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+{"text":"\\section{Introduction}\n\nEntanglement and its generation, besides its intrinsic interest as an unique quantum correlation and resource for quantum information processing \\cite{HHHH2009, NieChu2010}, continues to be vigorously investigated due to its relevance to a wide range of questions such as\nthermalization and the foundations of  statistical physics \\cite{Popescu06,JC05,DeutchLiSharma2013,Kaufman16,Rigol2016}, decoherence \\cite{Zurek91,Haroche1998,Zurek03,Davidovich_2016}, delocalization \\cite{WSSL2008,kim13} and quantum chaos in few and many-body systems \\cite{Miller99, Bandyopadhyay02, Wang2004,LS05,petitjean2006lyapunov, tmd08, Amico08,Chaudhary}. Some of these issues\nconcern the rate of entanglement production, the nature of multipartite entanglement sharing and distribution, and long-time saturation or indefinite growth. Details of entanglement production in integrable {\\it versus} nonintegrable systems is an\nactive area of study, and it is generally appreciated that concomitant with the production of\nnear random states of nonintegrable systems is the production of large entanglement that can lead to thermalization\nof subsystems.\n\nEntanglement produced from suddenly joining two spin chains each in its ground state produces entanglement growth $\\sim \\ln(t\/a)$ at long times~\\cite{Calabrese_2007}, whereas quenches starting from arbitrary states can produce large entanglement including a linear growth phase \\cite{JC05}. Similarly in an ergodic or eigenstate thermalized phase a system can show ballistic entanglement growth \\cite{kim13,Ho17}, whereas in the many-body localized phase\nit is known to have a logarithmic growth in time. The entanglement in almost all these studies relates to bipartite block entanglement between\nmacroscopically large subsystems in many-body systems.\n\nEven earlier works have explored the entanglement in Floquet or periodically forced systems such as the coupled kicked tops or standard maps, as a means to study the relationship between chaos and entanglement \\cite{FNP1998,Miller99,Lakshminarayan2001, Bandyopadhyay02,Fujisaki03,Bandyopadhyay04}.\nIt was seen that chaos in general increases bipartite entanglement and results in near\nmaximal entanglement as the states become typical, in the sense of Haar or uniform measure.\nThe initial states considered were mostly phase-space localized coherent states. In the case when the uncoupled systems are chaotic, and the interactions are weak, after a short Ehrenfest time scale\nthe growth of the entropy or entanglement is essentially as if the coherent states were initially random subsystem states.  In cases in which the subsystems are fully chaotic,\nthe growth of entanglement (beyond the Ehrenfest time) is dependent on the coupling strength rather than on measures of chaos such as the Kolmogorov-Sinai entropy or the Lyapunov exponent \\cite{Fujisaki03,Bandyopadhyay04} . A linear growth was observed in this case and perturbation theory \\cite{Fujisaki03} is successful in describing it, and the more extended time behavior was described by perturbation theory along with random matrix theory (RMT) \\cite{Bandyopadhyay04}. The linear growth leads to saturation values that are interaction independent, and in cases of moderately large coupling this is just the bipartite entanglement of typical or random states in the full Hilbert space.\n\nIn sharp contrast is the case for which the initial states are eigenstates of the non-interacting fully chaotic systems. This presents a very different scenario for weak couplings that to our knowledge has not been previously studied. The present paper develops a full theory in this case, starting\nfrom a properly regularized perturbation theory wherein a universal growth curve involving a suitably scaled time is derived. Importantly, this is further developed into a theory valid for non-perturbative strong couplings. We study these cases as an ensemble average over all uncoupled eigenstates, which clearly forms a special set of states in the Hilbert space for weak coupling.\n\n The entanglement production starts off linearly, as in the case of generic states, before saturating at much smaller entanglement values that are manifestly and strongly interaction dependent and reflect a ``memory\" of the initial ensemble to which it belongs. Interestingly while the linear regime is independent of whether the system possesses time-reversal symmetry or not, the subsequent behavior including the saturation value of the entanglement is larger for the case when time-reversal is broken. For generic initial states time-reversal symmetry has not played a significant\n role in the entanglement production or saturation.\nAs the interaction is increased this saturation approaches that of random or typical values and then the memory of starting off as a special initial state no longer persists. An essential aspect of this study is to elucidate at what interaction strength such a transition happens.\n\nThe interaction is properly measured by a scaled dimensionless transition parameter $\\Lambda$ that also determines transitions in the spectral statistics and eigenfunction entanglements of such systems \\cite{Srivastava16,Lakshminarayan16,Tomsovic18c}. If $\\Lambda=0$, the noninteracting case, although the two subsystems are chaotic, the quantum spectrum of the system has a Poisson level spacing statistic \\cite{Tkocz12} and therefore has many nearly degenerate levels which start to mix when the subsystems are weakly coupled. If $\\Lambda \\ll 1$, we are in a perturbative regime wherein the eigenstates with appreciable entanglements have a Schmidt rank of approximately two, i.e.\\ the reduced density matrix of the eigenstates has at most two principal nonzero eigenvalues. This circumstance carries over to time evolving states which are initially product eigenstates.  Universal features of the eigenstate entanglement depend only on the single scaled parameter $\\Lambda$.  For example, the linear entropy of the eigenstates $\\sim \\sqrt{\\Lambda}$ \\cite{Lakshminarayan16,Tomsovic18c}.  On the other hand, as shown in this paper, time evolving states develop a linear entropy $\\sim C(2,t)\\sqrt{\\Lambda}$, with $C(2,t)$ being a {\\it universal} function for a properly scaled time, independent of the details of the interaction or the chaotic subsystem dynamics, except for a slight dependence on whether the system is time-reversal symmetric or not.\n\n\nSuch a universality follows from the existence of underlying RMT models that\ndescribe the transition from uncoupled to strongly coupled systems, a RMT transition ensemble. Although it is standard to\napply RMT for stationary states and spectral statistics \\cite{Bohigas84, Brody81, Haake10}, as indeed done for strongly chaotic and weakly\ninteracting systems \\cite{Srivastava16,Lakshminarayan16,Tomsovic18c}, it is noteworthy that this is typically not valid for the time evolution, because RMT lacks correlations required for describing short time dynamics properly.  However the time scales\nover which the entanglement develops is much longer than the Ehrenfest time after which specific dynamical system features disappear.  Thus, universal behaviors can be derived from such RMT transition ensembles provided the time scales of interest remain much longer than the Ehrenfest time scale. It turns out that for $\\Lambda \\gtrsim 1$, the interaction is strong enough that the\nsystem has fluctuations that are typical of RMT of over the whole space, for example the consecutive neighbor spacing of eigenvalues is that of Wigner \\cite{Srivastava16}. This also signals the regime for which eigenstates have typical entanglement of random states \\cite{Lakshminarayan16,Tomsovic18c}  and as shown here, the time-evolving states lose memory of whether they initially belonged to special ensembles such as the noninteracting eigenstates.\n\nAlthough regularized perturbation theory, initially developed for studying symmetry breaking in\nstatistical nuclear physics~\\cite{French88a,Tomsovicthesis}, is used in the $\\Lambda \\ll 1$ regime, a novel recursive use of the perturbation theory allows for approximate, but very good extensions to the non-perturbative regime.  In fact, it covers well the full transition. This provides an impressive connection of the entanglement both as a function of time and as a function of the interaction strength to the RMT regime where nearly maximal entanglement is obtained and formulas such as Lubkin's for the linear entropy \\cite{Lubkin78} and Page's for the von Neumann entropy \\cite{Page93,Sen1996} are obtained.\nWe illustrate the general theory by specifically considering both time-reversal symmetric and violating RMT transition\nensembles, given respectively by subsystem Floquet operators chosen from the circular orthogonal ensemble (COE)\nand the circular unitary ensemble (CUE) respectively. These are classic RMT ensembles consisting of unitary\nmatrices that are uniformly chosen with densities that are invariant under orthogonal (COE) and unitary (CUE) groups \\cite{MehtaBook}.\n\nIn addition, we apply it to a dynamical system of coupled standard maps \\cite{Froeschle71,Lakshminarayan2001,Richter14}. The standard map is a textbook example of a chaotic Hamiltonian system and is simply a periodically kicked pendulum. There is a natural translation symmetry in\nthe angular momentum that makes it possible to consider the classical map on a torus phase space, with periodic boundary\nconditions in both position and momentum. This yields convenient finite dimensional models of quantum chaos, the dimension\nof the Hilbert space being the inverse scaled Planck constant. The model we consider is that of coupling two\nsuch maps which has proven useful in previous studies relating to entanglement \\cite{Lakshminarayan2001,Lakshminarayan16,Tomsovic18c}, spectral transitions \\cite{Srivastava16} and out-of-time-ordered correlators \\cite{RaviLak2019}.\n\nTwo possible concrete examples are\na pair of particles in a chaotic quantum dot with tunable interactions or two spin chains that are in the ergodic phase before being suddenly joined.\nRecent experiments have accessed information of time-evolving states of interacting few-body systems via\nstate tomography of single or few particles, facilitating the study of the role of entanglement in the\napproach to thermalization of closed systems.\nSpecifically, experiments that have\nstudied nonintegrable systems include, for example, few qubit kicked top implementations \\cite{Chaudhary,Neill2016} and the Bose-Hubbard Hamiltonian \\cite{Kaufman16}. Thus modifications of these to accommodate weakly interacting parts are conceivable. This work adds to the already voluminous contemporary research on thermalization in closed systems\nby looking in detail at the time evolution for a case when thermalization, in the sense of a typical subsystem entropy, is unlikely to occur \\cite{GWEG2018}, namely starting from product eigenstates and quenching the interactions by suddenly turning them on.\n\n\nThis paper is organized as follows:\nIn Sec.~\\ref{sec:background}\nthe necessary background material on entanglement in bipartite systems,\nthe random matrix transition ensemble and the unversal\ntransition parameter are given.\nSec.~\\ref{sec:universal-entanglment-dynamics-perturbation-regime}\nprovides the perturbation theory for the\nuniversal entanglement based on the eigenvalues of the reduced\ndensity matrix, ensemble averaging for the CUE or COE\nand invoking a regularization.\nBased on this the eigenvalue moments of the reduced density matrix\nare obtained in Sec.~\\ref{sec:eigenvalue-moments}\nleading to explicit expressions for the HCT entropies.\nIn particular it is shown that for small interaction\nbetween the subsystems the simultaneous\nre-scaling of time and of the entropies by their saturation\nvalues leads to a universal curve which is independent\nof the interaction.\nThe extension to the non-perturbative regime\nis done in Sec.~\\ref{sec:NonPerturbative}\nby using a recursively embedded perturbation theory\nto produce the full transition and the saturation values.\nA comparison with a dynamical system given by\na pair of coupled kicked rotors is done\nin Sec.~\\ref{sec:coupled-kicked-rotors}.\nFinally, a summary and outlook is given in\nSec.~\\ref{sec:summary-and-outlook}.\n\n\n\n\\section{Background}\n\\label{sec:background}\n\n\\subsection{Entanglement in bipartite systems}\n\nConsider pure states, $|\\psi\\rangle$, of a bipartite system whose Hilbert space is a tensor product space, $\\mathcal{H}^A \\otimes \\mathcal{H}^B$, with subsystem dimensionalities, $N_A$ and $N_B$, respectively.  Without loss of generality,  let $N_A \\leq N_B$.  The question to be studied is how much an initially unentangled state becomes entangled under evolution of some dynamics as a function of time.\n\nThe dynamics of a generic conservative system\ncould be governed by a Hamiltonian or by a unitary Floquet operator.\nSpecifically, a bipartite Hamiltonian system is of the form\n\\begin{equation}\nH(\\epsilon) = H_A \\otimes \\mathds{1}_B + \\mathds{1}_A \\otimes H_B + \\epsilon V_{AB} \\ , \\label{eq:GenericHamiltonian}\n\\end{equation}\nwhere the non-interacting limit is $\\epsilon = 0$.  In the case of a quantum map, the dynamics can be described by a unitary Floquet operator~\\cite{Srivastava16}\n\\begin{equation}\n\\mathcal{U}(\\epsilon) = (U_A \\otimes U_B) U_{AB}(\\epsilon)\\ ,\n\\label{eq:GenericFloquet}\n\\end{equation}\nfor which the non-interacting limit is $U_{AB}(\\epsilon\\rightarrow 0) = \\mathds{1}$.  We assume that both $V_{AB}$ and $U_{AB}(\\epsilon \\neq 0)$ are entangling interaction operators~\\cite{Lakshminarayan16}.\n\nThe Schmidt decomposition of a pure state is given by\n\\begin{equation}\n  \\ket{\\psi} = \\sum_{l=1}^{N_A} \\sqrt{\\lambda_l} \\, \\ket{\\phi^A_l}\\ket{\\phi^B_l}.\n  \\label{eq:GenericSchmidtDecomposition}\n\\end{equation}\nThe normalization condition on the state $\\ket{\\psi}$ gives\n\\begin{equation} \\label{eq:lambda-l-normalization}\n  \\sum_{l=1}^{N_A} \\lambda_l = 1 .\n\\end{equation}\nThe state is unentangled if and only if the largest eigenvalue $\\lambda_1 = 1$ (all others vanishing), and maximally entangled if $\\lambda_l = 1\/N_A$ for all $l$.  By partial traces, it follows that the reduced density matrices\n\\begin{equation}\n\\rho^A = \\tr_B(\\ket{\\psi}\\bra{\\psi}), \\qquad \\rho^B = \\tr_A(\\ket{\\psi}\\bra{\\psi})\\ ,\n\\end{equation}\nhave the property\n\\begin{equation}\n\\rho^A \\ket{\\phi^A_l} = \\lambda_l \\ket{\\phi^A_l}, \\quad \\text{and} \\quad \\rho^B \\ket{\\phi^B_l} = \\lambda_l \\ket{\\phi^B_l},\n\\end{equation}\n respectively.  They are positive semi-definite, share the same non-vanishing (Schmidt) eigenvalues $\\lambda_l$ and $\\{ \\ket{\\phi^A_l} \\}$, and $\\{ \\ket{\\phi^B_l} \\}$ form orthonormal basis sets in the respective Hilbert spaces.  For subsystem $B$ there are $N_B-N_A$ additional vanishing eigenvalues and associated eigenvectors.\n\nA very useful class of entanglement measures are given by the von Neumann entropy and Havrda-Charv\\'at-Tsallis (HCT) entropies~\\cite{Bennett96,Havrda67,Tsallis88,Bengtsson06}.  