}\n    \\bigg)\\|y\\|_{\\dot{\\mathcal{F}}^{\\frac72,1}_\\nu}.\n\\end{aligned}\n\\end{equation*}  \nSince $C_y(\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu})$ in \\eqref{Cy}, $C_{35}(\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu})$ and $C_{36}(\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu})$ in \\eqref{C35C36}, $\\big(C_{37}(\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu})\\big)^{-1}$ in \\eqref{C37C38}, $C_{39}(\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu})$ and $C_{40}(\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu})$ in \\eqref{C39C40}, are increasing functions of the argument, we can bound all of them by evaluating at the bigger quantity $C_S\\|y\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}_\\nu}$ since \n$\\|\\theta\\|_{\\dot{\\mathcal{F}}^{0,1}_\\nu} \\le \\|\\theta\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}_\\nu}\\le C_S\\|y\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}_\\nu}$.\nHere we suppress the dependency on $A_\\mu$, $|A_\\rho|R^2\/A_\\sigma$ from $C_S$. \n\nFor clarity of notation, in formulas \\eqref{dissipation} and \\eqref{C41} below, we understand that all of these functions are evaluated at $C_S\\|y\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}_\\nu}$.\nWe conclude that\n\\begin{equation}\\label{balance2}\n\\begin{aligned}\n    \\frac{d}{dt}\\|y\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}_\\nu}&\\leq -\\frac{A_\\sigma}{R^3} \\mathcal{D} \\|y\\|_{\\dot{\\mathcal{F}}^{\\frac72,1}_\\nu},\n\\end{aligned}\n\\end{equation}\nwith\n\\begin{equation}\\label{dissipation}\n    \\begin{aligned}\n       \\mathcal{D}\\Big(\\|y\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}_\\nu},\\frac{|A_\\rho|R^2}{A_\\sigma},\n       A_\\mu,\\frac{A_\\sigma}{R^3},\\nu\\Big)&=\\frac34C_y^{-1}-\\nu'(t)\\frac{R^3}{A_\\sigma}-C_{41}\\|y\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}_\\nu},\n    \\end{aligned}\n\\end{equation}\nwhere\n\\begin{multline}\\label{C41}\n    C_{41}=\n    \\\\\n    C_S^2\\Big(\n       2C_{40}+2(1+|A_\\mu|)\\frac{|A_\\rho|R^2}{A_\\sigma}C_{39}e^{\\nu(t)}+C_{S}C_{37}^{-1}\\Big(C_{35}C_{37}^{-2}\\!+\\!\\frac{|A_\\rho|R^2}{A_\\sigma}e^{\\nu(t)}C_{36}\\Big)\\Big).\n\\end{multline}\nNote that we can initially choose $\\nu_0>0$ in \\eqref{nu} to be arbitrarily small, and that $\\nu'(t) = \\frac{\\nu_0}{(1+t)^2}\\le \\nu_0$.  \n\nFinally, suppose that the following condition holds initially\n\\begin{equation}\\label{dissipationcond}\n     \\frac34C_y^{-1}-C_{41}\\|y\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}_\\nu}>0,\n\\end{equation}\nwhere $C_y$ was defined in \\eqref{Cy}.\nFor $\\nu_0$ small enough, using \\eqref{balance2} and the fact that $C_{41}$ decreases as $\\|y\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}_\\nu}$ decreases, then this condition will be propagated in time.   Thus it holds that\n\\begin{equation}\\label{auxbal}\n    \t\\|y\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}_\\nu}(t)+ \\frac{A_\\sigma}{R^3}  \\mathcal{D}\n    \n      \n       \\!\\int_0^t\\! \\|y\\|_{\\dot{\\mathcal{F}}^{\\frac72,1}_\\nu}(\\tau) d\\tau \\leq \\|y_0\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}}.\n\\end{equation}\nAbove $\\mathcal{D}=\\mathcal{D}\\Big(\\|y_0\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}_\\nu},\\frac{|A_\\rho|R^2}{A_\\sigma}, A_\\mu,\\frac{A_\\sigma}{R^3},\\nu\\Big)$.\nNow since $C_y$ and $C_{41}$ are monotonically increasing in $\\|y\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}_\\nu}$, the positivity condition \\eqref{dissipationcond} is equivalent to a medium-size condition on the initial data\n\\begin{equation*}\n    \\|y_0\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}}<h\\Big(\\frac{|A_\\rho|R^2}{A_\\sigma}, A_\\mu\\Big),\n\\end{equation*}\nwhere the function $h$ is implicitly defined via \\eqref{dissipationcond}. \nRecalling the bound for the inverse relation $\\widehat{\\theta}(k)=(S\\widehat{y})(k)$ in \\eqref{CSbounds}, and denoting \\begin{equation}\\label{Kdef}\n    K\\Big(\\frac{|A_\\rho|R^2}{A_\\sigma}, A_\\mu\\Big)=C_S\\big(A_\\mu,\\frac{|A_\\rho|R^2}{A_\\sigma}\\big) h\\Big(\\frac{|A_\\rho|R^2}{A_\\sigma}, A_\\mu\\Big),\n\\end{equation}\nwe obtain the medium-size condition for $\\theta_0$ given in \\eqref{condition}.  This completes the proof of \\eqref{condition} and \\eqref{estimatef12}.  The proof of \\eqref{decay} follows from applying \\eqref{embed} to \\eqref{balance2}.\n\n\n\nTo perform the estimate for $\\widehat{\\vartheta}(0)$ from \\eqref{zeroBoundUniform}, we use \\eqref{eqcurve3}$_1$ to obtain\n\\begin{equation*}\n    \\widehat{\\vartheta}_t(0)=\\frac{2\\pi}{L(t)}\\widehat{T}*\\widehat{(1+\\theta_\\alpha)}(0).\n\\end{equation*}\nNote that $\\widehat{U_\\alpha}(0)=0$.  We use the splitting for $T$ from \\eqref{Tsplit}, and notice from \\eqref{u0t0} with \\eqref{T0} that \n\\begin{equation*}\n    \\widehat{T_0}(k)=0 \\text{ for }|k|\\neq 1, \\qquad |\\widehat{T_0}(k)|\\leq \\frac{|A_\\rho|}2 \\text{ for }|k|= 1,\n\\end{equation*}\nthus $\\|T_0\\|_{\\mathcal{F}^{0,1}}\\leq A_\\rho$. Then we have\n\\begin{equation*}\n    |\\widehat{\\vartheta}(0)|\n    \\leq |\\widehat{\\vartheta}_0(0)|+\n    \\int_0^t \\frac{2\\pi}{L(\\tau)}\\Big(A_\\rho\\|\\theta\\|_{\\dot{\\mathcal{F}}^{1,1}}+\\big(\\|T_1\\|_{\\mathcal{F}^{0,1}}+\\|T_{\\geq2}\\|_{\\mathcal{F}^{0,1}}\\big)\\big(1+\\|\\theta\\|_{\\dot{\\mathcal{F}}^{1,1}}\\big)\\Big) d \\tau.\n\\end{equation*}\nThe estimates \\eqref{T1aux} and \\eqref{T2aux} with $s=0$ in Subsection \\ref{sec:NonLinearEst}, together with the bounds for $L(t)$ in \\eqref{Lbound}, give \n\\begin{equation*}\n    \\begin{aligned}\n    \\|T_1\\|_{\\mathcal{F}^{0,1}}&\\leq \\frac{A_\\sigma}{R^2}\\Big(2\\frac{|A_\\rho|R^2}{A_\\sigma}e^{\\nu(t)}\n    \\|\\theta\\|_{\\dot{\\mathcal{F}}^{1,1}}\\!+\\!C_{37}^{-2}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{2,1}}\\!+\\!2\\frac{|A_\\rho|R^2}{A_\\sigma}\\big((1\\!+\\! 2|A_\\mu|)\\!+\\!2{C_{\\mR}}\\big)e^{\\nu(t)}\\|\\theta\\|_{\\mathcal{F}^{0,1}}\\Big),\n    \\\\\n    \\|T_{\\geq2}\\|_{\\mathcal{F}^{0,1}}&\\leq \\frac{A_\\sigma}{R^2}\\Big(\\frac{|A_\\rho|R^2}{A_\\sigma}C_{34}e^{\\nu(t)}\\|\\theta\\|_{\\mathcal{F}^{0,1}}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{1,1}}\\!+\\!C_{37}^{-2}C_{33}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{1,1}}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{2,1}}\\Big).\n    \\end{aligned}\n\\end{equation*}\nTherefore, reordering and using \\eqref{auxbal}, \\eqref{interp} and \\eqref{embed}, we conclude that\n\\begin{equation*}\n    \\begin{aligned}\n    |\\widehat{\\vartheta}(0)|&\\leq |\\widehat{\\vartheta}_0(0)|+C_{42}\\|\\theta_0\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}},\n    \\end{aligned}\n\\end{equation*}\nwhere \n\\begin{multline}    \\label{C42}\n    C_{42}=\n    \\\\ \n    \\frac{1}{\\mathcal{D}}C_S^2C_{37}^{-1}\\Bigg(\\frac{|A_\\rho|R^2}{A_\\sigma}+C_{37}^{-2}+2\\frac{|A_\\rho|R^2}{A_\\sigma}e^{\\nu(t)}\\Big(1+\\big((1+2|A_\\mu|)+2{C_{\\mR}}\\big)\\Big)\n    \\\\+C_S^2\\|\\theta_0\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}}\\bigg(C_{37}^{-2}+C_{37}^{-2}C_{33}(1+C_S\\|\\theta_0\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}})\n    \\\\\n    +\n    2\\frac{|A_\\rho|R^2}{A_\\sigma}e^{\\nu(t)}\\Big(1+\\big((1+2|A_\\mu|)+2{C_{\\mR}}\\big)+\\frac{1}{2}C_{34}(1+C_S\\|\\theta_0\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}})\\Big)\\bigg)\\Bigg).\n\\end{multline} \nIn $C_{42}$ above  the quantities ($\\mathcal{D}$ in \\eqref{dissipation}, $C_{37}$ in \\eqref{C37C38}, $C_{33}$ and $C_{34}$ in \\eqref{C33C34}) are evaluated at\n$C_S\\|\\theta_0\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}}$, with $C_S$ defined in \\eqref{CSbounds}.\nThis completes the proof of \\eqref{zeroBoundUniform}.\n\\end{proof}\n\n\nWe will now prove Proposition \\ref{diagonalization}.\n\n\\begin{proof}[Proof of Proposition \\ref{diagonalization}]\nBecause $S^{-1}$ has trivial kernel, it suffices to prove that $S$ is a right inverse of $S^{-1}$, i.e., that\n\\begin{equation}\\label{rightinverse.proof}\n    \\sum_{j\\geq1}S^{-1}_{k,j}S_{j,m}=\\delta_{k,m},\n\\end{equation}\nwhere $\\delta_{k,m}$ is the Kronecker delta. \nThe cases $k=1$ and $k=2, m=1$ are straightforward. For the rest of cases, we have that\n\\begin{equation*}\n    \\begin{aligned}\n    \\sum_{j\\geq1}S^{-1}_{k,j}S_{j,m}&=\\sum_{j\\geq1}(-1)^{j-k} \\delta_{j\\geq k}\\prod_{l=1}^{j-k} \\frac{b(k-1+l)}{a(k)-a(k+l)}\\delta_{m\\geq j}\\prod_{n=1}^{m-j}\\frac{b(j-1+n)}{a(j-1+n)-a(m)}\\\\\n    &=\\sum_{j=k}^m (-1)^{j-k}\\prod_{n=1}^{m-j}\\prod_{l=1}^{j-k} \\frac{b(k-1+l)b(j-1+n)}{(a(k)-a(k+l))(a(j-1+n)-a(m))}.\n    \\end{aligned}\n\\end{equation*}\nIf $m=k$, then clearly $\\sum_{j\\geq1}S^{-1}_{k,j}S_{j,m}=1$. Now, we notice that\n\\begin{equation*}\n    \\begin{aligned}\n    \\prod_{n=1}^{m-j}\\prod_{l=1}^{j-k} b(k\\!-\\!1\\!+\\!l)b(j\\!-\\!1\\!+\\!n)&=b(k)b(k\\!+\\!1)\\dots b(j\\!-\\!2)b(j\\!-\\!1)b(j)b(j\\!+\\!1)\\dots b(m\\!-\\!1)\\\\\n    &=\\prod_{n=0}^{m-k-1}b(k+n)\n    \\end{aligned}\n\\end{equation*}\nis independent of $j$, and thus comes out of the sum as\n\\begin{equation*}\n    \\begin{aligned}\n    \\sum_{j\\geq1}S^{-1}_{k,j}S_{j,m}=\\prod_{n=0}^{m-k-1}b(k+n) \\sum_{j=k}^m (-1)^{j-k}A(j),\n    \\end{aligned}\n\\end{equation*}\nwhere we define\n\\begin{equation*}\n    A(j)=\\prod_{n=1}^{m-j}\\prod_{l=1}^{j-k} \\frac{1}{(a(k)\\!-\\!a(k\\!+\\!l))(a(j\\!-\\!1\\!+\\!n)\\!-\\!a(m))}.\n\\end{equation*}\nChanging $n\\leftarrow j-k-1+n$, it becomes\n\\begin{equation*}\n    A(j)=\\prod_{n=j-k}^{m-k-1}\\frac{1}{a(k+n)\\!-\\!a(m)}\\prod_{l=1}^{j-k} \\frac{1}{a(k)\\!-\\!a(k\\!+\\!l)}.\n\\end{equation*}\nThus, multiplying and dividing by the product of $a(k+n)-a(m)$ from $n=0$ to $n=j-k-1$, we rewrite it as follows\n\\begin{equation*}\n    \\begin{aligned}\n    A(j)=\\prod_{n=0}^{m-k-1}\\frac{1}{a(k+n)-a(m)}\\prod_{l=1}^{j-k}\\frac{a(k+l-1)-a(m)}{a(k)-a(k+l)}.\n    \\end{aligned}\n\\end{equation*}\nSince the first product does not depend on $j$, we obtain that\n\\begin{equation*}\n    \\begin{aligned}\n    \\sum_{j\\geq1}S^{-1}_{k,j}S_{j,m}=\\prod_{n=0}^{m-k-1}\\frac{b(k+n)}{a(k+n)-a(m)}\\sum_{j=k}^m (-1)^{j-k}\\prod_{l=1}^{j-k}\\frac{a(k+l-1)-a(m)}{a(k)-a(k+l)}.\n    \\end{aligned}\n\\end{equation*}\nChanging the index $j\\leftarrow j-k$ and denoting $\\tilde{m}=m-k$, we are left to show that, for all $\\tilde{m}\\geq1$,\n\\begin{equation*}\n    \\begin{aligned}\n    \\sum_{j=0}^{\\tilde{m}} (-1)^{j}\\prod_{l=1}^{j}\\frac{a(k+l-1)-a(\\tilde{m}+k)}{a(k)-a(k+l)}=0.\n    \\end{aligned}\n\\end{equation*}\nWriting the sum with a common denominator, the identity that we need to prove is\n\\begin{equation*}\n    \\begin{aligned}\n    W=\\sum_{j=0}^{\\tilde{m}} (-1)^{j}\\prod_{l=1}^{j}\\big(a(k+l-1)-a(\\tilde{m}+k)\\big)\\prod_{l=j+1}^{\\tilde{m}}\\big(a(k)-a(k+l)\\big)=0.\n    \\end{aligned}\n\\end{equation*}\nWe add the first two terms in the sum by noticing that they only differ in one term, which partially cancels as\n\\begin{equation*}\n    \\begin{aligned}\n    W&=-\\big(a(k\\!+\\!1)\\!-\\!a(k\\!+\\!\\tilde{m})\\big)\\prod_{l=2}^{\\tilde{m}}\\big(a(k)\\!-\\!a(k\\!+\\!l)\\big)\\\\\n    &+\n    \\sum_{j=2}^{\\tilde{m}} (-1)^{j}\\prod_{l=1}^{j}\\big(a(k+l-1)-a(\\tilde{m}+k)\\big)\\prod_{l=j+1}^{\\tilde{m}}\\big(a(k)-a(k+l)\\big).\n    \\end{aligned}\n\\end{equation*}\nWe now realize that we can repeat the same step with the new first two terms to obtain that\n\\begin{equation*}\n    \\begin{aligned}\n    W&=\\big(a(k\\!+\\!1)\\!-\\!a(k\\!+\\!\\tilde{m})\\big)\\big(a(k\\!+\\!2)\\!-\\!a(k\\!+\\!\\tilde{m})\\big)\\prod_{l=3}^{\\tilde{m}}\\big(a(k)\\!-\\!a(k\\!+\\!l)\\big)\\\\\n    &+\n    \\sum_{j=3}^{\\tilde{m}} (-1)^{j}\\prod_{l=1}^{j}\\big(a(k+l-1)-a(\\tilde{m}+k)\\big)\\prod_{l=j+1}^{\\tilde{m}}\\big(a(k)-a(k+l)\\big).\n    \\end{aligned}\n\\end{equation*}\nWe can continue the process to find that\n\\begin{equation*}\n    \\begin{aligned}\n    &W=\\!(-1)^r\\big(a(k\\!+\\!1)\\!-\\!a(k\\!+\\!\\tilde{m})\\big)\\big(a(k\\!+\\!2)\\!-\\!a(k\\!+\\!\\tilde{m})\\big)\\\\ &\\hspace{3cm}\\dots\\big(a(k\\!+\\!r)\\!-\\!a(k\\!+\\!\\tilde{m})\\big)\\!\\!\\!\\!\n    \\prod_{l=r+1}^{\\tilde{m}}\\!\\!\\!\\big(a(k)\\!-\\!a(k\\!+\\!l)\\big)\\\\\n    &\\hspace{1cm}+\n    \\sum_{j=r+1}^{\\tilde{m}} (-1)^{j}\\prod_{l=1}^{j}\\big(a(k+l-1)-a(\\tilde{m}+k)\\big)\\prod_{l=j+1}^{\\tilde{m}}\\big(a(k)-a(k+l)\\big).\n    \\end{aligned}\n\\end{equation*}\nThus for $r=\\tilde{m}-1$, we conclude that\n\\begin{equation*}\n    \\begin{aligned}\n    W\n    &=\\!(-1)^{\\tilde{m}-1}\\big(a(k\\!+\\!1)\\!-\\!a(k\\!+\\!\\tilde{m})\\big)\\dots \\big(a(k\\!+\\!\\tilde{m}\\!-\\!1)\\!-\\!a(k\\!+\\!\\tilde{m})\\big)\\big(a(k)\\!-\\!a(k\\!+\\!\\tilde{m})\\big)\\\\\n    &+ (-1)^{\\tilde{m}}\\prod_{l=1}^{\\tilde{m}}\\big(a(k+l-1)-a(\\tilde{m}+k)\\big)=0.\n    \\end{aligned}\n\\end{equation*}\nThis proves \\eqref{rightinverse.proof}.\n\nWe next prove that $S^{-1}$ in \\eqref{Sm1} diagonalizes \\eqref{linearZ}. Notice that $S^{-1}_{k,k}=1$ and $S^{-1}_{k,j}=0$ for $j<k$, $k\\geq2$. Then, for $k\\geq3$, we can write\n\\begin{equation*}\n    \\begin{aligned}\n    y_t(k)=\\sum_{j\\geq k}S^{-1}_{k,j}z_t(j)=-a(k)S^{-1}_{k,k}z(k)+\\sum_{j\\geq k+1}\\Big(-a(j)S^{-1}_{k,j}+b(j-1)S^{-1}_{k,j-1}\\Big)z(j).\n    \\end{aligned}\n\\end{equation*}\nFinally, notice that \n\\begin{equation*}\n    -a(j)S^{-1}_{k,j}+b(j-1)S^{-1}_{k,j-1}=-a(k)S^{-1}_{k,j},\\qquad j-1\\geq k\\geq 2,\n\\end{equation*}\nprecisely when $S^{-1}_{k,j}$ is given by \\eqref{Sm1}. The case $k=1$ is trivial, while for $k=2$ it holds that\n\\begin{equation*}\n    \\begin{aligned}\n    y_t(2)&=\\sum_{j\\geq 2}S^{-1}_{2,j}z_t(j)=c(1)z(1)\\!-\\!a(2)z(2)\\!+\\!\\sum_{j\\geq 3}\\!\\Big(\\!-a(j)S^{-1}_{2,j}\\!+\\!b(j\\!-\\!1)S^{-1}_{2,j-1}\\Big)z(j)\\\\\n    &=-a(2)\\frac{c(1)}{a(2)}z(1)-a(2)\\sum_{j\\geq2}S^{-1}_{2,j}z(j)=-a(2)y(2).\n    \\end{aligned}\n\\end{equation*}\nThis completes the proof.\n\\end{proof}\n\n\nWe are now ready to prove Lemma \\ref{lemmaS}.\n\n\n\n\\begin{proof}[Proof of Lemma \\ref{lemmaS}]\nDenote for $j \\ge k \\ge 2$\n\\begin{equation*}\n\\begin{aligned}\n    \\beta(k,j)=(-1)^{j-k}\\prod_{m=1}^{j-k} \\frac{b(k-1+m)}{a(k)-a(k+m)}.\n\\end{aligned}\n\\end{equation*}\nPlugging in the expressions for $b(k)$ and $a(k)$  in \\eqref{coeffs}, we find that\n\\begin{equation*}\n    \\begin{aligned}\n    \\beta(k,j)\\!=\\!\\Big(\\big(1\\!+\\!A_\\mu\\big)\\frac{A_\\rho R^2}{A_\\sigma}\\Big)^{j-k}\\!\\!\\!\\!e^{-i(j-k)\\widehat{\\vartheta}(0)}\\!\\!\\prod_{m=1}^{j-k}\\!\\!\\frac{(k\\!+\\!m)^2(k\\!-\\!2\\!+\\!m)}{(k\\!-\\!1\\!+\\!m)(k\\!+\\!1\\!+\\!m)}\\frac{1}{k^3\\!-\\!(k\\!+\\!m)^3\\!+\\!m}.\n    \\end{aligned}\n\\end{equation*}\nThen, we can write \n\\begin{equation}\\label{equivaux}\n    \\begin{aligned}\n    \\|y\\|_{\\ell^1}&=\\sum_{k\\geq1}|y(k)|\\leq \\Big(1+\\Big|\\frac{c(1)}{a(2)}\\Big|\\Big)|z(1)|+\\sum_{k\\geq2}\\sum_{j\\geq k}|\\beta(k,j)||z(j)|\\\\\n    &=\\Big(1+\\frac{1}{4}\\big|1\\!-\\!A_\\mu\\big|\\frac{|A_\\rho| R^2}{A_\\sigma}\\Big(\\frac34-\\log{2}\\Big)\\Big)|z(1)|+\\sum_{l\\geq2}\\sum_{j=0}^{l-2}|\\beta(l-j,l)||z(l)|\\\\\n    &\\leq \\Big(1+\\frac{1}{4}\\big|1\\!-\\!A_\\mu\\big|\\frac{|A_\\rho| R^2}{A_\\sigma}\\Big(\\frac34-\\log{2}\\Big)\\Big)|z(1)|+\\sum_{l\\geq2}\\lambda(l)|z(l)|,\n    \\end{aligned}\n\\end{equation}\nwhere\n\\begin{equation*}\n    \\lambda(l)=\\sum_{j=0}^{l-2}|\\beta(l-j,l)|.\n\\end{equation*}\nIn the following we will estimate $\\lambda(l)$ to control $\\beta$.\n\nThe identity $k^3-(k+m)^3+m=-m\\big(m^2+3m k+3k^2-1\\big)$ gives the bound\n\\begin{equation}\\label{index.lower}\n    |(l-j)^3-(l-j+m)^3+m|\\geq m\\big(m^2+2m(l-j)+(l-j)^2\\big)=m(m+l-j)^2.\n\\end{equation}\nThus\n\\begin{equation}\\label{betabound}\n    \\begin{aligned}\n    |\\beta(l-j,l)|&\\leq \\Big|\\big(1\\!+\\!A_\\mu\\big)\\frac{A_\\rho R^2}{A_\\sigma}\\Big|^{j}\\prod_{m=1}^{j}\\frac{l-j\\!-\\!2\\!+\\!m}{m(l-j\\!-\\!1\\!+\\!m)(l-j\\!+\\!1\\!+\\!m)}\\\\\n    &\\leq \\Big|\\big(1\\!+\\!A_\\mu\\big)\\frac{A_\\rho R^2}{A_\\sigma}\\Big|^{j}\\prod_{m=1}^{j}\\frac{1}{m(l-j\\!+\\!1\\!+\\!m)}\\\\\n    &\\leq \\Big(\\big|\\!+\\!A_\\mu\\big|\\frac{|A_\\rho| R^2}{A_\\sigma}\\Big)^{j}\\prod_{m=1}^{j}\\frac{1}{m(m+3)},\n    \\end{aligned}\n\\end{equation}\nwhere the last step is due to the fact that $j\\leq l-2$. Therefore, we have that\n\\begin{equation*}\n    \\begin{aligned}\n    \\lambda(l)&\\leq \\sum_{j=0}^{l-2}\\Big(\\big|1\\!+\\!A_\\mu\\big|\\frac{|A_\\rho| R^2}{A_\\sigma}\\Big)^j\\frac{6}{j!(j+3)!} \\leq \\sum_{j=0}^{\\infty}\\Big(\\big|1\\!+\\!A_\\mu\\big|\\frac{|A_\\rho| R^2}{A_\\sigma}\\Big)^j\\frac{6}{j!(j+3)!}\\\\\n    &=6\\frac{I_3\\Big(2\\sqrt{\\big|1\\!+\\!A_\\mu\\big|\\frac{|A_\\rho| R^2}{A_\\sigma}}\\Big)}{\\Big(\\big|1\\!+\\!A_\\mu\\big|\\frac{|A_\\rho| R^2}{A_\\sigma}\\Big)^{3\/2}},\n    \\end{aligned}\n\\end{equation*}\nwhere $I_3$ is the modified Bessel function of the first kind of order three as in \\eqref{bessel.In.def}.\nWe conclude thereby from \\eqref{equivaux} that\n\\begin{equation*}\n    \\begin{aligned}\n    \\|y\\|_{\\ell^1}&\\leq\n    \\Big(1\\!+\\!\\frac{1}{4}\\big|1\\!-\\!A_\\mu\\big|\\frac{|A_\\rho| R^2}{A_\\sigma}\\Big(\\frac34-\\log{2}\\Big)\\Big)|z(1)|+6\\frac{I_3\\Big(2\\sqrt{\\big|1\\!+\\!A_\\mu\\big|\\frac{|A_\\rho| R^2}{A_\\sigma}}\\Big)}{\\Big(\\big|1\\!+\\!A_\\mu\\big|\\frac{|A_\\rho| R^2}{A_\\sigma}\\Big)^{3\/2}}\\sum_{l\\geq2}|z(l)|\\\\\n    &\\leq C_{S}(A_\\mu,\\frac{|A_\\rho| R^2}{A_\\sigma})\\|z\\|_{\\ell^1},\n    \\end{aligned}\n\\end{equation*}\nwith $C_{S}$ given in \\eqref{CSbounds}. \n\nWe proceed to prove the reverse direction, $z=S y$. For $j \\ge k \\ge 2$, denote\n\\begin{equation*}\n\\begin{aligned}\n    \\alpha(k,j)=\\prod_{m=1}^{j-k} \\frac{b(k-1+m)}{a(k-1+m)-a(j)}\n.\n\\end{aligned}\n\\end{equation*}\nRecalling \\eqref{coeffs}, it becomes\n\\begin{equation*}\n    \\begin{aligned}\n    \\alpha(k,j)&=(-1)^{j-k}\\Big(\\big|1\\!+\\!A_\\mu\\big|\\frac{|A_\\rho| R^2}{A_\\sigma}\\Big)^{j-k}e^{-i(j-k)\\widehat{\\vartheta}(0)}\\\\\n    &\\quad\\cdot\\prod_{m=1}^{j-k}\\frac{(k\\!+\\!m)^2(k\\!-\\!2\\!+\\!m)}{(k\\!-\\!1\\!+\\!m)(k\\!+\\!1\\!+\\!m)}\\frac{1}{(k\\!-\\!1\\!+\\!m)^3\\!-\\!j^3\\!+\\!j\\!-\\!k\\!+\\!1\\!-\\!m}.\n    \\end{aligned}\n\\end{equation*}\nNext, proceeding as in \\eqref{equivaux}, we have\n\\begin{equation}\\label{equivaux2}\n    \\begin{aligned}\n    \\|z\\|_{\\ell^1}&=\\sum_{k\\geq1}|z(k)|\\leq \\big(1+\\Big|\\frac{c(1)}{a(2)}\\Big|\\big)|y(1)|+\\sum_{k\\geq2}\\sum_{j\\geq k}|\\alpha(k,j)||y(j)|\\\\\n    &\\leq \\Big(1+\\frac14\\big|1\\!-\\!A_\\mu\\big|\\frac{|A_\\rho| R^2}{A_\\sigma}\\Big(\\frac34-\\log{2}\\Big)\\Big)|y(1)|+\\sum_{l\\geq2}\\sigma(l)|y(l)|,\n    \\end{aligned}\n\\end{equation}\nwhere\n\\begin{equation*}\n    \\sigma(l)=\\sum_{j=0}^{l-2}|\\alpha(l-j,l)|.\n\\end{equation*}\nThe inequality below follows from \\eqref{index.lower} with $M+L-J = j$ and $L-J=m+k-1$\n\\begin{equation*}\n    \\begin{aligned}\n    |(k-1+m)^3-j^3+j-k+1-m|\\geq (j-k+1-m)j^2.\n    \\end{aligned}\n\\end{equation*}\nUsing the inequality above,  and $m \\le j \\le l -2$, we obtain that\n\\begin{equation*}\n    \\begin{aligned}\n    |\\alpha(l-j,l)|&\\leq \\Big|\\big(1\\!+\\!A_\\mu\\big)\\frac{A_\\rho R^2}{A_\\sigma}\\Big|^{j}\\prod_{m=1}^{j}\\frac{(l-j\\!+\\!m)^2(l-j-2+m)}{(l-j\\!-\\!1\\!+\\!m)(l-j\\!+\\!1\\!+\\!m)}\\frac{1}{l^2(j\\!+\\!1\\!-\\!m)}\\\\\n    &\\leq \\Big|\\big(1\\!+\\!A_\\mu\\big)\\frac{A_\\rho R^2}{A_\\sigma}\\Big|^{j}\\prod_{m=1}^{j}\\frac{1}{m(l-j\\!+\\!1\\!+\\!m)}\\\\\n    &\\leq \\Big(\\big|1\\!+\\!A_\\mu\\big|\\frac{|A_\\rho| R^2}{A_\\sigma}\\Big)^{j}\\prod_{m=1}^{j}\\frac{1}{m(m+3)},\n    \\end{aligned}\n\\end{equation*}\nwhich coincides with the bound found for $\\beta$ in \\eqref{betabound}. Therefore, \n\\begin{equation}\\label{sigmabound}\n    \\begin{aligned}\n    \\sigma(l)&\\leq6\\frac{I_3\\Big(2\\sqrt{\\big|1\\!+\\!A_\\mu\\big|\\frac{|A_\\rho| R^2}{A_\\sigma}}\\Big)}{\\Big(\\big|1\\!+\\!A_\\mu\\big|\\frac{|A_\\rho| R^2}{A_\\sigma}\\Big)^{3\/2}}.\n    \\end{aligned}\n\\end{equation}\nAbove we recall $I_3$ from \\eqref{bessel.In.def}.  Then going back to \\eqref{equivaux2} we have\n\\begin{equation*}\n    \\begin{aligned}\n    \\|z\\|_{\\ell^1}&\\leq \n     C_S\\left(A_\\mu,\\frac{|A_\\rho| R^2}{A_\\sigma}\\right)\\|y\\|_{\\ell^1},\n    \\end{aligned}\n\\end{equation*}\nwith $C_S$ defined in \\eqref{CSbounds}.   This completes the proof.\n\\end{proof}\n\n\n\n\n\n\n\\subsection{Nonlinear Estimates in $\\dot{\\mathcal{F}}^{s,1}_\\nu$} \\label{sec:NonLinearEst}\nWe recall that the nonlinear terms \\eqref{system} are given by\n\\begin{equation*}\n    N(\\alpha)=N_1+N_2+N_3,\n\\end{equation*}\nwith\n\\begin{equation}\\label{N1N2N3}\n\\begin{aligned}\n    N_1=(U_{\\geq 2})_\\alpha(\\alpha),\\quad N_2=T_{\\geq 2}(\\alpha)(1+\\theta_\\alpha(\\alpha)),\\quad\n    N_3=T_1(\\alpha)\\theta_\\alpha(\\alpha),\n\\end{aligned}\n\\end{equation}\nwhere $U$ and $T$ are described in \\eqref{Usplit} and \\eqref{Tsplit}.\n\nWe will first perform the estimates for $N_1$ for $s\\ge 0$.  From \\eqref{Usplit}, we have\n\\begin{multline}\\label{N1f0}\n    \\|N_1\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}=\\|U_{\\geq2}\\|_{\\dot{\\mathcal{F}}^{1+s,1}_\\nu}\n    \\\\\n    \\!\\leq\\! \\frac{\\pi}{L(t)}\\Big(\\|\\omega_{\\geq2}\\|_{\\dot{\\mathcal{F}}^{1+s,1}_\\nu}\\!+\\| \\mathcal{R}(\\omega_{\\geq1})\\|_{\\dot{\\mathcal{F}}^{1+s,1}_\\nu}\\!+\\| \\mathcal{S}(\\omega)\\|_{\\dot{\\mathcal{F}}^{1+s,1}_\\nu}\\Big).\n\\end{multline}\nUsing the bound \\eqref{Restimates}$_2$ and splitting $\\omega_{\\geq1}=\\omega_{1}+\\omega_{\\geq2}$ gives that\n\\begin{equation*}\n    \\begin{aligned}\n        \\| \\mathcal{R}(\\omega_{\\geq1})\\|_{\\dot{\\mathcal{F}}^{1+s,1}_\\nu}&\\leq   b(2,s){C_{\\mR}}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\omega_1\\|_{\\dot{\\mathcal{F}}^{1+s,1}_\\nu}+b(2,s){C_{\\mR}}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\omega_{\\geq2}\\|_{\\dot{\\mathcal{F}}^{1+s,1}_\\nu}\\\\\n        &\\quad+b(2,s){C_{\\mR}}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{1+s,1}_\\nu}\\|\\omega_1\\|_{\\mathcal{F}^{0,1}_\\nu}\n        +b(2,s){C_{\\mR}}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{1+s,1}_\\nu}\\|\\omega_{\\geq2}\\|_{\\mathcal{F}^{0,1}_\\nu}.\n    \\end{aligned}\n\\end{equation*}\nThen introducing the vorticity estimates \\eqref{omega1f0}, \\eqref{omega1fss}, \\eqref{omega2f0}, and \\eqref{omega2fss}, and ordering the terms in $A_\\rho$ and $A_\\sigma$, it follows that\n\\begin{equation*}\n    \\begin{aligned}\n        \\| \\mathcal{R}&(\\omega_{\\geq1})\\|_{\\dot{\\mathcal{F}}^{1+s,1}_\\nu}\\leq A_\\sigma\\frac{4\\pi}{L(t)}b(2,s){C_{\\mR}}\\Big(\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{3+s,1}_\\nu}\n        +|A_\\mu|C_{13}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^2\\|\\theta\\|_{\\dot{\\mathcal{F}}^{3+s,1}_\\nu}\\\\\n        &\\quad+\\|\\theta\\|_{\\dot{\\mathcal{F}}^{1+s,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{2,1}_\\nu}+|A_\\mu|C_{9}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{1+s,1}_\\nu}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{2,1}_\\nu}\\Big)\\\\\n        &\\quad+|A_\\rho|\\frac{L(t)}{\\pi}e^{\\nu(t)}b(2,s){C_{\\mR}}\\Big(2(1+2|A_\\mu|)\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{1+s,1}_\\nu}+ C_{14}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^2\\|\\theta\\|_{\\dot{\\mathcal{F}}^{1+s,1}_\\nu}\\\\\n        &\\quad +(1+2|A_\\mu|)\\|\\theta\\|_{\\dot{\\mathcal{F}}^{1+s,1}_\\nu} \\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}+C_{10}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{1+s,1}_\\nu}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^2\\Big).\n    \\end{aligned}\n\\end{equation*}\nInterpolation as in \\eqref{interpolation} yields that\n\\begin{equation}\\label{est1}\n    \\begin{aligned}\n        \\| \\mathcal{R}(\\omega_{\\geq1})\\|_{\\dot{\\mathcal{F}}^{1+s,1}_\\nu}&\\leq A_\\sigma\\frac{4\\pi}{L(t)}C_{21}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{3+s,1}_\\nu}\n        \\\\\n        &\\quad +|A_\\rho|\\frac{L(t)}{\\pi}e^{\\nu(t)}C_{22}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{1+s,1}_\\nu},\n    \\end{aligned}\n\\end{equation}\nwhere\n\\begin{equation}\\label{C21C22}\n\\begin{aligned}\n    C_{21}&=2b(2,s){C_{\\mR}}+|A_\\mu|b(2,s){C_{\\mR}}(C_{9}+C_{13})\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu},\\\\\n    C_{22}&=3b(2,s){C_{\\mR}}(1+2|A_\\mu|)+ b(2,s){C_{\\mR}}(C_{10}+C_{14})\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu},\n\\end{aligned}\n\\end{equation}\nand $C_\\mathcal{R}$ in \\eqref{CR}, $C_9$ and $C_{10}$ in \\eqref{C9C10}, $C_{13}$ and $C_{14}$ in \\eqref{C13C14} were defined previously.\n\n\n\n\nNext, we split $\\omega=\\omega_0+\\omega_1+\\omega_{\\geq2}$ as in \\eqref{omegasplit} and use the bound \\eqref{Sestimates}$_2$ to obtain \n\\begin{equation*}\n    \\begin{aligned}\n        \\|\\mathcal{S}(\\omega)\\|_{\\dot{\\mathcal{F}}^{1+s,1}_\\nu}&\\leq |A_\\rho|\\frac{L(t)}{\\pi}(C_3+C_4)e^{\\nu(t)}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{1+s,1}_\\nu}\n        +\n        C_3\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^2\\|\\omega_1\\|_{\\dot{\\mathcal{F}}^{1+s,1}_\\nu}\\\\\n        &\\quad+C_3\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^2\\|\\omega_{\\geq2}\\|_{\\dot{\\mathcal{F}}^{1+s,1}_\\nu}\n        +C_4\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{1+s,1}_\\nu}\\|\\omega_1\\|_{\\mathcal{F}^{0,1}_\\nu}\\\\\n        &\\quad+C_4\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{1+s,1}_\\nu}\\|\\omega_{\\geq2}\\|_{\\mathcal{F}^{0,1}_\\nu}.\n    \\end{aligned}\n\\end{equation*}\nSo again we introduce the vorticity estimates \\eqref{omega1f0}, \\eqref{omega1fss}, \\eqref{omega2f0}, and \\eqref{omega2fss}, and order the terms in $A_\\rho$ and $A_\\sigma$, to obtain \n\\begin{equation*}\n    \\begin{aligned}\n        \\|\\mathcal{S}&(\\omega)\\|_{\\dot{\\mathcal{F}}^{1+s,1}_\\nu}\\leq A_\\sigma\\frac{4\\pi}{L(t)}\\Big( C_3\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^2\\|\\theta\\|_{\\dot{\\mathcal{F}}^{3+s,1}_\\nu}+|A_\\mu|C_3C_{13}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^3\\|\\theta\\|_{\\dot{\\mathcal{F}}^{3+s,1}_\\nu} \\\\\n        &\\quad+C_4\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{1+s,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{2,1}_\\nu}+|A_\\mu|C_9C_4\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^2\\|\\theta\\|_{\\dot{\\mathcal{F}}^{1+s,1}_\\nu} \\|\\theta\\|_{\\dot{\\mathcal{F}}^{2,1}_\\nu}\\Big)\\\\\n        &\\quad+|A_\\rho|\\frac{L(t)}{\\pi}e^{\\nu(t)}\\Big( (C_3\\!+\\!C_4)\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{1+s,1}_\\nu}\\\\\n        &\\quad+2b(2,s) (1\\!+\\!2|A_\\mu|)C_3\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^2\\|\\theta\\|_{\\dot{\\mathcal{F}}^{1+s,1}_\\nu}+C_3C_{14}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^3\\|\\theta\\|_{\\dot{\\mathcal{F}}^{1+s,1}_\\nu}\\\\\n        &\\quad+C_4(1+2|A_\\mu|)\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^2\\|\\theta\\|_{\\dot{\\mathcal{F}}^{1+s,1}_\\nu}+C_4C_{10}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^3\\|\\theta\\|_{\\dot{\\mathcal{F}}^{1+s,1}_\\nu}\\Big).\n    \\end{aligned}\n\\end{equation*}\nApplying interpolation \\eqref{interpolation} and the  embedding \\eqref{embed}, \nwe obtain that\n\\begin{multline}\\label{est2}\n        \\|\\mathcal{S}(\\omega)\\|_{\\dot{\\mathcal{F}}^{1+s,1}_\\nu}\\leq A_\\sigma\\frac{4\\pi}{L(t)}C_{23}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^2\\|\\theta\\|_{\\dot{\\mathcal{F}}^{3+s,1}_\\nu}\n        \\\\\n          +|A_\\rho|\\frac{L(t)}{\\pi}e^{\\nu(t)}C_{24}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{1+s,1}_\\nu}\n\\end{multline}\nwhere\n\\begin{equation}\\label{C23C24}\n    \\begin{aligned}\n        C_{23}&=C_3+C_4+|A_\\mu|(C_3C_{13}+C_4C_9)\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu},\\\\\n        C_{24}&=C_3+C_4+(1+2|A_\\mu|)(2b(2,s)C_3+C_4)\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\n        \\\\\n        & \\quad +(C_3C_{14} +C_4C_{10})\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^2,\n    \\end{aligned}\n\\end{equation}\nwith $C_3$ and $C_4$ in \\eqref{C3}, $C_9$ and $C_{10}$ in \\eqref{C9C10}, $C_{13}$ and $C_{14}$ in \\eqref{C13C14} previously defined.     \n\nFinally, we combine estimates \\eqref{est1} and \\eqref{est2} together with vorticity estimate \\eqref{omega2fss} to close the bound \\eqref{N1f0}.  We obtain\n\\begin{equation}\\label{N1final}\n        \\|N_1\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\\leq \n        A_\\sigma\\Big(\\frac{2\\pi}{L(t)}\\Big)^2\n        C_{25}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{3+s,1}_\\nu}+|A_\\rho|e^{\\nu(t)}\n        C_{26}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{1+s,1}_\\nu},\n\\end{equation}\nwhere\n\\begin{equation}\\label{C25C26}\n    \\begin{aligned}\n        C_{25}=|A_\\mu| C_{13}+C_{21}+C_{23}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu},\\qquad       C_{26}=C_{14}+C_{22}+C_{24}.\n    \\end{aligned}\n\\end{equation}\nAbove  $C_{13}$ and $C_{14}$ are defined in \\eqref{C13C14}, $C_{21}$ and $C_{22}$ in \\eqref{C21C22}, and $C_{23}$ and $C_{24}$ in \\eqref{C23C24}.  This completes the estimate for $N_1$.\n\n\n\n\nWe proceed now to estimate the term $N_2$ from \\eqref{N1N2N3}.  For simplicity in this case we perform the estimate for $0\\le s \\le 1$.  The case $s>1$ would add several complications to the notation, but can also be proven similarly, as we will see in the following.  For $s\\ge 0$ we use \\eqref{bfcn.def} and \\eqref{s.inequality} to obtain\n\\begin{equation}\\label{N2aux}\n\\begin{aligned}\n    \\|N_2\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\\leq \\|T_{\\geq2}\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}(1+b(2,s) \\|\\theta\\|_{\\dot{\\mathcal{F}}^{1,1}_\\nu})+b(2,s)\\|T_{\\geq2}\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{1+s,1}_\\nu}.\n\\end{aligned}\n\\end{equation}\nWe recall that for a function $f(\\alpha)$ one has \\eqref{fourierCalcAlpha} and then for $s\\ge 0$ we have\n\\begin{equation}\\label{negative.norm.bound}\n    \\begin{aligned}\n        \\left|\\left|\\int_0^\\alpha f(\\eta)d\\eta-\\frac{\\alpha}{2\\pi}\\int_{-\\pi}^\\pi f(\\eta)d\\eta\\right|\\right|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}&\\leq \\left( 1 + 1_{s=0}\\right)\\|f-\\widehat{f}(0)\\|_{\\dot{\\mathcal{F}}^{-1+s,1}_\\nu} \\\\\n        &\\leq C(s)\\|f\\|_{\\dot{\\mathcal{F}}^{(s-1)^+,1}_\\nu},\n    \\end{aligned}\n\\end{equation}\nwhere $(s-1)^+ = s-1$ if $s\\ge 1$ and $(s-1)^+ = 0$ if $s \\le 1$ as usual.  We also define\n\\begin{equation}\\label{def.CS}\n    C(s) = \\left( 1 + 1_{s=0}\\right).\n\\end{equation}\nIf we performed the estimate below for $s>1$ we would need work with the norm of $\\dot{\\mathcal{F}}^{(s-1)^+,1}_\\nu$ instead of the simpler $\\mathcal{F}^{0,1}_\\nu$.  Thus we only do $0 \\le s \\le 1$ for the $N_2$ term.\n\n\n\nTherefore, for $0 \\le s \\le 1$ recalling the expression for $T_{\\geq2}$ in \\eqref{Tsplit}, we have\n\\begin{equation*}\n    \\begin{aligned}\n    \\|T_{\\geq2}\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}&\\leq C(s)\\|(1+\\theta_\\alpha)U_{\\geq2}\\|_{\\mathcal{F}^{0,1}_\\nu}+C(s)\\|\\theta_\\alpha U_{1}\\|_{\\mathcal{F}^{0,1}_\\nu}\\\\\n    &\\leq C(s)(1+\\|\\theta\\|_{\\dot{\\mathcal{F}}^{1,1}_\\nu})\\|U_{\\geq2}\\|_{\\mathcal{F}^{0,1}_\\nu}+C(s)\\|\\theta\\|_{\\dot{\\mathcal{F}}^{1,1}_\\nu}\\|U_1\\|_{\\mathcal{F}^{0,1}_\\nu}.\n    \\end{aligned}\n\\end{equation*}\nThen introducing $U_{\\geq2}$ and $U_1$  from\\eqref{Usplit} we obtain that\n\\begin{equation*}\n    \\begin{aligned}\n    \\|T_{\\geq2}\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}&\\leq  C(s)\\frac{\\pi}{L(t)}(1+\\|\\theta\\|_{\\dot{\\mathcal{F}}^{1,1}_\\nu})\\big(\\|\\omega_{\\geq2}\\|_{\\mathcal{F}^{0,1}_\\nu}+\\|\\mathcal{R}(\\omega_{\\geq1})\\|_{\\mathcal{F}^{0,1}_\\nu}+\\|\\mathcal{S}(\\omega)\\|_{\\mathcal{F}^{0,1}_\\nu}\\big)\n    \\\\\n    &\\quad+C(s)\\frac{\\pi}{L(t)}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{1,1}_\\nu}\\big(\\|\\omega_{1}\\|_{\\mathcal{F}^{0,1}_\\nu}+\\|\\mathcal{R}(\\omega_{0})\\|_{\\mathcal{F}^{0,1}_\\nu}\\big).\n    \\end{aligned}\n\\end{equation*}\nWe split the vorticity terms, $\\omega_{\\geq1} = \\omega_{1}+ \\omega_{\\geq2}$, and use $\\omega_0$ in \\eqref{omegasplit}.  We further use  the estimates for $\\mathcal{R}$ and $\\mathcal{S}$ in \\eqref{Restimates} and \\eqref{Sestimates} and use \\eqref{cosine.calc.FT} to obtain\n\\begin{equation*}\n    \\begin{aligned}\n    \\|T_{\\geq2}\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}&\\leq  C(s)\\frac{\\pi}{L(t)}(1+\\|\\theta\\|_{\\dot{\\mathcal{F}}^{1,1}_\\nu})\\Big((1+{C_{\\mR}}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}+C_1\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^2)\\|\\omega_{\\geq2}\\|_{\\mathcal{F}^{0,1}_\\nu}\\\\\n    &\\quad+({C_{\\mR}}+C_1\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu})\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\omega_{1}\\|_{\\mathcal{F}^{0,1}_\\nu}+A_\\rho\\frac{L(t)}{\\pi}e^{\\nu(t)}C_1\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^2\\Big)\n    \\\\\n    &\\quad+C(s)\\frac{\\pi}{L(t)}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{1,1}_\\nu}\\Big(\\|\\omega_{1}\\|_{\\mathcal{F}^{0,1}_\\nu}+A_\\rho\\frac{L(t)}{\\pi}e^{\\nu(t)}{C_{\\mR}}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\Big).\n    \\end{aligned}\n\\end{equation*}\nThen we introduce the vorticity estimates \\eqref{vorticityestimatess} and \\eqref{omega2f0} to obtain \n\\begin{equation*}\n    \\begin{aligned}    \\|T_{\\geq2}&\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\\leq  \n    A_\\sigma\\Big(\\frac{2\\pi}{L(t)}\\Big)^2C(s)\n\\Big(|A_\\mu|C_9\\big(1+C_\\mathcal{R}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}+C_1\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^2\\big)\\\\\n&\\quad+(C_\\mathcal{R}+C_1\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu})\\Big)(1+\\|\\theta\\|_{\\dot{\\mathcal{F}}^{1,1}_\\nu})\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{2,1}_\\nu}\n\\\\\n&\\quad+A_\\sigma\\Big(\\frac{2\\pi}{L(t)}\\Big)^2 C(s)\\|\\theta\\|_{\\dot{\\mathcal{F}}^{1,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{2,1}_\\nu} \\!+\\!|A_\\rho|C(s)\\Big(C_{10}(1\\!+\\!C_\\mathcal{R}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\!+\\!C_1\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^2)\\\\\n&\\quad+(1+2|A_\\mu|)(C_\\mathcal{R}+C_1\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu})+C_1\\Big)e^{\\nu(t)}(1+\\|\\theta\\|_{\\dot{\\mathcal{F}}^{1,1}_\\nu})\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^2\\\\\n&\\quad\n+|A_\\rho|C_\\mathcal{R}C(s)e^{\\nu(t)}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{1,1}_\\nu}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\n+(1+2|A_\\mu|)|A_\\rho|C(s)e^{\\nu(t)}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{1,1}_\\nu}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}.\n    \\end{aligned}\n\\end{equation*}\nThe above expression reduces to\n\\begin{equation}\\label{T2aux}\n    \\begin{aligned}    \\|T_{\\geq2}\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}&\\leq  \n    A_\\sigma\\Big(\\frac{2\\pi}{L(t)}\\Big)^2C_{33}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{1,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{2,1}_\\nu}+|A_\\rho|e^{\\nu(t)}C_{34}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{1,1}_\\nu},\n    \\end{aligned}\n\\end{equation}\nwhere\n\\begin{equation}\\label{C33C34}\n    \\begin{aligned}    \n    C_{33}&=C(s)+C(s)\\Big(|A_\\mu|C_9\\big(1+C_\\mathcal{R}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}+C_1\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^2\\big)\\\\\n    &\\qquad+(C_\\mathcal{R}+C_1\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}) \\Big)(1+\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}),\\\\\n    C_{34}&=C(s)\\Big(C_{10}(1+C_\\mathcal{R}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}+C_1\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}^2)\\\\\n    &\\quad+(1+2|A_\\mu|)(C_\\mathcal{R}+C_1\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu})+C_1\\Big)(1+\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu})\n    \\\\\n    &\\quad+C(s)C_\\mathcal{R}+C(s)(1+2|A_\\mu|),\n    \\end{aligned}\n\\end{equation}\nand $C_\\mathcal{R}$ is defined in \\eqref{CR}, $C(s)$ in \\eqref{def.CS}, $C_1$ in \\eqref{C1}, $C_9$ and  $C_{10}$ \\eqref{C9C10}.  \n\n\n\nThen, going back to \\eqref{N2aux}, we find that \n\\begin{equation}\\notag\n    \\begin{aligned}\n        \\|N_2\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}&\\leq \\Big(A_\\sigma\\Big(\\frac{2\\pi}{L(t)}\\Big)^2C_{33}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{1,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{2,1}_\\nu}\\\\\n        &\\quad+|A_\\rho|e^{\\nu(t)}C_{34}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{1,1}_\\nu}\\Big)(1+\\|\\theta\\|_{\\dot{\\mathcal{F}}^{1,1}_\\nu}+\\|\\theta\\|_{\\dot{\\mathcal{F}}^{1+s,1}_\\nu})\\\\\n        &\\leq A_\\sigma\\Big(\\frac{2\\pi}{L(t)}\\Big)^2C_{33}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{1,1}_\\nu} \\|\\theta\\|_{\\dot{\\mathcal{F}}^{2,1}_\\nu}\n        \\left( 2\\|\\theta\\|_{\\dot{\\mathcal{F}}^{1+s,1}_\\nu}+ 1\\right)\\\\\n        &\\quad+|A_\\rho|e^{\\nu(t)}C_{34}\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{1,1}_\\nu}\\left( 2\\|\\theta\\|_{\\dot{\\mathcal{F}}^{1+s,1}_\\nu}+ 1\\right).\n    \\end{aligned}\n\\end{equation}\nFinally, interpolation \\eqref{interpolation} gives that\n\\begin{equation*}\n    \\begin{aligned}\n        \\|\\theta\\|_{\\dot{\\mathcal{F}}^{1,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{1+s,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{2,1}_\\nu}\\leq \\|\\theta\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}^{\\frac{5+2s}{3}}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{3+s,1}_\\nu}^{\\frac{4-2s}{3}},\n    \\end{aligned}\n\\end{equation*}\nThus we have\n\\begin{equation}\\label{N2bound}\n    \\begin{aligned}\n        \\|N_2\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\n        &\\leq A_\\sigma\\Big(\\frac{2\\pi}{L(t)}\\Big)^2C_{33}\\left( 2\\|\\theta\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}^{\\frac{5+2s}{3}}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{3+s,1}_\\nu}^{\\frac{4-2s}{3}}+ \\|\\theta\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{3+s,1}_\\nu} \\right)\\\\\n        &\\quad+|A_\\rho|e^{\\nu(t)}C_{34}\\ \\left( 2\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{1,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{1+s,1}_\\nu}+ \\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{1,1}_\\nu}\\right).\n    \\end{aligned}\n\\end{equation}\nThis completes our estimates for $N_2$.  \n\n\n\nLastly, the estimate for $N_3$ in \\eqref{N1N2N3} for $s\\ge 0$ is obtained with \\eqref{bfcn.def} as follows\n\\begin{equation*}\n    \\begin{aligned}\n        \\|N_3\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\\leq b(2,s) \\left(\\|T_1\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{1,1}_\\nu}+\\|T_1\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{1+s,1}_\\nu} \\right).\n    \\end{aligned}\n\\end{equation*}\nThe bound for $T_1$ in \\eqref{Tsplit} for $s\\ge 0$ from \\eqref{negative.norm.bound} using \\eqref{u0t0} and \\eqref{cosine.calc.FT} is \n\\begin{equation}\\label{T1aux}\n    \\begin{aligned}\n        \\|T_1\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}&\\leq \\|U_1\\|_{\\dot{\\mathcal{F}}^{(s-1)^+,1}_\\nu}+\\|\\theta_\\alpha U_0\\|_{\\dot{\\mathcal{F}}^{(s-1)^+,1}_\\nu}\n        \\\\\n        &\\leq \\|U_1\\|_{\\dot{\\mathcal{F}}^{(s-1)^+,1}_\\nu}+2|A_{\\rho}| b(2,s) e^{\\nu(t)} \\|\\theta_\\alpha\\|_{\\dot{\\mathcal{F}}^{(s-1)^+,1}_\\nu}\n        \\\\\n        &\\leq \\frac{\\pi}{L(t)}\\big(\\|\\omega_1\\|_{\\dot{\\mathcal{F}}^{(s-1)^+,1}_\\nu}\n        +\\|\\mathcal{R}(\\omega_0)\\|_{\\dot{\\mathcal{F}}^{(s-1)^+,1}_\\nu}\\big)\n        \\\\\n        &\\quad +2|A_{\\rho}| b(2,s) e^{\\nu(t)} \\|\\theta_\\alpha\\|_{\\dot{\\mathcal{F}}^{(s-1)^+,1}_\\nu}.\n    \\end{aligned}\n\\end{equation}\nWe use \\eqref{omega1f0}, \\eqref{omega1fss} and \\eqref{Restimates} with \\eqref{omegasplit} and  \\eqref{cosine.calc.FT} to obtain\n\\begin{multline*}\n\\frac{\\pi}{L(t)}\\big(\\|\\omega_1\\|_{\\dot{\\mathcal{F}}^{(s-1)^+,1}_\\nu}\n        +\\|\\mathcal{R}(\\omega_0)\\|_{\\dot{\\mathcal{F}}^{(s-1)^+,1}_\\nu}\\big)\n        \\\\\n        \\leq \n        A_\\sigma\\Big(\\frac{2\\pi}{L(t)}\\Big)^2\n        \\|\\theta\\|_{\\dot{\\mathcal{F}}^{2+(s-1)^+,1}_\\nu}\n        +2b(2,s)|A_\\rho|\\Big((1+2|A_\\mu|)+2 C_\\mathcal{R}\\Big)e^{\\nu(t)}\n        \\|\\theta\\|_{\\dot{\\mathcal{F}}^{(s-1)^+,1}_\\nu}.\n\\end{multline*}\nThus\n\\begin{equation}\\label{N3boud}\n    \\begin{aligned}\n        \\|N_3\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}&\\leq A_\\sigma\\Big(\\frac{2\\pi}{L(t)}\\Big)^2 \n        b(2,s)\\|\\theta\\|_{\\dot{\\mathcal{F}}^{2+(s-1)^+,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{1+s,1}_\\nu}\\\\\n        &\\quad+|A_\\rho|e^{\\nu(t)}C_3^{\\prime}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{(s-1)^+,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{1+s,1}_\\nu}.\n    \\end{aligned}\n\\end{equation}\nHere\n\\begin{equation}\\label{C3prime.prime}\n        C_3^{\\prime}= 2b(2,s)^2\\left(1+(1+2|A_\\mu|)+2 C_\\mathcal{R} \\right).\n\\end{equation}\nThis completes our estimates for $N_3$.  \n\nIn summary, we can combine the bounds \\eqref{N1final}, \\eqref{N2bound}, and \\eqref{N3boud} to prove the following theorem.\n\n\\begin{thm}\\label{thm:nonLinear}\nFor $0\\le s \\le 1$ we have the estimate for $N$ from \\eqref{system} as\n\\begin{equation}\\label{thm:Nbound}\n    \\begin{aligned}\n        \\|N\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}&\\leq A_\\sigma\\Big(\\frac{2\\pi}{L(t)}\\Big)^2C_1(N)\\|\\theta\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{3+s,1}_\\nu}\\\\\n        &\\quad+\n        A_\\sigma\\Big(\\frac{2\\pi}{L(t)}\\Big)^2 2C_{33}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{s,1}_\\nu}^{\\frac{5+2s}{3}}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{3+s,1}_\\nu}^{\\frac{4-2s}{3}}\\\\\n        &\\quad+|A_\\rho|e^{\\nu(t)}C_2(N)\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{1+s,1}_\\nu},\n    \\end{aligned}\n\\end{equation}\nwhere \n\\begin{equation}\\label{C1NC2N}\n    \\begin{aligned}\n        C_1(N)&=C_{25}+C_{33}+b(2,s),\\\\\n        C_2(N)&=C_{26}+C_{34}(1+2\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu})+C_{3}^\\prime.\n    \\end{aligned}\n\\end{equation}\nFurther $C_{25}$ and $C_{26}$ are defined in \\eqref{C25C26}, $C_{33}$ and $C_{34}$ in \\eqref{C33C34}, and $C_{3}^\\prime$ is previously defined in \\eqref{C3prime.prime}.\n\\end{thm}\n\n\n\n\nPlugging in $s=1\/2$ in the bound for the nonlinear term in Theorem \\ref{thm:nonLinear} in  \\eqref{thm:Nbound},\nwe find that\n\\begin{equation}\\label{Nbound}\n    \\begin{aligned}\n        \\|N\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}_\\nu}&\\leq A_\\sigma\\Big(\\frac{2\\pi}{L(t)}\\Big)^2C_{35}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{\\frac72,1}_\\nu}\n        +|A_\\rho|e^{\\nu(t)}C_{36}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}_\\nu}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{\\frac72,1}_\\nu},\n    \\end{aligned}\n\\end{equation}\nwhere\n\\begin{equation}\\label{C35C36}\n    \\begin{aligned}\n        C_{35}&=C_{35}(\\|\\theta\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}_\\nu})=C_1(N)+2C_{33}\\|\\theta\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}_\\nu},\\\\\n        C_{36}&=C_{36}(\\|\\theta\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}_\\nu})=C_2(N),\n    \\end{aligned}\n\\end{equation}\nand $C_1(N)$ and $C_2(N)$ are defined in \\eqref{C1NC2N}, and  $C_{33}$ is defined in \\eqref{C33C34}. Notice that in the definition of $C_{35}$ and $C_{36}$ we can evaluated all the previous functions $C_i$ in the norm $\\|\\theta\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}_\\nu}$ instead of $\\|\\theta\\|_{\\mathcal{F}^{0,1}_\\nu}$, which could be done due to \\eqref{embed} and the fact that these $C_i$ are increasing functions of the norm.\n\n\n\n\\subsection{Regularization scheme and completion of the proof of Theorem \\ref{thm:global}}\\label{subregular}\n\nWe will now put all the pieces together to complete the proof of Theorem \\ref{thm:global}.\n\n\\begin{proof}[Proof of Theorem \\ref{thm:global}]\nThe proof then follows a standard regularization argument. Recall the high-frequency cut-off operator $\\mathcal{J}_N$ by \\eqref{CutOffHigh}.  Denote $f_N=\\mathcal{J}_N f$ and consider the regularized version of system \\eqref{finalsystem}\n:\n\\begin{equation*}\n    \\begin{aligned}\n     (\\vartheta_N)_t&=\\mathcal{J}_N\\big(\\frac{2\\pi}{L_N(t)} (U_N)_\\alpha+\\frac{2\\pi}{L_N(t)}T_N(1+(\\vartheta_N)_\\alpha)\\Big),\\\\\n     \\frac{L_N(t)}{2\\pi}&=R\\Big(1+\\frac{1}{2\\pi}\\text{Im}\\hspace{0.05cm} \\int_{-\\pi}^\\pi\\int_0^\\alpha e^{i(\\alpha-\\eta)} \\sum_{n\\geq1}\\frac{i^n}{n!