The von Neumann entropy is given by\n\\begin{equation} \\label{eq:GenericVonNeumannEntropy}\n\\begin{split}\nS_1 & = -\\tr_A(\\rho^A \\ln \\rho^A ) = -\\tr_B(\\rho^B \\ln \\rho^B )\\\\\n    &= -\\sum_{l=1}^{N_A} \\lambda_l \\ln \\lambda_l,\n\\end{split}\n\\end{equation}\nwhich vanishes if the state is unentangled and is maximized if all the nonvanishing eigenvalues are equal to $1\/N_A$.  The HCT entropies are obtained from moments of the Schmidt eigenvalues. Defining\n\\begin{equation}\n\\mu_\\alpha = \\tr_A[(\\rho^A)^\\alpha] = \\tr_B[(\\rho^B)^\\alpha] = \\sum_{l=1}^{N_A} \\lambda_l^\\alpha, \\quad \\alpha >0 \\ ,\n\\label{eq:GenericMoments}\n\\end{equation}\ngives the HCT entropies as\n\\begin{equation} \\label{eq:GenericHCTEntropy}\n  S_\\alpha = \\frac{1-\\mu_\\alpha}{\\alpha-1}.\n\\end{equation}\nNote that these differ from the R\\'enyi entropies \\cite{Ren1961wcrossref},\nwhich are defined by\n\\begin{equation*}\n  R_\\alpha = \\dfrac{\\ln \\mu_{\\alpha}}{1-\\alpha} .\n\\end{equation*}\nIn the limit $\\alpha \\to 1$ also $R_\\alpha$ turns into the\nvon Neumann entropy.\nIn this work we use the HCT entropies as performing ensemble averages\nis easier using $\\mu_{\\alpha}$ than $\\ln \\mu_{\\alpha}$.\n\n\n\n\\subsection{Quantum chaos, random matrix theory, and universality}\n\nMany statistical properties of strongly chaotic quantum systems are successfully modeled and derived with the use of RMT~\\cite{Brody81,Bohigas84}.  Generally speaking, the resulting properties are universal, and in particular, do not depend on any of the physical details of the system with the exception of symmetries that it respects.  Here the subsystems are assumed individually to be strongly chaotic.  Thus, the statistical properties of the dynamics, Eq.~(\\ref{eq:GenericFloquet}),  can be modeled with the operators $U_A$ and $U_B$ being one of the standard circular RMT ensembles~\\cite{Dyson62e}, orthogonal, unitary, or symplectic, depending on the fundamental symmetries of the system~\\cite{Porterbook}.\n\nWe concentrate on the orthogonal (COE) and unitary ensembles (CUE) depending on whether or not time reversal invariance is preserved, respectively.\nThe derivation of the typical entanglement production for some initial state relies on the dynamics governed by\nthe random matrix transition ensemble \\cite{Srivastava16,Lakshminarayan16}\n\\begin{equation}\n\\mathcal{U}_{\\text{RMT}}(\\epsilon) = (U_A^\\text{RMT} \\otimes U_B^\\text{RMT}) U_{AB}(\\epsilon)\\ .\n\\label{eq:GenericFloquetRMT}\n\\end{equation}\nThe operator $U_{AB}(\\epsilon)$ is assumed to be diagonal in the direct product basis of the two subsystem ensembles. Explicitly, the diagonal elements are considered to be of the form $\\exp(2 \\pi i \\epsilon  \\xi_{kl})$, where $\\xi_{kl}$ ($1\\le k,l\\le N_A,N_B$) is a random number uniformly distributed in $(-1\/2,1\/2]$.\n\n\n\\subsection{Symmetry breaking and the transition parameter}\n\nThe statistical properties of weakly interacting quantum chaotic bipartite systems have been studied recently, with the focus on spectral statistics, eigenstate entanglement, and measures of localization~\\cite{Srivastava16,Lakshminarayan16,Tomsovic18c}.  If the subsystems are not interacting, the spectrum of the full system is just the convolution of the two subsystem spectra giving an uncorrelated spectrum in the large dimensionality limit.  The eigenstates of the system are unentangled.  It is very fruitful to conceptualize this as a dynamical symmetry.  Upon introducing a weak interaction between the subsystems, this symmetry is weakly broken.  As the interaction strength increases, the spectrum becomes increasingly correlated, and the eigenstates entangled.\n\nHere $U_{AB}(\\epsilon)$ plays the role of a dynamical symmetry breaking operator.  For $\\epsilon=0$, the symmetry is preserved (the dynamics of the subsystems are completely independent), and as $\\epsilon$ gets larger, the more complete the symmetry is broken.  It is known that for sufficiently chaotic systems, there is a universal scaling given by a transition parameter which governs the influence of the symmetry breaking on the system's statistical properties~\\cite{French88a}.  The transition parameter is defined as~\\cite{Pandey83}\n\\begin{equation}\n\\Lambda = \\frac{v^2(\\epsilon)}{D^2},  \\label{eq:LambdaDef}\n\\end{equation}\nwhere $D$ is the mean level spacing and $v^2(\\epsilon)$ is the mean square matrix element in the eigenbasis of the symmetry preserving system, calculated locally in the spectrum.\n\nFor the COE and CUE the leading behavior in $N_A$ and $N_B$ is \\cite{Srivastava16,Tomsovic18c,HerKieFriBae2019:p}\n\\begin{align}\n\\label{eq:Lambda-RMT}\n& \\, \\Lambda = \\frac{N_A N_B}{4 \\pi^2} \\left[1-\\dfrac{\\sin^2 (\\pi \\epsilon)}{\\pi^2\\epsilon^2} \\right]\n\\sim \\frac{\\epsilon^2 N_AN_B}{12}\\ ,\n\\end{align}\nwhere the last result is in the limit of large $N_A$, $N_B$.\nThe transition parameter $\\Lambda$ ranges over  $0 \\le \\Lambda \\le N_A N_B\/4\\pi^2\\ (N_A, N_B  \\rightarrow \\infty)$, where the limiting cases are fully symmetry preserving, and fully broken, respectively.\nIn essence, the latter expression of Eq.~\\eqref{eq:Lambda-RMT}  illustrates the fact that as the system size grows, a symmetry breaking transition has a discontinuously fast limit in $\\epsilon$.\n\nThe transition parameter gives the relation necessary to compare the statistical properties of systems of any size and kind to each other.  As long as $\\Lambda$ has identical values, the systems have identical properties.  However, for a particular dynamical system, it can turn out to be rather difficult to calculate $\\Lambda$.  Although, the statistical properties are universal and independent of the nature of the system in this chaotic limit, properties such as whether the system is many-body or single particle, Fermionic or Bosonic, actually enter into its calculation.  For example, a method for calculating  $\\Lambda$ for highly excited heavy nuclei is given in Ref.~\\cite{French88b}.  The far simpler case of coupled kicked rotors is given in Ref.~\\cite{Srivastava16}, and is used ahead for illustration.  In extended systems, the issue of localization emerges, which must also be taken into account, and for them the term sufficiently chaotic is meant to exclude a localized regime.\n\n\\section{Universal entanglement production -- Perturbative regime}\n\\label{sec:universal-entanglment-dynamics-perturbation-regime}\n\nThe starting point of a derivation of the typical production rate of entanglement in initially unentangled states is the random matrix transition ensemble~\\eqref{eq:GenericFloquetRMT}.  Following a similar derivation sequence for the eigenstates in Refs.~\\cite{Lakshminarayan16,Tomsovic18c}, the first step is to derive expressions for the eigenvalues of the reduced density matrix, which can be obtained from the Schmidt decomposition of the time evolved state of the system. Applying a standard Rayleigh-Schr{\\\"o}dinger perturbation theory leads to perturbation expressions for the Schmidt eigenvalues.  However, due to the Poissonian fluctuations in the spectrum of the non-interacting system, near-degeneracies occur too frequently and cause divergences in the ensemble averages. It is therefore necessary first to regularize the eigenvalue expressions appropriately.  It also turns out that the perturbation expressions for the HCT entanglement measures can be further extended to a non-perturbative regime by recursively invoking the regularized perturbation theory leading to a differential equation, which is analytically solvable~\\cite{Tomsovic18c},\nsee Sec.~\\ref{sec:NonPerturbative}.\n\n\n\\subsection{Definitions}\n\nThe eigenvalues and corresponding eigenstates of the unitary operators\n$U_A$ and $U_B$ for the subsystems\nand of $\\mathcal{U}(\\epsilon) = (U_A \\otimes U_B) U_{AB}(\\epsilon)$\nof the full bipartite system \\eqref{eq:GenericFloquet} are given\nby the equations\n\\begin{eqnarray}\nU_A \\ket{j^A} &=& \\text{e}^{i \\theta_j^A}  \\ket{j^A},\\quad j=1,2,3,\\ldots,N_A \\nonumber \\\\\nU_B \\ket{k^B} &=& \\text{e}^{i \\theta_k^B} \\ket{k^B}, \\quad k=1,2,3,\\ldots,N_B\\\\\n\\mathcal{U}(\\epsilon) \\ket{\\Phi_{jk}} &=& \\text{e}^{i \\varphi_{jk}} \\ket{\\Phi_{jk}}\\ .\n\\nonumber\n\\end{eqnarray}\nTo simplify the notation, the superscripts $A$ and $B$ are dropped for both eigenkets, $\\ket{j^A}\\ket{k^B} \\equiv \\ket {jk}$, and the eigenvalues $\\theta_j^A \\equiv \\theta_j$ ($\\theta_k^B \\equiv \\theta_k$). It is understood that the labels $j$ and $k$ are reserved for the subsystems $A$ and $B$, respectively. Similarly for convenience, the subscript $AB$ is dropped from the operator $V_{AB}$.\n\nGiven the form \\eqref{eq:GenericFloquet} of the unitary operator $\\mathcal{U}(\\epsilon)$, in the limit $\\epsilon \\rightarrow 0$ one has $\\ket{\\Phi_{jk}} \\rightarrow \\ket{jk}$ which is a product eigenstate of the unperturbed system and forms a complete basis with spectrum $\\varphi_{jk} \\rightarrow \\theta_{jk} = \\theta_j + \\theta_k \\,\\,\\text{mod}\\,\\, 2\\pi$.  For non-vanishing $\\epsilon$ there is a unitary transformation $S$ between the eigenbases for the set $\\ket{\\Phi_{jk}}$ and $\\ket{jk}$ whose matrix elements can be identified using the relations\n\\begin{eqnarray}\n\\ket{\\Phi_{jk}} &=& \\sum_{j'k'} S_{jk,j'k'} \\ket{j'k'} = \\sum_{j'k'}  \\ket{j'k'} \\bra{j'k'}\\ket{\\Phi_{jk}} \\nonumber \\\\\n\\ket{jk} &=& \\sum_{j'k'} S^\\dagger_{jk,j'k'} \\ket{\\Phi_{j'k'}}\\ .\n\\label{eq:UnitaryTransform}\n\\end{eqnarray}\n\n\\subsection{Eigenvalues of the reduced density matrix}\n\nIn the limit $N_A \\rightarrow \\infty$, perturbation theory for unitary Floquet systems generates the same equations as for Hamiltonian systems up to vanishing corrections of $\\mathcal{O}((N_A N_B)^{-1})$ if one identifies $U_{AB}(\\epsilon)=\\exp(i\\epsilon V)$~\\cite{Tomsovic18c}.  For an initial unentangled state, begin by considering an eigenstate $\\ket{jk}$ of the non-interacting system.  Denote the time evolution of this initial state after $n$ iterations of the dynamics as $\\ket{j k(n;\\epsilon)}$ [$ = \\mathcal{U}^n(\\epsilon) \\ket{j k}$].  Upon the usual insertion of the completeness relation one gets\n\\begin{equation}\n\\ket{jk(n;\\epsilon)} =  \\sum_{j'k'} \\text{e}^{i n \\varphi_{j'k'}} S^\\dagger_{jk,j'k'}  \\ket{\\Phi_{j'k'}}. \\label{eq:GeneralTimeEvolvedState}\n\\end{equation}\nThis time evolved state has a standard Schmidt decomposed form\n\\begin{equation}\n\\ket{jk(n;\\epsilon)} = \\sum_{l = 1}^{N_A} \\sqrt{\\lambda_l(n;\\epsilon)} \\, \\ket{\\phi^A_l(n;\\epsilon)}\\ket{\\phi^B_l(n;\\epsilon)},\n\\label{eq:GeneralTimeEvolvedStateInSDForm}\n\\end{equation}\nwhere $\\lambda_1 \\geq \\lambda_2 \\geq \\ldots \\geq \\lambda_{N_A}$ are time-dependent Schmidt numbers (eigenvalues of the reduced density matrices) such that $\\sum_l \\lambda_l(n;\\epsilon) = 1$, and $\\{ \\ket{\\phi^A_l(n;\\epsilon)}\\}$, $\\{ \\ket{\\phi^B_l(n;\\epsilon)}\\}$ are the corresponding Schmidt eigenvectors of the $A$ and $B$ subspaces, respectively.\n\nIt was shown in Ref.~\\cite{Lakshminarayan16} that for weak perturbations, the Schmidt decomposition of the eigenstates to $\\mathcal{O}(N_A^{-1})$ corrections are given by the neighboring eigenstates of the unperturbed (non-interacting) system and the perturbation theory coefficients.  This can be considered as a kind of automatic Schmidt decomposition.\n The generalization to the time evolving states, $\\ket{j k(n;\\epsilon)}$, follows by another insertion of the unitary transformation $S$ to give\n\\begin{equation}\n\\ket{jk(n;\\epsilon)} = \\sum_{j''k''} \\sum_{j'k'} \\text{e}^{i n \\varphi_{j'k'}} S^\\dagger_{jk,j'k'} S_{j'k',j''k''} \\ket{j''k''}. \\label{eq:GeneralTimeEvolvedState2}\n\\end{equation}\nThis leads to the identification\n\\begin{equation}\n\\lambda_l(n;\\epsilon) = \\left| \\sum_{j'k'} \\text{e}^{i n \\varphi_{j'k'}} S^\\dagger_{jk,j'k'} S_{j'k',(jk)_l} \\right|^2,\\label{eq:SchmidtNumbersPT}\n\\end{equation}\nwhere $j''k''\\rightarrow (jk)_l$, meaning that fixing $l$ fixes a unique and distinct index pair $(jk)_l$; e.g.~$(jk)_1=jk$.  Only a small subset ($\\lesssim N_A$) of possible pairs $j''k''$ are related to a $(jk)_l$ due to the energy denominators in perturbation theory.  This is a direct result of the automatic Schmidt decomposition.\nFor $\\epsilon = 0$ one has $\\lambda_1(n;0) =1$, and the rest of the Schmidt eigenvalues vanish\nby the normalization \\eqref{eq:lambda-l-normalization}, as the initial state\nis a product state of eigenstates of the two subsystems.\n\nTo prepare for ensemble averaging, it is helpful to: i) assume that the $jk$ pairs are ordered by the order of the eigenvalues $\\varphi_{jk}$, ii) use the properties of $S$ so that the $n=0$ results are immediately evident, and iii) separate out the diagonal matrix element $S_{jk,jk}$ as a special case.  Let $\\Delta \\varphi_{jk,j^\\prime k^\\prime} = \\varphi_{jk} - \\varphi_{j^\\prime k^\\prime}$. For the largest eigenvalue, i.e.~Eq.~(\\ref{eq:SchmidtNumbersPT}) for $l = 1$, one finds\n\\begin{align}\n& \\lambda_1(n;\\epsilon) =  1 - 2  \\sum_{j'k', j''k''} \\left|S_{j'k',jk}\\right|^2 \\left|S_{j''k'',jk}\\right|^2 \\nonumber \\\\\n& \\qquad \\qquad \\qquad \\qquad \\qquad \\quad \\times \\sin^2 \\left(\\frac{n \\Delta \\varphi_{j'k',j''k''}}{2} \\right) \\nonumber \\\\\n& = 1 - 4 \\left|S_{jk,jk}\\right|^2\\sum_{j'k'}  \\left|S_{j'k',jk}\\right|^2  \\sin^2 \\left(\\frac{n \\Delta \\varphi_{jk,j'k'}}{2} \\right) \\nonumber \\\\\n& - 4  \\sum_{\\substack{j'k' \\le j''k'' \\\\ \\ne jk}} \\left|S_{j'k',jk}\\right|^2 \\left|S_{j''k'',jk}\\right|^2 \\sin^2 \\left(\\frac{n \\Delta \\varphi_{j'k',j''k''}}{2} \\right).  \\nonumber \\\\\n\\label{eq:LargestEigenvalueRaw}\n\\end{align}\nFor $l \\ge 2$, and thus $(jk)_l \\neq jk$, a similar manipulation gives\n\\begin{align}\n&\\lambda_l(n;\\epsilon) = - \\sum_{\\substack{j'k'\\\\j''k''}} \\Re\\left\\{S^\\dagger_{jk,j'k'} S_{j'k',(jk)_l} S^\\dagger_{(jk)_l,j''k''} S_{j''k'',jk} \\right\\}  \\nonumber \\\\\n&  \\qquad \\qquad \\qquad \\qquad \\times 2 \\sin^2 \\left(\\frac{n \\Delta \\varphi_{j'k',j''k''}}{2} \\right) \\nonumber \\\\\n& \\qquad \\qquad - \\sum_{\\substack{j'k'\\\\j''k''}} \\Im\\Big\\{S^\\dagger_{jk,j'k'} S_{j'k',(jk)_l} S^\\dagger_{(jk)_l,j''k''} S_{j''k'',jk} \\Big\\} \\nonumber \\\\\n& \\qquad \\qquad \\qquad \\qquad \\times \\sin \\left(n \\Delta \\varphi_{j'k',j''k''} \\right).\n\\label{eq:OtherEigenvaluesRaw}\n\\end{align}\nNote that summing Eq.~\\eqref{eq:OtherEigenvaluesRaw}\nover $l>1$ reproduces unity minus the expression of Eq.~\\eqref{eq:LargestEigenvalueRaw} as it must.  These Schmidt eigenvalue expressions are exact to order $\\mathcal{O}(N_A^{-1})$.\n\nLowest order Rayleigh-Schr\\\"{o}dinger perturbation theory is applied to the matrix elements of $S$ in order to obtain the complete $O(\\epsilon^2)$ terms of the corresponding expressions for the Schmidt eigenvalues.  Let $\\Delta \\theta_{jk,j'k'} = \\theta_{jk}-\\theta_{j'k'}$. The matrix elements $S_{jk,j'k'}$ are approximately\n\\begin{equation}\nS_{jk,j'k'} \\approx\n\\begin{cases}\n\\frac{1}{\\sqrt{\\mathcal{N}_{jk}}}, &  jk = j'k' \\\\\n\\frac{1}{\\sqrt{\\mathcal{N}_{jk}}} \\frac{\\epsilon \\, V_{j'k',jk}}{\\Delta \\theta_{jk,j'k'}},   & jk \\neq j'k',\n\\end{cases} \\label{eq:UnitaryTransformPT}\n\\end{equation}\nwhere $\\mathcal{N}_{jk}$ is the normalization factor and the perturbed quasienergy is\n\\begin{align}\n& \\varphi_{jk} = \\theta_{jk} +  \\epsilon^2 \\sum_{j'k' \\neq jk } \\frac{|V_{j'k',jk}|^2}{\\Delta \\theta_{jk,j'k'}}. \\label{eq:Unreg_quasienergy}\n\\end{align}\nNote first that in this derivation the diagonal matrix elements $V_{j'k',j'k'}$ are set to zero, because the energy shift due to the first order correction is a random number added to an uncorrelated spectrum giving another uncorrelated spectrum, and hence will not change the spectral statistics nor rotate the eigenvectors.  Secondly, the normalization factor is included even though its first correction is $O(\\epsilon^2)$ because it plays a significant role in determining the regularized expressions ahead, likewise for the perturbed eigenvalues multiplied by the time in the argument of the sine function.  For the largest eigenvalue Eq.~(\\ref{eq:LargestEigenvalueRaw}) becomes\n\\begin{align}\n&\\lambda_1(n;\\epsilon) \\approx 1 - \\frac{4}{\\mathcal{N}_{jk}} \\sum_{j'k' \\neq jk} \\bigg(\\frac{\\epsilon^2 \\, |V_{jk,j'k'}|^2}{\\mathcal{N}_{j'k'} \\, \\Delta \\theta_{j'k',jk}^2} \\bigg)  \\nonumber \\\\\n& \\qquad \\qquad \\qquad \\qquad \\times \\sin^2 \\left(\\frac{n \\Delta \\varphi_{j'k',jk}}{2}\\right),\n\\label{eq:LargestEigenvaluePT}\n\\end{align}\nand for $l \\ge 2$ the others, Eq.~\\eqref{eq:OtherEigenvaluesRaw}, read\n\\begin{align}\n& \\lambda_l(n;\\epsilon) \\approx \\frac{4}{\\mathcal{N}_{jk}}\n    \\frac{\\epsilon^2\\,|V_{(jk)_l,jk}|^2}{\\mathcal{N}_{(jk)_l}\\,\\Delta \\theta_{jk,(jk)_l}^2} \\sin^2 \\left(\\frac{n \\Delta \\varphi_{(jk)_l,jk}}{2}\\right),\n\\label{eq:OtherEigenvaluesPT}\n\\end{align}\nwhere $\\mathcal{N}_{jk}=\\mathcal{N}_{(jk)_l}=1 + \\mathcal{O}(\\epsilon^2)$.\n\n\\subsection{Ensemble averaging}\nBefore moving on to ensemble averaging, it is helpful to make some rescalings as follows:\n\\begin{align}\n& \\Delta \\theta_{j'k',jk} = D \\, s_{j'k'}\\\\\n& \\Delta \\varphi_{j'k',jk} = D \\, s_{j'k'}(\\epsilon) \\approx D s_{j'k'} \\bigg( 1 + \\frac{2 \\Lambda w_{j'k'}}{s^2_{j'k'}} \\bigg) \\label{eq:UnregSpacing}\\\\\n& \\epsilon^2 |V_{jk,j'k'}|^2 = \\Lambda D^2 \\, w_{j'k'}\\ ,\n\\end{align}\nwhere $D = 2\\pi\/(N_A N_B)$ is the mean level spacing of the full system, $s_{j'k'}=s_{j'k'}(0)$, and the $jk$ subscript is dropped where unnecessary.  The approximation in Eq.~\\eqref{eq:UnregSpacing} follows by considering only the matrix element that directly connects the two levels.  The other terms in Eq.~\\eqref{eq:UnregSpacing} move the levels back and forth and mostly cancel, but this term pushes the two levels away from each other and is dominant when the two levels are close lying where the correction may contribute.\nThe symmetry breaking (entangling) interaction matrix elements of $V$ represented in the eigenbasis of the unperturbed system behave as complex Gaussian random variable such that\n$ \\overline{ | V_{jk,j'k'} |^2 } = v^2 w_{j'k'} $, where $w_{j'k'}$ follow a Porter-Thomas distribution \\cite{PorTho1956} for the COE and an exponential one for the CUE:\n\\begin{equation}\n\\rho(w) =\n\\begin{cases}\n\\frac{1}{\\sqrt{2\\pi w}}\\exp(-w\/2) & \\qquad \\text{for COE} \\\\\n\\exp(-w)                          & \\qquad \\text{for CUE}.\n\\end{cases}\\label{PorterThomas}\n\\end{equation}\nIn both of the cases, $\\overline{w_{j'k'} }= 1$, which is consistent with $\\Lambda = \\epsilon^2 v^2 \/D^2$.  In real dynamical systems, deviations from Porter-Thomas distributions may occur as noted in Ref.~\\cite{Tomsovic18c}.\n\nThus, in the rescaled variables the Schmidt eigenvalues for $l \\ge 2$ are\n\\begin{equation}\n\\lambda_l(n;\\Lambda) \\approx \\frac{4}{\\mathcal{N}_{jk}}\\Bigg(\\frac{\\Lambda \\,w_{(jk)_l} }{\\mathcal{N}_{(jk)_l}\\, s_{(jk)_l}^2} \\Bigg) \\,\\sin^2\\bigg(\\frac{n Ds_{(jk)_l}{(\\epsilon)}}{2}\\bigg), \\label{eq:OtherEigenvaluesRescaled}\n\\end{equation}\nand the relation, following from the normalization\ncondition Eq.~\\eqref{eq:lambda-l-normalization},\n\\begin{equation}\n\\lambda_1(n;\\Lambda) = 1 - \\sum_{l\\ne 1}\\lambda_l(n;\\Lambda)\n\\end{equation}\nis exactly preserved to this order.  Next convert the expressions for the Schmidt eigenvalues into integrals, by making use of the function $R(s,w)$~\\cite{Tomsovic18c},\n\\begin{equation}\nR(s,w) = \\sum_{j'k' \\neq jk} \\delta(w-w_{j'k'}) \\delta(s-s_{j'k'}), \\label{eq:R-function}\n\\end{equation}\nwhich after ensemble averaging becomes the joint probability density of finding a level at a rescaled distance $s$ from $\\theta_{jk}$ and the corresponding scaled matrix element $w_{j'k'}$ at the value $w$.\nWith these definitions, scalings, and substitutions, Eq.~(\\ref{eq:LargestEigenvaluePT}) becomes\n\\begin{align}\n& \\lambda_1(n;\\epsilon) \\approx 1 - 4 \\Lambda \\int_{-\\infty}^\\infty \\dd{s} \\int_0^\\infty \\dd{w} \\frac{w}{s^2}\\,R(s,w)  \\nonumber \\\\\n&\\qquad \\qquad \\qquad \\times \\sin^2\\Big(\\frac{n D s}{2}\\big[\\,1+\\frac{2 \\Lambda w}{s^2}\\,\\big]\\Big).\n\\end{align}\nThe ensemble average of $\\lambda_1(n;\\epsilon)$ follows by substituting the ensemble average of $R(s,w)$ by\n\\begin{equation}\n\\overline{R(s,w)} = R_2(s) \\, \\rho(w),\n\\end{equation}\nwhere $R_2(s)$ is the two-point correlation function and $\\rho(w)$ is defined in Eq.~(\\ref{PorterThomas}).   For an uncorrelated spectrum $R_2(s)=1$ for $-\\infty < s < \\infty$.\nTherefore, the averaged largest Schmidt eigenvalue is\n\\begin{align}\n& \\overline{ \\lambda_1(n;\\Lambda)}\\approx 1 - 4 \\Lambda \\int_{-\\infty}^\\infty \\dd{s} \\int_0^\\infty \\dd{w} \\frac{w}{s^2}\\, \\rho(w) \\nonumber \\\\\n&\\qquad \\qquad \\qquad \\times \\sin^2\\Big(\\frac{n D s}{2}\\big[\\,1+\\frac{2 \\Lambda w}{s^2}\\,\\big]\\Big). \\label{eq:LargestEigenvalueUnreg}\n\\end{align}\nThis expression diverges due to the fact that too many spacings are vanishingly small across the ensemble, and the perturbation theory must account for spacings smaller than the matrix elements.  In the next subsection the expressions are regularized properly for small $s$.\n\nIt is worth noting that if the interest is in the ensemble average of some function of $\\lambda_1(n;\\Lambda)$, then one must consider the ensemble average of the same function of $R(s,w)$.  Perhaps, the simplest example is the ensemble average of the square of  $\\lambda_1(n;\\Lambda)$ for which the needed result is~\\cite{Tomsovic18c}\n\\begin{eqnarray}\n\\overline{R(s_1,w_1)R(s_2,w_2)} &=& R_3(s_1,s_2) \\rho(w_1)\\rho(w_2) + \\nonumber \\\\\n&&\\hspace*{-2cm}\n   \\delta\\left(w_1-w_2\\right)\\rho(w_1) \\delta\\left(s_1-s_2\\right)R_2(s_1)\n\\label{threepoint}\n\\end{eqnarray}\nwhich involves both the $2$-point and $3$-point spectral correlation functions.  However, it turns out that the leading correction depends on $R_2(s)$, as the $R_3(s_1,s_2)$ term gives a contribution that is $\\mathcal{O}(\\sqrt{\\Lambda})$ smaller in comparison, and for example, generating the leading correction of high order moments depends only on the $2$-point spectral correlation function.  This circumstance is helpful ahead in the next section.\n\nFollowing the same sequence of steps for the second largest eigenvalue $\\lambda_2$ requires, in addition, the probability density of the closest scaled energy of one of the $\\ket{(jk)_l}$.  For uncorrelated spectra it is given by $\\rho_{\\text{CN}}(s) = 2 \\exp(-2 s)$ for $0 \\le s < \\infty$~\\cite{Tomsovic18c,Srivastava19}.  One finds for the ensemble average of second largest eigenvalue,\n\\begin{align}\n& \\overline{\\lambda_2(n;\\Lambda)} \\approx 4 \\Lambda \\int_{-\\infty}^\\infty \\dd{s} \\int_0^\\infty \\dd{w} \\frac{w}{s^2} \\,\\rho(w) \\, \\rho_{\\text{CN}}(s)\\nonumber \\\\\n&\\qquad \\qquad \\qquad \\times \\sin^2\\Big(\\frac{n D s}{2}\\big[\\,1+\\frac{2 \\Lambda w}{s^2}\\,\\big]\\Big),\\label{eq:OtherEigenvalueUnreg}\n\\end{align}\nwhich is also divergent for small $s$.  It turns out that the apparent order of corrections, $\\mathcal{O}(\\Lambda)$, seen in Eqs.~(\\ref{eq:LargestEigenvalueUnreg}, \\ref{eq:OtherEigenvalueUnreg}), is not correct.  The regularization required to deal with the small energy denominators in the perturbation expressions alters the leading order to $\\mathcal{O}(\\sqrt{\\Lambda})$.\n\n\\subsection{Regularized perturbation theory}\n\nThe method for regularizing the perturbation expressions was introduced in Refs.~\\cite{Tomsovicthesis,French88a} and developed for the Schmidt eigenvalues pertaining to the eigenstates of interacting quantum chaotic systems in Refs.~\\cite{Lakshminarayan16,Tomsovic18c}. The standard Rayleigh-Schr\\\"{o}dinger perturbation expressions break down when the unperturbed spectrum has nearly degenerate levels due to the small energy denominators.  However, there is an infinite sub-series of terms within the perturbation series of a quantity of interest which involve only two levels that are diverging due to near-degeneracy. This subseries can be resummed to get the corresponding regularized expressions.  These results are equivalent to the two-dimensional degenerate perturbation theory results.\n\nThe regularized expressions for the Schmidt eigenvalues upon resummation of the two-level like terms of the perturbation series boil down to essentially replacing\n\\begin{align}\n\\frac{1}{\\mathcal{N}_{jk}}\\Bigg(\\frac{\\Lambda\\,w_{j'k'} }{\\mathcal{N}_{j'k'}\\, s_{j'k'}^2} \\Bigg) \\mapsto \\frac{\\Lambda \\, w_{j'k'}}{s_{j'k'}^2+ 4 \\, \\Lambda \\, w_{j'k'}} , \\label{eq:RegularizedNormTimesMatrixElement}\n\\end{align}\nalong with the energy spacing~\\cite{Srivastava16} in Eq.~(\\ref{eq:UnregSpacing}) as\n\\begin{equation}\ns_{j'k'}(\\epsilon)  \\mapsto \\sqrt{s_{j'k'}^2 + 4 \\Lambda w_{j'k'} } \\label{eq:RegularizedSpacing}\n\\end{equation}\nin Eqs.~(\\ref{eq:LargestEigenvaluePT}, \\ref{eq:OtherEigenvaluesPT}).  To verify the result in Eq.~(\\ref{eq:RegularizedNormTimesMatrixElement}), the Schmidt eigenvalues in Eqs.~(\\ref{eq:LargestEigenvalueRaw}, \\ref{eq:OtherEigenvaluesRaw}) were expanded using perturbation theory of the matrix elements $S_{jk,j'k'}$ up to and including order $\\mathcal{O}(\\epsilon^4)$.  The details for this are given in App.~\\ref{app:RegularizationDerivation}.  For a two-level system the normalization is\n\\begin{equation}\n|S_{jk,jk}|^2 = \\frac{1}{\\mathcal{N}_{jk}} = \\frac{1}{2}\\Bigg( 1 + \\frac{|s_{j'k'}|}{\\sqrt{s_{j'k'}^2+4 \\Lambda w_{j'k'}}} \\Bigg)\n\\end{equation}\nand the matrix element\n\\begin{equation}\n|S_{j'k',jk}|^2 = \\frac{1}{2}\\Bigg( 1 - \\frac{|s_{j'k'}|}{\\sqrt{s_{j'k'}^2+4 \\Lambda w_{j'k'}}} \\Bigg).\n\\end{equation}\nUsing these results, we get Eq.~(\\ref{eq:RegularizedNormTimesMatrixElement}). This gives the regularized Schmidt eigenvalues for $l \\neq 1$ as\n\\begin{align}\n& \\lambda_l(n;\\Lambda) = \\frac{4 \\, \\Lambda \\, w_{(jk)_l}}{s^2_{(jk)_l} + 4 \\, \\Lambda \\, w_{(jk)_l} } \\nonumber \\\\\n& \\qquad \\qquad \\qquad \\times \\sin^2 \\Big( \\frac{n D}{2} \\sqrt{s^2_{(jk)_l} + 4 \\, \\Lambda \\, w_{(jk)_l}} \\,\\Big). \\label{eq:SchmidtEigenvaluelneq1Reg}\n\\end{align}\nRescaling the spacing $z = s\/\\sqrt{\\Lambda}$ and time\n\\begin{equation}  \\label{eq:time-rescaling}\n   t = n D\\sqrt{\\Lambda} ,\n\\end{equation}\nthe ensemble average of the first two Schmidt eigenvalues is given by\n\\begin{align}\n& \\overline{ \\lambda_1(t;\\Lambda)} = 1- \\sqrt{\\Lambda}\\int_0^\\infty \\dd{w} \\int_{-\\infty}^\\infty \\dd{z} \\frac{4 w}{z^2 + 4 w} \\, \\rho(w)  \\nonumber \\\\\n& \\qquad \\qquad \\qquad \\times  \\sin^2\\Big(\\frac{t}{2} \\sqrt{z^2 + 4 w} \\Big) \\nonumber \\\\\n& \\qquad \\quad \\, = 1 - C_2(1;t)\\,\\sqrt{\\Lambda},\n\\end{align}\nwhere $C_2(1;t)$ is a short-hand for the integral\n(the general notation for arbitrary moments is given in\nEq.~(\\ref{eq:C2Integral}) ahead) and,\n\\begin{align}\n& \\overline{ \\lambda_2(t;\\Lambda)} = \\sqrt{\\Lambda}\\int_0^\\infty \\dd{w} \\int_0^\\infty \\dd{z} \\frac{4 w}{z^2 + 4 w} \\nonumber \\\\\n& \\qquad \\qquad \\qquad \\times  \\, \\rho(w) \\,\\big(2 \\,\\text{e}^{-2 z \\sqrt{\\Lambda}}\\,\\big) \\sin^2\\Big(\\frac{t}{2} \\sqrt{z^2 + 4 w} \\Big) \\nonumber \\\\\n& \\, \\quad \\qquad = C_{2}(1;t) \\sqrt{\\Lambda} + \\mathcal{O}(\\Lambda \\ln \\Lambda).\n\\end{align}\nFor sufficiently small $\\Lambda$ it turns out that for both the COE and CUE cases, only (unperturbed) eigenstates corresponding to the first two largest Schmidt eigenvalues contribute largely to the state $\\ket{jk(t;\\Lambda)}$, i.e.,\n\\begin{equation}\n\\overline{ \\lambda_1(t;\\Lambda) + \\lambda_2(t;\\Lambda)} = 1 + \\mathcal{O}(\\Lambda \\ln \\Lambda ),\n\\end{equation}\nand other Schmidt eigenvalues ($l>2$) contribute in higher orders than $\\sqrt{\\Lambda}$.\nThis is crucial for extending the perturbation theory of the Schmidt eigenvalue moments to the non-perturbative regime, which is done in Sec.~\\ref{sec:NonPerturbative}.\nIt should be noted that as the unperturbed spectrum is uncorrelated, there is a non-zero probability of three-level, four-level and so-forth near-degeneracy occurrences, but with lower probability from the two-level case, and hence their contributions are higher of order than $\\sqrt{\\Lambda}$.\n\nMoreover note that the perturbation expressions for the Schmidt eigenvalues of the eigenstates $\\{\\ket{\\Phi_{jk}}\\}$ given in Refs.~\\cite{Lakshminarayan16, Tomsovic18c} for the largest eigenvalue and the other eigenvalues are $1 -  \\sum_{j'k \\neq jk} \\epsilon^2 |V_{(j'k',jk}|^2\/\\Delta\\theta^2_{jk,j'k'}$ and $\\epsilon^2 |V_{(jk)_l,jk}|^2\/\\Delta\\theta^2_{jk,(jk)_l}$, respectively, in contrast to the expressions for the Schmidt eigenvalues of a time evolving state $\\ket{jk(n;\\epsilon)}$ presented in Eqs.~(\\ref{eq:LargestEigenvaluePT}, \\ref{eq:OtherEigenvaluesPT}). Due to an extra normalization factor in the denominators of Eqs.~(\\ref{eq:LargestEigenvaluePT}, \\ref{eq:OtherEigenvaluesPT}), the expression for the regularization, although related, takes on a different form than that in Refs.~\\cite{Lakshminarayan16, Tomsovic18c}.\n\n\n\\section{Eigenvalue moments of the reduced density matrix}\n\\label{sec:eigenvalue-moments}\n\nTo fully characterize the entanglement of the evolving state, the Schmidt eigenvalue expression in Eq.~(\\ref{eq:SchmidtEigenvaluelneq1Reg}) is used to compute the leading order of general moments analytically and thereby the HCT entropies, good up to and including $\\mathcal{O}(\\sqrt{\\Lambda})$.\n\n\\subsection{General moments} \\label{subsec:GeneralMomentsCalc}\n\nConsider the ensemble average of the general moments $\\mu_\\alpha$, Eq.~\\eqref{eq:GenericMoments}, of the Schmidt eigenvalues.  The largest eigenvalue must be separated out from the others and two integrals considered.  First, consider general moments of the sum of all the Schmidt eigenvalues other than the largest, i.e.\n\\begin{equation}\n\\overline{ \\sum_{l \\neq 1} \\lambda_l^\\alpha(t;\\Lambda) } = C_2(\\alpha;t) \\, \\sqrt{\\Lambda} \\,,\n\\end{equation}\nwhere after rescaling $s$ to $z$ in Eq.~(\\ref{eq:R-function})\n\\begin{align}\n& C_2(\\alpha;t) = \\int_{-\\infty}^\\infty \\dd{z} \\int_0^\\infty \\dd{w} \\overline{R(z,w)} \\frac{4^\\alpha w^\\alpha}{(z^2+4w)^\\alpha}  \\nonumber \\\\\n& \\qquad \\qquad \\quad \\times \\sin^{2\\alpha}\\bigg( \\frac{t}{2} \\sqrt{z^2+4w}\\,\\bigg) \\nonumber \\\\\n& \\qquad \\qquad = \\int_{-\\infty}^\\infty \\dd{z} \\int_0^\\infty \\dd{w} \\rho(w) \\frac{4^\\alpha w^\\alpha}{(z^2+4w)^\\alpha}  \\nonumber \\\\\n& \\qquad \\qquad \\quad \\times \\sin^{2\\alpha}\\bigg( \\frac{t}{2} \\sqrt{z^2+4w}\\,\\bigg). \\label{eq:C2Integral}\n\\end{align}\nThe evaluation of this integral is discussed in the next subsection and\nApp.