}(\\theta_N(\\alpha)-\\theta_N(\\eta))^n d\\eta d\\alpha\\Big)^{-\\frac12},\\\\\n     0&=\\int_{-\\pi}^{\\pi} e^{i(\\alpha+\\theta_N(\\alpha))}d\\alpha.\n    \\end{aligned}\n\\end{equation*}\nWe abused notation in the definition of $L_N(t)$ above since we are not using $\\mathcal{J}_N$ from \\eqref{CutOffHigh}. Solving the last constraint by the implicit function theorem (see Proposition \\ref{IFTprop} in Section \\ref{IFTSection}) gives $F(\\theta_N)=(\\text{Re}\\hspace{0.05cm}\\hat{\\theta}(1), \\text{Im}\\hspace{0.05cm} \\hat{\\theta}(1))$ which can be solved for $\\widehat{\\theta}(\\pm 1)$.   \nThus substituting in as well the expression for $L_N(t)$, we obtain one equation for $\\varphi_N=\\widehat{\\vartheta}(0)+\\mathcal{F}^{-1}(1_{|k|\\neq1}\\widehat{\\theta_N}(k))$. We thereby have the system written as an ODE of the form \n\\begin{equation*}\n    \\dot{\\varphi}_N=\\mathcal{J}_N \\mathcal{G}\\big(\\varphi_N\\big),\\qquad \\varphi_N(0)=\\widehat{\\vartheta}_0(0)+\\mathcal{F}^{-1}(1_{|k|\\neq1}\\widehat{\\theta_{N,0}}(k)),\n\\end{equation*}\nfor a certain nonlinear function $\\mathcal{G}$.  Here $\\theta_{N,0}=\\mathcal{J}_N\\theta_{0}$ is the initial condition.  Therefore, Picard's theorem on Banach spaces yields the local existence of regularized solutions $\\varphi_N\\in C^1([0,T_N);H_N^m)$, where the space $H_N^m$ is defined by $H_N^m=\\{f\\in H^m(\\mathbb{T}): \\text{supp}(\\widehat{f})\\subset [-N,N]\\}.$ Furthermore, is clear that the \\textit{a priori} estimates in Subsection \\ref{subsecGlobal}, in particular the energy balance \\eqref{estimatef12}, hold for the regularized system, which provides uniform bounds for $\\varphi_N$ in the space $L^\\infty(\\mathbb{R}_+; \\dot{\\mathcal{F}}^{\\frac12,1}_\\nu)\\cap L^1(\\mathbb{R}_+; \\dot{\\mathcal{F}}^{\\frac72,1}_\\nu)$. One can then apply a version of the Aubin-Lions lemma to get the strong convergence to the full system, up to a subsequence, of the approximated problems and thereby conclude the existence result. We refer Section 5 of \\cite{GG-BS2019} for further details of such an approximation argument.   \n\\end{proof}\n\n\n\n\n\\section{Uniqueness}\\label{sec:uniqueness}\n\n\nIn this last section we will prove uniqueness in $\\dot{\\mathcal{F}}^{\\frac12,1}$ of solutions to \\eqref{finalsystem} with initial data of the size given by the constraint in \\eqref{condition}.  In particular the main result of this section is  Theorem \\ref{uniquenessproposition} just below.  To prove this theorem, in Subsection \\ref{sec:unique.length} we prove the required estimates for the differences of the lengths.  Then in Subsection \\ref{subsec:uniq.vort} we prove the estimates on the differences of the vorticity strength.  After that in Subsection \\ref{subsec:uniq.nonlinear} we prove the main estimates on the differences of the non-linear terms.  Lastly in Subsection \\ref{subsec:uniq.proof} we collect all the previous estimates to prove the uniqueness of the solutions to \\eqref{system} as in Theorem \\ref{uniquenessproposition}.\n\n\n\n\n\nThroughout the proof of Theorem \\ref{uniquenessproposition}, we define coefficients that will be used in the rest of this section.\n\n\\begin{defn}\\label{uniquenessimplicitdef}\nWe use the symbol $\\mathcal{E}>0$ to denote any coefficient that is integrable in time and may depend upon $\\|\\theta_{1},\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}}$, recalling \\eqref{max.fcns}, which is bounded and  $\\|\\theta_{1},\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac72,1}}$ which is time integrable.\n\nThe symbol $\\mathcal{C}>0$ will denote any coefficient that is bounded and can depend upon $\\|\\theta_{1},\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}}$.  \n\nThe symbol $\\mathcal{C}_{s}>0$ will denote any coefficient that is bounded and can depend upon $\\|\\theta_{1},\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{s,1}}$.  \n\nThe symbol $\\mathcal{E}_{s}>0$ will denote any coefficient that may depend upon $\\mathcal{C}\\|\\theta_{1},\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{2+s,1}}$.\n\\end{defn}\n\n\n\n\\begin{thm}\n\\label{uniquenessproposition}\nConsider two solutions $\\vartheta_{1}$ and $\\vartheta_{2}$ of \\eqref{system} with the same initial data satisfying the medium size condition, as in Theorem \\ref{thm:global}.   Then these solutions satisfy the following differential inequality\n\\begin{multline}\\label{uniqineq}\n\\frac{d}{dt}\\Big(|\\widehat{\\vartheta}_{1}(0) - \\widehat{\\vartheta}_{2}(0)| + \\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}}\\Big) \n\\\\\n\\leq (\\mathcal{C}+\\mathcal{E})(|\\widehat{\\vartheta}_{1}(0) - \\widehat{\\vartheta}_{2}(0)| + \\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}}).\n\\end{multline}\n\\end{thm}\n\n\nWith Theorem \\ref{uniquenessproposition}, we can conclude by Gronwall's inequality that for any $T>0$, we have\n\\begin{multline*}\n \\Big(|\\widehat{\\vartheta}_{1}(0) - \\widehat{\\vartheta}_{2}(0)| + \\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}}\\Big) \\Big|_{t=T} \n \\\\\n \\leq \\exp\\left(\\int_{0}^{T} (\\mathcal{C}+\\mathcal{E}) dt\\right)\n \\Big(|\\widehat{\\vartheta}_{1}(0) - \\widehat{\\vartheta}_{2}(0)| + \\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}}\\Big)\\Big|_{t=0}= 0.\n \\end{multline*}\nThis holds since $\\mathcal{E}=\\mathcal{C}\\|\\theta_{1},\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac72,1}}$ with \\eqref{max.fcns}, is time integrable by Theorem \\ref{thm:global}.\n\n\nThe rest of this section is devoted to proving Theorem \\ref{uniquenessproposition}.  Given two solutions $\\vartheta_{1}(\\alpha,t)$ and $\\vartheta_{2}(\\alpha,t)$ with initial data $\\vartheta_{1}(\\alpha,0) = \\vartheta_{2}(\\alpha,0)$ that satisfies \\eqref{condition}; their respective evolution equations are given by \\eqref{system} as follows\n$$(\\vartheta_{i})_t(\\alpha)=\\frac{2\\pi}{L_{i}(t)}\\Big(\\mathcal{L}_{i}(\\alpha)+N_{i}(\\alpha)\\Big),$$\nwhere the vorticity terms are denoted by $\\omega_{1}$ and $\\omega_{2}$ respectively. The evolution of $\\vartheta_{1} - \\vartheta_{2}$ is then given by\n\\begin{align}\\notag\n (\\vartheta_{1}-\\vartheta_{2})_t(\\alpha)\n &= \\Big(\\frac{2\\pi}{L_{1}(t)} -\\frac{2\\pi}{L_{2}(t)}\\Big) \\Big(\\mathcal{L}_{1}(\\alpha)+N_{1}(\\alpha)\\Big) + \\frac{2\\pi}{L_{2}(t)}(\\mathcal{L}_{1}(\\alpha) - \\mathcal{L}_{2}(\\alpha))  \\\\&\\hspace{2in}+ \\frac{2\\pi}{L_{2}(t)}(N_{1}(\\alpha) - N_{2}(\\alpha)). \\label{theta1theta2}\n\\end{align}\nUsing the evolution equation \\eqref{theta1theta2}, \nwe will prove \\eqref{uniqineq}. In Proposition \\ref{lengthdifferenceprop}, we give an estimate to control the length difference in the first term on the right hand side of \\eqref{theta1theta2}. By the estimates from Section \\ref{sec:NonLinearEst}, the coefficient, $\\mathcal{L}_{1}(\\alpha)+N_{1}(\\alpha)$, of the length difference in the first term is bounded by $\\mathcal{C}+\\mathcal{E}$.  The bound on the length difference is shown in in Proposition \\ref{lengthdifferenceprop}. The second term on the RHS of \\eqref{theta1theta2} gives a linear coercive estimate for the time evolution. Lastly, Section \\ref{subsec:uniq.nonlinear} is dedicated to controlling the third terms on the RHS of \\eqref{theta1theta2} using the idea of Proposition \\ref{uniquenessprop} and the non-linear estimates as in Section \\ref{sec:NonLinearEst}.\n\n\n\nAdditionally, we will use the following idea repeatedly:\n\\begin{prop}\\label{uniquenessprop}\nConsider two functions $f$ and $g$ in $\\dot{\\mathcal{F}}^{s,1}$ for some $s \\geq 0$.  We also consider some operator $T$. Then, for any $n \\in \\mathbb{N}$ we have\n\\begin{multline}\\label{uniqueoperator}\n\\|   f(\\alpha)^{n}Tf(\\alpha)  - g(\\alpha)^{n} Tg(\\alpha)\\|_{\\mathcal{F}^{0,1}} \\\\ \\leq \\|f\\|_{\\mathcal{F}^{0,1}}^{n}\\|Tf-Tg\\|_{\\mathcal{F}^{0,1}} + \\Big(\\sum_{k=0}^{n-1}\\|f\\|_{\\mathcal{F}^{0,1}}^{n-k-1}\\|g\\|_{\\mathcal{F}^{0,1}}^{k}\\Big)\\|f-g\\|_{\\mathcal{F}^{0,1}}\\|Tg\\|_{\\mathcal{F}^{0,1}}.\n\\end{multline}\nFor $s>0$, we have\n\\begin{multline}\\label{uniqueoperatorsnorm}\n\\|   f(\\alpha)^{n}Tf(\\alpha)  - g(\\alpha)^{n} Tg(\\alpha)\\|_{\\dot{\\mathcal{F}}^{s,1}} \\\\ \\leq b(n+1,s)\\Big(\\|f\\|_{\\mathcal{F}^{0,1}}^{n}\\|Tf-Tg\\|_{\\dot{\\mathcal{F}}^{s,1}}  + n\\|f\\|_{\\mathcal{F}^{0,1}}^{n-1}\\|f\\|_{\\dot{\\mathcal{F}}^{s,1}}\\|Tf-Tg\\|_{\\mathcal{F}^{0,1}} \\\\+  n\\|f,g\\|_{\\mathcal{F}^{0,1}}^{n-1}\\|f-g\\|_{\\dot{\\mathcal{F}}^{s,1}}\\|Tg\\|_{\\mathcal{F}^{0,1}} + n\\|f,g\\|_{\\mathcal{F}^{0,1}}^{n-1}\\|f-g\\|_{\\mathcal{F}^{0,1}}\\|Tg\\|_{\\dot{\\mathcal{F}}^{s,1}} \\\\+ n(n-1)\\|f,g\\|_{\\mathcal{F}^{0,1}}^{n-2}\\|f,g\\|_{\\dot{\\mathcal{F}}^{s,1}}\\|f-g\\|_{\\mathcal{F}^{0,1}}\\|Tg\\|_{\\mathcal{F}^{0,1}}\\Big),\n\\end{multline}\nwhere we recall the definition \\eqref{max.fcns}.  In the special case where $T = \\frac{d}{d\\alpha^{j}}$ for some $j\\in \\mathbb{N}$, we obtain\n\\begin{multline}\\label{uniquenessestimate}\n\\|   f(\\alpha)^{n}\\frac{d^{j}}{d\\alpha^{j}}f(\\alpha)  - g(\\alpha)^{n} \\frac{d^{j}}{d\\alpha^{j}}g(\\alpha)\\|_{\\mathcal{F}^{0,1}} \\\\ \\leq \\|f\\|_{\\mathcal{F}^{0,1}}^{n}\\|f-g\\|_{\\mathcal{F}^{j,1}} + \\Big(\\sum_{k=0}^{n-1}\\|f\\|_{\\mathcal{F}^{0,1}}^{n-k-1}\\|g\\|_{\\mathcal{F}^{0,1}}^{k}\\Big)\\|f-g\\|_{\\mathcal{F}^{0,1}}\\|g\\|_{\\mathcal{F}^{j,1}}.\n\\end{multline}\n\\end{prop}\n\n\\begin{proof}\nSince\n\\begin{align*}\n f(\\alpha)^{n}Tf(\\alpha)  - g(\\alpha)^{n} Tg(\\alpha) &=   f(\\alpha)^{n}Tf(\\alpha)  - f(\\alpha)^{n} Tg(\\alpha) \\\\&\\hspace{0.5in}+  f(\\alpha)^{n}Tg(\\alpha)  - g(\\alpha)^{n} Tg(\\alpha)\\\\ &= f(\\alpha)^{n}(Tf(\\alpha)  -  Tg(\\alpha)) +  (f(\\alpha)^{n}-g(\\alpha)^{n})Tg(\\alpha).\n\\end{align*}\nWe obtain\n\\begin{multline*}\n\\| f(\\alpha)^{n}Tf(\\alpha)  - g(\\alpha)^{n} Tg(\\alpha)\\|_{\\mathcal{F}^{0,1}} \\\\ \\leq \\|f\\|_{\\mathcal{F}^{0,1}}^{n}\\|Tf-Tg\\|_{\\mathcal{F}^{0,1}} + \\|f(\\alpha)^{n} - g(\\alpha)^{n}\\|_{\\mathcal{F}^{0,1}}\\|Tg\\|_{\\mathcal{F}^{0,1}}.\n\\end{multline*}\nNext, we have\n\\begin{align*}\n\\|f(\\alpha)^{n} - g(\\alpha)^{n}\\|_{\\mathcal{F}^{0,1}} &= \\Big\\|\\sum_{k=0}^{n-1}f(\\alpha)^{n-k} g(\\alpha)^{k} -f(\\alpha)^{n-k-1} g(\\alpha)^{k+1} \\Big\\|_{\\mathcal{F}^{0,1}}\\\\ &\\leq \\sum_{k=0}^{n-1}\\|f\\|_{\\mathcal{F}^{0,1}}^{n-k-1}\\|g\\|_{\\mathcal{F}^{0,1}}^{k}\\|f-g\\|_{\\mathcal{F}^{0,1}}.\n\\end{align*}\nThis yields \\eqref{uniqueoperator}.  Then \\eqref{uniqueoperatorsnorm} is proven similarly.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Estimates for the differences of the lengths}\\label{sec:unique.length}\n\nWe need to control the difference in the lengths $L_{1}(t)$ and $L_{2}(t)$, for example, to control the first term on the right hand side of \\eqref{theta1theta2}. In this section, we prove the following proposition on the differences of the lengths.\n\n\\begin{prop}\\label{lengthdifferenceprop}\nConsider the lengths, $L_{1}(t)$ and $L_{2}(t)$, of two solutions, $\\vartheta_{1}$ and $\\vartheta_{2}$ respectively, to \\eqref{system} as defined by \\eqref{modul}. Then, we have\n\\begin{equation}\\label{lengthdifference}\n\\left|\\frac{L_{1}(t)}{2\\pi}-\\frac{L_{2}(t)}{2\\pi}\\right| \\leq  C_L \\|\\theta_{1}-\\theta_{2}\\|_{\\mathcal{F}^{0,1}}\n\\end{equation}\nwith $C_{L}$ defined by \\eqref{CL}.  \n\\end{prop}\n\n\n\\begin{proof}\nWe recall equation \\eqref{Lequation}.  Thus, denoting $\\Delta f=f(\\alpha)-f(\\eta)$, we have that \n\\begin{multline}\\label{lenghtaux}\n    \\Big(\\frac{L_1(t)}{2\\pi}\\Big)^2-\\Big(\\frac{L_2(t)}{2\\pi}\\Big)^2\n    \\\\\n    =\\frac{\\frac{R^2}{2\\pi}\\Big(\\text{Im}\\hspace{0.05cm} \\int\\limits_{-\\pi}\\limits^\\pi\\!\\int\\limits_{0}\\limits^\\alpha e^{i(\\alpha-\\eta)} \\sum\\limits_{n\\geq1}\\frac{i^n}{n!}(\\Delta\\theta_2)^n d\\eta d\\alpha\\!-\\!\\text{Im}\\hspace{0.05cm} \\int\\limits_{-\\pi}\\limits^\\pi\\!\\int\\limits_{0}\\limits^\\alpha e^{i(\\alpha-\\eta)} \\sum\\limits_{n\\geq1}\\frac{i^n}{n!}(\\Delta\\theta_1)^n d\\eta d\\alpha\\Big)}{\\Big(1\\!+\\!\\text{Im}\\hspace{0.05cm} \\!\\int\\limits_{-\\pi}\\limits^\\pi\\!\\int\\limits_{0}\\limits^\\alpha e^{i(\\alpha-\\eta)}\\!\\! \\sum\\limits_{n\\geq1}\\frac{i^n}{n!}(\\Delta\\theta_1)^n d\\eta \\frac{d\\alpha}{2\\pi}\\!\\Big)\\!\\Big(1\\!+\\!\\text{Im}\\hspace{0.05cm} \\!\\int\\limits_{-\\pi}\\limits^\\pi\\!\\int\\limits_{0}\\limits^\\alpha e^{i(\\alpha-\\eta)}\\!\\! \\sum\\limits_{n\\geq1}\\frac{i^n}{n!}(\\Delta\\theta_2)^n d\\eta \\frac{d\\alpha}{2\\pi}\\!\\Big)}.\n\\end{multline}\nRecalling the estimates in \\eqref{lengthauxbound} - \\eqref{C37C38}, the denominator is bounded by\n\\begin{multline*}\n\\bigg(\\prod_{m=1}^2\\Big(\\!1\\!+\\!\\text{Im}\\hspace{0.05cm} \\!\\int\\limits_{-\\pi}\\limits^\\pi\\!\\int\\limits_{0}\\limits^\\alpha e^{i(\\alpha-\\eta)}\\!\\! \\sum\\limits_{n\\geq1}\\frac{i^n}{n!}(\\Delta\\theta_m)^n d\\eta \\frac{d\\alpha}{2\\pi}\\!\\Big)\\!\\bigg)^{-1}\n\\\\\n\\leq \n\\bigg(\\prod_{m=1}^2\n\\Big(1-\\frac{\\pi}{2}\\big(e^{2\\|\\theta_m\\|_{\\mathcal{F}^{0,1}}}-1\\big)\\Big)\n\\bigg)^{-1}.\n\\end{multline*}\nFurther the numerator has the upper bound\n\\begin{equation*}\n\\begin{aligned}\n&\\text{Im}\\hspace{0.05cm} \\int\\limits_{-\\pi}\\limits^\\pi\\!\\int\\limits_{0}\\limits^\\alpha e^{i(\\alpha-\\eta)} \\sum\\limits_{n\\geq1}\\frac{i^n}{n!}(\\Delta\\theta_2)^n d\\eta d\\alpha\\!-\\!\\text{Im}\\hspace{0.05cm} \\int\\limits_{-\\pi}\\limits^\\pi\\!\\int\\limits_{0}\\limits^\\alpha e^{i(\\alpha-\\eta)} \\sum\\limits_{n\\geq1}\\frac{i^n}{n!}(\\Delta\\theta_1)^n d\\eta d\\alpha\\\\\n&=\\text{Im}\\hspace{0.05cm} \\int\\limits_{-\\pi}\\limits^\\pi\\!\\int\\limits_{0}\\limits^\\alpha e^{i(\\alpha-\\eta)} \\sum\\limits_{n\\geq1}\\frac{i^n}{n!}(\\Delta\\theta_2-\\Delta\\theta_1)\\sum\\limits_{m=0}\\limits^{n-1}\\big(\\Delta\\theta_1\\big)^m\\big(\\Delta\\theta_2\\big)^{n-m-1} d\\eta d\\alpha\\\\\n&\\leq  \\pi^2  \\sum\\limits_{n\\geq1}\\frac{2\\|\\theta_1-\\theta_2\\|_{\\mathcal{F}^{0,1}}}{n!}\\sum\\limits_{m=0}\\limits^{n-1}\\big(2\\|\\theta_1\\|_{\\mathcal{F}^{0,1}}\\big)^m\\big(2\\|\\theta_2\\|_{\\mathcal{F}^{0,1}}\\big)^{n-m-1}.\n\\end{aligned}\n\\end{equation*}\nThus we further obtain the estimate\n\\begin{equation*}\n\\begin{aligned}\n&\n\\left| \\text{Im}\\hspace{0.05cm} \\int\\limits_{-\\pi}\\limits^\\pi\\!\\int\\limits_{0}\\limits^\\alpha e^{i(\\alpha-\\eta)} \\sum\\limits_{n\\geq1}\\frac{i^n}{n!}(\\Delta\\theta_2)^n d\\eta d\\alpha\\!-\\!\\text{Im}\\hspace{0.05cm} \\int\\limits_{-\\pi}\\limits^\\pi\\!\\int\\limits_{0}\\limits^\\alpha e^{i(\\alpha-\\eta)} \\sum\\limits_{n\\geq1}\\frac{i^n}{n!}(\\Delta\\theta_1)^n d\\eta d\\alpha\n\\right|\n\\\\\n&\\leq \\frac{2\\pi^2\\|\\theta_1-\\theta_2\\|_{\\mathcal{F}^{0,1}}}{2\\|\\theta_1\\|_{\\mathcal{F}^{0,1}}-2\\|\\theta_2\\|_{\\mathcal{F}^{0,1}}}\\sum\\limits_{n\\geq1}\\frac{1}{n!}\\Big(\\big(2\\|\\theta_1\\|_{\\mathcal{F}^{0,1}}\\big)^n-\\big(2\\|\\theta_2\\|_{\\mathcal{F}^{0,1}}\\big)^n\\Big)\\\\\n&=2\\pi^2\\|\\theta_1-\\theta_2\\|_{\\mathcal{F}^{0,1}}\\frac{e^{2\\|\\theta_1\\|_{\\mathcal{F}^{0,1}}}-e^{2\\|\\theta_2\\|_{\\mathcal{F}^{0,1}}}}{2\\|\\theta_1\\|_{\\mathcal{F}^{0,1}}-2\\|\\theta_2\\|_{\\mathcal{F}^{0,1}}}.\n\\end{aligned}\n\\end{equation*}\nWe substitute this back into \\eqref{lenghtaux} to obtain that\n\\begin{equation}\\label{diflength}\n\\begin{aligned}\n\\Big|\\Big(\\frac{L_1(t)}{2\\pi}\\Big)^2-\\Big(\\frac{L_2(t)}{2\\pi}\\Big)^2\\Big|\\leq  R^2 C_{L,2} \\|\\theta_1-\\theta_2\\|_{\\mathcal{F}^{0,1}},\n\\end{aligned}\n\\end{equation}\nwhere\n\\begin{equation}\\notag\n    \\begin{aligned}\n    C_{L,2}=\n    \\pi\\frac{e^{2\\|\\theta_1\\|_{\\mathcal{F}^{0,1}}}\\!-\\!e^{2\\|\\theta_2\\|_{\\mathcal{F}^{0,1}}}}{2\\|\\theta_1\\|_{\\mathcal{F}^{0,1}}\\!-\\!2\\|\\theta_2\\|_{\\mathcal{F}^{0,1}}}\n    \\bigg(\\prod_{m=1}^2\n\\Big(1-\\frac\\pi2\\big(e^{2\\|\\theta_m\\|_{\\mathcal{F}^{0,1}}}-1\\big)\\Big)\n\\bigg)^{-1}.\n    \\end{aligned}\n\\end{equation}\nThe estimate \\eqref{diflength} allows to easily bound terms like $L_1(t)-L_2(t)$ or $L_1(t)^{-1}-L_2(t)^{-1}$. In fact, using \\eqref{Lbound}, we obtain that\n\\begin{equation*}\n    \\begin{aligned}\n    \\Big|\\frac{L_1(t)}{2\\pi}-\\frac{L_2(t)}{2\\pi}\\Big|=\\frac{1}{\\frac{L_1(t)}{2\\pi}+\\frac{L_2(t)}{2\\pi}}\\Big|\\Big(\\frac{L_1(t)}{2\\pi}\\Big)^2-\\Big(\\frac{L_2(t)}{2\\pi}\\Big)^2\\Big|\\leq  C_L\\|\\theta_1-\\theta_2\\|_{\\mathcal{F}^{0,1}},\n    \\end{aligned}\n\\end{equation*}\nwhere, with $C_{37}$ is defined in \\eqref{C37C38}, we have\n\\begin{equation}\\label{CL}\n    \\begin{aligned}\n    C_{L}=R\\frac{C_{L,2}}{2C_{37}}.\n    \\end{aligned}\n\\end{equation}\nThis completes the proof of \\eqref{lengthdifference}.\n\\end{proof}\n\n\n\n\n\\subsection{Estimates for the differences of the vorticity strength}\\label{subsec:uniq.vort}\nIn this subsection we will estimate the differences of the vorticity strength terms.\nWe use the splitting in \\eqref{omegasplit} as\n\\begin{multline*}\n\\omega_{1}(\\alpha)-\\omega_{2}(\\alpha) \n\\\\\n=\n(\\omega_{1})_{0}(\\alpha)-(\\omega_{2})_{0}(\\alpha) + (\\omega_{1})_{1}(\\alpha)-(\\omega_{2})_{1}(\\alpha) + (\\omega_{1})_{\\geq 2}(\\alpha)-(\\omega_{2})_{\\geq 2}(\\alpha).\n\\end{multline*}\nAbove $(\\omega_{i})_{0}(\\alpha)$ is the zero component, as defined in \\eqref{omegasplit}, of the vorticity term $\\omega_{i}$ for $i=1,2$ etc.  In this section, we prove the following estimates on each difference in the vorticity decomposition.\n\n\\begin{prop}\\label{prop:diff.vorticity.ests}\nFor $s \\ge 0$, we have the estimates\n\\begin{equation}\\label{omega0diffest}\n\\|(\\omega_{1})_{0}-(\\omega_{2})_{0}\\|_{\\dot{\\mathcal{F}}^{s,1}} \\leq   2 \\left|A_\\rho \\right| C_{L}\\|\\theta_{1}-\\theta_{2}\\|_{\\mathcal{F}^{0,1}} + \\frac{A_\\rho}{\\pi}L_{2}(t)|\\widehat{\\vartheta}_{1}(0)-\\widehat{\\vartheta}_{2}(0)|,\n\\end{equation}\nand\n\\begin{multline}\\label{linearomegadiffnorm}\n\\|(\\omega_{1})_{1}-(\\omega_{2})_{1}\\|_{\\dot{\\mathcal{F}}^{s,1}}\\leq \n\\mathcal{C}_{s}(\\|\\theta_{1}-\\theta_{2}\\|_{\\mathcal{F}^{0,1}} +|\\widehat{\\vartheta}_{1}(0)-\\widehat{\\vartheta}_{2}(0)|)\\\\ + \\mathcal{E} \\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{s,1}}  + \\frac{4A_{\\sigma}\\pi}{L_{2}(t)}\\|\\theta_{1} - \\theta_{2}\\|_{\\mathcal{F}^{s+2,1}}.\n\\end{multline}\nFor $s>0$ \nwe further have\n\\begin{multline}\\label{omeganonlinearest}\n\\|(\\omega_{1})_{\\geq 2}-(\\omega_{2})_{\\geq 2}\\|_{\\dot{\\mathcal{F}}^{s,1}} \\leq \\mathcal{E}_{s}(\\|\\theta_{1}-\\theta_{2}\\|_{\\mathcal{F}^{0,1}} +|\\hat{\\vartheta_{1}}(0)-\\hat{\\vartheta_{2}}(0)|) \\\\+ \\mathcal{E}_{0}\\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{s,1}} + \\mathcal{C}_{s}\\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{2,1}} + \\tilde{\\Gamma}\\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{2+s,1}}\n\\end{multline}\nwhere $\\tilde{\\Gamma}$ is given by \\eqref{tildeGamma}.\n\\end{prop}\n\nWe further give the estimate of the form \\eqref{omeganonlinearest} when $s=0$ in \\eqref{omega.diff.zero.est}.  It is important to notice that $\\tilde{\\Gamma}$ given by \\eqref{tildeGamma} is smaller than the corresponding coefficient of $\\|\\theta\\|_{\\dot{\\mathcal{F}}^{s+2,1}}$ in the estimate of \\eqref{omega2fss}.\n\n\\begin{proof}\nFor the zero-th order term in the splitting we have\n\\begin{align*}\n(\\omega_{1})_{0}(\\alpha)-(\\omega_{2})_{0}(\\alpha) &= -2A_\\rho\\Big(\\frac{L_{1}(t)}{2\\pi}-\\frac{L_{2}(t)}{2\\pi} \\Big)\\sin{(\\alpha+\\widehat{\\vartheta}_{1}(0))}\\\\ &\\hspace{0.5in}- 2A_\\rho\\frac{L_{2}(t)}{2\\pi}(\\sin{(\\alpha+\\widehat{\\vartheta}_{1}(0))}-\\sin{(\\alpha+\\widehat{\\vartheta}_{2}(0))}).\n\\end{align*}\nHence, for $s \\ge 0$, \nwe have the estimate\n\\begin{align*}\n\\|(\\omega_{1})_{0}-(\\omega_{2})_{0}\\|_{\\dot{\\mathcal{F}}^{s,1}} &\\leq \\frac{\\left| A_\\rho \\right| }{\\pi} |L_{1}(t)-L_{2}(t)|\\|\\sin{(\\alpha+\\widehat{\\vartheta}_{1}(0))}\\|_{\\dot{\\mathcal{F}}^{s,1}} \\\\&\\hspace{0.25in}+  \\frac{\\left| A_\\rho \\right|}{\\pi}L_{2}(t)\\|\\sin{(\\alpha+\\widehat{\\vartheta}_{1}(0))}-\\sin{(\\alpha+\\widehat{\\vartheta}_{2}(0))}\\|_{\\dot{\\mathcal{F}}^{s,1}}.\n\\end{align*}\nWe have\n$\\|\\sin{(\\alpha+\\widehat{\\vartheta}_{1}(0))}\\|_{\\dot{\\mathcal{F}}^{s,1}} \\leq 1$\nand\n\\begin{multline*}\n\\|\\sin{(\\alpha+\\widehat{\\vartheta}_{1}(0))}-\\sin{(\\alpha+\\widehat{\\vartheta}_{2}(0))}\\|_{\\dot{\\mathcal{F}}^{s,1}} \\leq 2\\Big|\\sin\\Big(\\frac{\\widehat{\\vartheta}_{1}(0)-\\widehat{\\vartheta}_{2}(0)}{2} \\Big)\\Big|\\\\ \\cdot \\Big\\|\\cos\\Big(\\alpha +\n\\frac{\\widehat{\\vartheta}_{1}(0)+\\widehat{\\vartheta}_{2}(0)}{2} \\Big)\\Big\\|_{\\dot{\\mathcal{F}}^{s,1}}\n\\end{multline*}\nWe have \n$\\Big\\|\\cos\\Big(\\alpha +\\frac{\\widehat{\\vartheta}_{1}(0)+\\widehat{\\vartheta}_{2}(0)}{2} \\Big)\\Big\\|_{\\dot{\\mathcal{F}}^{s,1}} \\leq 1$\nand since we assume the difference is small we have\n\\begin{equation}\\label{sinezero}\n\\Big|\\sin\\Big(\\frac{\\widehat{\\vartheta}_{1}(0)-\\widehat{\\vartheta}_{2}(0)}{2} \\Big)\\Big| \\leq \\frac{1}{2} |\\widehat{\\vartheta}_{1}(0)-\\widehat{\\vartheta}_{2}(0)|.\n\\end{equation}\nHence, we obtain \\eqref{omega0diffest}, which completes the difference estimates for the zero order vorticity strength terms.\n\nNext, the linear difference in the vorticity strength terms from \\eqref{omegasplit} is\n\\begin{equation}\\label{linearomegadiff}\n(\\omega_{1})_{1}(\\alpha)-(\\omega_{2})_{1}(\\alpha) = \\frac{A_\\mu}{\\pi}W_{1} + 4A_\\sigma\\pi W_{2} -\\frac{A_\\rho}{\\pi}  W_{3}\n\\end{equation}\nwhere\n\\begin{equation}\\notag\nW_{1} =L_{1}(t)\\mathcal{D}_1((\\omega_{1})_0)(\\alpha) - L_{2}(t)\\mathcal{D}_1((\\omega_{2})_0)(\\alpha)\n\\end{equation}\nand\n\\begin{equation}\\notag\nW_{2} =\\Big(\\frac{1}{L_{1}(t)} -\\frac{1}{L_{2}(t)}\\Big) (\\theta_{1})_{\\alpha\\alpha} + \\frac{1}{L_{2}(t)}((\\theta_{1})_{\\alpha\\alpha} - (\\theta_{2})_{\\alpha\\alpha})\n\\end{equation}\nand\n\\begin{multline}\\notag\nW_{3} =(L_{1}(t)-L_{2}(t))\\cos{(\\alpha+\\widehat{\\vartheta}_{1}(0))}\\theta_{1}(\\alpha) \\\\+ L_{2}(t)[\\cos{(\\alpha+\\widehat{\\vartheta}_{1}(0))}-\\cos{(\\alpha+\\widehat{\\vartheta}_{2}(0))}]\\theta_{1}(\\alpha)\\\\ + L_{2}(t)\\cos{(\\alpha+\\widehat{\\vartheta}_{2}(0))}[\\theta_{1}(\\alpha)-\\theta_{2}(\\alpha)].\n\\end{multline}\nWe will estimate each of the terms  $W_{1}$,  $W_{2}$ and  $W_{3}$ in the following.