~\\ref{app:C-2-alpha-t-derivation}.\nNow focusing on the ensemble average of the largest Schmidt eigenvalue,\n\\begin{align}\n& \\overline{\\lambda_1^\\alpha(t;\\Lambda)} = \\overline{\\bigg( 1 - \\sum_{l \\neq 1} \\lambda_l(t;\\Lambda)\\bigg)^\\alpha} \\nonumber \\\\\n& \\qquad\\,\\,\\quad= \\overline{\\Bigg[ 1 - \\int_{-\\infty}^\\infty \\dd{z} \\int_0^\\infty \\dd{w} R(z,w) \\frac{4 w}{z^2+4w}} \\nonumber \\\\\n& \\qquad \\qquad \\quad \\overline{\\times \\sin^{2\\alpha}\\bigg( \\frac{t}{2} \\sqrt{z^2+4w}\\,\\bigg) \\Bigg]^\\alpha}\\,. \\label{eq:LargestEigenvalueExactIntegral}\n\\end{align}\nBefore computing the above for a general power $\\alpha$, consider the $\\alpha =2$ case.  Expanding the above expression gives a square of the integral term, which is a quadruple integral containing the product $R(z_1,w_1) R(z_2,w_2)$.  Equation (\\ref{threepoint}) has two contributions, the diagonal term, where $(z_1,w_1) = (z_2,w_2)$, and the off-diagonal one.   For an uncorrelated spectrum, any multi-point spectral correlation function is unity.  Thus\n\\begin{align}\n\\overline{\\lambda_1^2(t;\\Lambda)} = &\\; 1 - 2 C_2(1;t)\\sqrt{\\Lambda} + C_2(2;t) \\sqrt{\\Lambda} \\nonumber \\\\\n&  + C^2_2(1;t)\\, \\Lambda,\n\\end{align}\nwhere the off-diagonal term, $R_3(z_1,z_2)$, is responsible for the $\\mathcal{O}(\\Lambda)$ term.\nThis illustrates that to the leading $\\mathcal{O}(\\sqrt{\\Lambda})$, the diagonal term alone suffices, and the other terms contribute to higher than leading order.  This simplifies the $\\overline{\\lambda_1^\\alpha}$ computation, where after the binomial expansion of Eq.~(\\ref{eq:LargestEigenvalueExactIntegral}), keeping only the terms contributing to the leading order gives,\n\\begin{align}\n& \\overline{ \\lambda_1^\\alpha(t;\\Lambda) } = 1 + \\sum_{p=1}^\\infty (-1)^p \\binom{\\alpha}{p} \\overline{ \\sum_{l\\neq 1} \\lambda_l^p }.\n\\end{align}\nFinally, the general moments Eq.~\\eqref{eq:GenericMoments}\nof the Schmidt eigenvalues for $\\alpha >1\/2$ are given by\n\\begin{equation}\n\\overline{\\mu_{\\alpha}(t;\\Lambda) } = 1 + \\sum_{p=1}^\\infty (-1)^p \\binom{\\alpha}{p} \\overline{ \\sum_{l\\neq 1} \\lambda_l^p }\\, +\\, \\overline{ \\sum_{l\\neq 1} \\lambda_l^\\alpha } .\n\\end{equation}\nThese can be written as\n\\begin{equation} \\label{eq:mu-alpha-t---C-alpha-t}\n\\overline{ \\mu_{\\alpha}(t;\\Lambda) } = 1 - C(\\alpha;t) \\sqrt{\\Lambda},\n\\end{equation}\nwhere\n\\begin{equation} \\label{eq:C-alpha-t-via-C-2}\nC(\\alpha;t) = \\sum_{p=1}^\\infty (-1)^{p+1} \\binom{\\alpha}{p} C_{2}(p;t) - C_{2}(\\alpha;t).\n\\end{equation}\nThese functions are central for the analytical\ndescription of the entropies and shown\nin Fig.~\\ref{fig:Calpha} for the COE and the CUE.\nA particular feature is the overshooting before the saturation sets in.\nFor the CUE the location of the maxima occurs slightly later in $t$\nthan for the COE and also the saturation regime is reached\nslightly later.\nMoreover, the saturation value is slightly larger than in the COE case.\n\n\\begin{figure}\n\\includegraphics[width=8.6cm]{fig_Calpha_COE.pdf}\n\n\\includegraphics[width=8.6cm]{fig_Calpha_CUE.pdf}\n\n\\caption{Plot of $C(\\alpha;t)$ for $\\alpha=2$ (solid),\n $\\alpha=3$ (dashed), $\\alpha=4$ (dot-dashed),\nand $\\pdv*{C(\\alpha;t)}{\\alpha}|_{\\alpha \\rightarrow 1}$ (dotted)\nfor (a) COE and (b) CUE.}\n\\label{fig:Calpha}\n\\end{figure}\n\n\nIn addition, it can be shown that $\\overline{ \\mu_\\alpha (t;\\Lambda) }$ is evaluated up to and including $\\mathcal{O}(\\sqrt{\\Lambda})$ by the first and second largest Schmidt eigenvalues\n\\begin{equation}\n  \\overline{ \\mu_\\alpha (t;\\Lambda) }\n   = \\overline{\\lambda_1^\\alpha(t;\\Lambda)} +\n     \\overline{\\lambda_2^\\alpha(t;\\Lambda)},\n\\end{equation}\nas other Schmidt eigenvalues ($l > 2$) do not contribute to $ \\mathcal{O} ( \\sqrt{ \\Lambda } ) $.\nThis relation is vital for recursively invoking perturbation theory\nin Sec.~\\ref{sec:NonPerturbative}\nin order to extend the results beyond the perturbative regime.\n\n\\subsection{Entropies}\n\nThe HCT entropies can be computed in the perturbation regime using the results for the average eigenvalue moments. For $\\alpha \\neq 1$ one has\n\\begin{equation}\n\\overline{ S_\\alpha(t;\\Lambda)} = \\frac{C(\\alpha;t)}{\\alpha -1} \\sqrt{\\Lambda},\n\\end{equation}\nwhere $C(\\alpha; t)$ is given by Eq.~\\eqref{eq:C-alpha-t-via-C-2}.\nThis requires the computation of $C_2(\\alpha; t)$,\nwhich is done in App.~\\ref{app:C-2-alpha-t-derivation},\nand leads to\n\\begin{equation} \\label{eq:C-2-alpha-t}\n  C_{2}(\\alpha;t) = 2^\\alpha \\sum_{q=0}^\\infty \\sum_{m=0}^q (-1)^q\n                   \\binom{\\alpha}{q} a_{qm} f_{m}(\\alpha;t),\n\\end{equation}\nwhere\n\\begin{equation}\na_{qm} = \\binom{q}{\\frac{q-m}{2}}\n        \\left[ \\frac{1+(-1)^{q-m}}{2^q(1+\\delta_{m,0})} \\right]\n\\end{equation}\nand\n\\begin{align}\n  f_m(\\alpha;t) = & \\int_{-\\infty}^\\infty \\dd{z} \\int_0^\\infty\n              \\dd{w} \\frac{w^\\alpha \\rho(w)}{(z^2+4w)^\\alpha} \\nonumber \\\\\n                  & \\qquad \\times \\cos(m t \\sqrt{z^2+4w}\\,).\n\\end{align}\nExplicit expressions for $f_m(\\alpha;t)$\nfor the COE and CUE are derived in App.~\\ref{app:C-2-alpha-t-derivation-COE}\nand \\ref{app:C-2-alpha-t-derivation-CUE}, respectively.\n\n\n\n\\subsection{Discussion}\n\nTo discuss some qualitative features, the case $\\alpha = 2$, which corresponds\nto the linear entropy, is considered here.\nBy Eqs.~(\\ref{eq:GenericHCTEntropy}, \\ref{eq:mu-alpha-t---C-alpha-t})\none has $S_2(t;\\Lambda) = C(2; t) \\sqrt{\\Lambda}$.\nIn case of the COE\n\\begin{align}\n& C(2;t) = 4 \\pi t \\big( \\text{e}^{-t^2} [ \\{ 1 + 2 t^2 \\} I_0(t^2) + 2 t^2 I_1(t^2) ] \\nonumber \\\\\n& \\qquad \\qquad - 4 t^2 \\text{e}^{-4 t^2} [ I_0(4 t^2) + I_1(4 t^2) ] \\, \\big),\n\\end{align}\nwhere $I_n(z)$ is the modified Bessel function of the first kind\n\\cite[Eq.~10.25.2]{DLMF}.\nWhereas for the CUE case\n\\begin{align}\n& C(2;t) = \\pi t \\Big( 3 \\text{e}^{-t^2} - \\frac{1}{2}\\text{e}^{-4 t^2} \\Big) + \\pi^{3\/2} \\text{erf}(t) \\Big( \\frac{1}{2} + 3 t^2 \\Big) \\nonumber \\\\\n& \\qquad \\qquad + \\pi^{3\/2} \\text{erf}(2 t) \\Big( \\frac{1}{8} - 3 t^2 \\Big),\n\\end{align}\nwhere $\\text{erf}(z)$ is the error function.\nFor both COE and CUE cases, $C(2;t)$ for small $t$ has the expansion\n\\begin{align} \\label{eq:C-2-t}\n  & C(2;t) = 4 \\pi t + \\mathcal{O}(t^3)\n\\end{align}\nand naturally gives linear-in-time entropy growth for short time $t$.\nThe same is true for other $\\alpha$-entropies, except for $\\alpha = 1$,\nfor which the leading term is of the order $\\mathcal{O}(t\\,\\ln t)$. In fact, it can be shown that for both COE and CUE cases, with $\\alpha > 1$ and short time $t$,\n\\begin{equation}\n\\dv{}{t} C(\\alpha;t) \\big|_{t \\rightarrow 0} = 2 \\pi \\alpha. \\label{eq:InitialRateCalpha}\n\\end{equation}\n\nIn the limit $t \\rightarrow \\infty$, saturation values of the entropies\ncan be obtained from\n\\begin{equation}\n   S_2(\\infty; \\Lambda) = \\frac{C(\\alpha; \\infty)}{\\alpha-1} \\sqrt{\\Lambda},\n\\end{equation}\nwhich are of the order $\\mathcal{O}(\\sqrt{\\Lambda}\\,)$.\nUsing the explicit expressions for $f_m(\\alpha;t)$\nderived in App.~\\ref{app:C-2-alpha-t-derivation-COE}, \\ref{app:C-2-alpha-t-derivation-CUE}\none sees that in the limit $t \\rightarrow \\infty$,\n$f_m(\\alpha;t)$ vanish for all $m \\neq 0$.\n Using this fact, an expression for saturation value\nof the $\\alpha$-entropies ($\\alpha \\neq 1$) can be derived as\n\\begin{align}\n\\overline{S_\\alpha(\\infty, \\Lambda)} & = \\Bigg(\\alpha \\,\\,  _{3}F_2(1\/2,3\/2,1-\\alpha\\,;\\,2,2\\,;\\,1) \\nonumber \\\\\n&\\; \\qquad \\quad - \\frac{2}{\\pi} \\,\\,\\frac{\\Gamma(\\alpha-1\/2)\\,\\Gamma(\\alpha+1\/2)}{\\Gamma(\\alpha)\\,\\Gamma(\\alpha+1)} \\Bigg) \\nonumber \\\\\n& \\; \\qquad \\times \\frac{\\sqrt{\\Lambda}}{\\alpha-1}\\begin{cases}\n\t\t\t\t\t\t\t\\sqrt{2\\pi} \\; \\quad \\quad  \\text{for COE,} \\\\\n\t\t\t\t\t\t\t\\pi^{3\/2}\/2 \\, \\; \\quad \\text{for CUE.}\n\t\t\t\t\t\t \\end{cases}\n  \\label{eq:sat-S-alpha}\n\\end{align}\nHere $_{m}F_n$ is a generalized hypergeometric function\n\\cite[Eq.~35.8.1]{DLMF} defined by\n\\begin{align}\n& _{m}F_n ( a_1 ,\\ldots, a_m ; b_1 ,\\ldots, b_n ; z) = \\sum_{k=0}^\\infty \\frac{(a_1)_k \\ldots (a_m)_k}{(b_1)_k \\ldots (b_n)_k} \\frac{z^k}{k!},\n\\end{align}\nwhere $(a)_k = \\Gamma(a+k)\/\\Gamma(a)$ is Pochhammer's symbol.\nEquation~\\eqref{eq:sat-S-alpha} shows that the saturation values\nfor both COE and CUE scale with $\\sqrt{\\Lambda}$\nand that the CUE case leads to a slightly (11\\%) larger value.\n\n\n\nFor the linear entropy, $\\alpha=2$, Eq.~\\eqref{eq:sat-S-alpha} simplifies to\n\\begin{align} \\label{eq:S2-sat-COE-CUE}\n\\overline{S_2(\\infty;\\Lambda)} = \\sqrt{\\Lambda}\n  \\begin{cases}\n  5 \\sqrt{\\pi\/8}   & \\text{for COE}, \\\\\n  5 \\pi^{3\/2}\/8   & \\text{for CUE}.\n  \\end{cases}\n\\end{align}\nFor the special case of the von Neumann entropy for $\\alpha=1$,\n$\\lim_{t\\rightarrow \\infty} \\pdv*{C(\\alpha;t)}{\\alpha}|_{\\alpha \\rightarrow 1}$\nneeds to be computed. It can be shown that\n\\begin{align} \\label{eq:saturation-perturbatively}\n\\overline{S_1(\\infty;\\Lambda)}\n= & \\Big(\n 4 \\ln 2- \\frac{3}{16}\\, _{4}F_3(1,1,3\/2,5\/2 \\,;\\,2,3,3\\,;\\,1)   \\Big)\n\\nonumber \\\\\n& \\times \\sqrt{\\Lambda} \\begin{cases}\n\t\t\t\t\t\t\t\\sqrt{2\\pi} \\; \\quad \\quad  \\text{for COE,} \\\\\n\t\t\t\t\t\t\t\\pi^{3\/2}\/2 \\, \\; \\quad \\text{for CUE.}\n\t\t\t\t\t\t \\end{cases}\n\\end{align}\nAn extension to the non-perturbative result will be discussed\nin Sec.~\\ref{sec:long-time-saturation}.\n\nIn the perturbative regime, if the entropies $\\overline{ S_\\alpha (t;\\Lambda)}$ are scaled with respect to\ntheir saturation values,\n\\begin{equation}\n\\overline{ \\mathcal{S}_\\alpha (t) }\n    = \\frac{\\overline{S_\\alpha(t; \\Lambda)}}\n           {\\overline{S_\\alpha(\\infty; \\Lambda)}},\n\\label{eq:Universal-Scaling}\n\\end{equation}\nthey do not depend on the transition parameter,\nleading to one universal curve for each $\\alpha$ described\nby the prediction\n\\begin{equation}\n\\overline{ \\mathcal{S}_\\alpha (t) } = \\frac{C(\\alpha;t)}{C(\\alpha;\\infty)}.\n  \\label{eq:UniversalCurveTheory}\n\\end{equation}\nThis universal property is depicted for the linear entropy in Fig.~\\ref{fig:UniversalCurvePlot} for various $\\Lambda$-values. As $\\Lambda$ goes beyond the perturbation regime, departure from the universal curve is seen due to the breakdown of the perturbation theory. In the forthcoming section, the extension of the theory to the non-perturbative regime is discussed.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=8.6cm]{fig_CUE_universal.pdf}\n\n    \\caption{\\label{fig:UniversalCurvePlot} Scaled linear entropy\n      $\\overline{ \\mathcal{S}_2(t) }$, Eq.~\\eqref{eq:Universal-Scaling}, for the\n      random matrix transition ensemble in Eq.~(\\ref{eq:GenericFloquetRMT}) for\n      the CUE case for $\\Lambda = 10^{-6}$ (magenta circles),\n      $\\Lambda = 10^{-4}$ (red triangles), $\\Lambda = 10^{-2}$ (blue squares),\n      and $\\Lambda = 1$ (green diamonds); The theoretical prediction\n      Eq.~(\\ref{eq:UniversalCurveTheory}) for $\\alpha =2$ is shown as solid curve.}\n\\end{figure}\n\n\n\\section{Non-perturbative regime} \\label{sec:NonPerturbative}\n\nThe results obtained from the perturbation theory can be extended to the\nnon-perturbative regime to produce the full transition and the saturation\nvalues by employing the recursively embedded perturbation theory technique as\ndone in Ref.~\\cite{Lakshminarayan16,Tomsovic18c} for the eigenstates.\n\n\n\\subsection{Full transition}\n\n\nFor small enough $\\Lambda$, the time evolved state $\\ket{jk(n;\\epsilon)} \\equiv \\ket{jk(t;\\Lambda)}$ can be Schmidt decomposed as\n\\begin{equation}\n\\ket{jk(t;\\Lambda)} = \\sqrt{\\lambda_1(t;\\Lambda)} \\,\\ket{(jk)_1} + \\sqrt{\\lambda_2(t;\\Lambda)}  \\,\\ket{(jk)_2},\n\\end{equation}\nsuch that $\\lambda_1 + \\lambda_2 = 1$, where the time-dependent phase-factor is absorbed into the definition of the Schmidt eigenvectors $\\ket{ (jk)_l }$. Now increasing the interaction strength, another unperturbed state energetically close to $\\ket{(jk)_1}$ will contribute to $\\ket{jk(n;\\epsilon)}$,\n\\begin{align}\n& \\ket{jk(t;\\Lambda)} = \\sqrt{\\lambda_1'(t;\\Lambda)} \\,\\bigg(\\sqrt{\\lambda_1(t;\\Lambda)} \\,\\ket{(jk)_1} + \\nonumber \\\\\n& \\qquad \\qquad \\quad \\sqrt{\\lambda_2(t;\\Lambda)}  \\,\\ket{(jk)_2}\\bigg) + \\sqrt{\\lambda_2'(t;\\Lambda)} \\, \\ket{(jk)_3},\n\\end{align}\nwhere $\\lambda_{1,2}'$ follow same statistical properties as the unprimed ones. Thus the purity is $\\mu_2' = \\lambda_1^{'2} \\lambda_1^2 + \\lambda_1^{'2} \\lambda_2^2 + \\lambda_2^{'2}$ giving\n\\begin{align}\n\\mu_2' - \\mu_2 = -(1-\\lambda_1^{'2} - \\lambda_2^{'2})\\mu_2 + \\lambda_2^{'2} (1-\\mu_2).\n\\end{align}\n\n\n\\begin{figure}\n\\includegraphics[width=8.4cm]{fig_COE_TP_1e-06_N_50.pdf}\n\n\\includegraphics[width=8.4cm]{fig_COE_TP_0p0001_N_50}\n\n\\includegraphics[width=8.4cm]{fig_COE_TP_0p01_N_50.pdf}\n\n\\includegraphics[width=8.4cm]{fig_COE_TP_1_N_50}\n\\caption{\\label{fig:COE_Salpha} Entropies $\\overline{ S_\\alpha }$ for the COE\n  case with $N_A=N_B=50$ for (a) $\\Lambda=10^{-6}$, (b) $\\Lambda=10^{-4}$, (c)\n  $\\Lambda=10^{-2}$, and (d) $\\Lambda=1$ for $\\alpha=1$ (green diamonds),\n  $\\alpha=2$ (magenta circles), $\\alpha=3$ (red triangles), and $\\alpha=4$\n  (blue squares). Black lines show the corresponding theory curves,\n  Eq.~(\\ref{eq:alphaEntropyTheory}).}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[width=8.4cm]{fig_CUE_TP_1e-06_N_50.pdf}\n\n\\includegraphics[width=8.4cm]{fig_CUE_TP_0p0001_N_50}\n\n\\includegraphics[width=8.4cm]{fig_CUE_TP_0p01_N_50.pdf}\n\n\\includegraphics[width=8.4cm]{fig_CUE_TP_1_N_50}\n\\caption{\\label{fig:CUE_Salpha} Entropies $\\overline{ S_\\alpha }$ for the CUE\n  case with $N_A=N_B=50$ for (a) $\\Lambda=10^{-6}$, (b) $\\Lambda=10^{-4}$, (c)\n  $\\Lambda=10^{-2}$, and (d) $\\Lambda=1$ for $\\alpha=1$ (green diamonds),\n  $\\alpha=2$ (magenta circles), $\\alpha=3$ (red triangles), and $\\alpha=4$\n  (blue squares). Black lines show the corresponding theory curves,\n  Eq.~(\\ref{eq:alphaEntropyTheory}).}\n\\end{figure}\n\nFor a given $\\alpha$, following this technique and replacing $\\lambda_{1,2}^{'\\alpha}$ with their ensemble average, a differential equation for the moments $\\overline{ \\mu_\\alpha (t;\\Lambda) }$ can be derived, good up to $\\mathcal{O}(\\sqrt{\\Lambda})$,\n\\begin{equation}\n\\pdv{\\overline{ \\mu_\\alpha(t;\\Lambda) }}{\\sqrt{\\Lambda}} = - C(\\alpha;t) \\overline{ \\mu_\\alpha (t;\\Lambda)}.\n\\end{equation}\nThis has a solution of the form (valid for the infinite dimensional case)\n\\begin{equation}\n\\overline{ \\mu_\\alpha(t;\\Lambda)} \\approx \\exp(-C(\\alpha;t) \\sqrt{\\Lambda}).\n\\end{equation}\nIn the limit $\\Lambda \\rightarrow \\infty$, and for large (but finite)\ndimensionality $N=N_A =N_B$, the moments tend to the random matrix result\n\\begin{equation}\n\\overline{ \\mu_\\alpha^\\infty } = \\mathcal{C}_\\alpha\/N^{\\alpha-1},\n\\end{equation}\nwhere the $\\mathcal{C}_\\alpha$ are Catalan numbers \\cite[\\S26.5.]