\n\nFor $W_{1}$ we have using \\eqref{mdsplit} that\n\\begin{multline*}\nW_{1} = \\theta_{1}(\\alpha)\\mathcal{H}  ( (\\omega_{1})_0)(\\alpha)+\\text{Im}\\hspace{0.05cm}\\hspace{0.05cm}\\mathcal{R}((\\omega_{1})_0)(\\alpha)- \\theta_{2}(\\alpha)\\mathcal{H}  ((\\omega_{2})_0)(\\alpha)+\\text{Im}\\hspace{0.05cm}\\hspace{0.05cm}\\mathcal{R}((\\omega_{2})_0)(\\alpha).\n\\end{multline*}\nIt can be shown by the estimates in $\\mathcal{R}$ and $L_{1}-L_{2}$ that for $s \\ge 0$ we have\n\\begin{equation}\\label{D1est}\n\\|W_{1}\\|_{\\dot{\\mathcal{F}}^{s,1}} \\leq \\mathcal{C}_{s}(\\|\\theta_{1}-\\theta_{2}\\|_{\\mathcal{F}^{0,1}} + |\\widehat{\\vartheta}_{1}(0)-\\widehat{\\vartheta}_{2}(0)|)+ \\mathcal{E}\\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{s,1}}.\n\\end{equation}\nFor $W_{2}$, we have\n\\begin{equation}\\label{W2est}\n\\|W_{2}\\|_{\\dot{\\mathcal{F}}^{s,1}} \\leq \\frac{1}{L_{1}(t)L_{2}(t)}C_{L}\\|\\theta_{1}-\\theta_{2}\\|_{\\mathcal{F}^{0,1}}\\|\\theta_{1}\\|_{\\mathcal{F}^{s+2,1}}+ \\frac{1}{L_{2}(t)}\\|\\theta_{1} - \\theta_{2}\\|_{\\mathcal{F}^{s+2,1}}.\n\\end{equation}\nThe important term in $W_{2}$ for the purposes of the uniqueness argument is the second term that has the difference $\\|\\theta_{1} - \\theta_{2}\\|_{\\mathcal{F}^{s+2,1}}$.  For $W_{3}$, using \\eqref{lengthdifference}, we obtain that\n\\begin{equation}\\label{W3est2}\n    \\|W_{3}\\|_{\\dot{\\mathcal{F}}^{s,1}} \\leq \\mathcal{C}_{s}(\\|\\theta_{1}-\\theta_{2}\\|_{\\mathcal{F}^{0,1}}+|\\widehat{\\vartheta}_{1}(0)-\\widehat{\\vartheta}_{2}(0)|)+ \\mathcal{E}\\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{s,1}}.\n\\end{equation}\nHence, from \\eqref{D1est}, \\eqref{W2est} and \\eqref{W3est2}, we obtain from \\eqref{linearomegadiff} that\n\\begin{multline*}\n\\|(\\omega_{1})_{1}-(\\omega_{2})_{1}\\|_{\\dot{\\mathcal{F}}^{s,1}}\\leq \n\\mathcal{C}_{s}(\\|\\theta_{1}-\\theta_{2}\\|_{\\mathcal{F}^{0,1}} +|\\widehat{\\vartheta}_{1}(0)-\\widehat{\\vartheta}_{2}(0)|)\\\\ + \\mathcal{E} \\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{s,1}}  + \\frac{4A_{\\sigma}\\pi}{L_{2}(t)}\\|\\theta_{1} - \\theta_{2}\\|_{\\mathcal{F}^{s+2,1}}.\n\\end{multline*}\nNote that the coefficient in front of $\\|\\theta_{1} - \\theta_{2}\\|_{\\mathcal{F}^{s+2,1}}$ is the same as that in \\eqref{omega1fss} in front of $\\|\\theta\\|_{\\mathcal{F}^{s+2,1}_{\\nu}}$.\nThis completes the difference estimates for the linear terms in the vorticity strength given by \\eqref{linearomegadiffnorm}.\n\nNext we estimate differences of the nonlinear terms in the vorticity strength from \\eqref{omegasplit}.  We decompose the terms as\n\\begin{equation}\\label{omegadiff2}\n(\\omega_{1})_{\\geq 2}-(\\omega_{2})_{\\geq 2} = W_{21} + W_{22} + W_{23}\n\\end{equation}\nwhere the terms $W_{21}$, $W_{22}$, and $W_{23}$ are given by\n\\begin{equation}\\notag\nW_{21} = A_\\mu\\frac{L_{1}(t)}{\\pi}\\mathcal{D}_{\\geq2}(\\omega_{1})(\\alpha) - A_\\mu\\frac{L_{2}(t)}{\\pi}\\mathcal{D}_{\\geq2}(\\omega_{2})(\\alpha),\n\\end{equation}\nand\n\\begin{multline}\\notag\nW_{22} = A_\\rho\\frac{L_{2}(t)}{\\pi}\\sin{(\\alpha\\!+\\!\\widehat{\\vartheta}_{2}(0))}\\sum_{j\\geq1}\\!\\!\\frac{(-1)^j(\\theta_{2}(\\alpha))^{2j}}{(2j)!}\\\\ - A_\\rho\\frac{L_{1}(t)}{\\pi}\\sin{(\\alpha\\!+\\!\\widehat{\\vartheta}_{1}(0))}\\sum_{j\\geq1}\\!\\!\\frac{(-1)^j(\\theta_{1}(\\alpha))^{2j}}{(2j)!},\n\\end{multline}\nand\n\\begin{multline}\\notag\nW_{23} = A_\\rho\\frac{L_{2}(t)}{\\pi}\\cos{(\\alpha+\\widehat{\\vartheta}_{2}(0))}\\sum_{j\\geq1}\\frac{(-1)^j(\\theta_{2}(\\alpha))^{1+2j}}{(1+2j)!}\\\\-A_\\rho\\frac{L_{1}(t)}{\\pi}\\cos{(\\alpha+\\widehat{\\vartheta}_{1}(0))}\\sum_{j\\geq1}\\frac{(-1)^j(\\theta_{1}(\\alpha))^{1+2j}}{(1+2j)!}.\n\\end{multline}\nFirst, we use \\eqref{mdsplit} to observe that\n\\begin{multline}\\label{W21expansion}\nW_{21} = -A_{\\mu}\\Big(\\theta_{1}(\\alpha)\\mathcal{H}  ( (\\omega_{1})_{\\geq1})(\\alpha)+\\text{Im}\\hspace{0.05cm}\\hspace{0.05cm} \\mathcal{R}((\\omega_{1})_{\\geq1})(\\alpha)+\\text{Im}\\hspace{0.05cm}\\hspace{0.05cm}\\mathcal{S}(\\omega_{1})(\\alpha)\\\\ -\\theta_{2}(\\alpha)\\mathcal{H}  ( (\\omega_{2})_{\\geq1})(\\alpha)-\\text{Im}\\hspace{0.05cm}\\hspace{0.05cm} \\mathcal{R}((\\omega_{2})_{\\geq1})(\\alpha)-\\text{Im}\\hspace{0.05cm}\\hspace{0.05cm}\\mathcal{S}(\\omega_{2})(\\alpha) \\Big)\\\\\n= -A_{\\mu}(W_{211}+W_{212}+W_{213}),\n\\end{multline}\nwhere the differences of like terms with either subscript $1$ or $2$ are combined in each $W_{21j}$. Then for $s \\ge 0$ we have\n\\begin{multline}\\notag\n\\|W_{211}\\|_{\\dot{\\mathcal{F}}^{s,1}} \\leq b(2,s) \\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{s,1}}\\|(\\omega_{1})_{\\geq 1}\\|_{\\mathcal{F}^{0,1}} + b(2,s)\\|\\theta_{1}-\\theta_{2}\\|_{\\mathcal{F}^{0,1}}\\|(\\omega_{1})_{\\geq 1}\\|_{\\dot{\\mathcal{F}}^{s,1}}\\\\ + b(2,s)\\|\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{s,1}}\\|(\\omega_{1})_{1} -(\\omega_{2})_{1}\\|_{\\mathcal{F}^{0,1}} +  b(2,s)\\|\\theta_{2}\\|_{\\mathcal{F}^{0,1}}\\|(\\omega_{1})_{1} -(\\omega_{2})_{1}\\|_{\\dot{\\mathcal{F}}^{s,1}} \\\\ + b(2,s)\\|\\theta_{2}\\|_{\\mathcal{F}^{0,1}}\\|(\\omega_{1})_{\\geq 2} -(\\omega_{2})_{\\geq 2}\\|_{\\dot{\\mathcal{F}}^{s,1}} +b(2,s)\\|\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{s,1}}\\|(\\omega_{1})_{\\geq 2} -(\\omega_{2})_{\\geq 2}\\|_{\\mathcal{F}^{0,1}}.\n\\end{multline}\nThus, using \\eqref{omega1f0}, \\eqref{omega1fss}, \\eqref{omega2f0}, \\eqref{omega2fss}  and \\eqref{linearomegadiffnorm} we have\n\\begin{multline}\\label{W211est}\n\\|W_{211}\\|_{\\dot{\\mathcal{F}}^{s,1}} \\leq\n\\mathcal{E}_{s}(\\|\\theta_{1}-\\theta_{2}\\|_{\\mathcal{F}^{0,1}}+|\\widehat{\\vartheta}_{1}(0)-\\widehat{\\vartheta}_{2}(0)|) +\\mathcal{E}_{0}\\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{s,1}}\n\\\\\n+\\mathcal{C}_{s}\\|\\theta_{1}-\\theta_{2}\\|_{\\mathcal{F}^{2,1}} \n+\n\\frac{4A_{\\sigma}\\pi}{L_{2}(t)}b(2,s)\\|\\theta_{1}-\\theta_{2}\\|_{\\mathcal{F}^{2+s,1}} \n\\\\\n+ \nb(2,s)\\|\\theta_{2}\\|_{\\mathcal{F}^{0,1}}\\|(\\omega_{1})_{\\geq 2} -(\\omega_{2})_{\\geq 2}\\|_{\\dot{\\mathcal{F}}^{s,1}}\n+b(2,s)\\|\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{s,1}}\\|(\\omega_{1})_{\\geq 2} -(\\omega_{2})_{\\geq 2}\\|_{\\mathcal{F}^{0,1}},\n\\end{multline}\nwhere $\\mathcal{E}_{s} = \\|\\theta_{1},\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{2+s,1}}\\mathcal{E}$  and some bounded constant $\\mathcal{E}$ and $C$ is a bounded constant depending on $\\|\\theta_{1},\\theta_{2}\\|_{\\mathcal{F}^{0,1}}$.\n\nWe now consider the term $W_{212}$.  We have from \\eqref{Restimates} that\n\\begin{multline}\\label{Rdiffgeq1}\n\\|\\mathcal{R}((\\omega_{1})_{\\geq 1}) - \\mathcal{R}((\\omega_{2})_{\\geq 1})\\|_{\\dot{\\mathcal{F}}^{s,1}}\\\\ \\leq b(2,s){C_{\\mR}}(\\|(\\omega_{1})_{\\geq 1}-(\\omega_{2})_{\\geq 1}\\|_{\\dot{\\mathcal{F}}^{s,1}} \\|\\theta_{1}\\|_{\\mathcal{F}^{0,1}} + \\|(\\omega_{1})_{\\geq 1}-(\\omega_{2})_{\\geq 1}\\|_{\\mathcal{F}^{0,1}} \\|\\theta_{1}\\|_{\\dot{\\mathcal{F}}^{s,1}}\\\\+\\|(\\omega_{2})_{\\geq 1}\\|_{\\dot{\\mathcal{F}}^{s,1}} \\|\\theta_{1}-\\theta_{2}\\|_{\\mathcal{F}^{0,1}}+\\|(\\omega_{2})_{\\geq 1}\\|_{\\mathcal{F}^{0,1}} \\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{s,1}}).\n\\end{multline}\nHence, using \\eqref{omega1f0}, \\eqref{omega1fss}, \\eqref{omega2f0}, \\eqref{omega2fss} and \\eqref{linearomegadiffnorm}, we obtain\n\\begin{multline}\\label{W212est}\n\\|W_{212}\\|_{\\dot{\\mathcal{F}}^{s,1}}\\leq  \\mathcal{E}_{s}(\\|\\theta_{1}-\\theta_{2}\\|_{\\mathcal{F}^{0,1}}+|\\widehat{\\vartheta}_{1}(0)-\\widehat{\\vartheta}_{2}(0)|) +\\mathcal{E}_{0}\\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{s,1}}\\\\\n+ \\mathcal{E}\\|\\theta_{1},\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{s,1}}\\|\\theta_{1}-\\theta_{2}\\|_{\\mathcal{F}^{2,1}}\n+\\frac{4A_{\\sigma}{C_{\\mR}} \\pi}{L_{2}(t)}b(2,s)\\|\\theta_{1}-\\theta_{2}\\|_{\\mathcal{F}^{2+s,1}}\\|\\theta_{1},\\theta_{2}\\|_{\\mathcal{F}^{0,1}}\n\\\\ \n+ \nb(2,s){C_{\\mR}} (\\|\\theta_{1}\\|_{\\mathcal{F}^{0,1}}\\|(\\omega_{1})_{\\geq 2} -(\\omega_{2})_{\\geq 2}\\|_{\\dot{\\mathcal{F}}^{s,1}} + \\|\\theta_{1}\\|_{\\dot{\\mathcal{F}}^{s,1}}\\|(\\omega_{1})_{\\geq 2} -(\\omega_{2})_{\\geq 2}\\|_{\\mathcal{F}^{0,1}}).\n\\end{multline}\nThis is the estimate for $W_{212}$.\n\n\n\nNext for $W_{213}$ containing the difference in $\\mathcal{S}$, we actually have to consider two differences.\nWe recall the splitting of $\\mathcal{S}$ from \\eqref{Ssum} and we will use $f_i = \\omega_i$ below. First, it can be shown from  \\eqref{tildeS} that\n\\begin{align}\\label{Boperatordiff}\n\\|{\\mathcal{B}}(f_{1}) - {\\mathcal{B}}(f_{2})\\|_{\\dot{\\mathcal{F}}^{s,1}}\n&\\leq  B_{1} + B_{2}\n\\end{align}\nwhere\n\\begin{equation}\\notag\nB_{1} = \\Big\\||k|^{s}\\sum_{\\substack{n,l\\geq 0 \\\\ n+l\\geq 2}}\\frac{(-1)^ni^{l+n+1}(\\ast^{l}\\widehat{\\theta}_{1}(k))-\\ast^{l}\\widehat{\\theta_{2}}(k))}{l!} \\ast \\widehat{\\mathcal{S}_{n}(f_{1})}(k) \\Big\\|_{\\ell^{1}}\n\\end{equation}\nand\n\\begin{equation}\\notag\nB_{2} = \\Big\\||k|^{s}\\sum_{\\substack{n,l\\geq 0 \\\\ n+l\\geq 2}}\\frac{(-1)^{n} i^{l+n+1}\\ast^{l}\\widehat{\\theta}_{2}(k)}{l!} \\ast (\\widehat{\\mathcal{S}_{n}(f_{1})}(k)-\\widehat{\\mathcal{S}_{n}(f_{2})}(k)) \\Big\\|_{\\ell^{1}}.\n\\end{equation}\nFor $B_{1}$, we obtain using similar arguments to Proposition \\ref{uniquenessprop} that\n\\begin{multline}\\notag\nB_{1} \\leq \n\\sum_{\\substack{n,l\\geq 0 \\\\ n+l\\geq 2}} \\frac{b(l+1,s)}{(l-1)!} \\Big( \n \\|\\theta_{1},\\theta_{2}\\|_{\\mathcal{F}^{0,1}}^{l-1}\\|\\mathcal{S}_{n}(f_{1})\\|_{\\dot{\\mathcal{F}}^{s,1}} \\|\\theta_{1}-\\theta_{2}\\|_{\\mathcal{F}^{0,1}}\n \\\\\n+ \n(l-1)\\|\\theta_{1},\\theta_{2}\\|_{\\mathcal{F}^{0,1}}^{l-2}\\|\\theta_{1},\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{s,1}}\\|\\mathcal{S}_{n}(f_{1})\\|_{\\mathcal{F}^{0,1}}\n\\|\\theta_{1}-\\theta_{2}\\|_{\\mathcal{F}^{0,1}}\n\\\\ \n+ \\|\\theta_{1},\\theta_{2}\\|_{\\mathcal{F}^{0,1}}^{l-1}\\|\\mathcal{S}_{n}(f_{1})\\|_{\\mathcal{F}^{0,1}}\\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{s,1}}\\Big).\n\\end{multline}\nFor $B_{2}$, we first consider the difference in the operator $\\mathcal{S}_{n}$. By \\eqref{Snfourier}, we have for two functions $f_{1}$ and $f_{2}$, with $a_{n}$ given by \\eqref{an.def}, that\n\\begin{align*}\n|i^{n}(\\widehat{\\mathcal{S}_{n}(f_{1})}-\\widehat{\\mathcal{S}_{n}(f_{2})})(k)| &= \\sum_{k_{2},\\ldots, k_{n+1}\\in \\mathbb{Z}}|I(k_{1},\\ldots,k_{n+1})|\\\\\n&\\cdot\\Big(|\\hat{f_{1}}(k_{n+1})\\prod_{j=1}^{n}P_{1}(k_{j}-k_{j+1}) - \\hat{f_{2}}(k_{n+1})\\prod_{j=1}^{n}P_{2}(k_{j}-k_{j+1})|\\Big)\\\\\n&\\leq a_{n} |\\hat{f}_{1}-\\hat{f}_{2}|\\ast^{n} |P_{1}| + a_{n} |\\hat{f}_{2}|\\ast|\\ast^{n} P_{1}-\\ast^{n} P_{2}|.\n\\end{align*}\nHence using the estimates as in \\eqref{S1S2S3} we have\n\\begin{align}\\label{B2est}\n\\begin{split}\nB_{2} &\\leq \\sum_{\\substack{n,l\\geq 0 \\\\ n+l\\geq 2}} a_{n} \\Big\\||k|^{s}\\frac{\\ast^{l}\\widehat{\\theta}_{2}(k)}{l!} \\ast(|\\hat{f}_{1}-\\hat{f}_{2}|\\ast^{n} |P_{1}|) \\Big\\|_{\\ell^{1}} + \\tilde{B}_{2}\\\\\n&\\leq \\tilde{C}_3\\|\\theta_{2}\\|_{\\mathcal{F}^{0,1}}^2\\|f_{1}-f_{2}\\|_{\\dot{\\mathcal{F}}^{s,1}}+\\tilde{C}_4\\|\\theta_{2}\\|_{\\mathcal{F}^{0,1}}\\|\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{s,1}}\\|f_{1}-f_{2}\\|_{\\mathcal{F}^{0,1}} + \\tilde{B}_{2}\n\\end{split}\n\\end{align}\nwhere\n\\begin{align}\\notag\n\\tilde{B}_{2} = \\sum_{\\substack{n,l\\geq 0 \\\\ n+l\\geq 2}} a_{n} \\Big\\||k|^{s}\\frac{\\ast^{l}\\widehat{\\theta}_{2}(k)}{l!} \\ast(|\\hat{f}_{2}|\\ast|\\ast^{n} P_{1}-\\ast^{n} P_{2}|) \\Big\\|_{\\ell^{1}}.\n\\end{align}\nWe have for $s > 0$ that\n\\begin{multline}\\notag\n\\||k|^{s}(P_{1}-P_{2})\\|_{\\ell^{1}} \\leq \\sum_{m\\geq 1} \\frac{b(m,s)}{(m-1)!}(\\|\\theta_{1},\\theta_{2}\\|_{\\mathcal{F}^{0,1}}^{m-1}\\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{s,1}}\\\\ + (m-1)\\|\\theta_{1},\\theta_{2}\\|_{\\mathcal{F}^{0,1}}^{m-2}\\|\\theta_{1},\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{s,1}}\\|\\theta_{1}-\\theta_{2}\\|_{\\mathcal{F}^{0,1}})\n\\end{multline}\nwith an analogous estimate holding in the case $s=0$.\nHence we have\n\\begin{multline}\\notag\n\\tilde{B}_{2} \\leq \\mathcal{E}\\Big[(\\|\\theta_{1},\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{s,1}}+\\|f_{i}\\|_{\\dot{\\mathcal{F}}^{s,1}})\\|\\theta_{1}-\\theta_{2}\\|_{\\mathcal{F}^{0,1}} + (\\|\\theta_{1},\\theta_{2}\\|_{\\mathcal{F}^{0,1}}+\\|f_{i}\\|_{\\mathcal{F}^{0,1}})\\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{s,1}}\\Big].\n\\end{multline}\nWe also similarly estimate ${\\mathcal{A}}$ as in \\eqref{breveSsbound} to obtain\n\\begin{multline}\\notag\n\\|{\\mathcal{A}}(f_{1})(\\alpha) -\t{\\mathcal{A}}(f_{2})(\\alpha)\\|_{\\dot{\\mathcal{F}}^{s,1}} \\leq \\mathcal{E}\\Big[\\|f_{i}\\|_{\\dot{\\mathcal{F}}^{s,1}}\\|\\theta_{1}-\\theta_{2}\\|_{\\mathcal{F}^{0,1}} + \\|f_{i}\\|_{\\mathcal{F}^{0,1}}\\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{s,1}}\\Big]\\\\ + \\mathcal{C}_{\\mathcal{R}} C_{4}\\|\\theta_{1},\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{s,1}}\\|\\theta_{1},\\theta_{2}\\|_{\\mathcal{F}^{0,1}}\\|f_{1}-f_{2}\\|_{\\mathcal{F}^{0,1}} + \\mathcal{C}_{\\mathcal{R}} C_{3}\\|\\theta_{1},\\theta_{2}\\|_{\\mathcal{F}^{0,1}}^{2}\\|f_{1}-f_{2}\\|_{\\dot{\\mathcal{F}}^{s,1}}.\n\\end{multline}\nIn summary, for $s>0$, using \\eqref{omega0diffest} and \\eqref{linearomegadiffnorm}, we obtain\n\\begin{multline}\\label{W213est}\n    \\|W_{213}\\|_{\\dot{\\mathcal{F}}^{s,1}} \\leq \\mathcal{E}_{s}(\\|\\theta_{1}-\\theta_{2}\\|_{\\mathcal{F}^{0,1}} +|\\hat{\\vartheta_{1}}(0)-\\hat{\\vartheta_{2}}(0)|)+ \\mathcal{E}_{0}\\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{s,1}}\\\\ + \\mathcal{C}_{s}\\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{2,1}} + C_{3}\\|\\theta_{1},\\theta_{2}\\|_{\\mathcal{F}^{0,1}}^{2}\\frac{4A_{\\sigma}\\pi}{L_{2}(t)}\\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{2+s,1}}\\\\ + C_{4}\\|\\theta_{1},\\theta_{2}\\|_{\\mathcal{F}^{0,1}}\\|\\theta_{1},\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{s,1}}\\|(\\omega_{1})_{\\geq 2}-(\\omega_{1})_{\\geq 2}\\|_{\\mathcal{F}^{0,1}} + C_{3}\\|\\theta_{1},\\theta_{2}\\|_{\\mathcal{F}^{0,1}}^{2}\\|(\\omega_{1})_{\\geq 2}-(\\omega_{2})_{\\geq 2}\\|_{\\dot{\\mathcal{F}}^{s,1}},\n\\end{multline}\nwhere $C_{3}$ and $C_{4}$ are given by \\eqref{C3} and ${C_{\\mR}}$ is given by \\eqref{CR}.\n\n\n\nFurther using Proposition \\ref{uniquenessprop}, then $W_{22}$ and $W_{23}$ can be estimated by a bound of the type:\n\\begin{equation}\\label{W22W23}\n\\|W_{2j}\\|_{\\dot{\\mathcal{F}}^{s,1}} \\leq \\mathcal{C}_{s}(\\|\\theta_{1}-\\theta_{2}\\|_{\\mathcal{F}^{0,1}} +|\\widehat{\\vartheta}_{1}(0)-\\widehat{\\vartheta}_{2}(0)|)+\\mathcal{E}\\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{s,1}}\n\\end{equation}\nfor $j=2,3$.\nHence, using \\eqref{W211est}, \\eqref{W212est}, \\eqref{W213est}, we obtain from \\eqref{omegadiff2} that\n\\begin{multline}\\label{omeganonlinearestinitial}\n\\|(\\omega_{1})_{\\geq 2}-(\\omega_{2})_{\\geq 2}\\|_{\\dot{\\mathcal{F}}^{s,1}} \\leq \\mathcal{E}_{s}(\\|\\theta_{1}-\\theta_{2}\\|_{\\mathcal{F}^{0,1}} +|\\hat{\\vartheta_{1}}(0)-\\hat{\\vartheta_{2}}(0)|) \\\\+ \\mathcal{E}_{0}\\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{s,1}} + \\mathcal{C}_{s}\\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{2,1}} + \\tilde{\\Gamma}\\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{2+s,1}} \\\\\n+|A_{\\mu}|C_{12}\\|\\theta_{1},\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{s,1}}\\|(\\omega_{1})_{\\geq 2}-(\\omega_{2})_{\\geq 2}\\|_{\\mathcal{F}^{0,1}} +|A_{\\mu}|C_{2}\\|\\theta_{1},\\theta_{2}\\|_{\\mathcal{F}^{0,1}}\\|(\\omega_{1})_{\\geq 2}-(\\omega_{2})_{\\geq 2}\\|_{\\dot{\\mathcal{F}}^{s,1}}\n\\end{multline}\nwhere\n\\begin{equation}\\label{tildeGamma}\n\\tilde{\\Gamma} =|A_{\\mu}| \\frac{4A_{\\sigma}\\pi}{L_{2}(t)} C_{2}\\|\\theta_{1},\\theta_{2}\\|_{\\mathcal{F}^{0,1}}\n\\end{equation}\nwith $C_{2}$ given by \\eqref{C12}.\n\nComputing an estimate analogous to \\eqref{omeganonlinearestinitial} for $s=0$ yields the estimate\n\\begin{multline}\\label{omega.diff.zero.est}\n\\|(\\omega_{1})_{\\geq 2}-(\\omega_{2})_{\\geq 2}\\|_{\\dot{\\mathcal{F}}^{s,1}} \\leq \\mathcal{E}_{s}(\\|\\theta_{1}-\\theta_{2}\\|_{\\mathcal{F}^{0,1}} +|\\hat{\\vartheta_{1}}(0)-\\hat{\\vartheta_{2}}(0)|) \\\\+ \\mathcal{E}_{0}\\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{s,1}} + \\mathcal{C}_{s}\\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{2,1}} + \\Gamma\\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{2+s,1}}\n\\end{multline}\nwhere\n\\begin{equation}\\label{Gamma}\n\\Gamma = |A_{\\mu}|\\tilde{C}_{8}\\frac{4A_{\\sigma}\\pi}{L_{2}(t)}\n\\end{equation}\nfor $\\tilde{C}_{8}$ given by \\eqref{tildeC8} and the other constants given by Definition \\ref{uniquenessimplicitdef}. \n\\end{proof}\n\n\n\n\\subsection{Estimates for the differences of the main nonlinear term}\\label{subsec:uniq.nonlinear}\nIn this section, we show the following bound on the nonlinear difference term of \\eqref{theta1theta2}.\n\n\\begin{prop}\\label{nonlineardiffprop}\nWe have the following estimate for $\\delta>0$ given by \\eqref{delta} and for $\\epsilon(\\|\\theta_{1},\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}})$ that can be chosen to be arbitrarily small:\n\\begin{equation*}\n\\|N_{1}-N_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}} \\leq {\\mathcal{E}}(\\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}} + |\\hat{\\vartheta}_{1}(0)-\\hat{\\vartheta}_{2}(0)|) + (\\delta+\\epsilon) \\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac72,1}}\n\\end{equation*}\nwhere ${\\mathcal{E}}>0$ is a time integrable coefficient depending on $\\|\\theta_{1},\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}}$ and $\\|\\theta_{1},\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac72,1}}$.  Further $\\epsilon = \\epsilon(\\|\\theta_{1},\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}})>0$ can be chosen arbitrarily small.\n\\end{prop}\n\nWe remark that all the terms involving the upper bound of ${\\mathcal{E}}(\\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}} + |\\hat{\\vartheta}_{1}(0)-\\hat{\\vartheta}_{2}(0)|)$ follow similarly to the estimates from Section \\ref{secanalytic} also using the idea of Proposition \\ref{uniquenessprop} and the vorticity estimates in Proposition \\ref{prop:diff.vorticity.ests}.  Our proof below is focused on the estimates of the term $\\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac72,1}}$ and the constant $\\delta>0$. In the proof of Proposition \\ref{nonlineardiffprop}, when we compute the difference of nonlinear terms in $\\dot{\\mathcal{F}}^{\\frac12,1}$, any terms where the difference occurs as $\\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{s,1}}$ for $s<7\/2$ can be absorbed by interpolation and the Young's inequality into the term ${\\mathcal{E}}\\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}}$ and contributes an arbitrarily small term $\\epsilon \\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac72,1}}$ to be absorbed into the linear decay, e.g. see the estimate to obtain \\eqref{samplediff1}. This term is then taken care of by the Gronwall argument described in the comment below Theorem \\ref{uniquenessproposition}. In this way, most of the nonlinear terms can be easily estimated and only the few terms of order $\\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac72,1}}$ need be computed.\n\n\\begin{proof}\nFor the nonlinear terms, we will denote the decomposition given in \\eqref{N1N2N3} of $N_1$ for $\\theta_{1}$ and $N_2$ for $\\theta_{2}$ respectively by \n$$ N_{1} = N_{11}+N_{12}+N_{13},\n\\quad\nN_{2} = N_{21}+N_{22}+N_{23}.$$\nWe now consider the differences of $N_{1j}- N_{2j}$ for $j=1,2,3$. We will only explicitly compute the constant in front of terms with difference $\\| \\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac72,1}}$.  We make this idea clear in the following. Denoting $(U_{i})_{\\geq 2}$ for the term containing $\\theta_{i}$, we have\n\\begin{multline}\\label{N1diff}\n\\|N_{11}-N_{21}\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}} \\leq \\Big|\\frac{\\pi}{L_{1}(t)}-\\frac{\\pi}{L_{2}(t)}\\Big|\\|\\frac{L_{1}(t)}{\\pi}(U_{1})_{\\geq 2}\\|_{\\dot{\\mathcal{F}}^{\\frac32,1}} \\\\+ \\frac{1}{L_{2}(t)}\\|\\frac{L_{1}(t)}{\\pi}(U_{1})_{\\geq 2}-\\frac{L_{2}(t)}{\\pi}(U_{2})_{\\geq 2}\\|_{\\dot{\\mathcal{F}}^{\\frac32,1}}.\n\\end{multline}\nIn the latter term of \\eqref{N1diff}, we have a term of the form\n\\begin{multline}\\label{Romegadiff}\n    \\|\\mathcal{R}_{1}((\\omega_{1})_{\\geq 1})-\\mathcal{R}_{2}((\\omega_{2})_{\\geq 1})\\|_{\\dot{\\mathcal{F}}^{\\frac32,1}} \n    \\\\\n    \\leq \\ldots + \\frac{4A_{\\sigma}\\pi}{L_{2}(t)}\\|\\mathcal{R}_{1}((\\theta_{1})_{\\alpha\\alpha})-\\mathcal{R}_{2}((\\theta_{2})_{\\alpha\\alpha})\\|_{\\dot{\\mathcal{F}}^{\\frac32,1}}\n    + \\|\\mathcal{R}_{1}((\\omega_{1})_{\\geq 2})-\\mathcal{R}_{2}((\\omega_{2})_{\\geq 2})\\|_{\\dot{\\mathcal{F}}^{\\frac32,1}}.\n\\end{multline}\nAbove the dots ``$\\ldots$'' represent that other terms are present which turn out to be lower order.  We similarly denote $\\mathcal{R}_{i}$ for the term \\eqref{R} which contains $\\theta_{i}$.   Then for the first term present in the upper bound above we have the estimate\n\\begin{align}\\notag\n\\begin{split}\n\\|\\mathcal{R}_{1}((\\theta_{1})_{\\alpha\\alpha})-\\mathcal{R}_{2}((\\theta_{2})_{\\alpha\\alpha})\\|_{\\dot{\\mathcal{F}}^{\\frac32,1}} &\\leq  \\|\\mathcal{R}_{1}((\\theta_{1})_{\\alpha\\alpha}-(\\theta_{2})_{\\alpha\\alpha})\\|_{\\dot{\\mathcal{F}}^{\\frac32,1}}\\\\ &\\hspace{0.5in}+ \\|\\mathcal{R}_{1}((\\theta_{2})_{\\alpha\\alpha})-\\mathcal{R}_{2}((\\theta_{2})_{\\alpha\\alpha})\\|_{\\dot{\\mathcal{F}}^{\\frac32,1}}\\\\\n\\leq \\sqrt{2}{C_{\\mR}}&(\\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac72,1}}\\|\\theta_{1}\\|_{\\mathcal{F}^{0,1}} + \\|\\theta_{1}-\\theta_{2}\\|_{\\mathcal{F}^{2,1}}\\|\\theta_{1}\\|_{\\dot{\\mathcal{F}}^{\\frac32,1}}\\\\ &+\\|\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac72,1}}\\|\\theta_{1}-\\theta_{2}\\|_{\\mathcal{F}^{0,1}} + \\|\\theta_{2}\\|_{\\mathcal{F}^{2,1}}\\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac32,1}}).