{DLMF}.\nFor the $N_A \\neq N_B$ case such an expression can be found following Ref.~\\cite{Sommers04}. Incorporating this limit\ninto the above differential equation solution gives an approximate expression for the moments valid for any $\\Lambda$,\n\\begin{equation}\n\\overline{ \\mu_\\alpha (t;\\Lambda) } \\approx \\exp(- \\frac{C(\\alpha;t)}{1-\\overline{\\mu_\\alpha^\\infty}}\\sqrt{\\Lambda})(1-\\overline{\\mu_\\alpha^\\infty}) + \\overline{ \\mu_\\alpha^\\infty}.\n\\end{equation}\nUsing the definition of the HCT entropies (\\ref{eq:GenericHCTEntropy})\ngives\n\\begin{equation}\n\\overline{S_\\alpha(t;\\Lambda) } \\approx \\bigg[ 1 - \\exp(-\\frac{C(\\alpha;t)}{(\\alpha-1)\\overline{ S_\\alpha^\\infty}}\\sqrt{\\Lambda})\\bigg] \\overline{S_\\alpha^\\infty}, \\label{eq:alphaEntropyTheory}\n\\end{equation}\nwhere\n\\begin{equation}\n\\overline{ S_\\alpha^\\infty} = \\frac{1-\\mathcal{C}_\\alpha N^{1-\\alpha}}{\\alpha-1}.\n\\end{equation}\nTo apply Eq.~\\eqref{eq:alphaEntropyTheory}\none has to use for $C(\\alpha; t)$ the results\ncorresponding to the CUE or the COE,\nas given by Eq.~\\eqref{eq:C-alpha-t-via-C-2}.\nWhen $\\Lambda$ is large, however,\nthere is no difference between CUE and COE\ndue to the same scaling of $C(\\alpha; t)$,\nas for example in Eq.~\\eqref{eq:C-2-t} for $\\alpha=2$.\n\n\nThe result Eq.~(\\ref{eq:alphaEntropyTheory}) is in agreement with\nnumerical computations for both the COE, see Fig.~\\ref{fig:COE_Salpha},\nand the CUE, see Fig.~\\ref{fig:CUE_Salpha}.\nFor these numerical calculations, 20 realizations of the random matrix model Eq.~\\eqref{eq:GenericFloquetRMT} for $N_A=N_B=50$ have been used, leading to a total of $5 \\times 10^4$ initially unentangled eigenstates $\\ket{jk}$ used for averaging. This amount of averaging is\nparticularly relevant for small values of $\\Lambda$\nfor which the time evolution of the entanglement of the individual\nstates shows strong fluctuations from one state to another.\nThese are also the origin of the small fluctuations\nseen in both figures for $\\Lambda=10^{-6}$\nfor the von Neumann entropy $\\overline{S_1(t; \\Lambda)}$,\nwhich is the most sensitive of the considered entropies.\nMoreover, at small $\\Lambda$, finite $N$ effects\nbecome visible, in particular for the COE,\ndue to the small overall amount of entanglement.\nIncreasing the matrix dimension of the subsystems\nto $N=100$ improves the agreement with the theoretical prediction\n(not shown).\nFor $\\Lambda=10^{-4}$ and $\\Lambda=10^{-2}$ excellent\nagreement of the numerically computed entropies and\nthe theory is found.\nFor $\\Lambda=1$ again the von Neumann entropy shows\nsmall deviations from the theoretical prediction.\n\n\\subsection{Long-time saturation}\n\\label{sec:long-time-saturation}\n\n\\begin{figure}[b]\n\\centering\n \\includegraphics[width=8.4cm]{fig_saturation_logscale.pdf}\n\\caption{\\label{fig:saturation}\nSaturation values of the linear entropy, $\\overline{S_2(\\infty;\\Lambda)}$, as a function of $\\Lambda$\nfor the COE (blue squares) and CUE (red diamonds) in comparison\nwith the prediction Eq.~\\eqref{eq:sat-alphaEntropyTheory}\n(dashed and solid black lines representing COE and CUE, respectively).}\n\\end{figure}\n\n\nUsing the result Eq.~\\eqref{eq:alphaEntropyTheory}\none can also perform the long-time limit to\nobtain a prediction for the saturation\nvalues $\\overline{S_\\alpha(\\infty;\\Lambda) }$\ngoing beyond the perturbative result Eq.~\\eqref{eq:saturation-perturbatively}.\nThus one gets\n\\begin{equation}\n   \\overline{S_\\alpha(\\infty;\\Lambda) } = \\bigg[ 1 - \\exp(-\\frac{C(\\alpha;\\infty)}{(\\alpha-1)\\overline{ S_\\alpha^\\infty}}\\sqrt{\\Lambda})\\bigg] \\overline{S_\\alpha^\\infty}. \\label{eq:sat-alphaEntropyTheory}\n\\end{equation}\nFor large $\\Lambda$ the exponential becomes very small\nso that the saturation reaches $\\overline{S_\\alpha^\\infty}$.\nHowever, for small $\\Lambda$ a reduced saturation value is obtained.\nFigure~\\ref{fig:saturation} illustrates\nthis for the linear entropy for the COE and CUE\nwhere Eq.~\\eqref{eq:S2-sat-COE-CUE} is used for $C(\\alpha, \\infty)$.\nVery good agreement of the prediction with the numerical results\nis found.\nUp to $\\Lambda=10^{-1}$ the saturation value\nfollows Eq.~\\eqref{eq:S2-sat-COE-CUE}\nand then the behavior given by Eq.~\\eqref{eq:sat-alphaEntropyTheory}\nsets in.\nThe saturation values for the COE are below\nthose of the CUE but eventually both approach $\\overline{S_2^\\infty}$.\n\n\\section{Coupled kicked rotors}\n\\label{sec:coupled-kicked-rotors}\n\n\\begin{figure}\n\\includegraphics[width=8.4cm]{fig_KR_TP_1e-06_N_100.pdf}\n\n\\includegraphics[width=8.4cm]{fig_KR_TP_0p0001_N_100.pdf}\n\n\\includegraphics[width=8.4cm]{fig_KR_TP_0p01_N_100.pdf}\n\n\\includegraphics[width=8.4cm]{fig_KR_TP_1_N_100.pdf}\n\n\\caption{\\label{fig:KR_Salpha} Entropies $\\overline{ S_\\alpha }$ for the coupled kicked rotors with completely broken time-reversal invariance and $N=100$, for (a) $\\Lambda=10^{-6}$, (b) $\\Lambda=10^{-4}$, (c)\n  $\\Lambda=10^{-2}$, and (d) $\\Lambda=1$ for $\\alpha=1$ (green diamonds),\n  $\\alpha=2$ (magenta circles), $\\alpha=3$ (red triangles), and $\\alpha=4$\n  (blue squares). Black lines show the corresponding theory curves,\n  Eq.~(\\ref{eq:alphaEntropyTheory}).}\n\\end{figure}\n\nA bipartite system whose subsystems exhibit classical chaotic motion is considered here to compare against the universal entanglement dynamics results derived from random matrix theory. The knowledge of $\\Lambda$ and its relation to the system dependent details is crucial for the comparison.\nIn case of a system whose subsystems are kicked rotors quantized on the unit torus, it is possible to analytically find $\\Lambda$ as a function of system dependent details as shown in Refs.~\\cite{Srivastava16,Tomsovic18c}. The Floquet unitary operator of the system has the form given by Eq.~(\\ref{eq:GenericFloquet}) where the subsystem Floquet operator for one kicked rotor is\n\\begin{equation}\nU_A = \\exp[- i p_A^2\/(2\\hbar)] \\exp(-i V_A\/\\hbar),\n\\end{equation}\nwith kicking potential given by\n\\begin{equation}\nV_A = K_A \\cos(2\\pi q_A)\/4\\pi^2,\n\\end{equation}\nwhere $K_A$ is the kicking strength. Similarly for subsystem $B$. The entangling operator is\n\\begin{equation}\nU_{AB}(b) = \\exp(-i b V_{AB}\/\\hbar),\n\\end{equation}\nwhere the interaction potential is\n\\begin{equation}\nV_{AB} = \\frac{1}{4\\pi^2} \\cos[2\\pi(q_A+q_B)].\n\\end{equation}\nThe angle variables $q_j$ is restricted to the interval $[0,1)$, and similarly for the momenta $p_j$. This restriction leads to a 4-dimensional torus phase space for the corresponding classical system \\cite{Froeschle71,Lakshminarayan2001,Richter14}. The kicking strengths $(K_A,\\, K_B) = (10,14),\\,(18,22),\\, \\ldots$ with up to 20 realizations are chosen such that the classical dynamics is chaotic. The boundary conditions are chosen such that both time-reversal invariance and parity symmetry are broken. Thus the subsystem spectral fluctuations are approximately like those of the CUE. In addition we use $N=N_A = N_B = 100$ for the numerical computations. The transition parameter for the coupled kicked rotors is \\cite{Srivastava16,Tomsovic18c,HerKieFriBae2019:p}\n\\begin{equation}\n\\Lambda_{\\text{KR}} \\simeq \\frac{N^2}{4\\pi^2} \\left(1-J_0^2(Nb\/2\\pi) \\right) \\approx \\frac{N^4 b^2}{32 \\pi^4},\n\\end{equation}\nwhere $J_0(\\cdot)$ is the Bessel function of first kind\n\\cite[Eq.~10.2.2]{DLMF},\nand the approximation is true when $Nb \\ll 1$. In Fig.~\\ref{fig:KR_Salpha}, the entanglement dynamics for various $\\Lambda$-values of the coupled kicked rotors is shown against the theory given by Eq.~(\\ref{eq:alphaEntropyTheory}).\nOverall good agreement is found\nwith some small deviations for the von Neumann entropy\nwhich are similar to those found for the\nCUE case shown in Fig.~\\ref{fig:CUE_Salpha}.\n\n\\section{Summary and outlook}\n\\label{sec:summary-and-outlook}\n\nAn analytic theory is given in this paper for the rate of entanglement production for a quenched system as a function of the interaction strength between chaotic subsystems.  In particular, all the expressions are given in terms of the universal transition parameter $\\Lambda$.  It is shown that in the perturbative regime for an initial product of subsystem eigenstates (a so-called quench), the entanglement saturates at very small values proportional to $\\sqrt{\\Lambda}$.  Furthermore, in the same regime, once the appropriate time scale is properly identified and the entanglement entropies are scaled by their saturation value, there exists a single universal entropy production curve: for a given system size, the interaction strength determines $\\Lambda$, which determines the time scale and saturation values, and there is no other dependence in the entropy production beyond that.  The universal curve has an overshoot, which is slightly more pronounced for the time reversal non-invariant case, and then it settles down to a saturation value.  As $\\Lambda$ increases, the perturbation regime eventually breaks down, roughly for $\\Lambda \\gtrsim 10^{-2}$, as illustrated in Fig.~\\ref{fig:UniversalCurvePlot}.\n\nAs for the full eigenstates of the interacting system \\cite{Tomsovic18c}, it was also possible here to recursively embed the perturbation theory.  This enables a description of the full transition in entropy production behaviors as a function of subsystem interaction strength and size, the limiting behaviors being no entanglement entropy production for non-interacting systems, and for strongly interacting systems the production behavior seen for initially random product states.  The expressions are uniformly valid for all times and interaction strengths.  It also turns out that the initial entropy production rate is even independent of whether time reversal symmetry is preserved or not.\n\nThe present study also raises various interesting\nquestions to be addressed in the future:\nthe considered case of initial states given by direct products of subsystem eigenstates has the crucial property\nthat the automatic Schmidt decomposition holds,\nwhich allows for a perturbative treatment.\nIf one considers instead,\nfor example, sums of such eigenstates or\ndirect products of subsystem random vectors, then a much faster entanglement\ngeneration occurs, which requires a completely different\ntheoretical description.\nMoreover, although not shown, the fluctuations of the entropies seen from one initial\nstate to another depend dramatically on whether it is subsystem eigenvectors or random states which are being considered.  This should also be reflected in the statistics\nof Schmidt eigenvalues, which are expected to show\nheavy-tailed distributions as was found before in the case of eigenstates.\nAnother interesting ensemble for the case of a dynamical system\nlike the coupled kicked rotors are coherent states\nas initial states. There one will have an initial phase\nfor which the entanglement only grows very slowly\nup to the Ehrenfest time beyond which a fast increase\nof entanglement occurs.\nFinally, bipartite many-body systems, like an interacting spin-chain,\nshould share many of the features\nof the entanglement production demonstrated here,\nand at the same time also allow for even more possibilities\nof initial states.\n\n\n\\acknowledgments\n\nWe would like to thank Maximilian Kieler for useful discussions.\nOne of the authors (ST) gratefully acknowledges support for visits\nto the Max-Planck-Institut f\\\"ur Physik komplexer Systeme.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Figure Captions\\markboth\n        {FIGURECAPTIONS}{FIGURECAPTIONS}}\\list\n        {Figure \\arabic{enumi}:\\hfill}{\\settowidth\\labelwidth{Figure\n999:}\n        \\leftmargin\\labelwidth\n        \\advance\\leftmargin\\labelsep\\usecounter{enumi}}}\n\\let\\endfigcap\\endlist \\relax\n\\def\\tablecap{\\section*{Table Captions\\markboth\n        {TABLECAPTIONS}{TABLECAPTIONS}}\\list\n        {Table \\arabic{enumi}:\\hfill}{\\settowidth\\labelwidth{Table\n999:}\n        \\leftmargin\\labelwidth\n        \\advance\\leftmargin\\labelsep\\usecounter{enumi}}}\n\\let\\endtablecap\\endlist \\relax\n\\def\\reflist{\\section*{References\\markboth\n        {REFLIST}{REFLIST}}\\list\n        {[\\arabic{enumi}]\\hfill}{\\settowidth\\labelwidth{[999]}\n        \\leftmargin\\labelwidth\n        \\advance\\leftmargin\\labelsep\\usecounter{enumi}}}\n\\let\\endreflist\\endlist \\relax\n\\def\\list{}{\\rightmargin\\leftmargin}\\item[]{\\list{}{\\rightmargin\\leftmargin}\\item[]}\n\\let\\endquote=\\endlist\n\\makeatletter\n\\newcounter{pubctr}\n\\def\\@ifnextchar[{\\@publist}{\\@@publist}{\\@ifnextchar[{\\@publist}{\\@@publist}}\n\\def\\@publist[#1]{\\list\n        {[\\arabic{pubctr}]\\hfill}{\\settowidth\\labelwidth{[999]}\n        \\leftmargin\\labelwidth\n        \\advance\\leftmargin\\labelsep\n        \\@nmbrlisttrue\\def\\@listctr{pubctr}\n        \\setcounter{pubctr}{#1}\\addtocounter{pubctr}{-1}}}\n\\def\\@@publist{\\list\n        {[\\arabic{pubctr}]\\hfill}{\\settowidth\\labelwidth{[999]}\n        \\leftmargin\\labelwidth\n        \\advance\\leftmargin\\labelsep\n        \\@nmbrlisttrue\\def\\@listctr{pubctr}}}\n\\let\\endpublist\\endlist \\relax\n\\makeatother\n\\newskip\\humongous \\humongous=0pt plus 1000pt minus 1000pt\n\\def\\mathsurround=0pt{\\mathsurround=0pt}\n\\def\\eqalign#1{\\,\\vcenter{\\openup1\\jot \\mathsurround=0pt\n        \\ialign{\\strut \\hfil$\\displaystyle{##}$&$\n        \\displaystyle{{}##}$\\hfil\\crcr#1\\crcr}}\\,}\n\\newif\\ifdtup\n\\def\\panorama{\\global\\dtuptrue \\openup1\\jot \\mathsurround=0pt\n        \\everycr{\\noalign{\\ifdtup \\global\\dtupfalse\n        \\vskip-\\lineskiplimit \\vskip\\normallineskiplimit\n        \\else \\penalty\\interdisplaylinepenalty \\fi}}}\n\\def\\eqalignno#1{\\panorama \\tabskip=\\humongous\n        \\halign to\\displaywidth{\\hfil$\\displaystyle{##}$\n        \\tabskip=0pt&$\\displaystyle{{}##}$\\hfil\n        \\tabskip=\\humongous&\\llap{$##$}\\tabskip=0pt\n        \\crcr#1\\crcr}}\n\\relax\n\n\n\n\\def\\begin{equation}{\\begin{equation}}\n\\def\\end{equation}{\\end{equation}}\n\\def\\begin{eqnarray}{\\begin{eqnarray}}\n\\def\\end{eqnarray}{\\end{eqnarray}}\n\\def\\bar{\\partial}{\\bar{\\partial}}\n\\def\\bar{J}{\\bar{J}}\n\\def\\partial{\\partial}\n\\def\\noindent{\\noindent}\n\n\n\n\\def\\kappa{\\kappa}\n\\def\\rho{\\rho}\n\\def\\alpha{\\alpha}\n\\def\\Alpha{\\Alpha}\n\\def\\beta{\\beta}\n\\def\\Beta{\\Beta}\n\\def\\gamma{\\gamma}\n\\def\\Gamma{\\Gamma}\n\\def\\delta{\\delta}\n\\def\\Delta{\\Delta}\n\\def\\epsilon{\\epsilon}\n\\def\\Epsilon{\\Epsilon}\n\\def\\p{\\pi} \n\\def\\Pi{\\Pi}\n\\def\\chi{\\chi}\n\\def\\Chi{\\Chi}\n\\def\\theta{\\theta}\n\\def\\Theta{\\Theta}\n\\def\\mu{\\mu}\n\\def\\nu{\\nu}\n\\def\\omega{\\omega}\n\\def\\Omega{\\Omega}\n\\def\\lambda{\\lambda}\n\\def\\Lambda{\\Lambda}\n\\def\\s{\\sigma} \n\\def\\Sigma{\\Sigma}\n\\def\\varphi{\\varphi}\n\n\n\\def\\relax{\\rm I\\kern-.18em R}{\\relax{\\rm I\\kern-.18em R}}\n\\def\\relax{\\rm 1\\kern-.35em1}{\\relax{\\rm 1\\kern-.35em1}}\n\n\\renewcommand{\\thesection.\\arabic{equation}}}{\\thesection.