\n\\end{split}\n\\end{align}\nWe therefore have\n\\begin{multline}\\label{r1r2diffaa}\n\\|\\mathcal{R}_{1}((\\theta_{1})_{\\alpha\\alpha})-\\mathcal{R}_{2}((\\theta_{2})_{\\alpha\\alpha})\\|_{\\dot{\\mathcal{F}}^{\\frac32,1}}\n\\\\\n\\leq \\sqrt{2}{C_{\\mR}}(\\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac72,1}}\\|\\theta_{1}\\|_{\\mathcal{F}^{0,1}}+\\|\\theta_{1}\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}}^{2\/3} \\|\\theta_{1}\\|_{\\dot{\\mathcal{F}}^{\\frac72,1}}^{1\/3}\\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac72,1}}^{1\/2}\\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}}^{1\/2})\n\\\\\n+\\sqrt{2}{C_{\\mR}}(\\|\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac72,1}}\\|\\theta_{1}-\\theta_{2}\\|_{\\mathcal{F}^{0,1}} +\\|\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}}^{1\/2} \\|\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac72,1}}^{1\/2}\\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac72,1}}^{1\/3}\\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}}^{2\/3}).\n\\end{multline}\nHence, if we apply Young's inequality, e.g.\n\\begin{multline*}\n\\|\\theta_{1}\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}}^{2\/3} \\|\\theta_{1}\\|_{\\dot{\\mathcal{F}}^{\\frac72,1}}^{1\/3}\\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac72,1}}^{1\/2}\\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}}^{1\/2}\\\\ \\leq \\frac{1}{4\\epsilon}\\|\\theta_{1}\\|_{\\dot{\\mathcal{F}}^{\\frac72,1}}^{2\/3}\\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}} + \\epsilon \\|\\theta_{1}\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}}^{4\/3} \\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac72,1}}\n\\end{multline*}\nwe obtain\n\\begin{multline}\\label{samplediff1}\n\\|\\mathcal{R}_{1}((\\theta_{1})_{\\alpha\\alpha})-\\mathcal{R}_{2}((\\theta_{2})_{\\alpha\\alpha})\\|_{\\dot{\\mathcal{F}}^{\\frac32,1}} \\\\ \\leq \\mathcal{E}(\\|\\theta_{1},\\theta_{2}\\|_{_{\\dot{\\mathcal{F}}^{\\frac72,1}}}\\|\\theta_{1}-\\theta_{2}\\|_{\\mathcal{F}^{0,1}} +c_{\\epsilon}(\\|\\theta_{1},\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac72,1}}^{2\/3}+\\|\\theta_{1},\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac72,1}}^{3\/4})\\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}}) \\\\+\\epsilon (\\|\\theta_{1}\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}}^{4\/3}+\\|\\theta_{1}\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}}^{3\/2})\\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac72,1}}+ \\sqrt{2}{C_{\\mR}}\\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac72,1}}\\|\\theta_{1}\\|_{\\mathcal{F}^{0,1}}\n\\end{multline}\nfor a constant $c_{\\epsilon}$. The first two terms in \\eqref{samplediff1} are linear in $\\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{s,1}}$ for $0 \\leq s \\leq 1\/2$ and the constants are time integrable on $[0,T]$ for any $T>0$ since $\\|\\theta_{1},\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac72,1}}$ is integrable in time. The final two terms with the difference $\\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac72,1}}$ needs to be absorbed in the linear decay coming from $\\mathcal{L}_{1}-\\mathcal{L}_{2}$. For the other term in \\eqref{Romegadiff}, we have similarly,\n\\begin{multline}\n\\|\\mathcal{R}_{1}((\\omega_{1})_{\\geq 2})-\\mathcal{R}_{2}((\\omega_{2})_{\\geq 2})\\|_{\\dot{\\mathcal{F}}^{\\frac32,1}} \\leq \\ldots + \\sqrt{2}{C_{\\mR}}\\|\\theta_{1},\\theta_{2}\\|_{\\mathcal{F}^{0,1}}\\Gamma\\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac72,1}}\\\\ + \\epsilon\\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac72,1}},\n\\end{multline}\nwhere we use \\eqref{Gamma} and $\\epsilon=\\epsilon(\\|\\theta_{1},\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}})$ is a constant that can be chosen arbitrarily small.\n\nThe only other terms containing a term like $\\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac72,1}}$ are the other two terms that also come from $N_{11}-N_{12}$, as can be observed from the terms which contain $\\|\\theta\\|_{\\dot{\\mathcal{F}}^{\\frac72,1}}$ in the estimates of Section \\ref{secanalytic}. The first term is\n\\begin{align}\n\\frac{\\pi}{L_{2}(t)}\\|\\mathcal{H}  ( (\\omega_{1})_{\\geq2} )-\\mathcal{H}  ((\\omega_{2})_{\\geq2})\\|_{\\dot{\\mathcal{F}}^{\\frac32,1}} &= \\frac{\\pi}{L_{2}(t)}\\| (\\omega_{1})_{\\geq2}-(\\omega_{1})_{\\geq2}\\|_{\\dot{\\mathcal{F}}^{\\frac32,1}} \\nonumber \\\\\n&\\leq \\ldots + (\\frac{\\pi\\Gamma}{L_{2}(t)} + \\epsilon)\\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac72,1}}, \\label{Hdiff}\n\\end{align}\nwhere the unwritten terms are lower order due to being linear in $\\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{s,1}}$ for $0 \\leq s \\leq 1\/2$ with time integrable coefficients as done in \\eqref{samplediff1}. Again, $\\epsilon=\\epsilon(\\|\\theta_{1},\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}})$ is a constant that can be chosen arbitrarily small. Similarly, the final term that we need to compute is\n\\begin{align}\n\\frac{\\pi}{L_{2}(t)}\\|\\mathcal{S}(\\omega_{1})-\\mathcal{S}(\\omega_{2})\\|_{\\dot{\\mathcal{F}}^{\\frac32,1}} &\\leq \\ldots +\\frac{\\pi}{L_{2}(t)}\\|\\mathcal{S}((\\omega_{1})_{1})-\\mathcal{S}((\\omega_{2})_{1})\\|_{\\dot{\\mathcal{F}}^{\\frac32,1}} \\nonumber\\\\\n&\\hspace{0.5in}+\\frac{\\pi}{L_{2}(t)}\\|\\mathcal{S}((\\omega_{1})_{\\geq 2})-\\mathcal{S}((\\omega_{2})_{\\geq 2})\\|_{\\dot{\\mathcal{F}}^{\\frac32,1}} \\nonumber \\\\\n\\leq \\ldots& + (\\frac{\\pi\\Gamma C_{3}\\|\\theta_{1},\\theta_{2}\\|_{\\mathcal{F}^{0,1}}^{2}}{L_{2}(t)}+\\epsilon(\\|\\theta_{1},\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}}))\\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac72,1}}, \\label{Sneededdiff}\n\\end{align}\nwhere we use the estimate on $\\mathcal{S}(f_{1})-\\mathcal{S}(f_{2})$ computed to give \\eqref{W213est} and so $C_{3}$ is given by \\eqref{C3} and $\\Gamma$ is given by \\eqref{Gamma}.\n\n\nAll remaining nonlinear terms in the estimates of the evolution of $\\theta_{1}-\\theta_{2}$ are lower order due to having no futher upper bounds in terms of the highest order difference $\\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac72,1}}$.  Hence, in total the difference of nonlinear terms yields\n\\begin{equation}\\notag\n\\|N_{1}-N_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}} \\leq {\\mathcal{E}}(\\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}} + |\\hat{\\vartheta}_{1}(0)-\\hat{\\vartheta}_{2}(0)|) + (\\tilde{\\delta}+\\epsilon) \\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac72,1}},\n\\end{equation}\nwhere $\\epsilon=\\epsilon(\\|\\theta_{1},\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}})$ is a constant that can be chosen arbitrarily small; and for $\\Gamma= |A_{\\mu}|\\tilde{C}_{8}\\frac{4A_{\\sigma}\\pi}{L_{2}(t)}$ given by \\eqref{Gamma}, we have $\\tilde{\\delta}$ given by\n\\begin{equation*}\n\\tilde{\\delta} = \\frac{\\pi}{L_{2}(t)}(\\sqrt{2}{C_{\\mR}}\\frac{4A_{\\sigma}\\pi}{L_{2}(t)} + b(2,3\/2){C_{\\mR}}\\|\\theta_{1}, \\theta_{2}\\|_{\\mathcal{F}^{0,1}}\\Gamma + \\Gamma + \\Gamma C_{3}\\|\\theta_{1}, \\theta_{2}\\|_{\\mathcal{F}^{0,1}}^{2})\n\\end{equation*}\nand so by \\eqref{Lbound}, setting\n\\begin{multline}\\label{delta}\n\\delta =\n\\\\\n\\frac{A_{\\sigma}}{C_{37}^{2}R^{2}}(\\sqrt{2}{C_{\\mR}}+ b(2,3\/2){C_{\\mR}}\\|\\theta_{1}, \\theta_{2}\\|_{\\mathcal{F}^{0,1}}|A_{\\mu}|\\tilde{C}_{8}+ |A_{\\mu}|\\tilde{C}_{8} +|A_{\\mu}|\\tilde{C}_{8} C_{3}\\|\\theta_{1}, \\theta_{2}\\|_{\\mathcal{F}^{0,1}}^{2}),\n\\end{multline}\nwe obtain the result of Proposition \\ref{nonlineardiffprop}.\n\\end{proof}\n\n\\subsection{Proof of uniqueness}\\label{subsec:uniq.proof}\nWe are now ready to prove Theorem \\ref{uniquenessproposition}. \n\n \n\n\\begin{proof}[Proof of Theorem \\ref{uniquenessproposition}]\nFrom Proposition \\ref{linearfourier}, the difference of the linear terms is\n\\begin{multline}\\label{lineardifference}\n\\widehat{\\mathcal{L}}_{1}(k) - \\widehat{\\mathcal{L}}_{2}(k) \n=-A_\\sigma \\frac{4\\pi^2}{L_{2}(t)^2}k(k^2-1)(\\widehat{\\theta}_{1}(k)- \\widehat{\\theta}_{2}(k))\n\\\\\n-(1+A_\\mu)A_\\rho\\frac{(k^2-1)(k+1)}{k(k+2)}e^{-i\\widehat{\\vartheta}_{2}(0)} (\\widehat{\\theta}_{1}(k+1)- \\widehat{\\theta}_{2}(k+1)) \n\\\\\n+ \\tilde{L}_{1}(k) \n+ \\tilde{L}_{2}(k),\n\\end{multline}\nwhere\n\\begin{equation*}\n\\tilde{L}_{1}(k) = -4\\pi^2A_\\sigma k(k^2-1)\\widehat{\\theta}_{1}(k)\\Big(\\frac{1}{L_{1}(t)^2}-\\frac{1}{L_{2}(t)^2}\\Big)\n\\end{equation*}\nand \n\\begin{equation*}\n\\tilde{L}_{2}(k) = -(1+A_\\mu)A_\\rho\\frac{(k^2-1)(k+1)}{k(k+2)}(e^{-i\\widehat{\\vartheta}_{1}(0)}-e^{-i\\widehat{\\vartheta}_{2}(0)})\\widehat{\\theta}_{1}(k+1).\n\\end{equation*}\nAnd similarly for $k=2$.\nFor $\\tilde{L}_{1}$, \nwe have\n\\begin{equation}\\label{L1unique}\n\\|\\tilde{L}_{1}\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}}\\leq 4\\pi^{2}A_\\sigma \\Big|\\frac{1}{L_{1}(t)^{2}} - \\frac{1}{L_{2}(t)^{2}}\\Big|\\||k|^{3\/2}(k^{2}-1)\\widehat{\\theta}_{1}(k)\\|_{\\ell^{1}}.\n\\end{equation}\nFor $\\tilde{L}_{2}$, we have\n\\begin{multline}\\label{L2unique}\n\\|\\tilde{L}_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}}\n\\\\\n\\leq |(1+A_\\mu)A_\\rho| |e^{-i\\widehat{\\vartheta}_{1}(0)}-e^{-i\\widehat{\\vartheta}_{2}(0)}|\\Big\\|\\frac{|k|^{1\/2}|k^2-1||k+1|}{2|k||k+2|}|\\widehat{\\theta}_{1}(k+1)|  \\Big\\|_{\\ell^{1}}.\n\\end{multline}\nUsing similar arguments as earlier, we obtain from \\eqref{L1unique} and \\eqref{L2unique} that\n\\begin{equation}\\label{linearnonlinearparts}\n\\|\\tilde{L}_{1},\\tilde{L}_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}} \\leq {\\mathcal{E}}(\\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}} + |\\hat{\\vartheta}_{1}(0)-\\hat{\\vartheta}_{2}(0)|)\n\\end{equation}\nwhere ${\\mathcal{E}}$ is a time integrable coefficient. Hence, the new quantities from \\eqref{linearnonlinearparts} do not need to be absorbed by the linear decay. The coefficient $\\delta$ of the norm $\\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac72,1}}$ is less than the coefficients in \\eqref{Nbound}, and hence, $\\|\\theta_{1},\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac12,1}}$ satisfying \\eqref{condition} and taking $\\epsilon$ arbitrarily small is sufficient for $(\\delta+\\epsilon)\\|\\theta_{1}-\\theta_{2}\\|_{\\dot{\\mathcal{F}}^{\\frac72,1}}$ from Proposition \\ref{nonlineardiffprop} to be absorbed into the linear decay terms of \\eqref{lineardifference} by following the similar procedure to Section \\ref{subsecGlobal}.\n\\end{proof}\n\n\n\n\n\\subsection*{Acknowledgements}\nFG and EGJ were partially supported by the grant MTM2017-89976-P (Spain). FG, EGJ and NP were partially supported by the ERC through the Starting Grant project H2020-EU.1.1.-639227. RMS was partially supported by the NSF grant DMS-1764177 (USA).\n\n\n\n\n\\providecommand{\\bysame}{\\leavevmode\\hbox to3em{\\hrulefill}\\thinspace}\n\\providecommand{\\href}[2]{#2}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nApplications of random matrix theory cover many\nbranches of physics and cross-disciplinary\nfields~\\cite{RMTGENERAL} involving multivariate analysis of large\nand noisy data sets~\\cite{MULTIVARIATE}. The standard random matrix\nformulation belongs to the Gaussian basin, with a measure that is\nGaussian or polynomial with finite second moment. The ensuing\nmacroscopic spectral distribution is localized with finite supports\non the real axis. The canonical distribution for a Gaussian measure\nis Wigner's semi-circle.\n\nThe class of stable (L\\'{e}vy) distributions~\\cite{LEVY} is however much\nlarger (the Gaussian class represents only one fixed point in the\nstability basin of the L\\'{e}vy class), and one is tempted to ask, why\nthe theory of random L\\'{e}vy matrices is not so well established.\nThe case of L\\'{e}vy randomness is far from being academic, and many\ndistributions in physics and outside (finance, networks) exhibit\npower-like behavior referred to as {\\it fat} or\n{\\it heavy tails}~\\cite{FATTAIL}.\n\nOne of the chief reasons for why the theory of\nrandom L\\'{e}vy matrices is not yet well understood is the\ntechnical difficulty inherent to these distributions. First,\neven for one-dimensional stable distributions,\nthe explicit form of the probability distribution\nfunctions (pdf) is known analytically only in\nfew cases~\\cite{FELLER,ZOLOTARIEV}.\nSecond, L\\'{e}vy distributions have divergent moments, and the\nfiniteness of the second moment (condition for the Gaussian stability\nclass) is usually a key for many of the techniques established in\nRandom Matrix Theory. Third, numerical studies involving\npower-like behavior on infinite supports require enormous statistics\nand are very sensitive to systematic errors.\n\nIn 1994, Bouchaud and Cizeau~\\cite{BC} (hereafter BC)\nconsidered large $N\\times N$  random symmetric\nmatrices with entries sampled from one-dimensional, stable\ndistributions. In the large $N$ limit they obtained analytical\nequations for the entries of the resolvent, and then\nchecked their predictions for the spectra using numerically\ngenerated spectra by random sampling. The agreement was fair, although not\nperfect. Contrary to the standard Gaussian-like ensembles, the\nmeasure in the BC approach was not rotationally invariant.\n\nIn 2002, following the work in~\\cite{BERVOIC}, we have suggested\nanother L\\'{e}vy-type ensemble~\\cite{FREELEVY} (hereafter FRL).\nBy construction its measure is rotationally invariant.\nThe average spectral distribution in this ensemble is stable\nunder matrix the convolution of two independent but identical\nensembles. It is similar to the stability property of \none-dimensional L\\'{e}vy distributions. \nThe measure is non-analytic in the matrix $H$ and\nuniversal at large $H$ with a potential $V(H)\\approx {\\rm ln}\\,H^2$. This\nweak logarithmic rise in the asymptotic potential is at the origin\nof the long tail in the eigenvalue spectra. \n\nThe present work will compare the WL and FRL results as advertised\nin~\\cite{KRACOW}. In section~2 we reanalyze and correct the original\narguments for the resolvent presented in~\\cite{BC}. Our integral\nequations for the resolvent and spectral density are different from\nthe ones in~\\cite{BC}. We carry explicit analytical\ntransformations and expansions to provide insights to the spectrum.\nWe show that there is a perfect agreement between the analytical and\nnumerical results obtained by sampling large L\\'{e}vy matrices. \nWe also discuss the relation of ours and BC's results. \nIn section~3 we recall the key concepts behind FRL ensembles. In large\n$N$ the resolvent obeys a simple analytic equation. The resulting\nspectra are compared to the spectra following from the corrected BC\nanalysis. \nThe WL and FRL matrices represent two types of stability under\nmatrix convolution. In both cases we have a power behavior in the\ntails of the spectrum. By a pertinent rescaling we may in fact\nenforce the same tail behavior and compare the spectra. The observed\ndifferences disappear in the Gauss limit and become more pronounced\nin the Cauchy limit. In sectin 4 we explain the relation between \nthe two types of stability on a simple example: we construct sums \nof WL matrices rotated by random $O(N)$ matrices and show that \nthe spectrum of these sums converges by a matrix central \nlimit theorem to the pertinent symmetric FRL spectrum. Our \nconclusions are in section 5.\n\n\\section{Wigner-L\\'{e}vy Matrices}\n\n\\subsection*{Definition of the ensemble}\nIn a pioneering study on random L\\'{e}vy matrices, Bouchaud and\nCizeau~\\cite{BC} discussed a Wigner ensemble of $N\\times N$\nreal symmetric random matrices with elements being iid random variables:\nwith probability density function following a L\\'{e}vy\ndistribution $P(x)\\equiv N^{1\/\\mu} L_\\mu^{C,\\beta}\\left(N^{1\/\\mu} x\\right)$,\nwith $\\mu$ being the stability index, $\\beta$ -- the asymmetry parameter,\nand $C$ -- the range of the distribution (see below). We shall\ncall these matrices Wigner-L\\'evy (WL) or Bouchaud-Cizeau (BC) matrices.\nThe probability measure for the ensemble of such matrices \nis given by:\n\\begin{eqnarray}\nd\\mu_{WL}(H) =\\prod_{i\\le j}\\,P(H_{ij})\\,dH_{ij}\n\\label{measure}\n\\end{eqnarray}\nThe scaling factor $N^{1\/\\mu}$ in pdf makes the\nlimiting eigenvalue density independent of the matrix \nsize $N$ when $N\\rightarrow \\infty$. \nAlternatively one can think of the matrix elements\n$H_{ij}$ as if they were calculated as \n$H_{ij} = h_{ij}\/N^{1\/\\mu}$ with $h_{ij}$ being \niid random numbers independent of $N$:\n$p(x) = L_\\mu^{C,\\beta}(x)$.\n\nL\\'{e}vy distributions are notoriously hard to write explicitly\n(except in few cases), but their characteristic functions are more\nuser friendly~\\cite{FELLER}\n\\begin{eqnarray}\nL_{\\mu}^{C,\\,\n\\beta}(x)=\\frac{1}{2\\pi} \\int dk \\hat{L}(k)e^{ikx}\n\\label{levydef}\n \\end{eqnarray}\nwhere  the characteristic function is given by\n\\begin{eqnarray}\n\\log \\hat{L}(k)= -C|k|^{\\mu} (1+i \\beta ~{\\rm sign}(k) ~\\tan\n(\\pi \\mu \/2)).\n\\label{logchar}\n\\end{eqnarray}\nThe parameters $\\mu$, $\\beta$ and $C$ are related\nto the asymptotic behavior of $L_{\\mu}^{C,\\beta}(x)$\n\\begin{eqnarray}\n \\lim_{x \\rightarrow \\pm \\infty}  L_{\\mu}^{C ,\\beta}(x)=\n\\gamma(\\mu) \\frac{C (1\\pm \\beta)}\n{|x|^{\\mu+1}}\n\\label{levy1}\n\\end{eqnarray}\nwith  the $\\mu$-dependent parameter $\\gamma(\\mu)$ given by\n\\begin{eqnarray}\n\\gamma(\\mu)=\\Gamma(1+\\mu)\\sin(\\frac{\\pi\\mu}{2})\n\\end{eqnarray}\nHere $\\mu$ is the stability index defined in the interval $(0,2]$,\n$-1 \\le\\beta\\le 1$ measures the asymmetry of the distribution\nand the range $C > 0$ is the analogue of the variance, in a sense\nthat a typical value of $x$ is $C^{1\/\\mu}$. A standard choice\ncorresponds to $C=1$.\n\nWe shall consider here only the stability index in the range $(1,2)$,\nalthough as will be shown later, results obtained in this range seem to be\nvalid also for $\\mu=1$. We also assume that all random variables have zero\nmean.\n\n\\subsection*{Determination of the eigenvalue density}\n\nA method of calculating the eigenvalue density of the Wigner-L\\'evy\nmatrices was invented by Bouchaud and Cizeau \\cite{BC}. Let us \nin this section briefly recall the main steps of the method. \nIt is convenient to introduce the resolvent, called also Green's \nfunction:\n\\begin{eqnarray}\ng(z) =\\frac{1}{N}\\langle {\\rm Tr} \\; G(z) \\rangle\n\\end{eqnarray}\nwhere elements of the matrix $G(z)$ are\n\\begin{eqnarray} G_{ij}(z)=\n(z-H)_{ij}^{-1}\n\\label{resolvent}\n\\end{eqnarray}\nand the averaging is carried out using the measure (\\ref{measure}).\nThe resolvent contains the same information as the eigenvalue density \n$\\rho(\\lambda)$. Indeed if one approaches the real axis one finds\nthat $\\rho(\\lambda) = -1\/\\pi \\lim_{\\epsilon \\rightarrow 0^+}\n{\\rm Im} \\; g(\\lambda + i\\epsilon)$. This is\nhow one usually calculates $\\rho(\\lambda)$ from $g(z)$. \nFor the Wigner-L\\'evy ensemble, where individual matrix\nelements have large scale-free statistical fluctuations\na slightly different method turns out to be more practical --\na method which allows one to avoid problems in\ntaking the double limit (first $N\\rightarrow \\infty$ \nand $\\epsilon \\rightarrow 0^+$) in which the\nfluctuations are suppressed in an uncontrollable way in\nthe presence of an imaginary part of $z$.\nIn the BC method $z$ is kept on the real axis and\nfluctuations are not suppressed, so one can safely take\nthe large $N$ limit.\n\nThe method goes as follows \\cite{BC}. \nOne first generates a symmetric $N\\times N$ random matrix $H$\nusing the measure (\\ref{measure}) \nand then by inverting $H\\!-\\!z$ one\ncalculates the resolvent $G(z)$ (\\ref{resolvent}).\nNext one adds a new row (and a symmetric column) \nof independent numbers identically distributed as those in the old matrix $H$.\nOne obtains a new $(N\\!+\\!1)\\times (N\\!+\\!1)$ matrix $H^{+1}$, \nwhere $+1$ emphasizes that it has one more row and \none more column than $H$. \nIt is convenient to number elements of the original\nmatrix $H_{ij}$ by indices running over the range\n$i,j=1,\\dots, N$, and assign the index $0$ to the new row and the new column \nso that now indices of $H^{+1}$ run over the range $0,\\dots, N$.\nIf one inverts the matrix $H^{+1}\\! - \\!z$ one obtains a new \n$(N\\!+\\!1)\\times (N\\!+\\!1)$ resolvent $G^{+1}(z)$.\nOne can show that it obeys a recursive relation~\\cite{BC}\n\\begin{eqnarray}\nz-\\frac{1}{G_{00}^{+1}(z)}= H_{00} + \\sum_{i,j}^N H_{0i}\nH_{0j} G_{ij}(z) = \\frac{h_{00}}{N^{1\/\\mu}} + \\sum_{i,j}^N\n\\frac{h_{0i}h_{0j} G_{ij}(z)}{N^{2\/\\mu}}.\n\\label{diagoo}\n\\end{eqnarray}\nwhich relates the element $G_{00}^{+1}(z)$ \nof the $(N\\!+\\!1)\\times (N\\!+\\!1)$ resolvent to \nthe elements of the old $N\\times N$ resolvent $G(z)$.\nOne can also derive similar equations for off-diagonal elements\nof $G^{+1}(z)$:\n\\begin{eqnarray}\n\\frac{G_{0j}^{+1}(z)}{G_{00}^{+1}(z)}=\n\\sum_i^N \\frac{h_{0i}G_{ij}(z)}{N^{2\/\\mu}}\n\\label{off-diag}\n\\end{eqnarray}\nThe difference between the probability distribution of the \nelements of $G^{+1}(z)$ and of $G(z)$ dissapears \nin the limit $N\\rightarrow \\infty$. \nThe diagonal elements\nof the matrix $G^{+1}(z)$ are identically distributed as the diagonal \nelements of the matrix $G(z)$. The same holds\nfor off-diagonal ones. Moreover in this limit\nall elements of the $G$ matrix\nbecome independent of each other. In particular\nall diagonal elements of $G(z)$ \nbecome independent identically distributed (iid)\nrandom variables as $N\\rightarrow \\infty$.\nOne can use equations (\\ref{diagoo}) and\n(\\ref{off-diag}) to derive self-consistency equations \nfor the probability density function (pdf)\nfor diagonal and the pdf for off-diagonal\nelements. We are here primarily interested in \nthe distribution of the diagonal elements, as we shall\nsee below. The self-consistency\nequation for the probability distribution of diagonal\nelements follows from the equation (\\ref{diagoo}) \nand is independent of the distribution of the off-diagonal elements.\nThis can be seen as follows.\nLet us first define after Bouchaud and Cizeau~\\cite{BC} \na quantity:\n\\begin{eqnarray}\nS_0(z) = z - \\frac{1}{G^{+1}_{00}(z)}\n\\label{SG}\n\\end{eqnarray}\nIt is merely a convenient change of variables  suited\nto the left hand side of equation (\\ref{diagoo}).\nIt is clear that if one determines the pdf for $S=S_0(z)$,\none will also be able to determine\nthe pdf for $G=G^{+1}_{00}(z)$ since the two pdfs can\nbe obtained from each other by the \nchange of variables (\\ref{SG}):\n\\begin{eqnarray}\nP_G(G) = \\frac{1}{G^2} P_S\\left(z-\\frac{1}{G}\\right)\n\\label{vchange}\n\\end{eqnarray}\nwhere $P_G$ and $P_S$ are pdfs for $G^{+1}_{00}(z)$\nand $S_0(z)$ respectively. Since the probability distribution\n$P_G$ is identical for all diagonal elements of $G$,\nit remains to determine \nthe probability distribution $P_S$ for the quantity $S_0(z)$.\n\nLet us sketch how to do that.\nFirst observe that the first term in the equation (\\ref{diagoo})\ncan be neglected at large $N$, so the equation assumes the\nform:\n\\begin{eqnarray}\nS_0(z)=  \\sum_{i}^N\n\\frac{h^2_{0i} G_{ii}(z)}{N^{2\/\\mu}}\n+\\sum_{i\\neq j}\n\\frac{h_{0i}h_{0j} G_{ij}(z)}{N^{2\/\\mu}}\\,\\,.\n\\label{SELF1}\n\\end{eqnarray}\nNow observe that by construction $h_{0i}$ are independent\nof $G_{ij}(z)$. We shall now show that for large $N$\nthe first term in (\\ref{SELF1}) dominates over the\nsecond one. We shall modify here the  \nargument used in \\cite{BC}, where it was assumed that the off-diagonal \nelements $G_{ij}N(z),~i\\ne j$ are suppressed by a factor $1\/N^{1\/\\mu}$\nwith respect to the diagonal\nelements. Instead we note that by construction the quantities\n$G_{ij}(z)$ are statistically independent of $h_{0i},~i=0,\\dots,N$.\nSince we shall only be interested by the\ndiagonal elements of the resolvent matrix, we may replace the contribution\nof the off-diagonal elements $h_{0i}h_{0j}G_{ij}(z),~i\\ne j$ by their\naveraged values. As a result, the contribution of the off-diagonal terms\naverages out. Note that this is also true in the Gaussian limit $\\mu=2$\nwhere following \\cite{BC} we may replace all elements of (\\ref{SELF1}) by\ntheir respective averages. Taking this into account and omitting the\nsubleading contribution $H_{00}$, we get\n\\begin{equation}\nS_0(z) = \\sum_i^N \\frac{h_{0i}^2 G_{ii}(z)}{N^{2\/\\mu}}.\n\\label{SELF2}\n\\end{equation}\nThus the problem was simplified to \nan equation where the left hand side ($S_0=z-1\/G^{+1}_{00}$)\nand the right hand side depend only of the diagonal elements of \nthe $G$-matrix, which as we mentioned before, are \nidentically distributed in the limit $N \\rightarrow \\infty$.\nUsing (\\ref{SELF2}) one can derive a self-consistency equation for the \nprobability density function (pdf) $P_G$ for diagonal elements\nof the matrix $G$.\n\n\\subsection*{Generalized central limit theorem}\n\nTo proceed further we apply with Bouchaud and Cizeau~\\cite{BC}\nthe {\\it generalized central limit theorem} to derive\nthe universal behavior of the sum on the right hand side of\n(\\ref{SELF2}) in the limit $N\\rightarrow \\infty$:\n\\begin{itemize}\n\\item {\\bf i.} If the $h_{0 i}$'s are sampled from\nthe L\\'{e}vy distribution $L_{\\mu}^{C,\\beta}$, the squares $t_i=h_{0i}^2$\nfor large $t_i$ are distributed solely along the positive real axis,\nwith a heavy tail distribution:\n\\begin{equation}\n\\propto \\gamma(\\mu) \\frac{C dt_i}{t_i^{1+\\mu\/2}}\n\\end{equation}\nirrespective of $\\beta$. The sum\n\\begin{equation}\n\\sum_i^N \\frac{t_i}{N^{2\/\\mu}}\n\\end{equation} is distributed following\n$L_{\\mu\/2}^{C',1}(t_i)$. The range parameters $C$ and\n$C'$ are related by~\\rf{logchar}\n\\begin{equation}\n2 C' \\gamma(\\mu\/2) = C \\gamma(\\mu)\n\\label{scaling}\n\\end{equation}\nThe factor 2 on the left hand side corresponds to the sum $(1+\\beta)+(1-\\beta)$\nappearing as a contribution from positive and negative values of the original\ndistribution of $h_{0i}$.\nThis relation  is important when\ncomparing to the numerical results below where $C=1$ is used.\nFrom now on (and to simplify the equations) we assume instead that $C'=1$.\n\\item {\\bf ii.} By virtue of the central limit theorem \nthe following sum \n\\begin{equation}\n\\sum_i^N \\frac{G_{ii}(z) t_i}{N^{2\/\\mu}}\n\\end{equation}\nof iid heavy tailed numbers $t_i$ is \nfor $G_{ii}(z)={\\cal O}(N^0)$ and $N\\rightarrow \\infty$\nL\\'evy distributed with the pdf:\n$L_{\\mu\/2}^{C(z),\\beta(z)}$, which has the stability index $\\mu\/2$\nand the effective range $C(z)$ and the asymmetry parameter $\\beta(z)$\ncalculated from the equations:\n\\begin{equation}\nC(z)=\\frac{1}{N}\\sum_i^N |G_{ii}(z)|^{\\mu\/2}\n\\label{ceff}\n\\end{equation}\nand\n\\begin{equation}\n\\beta(z)=\\frac{\\frac{1}{N}\\sum_i^N |G_{ii}(z)|^{\\mu\/2}{\\rm sign}(G_{ii}(z))}\n{\\frac{1}{N}\\sum_i^N |G_{ii}(z)|^{\\mu\/2}}\\label{beta}\\,\\,.\n\\end{equation}\nwhich follow from the composition rules for the tail amplitudes\nof iid heavy tailed numbers $t_i$ defined above.\n\\end{itemize}\n\n\\subsection*{Integral Equations}\n\nWe saw in the previous section that \nthe generalized central limit theorem implies for large\n$N$ that the ``self-energy'' $S=S_0(z)$ is distributed according to the\nL\\'evy law $P_S(S) = L_{\\mu\/2}^{C(z),\\beta(z)}(S)$ with the\nstability index $\\mu\/2$ being a half of the stability index\nof the L\\'evy law\ngoverning the distribution of individual elements\nof the matrix $H$, and with the\neffective range parameter $C(z)$ and the\nasymmetry parameter $\\beta(z)$ which can be calculated\nfrom the equations (\\ref{ceff}) and (\\ref{beta}), respectively.\nOne should note that the effective parameters $C(z),\\beta(z)$\nof the distribution $P_S(S)$ are calculated\nfor $C=1$ and that they are independent of $\\beta$ of\nthe probability distribution: $L_\\mu^{C,\\beta}(h_{ij})$ of \nthe $H$-matrix elements.\n\nThe sums on the right hand side of the two equations \n(\\ref{ceff}), (\\ref{beta}) for $C(z)$ and $\\beta(z)$ \nhave a common form $\\frac{1}{N} \\sum_i f(G_{ii}(z))$.\nSince in the limit $N\\rightarrow \\infty$, the diagonal\nelements become iid, the sums on can be substituted\nby integrals over the probability density for $G_{ii}$:\n\\begin{eqnarray}\n\\frac{1}{N}\\sum_i^N \\langle f(G_{ii}(z))\\rangle = \n\\int dG \\; P_G(G) \\; f(G)  = \n\\int \\frac{dG}{G^2} P_S\\left (z-\\frac{1}{G}\\right) \\; f(G)\n\\end{eqnarray}\nwhere in the second step we used the equation (\\ref{vchange}).\nSince the distribution\n$P_S(S) = L_{\\mu\/2}^{C(z),\\beta(z)}(S)$ is known\nup to the values of two effective parameters\n$C(z)$ and $\\beta(z)$\nthe equations (\\ref{ceff}) and (\\ref{beta}) can be written\nas self-consistency relations for $\\beta(z)$ and $C(z)$:\n\\begin{eqnarray}\nC(z) &=& -\\hspace{-11pt}\\int_{-\\infty}^\\infty \\frac{dG}{G^2}\n|G|^{\\mu\/2}L_{\\mu\/2}^{C(z),\\beta(z)}\n(z-1\/G)\\nonumber \\\\\n\\beta(z)&=&\\frac{-\\hspace{-10pt}\\int_{-\\infty}^\\infty \\frac{dG}{G^2}\n|G|^{\\mu\/2}{\\rm sign}(G) L_{\\mu\/2}^{C(z),\\beta(z)}(z-1\/G)}\n{-\\hspace{-10pt}\\int_{-\\infty}^\\infty \\frac{dG}{G^2} |G|^{\\mu\/2}\nL_{\\mu\/2}^{C(z),\\beta(z)}(z-1\/G)}.\\label{correct}\n\\end{eqnarray}\nThe symbol $-\\hspace{-11pt}\\int$ stands for principal value of the\nintegral.  Notice here the difference between our second equation\nand that in~\\cite{BC}. In addition to~\\cite{BC} we also note\nthat the resolvent takes the form:\n\\begin{equation}\ng(z) = \\frac{1}{N} \\sum_i G_{ii}(z) \\ \\longrightarrow \\ \ng(z) = -\\hspace{-11pt}\\int_{-\\infty}^\\infty  \\frac{dG}{G^2}~G~ L_{\\mu\/2}^{C(z),\\beta(z)} (z-1\/G)\n\\label{G}\n\\end{equation}\nThe integrals \\rf{correct} and \\rf{G} can be rewritten using the\nnew integration variable $x=1\/G$ as\n\\begin{eqnarray}\nC(z)&=&\\int^{+\\infty}_{-\\infty} \ndx |x|^{-\\mu\/2}L_{\\mu\/2}^{C(z),\\beta(z)}(z-x)\\nonumber\\\\\n\\beta(z)&=&\n\\frac{\\int^{+\\infty}_{-\\infty} \ndx ~{\\rm sign}(x)|x|^{-\\mu\/2}L_{\\mu\/2}^{C(z),\\beta(z)}(z-x)}\n{\\int^{+\\infty}_{-\\infty} dx |x|^{-\\mu\/2}L_{\\mu\/2}^{C(z),\\beta(z)}(z-x)}\n\\label{correct1}\n\\end{eqnarray}\nand\n\\begin{equation}\ng(z)=-\\hspace{-11pt}\\int_{-\\infty}^\\infty  \\frac{dx}{x}L_{\\mu\/2}^{C(z),\\beta(z)}(z-x)\n\\end{equation}\nAll the steps above require both $z$ and $G_{ii}(z)$\nto be strictly real. The argument cannot be extended to the complex $z$ plane.\nSo all integrals above should be interpreted as principal value integrals, \nwherever it is necessary.\nSince $\\mu < 2$ all integrals in \\rf{correct1} are convergent also\nin the usual sense.\n\nThe equation for $g(z)$ can be rewritten as\n\\begin{equation} g(z)= -\\hspace{-11pt}\\int_{-\\infty}^\\infty \\frac{dx}{z-x}L_{\\mu\/2}^{C(z),\\beta(z)}(x).\n\\label{gdef1} \\end{equation} \nNotice the nontrivial dependence on $z$ in the\nparameters of the L\\'{e}vy distribution. Above equation can be a\nsource of  confusion, since its structure  resembles another\nrepresentation of $g(z)$\n\\begin{equation}\ng(z) =  -\\hspace{-11pt}\\int_{-\\infty}^\\infty  \\frac{d\\lambda}{z-\\lambda}\\rho(\\lambda)\n\\label{hilbert}\n\\end{equation}\nwhich superficially looks as if one could identify in \\rf{gdef1} $x$ and\n$\\lambda$ and $L_{\\mu\/2}^{C(z),\\beta(z)}(x)$ with $\\rho(\\lambda)$. This is\nnot the case, and one should instead invert \n(\\ref{gdef1}) using the inverse Hilbert transform:\n\\begin{equation}\n\\rho(\\lambda) = \\frac{1}{\\pi^2} -\\hspace{-11pt}\\int_{-\\infty}^\\infty  \\frac{dz}{z-\\lambda} g(z).\n\\label{disp}\n\\end{equation}\nIn other words, one first has to reconstruct numerically  the real\npart of the resolvent, and only then  compute numerically the\nspectral function  $\\rho(\\lambda)$, using the \"dispersive relation\"\n(\\ref{disp}). This is a difficult and rather subtle procedure.\nIn the next section we give some analytical insights to the\nintegral equations that would help solve them and extract the\nspectral function.\n\n\\subsection*{Analytical properties useful for numerics}\n\nOne cannot do the integrals from previous section analytically.\nAs mentioned, one cannot even write down an explicit form\nof the L\\'evy distribution. It is a great numerical challenge\nto solve the problem numerically even if all the expressions\nare given. One realizes that already\nwhen one tries to compute the Fourier integral (\\ref{levydef})\nof the characteristic function since one immediately\nsees that the integrand in the form (\\ref{levydef})\nis a strongly oscillating function making the numerics\nunstable. Fortunately using the power of the complex\nanalysis one can change this integral to a form which\nis numerically stable. So in this section we present \nsome analytic tricks which allow one to reduce the problem \nof computing the eigenvalue density \nas formulated in the previous section \nto a form which is well suited to \nthe numerical computation.\n\nTo simplify \\rf{correct1} we proceed in steps. First, we make\nuse of the L\\'{e}vy distribution through its characteristic\n\\begin{equation}\nL_{\\mu\/2}^{C,\\beta}(x)=\\frac{1}{2\\pi}\\int_{-\\infty}^{\\infty}dk\ne^{ikx} e^{-C|k|^{\\mu\/2}(1+i\\zeta ~{\\rm sign}(k))}. \\label{Levyf}\n\\end{equation}\nwith $\\zeta=\\beta~\\tan(\\pi\\mu\/4)$. By rescaling through\n\n\\begin{eqnarray} k &=& C^{-2\/\\mu}k',\\\\\n\\nonumber x &=& C^{2\/\\mu}x', \\\\ \\label{scale1} z &=& C^{2\/\\mu}z',\n\\nonumber\n\\end{eqnarray}\nwe can factor out the range\n$L_{\\mu\/2}^{C,\\beta}(x)=C^{-2\/\\mu}L_{\\mu\/2}^{1,\\beta}(x')$.\nSecond, we make use of the following integrals,\n\n\\begin{eqnarray}\n-\\hspace{-11pt}\\int_{-\\infty}^\\infty  \\frac{dx}{z-x}e^{ikx}&=&-2ie^{ikz}{\\rm sign}k,\\nonumber \\\\\n\\int_0^{\\infty}dx'~\\frac{\\cos(k'x')}{x'^{\\mu\/2}}&=&|k'|^{\\mu\/2-1}\\Gamma(1-\\mu\/2)\n\\sin(\\pi\\mu\/4),\\\\ \\nonumber\n\\int_0^{\\infty}dx'~\\frac{\\sin(k'x')}{x'^{\\mu\/2}}&=&|k'|^{\\mu\/2-1}{\\rm\nsign}(k') \\Gamma(1-\\mu\/2)\\cos(\\pi\\mu\/4).\n\\end{eqnarray}\nLast, we make use of the change of variables $p=k'^{\\mu\/2}$.\nWith this in mind, we obtain\n\n\\begin{eqnarray}\nC^2(z')&=&\\frac{4}{\\pi\\mu}\\Gamma\\left(1-\\frac{\\mu}{2}\\right)\n\\sin\\left(\\frac{\\pi\\mu}{4}\\right) \\int_0^{\\infty} dp\n\\cos(p^{2\/\\mu}z'-\\zeta(z')~ p)e^{-p},\\\\ \\label{C}\n\\zeta(z')&=&\n\\frac{\\int_0^{\\infty} dp \\sin(p^{2\/\\mu}z'-\\zeta(z')~ p)e^{-p}}\n{\\int_0^{\\infty} dp \\cos(p^{2\/\\mu}z'-\\zeta(z')~ p)e^{-p}}, \\label{bet}\n\\end{eqnarray}\nwith $\\zeta(z')=\\tan(\\pi\\mu\/4)~\\beta(z')$.\nFor every $z'$ we can iteratively solve the equation for $\\zeta(z')$,\nthen we determine $C(z')$ and use $z = C^{2\/\\mu}(z') z'$ to express\neverything in terms of $z$. These transformations solve~\\rf{correct1}.\nUsing the same method we rewrite the equation for $g(z)$ as\n\n\\begin{equation}\n\\bar{g}(z')=C(z')^{2\/\\mu}g(z) =\\frac{2}{\\mu}\\int_0^{\\infty}dp~p^{(2-\\mu)\/\\mu}\n\\sin(p^{2\/\\mu}z' - p~\\zeta(z'))~e^{-p}.\n\\end{equation}\n\nThese integral forms are useful to study the small-$z'$ limit. In this\ncase $\\zeta(z')$ is an antisymmetric function of $z'$ and has an expansion\nin powers of $z'$. Using\n$\\zeta(z')=k_1 z'+ {\\cal O}(z'^3) $ we can recursively obtain\nthe coefficients of this expansion. The first term is\n$k_1=\\Gamma(1+2\/\\mu)\/2$. Similarly $C(z')$ is a symmetric function in $z'$\n\n\\begin{equation}\nC^2(z')=\\frac{4}{\\pi\\mu}\\Gamma\\left(1-\\frac{\\mu}{2}\\right)\n\\sin\\left(\\frac{\\pi\\mu}{4}\\right)+{\\cal O}(z'^2)\n\\end{equation}\n\nFor $|z'|$ large ($z' \\to \\pm \\infty$) a different approach\nis needed. In this case we  follow\nNolan~\\cite{NOLAN} and treat the two integrals (numerator and\ndenominator) of \\rf{bet} together, i.e. we  consider the integral\n\n\\begin{equation} \\int_0^{\\infty} dp e^{-h(p)}, \\end{equation} where \\begin{equation} h(p) =\n(1-i\\zeta)p + iz' p^{2\/\\mu}.\n\\end{equation}\nNolan's idea is to close the contour of integration in the\ncomplex $p$ plane in the\nfollowing way: at $p \\to \\infty$ we add an arc and afterwards\ncontinue until $p=0$ along the line where $\\Im h(p)$=0. Using the\nparameterization $p=re^{i\\theta}$ we  get the parametric equation\nfor $r(\\theta)$ along this line, valid for $z'>0$\n\n\\begin{equation} r(\\theta)=\\left(\\frac{\\sin(\\theta_0-\\theta)}{z' \\cos(\\theta_0)\n\\cos(\\frac{2}{\\mu}\\theta)}\\right)^{\\mu\/(2-\\mu)}.\n\\end{equation}\nThe angle\n$\\theta$ for the curve we need ($\\mu \\ge 1$) is bounded between\n$\\theta_0 = {\\rm arctan}(\\zeta(z'))$, where $r=0$ and\n$-\\theta_1=-\\pi\\mu\/4$, where $\\cos(\\frac{2}{\\mu}\\theta)$ is zero.\nLet us now  introduce a new variable $\\psi$ through\n$\\theta=\\theta_0-\\psi$ and $0 \\le \\psi \\le \\theta_0+\\theta_1$. In\nthis range we have \\begin{eqnarray}\nV(\\psi)&=&\\left(\\frac{\\sin(\\psi)}{\\cos(\\theta_0)\n\\cos(\\frac{2}{\\mu}(\\psi-\\theta_0))} \\right)^{\\mu\/(2-\\mu)}, \\\\\n\\nonumber\n\\Re h(\\psi)&=&\\left(\\frac{1}{z'}\\right)^{\\mu\/(2-\\mu)}V(\\psi)\n\\frac{\\cos(\\frac{2-\\mu}{\\mu}\\psi\n-\\frac{2}{\\mu}\\theta_0)}{\\cos(\\theta_0)\n\\cos(\\frac{2}{\\mu}(\\psi-\\theta_0))}. \\end{eqnarray}\nAfter some manipulations we obtain\n\n\\begin{eqnarray} \\int \\sin(p^{2\/\\mu}z'-\\zeta(z')~ p)e^{-p}\n&=&\\int_0^{\\theta_0+\\theta_1}d\\psi\n\\left(\\frac{1}{z'}\\right)^{\\mu\/(2-\\mu)}V(\\psi)e^{-\n\\Re h(\\psi)}\\\\ \\nonumber\n&\\times&\\left(\\frac{2}{2-\\mu}\\frac{\\cos(\\frac{2-\\mu}{\\mu}(\\psi-\\theta_0))}\n{\\cos(\\frac{2}{\\mu}(\\psi-\\theta_0)}-\\frac{\\mu}{2-\\mu}\\frac{\\sin\\theta_0}\n{\\sin\\psi}\\right)\\\\ \\nonumber \\int \\cos(p^{2\/\\mu}z'-\\zeta(z')~ p)e^{-p}\n&=&\\int_0^{\\theta_0+\\theta_1}d\\psi\n\\left(\\frac{1}{z'}\\right)^{\\mu\/(2-\\mu)}V(\\psi)e^{-\n\\Re h(\\psi)}\\\\ \\nonumber\n&\\times&\\left(\\frac{2}{2-\\mu}\\frac{\\sin(\\frac{2-\\mu}{\\mu}(\\psi-\\theta_0))}\n{\\cos(\\frac{2}{\\mu}(\\psi-\\theta_0))}+\\frac{\\mu}{2-\\mu}\\frac{\\cos\\theta_0}\n{\\sin\\psi}\\right). \\label{integrals} \\end{eqnarray}\n\nThe resulting integrals look complicated, however they contain\nboth the small-$z'$ and the large-$z'$ asymptotics. For $z' \\to 0$\nwe have $\\psi = z' p^{2\/\\mu-1}$, which reproduces the small-$z'$\nexpansion presented above. For $z' \\to \\infty$ we have $\\psi =\n\\theta_0+\\theta_1-u\/z'^{\\mu\/2}$. Note that in this limit \\begin{equation}\n\\cos(\\frac{2}{\\mu}(\\psi-\\theta_0))=\\sin(\\frac{2}{\\mu}\\frac{u}{z'^{\\mu\/2}})\n\\end{equation} and the $z'$ dependence in ${\\rm Re}h(\\psi)$ vanishes in\nleading order.\n\nThe large-$z'$ asymptotics requires some work. For the\nleading orders we have\n\n\\begin{eqnarray}\n\\int \\sin(p^{2\/\\mu}z'-\\zeta(z')~ p)e^{-p}\n&\\approx&\\Gamma(1+\\mu\/2)\\sin\\theta_1~(z')^{-\\mu\/2}\\\\ \\nonumber \\int \\cos\n(p^{2\/\\mu}z'-\\zeta(z')~ p)e^{-p} &\\approx&\\Gamma(1+\\mu\/2)\n\\cos\\theta_1~(z')^{-\\mu\/2}.\n\\end{eqnarray}\nThus for $z' \\to \\infty$ we have\n\n\\begin{equation}\n\\zeta(z')=\\tan(\\theta_1)+{\\cal O}(1\/z'^{\\mu\/2}) \\end{equation} and \\begin{eqnarray}\nC(z')&=& z'^{-\\mu\/4}(1+{\\cal O}(1\/z'^{\\mu\/2}))\\\\ \\nonumber z\n&=&\\sqrt{z'}(1+{\\cal O}(1\/z'^{\\mu\/2})).\n\\end{eqnarray}\nAll the formulas above apply to the case $z'>0$. One can also derive similar\nformulas for $z'<0$, it is however more practical to use the symmetry\nproperties of the functions $C(z')$ and $\\zeta(z')$.\n\nAs a final check let us compute $\\bar{g}(z')$. A rerun of the\nabove transformations on the integrals give\n\n\\begin{eqnarray}\n\\bar{g}(z')&=&\\frac{2}{2-\\mu}\\int_0^{\\theta_0+\\theta_1}d\\psi\n\\left(\\frac{1}{z'}\\right)^{2\/(2-\\mu)}V(\\psi)^{2\/\\mu}e^{-{\\rm\nRe}h(\\psi)}\n\\\\ \\nonumber\n&\\times&\\left(\\frac{2}{\\mu}\\frac{1}{\\cos(\\frac{2}{\\mu}(\\psi-\\theta_0))}\n+\\frac{\\sin(\\frac{2-\\mu}{\\mu}\\psi-\\frac{2}{\\mu}\\theta_0)}\n{\\sin\\psi}\\right). \\end{eqnarray}\nNotice that both asymptotics follow from this representation.\nFor $z' \\to \\infty$ we have $\\bar{g}(z')=1\/z' + \\cdots$.\nwhich implies $g(z) = 1\/z +\\cdots$. This can be viewed as a check\nof the correct normalization of the eigenvalue density distribution.\n\n\\begin{figure\n\\psfrag{L}{\\bf{\\Large $\\lambda$}}\n\\psfrag{R}{\\bf{\\Large $\\rho(\\lambda)$}}\n\\centerline{\\scalebox{0.6}{\\rotatebox{0}{\\includegraphics{Large1.95.ps}}}}\n\\caption[phased]{{\\small Theoretical (black)\nand numerical (red) eigenvalue distributions for $\\mu=1.95$}}\n\\label{fig1}\n\\end{figure}\n\\begin{figure\n\\psfrag{L}{\\bf{\\Large $\\lambda$}}\n\\psfrag{R}{\\bf{\\Large $\\rho(\\lambda)$}}\n\\centerline{\\scalebox{0.6}{\\rotatebox{0}{\\includegraphics{Large1.75.ps}}}}\n\\caption[phased]{{\\small Theoretical (black)\nand numerical (red)  eigenvalue distribution for $\\mu=1.75$}} \\label{fig2}\n\\end{figure}\n\\begin{figure\n\\psfrag{L}{\\bf{\\Large $\\lambda$}}\n\\psfrag{R}{\\bf{\\Large $\\rho(\\lambda)$}}\n\\centerline{\\scalebox{0.6}{\\rotatebox{0}{\\includegraphics{Large1.50.ps}}}}\n\\caption[phased]{{\\small Theoretical (black)\nand numerical (red) eigenvalue distribution for $\\mu=1.50$}} \\label{fig3}\n\\end{figure}\n\\begin{figure\n\\psfrag{L}{\\bf{\\Large $\\lambda$}}\n\\psfrag{R}{\\bf{\\Large $\\rho(\\lambda)$}}\n\\centerline{\\scalebox{0.6}{\\rotatebox{0}{\\includegraphics{Large1.25.ps}}}}\n\\caption[phased]{{\\small Theoretical (black)\nand numerical (red) eigenvalue distribution for $\\mu=1.25$}} \\label{fig4}\n\\end{figure}\n\\begin{figure\n\\psfrag{L}{\\bf{\\Large $\\lambda$}}\n\\psfrag{R}{\\bf{\\Large $\\rho(\\lambda)$}}\n\\centerline{\\scalebox{0.6}{\\rotatebox{0}{\\includegraphics{Large1.00.ps}}}}\n\\caption[phased]{{\\small Theoretical (black)\nand numerical (red) eigenvalue distribution for $\\mu=1.00$}} \\label{fig5}\n\\end{figure}\n\n\n\n\\subsection*{Numerical Comparison}\n\nIn this section we show perfect agreement between the theoretical\nanalysis for the eigenvalue distribution~\\rf{disp}\nand the numerically generated eigenvalue distribution. The latter\nis obtained by diagonalizing $N\\times N$ random L\\'{e}vy matrices\nsampled using the measure~\\rf{measure}. The former was generated\nby calculating numerically $g(z)$ as detailed above.\nWe performed numerically the inverse Cauchy transform~\\rf{disp}.\nIt is important to note that the integral transforms entering into\nthe definition of the eigenvalue density $\\rho(\\lambda)$ in\n\\rf{disp} converge slowly for $z\\to \\infty$. For that, we have used\nthe asymptotic expansion of $g(z)$ to perform the large-$z$ part of\nthe integrals.\n\nAs noted above, all the above analytical results were obtained\nusing a specific choice of the scale factor~\\rf{scaling}. The\ncomparison with the numerically generated eigenvalue distribution\ngenerated using $L_{\\mu}^{1,0}$ distribution requires rescaling\nthrough $\\lambda \\to \\phi\\lambda$ and\n$\\rho(\\lambda)\\to\\rho(\\lambda)\/\\phi$ with\n\\begin{equation} \\phi=\n\\left(\\frac{\\Gamma(1+\\mu)\n\\cos(\\frac{\\pi\\,\\mu}{4})}{\\Gamma(1+\\frac{\\mu}{2})}\\right)^{1\/\\mu}\n\\label{phi}\n\\end{equation}\nBelow we  show a sequence of results for\n$\\mu=1.95,~1.75,~1.50,~1.25$ and $1.00$ with this rescaling.\nThe comparison is for high statistics $400\\times 400$ samples (red).\nWe have checked that the convergence is good already for $100\\times 100$\nsamples, with no significant difference between $N=100, 200, 400$.\nThe numerical results are also not sensitive to the choice $\\beta\\neq 0$.\nThe agreement between the results following from the integral equations\nand the numerically generated spectra is perfect.\nThis is true even for $\\mu=1$, where in principle the arguments used in\nthe derivation may not be valid.\n\n\n\\subsection*{Numerical observation}\n\nIn the mean-field approximation \\cite{BC} one can assume that\nthere are no correlations between large eigenvalues of the \nWigner-L\\'evy random matrix. In this case the eigenvalue \ndensity takes the form:\n\\begin{equation} \n\\widehat{\\rho}_\\mu(\\lambda) = L_{\\mu\/2}^{C(\\lambda),\\beta(\\lambda)}(\\lambda).\n\\label{gdef1hat} \n\\end{equation} \nIt is natural to ask how good this mean-field approximation is.\nThis can be done by comparing the mean-field eigenvalue \ndistribution $\\widehat{\\rho}(\\lambda)$ (\\ref{gdef1hat})\nto the eigenvalue distribution $\\rho(\\lambda)$ calculated \nby the inverse Hilbert transform (\\ref{disp}) of \nthe resolvent $g(z)$ (\\ref{gdef1}) as we did\nin previous section. We made this comparison numerically. \nThe result of this numerical experiment was that\nwithin the numerical accuracy which we achieved \nthe two curves representing $\\widehat{\\rho}(\\lambda)$\nand $\\rho(\\lambda)$ lied on top of each other \nin the whole studied range of $\\lambda$. Since our numerical\ncodes are written in {\\em Mathematica} we could push the \nnumerical accuracy very far, being only limited by the\nexecution time of the code. We have not seen any sign \nof deviation between the shapes of the two curves. \nThis provides us with a strong numerical evidence that\nthe mean-field argument \\cite{BC} gives \nan exact result but so far we have not managed to prove it.