\\arabic{equation}}\n\\csname @addtoreset\\endcsname{equation}{section}\n\n\n\n\n\n\n\\def${\\cal N}=4${${\\cal N}=4$}\n\\def\\alpha _S{\\alpha _S}\n\\def\\boldsymbol {\\boldsymbol }\n\n\\newcommand{\\Feyn}[1]{#1\\kern-0.45em\/}\n\n\\headheight 10 pt\n\n\n\n\n\\begin{document}\n\n\\title{\\Large \\bf Dual conformal invariance in the Regge limit}\n\\author{\\large C{\\'e}sar~G{\\'o}mez, Johan~Gunnesson, Agust{\\'i}n~Sabio~Vera \\\\\n{\\it Instituto de F{\\' i}sica Te{\\' o}rica UAM\/CSIC,}\\\\ \n{\\it Universidad Aut{\\' o}noma de Madrid, E-28049 Madrid, Spain}}\n\n\\maketitle\n\n\\vspace{-9cm}\n\\begin{flushright}\n{\\small IFT--UAM\/CSIC--09--38}\n\\end{flushright}\n\n\\vspace{7cm}\n\\begin{abstract}\n\\noindent\n\nA dual conformal symmetry, analogous to the dual conformal symmetry observed for the scattering amplitudes of ${\\cal N}=4$ Super Yang-Mills theory, is identified in the Regge limit of QCD. Combined with the original two-dimensional conformal symmetry of the theory, this dual symmetry can potentially explain the integrability of the BFKL Hamiltonian. We also give evidence that the symmetry survives when a subset of unitarity corrections are taken into account by studying briefly the non-planar $2$ to $m$ reggeon transition vertices.\n   \n\\end{abstract}\n\n\n\\section{Introduction}\n\nIn the last few years there has been a great deal of progress in the study of gluon scattering amplitudes in \nthe maximally supersymmetric gauge theory in four dimensions, ${\\cal N}=4$ Super Yang-Mills (SYM). One of the most surprising \ndevelopments has been the discovery of a hidden symmetry in the planar ($N_c \\rightarrow \\infty$) limit, coined \nas ``dual super-conformal symmetry''~\\cite{dualconformal, korchemskyconf,confward}, different from the original \nsuper-conformal symmetry of the Lagrangian. This symmetry was uncovered by introducing a new set of variables \n$x_i$, related to the external (all taken as incoming) gluon momenta $p_i$, $i=1\\ldots n$, through\n\\begin{equation}\nx_i - x_{i+1}=p_i \\ , \\label{eq:xi}\n\\end{equation}\nand acts on the $x_i$ just as a four-dimensional conformal symmetry acts on spatial coordinates. The presence \nof this dual symmetry can be understood through the AdS\/CFT correspondence~\\cite{AdSCFT} since it was \nshown~\\cite{fermionicT} that the problem of calculating a given scattering amplitude can be mapped, \nthrough a fermionic T-duality, to that of calculating a light-like Wilson loop with corners at coordinates \ngiven by the $x_i$. This fermionic T-duality maps the string $\\sigma$-model to itself, and the dual conformal \nsymmetry becomes the ordinary symmetry of the space in which the Wilson loop lives. \n\nLike the ordinary conformal symmetry, the dual symmetry is broken by infrared divergences, arising as cusp \ndivergences in the language of Wilson loops. However, the cusp divergences are known to exponentiate, which \nallows the use of the broken symmetry to impose powerful constraints on the amplitudes in the form of anomalous \nWard identities. These identities fix the 4 and 5 point amplitudes completely while the undetermined parts of \nhigher-point amplitudes can only depend on dual-conformal invariants. Also, taken together, the original and \ndual conformal symmetries generate an infinite-dimensional Yangian symmetry~\\cite{plefka}, ordinarily characteristic \nof exactly solvable models.\n\nInteresting properties of ${\\cal N}=4$ amplitudes appear also in their high energy (Regge) limit. In a nutshell, Regge theory \nestablishes the structure of scattering amplitudes when the momentum transfer is small compared to the total \ncenter-of-mass energy. It turns out that ${\\cal N}=4$ amplitudes exhibit Regge-like behaviour at all orders in the 't \nHooft coupling even outside of the Regge limit. In fact, the 4 and 5 point amplitudes are Regge \nexact~\\cite{korchemskyconf, AgustinBartelsLipatov}, \nmeaning that they can always be written in a factorized form characteristic of high energies, irrespective of \nthe values of the kinematical invariants\\footnote{Also, a proposal for the undetermined part of the 6 point amplitude, having the correct Regge behaviour and given in terms of conformal cross-ratios, is given in the latest version of \\cite{AgustinBartelsLipatov2}. }. Furthermore, in the Leading Logarithmic Approximation (LLA) of gluon \namplitudes the Regge limit is independent of the gauge theory, so ${\\cal N}=4$ can give insight into the high energy \nbehaviour of QCD.\n\nIn the Regge limit amplitudes are dominated by the $t$-channel exchange of reggeized gluons (a reggeized gluon is \na collective state of ordinary gluons projected on a colour octet). The bound state of two reggeized gluons when \nprojected on a colour singlet in the $t$-channel is known as the hard (or perturbative) pomeron. The interaction between \nreggeized gluons is governed by the Schr{\\\"o}dinger-like BFKL integral equation~\\cite{BFKL} where the invariant mass of \n$s$-channel gluons can be interpreted as the time variable and its kernel as an effective Hamiltonian living on the \ntwo-dimensional transverse space. This Hamiltonian is free from infrared singularities and carries a yet to be understood \nintegrability~\\cite{BFKLintegrable} \\footnote{Integrability also appears when the gluon composite states are projected onto the\nadjoint representation~\\cite{Lipatov:2009nt}.}.\n\nThe question that arises is if the integrable structures present in ${\\cal N}=4$ SYM can shed some light on the integrability \nfound in the Regge limit. In this region the dynamics of the theory is reduced to the transverse plane, where a two-dimensional \nconformal symmetry was found in the effective Hamiltonian~\\cite{Lipatov1986}. Given the emergence of the Yangian in the \nfour-dimensional case, and that one can heuristically interpret this $SL(2,C)$ as a reduction of the four-dimensional ordinary conformal \nsymmetry, it would then seem natural to look for a dual $SL(2,C)$ symmetry in the high-energy limit\\footnote{Hints in this direction have already appeared in the literature. A similar dual symmetry was exploited in \\cite{Lipatov:2009nt} in order to map supersymmetric multiparticle amplitudes in multiregge kinematics to an integrable open spin chain. In fact, the octet kernel, after subtraction of infrared divergences, can be written in a form manifestly invariant under this symmetry. Also, the BFKL Hamiltonian has holomorphic separability into two pieces which can be written such that they are invariant under the duality transformation $p_i \\to \\rho_i - \\rho_{i+1}\n\\to p_{i+1}$, similar to the change of variables \\eqref{eq:xi}, with the $\\rho$ being the gluon transverse coordinates in\ncomplex notation~\\cite{Lipatov:1998as}.}. It is this question that we address in this Letter, showing that BFKL indeed exhibits covariance under such a dual symmetry.\n\nWe will also study a set of corrections to BFKL, in the form of $2\\rightarrow m$ reggeized gluon transition \nvertices, which also turn out to be dual $SL(2,C)$-covariant. This result is important for high-energy QCD, since the inclusion of such vertices is necessary, at sufficiently high energies, to fulfill unitarity in all channels. \n\n\n\\section{The dual $SL(2,C)$ symmetry}\n\n\n\\begin{figure}[ht]\n\\psfrag{+}{$+$} \\psfrag{=}{$=$} \\psfrag{ka}{$k_A$} \\psfrag{kb}{$k_B$}\n\\psfrag{kp}{$k'$} \\psfrag{F}{$F$} \\psfrag{kamq}{$k_A - q$}\n\\psfrag{kbmq}{$k_B - q$} \\psfrag{x1}{$x_1$} \\psfrag{x2}{$x_2$}\n\\psfrag{x3}{$x_3$} \\psfrag{x4}{$x_4$}\n\\begin{center}\n\\includegraphics{BFKL2.eps}\n\\caption{\\small The BFKL integral equation for the four-point reggeized gluon\nGreen function.} \\label{fig:BFKL}\n\\end{center}\n\\end{figure}\n\nThe scattering amplitude for the 2 to 2 reggeized gluons process in the Regge limit has an iterative structure \ndominated by the exchange in the $t$-channel of a colour singlet. This implies that in the LLA the corresponding \n4 point gluon Green function can be written as the solution to an integral \nequation, the BFKL equation \\footnote{For an introductory treatment of the BFKL equation, see \\cite{rossforshaw}}, shown in Fig.~\\ref{fig:BFKL}. Written in terms of $\\omega$, the Mellin conjugate variable of the \ncenter-of-mass energy (which can be translated into \nthe rapidity, $Y$, of the emitted particles in the $s$-channel)), and the incoming two dimensional momenta it reads   \n\\begin{equation}\n\\omega F(\\omega , \\, \\boldsymbol {k}_A,\\, \\boldsymbol {k}_B,\\, \\boldsymbol {q}) = \\delta^{(2)}(\\boldsymbol {k}_A - \\boldsymbol {k}_B) + \\int d^2\\boldsymbol {k}' K(\\boldsymbol {k}_A,\\,\n \\boldsymbol {k}_A-\\boldsymbol {q};\\, \\boldsymbol {k}',\\, \\boldsymbol {k}'-\\boldsymbol {q}) F(\\omega , \\, \\boldsymbol {k}',\\, \\boldsymbol {k}_B ,\\, \\boldsymbol {q}) \\, \\label{eq:BFKLconK} \n\\end{equation}\nwhere the kernel $K(\\boldsymbol {k}_A,\\, \\boldsymbol {k}_A-\\boldsymbol {q};\\, \\boldsymbol {k}',\\, \\boldsymbol {k}'-\\boldsymbol {q})$ is given by\n\\begin{align}\n\\frac{K_R (\\boldsymbol {k}_A,\\, \\boldsymbol {k}_A-\\boldsymbol {q}; -\\boldsymbol {k}'+\\boldsymbol {q}, \\, -\\boldsymbol {k}')}{8\\pi^3\\boldsymbol {k}_A^2(\\boldsymbol {k}' - \\boldsymbol {q})^2} \n+ \\left[  \\omega (\\boldsymbol {k}_A^2) + \\omega ((\\boldsymbol {k}_A-\\boldsymbol {q})^2) \\right]\\delta ^{(2)}(\\boldsymbol {k}_A-\\boldsymbol {k}') \\ . \\label{eq:Kernel}\n\\end{align}\nThe ``real emission'' part\\footnote{Due to the optical theorem when the forward ${\\bf q}=0$ limit is taken this piece in \nthe kernel corresponds to the contribution to multiparticle production from on-shell gluons in the $s$-channel.} \nhas the following structure\n\\begin{equation}\nK_R (\\boldsymbol {p}_1,\\, \\boldsymbol {p}_2;\\, \\boldsymbol {p}_3,\\, \\boldsymbol {p}_4) = -N_cg^2\\left[ \\left( \\boldsymbol {p_3}+\\boldsymbol {p_4} \\right)^2 - \\frac{\\boldsymbol {p}_2^2\\boldsymbol {p}_4^2}{(\\boldsymbol {p}_2+\\boldsymbol {p}_3)^2}- \\frac{\\boldsymbol {p}_1^2\\boldsymbol {p}_3^2}{(\\boldsymbol {p}_1+\\boldsymbol {p}_4)^2}  \\right]. \\label{eq:KernelR}\n\\end{equation}\nThis notation, with $p_1,\\, \\ldots ,\\, p_4$ being the cyclically ordered reggeized gluon momenta taken as incoming, \nwill be convenient for the generalization of this vertex to the $2 \\rightarrow m$ reggeized gluon transition case \nas we will see below. \n\nThe gluon Regge trajectory reads\n\\begin{equation}\n\\omega (\\boldsymbol {q}^2) = -\\frac{g^2N_c}{16 \\pi ^3 }\\int d^2\\boldsymbol {k}' \\frac{\\boldsymbol {q}^2}{\\boldsymbol {k}'^2(\\boldsymbol {k}'-\\boldsymbol {q})^2} \\ . \n\\label{eq:trajectoryreggegluon}\n\\end{equation}\nThe trajectory is IR divergent, requiring it to be regularized in general.\n\nWe will now show that the BFKL equation in Eq.~\\eqref{eq:BFKLconK} exhibits formally a dual $SL(2,C)$ symmetry, which, \nin contrast with the original $SL(2,C)$ symmetry of BFKL, uncovered by Fourier transforming into a coordinate \nrepresentation, is realized in the transverse momentum space. This new symmetry is closely analogous to the dual \nconformal symmetry observed in $\\mathcal{N}=4$ SYM for gluon scattering amplitudes, and we will see that it turns out \nto be broken by infrared effects just as in the four dimensional gauge theory.\n\nLet us now rewrite Eq.~\\eqref{eq:BFKLconK} in terms of dual variables. Taken as incoming, the external momenta are \n$\\boldsymbol {k}_A$, $-\\boldsymbol {k}_A+ \\boldsymbol {q}$, $\\boldsymbol {k}_B - \\boldsymbol {q}$ and $-\\boldsymbol {k}_B$ so, introducing the notation \n$x_{i,j}\\equiv x_i - x_j$, we define the new set of variables as\n\\begin{equation}\n\\boldsymbol {p}_1 = x_{1,2} = \\boldsymbol {k}_A, \\, \\boldsymbol {p}_2 = x_{2,3} = \\boldsymbol {q} -\\boldsymbol {k}_A, \\,\n\\boldsymbol {p}_3 = x_{3,4} = \\boldsymbol {k}_B - \\boldsymbol {q}, \\, \\boldsymbol {p}_4=x_{4,1} = -\\boldsymbol {k}_B. \n\\end{equation}\nEquivalently, we could have written $\\boldsymbol {k}_A = x_{1,2}, \\, \\boldsymbol {k}_B = x_{1,4}, \\, \n\\boldsymbol {q} = x_{1,3}$ with $x_1$ then being a simple shift of the origin for the external momenta.\n\nIn these new variables the gluon Regge trajectory is\n\\begin{equation}\n\\omega (\\boldsymbol {k}_A^2) = \\omega (x^2_{1,2})=-\\frac{g^2N_c}{16 \\pi ^3 }\\int d^2 x_I \\frac{x^2_{1,2}}{x^2_{I,1}x^2_{I,2}} \\ , \\label{eq:trajectoryreggegluonk1x}\n\\end{equation}\nwhere we have introduced $x_I$ through $\\boldsymbol {k}' = x_{I,2}$. Ignoring for the moment that this expression is divergent and thus ill-defined, we see that is has a formal two-dimensional conformal symmetry. It is formally invariant under translations, rotations and scalings of the $x_i$, and also under the conformal inversions $x_i \\rightarrow \\frac{x_i}{x_i^2}$, since they would imply\n\\begin{equation}\nd^2 x_I \\rightarrow \\frac{d^2 x_I}{x_I^4} \\ , \\;\\; x_{i,j}^2 \\rightarrow \\frac{x_{i,j}^2}{x_i^2x_j^2} \\ . \\label{eq:transfxij} \n\\end{equation}\nIn the same way $\\omega ((\\boldsymbol {k}_B-\\boldsymbol {q})^2) = \\omega (x^2_{2,3})$ is also formally conformally invariant. \nNow, given that the trajectory is infrared divergent one would expect this symmetry to be broken by the introduction \nof a regulator, an issue which is discussed in the next section.\n\nRewriting the full kernel~\\eqref{eq:Kernel} in terms of the $x_i$, with $\\boldsymbol {k}'=x_{1,I}$, we get\n\\begin{equation}\nK(x_{1,2},\\, x_{3,2};\\, x_{1,I},\\, x_{3,I}) = \n\\frac{K_R (x_{1,2},\\, x_{2,3}; x_{3,I}, \\, x_{I,1})}{8\\pi^3x_{1,2}^2x_{I,3}^2} \n+ \\left[  \\omega (x_{1,2}^2) + \\omega (x_{2,3}^2) \\right]\\delta ^{(2)}(x_{2,I}) \\ , \\label{eq:Kernelx}\n\\end{equation}\nwhere\n\\begin{equation}\nK_R (x_{1,2},\\, x_{2,3}; x_{3,I}, \\, x_{I,1}) = -N_cg^2\\left[  x_{1,3}^2 - \\frac{x_{2,3}^2 x_{I,1}^2}{x_{2,I}^2}- \\frac{x_{1,2}^2x_{I,3}^2}{x_{2,I}^2}  \\right]  \\ . \\label{eq:KernelRx}\n\\end{equation}\nUsing that $\\delta ^{(2)}(x_{2,I}) \\rightarrow x_2^2x_I^2 \\delta ^{(2)}(x_{2,I})$ under conformal inversions one then finds immediately that the kernel transforms covariantly \\footnote{This is again similar to the (conjectured) dual conformal symmetry of scattering amplitudes in ${\\cal N}=4$ SYM, under which the amplitudes transform covariantly, as opposed to the ordinary conformal symmetry which leave them invariant.}\n\\begin{equation}\nK(x_{1,2},\\, x_{3,2};\\, x_{1,I},\\, x_{3,I}) \\rightarrow  x^2_2x^2_I K(x_{1,2},\\, x_{3,2};\\, x_{1,I},\\, x_{3,I}) \\ .