\nThe value of the eigenvalue density $\\widehat{\\rho}_\\mu(0)$ \nfor $\\lambda=0$ can be calculated analytically for the mean-field\ndensity (\\ref{gdef1hat}). Rescaling the density \n$\\widehat{\\rho}_\\mu(\\lambda) \\rightarrow \\widehat{\\rho}_\\mu(\\phi \\lambda)\/\\phi$ \nby the factor $\\phi$ (\\ref{phi}) we eventually obtain\n\\begin{equation}\n\\widehat{\\rho}_\\mu(0) = \n\\frac{\\Gamma(1+2\/\\mu)}{\\pi}\\left(\\frac{\\Gamma(1+\\mu\/2)^2}{\\Gamma(1+\\mu)}\\right)^\n{1\/\\mu}\n\\label{r0bc}\n\\end{equation}\nWe draw this function in Fig.\\ref{fig6}. In the same figure \nwe also show points representing numerically evaluated values of \nthe corresponding density $\\rho_\\mu(0)$ (\\ref{disp}) at some values\nof $\\mu$. Within the numerical accuracy $\\rho_\\mu(0)$ and \n$\\widehat{\\rho}_\\mu(0)$ assume the same values.\n\\begin{figure\n\\psfrag{M}{\\bf{\\Large $\\mu$}}\n\\psfrag{RH}{\\bf{\\Large $\\rho(0)$}}\n\\centerline{\\scalebox{0.6}{\\rotatebox{0}{\\includegraphics{rh000.ps}}}}\n\\caption[phased]{{\\small The line represents the function\n$\\widehat{\\rho}_\\mu(0)$ (\\ref{r0bc}) while the circles the values of  \n$\\rho_\\mu(0)$ computed numerically for $\\mu=1.0,1.25,1.5,1.75,1.95,2.0$ \nfrom the equation (\\ref{disp}).}} \n\\label{fig6}\n\\end{figure} \n\nThe physical meaning of the mean-field argument\nis that indeed one can think of the large eigenvalues as \nindependent of each other. A similar observation has\nbeen made recently \\cite{SF}. The mathematical meaning \nof the mean-field argument \nis more complex as we shall discuss below.\n\nLet us for brevity denote the function \n$L_{\\mu\/2}^{C(z),\\beta(z)}(x)$, which is a function \nof two real arguments, by \n$f(z,x) \\equiv L_{\\mu\/2}^{C(z),\\beta(z)}(x)$.\nThe equation (\\ref{gdef1}) can be now written as:\n\\begin{equation} \ng(z)= -\\hspace{-11pt}\\int_{-\\infty}^\\infty  d x \\frac{f(z,x)}{z-x}\n\\label{gdef2} \n\\end{equation} \nand (\\ref{disp}) as\n\\begin{equation}\n\\rho(\\lambda) = \\frac{1}{\\pi^2} -\\hspace{-11pt}\\int_{-\\infty}^\\infty  dz \\frac{g(z)}{z-\\lambda} .\n\\end{equation} \nWhen we insert (\\ref{gdef2}) into the last equation we have:\n\\begin{equation}\n\\rho(\\lambda) = \n\\frac{1}{\\pi^2} -\\hspace{-11pt}\\int_{-\\infty}^\\infty  dz -\\hspace{-11pt}\\int_{-\\infty}^\\infty  dx \\frac{f(z,x)}{(z-\\lambda)(z-x)}.\n\\end{equation}\nA question is when this exact expression for $\\rho(\\lambda)$ is\nequal to the mean-field solution: \n$\\widehat{\\rho}(\\lambda) = f(\\lambda,\\lambda)$.\nRecall the Poincar\\'e-Bertrand theorem.\nIt tells us that the following equation holds\n\\begin{equation}\nf(\\lambda,\\lambda) = \n\\frac{1}{\\pi^2} -\\hspace{-11pt}\\int_{-\\infty}^\\infty  dz -\\hspace{-11pt}\\int_{-\\infty}^\\infty  dx \\frac{f(z,x)}{(z-\\lambda)(z-x)} -\n\\frac{1}{\\pi^2} -\\hspace{-11pt}\\int_{-\\infty}^\\infty  dx -\\hspace{-11pt}\\int_{-\\infty}^\\infty  dz \\frac{f(z,x)}{(z-\\lambda)(z-x)}.\n\\label{pb}\n\\end{equation} \nWe see that the density $\\rho(\\lambda)$ is given by the mean-field\nresult: $\\rho(\\lambda)=\\widehat{\\rho}(\\lambda) \\equiv f(\\lambda,\\lambda)$\nif the second term on the right-hand side of (\\ref{pb}) vanishes.\nUnfortunately we have not managed to show that this is really\nthe case for $f(z,x) = L_{\\mu\/2}^{C(z),\\beta(z)}(x)$.\nOne can however trivially observe that it would be the case if\n$f(z,x)$ had the following form \n$f(z,x)=f(x,x)=L_{\\mu\/2}^{C(x),\\beta(x)}(x)$,\nand probably also if\n$f(z,x)$ were a slowly varying function of $z$ for $z$ \nclose to $x$, in which case the integral (\\ref{gdef2}) would\npick up only the contribution from $f(x,x)$ leading to\n(\\ref{gdef1hat}).\n\n\\section{Free Random L\\'{e}vy Matrices}\n\n\\subsection*{Rotationally invariant measure}\n\nClearly Wigner L\\'evy matrices are not rotationally invariant.\nIn this section we shall discuss orthogonally (or unitary) invariant\nensembles of L\\'evy matrices. It can be shown that maximally\nrandom measures for such matrices have the form \\cite{FREELEVY,BAL}:\n\\begin{eqnarray}\nd\\mu_{FR}(H)=\\prod_{i\\le j}\\,dH_{ij}\\,e^{-\\,N{\\rm Tr}V(H)}\n\\label{frmeasure}\n\\end{eqnarray}\nWe shall be interested only in potentials with have tails\nwhich lead to eigenvalue distributions (spectral densities)\nwith heavy tails $\\rho(\\lambda) \\sim \\lambda^{-1-\\mu}$ \nbelonging to the L\\'{e}vy domain of attraction.\nA generic form of $V(\\lambda)$ at asymptotic eigenvalues $\\lambda$\nis in this case\n\\begin{eqnarray}\nV(\\lambda)={\\rm ln}\\lambda^2+{\\cal O}(1\/\\lambda^\\mu)\n\\end{eqnarray}\nIn general the potential does not have to be an analytic function.\nWe shall be interested here only in stable ensembles in\nthe sense that the spectral measure~\\rf{frmeasure} \nfor the convolution of two independent and identical ensembles \nhas the same form as the measure of the individual\nensembles. In other words, the spectral measure for\na matrix constructed as a sum of two independent matrices\ntaken from the ensemble has exactly the same spectral measure\n(eigenvalue density) modulo linear transformations.\n\nIt turns out that one can classify all the stable spectral\nmeasures thanks to the relation of the problem to free probability\ncalculus. The matrix ensemble (\\ref{frmeasure}) is in the large\n$N$ limit a realization of free random variables\n\\cite{FREELEVY}, so one can use theorems developed in free random\nprobability~\\cite{BERVOIC}. In particular we can use the fact that\nin free probability theory stable laws are classified. They actually\nparallel stable laws (\\ref{logchar})\nof classical probability theory.\nIn free probability the analogue of the {\\em logarithm} of the\ncharacteristic function (\\ref{logchar}) is the\nR-transform, introduced  by Voiculescu~\\cite{VOICULESCU}.\nThe R-transform linearizes the matrix convolution, generating \nspectral cumulants, which are additive under convolution.\n\\def\\begin{eqnarray}{\\begin{eqnarray}}\n\\def\\end{eqnarray}{\\end{eqnarray}}\n\\def\\begin{eqnarray}{\\begin{eqnarray}}\n\\def\\end{eqnarray}{\\end{eqnarray}}\n\\def\\langle{\\langle}\n\\def\\rangle{\\rangle}\n\\def{\\textstyle \\frac{1}{2}}{{\\textstyle \\frac{1}{2}}}\n\\def{\\cal O}{{\\cal O}}\n\\def{\\cal B}{{\\cal B}}\n\\newcommand{{\\mbox{e}}}{{\\mbox{e}}}\n\\def\\partial{\\partial}\n\\def{\\vec r}{{\\vec r}}\n\\def{\\vec k}{{\\vec k}}\n\\def{\\vec q}{{\\vec q}}\n\\def{\\vec p}{{\\vec p}}\n\\def{\\vec P}{{\\vec P}}\n\\def{\\vec \\tau}{{\\vec \\tau}}\n\\def{\\vec \\sigma}{{\\vec \\sigma}}\n\\def{\\vec J}{{\\vec J}}\n\\def{\\vec B}{{\\vec B}}\n\\def{\\hat r}{{\\hat r}}\n\\def{\\hat k}{{\\hat k}}\n\\def\\roughly#1{\\mathrel{\\raise.3ex\\hbox{$#1$\\kern-.75em%\n\\lower1ex\\hbox{$\\sim$}}}}\n\\def\\roughly<{\\roughly<}\n\\def\\roughly>{\\roughly>}\n\\def{\\mbox{fm}}{{\\mbox{fm}}}\n\\def{\\vec x}{{\\vec x}}\n\\def{\\rm EM}{{\\rm EM}}\n\\def{\\mbox{\\boldmath $\\pi$}}{{\\mbox{\\boldmath $\\pi$}}}\n\\def{\\mbox{\\boldmath $\\rho$}}{{\\mbox{\\boldmath $\\rho$}}}\n\\def{\\mbox{\\boldmath $\\alpha$}}{{\\mbox{\\boldmath $\\alpha$}}}\n\\def{\\mbox{\\boldmath $\\Sigma$}}{{\\mbox{\\boldmath $\\Sigma$}}}\n\\def{\\mbox{\\boldmath $\\Pi$}}{{\\mbox{\\boldmath $\\Pi$}}}\n\\def{\\mbox{\\boldmath $\\Gamma$}}{{\\mbox{\\boldmath $\\Gamma$}}}\n\\def{\\rm Tr}\\,{{\\rm Tr}\\,}\n\\def{\\bar p}{{\\bar p}}\n\\def{z \\bar z}{{z \\bar z}}\n\\def{\\cal M}_s{{\\cal M}_s}\n\\def\\abs#1{{\\left| #1 \\right|}}\n\\def{\\vec \\epsilon}{{\\vec \\epsilon}}\n\\def\\nlo#1{{\\mbox{N$^{#1}$LO}}}\n\\def{\\mbox{M1V}}{{\\mbox{M1V}}}\n\\def{\\mbox{M1S}}{{\\mbox{M1S}}}\n\\def{\\mbox{E2S}}{{\\mbox{E2S}}}\n\\def{\\cal R}_{\\rm M1}}\\def\\rE{{\\cal R}_{\\rm E2}{{\\cal R}_{\\rm M1}}\\def\\rE{{\\cal R}_{\\rm E2}}\n\\def\\J#1#2#3#4{ {#1} {\\bf #2} (#4) {#3}. }\n\\defPhys. Rev. Lett.{Phys. Rev. Lett.}\n\\defPhys. Lett.{Phys. Lett.}\n\\defPhys. Lett. B{Phys. Lett. B}\n\\defNucl. Phys.{Nucl. Phys.}\n\\defNucl. Phys. A{Nucl. Phys. A}\n\\defNucl. Phys. B{Nucl. Phys. B}\n\\defPhys. Rev.{Phys. Rev.}\n\\defPhys. Rev. C{Phys. Rev. C}\n\\def{\\cal M}{{\\cal M}}\n\\newcommand{\\begin{equation}}{\\begin{equation}}\n\\newcommand{\\end{equation}}{\\end{equation}}\n\\newcommand{\\begin{eqnarray}}{\\begin{eqnarray}}\n\\newcommand{\\end{eqnarray}}{\\end{eqnarray}}\n\\newcommand{\\f}[2]{\\frac{#1}{#2}}\n\\newcommand{\\lambda}{\\lambda}\n\\newcommand{\\Lambda}{\\Lambda}\n\\newcommand{\\longrightarrow}{\\longrightarrow}\n\\newcommand{\\mbox{\\rm tr}}{\\mbox{\\rm tr}}\n\\newcommand{\\alpha}{\\alpha}\n\\newcommand{\\beta}{\\beta}\n\\newcommand{\\varepsilon}{\\varepsilon}\n\\newcommand{\\delta}{\\delta}\n\n\\subsection*{Stable laws in free probability}\nThe remarkable achievement by Bercovici and Voi\\-cu\\-les\\-cu~\\cite{BERVOIC}\nis an explicit derivation of all R-transforms defined by the equation\n$R(G(z))=z-1\/G(z)$ where $G(z)$ is the resolvent  for all free\nstable distributions. \nWe just note that $R(G(z))$ is a sort of self-energy for rotationally\nsymmetric FRL ensembles which is the analogue of (\\ref{SELF2}) which\nwe previously defined for Wigner-L\\'evy ensembles. \nIt is self-averaging and additive.\n\nFor stable laws $R(z)$ is known. It \ncan has either the trivial form $R(z)=a$ or\n\\begin{eqnarray}\nR(z) = b z^{\\mu-1}\n\\end{eqnarray}\nwhere $0<\\mu<2$, $b$ is a parameter which can be related to the\nstability index $\\mu$, the asymmetry parameter $\\beta$, and the range $C$ \nknown from the corresponding stable laws (\\ref{logchar}) of\nclassical probability \\cite{BERVOIC,PATA}\n\\indent{\\begin{eqnarray}\nb = \\left\\{ \\begin{array}{cl}\nC\\ e^{i(\\frac{\\mu}2-\\!1)(1\\!+\\!\\beta)\\pi}\n     {\\rm ~~for~~} 1 <\\mu<2 \\nonumber \\\\\n C \\ e^{i[\\pi+\\frac{\\mu}2(1\\!+\\!\\beta)\\pi]}\n      {\\rm ~~for~~} 0 <\\mu <1 \\nonumber\n    \\end{array} \\right. \\, .\n\\end{eqnarray}}\nIn the marginal case: $\\mu=1$, $R(z)$ reads:\n\\begin{eqnarray}\nR(z) = - i C(1+\\beta) -\n\\frac{2\\beta C}{\\pi} \\ln C z\n\\end{eqnarray}\nThe branch cut structure of $R(z)$ is chosen in such a way that the\nupper complex half plane is mapped to itself.\nRecalling that $R=z-1\/G$ in the large $N$ limit,\none finds that for the trivial case $R(z)=0$,\nthe resolvent: $G(z)=z^{-1}$\nand the spectral distribution\nis a Dirac delta, $\\rho(\\lambda)=\\delta(\\lambda)$.\nOtherwise, on the upper half-plane, the resolvent fulfills\nan algebraic equation\n\\begin{eqnarray}\nbG^{\\mu}(z)-zG(z)+1=0\\,\\,,\n\\label{Levygreen}\n\\end{eqnarray}\nor in the marginal case ($\\mu=1$):\n\\begin{eqnarray}\n\\bigg(\\!z\\!+i C(1\\!+\\!\\beta)\\!\\bigg) G(z) +\\frac{2\\beta C}{\\pi}\nG(z)\\ln C G(z) - 1\\! = \\!0.\n\\end{eqnarray}\nOn the lower half-plane $G(\\bar{z})=\\bar{G}(z)$ \\cite{BERVOIC}.\n\nThe equation for the resolvent (\\ref{Levygreen})\nhas explicit solutions only for the following values:\n$\\mu=1\/4,1\/3,1\/2, 2\/3,3\/4,4\/3,3\/2$ and $2$.\nIn all other cases the equation is transcendental and one\nhas to apply numerical procedures to unravel the spectral distribution.\nAgain the form of the potential generating\nstable free L\\'{e}vy ensembles is highly non-trivial and is\nonly known in few cases~\\cite{FREELEVY}. We refer to~\\cite{FREELEVY}\nfor further references and discussions.\n\n\\subsection*{Comparison of free L\\'evy and Wigner-L\\'evy spectra}\n\n\\begin{figure\n\\psfrag{L}{\\bf{\\Large $\\lambda$}}\n\\psfrag{R}{\\bf{\\Large $\\rho(\\lambda)$}}\n\\centerline{\\scalebox{0.6}{\\rotatebox{0}{\\includegraphics{comp1.95new.ps}}}}\n\\caption[phased]{{\\small WL (black) versus FRL (red)\nfor $\\mu=1.95$}} \\label{fig7}\n\\end{figure}\n\\begin{figure\n\\psfrag{L}{\\bf{\\Large $\\lambda$}}\n\\psfrag{R}{\\bf{\\Large $\\rho(\\lambda)$}}\n\\centerline{\\scalebox{0.6}{\\rotatebox{0}{\\includegraphics{comp1.75new.ps}}}}\n\\caption[phased]{{\\small WL (black) versus FRL (red)\nfor $\\mu=1.75$}} \\label{fig8}\n\\end{figure}\n\\begin{figure\n\\psfrag{L}{\\bf{\\Large $\\lambda$}}\n\\psfrag{R}{\\bf{\\Large $\\rho(\\lambda)$}}\n\\centerline{\\scalebox{0.6}{\\rotatebox{0}{\\includegraphics{comp1.50new.ps}}}}\n\\caption[phased]{{\\small WL (black) versus FRL (red)\nfor $\\mu=1.50$}} \\label{fig9}\n\\end{figure}\n\\begin{figure\n\\psfrag{L}{\\bf{\\Large $\\lambda$}}\n\\psfrag{R}{\\bf{\\Large $\\rho(\\lambda)$}}\n\\centerline{\\scalebox{0.6}{\\rotatebox{0}{\\includegraphics{comp1.25new.ps}}}}\n\\caption[phased]{{\\small WL (black) versus FRL (red)\nfor $\\mu=1.25$}} \\label{fig10}\n\\end{figure}\n\\begin{figure\n\\psfrag{L}{\\bf{\\Large $\\lambda$}}\n\\psfrag{R}{\\bf{\\Large $\\rho(\\lambda)$}}\n\\centerline{\\scalebox{0.6}{\\rotatebox{0}{\\includegraphics{comp1.00new.ps}}}}\n\\caption[phased]{{\\small WL (black) versus FRL (red)\nfor $\\mu=1.00$}} \\label{fig11}\n\\end{figure}\nWe present in Figures 7-11 several comparisons between\nthe free random L\\'{e}vy spectra (FRL) following from the\nsolution to the transcendental equation (red) and the\nrandom L\\'{e}vy spectra (BC) obtained by solving the coupled\nintegral equations (black), for zero asymmetry ($\\beta=0$)\nand different tail indices $\\mu$. The FRL spectra are normalized\nto agree with the BC spectra in the tails of the distributions. \nWe recall that the FRL spectra asymptote\n$\\rho(\\lambda)\\approx {\\rm sin}(\\pi \\mu\/2)\/\\pi$. \nThe comparison in bulk shows that the spectra are similar, in particular \nclose to the Gaussian limit $\\mu=2$, where both approaches become equivalent.\nFor smaller $\\mu$ there are differences.\n\nWL and FRL matrices represent two types of random matrices \nspectrally stable under matrix\naddition. For the WL matrices it follows from the measure, since each\nmatrix elements is generated from a stable L\\'{e}vy distribution and therefore\nthe sum of $\\cal N$ WL matrices, scaled by $1\/{\\cal N}^{1\/\\mu}$ is equivalent \nto the original WL ensemble. The important point is that the WL measure is\nnot symmetric while the FRL one is. \n\n\\section{Spectral stability and maximal entropy principle}\n\nThe matrix ensembles discussed in this paper are\nstable with respect to matrix addition in the sense\nthat the eigenvalue distribution for the \nmatrix constructed as a sum of \ntwo independent matrices from the original ensemble \n$H=H_1+H_2$ is identical as the the original one up to \na trivial rescaling.\nWigner-L\\'evy matrices are obviously stable, since\nthe probability distribution for individual matrix elements\nof the sum $H_{ij} = H_{1,ij}+H_{2,ij}$ is stable. \nA sum of two Wigner-L\\'evy matrices is again a Wigner-L\\'evy\nmatrix. The Wigner-L\\'evy matrices are not rotatationally \ninvariant. This means in particular \nthat the eigenvalue distribution itself\ndoes not provide the whole information about the underlying\nmatrix ensemble. Indeed, if $O$ is a fixed orthogonal matrix,\nand $H$ is a Wigner-L\\'evy matrix, then the matrix $OHO^T$\nis not anymore a Wigner-L\\'evy matrix but it has\nexactly the same eigenvalue distribution as $H$.\nIn other words, an ensemble of Wigner matrices is not \nmaximally random among ensembles with the same\neigenvalue distribution.\nOne expects that maximally random ensemble with the given\nspectral properties should be rotationally invariant. In this\ncase one also expects that the stability holds\nnot only for the sum $H=H_1+H_2$ but also for the sum \nof relatively rotated matrices: \n$H= H_1 + O H_2 O^T$,\nwhere $O$ is an arbitrary orthogonal matrix.\nIt can be shown \\cite{BAL} that the ensemble of random matrices\nwhich maximizes randomness (Shannon's entropy) for \na given spectral density has the probability measure\nexactly of the form (\\ref{frmeasure}) as discussed here.\n\nStable laws are important because they define domains\nof attractions. For example, if one thinks of a matrix addition \none expects that a sum of many independent\nidentically distributed random matrices \n$H = H_1 + \\dots + H_n$ should for $n \\rightarrow \\infty$ \nbecome a random matrix from a stable ensemble.\n\nMaximally random spectrally stable ensembles which we discussed\nin the section on free random matrices play a special role \nsince they can serve as an attraction point for the sums of\niid rotationally invariant matrices. Moreover one expects \nthat even for not rotationally invariant random matrices \n$H_i$, the sums of the form \n$B = O_1 H_1 O_1^T  + \\dots + O_n H_n O_n^T$ where\n$O_i$ are random orthogonal matrices, will for large $n$\ngenerate a maximally random matrix $B$ from a spectrally\nstable ensemble. In this spirit one can expect that\nif ones adds many randomly rotated Wigner-L\\'evy matrices:\n\\begin{eqnarray}\nB = \\frac{1}{{\\cal N}^{1\/\\mu}}\\sum_i^{\\cal N} O_i A_i O_i^T\n\\end{eqnarray}\nthat for ${\\cal N}\\to \\infty$ the matrices $B$ should \nbecome rotationally invariant, maximally random with\na distribution governed by the FRL symmetric distribution. \nIn Figures 12-14 we show that this is indeed the case.\nThe plots illustrate the two types of stability discussed above. \nIn each case we generate $N=100$ WL matrices \nand combine ${\\cal N}=100$ of them either as a simple sum (black) or a \nrotated sum (red) with the appropriate scale factor. The plots\nrepresent the numerically measured spectra for the two cases.\nWe present results for $\\mu=1.5,~1.25$ and 1, which all show that a simple sum\nreproduces the BC result, while the rotated sum reproduces the symmetric\nFRL distribution.\n\\begin{figure\n\\psfrag{L}{\\bf{\\Large $\\lambda$}}\n\\psfrag{R}{\\bf{\\Large $\\rho(\\lambda)$}}\n\\centerline{\\scalebox{0.6}{\\rotatebox{0}{\\includegraphics{fr1.50.ps}}}}\n\\caption[phased]{{\\small WL (black) versus FRL (red) stability\nfor $\\mu=1.5$}} \\label{fig12}\n\\end{figure}\n\\begin{figure\n\\psfrag{L}{\\bf{\\Large $\\lambda$}}\n\\psfrag{R}{\\bf{\\Large $\\rho(\\lambda)$}}\n\\centerline{\\scalebox{0.6}{\\rotatebox{0}{\\includegraphics{fr1.25.ps}}}}\n\\caption[phased]{{\\small WL (black) versus FRL (red) stability\nfor $\\mu=1.25$}} \\label{fig13}\n\\end{figure}\n\\begin{figure\n\\psfrag{L}{\\bf{\\Large $\\lambda$}}\n\\psfrag{R}{\\bf{\\Large $\\rho(\\lambda)$}}\n\\centerline{\\scalebox{0.6}{\\rotatebox{0}{\\includegraphics{fr1.00.ps}}}}\n\\caption[phased]{{\\small WL (black) versus FRL (red) stability\nfor $\\mu=1$}} \\label{fig14}\n\\end{figure}\n\\section{Conclusions}\n\nWe have given a detailed analysis of the macroscopic limit of\ntwo distinct random matrix theories based on L\\'{e}vy type\nensembles. The first one was put forward by Bouchaud and Cizeau\n\\cite{BC} and uses a non-symmetric measure under the orthogonal\ngroup, and the second one was suggested by us~\\cite{FREELEVY} and\nuses a symmetric measure.\n\nAfter correcting the original analysis\nin~\\cite{BC}, in particular our formulae\n(\\ref{correct1}), (\\ref{correct}) replace (10b) and (12b) in~\\cite{BC},\nand their eq. (15) is replaced by the pair of ``dispersion relations''\n(\\ref{G}) {\\em and} (\\ref{disp}), we  found perfect agreement between\nthe analytical and numerical spectra. The WL measure is easy to\nimplement numerically for arbitrary asymmetry parameter $\\beta$\nin the L\\'{e}vy distributions. The spectrum of\nWL matrices does not depend on $\\beta$ and remains symmetric and universal,\ndepending only on $\\mu$.\n\nWe have also shown that the spectra generated analytically for symmetric\nFRL matrices are similar to the ones generated from WL matrices.\nUnlike the WL ensemble, the FRL ensemble allows for\nboth symmetric and asymmetric L\\'{e}vy distributions. Both ensembles\nare equally useful for addressing issues of recent\ninterest~\\cite{LEVYFIN,BPBOOK2}.\n\nLet us finish the paper with two remarks. \n\\begin{itemize}\n\\item {\\bf i.} The application of free probability calculus to asymptotically free\nmatrix realizations allows one to derive spectral density of the\nmatrices from the underlying matrix ensemble\nbut it does not tell one how to calculate eigenvalue correlation\nfunctions, or joint probabilities for many eigenvalues. Actually\ndifferent matrix realizations of free random variables may have\ncompletely different structure of eigenvalue correlations even\nif they are realizations of the same free random variables.\nTo fix correlations or joint probabilities for \ntwo or more eigenvalues, one has to introduce the concept\nof higher order freeness \\cite{2F}. We think however that \nif one imposes on a matrix\nrealization of free random variables an additional requirement\nthat it has to be maximally random in the sense of maximizing\nShannon's entropy \\cite{BAL} then this aditional\nrequirement automatically fixes the probability\nmeasure (\\ref{frmeasure}) for the ensemble\nand thus also all multi-eigenvalues correlations.\n\n\\item {\\bf ii.} Large eigenvalues behave differently for Wigner-L\\'evy and\nmaximally random free random L\\'evy matrices\ndiscussed in this paper. As pointed out recently \\cite{SF},\nthe largest eigenvalues fluctuate independently \nfor Wigner-L\\'evy ensemble, a little bit like in the mean-field\nargument \\cite{BC} mentioned before, while for the maximally \nrandom matrix ensemble (\\ref{frmeasure}) even large eigenvalues\nare correlated \\cite{FREELEVY}. \n\n\\end{itemize}\n\n\n\\begin{acknowledgments}\nWe thank Jean-Philippe Bouchaud for discussions\nwhich prompted us to revisit and compare the two approaches\npresented in this paper. This work was partially supported\nby the Polish State Committee for Scientific Research (KBN) grant\n2P03B 08225 22 (MAN, ZB), Marie Curie TOK programme \"COCOS -\nCorrelations in Complex Systems\" (Contract MTKD-CT-2004-517186),\n(MAN, JJ, ZB, GP), the National Office for Research and Technology\ngrant RET14\/2005 (GP) and by US-DOE grants DE-FG02-88ER40388 and\nDE-FG03-97ER4014 (IZ).\n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction:}\nBiomolecules ($DNA$ \\& $proteins$) live in the crowded, constrained regime. Such molecules are soft object and therefore\ncan be easily squeezed into spaces that are much smaller than the natural size of the molecule in the bulk. For instance,\n$actin$ filaments in $eukaryotic$ cell \\cite{1} or $protein$ encapsulated in $Ecoli$ \\cite{2} is found in nature are\nexamples of confined biomolecules serves the basis for understanding numerous phenomenon observed in polymer technology,\nbio-technology and many other molecular processes occurring in the living cells. The conformational properties of single\nbiomolecules have attracted considerable attention in recent years due to developments in single molecule based observations\nand experiments \\cite{3,4,5,6,7,8}. Under confined geometrical condition, excluded volume effects and effect of geometrical constraint\ncompete with entropy of the molecule. Therefore, constraints can modify conformational properties and\nadsorption desorption transition behaviours of macromolecules.\n\nBehaviour of the flexible polymer molecule in a good solvent, confined to different geometries have been studied \nfor past few years \\cite{9,10}. For example, Brak and his coworkers \\cite{11} used directed self avoiding\nwalk model to study behaviour of flexible polymer chain confined between two parallel walls on square lattice. However,\nin present investigation we have considered a linear semiflexible copolymer chain on square lattice constrained by a one dimensional\nstair shaped surface. For two dimensional space, surface is a line and copolymer chain is constrained by the such surface. The chain\nis made of two type of monomers $A$ and $B$ distributed along the back bone of chain in a sequence $A-B-A-B\\dots$. \nTherefore, it is a sequential or alternating copolymer chain.\nWe have considered sequential copolymer chain because it serve as a paradigmatic \nmodel of actually disordered macromolecule {\\it e. g.} $proteins$.\n\nTo analyze the effect of constraint, we have used\nfully directed self avoiding walk model introduced by Privmann {\\it et. al} \\cite{12} and used generating function technique \nto solve the model analytically.  \nThe results so obtained is used to discuss the behaviour of homopolymer chain in constrained geometry and also to compare results obtained\nfor conformational properties of the polymer chain for the case, when chain is in the bulk, in absence of one \ndimensional stair to the case when there is stair to constrain the chain. \nIf constraint is an attractive surface, it contributes an energy $\\epsilon_s$\n($<0$) for each step of walk making on the surface. This leads\nto an increased probability defined by a Boltzmann weight\n$\\omega=\\exp(-\\epsilon_s\/k_BT)$ of making a step on the surface\n($\\epsilon_s < 0$ or $\\omega > 1$, $T$ is temperature and\n$k_B$ is the Boltzmann constant). \nThe chain gets adsorbed on the surface at an appropriate  value of $\\omega$ or $\\epsilon_s$. \nTherefore, transition\nbetween adsorbed and desorbed phases is marked by a critical value of adsorption\nenergy $\\epsilon_s$ or $\\omega_c$. \nThe crossover\nexponent $\\phi$ at the adsorption transition point is defined as, $N_s \\sim N^{\\phi}$, where $N$ is the total\nnumber of monomers in the chain while $N_s$ is the number of monomers adsorbed onto the surface. \nA comparison is also made to discuss effect of constraint on adsorption transition point. \n\n\nThe paper is organized as follows: In Sec. 2 lattice model of\ndirected self avoiding walk is described for a linear semiflexible sequential copolymer chain \nin good solvent. In sub-section 2.1, we have discussed behaviour \nof the copolymer chain in the bulk in presence of one dimensional stair shaped surface. The results\nobtained in constrained geometry case is compared with those obtained in absence of constraint \nfor semiflexible homopolymer chain.\nSub-section 2.2, is devoted to discuss adsorption of copolymer chain on a one dimensional flat surface \nin presence of constraint and results obtained is compared for the case when there is no constraint near the \nhomopolymer and copolymer chains. In Sub-section 2.3, adsorption of copolymer chain on constraint is discussed. While\nin sub-section 2.4, general expression of the partition function of copolymer chain which\nis interacting with constraint and flat surface is discussed.\nFinally, in Sec. 3 we summarize the results obtained.\n\n\\section{Model and method}\nA lattice model of fully directed self-avoiding walk $\\cite{12}$ is used to\nmodel a semiflexible alternating copolymer chain in two dimensions and generating function technique has\nbeen used to calculate partition function of the copolymer chain in presence of an impenetrable surface (line) having\nshape like a stair (as shown schematically in figure 1). Since copolymer chain is fully directed \ntherefore, walker is allowed to take steps along $+x$, $+y$ directions on the square lattice.\nThe stiffness of the chain is accounted by introducing an energy barrier for each bend in the walk of the chain. \nThe stiffness weight $k=exp(-\\beta\\epsilon_{b})$ where $\\beta=(k_BT)^{-1}$ is \ninverse of the temperature and $\\epsilon_b(>0)$ is the energy associated with \neach bend of the walk of copolymer chain. For $k=1$ or $\\epsilon_{b}=0$ the chain is said to be flexible and for \n$0<k<1$ or $0<\\epsilon_{b} <\\infty$ the chain is  \nsemiflexible. However, when $\\epsilon_{b}\\to\\infty$ or $k\\to0$, \nthe copolymer chain is like a rigid rod.\n\nDirected self avoiding walk model can be solved \nanalytically and therefore gives exact value of conformational properties and \nadsorption transition point of the chain. However,  \ndirected walk model is restrictive in the sense that bending angle has same value {\\it i. e.}\n$90^0$ (for present study) for each bend and directedness of the walk introduces certain extent of stiffness in the \npolymer chain because all directions of the space are not treated equally by the walker. \n\n\\begin{figure}[htbp] \n\\centering \n\\epsfxsize=14cm\\epsfbox{fig01.eps} \n\\caption{In this figure polymer chain is shown (I) in absence of constraint (one dimensional stair shaped surface) while in (II) there\nis constraint. Walk of the chain starts from point O and directed along both the directions of space.}\n\\label{Figure1}\n\\end{figure} \n\nThe partition function of a semiflexible sequential copolymer chain made of two type of monomers (A \\& B) can be written as, \n\n\\begin{equation}\nZ(k,x_1,x_2)={\\sum}^{N=\\infty}_{N=0}\\sum_{ all\\hspace{0.07cm}walks\\hspace{0.07cm}of\\hspace{0.05cm}N\\hspace{0.05cm}steps} {x_1}^{N\/2}{x_2}^{N\/2}k^{N_b}\n\\end{equation}\nwhere, $N_b$ is the total number of bends in a self avoiding walk of $N$ steps (monomers),\n$x_1$ and $x_2$ is the step fugacity of each of the A and B type monomers, respectively.\nFor the sake of mathematical simplicity we assume here and onwards $x_1=x_2=x$. Method\nof analysis discussed in this paper can also be extended to the case when $x_1\\ne x_2$. \n\nWe consider an impenetrable line, located in the $x-y$ plane (as shown schematically in the figure 1) has shape like a stair, one end\nof the copolymer chain is grafted to origin $O$. Since stair is impenetrable, therefore, it constrains the\npolymer chain. We are interested in analyzing effect of this constrain on the conformation properties\nof the copolymer chain in the bulk and also in calculating adsorption desorption phase transition point of the\ncopolymer chain in presence of constraint.\n\n\\subsection{\\bf \\it Copolymer chain in the bulk}\n\nThe partition function of a sequential semiflexible copolymer chain in the bulk is calculated using the\nmethod of analysis discussed by Mishra {\\it et al.}, $\\cite{13}$ and \ncomponents of the partition function $Z(k,x)$ of the copolymer chain in the bulk is written as follows:\n\\begin{equation}\nX_0(k,x)= x+x^2k(1+Y+kX_0)\n\\end{equation}\nwhere, $X_n(k,x)$ (here and onwards $n$=0,1,2,3,4...) is the sum of Boltzmann weight of all walks with \nfirst step along $+x$-direction and $n$ step distant from the stair\nwhile $Y(k,x)$ is sum of Boltzmann weight of all the walks of the copolymer chain having first step along $+y$ direction.\nAbove equation can be re-written as,\n\\begin{equation}\nX_0(k,x)= \\frac{x(1+kx)}{1-x^2k^2}+\\frac{x^2kY}{1-x^2k^2}\n\\end{equation}\nSimilarly, we have\n\\begin{equation}\n\\hspace{-2.5cm}X_n(k,x)=x+x^2+..+x^n+x^n\\frac{x(1+kx)}{1-x^2k^2}+\\large(kx+kx^2+...+kx^n+x^n\\frac{x^2k}{1-x^2k^2}\\large)Y\n\\end{equation}\nand substituting $n=\\infty$, we recover $X_\\infty(k,x)=X(k,x)$ \\cite{13},\n\\begin{equation}\nX_\\infty(k,x)= \\frac{x}{1-x}+\\frac{xkY'}{1-x}\n\\end{equation}\nWhere, $Y'$ is the other component of the partition function in absence of constraint. \nBut, another component of the partition function of the semiflexible copolymer chain in presence of constraint is,\n\\begin{equation}\nY(k,x)=x+x(x+kX_1)+x^2(x+kX_2)+x^3(x+kX_3)+.......\n\\end{equation}\nso that,\n\\begin{equation}\n\\hspace{-2.5cm}Y(k,x)=\\frac{x(1-x^2)(1-x^2k^2)+kx^2(1-k^2x^2)+kx^2(1-x)(x+kx^2)}{(1-x)(1-x^2)(1-k^2x^2)-k^2x^2(1-x^2k^2)-k^2x^4(1-x)}\n\\end{equation}\nThus, the partition function of the copolymer chain \nin the bulk is written as,\n\\begin{equation}\nZ(k,x)=X_0(k,x)+Y(k,x)\n\\end{equation}\nsuch that\n\\begin{equation}\n\\hspace{-2.5cm}Z(k,x)=\\frac{x(2+(-1+2k)x+(-2+k-2k^2)x^2+(1-2k+2k^2-2k^3)x^3)}{1-x-(1+2k^2)x^2+(1+k^2)x^3+k^4x^4}\n\\end{equation}\nFrom singularity of the partition function $Z(k,x)$, we obtain critical value of step fugacity, $x_c$, needed for\npolymerization of an infinitely long linear semiflexible copolymer chain. Since, we have considered $x_1=x_2=x$, we can \ntreat this result for semiflexible homopolymer chain {\\it i. e.} both the monomers of copolymer chain are identical. \nFor, $k=1$, we have $x_c$(with constraint)=0.532089 while $x_c$(without constraint)=0.5 \\cite{13} for flexible\nhomopolymer chain for fully directed self avoiding walk model on square lattice. Variation of $x_c$ with $k$ in\npresence as well as absence of stair (constraint) is shown in figure 2.\n\nPersistent length, $l_p$, of the chain can be calculated using definition of Mishra {\\it et. al} \\cite{13}, \n$l_p${\\large $(=\\frac{x\\frac {\\partial Log Z(k,x)}{\\partial x}}{k\\frac{\\partial Log Z(k,x)}{\\partial k}})$}, as average distance \nof the polymer chain between its two successive bends. The variation of the persistent length \nwith $k$ is shown in figure 2. A comparison of $l_p$ is also made for the case when there no constraint \\cite{13}.\n\n\\subsection{Adsorption on one dimensional flat line}\nIn order to analyze the effect of presence of constraint on adsorption transition point, we consider an impenetrable line\nlocated at $x$=0 and calculated components of the partition function of the sequentional copolymer chain, which is interacting with\nimpenetrable flat line (while there is no attraction between monomer and constraint) \nusing method of analysis used by Mishra {\\it et. al}, we have component of the partition function along the surface as,\n \n\\begin{equation}\n\\hspace{-2.5cm}S^f(k,\\omega_1,\\omega_2,x)=s_1(1+s_2+kX_1)+s_1s_2(1+s_2+kX_2)+s_1^2s_2(1+s_2+kX_3)+\\dots\n\\end{equation}\nhere $A$ type monomer of the copolymer chain is grafted at $O$, we have $s_1(=\\omega_1x)$, $s_2=(\\omega_2x)$ is the \nBoltzmann weight of $A$ and $B$ type monomers of the copolymer chain\nwith flat surface, respectively. While $X_n(k,x)$ is the component of the partition function\nperpendicular to the flat surface {\\it i. e.} along $x$-axis. The value of $X_n(k,x)$ we have used from Eq. (4) to calculate partition function,\n$Z^f(k,\\omega_1,\\omega_2,x)$, of the copolymer chain (provided $s_1s_2<1$) as follows,\n\\begin{equation}\n\\hspace{-2.5cm}S^f(k,\\omega_1,\\omega_2,x)=\\frac{s_1+s_1s_2}{1-s_1s_2}+\\frac{s_1kx(1+s_2)}{(1-s_1s_2)(1-s_1s_2x^2)}+\\frac{s_1s_2kx^2(1+s_1)}{(1-s_1s_2)(1-s_1s_2x^2)}+u_1+u_2\n\\end{equation}\n\n\\begin{figure}[htbp]\n\\epsfxsize=16cm\n\\centerline{\\epsfbox{fig02.eps}}\n\\caption{This figure compares the value of conformational properties of semiflexible\nhomopolymer chain when there is constraint (an impenetrable stair shaped surface) to that of the case if there \nis no constraint. We have also compared the value of adsorption desorption transition point of the semiflexible \nhomopolymer chain on a flat surface in presence and absence of the constraint.}\n\\label{Figure2}\n\\end{figure}\nwhere,\n\n$u_1$={\\Large$\\frac{s_1kx(1+s_2x)(x+kx^2)}{(1-s_1s_2x^2)(1-x^2k^2)}$}\n\nand\n\n$u_2$={\\Large$(\\frac{s_1k^2x(1+s_2)}{(1-s_1s_2)(1-s_1s_2x^2)}+\\frac{s_1s_2k^2x^2(1+s_1)}{(1-s_1s_2)(1-s_1s_2x^2)}+\\frac{s_1(1+s_2x)k^2x^3}{(1-s_1s_2x^2)(1-k^2x^2)})$}Y\nTherefore, partition function of the copolymer chain with $A$ type monomer grafted at $O$ can be calculated as,\n\n\\begin{equation}\nZ^f(k,\\omega_1,\\omega_2,x)=S^f(k,\\omega_1,\\omega_2,x)+X_0(k,x)\\hspace{2cm} (s_1s_2<1)\n\\end{equation}\n\\begin{equation}\n\\hspace{-2.5cm}Z^f(k,\\omega_1,\\omega_2,x)=\\frac{f(k,\\omega_1,\\omega_2,x)}{(u_3(1-x-(1+2k^2)x^2+(1+k^2)x^3+k^4x^4)))}\\hspace{1cm} (s_1s_2<1)\n\\end{equation}\n\nwhere,\n\n$u_3=(-1+s_1s_2)(-1+s_1s_2x^2)$\n\nWe are interested in the singularity of the partition function. Therefore, comparing the denominator of the partition\nfunction $Z^f(k,\\omega_1,\\omega_2,x)$ equal to zero, we obtain critical value of monomer surface attraction\nrequired for adsorption of copolymer chain, $\\omega_{c1}=\\frac{1}{\\omega_{c2}x^2}$ on the flat surface. Effect of\nconstraint on adsorption transition of copolymer chain on flat surface is analyzed through figure (3). \nReplacing $s_1$ with $s_2$ in Eq. (13) we\ncould obtain partition function for copolymer having first monomer of $B$ type. The value of adsorption transition\npoint remain unchanged when grafted monomer of the copolymer chain is of $A$ or $B$ type. \nSubstituting $\\omega_{c1}=\\omega_{c2}=\\omega_c$, we get value of $\\omega_c$ required for adsorption of homopolymer\nsemiflexible chain on an impenetrable line in presence of constraint. This result is shown in figure 2 and comparison\nof this result with those obtained in absence of constraint \\cite{13} is shown for different values of $k$.\n\\begin{figure}[htbp]\n\\epsfxsize=16cm\n\\centerline{\\epsfbox{fig03.eps}}\n\\caption{This figure compares the value of $\\omega_{c1}$ required for adsorption of copolymer chain\non the flat surface for $\\omega_{c2}=0.5,0.9$ in presence and absence of the constraint.}\n\\label{Figure3}\n\\end{figure}\n\n\\subsection{Adsorption of copolymer chain on constraint (one dimensional stair)}\nThe stair shaped one dimensional line (surface) is located in the $x-y$ plane as shown schematically in figure (1). The copolymer chain is interacting\nwith this surface. While, flat surface (line) located at $x=0$ is neutral for copolymer chain. The components of the partition function of\nsurface interacting copolymer chain can be calculated using method discussed above, as,\n\n\\begin{equation}\n\\hspace{-2.5cm}S_{0A}(k,\\omega_3,\\omega_4,x)= s_3+s_3s_4k(1+Y_{A}+s_3k)+s_3^2s_4k^2(1+Y_{A}+s_4k)+s_3^2s_4^2k^3(1+Y_{A}+s_3k)+\\dots\n\\end{equation}\n\nwhere, $s_3(=\\omega_3x)$, $s_4(=\\omega_4x)$ is the Boltzmann weight of interaction energy of $A$ and $B$ type monomers with stair shaped surface\nrespectively. $S_{nA}(k,\\omega_3,\\omega_4,x)$ [$A$ indicates that first monomer grafted at $O$ of the copolymer chain is of $A$ type] \nis the sum of Boltzmann weight of all walks having first step along $+x$-direction and $n$ step distant from the stair\nwhile $Y_A(k,\\omega_3,\\omega_4,x)$ is sum of Boltzmann weight of all the walks of the surface interacting copolymer chain having first step \nalong $+y$ direction. In the same fashion one can define $S_{nB}(k,\\omega_3,\\omega_4,x)$ and $Y_B(k,\\omega_3,\\omega_4,x)$. \nWe assume that first monomer of copolymer chain is of $A$ type therefore for $s_3s_4<1$, we have,\n\\begin{equation}\nS_{0A}(k,\\omega_3,\\omega_4,x)= \\frac{s_3(1+s_4k)}{1-s_3s_4k^2}+\\frac{s_3s_4kY_A}{1-s_3s_4k^2}\n\\end{equation}\nalso, for odd $n$=$p$,\n\\begin{equation}\n\\hspace{-2.5cm}S_{pA}(k,\\omega_3,\\omega_4,x)=x+x^2+..+x^n+x^n\\frac{s_3(1+ks_4)}{1-s_3s_4k^2}+\\large(kx+kx^2+...+kx^n+x^n\\frac{s_3s_4k}{1-s_3s_4k^2}\\large)Y_A\n\\end{equation}\nand for even $n$=$q$,\n\\begin{equation}\n\\hspace{-2.5cm}S_{qA}(k,\\omega_3,\\omega_4,x)=x+x^2+..+x^n+x^n\\frac{s_4(1+ks_3)}{1-s_3s_4k^2}+\\large(kx+kx^2+...+kx^n+x^n\\frac{s_3s_4k}{1-s_3s_4k^2}\\large)Y_A\n\\end{equation}\nThe component of the partition function of surface interacting semiflexible copolymer chain along $y$-axis is,\n\\begin{equation}\nY_A(k,\\omega_3,\\omega_4,x)=x+x(x+kS_{1B})+x^2(x+kS_{2A})+x^3(x+kS_{3B})+\\dots\n\\end{equation}\nsubstituting the value of $S_{qA}(k,\\omega_3,\\omega_4,x)$ and $S_{pB}(k,\\omega_3,\\omega_4,x)$ in Eq. (18) we get,\n\\begin{equation}\n\\hspace{-2.5cm}Y_A(k,\\omega_3,\\omega_4,x)=\\frac{x(1-x^2)(1-s_3s_4k^2)+kx^2(1-k^2s_3s_4)+kx^2(1-x)(s_3+ks_3s_4)}{(1-x)(1-x^2)(1-k^2s_3s_4)-k^2x^2(1-s_3s_4k^2)-k^2s_3s_4x^2(1-x)}\n\\end{equation}\nThus, we calculated partition function of copolymer chain interacting with a stair shaped impenetrable surface as,\n\\begin{equation}\nZ^s(k,\\omega_3,\\omega_4,x)=S_{0A}(k,\\omega_3,\\omega_4,x)+Y_A(k,\\omega_3,\\omega_4,x)\n\\end{equation}\ntherefore, partition function of the copolymer chain which is interacting with an impenetrable stair shaped line is,\n\\begin{equation}\n\\hspace{-2.5cm}Z^s(k,\\omega_3,\\omega_4,x)=\\frac{N^r(k,\\omega_3,\\omega_4,x)}{D^r(k,\\omega_3,\\omega_4,x)}\n\\end{equation}\n\nwhere,\n\n$N^r$=$x+kx^2-x^3-s_3(2k^3s_4x^2-(-1+x)^2(1+x)+k^2x(s_4+x-2s_4x)+k(-1+x)(s_4+s_4x+x^2))$\n\nand,\n\n$D^r=k^4s_3s_4x^2+(-1+x)^2(1+x)+k^2(s_3s_4(-1+x)-x^2)$\n\nFrom singularity of the partition function $Z^s(k,\\omega_3,\\omega_4,x)$, we \nfound that $\\omega_3=\\frac{-1+x+x^2+k^2x^2-x^3}{k^2\\omega_4x^2(-1+x+k^2x^2)}$ \n({\\it i. e.} on comparing $D^r(k,\\omega_3,\\omega_4,x)=0$) required for adsorption of semiflexible copolymer chain and it is independent of bending \nenergy or stiffness $k$ of the polymer chain. \nThis we have discussed through numerical results shown in figure (4) for $\\omega_{c2}=0.5, 0.9 \\& 1.0$. \nThis is because, the polymer chain is directed towards the surface (constraint). \nReplacing $\\omega_3$ and $\\omega_4$ with $\\omega_c$, we have adsorption \ntransition point for homopolymer chain. The value of $\\omega_c$ for homopolymer chain is close to unity (1$\\pm$0.0001) \nfor the values of $k$, $0.1\\le k<1$.\n\n\\begin{figure}[htbp]\n\\epsfxsize=16cm\n\\centerline{\\epsfbox{fig04.eps}}\n\\caption{This figure is used to show that copolymer chain adsorption on constraint appears independent\nof stiffness of the copolymer chain. However, in the very stiff chain limit adsorption of the chain on \ncontsraint may not be possible.}\n\\label{Figure4}\n\\end{figure}\n\n\\subsection{General expression of the partition function}\n\nWe now derive expression of the partition function of the  sequential semiflexible linear \ncopolymer chain for the case when copolymer chain is interacting with flat surface and constraint. \nThe flat impenetrable surface (line) located at $x=0$ and another\nsurface is constraint (an impenetrable stair shaped line located along the line $x=y$; schematically shown in the figure (1)).\nWe assume that $A$ type monomer of copolymer chain is first monomer of the chain and one end of the chain is grafted at (0,0) or $O$. \nLet $S_A$ is the component of the partition function having first step on to the flat surface, then,\n\\begin{equation}\n\\hspace{-2.5cm}S_{A}(k,\\omega_1,\\omega_2,\\omega_3,\\omega_4,x)= s_1+s_1s_2k(1+S_{1B}+s_2k)+s_1^2s_2k^2(1+S_{2A}+s_2k)+\\dots\n\\end{equation}\n\nSubstituting value of $S_{qA}$, $S_{pB}$ from Eqs. (16\\& 17) and using following relation we have general expression of the partition function,\n\\begin{equation}\n\\hspace{-2.5cm}Z^{s\\&f}(k,\\omega_1,\\omega_2,\\omega_3,\\omega_4,x)=S_{A}(k,\\omega_1,\\omega_2,\\omega_3,\\omega_4,x)+S_{0A}(k,\\omega_3,\\omega_4,x)\n\\end{equation}\n\\begin{equation}\n\\hspace{-2.5cm}Z^{s\\&f}(k,\\omega_1,\\omega_2,\\omega_3,\\omega_4,x)=\\frac{s_1(1+s_2)}{1-s_1s_2}+u_4+u_5-\\frac{u_6}{u_7}+\\frac{u_8}{u_{9}}\\hspace{.5cm}(s_1s_2<1 ; s_3s_4<1)\n\\end{equation}\n\nwhere,\n\n\\vspace{.2cm}\n\n$u_4$={\\Large$\\frac{ks_3(1+ks_4)s_1x(1+s_2x)}{(-1+k^2s_3s_4)(-1+s_1s_2x^2)}$}\n\n\\vspace{.2cm}\n\n\n$u_5$={\\Large$\\frac{ks_1(1+s_2)x(1+s_2x)}{(-1+s_1s_2)(-1+s_1s_2x^2)}$}\n\n\\vspace{.2cm}\n\n\n$u_6=s_3(-(k^2(-1+s_4)x^2)+k^3s_4x^2-(-1+x)^2(1+x)+ks_4(-1+x^2))$\n\n\\vspace{.2cm}\n\n\n$u_7=k^4s_3s_4x^2+(-1+x)^2(1+x)+k^2(s_3s_4(-1+x)-x^2)$\n\n\\vspace{.2cm}\n\n\n$u_8=(k^2s_1x^2(-1-k^2s_3s_4(-1+x)+k^3s_3s_4x+k(-1+s_3(-1+x))x+x^2)(-1-s_2(1+x+s_1x)+s_3s_4((-1+s_1s_2)(1+s_2x)+k^2(1+s_2+s_2x+s_1s_2x))))$\n\n\\vspace{.2cm}\n\n\nand\n\n\\vspace{.2cm}\n\n\n$u_{9}=((-1+k^2s_3s_4)(-1+s_1s_2)(-1+s_1s_2x^4)(k^4s_3s_4x^2+(-1+x)^2(1+x)+k^2(s_3s_4(-1+x)-x^2)))$\n\n\\vspace{.2cm}\n\n\nWe call this relation (Eq. 24) as general expression of the partition function because from this expression we can recover \nexpression of partition function discussed in above sub-sections by suitable substitutions. For example, replacing $s_1$, $s_2$,\n$s_3$ and $s_4$ by x we have partition function of the copolymer chain in the bulk {\\it i. e.} $Z(k,x)$. \n\n\\section{Result and discussion}\n\nWe have considered a linear semiflexible sequential copolymer chain made of two type of monomers and fully directed\nself avoiding walk model is used to model the copolymer chain on square lattice. \nTechnique of generating function is used to calculate partition function of the chain in presence of constraint (an impenetrable one\ndimensional line) analytically. \nThe variation of\ncritical value of step fugacity required for polymerization of an infinitely long linear copolymer chain and \npersistent length under good solvent have been discussed. \nThe results obtained with constraint is compared to case when there is no constraint. \nWe have also compared the value of monomer surface attraction required for\nadsorption of semiflexible copolymer chain on a flat surface and constraint. The surface is a one dimensional line\nand walks of the chain is directed along both the directions of the space.\n\nWe have found that adsorption of the homopolymer chain on constraint occurs at a very small value of monomer and surface attraction. \nThis is because, walks of the polymer\nchain is directed along the stair shaped surface, walks of the chain follows curvature of the constraint and major \ncontribution in the partition function of copolymer chain is due to walks lying on the constraint. \nIn the case of stair shaped impenetrable surface, critical value of\nmonomer surface attraction required for the copolymer\nchain adsorption appears independent of bending energy\nof the chain (excluding the case of very stiff chains). While adsorption of copolymer and homopolymer chain on the flat surface \nis function of semiflexibility of the chain. The stiffer chain adsorption occurs at a smaller\nvalue of monomer surface attraction than the flexible polymer chain.\nThese features of adsorption\non an impenetrable flat and curved surfaces are similar to the semiflexible polymer adsorption $\\cite{13,14,15,16,17}$.\n\n\\small\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}