\n\\end{equation}\nTogether with translations, rotations and dilatations this forms a dual $SL(2,C)$ symmetry, different from the one \npreviously known. More precisely, dilatations and rotations coincide with the original $SL(2,C)$-symmetry, while \ntranslations and inversions will be different. \n\nApplied to the BFKL equation~\\eqref{eq:BFKLconK} and using that the integration measure transforms according \nto~\\eqref{eq:transfxij} one finds that a factor of $\\frac{x_2^2}{x_I^2}$ is produced inside the integral. \nConsequently, if the Green function $F(\\omega , \\, x_{1,2},\\, x_{1,4},\\, x_{1,3})$ were to produce a factor of \n$x_2^2$ upon inversion, then its convolution with the kernel $K \\otimes F$, would transform in the same way as $F$ \nitself. Now, at lowest order, $F$ is simply given by the delta function, which indeed transforms in this way, \n$\\delta^{(2)} (\\boldsymbol {k}_A - \\boldsymbol {k}_B)=\\delta ^{(2)} (x_{2,4}) \\rightarrow x_2^2 x_4^2 \\delta ^{(2)} (x_{2,4})$. \nSince $F$ can be constructed through iterated convolution with the kernel, it follows that the Green function \nshould have the same conformal properties as the delta function. \n\nWe can obtain a formal expression for the Green function having the correct conformal properties by iteration. \nIntroducing the short-hand notation\n\\begin{equation}\n\\omega_0 \\left(\\bf{k}_A,\\, \\bf{q} \\right) \\equiv \\omega (\\boldsymbol {k}_A^2) + \\omega ((\\boldsymbol {k}_A-\\boldsymbol {q})^2),  \n\\xi \\left(\\bf{k},\\, \\bf{k}_A,\\, \\bf{q}\\right) \\equiv \n\\frac{K_R (\\boldsymbol {k}_A,\\, \\boldsymbol {k}_A-\\boldsymbol {q}; -\\boldsymbol {k}+\\boldsymbol {q}, \\, -\\boldsymbol {k})}{8\\pi^3 \\boldsymbol {k}_A^2(\\boldsymbol {k} - \\boldsymbol {q})^2} \\ \n\\end{equation}\none finds (with $\\bf{k}_0 \\equiv \\bf{k}_A$):\n\\begin{eqnarray}\nF \\left(\\omega,\\bf{k}_A,\\bf{k}_B,\\bf{q}\\right) =\n{\\delta^{(2)} \\left(\\bf{k}_A-\\bf{k}_B\\right) + \\sum_{n=1}^\\infty \\prod_{i=1}^n \n\\int d^2 \\bf{k}_i \\, {\\xi \\left(\\bf{k}_i,\\bf{k}_{i-1},\\bf{q}\\right) \\over \n\\omega - \\omega_0 \\left(\\bf{k}_i,\\bf{q}\\right)} \\delta^{(2)}\n\\left(\\bf{k}_n-\\bf{k}_B\\right) \\over \\omega - \\omega_0 \\left(\\bf{k}_A,\\bf{q}\\right)}.  \n\\end{eqnarray}\n\nRather than $\\omega$ it is more natural to use the rapidity difference, $Y$, between the external \nparticles as the evolution variable. To this end we perform the inverse Mellin transform\n\\begin{eqnarray}\n{\\cal F} \\left(\\bf{k}_A,\\bf{k}_B,\\bf{q},Y\\right) &=&\n\\int_{a-i \\infty}^{a+i \\infty} {d \\omega \\over 2 \\pi i} e^{\\omega Y}\nF \\left(\\omega,\\bf{k}_A,\\bf{k}_B,\\bf{q}\\right).\n\\end{eqnarray}\nThe formula $\\int_{a-i \\infty}^{a+i \\infty} {d \\omega \\over 2 \\pi i} e^{\\omega Y}\n\\prod_{i=0}^n \\frac{1}{\\omega-\\omega_i} = e^{\\omega_0 Y} \\prod_{i=1}^n\n\\int_0^{y_{i-1}} d y_i e^{\\omega_{i,i-1} y_i}$ for $n > 0$, with $\\omega_{i,j} \\equiv \n\\omega_i - \\omega_j, y_0 \\equiv Y$, is useful to obtain the final expression, written in dual $x$-variables:\n\\begin{equation}\n{\\cal F} \\left(x_{12}, x_{14},x_{13},Y\\right) = e^{\\omega _{2,1} Y} \\Bigg\\{\\delta^{(2)} \\left(x_{24}\\right)\n+ \\sum_{n=1}^\\infty \\prod_{i=1}^n \\int_0^{y_{i-1}} \\hspace{-0.3cm} d y_i\n\\int d^2 x_i \\, \\xi_{i,i-1} e^{\\omega_{i,i-1} y_i} \\delta^{(2)}\n\\left(x_{4,n} \\right) \\Bigg\\}, \n\\label{eq:Frapidityx}\n\\end{equation}\nwhere\n\\begin{eqnarray}\n\\omega_{i,i-1} &=& \\omega _0 \\left(x_{1,i},x_{13}\\right)-\\omega_0 \\left(x_{1,i-1},x_{13}\\right) \\ ,\\\\\n\\xi_{i,i-1} &=& {{\\bar \\alpha}_s \\over 2 \\pi} \\Bigg\\{{ x^2_{i,3} x^2_{i-1,1} +\nx^2_{i-1,3} x^2_{i,1} - x^2_{i-1,i} x^2_{13}\\over x^2_{i,3} x^2_{i-1,i} x^2_{i-1,1}}\\Bigg\\} \\ .\n\\end{eqnarray}\nThis representation preserves the transformation properties of the original equation. In the forward case, where the \nmomentum transfer is zero, the same structure remains with  \n\\begin{equation}\n\\omega_{i,i-1} = 2\\left(\\omega \\left(x_{1,i}\\right)-\\omega \\left(x_{1,i-1}\\right)\\right) \\ ,\n\\xi_{i,i-1} = {{\\bar \\alpha}_s \\over \\pi} { 1 \\over x^2_{i-1,i}} \\ .\n\\end{equation}\nIn this case the solution also has a formal dual $SL(2,C)$ covariance. This should be contrasted with \nthe original $SL(2,C)$-invariance of the BFKL kernel, which does not appear in the forward case.\n\nBefore ending this section, it is noteworthy to mention that this formal dual $SL(2,C)$ covariance is \npresent in the same form for all color projections in the $t$-channel since they only differ by a different factor in front of $K_R$: with $N_c=3$, $c_1 = 1,\\, c_{8_a} = c_{8_s} = 1\/2,\\, c_{10} = c_{\\overline{10}} = 0$, etc.\n\n\\section{The effect of IR divergences}\n \nIn ${\\cal N}=4$ SYM infrared divergences break the dual conformal symmetry. For BFKL, such divergences cancel, opening the \npossibility that the dual $SL(2,C)$-symmetry remains exact. However, this turns out not to be the case. Perhaps the \nsimplest way to see this is by studying the forward case. If $F$ has the transformation properties of the delta \nfunction it can be written as\n\\begin{equation}\nF = F_1 \\delta ^{(2)}(\\boldsymbol {k}_A - \\boldsymbol {k}_B) + \\frac{1}{(\\boldsymbol {k}_A - \\boldsymbol {k}_B)^2}F_2 \\ ,\n\\end{equation}\nwhere $F_1$ and $F_2$ are dual conformally invariant, since $(\\boldsymbol {k}_A - \\boldsymbol {k}_B)^{-2}$ is the only other function \nthat transforms correctly. When $\\boldsymbol {q}=0$, $x_1=x_3$ and no non-trivial conformal invariant can be formed from the \nthree remaining $x_i$. $F_2$ can thus only be a function of $\\omega$ (or equivalently the rapidity $Y$), and the \ncoupling. But when forming physical quantities one integrates over $\\boldsymbol {k}_A$ and $\\boldsymbol {k}_B$ and the divergences \nat $\\boldsymbol {k}_A=\\boldsymbol {k}_B$ must cancel between $F_1$ and $F_2$. The factor $(\\boldsymbol {k}_A - \\boldsymbol {k}_B)^{-2}$ is singular \nenough to cancel one factor of the trajectory, but $F_1$ is obtained by repeated application of the trajectory \npart of the kernel so, starting from the second iteration, products of two or more trajectories will appear and \nthe divergences will fail to cancel.\n\nOne can also observe the breakdown of the dual $SL(2,C)$ symmetry directly by regularizing the integrals and \ncancelling the divergences explicitly when performing the iteration. One then finds that the first iteration \nrespects the symmetry, while the second iteration produces a contribution to $F_2$ (when $\\boldsymbol {q}=0$) proportional \nto an anomalous factor of the form $\\ln \\left( \\frac{(\\boldsymbol {k}_A-\\boldsymbol {k}_B)^4}{\\boldsymbol {k}_A^2 \\boldsymbol {k}_B^2}\\right)$, which \nbreaks the symmetry under inversions. The origin of this factor is the regularization of infrared divergences. \nFor example, using dimensional regularization with $D=4-2\\epsilon$\n\\begin{equation}\n\\omega (x^2_{1,2})=-\\frac{g^2N_c}{16 \\pi ^3 }(4 \\pi \\mu)^{2\\epsilon}\\int d^{2-2\\epsilon} x_I \\frac{x^2_{1,2}}{x^2_{I,1}x^2_{I,2}}\\approx -\\frac{g^2N_c}{8 \\pi ^2 }(4\\pi e^{-\\gamma})^\\epsilon  \\left( \\ln \\frac{x^2_{1,2}}{\\mu ^2}- \\frac{1}{\\epsilon} \\right) \\ .\n\\end{equation}\nThe divergences will cancel between the trajectories and the real emission part of the kernel, but factors such \nas $\\ln x^2_{1,2}$ will add up giving a non-vanishing anomalous term. So, even though BFKL is infra-red finite, \na remnant of the divergences remains in the form of the breaking of the dual $SL(2,C)$ symmetry in the form given here.\n\nFurther insight can be gained by studying a standard representation of the Green function in the forward case, \nobtained by diagonalizing the BFKL kernel. It is\n\\begin{equation}\n{\\cal F} \\left(x_{1,2},x_{1,4},Y\\right) = \n\\sum_{n=-\\infty}^\\infty \\int \\frac{d \\gamma}{2 \\pi i}\n\\left({x^2_{1,2} \\over x^2_{1,4}}\\right)^{\\gamma-\\frac{1}{2}}\n\\frac{e^{{\\bar{\\alpha}_s}\\chi_n\\left(\\gamma\\right) Y + i n \\theta_{2,4}}}{\\pi\\sqrt{x_{1,2}^2 x_{1,4}^2}}\\ ,\n\\end{equation}\nwith $\\chi_n  \\left(\\gamma\\right) =  2 \\Psi\\left(1\\right)- \\Psi\\left(\\gamma+\\frac{|n|}{2}\\right)\n- \\Psi\\left(1-\\gamma+\\frac{|n|}{2}\\right)$ and \n$\\cos{\\theta_{2,4}} = { x_{1,2} \\cdot x_{1,4} \\over \\sqrt{x^2_{1,2} x^2_{1,4}}}$.\nIn this representation any dependence on an IR cutoff has canceled explicitly, and one can check that the covariance under conformal inversions is lost.\n\nAn important issue is whether the dual $SL(2,C)$ symmetry is broken beyond repair or whether it can be deformed to take into consideration the anomalous terms. In a best case scenario the symmetry would obey to all orders a simple relation such as the anomalous Ward identity satisfied by the dual conformal symmetry of $\\mathcal{N}=4$ scattering amplitudes \\cite{confward}. This issue is studied in \\cite{johan}, with the result that the dual $SL(2,C)$ does not obey such a simple all-order relation, but can still be deformed so that it becomes exact, at least up to the order studied. The representation then becomes coupling-dependent, but encouragingly, it seems to do so in such a way that the algebra generated by the original and dual $SL(2,C)$ symmetries remains coupling-independent.\n\n\\section{$2\\rightarrow m$ reggeized gluon vertex}\n\n\nThe BFKL amplitude will violate bounds imposed by unitarity at sufficiently high energies. \nIn order to restore unitarity, one of the new elements that must be introduced is a vertex in which the number \nof reggeized gluons in the $t$-channel is not conserved. As shown in Fig.~\\ref{fig:vertex} we choose to write this \n$2\\rightarrow m$ vertex (see, for example, Eq.~(3.57) of \\cite{BartelsEwerz}) using a convenient assignment \nof the momentum indices\n\\begin{align}\n&K_{2\\rightarrow m}^{\\{ b \\} \\rightarrow \\{a \\}}(\\boldsymbol {p}_2,\\, \\boldsymbol {p}_3;\\, \\boldsymbol {p}_4,\\, \\ldots ,\\,  \\boldsymbol {p}_{m+2},\\, \\boldsymbol {p}_1 ) = f_{a_1b_1c_1}f_{c_1a_2c_2}\\cdots f_{c_{m-1}a_mb_2} g^m \\nonumber \\\\\n& \\times \n\\left[ (\\boldsymbol {p}_4 + \\cdots + \\boldsymbol {p}_1)^2 - \\frac{\\boldsymbol {p}_3^2(\\boldsymbol {p}_5 + \\cdots + \\boldsymbol {p}_1)^2}{(\\boldsymbol {p}_3 + \\boldsymbol {p}_4)^2}\n -  \\frac{\\boldsymbol {p}_2^2(\\boldsymbol {p}_4 + \\cdots + \\boldsymbol {p}_{m+2})^2}{(\\boldsymbol {p}_1 + \\boldsymbol {p}_2)^2}  + \\frac{\\boldsymbol {p}_1^2\\boldsymbol {p}_3^2(\\boldsymbol {p}_5 + \\cdots + \\boldsymbol {p}_{m+2})^2}{(\\boldsymbol {p}_1 + \\boldsymbol {p}_2)^2(\\boldsymbol {p}_3 + \\boldsymbol {p}_4)^2} \\right] \\label{eq:K2m} \\ ,\n\\end{align}\nwhere the $a_1,\\, b_1$ etc. are the color indices of the reggeized gluons and $f_{ijk}$ the structure constants of \n$SU(N_c)$. \n\n\\begin{figure}[ht]\n\\psfrag{x1}{$x_1$} \\psfrag{x2}{$x_2$} \\psfrag{x3}{$x_3$} \\psfrag{x4}{$x_4$} \\psfrag{x5}{$x_5$} \\psfrag{p1}{$\\boldsymbol {p}_1$} \\psfrag{p2}{$\\boldsymbol {p}_2$} \\psfrag{p3}{$\\boldsymbol {p}_3$} \\psfrag{p4}{$\\boldsymbol {p}_4$} \\psfrag{p5}{$\\boldsymbol {p}_5$} \\psfrag{pmp2}{$\\boldsymbol {p}_{m+2}$} \\psfrag{a1}{$a_1$} \\psfrag{a2}{$a_2$}  \\psfrag{amm1}{$a_{m-1}$} \\psfrag{am}{$a_m$} \\psfrag{b1}{$b_1$} \\psfrag{b2}{$b_2$}\n\\begin{center}\n\\includegraphics{reggluonvertex.eps}\n\\caption{\\small The $2\\rightarrow m$ reggeized gluon vertex. All momenta are taken as ingoing.} \\label{fig:vertex}\n\\end{center}\n\\end{figure}\n\nWritten in terms of $x$ variables this becomes\n\\begin{align}\n&K_{2\\rightarrow m}^{\\{ b \\} \\rightarrow \\{a \\}}(x_{23},\\, x_{34};\\, x_{45},\\, \\ldots ,\\,  x_{m+2,1},\\, x_{12}) = \\nonumber \\\\ &f_{a_1b_1c_1}f_{c_1a_2c_2}\\cdots f_{c_{m-1}a_mb_2} g^m \\left[ x_{24}^2 - \\frac{x_{34}^2x_{25}^2}{x_{35}^2} -\\frac{x_{23}^2x_{14}^2}{x_{13}^2} +\\frac{x_{23}^2x_{34}^2x_{15}^2}{x_{13}^2 x_{35}^2} \\right] \\ ,\n\\end{align}\nand is manifestly conformally covariant. The assignment of the momenta in \\eqref{eq:K2m} was chosen so that \nthe vertex takes a form independent of $m$ when written in terms of the $x_i$. Note that the last term vanishes \nwhen $m=2$ since then $x_1 = x_5$, and one recovers the corresponding term in the BFKL kernel. \n\nIn \\cite{reggevertex2a4} it was shown that the $2\\rightarrow 4$ reggeized gluon vertex exhibited the same coordinate representation $SL(2,C)$-invariance as the BFKL equation. This was taken to indicate that a unitary, two-dimensional CFT describing scattering amplitudes in the Regge limit should exhibit this $SL(2,C)$-invariance. Our results would seem to indicate that such a theory should also be covariant under the dual $SL(2,C)$.\n\n\\section{Conclusions}\n\n\nWe have shown that not only does the LLA BFKL kernel, and its extension in the form of the $2 \\rightarrow m$ reggeized gluon vertex, exhibit the ordinary $SL(2,C)$-symmetry, found by Lipatov but also a dual $SL(2,C)$, analogous to the dual conformal symmetry of $\\mathcal{N}=4$. It is tempting to interpret these symmetries as reductions to the transverse plane of the conformal and dual conformal symmetries of the supersymmetric theory, although it is not clear exactly how such a reduction should be carried out. Purely transverse versions of the conformal algebras are not symmetries of the 4-dimensional gauge theory amplitudes, but seem to emerge in the Regge limit. \n\nAlso, the dual invariance of the reggeized gluon vertex suggests that a unitary two-dimensional CFT describing high-energy gauge theory should have both $SL(2,C)$ groups. In future work, having identified the dual $SL(2,C)$ one can try to understand the origin of the integrability of the Regge limit in terms of the integrability of $N=4$ SYM.\n\n\n\n\\vspace{5mm}\n\\centerline{\\bf Acknowledgments}\n\nWe would like to thank Lev Lipatov for useful discussions. The work of C. G. has been partially\nsupported by the Spanish DGI contract FPA2003-02877 and the CAM grant HEPHACOS\nP-ESP-00346. The work of J. G. is supported by a Spanish FPU grant.\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIn this paper, we are concerned with the $L^p$ mapping properties of \nthe pseudodifferential operators in the form \n\\begin{equation}\n\\label{eq:1}\nT_\\sigma f(x)=\\int\\limits_{{\\mathbf R}^n} \\sigma(x,\\xi) e^{2\\pi i \\xi x} \\hat{f}(\\xi) d\\xi.\n\\end{equation} \nThe operators $T_\\sigma$  have been subject of continuous interest \nsince the sixties. We should mention that their usefullness in \n the study of partial differential equations have been realized  much \nearlier, but it seems that their systematic study began with the \nfundamental works of Kohn and Nirenberg, \\cite{KN} and  H\\\"ormander, \n\\cite{Hor}. \n\nTo describe the \nresults obtained in these early papers, \ndefine the H\\\"ormander's class $S^m$, which consists \nof all functions $\\sigma(x,\\xi)$, so that \n\\begin{equation}\n\\label{eq:2}\n|D^\\beta_x D^\\alpha_\\xi \\sigma(x, \\xi)|\\leq C_{\\alpha, \\beta} (1+|\\xi|)^{m-|\\alpha|}.\n\\end{equation}\nfor all multiindices $\\alpha, \\beta$. \nA classical theorem in \\cite{Hor} then states that \n$Op(\\sigma):H^{s+m,p}\\to H^{s, p}$  for all $s\\geq 0$ and \n$1<p<\\infty$. In particular, $Op(\\sigma):L^p\\to L^p$,  $1<p<\\infty$, whenever the symbol $\\sigma\\in S^{m}$. \nSubsequent improvements of these methods established the \nboundedness of $Op(\\sigma)$ (basically under the assumption $\\sigma\\in S^m$ for \nappropriate $m$) \n to various related function spaces, like \nBesov, Triebel-Lizorkin spaces to name a few, but we will not review \nthose here, since they fall  \noutside of the scope of this paper. \n\nIt is worth mentioning however, that the simple to verify condition \n\\eqref{eq:2} is the one arising in many  applications. \nThe $L^2$ boundedness plays special role in the theory and that is why\nwe discuss it separately. \n\nThe class of symbols $S^m_{\\rho, \\delta}$, \ndefined via \n\\begin{equation}\n\\label{eq:3}\n|D^\\beta_x D^\\alpha_\\xi \\sigma(x, \\xi)|\\leq C_{\\alpha, \\beta} \n(1+|\\xi|)^{m-\\rho|\\alpha|+\\delta|\\beta|}.\n\\end{equation}\nrepresents a larger set of symbols than $S^m=S^m_{1, 0}$, which has \nsubsequently found \n applications in  local \nsolvability for linear PDE's, \\cite{BF}.\n\nHere, we have to mention the celbrated  result of \nCalder\\'on-Vaillancourt, \n\\cite{CV1}, \\cite{CV2} which  states  that $L^2$ boundedness for $T_\\sigma$ \nholds, whenever  \n$\\sigma\\in S^{0}_{\\rho, \\rho}$, $0\\leq \\rho<1$, \n whereas $S^{0}_{1,1}$ is a ``forbidden'' class, \nin the sense that there are symbols in that class, \nwhich give rise to unbounded on \n$L^2$ operators. We should mention here the work of Cordes \\cite{Cor}, \n who improved the \nresult for $S^0_{0, 0}$ by requiring that \\eqref{eq:3} holds only for \n$|\\alpha|, |\\beta|\\leq [n\/2]+1$. \n\nRegarding less regular in $x$ symbols,  \nfor any modulus of continuity \n$\\omega:R_+\\to R_+$ (that is,  an  increasing and  continuous  function), \ndefine the space $C^\\omega$ of all uniformly continuous and bounded functions \n$u:{\\mathbf R}^n\\to \\mathcal C$,  satisfying \n$$\n|u(x+y)-u(x)|\\leq \\omega(|y|). \n$$\nThe following class of symbols was introduced and \n studied by  Coifman-Meyer, \\cite{CM}. More precisely, \nlet $\\sigma(x,\\xi)\\in C^\\omega S^{0}_{1,0}$, which means  it  satisfies \n$$\\sup_{x, y, \\xi} <|\\xi|>^{|\\alpha|} \n|D_\\xi^\\alpha [\\sigma(x+y, \\xi)-\\sigma(x, \\xi)]| \\leq C_\\alpha \\omega(|y|)\n$$\nand assume that $\\sum_{j>0} \\omega(2^{-j})^2<\\infty$. Then for \nall $1<p<\\infty$, $\nOp(\\sigma):L^p\\to L^p. \n$\nThe condition $\\sum_j \\omega(2^{-j})^2<\\infty$   is clearly very mild continuity assumption for the function $x\\to \\sigma(x, \\xi)$. \nIn particular, one sees that $\\cup_{\\gamma>0} \nC^\\gamma S^0\\subset C^\\omega S^{0}_{1,0}$. \nRelated results can be found in the work of M. Taylor, \\cite{Taylor} (see Proposition 2.4, p. 23) and J. Marschall, \\cite{Marschall} where the \nspaces $C^\\omega$ are replaced by $H^{\\varepsilon,p}$ spaces with $p$ as large as one wish and $0<\\varepsilon=\\varepsilon(p)<<1$  (see also \\cite{Taylor}, p. 61)\n\nOne of the purposes of this work is  to get away from the \ncontinuity requirements  on $x\\to \\sigma(x, \\xi)$. Even more importantly, \nwe would like  to replace the pointwise conditions on the derivatives of $\\xi$ \nby averaged ones. This particular point has not been thoroughly \nexplored appropriately  \n in the literature in the author's opinion, \nsee Theorem \\ref{theo:1} below.  \n\nOn the other hand, a particular motivation for such considerations is \nprovided by the recent  papers of Rodnianski-Tao \\cite{RT} and \nthe author  \\cite{Stefanov}, where concrete parametrices \n(i.e. pseudodifferential operators,  representing \napproximate solutions to certain PDE's) \nwere constructed for the \nsolutions of certain first order perturbation of the wave \nand Schr\\\"odinger equations.  \nA very quick inspection of these examples shows that\\footnote{Most \nreaders are \n likely to have their own  fairly long list \nwith  favorite examples, for which \n the H\\\"ormander condition fails.} \n {\\it \nthey do not obey\npointwise conditions on the derivatives on the \nsymbols}   and thus, these methods fail \nto imply  $L^2$ bounds for \nthese (and related problems). Moreover, one often times has to deal with the \nsituation, where the maps $\\xi\\to \\sigma(x, \\xi)$ \nare not smooth in a pointwise sense. \nOn the other hand, one may still be able to control averaged quantities  like \n\\begin{equation}\n\\label{eq:5}\n\\sup_x \\norm{\\sigma(x, \\xi)}{H^{n\/2}_\\xi}<\\infty.\n\\end{equation}\nThis will be our treshold condition for $L^2$ boundedness, \nwhich we try to achieve. \\\\ Heuristically at least, \\eqref{eq:5} \nmust be ``enough'' in some sense, \n since if we had simple symbols like $\\sigma(x,\\xi)=\\sigma_1(x) \\sigma_2(\\xi)$, \nthen the  $L^2$ boundedness of $Op(\\sigma)$ is equivalent to \n   $\\norm{\\sigma_1}{L^\\infty_x}<\\infty, \n\\norm{\\sigma_2}{L^\\infty_\\xi}<\\infty$. Clearly, \n$\\norm{\\sigma_2}{L^\\infty_\\xi({\\mathbf R}^n)}$ just fails to be controlled by \n\\eqref{eq:5}, but on the other hand, the quanitity in \\eqref{eq:5} \nis controlled by the \nappropriate Besov space  $B^{n\/2}_{2,1}$ norm. \n\n A final motivation for \nthe current study is  to achieve a scale\n invariant condition, which gives an estimate of the \n$L^2\\to L^2$ ($L^p\\to L^p$) norm of \n$Op(\\sigma)$ in terms of a {\\it scale invariant quantity}, \nthat is,  we aim at showing \nan estimate, \n$$\n\\norm{Op(\\sigma)}{L^p\\to L^p}\\leq C \\norm{\\sigma}{Y} \\norm{f}{L^p}, \n$$\nwhere for every $\\lambda\\neq 0$, one has $\\norm{\\sigma(\\lambda \\cdot,\\lambda^{-1} \n\\cdot) }{Y}=\\norm{\\sigma}{Y}$. \n\nIn that regard, note that the  condition (which is one of the requirements of \n the  H\\\"ormander class $S^0$)  \n\\begin{equation}\n\\label{eq:6}\n\\sup_{x}| D_\\xi^\\alpha \\sigma(x, \\xi)|\\leq  C_\\alpha |\\xi|^{-|\\alpha|}\n\\end{equation}\nis   scale invariant in the sense described above. Moreover, \nby the standard Calder\\'on-Zygmund theory (see \\cite{Stein}), \nthe pointwise condition \\eqref{eq:6} together with  \n$\\|T_\\sigma\\|_{L^2\\to L^2}<\\infty$  implies \n$$\nT_\\sigma f(x)=\\int K(x, x-y) f(y) dy,\n$$\nwhere $K(x,\\cdot)$ satisfies the H\\\"ormander-Mihlin conditions, namely $|K(x, z)|\\leq C|z|^{-n}$ and \n$|\\nabla_z K(x,z)|\\leq C|z|^{-n-1}$, where the constant $C$ depends on \nthe constants \n$C_\\alpha: |\\alpha|<[n\/2]+1$ in \\eqref{eq:6}.  \nThis  in turn is enough to conclude  \nthat $T_\\sigma:L^p \\to L^p$ for all $1<p\\leq 2$ and in fact there is the \nendpoint estimate  $T_\\sigma:L^1\\to L^{1, \\infty}$. \n\n\\subsection{$L^p$ estimates for PDO with rough symbols - statement of results} \nWe start now with our main theorems, which concern the  $L^2$ and the $L^p$ boundedness for pseudodifferential \noperators $Op(\\sigma)$ with rough symbols. Our first result establishes that a Besov space version of \\eqref{eq:5} is enough for $L^2$ boundedness and \nthe result is sharp. \n\\begin{theorem}($L^2$ bounds) \n\\label{theo:1}\nLet $\\sigma(x, \\xi):{\\mathbf R}^n\\times {\\mathbf R}^n\\to \\mathcal C$ and $T_\\sigma$ is the \ncorresponding pseudodifferential operator. Then \n\\begin{equation}\n\\label{eq:7}\n\\norm{T_\\sigma }{L^2({\\mathbf R}^n)\\to L^2({\\mathbf R}^n)}\\leq C (\\sum\\limits_l 2^{l n\/2} \\sup\\limits_x \\|P_l^\\xi \\sigma(x, \\cdot)\\|_{L^2({\\mathbf R}^n)}), \n\\end{equation} \nwhere $P_l^\\xi$ is the Littlewood-Paley operator in the $\\xi$ variable. \n\nMoreover, the result is sharp in the  following sense:  \nfor every $p>2$, \nthere exists $\\sigma(x,\\xi)$ so that $\\sup_{x} | D_\\xi^{\\alpha}\\sigma(x, \\xi)|\\leq C_\\alpha |\\xi|^{-|\\alpha|}$ and $\\sup_x \\norm{\\sigma(x, \\cdot)}{W^{p, n\/p}}<\\infty$,  \nbut  $T_\\sigma$ fails to be bounded on $L^2({\\mathbf R}^n)$. \n\\end{theorem}\n{\\bf Remark:} \n\\begin{enumerate} \n\\item Note that the estimate on $T_\\sigma$ is scale invariant. \n\\item The sharpness claim of the theorem, roughly speaking, \nshows that in the scale of spaces\\footnote{Note that \nthese  spaces scale the same and moreover \nby Sobolev embedding these are strictly decreasing sequence, at least  \nfor $2\\leq p<\\infty$.} $W^{p, n\/p}$, $\\infty\\geq p\\geq 2$, one may \nnot require anything less than $W^{2,n\/2}=H^{n\/2}$ of the symbol in \norder  to ensure $L^2$ boundedness.\n\\item The counterexample to which we refer in Theorem \\ref{theo:1} \nis a simple variation of the well-known example of $\\sigma\\in \nS^{0}_{1,1}$, the ``forbidden class'', \n which fails to be $L^2$ bounded, see \\cite{Stein}, p. 272 and Section \n\\ref{sec:counter} below. \n \\end{enumerate}\nOur next result concerns $L^p$ boundedness for $T_\\sigma$. \n\\begin{theorem}($L^p$ bounds) \n\\label{theo:3}\nFor the pseudodifferential operator $T_\\sigma$ there \nis the estimate for all $2<p\\leq \\infty$, \n\\begin{equation}\n\\label{eq:22}\n\\norm{T_\\sigma }{L^p({\\mathbf R}^n)\\to L^p({\\mathbf R}^n)}\\leq \nC (\\sum\\limits_l 2^{l n\/2} \\sup\\limits_x \\|P_l^\\xi \\sigma(x, \\cdot)\\|_{L^2({\\mathbf R}^n)}), \n\\end{equation}\nFor the range $1<p<2$ and indeed for the weak type $(1,1)$, there is \n\\begin{equation}\n\\label{eq:8}\n\\norm{T_\\sigma}{L^p\\to L^p}+\\norm{T_\\sigma}{L^1\\to L^{1, \\infty}}\\leq C \n\\sum\\limits_l  2^{ l n} \\sup\\limits_{x}  \\|P_l^\\xi \\sigma(x, \\cdot)\\|_{L^1({\\mathbf R}^n)}. \n\\end{equation}\nAlternatively, if one assumes the $L^2$ bound, together with \n\\eqref{eq:6}, one still gets $L^p\\to L^p$, $1<p\\leq 2$, and \nin fact weak type $(1,1)$ \n  bounds. Moreover, \n$$\n\\norm{T_\\sigma}{L^p\\to L^p}\\leq C (\n\\sum\\limits_l  2^{ l n\/2} \\sup\\limits_{x}  \\|P_l^\\xi \\sigma(x, \\cdot)\\|_{L^2({\\mathbf R}^n)}+\n\\sup_{|\\alpha|<[n\/2]+1} \\sup_{x,\\xi} |\\xi|^{|\\alpha|} | D_\\xi^\\alpha \\sigma(x, \\xi)|).\n$$\n\\end{theorem}\nAs we pointed out in Theorem \\ref{theo:1}, the estimates are \nessentially sharp for $L^p$,  $2\\leq p<\\infty$ boundedness. \nThe following  corollary gives even more precise \ncondition under which a symbol $\\sigma$ will give rise to a bounded \noperator on $L^q$ in the case of a given $1<q<2$\n\\begin{corollary}\n\\label{cor:1}\nLet $1<q<2$.  Then \n$$\n\\norm{T_\\sigma}{L^q\\to L^q}\\leq C\\sum\\limits_l 2^{ln\/q} \\sup\\limits_x \n\\|P_l^\\xi \\sigma(x, \\cdot)\\|_{L^{q}({\\mathbf R}^n)}. \n$$\n\\end{corollary}\nClearly the proof follows by  interpolation \nfrom  the $L^2$ estimates  in Theorem \\ref{theo:1} and the weak type $(1,1)$ \nestimates of Theorem \\ref{theo:3}. \n\n\n\\subsection{PDO's with  homogeneous of degree \nzero symbols - statement of  results}\n\nRegarding symbols that are  homogeneous of degree zero, \ni.e. $\\sigma(x, \\xi)=q(x, \\xi\/|\\xi)$, where $q:{\\mathbf R}^n\\times \n\\mathbf S^{n-1}\\to \\mathcal C$, we  obtain more precise results in terms of \nthe smoothness of $q$. \\\\\nNote that the classical H\\\"ormander condition  requires \npointwise smoothness of the function $q$ in both variables. \nOur result on the other hand requires much less than that.  \n\\begin{theorem}($L^p$ bounds for  homogeneous of degree zero symbols)\\\\\n\\label{theo:4}\nLet $q:{\\mathbf R}^n\\times \\mathbf S^{n-1}\\to \\mathcal C$. Let  \n$$\nT_q f(x)= \\int\\limits_{{\\mathbf R}^n} q (x,\\xi\/|\\xi|) e^{2\\pi i \\xi x} \n\\hat{f}(\\xi) d\\xi.\n$$\nThen $T_q :L^2\\to L^2$, if $\\sum_l 2^{l(n-1)\/2} \n\\|P_l^{\\xi\/|\\xi} q\\|_{L^2(\\mathbf S^{n-1})}<\\infty$ and in fact \n\\begin{equation}\n\\label{eq:9}\n\\norm{T_q}{L^2\\to L^2}\\leq C \\sum_l 2^{l(n-1)\/2} \n\\sup_x \\|P_l^{\\xi\/|\\xi|} q(x, \\cdot)\\|_{L^2(\\mathbf S^{n-1})}.\n\\end{equation}\nConcerning $L^p$ bounds, we have for every  $2\\leq p\\leq  \\infty$. \n\\begin{equation}\n\\label{eq:10}\n\\norm{T_q}{B^0_{p,1}\\to L^p}\\leq C_n (\\sum_l 2^{l(n-1)\/p'} \n\\sup_x \\|P_l^{\\xi\/|\\xi|} q\\|_{L^{p'}(\\mathbf S^{n-1})}).\n\\end{equation}\nNote that in \\eqref{eq:10}, \nthe constant $C_n$  is independent of  $p,r$. \n\\end{theorem} \n{\\bf Remark:}\n\\begin{enumerate}\n\\item It would be interesting to see whether the usual $L^p\\to L^p$  \n boundedness  holds true. \n\\item \nNote that there is no  weak type $(1,1)$ statement in \n Theorem \\ref{theo:4}. This is a  difficult \nissue even for multipliers. \n\\end{enumerate}\n\\noindent The sharpness statement associated with Theorem \\ref{theo:4} is \n\\begin{proposition}\n\\label{prop:5}\nFor every $N>1$, there exists a homogeneous of degree zero  symbol \n$\\sigma(x, \\xi):\\mathbf R^2\\times \\mathbf R^2\\to \\mathbf R^1$, \nso that $\\sup_{x, \\xi} |\\sigma(x, \\xi)|<\\infty$ and \n$\\sup_x \\norm{\\sigma(x,\\xi)}{W^{1, 1}(\\mathbf S^1)}<\\infty$, \nand so that $\\norm{T_\\sigma}{L^2\\to L^2}>N$. \n\\end{proposition}\nThe counterexample considered here is a smoothed out version  of the \nmaximal directional Hilbert transform in the plane $H_* f(x)=\\sup_{u\\in \\mathbf S^1} \n|H_u f(x)|$. We mention the spectacular recent result of \nLacey and Li, \\cite{LL} showing the boundedness of  \n$H_*$ on $L^p(\\mathbf R^2): 2<p<\\infty$ with a \n$H_*: L^2(\\mathbf R^2)\\to L^{2, \\infty}(\\mathbf R^2)$ as an endpoint \nestimate. Note that the $L^2\\to L^2$ bound fails, as elementary examples show, see \\cite{LL}.  \nWe verify later that the condition \\eqref{eq:10} \njust fails  for the (smoothed out) multiplier $\\sigma$  of $H_*$  in two \ndimensions, \nbut on the other hand the condition \n$\\norm{\\sigma(x,\\xi)}{W^{1, 1}(\\mathbf S^1)}<\\infty$ holds. \n\nThis example will show that the Besov spaces requirements for $\\sigma$ in \n\\eqref{eq:9} and \\eqref{eq:10} cannot be replaced by \nSobolev spaces and\/or spaces with less derivatives.  \n\n\\subsection{PDO's with  radial symbols.}\nFinally, we consider the case of radial symbols. That is for \n$\\rho:{\\mathbf R}^n\\times R^1_+\\to \\mathcal C$ and \n$$\nT_\\rho f(x)=\\int_{{\\mathbf R}^n} \\rho(x, |\\xi|) e^{2\\pi i \\xi x} \\hat{f}(\\xi) d\\xi.\n$$\n\\begin{theorem}\n\\label{theo:radial}\nThe operator $T_\\rho:L^2\\to L^2$, if $\\sum_l 2^{l\/2} \\sup\\limits_x \\|P^{|\\xi|}_l\n\\rho(x, \\cdot)\\|_{L^2(\\mathbf R^1)}<\\infty$. In fact, \n$$\n\\|T_\\rho\\|_{L^2\\to L^2}\\leq C \\sum_l 2^{l\/2} \\sup\\limits_x \\|P^{|\\xi|}_l\n\\rho(x, \\cdot)\\|_{L^2(\\mathbf R^1)}.\n$$\n\n\\end{theorem}\nClearly, establishing  $L^p, p\\neq 2$ bounds for simple radial {\\it symbols} \nis already a notoriously difficult problem. \nOne only needs to point out to the Bochner-Riez multiplier \n$(1-|\\xi|^2)^{\\delta}_+$ (which satisfy $L^p$ bounds  only in certain range of\n$p's$, \ndepending on the dimension and\n$\\delta$) or even the simpler ``thin annulus'' multiplier \n$\\varphi(2^m(1-|\\xi|^2))$ to understand the difficulty of the problem in general. \n\n\\section{Applications}\nIn this section, we demonstrate the effectiveness of the $L^p$ boundedness\ntheorems for rough PDO's. We will mostly concentrate on application to \nmaximal functions and operators\\footnote{In addition, the author has also \nidentified   several applications to bilinear\/multilinear  operators of \n  importance to certain\ndispersive PDE's, which  will be addressed in a future publication.} \nSome of our examples will be well-known results for  maximal operators, \nwhile others will be a higher dimensional extensions of such results. \n\n\\subsection{Almost everywhere convergence for Cesaro sums of $L^p$ functions  in 1 D} \nWe start with Cesaro's sum for Fourier series in one space dimension. For any\n$\\delta>0$, define \n$$\n\\mathcal C_\\delta f(x)=\\sup_{u>0} \\int_{\\mathbf R^1} (1-\\xi^2\/u^2)^{\\delta}_+  \ne^{2\\pi i \\xi x} \\hat{f}(\\xi) d\\xi.\n$$ \nClearly, as a limit as $\\delta\\to 0$, we get the Carleson's operator. \nUnfortunately, one cannot  conclude that \n$\\sup_{\\delta>0}\\|C_\\delta\\|_{L^p}<\\infty$, for\nthat would imply the famous Carleson-Hunt theorem. \nOn the other hand, define the maximal ''thin\ninterval operator''\n$$\nT_m f(x)=\\sup_{u>0} \\int_{\\mathbf R^1} \\varphi(2^m(1-\\xi^2\/u^2))  \ne^{2\\pi i \\xi x} \\hat{f}(\\xi) d\\xi.\n$$\nA \nsimple argument based on (the proof of) Theorem \\ref{theo:3} yields \n\\begin{proposition}\n\\label{prop:Carl}\nFor any $\\varepsilon>0, 1<p<\\infty$, there exists $C_{p, \\varepsilon}$, so that \n\\begin{equation}\n\\label{eq:908}\n\\sup_m \\|T_m \\|_{L^p(\\mathbf R^1)\\to L^p(\\mathbf R^1)}\\leq C_p. \n\\end{equation}\nIn fact, there is the more general pointwise \nbound $\\sup\\limits_m |T_m f(x)|\\leq C M (\\sup_k |P_k f|)(x)$, \nwhich implies \\eqref{eq:908} as well as \n\\begin{equation}\n\\label{eq:909}\n\\|C_\\delta\\|_{L^p\\to L^p}+\n\\|C_\\delta\\|_{F^0_{1, \\infty}(\\mathbf R^1)\\to L^{1,\\infty}(\\mathbf R^1)}\\leq C_{\\delta,p}. \n\\end{equation}\n\\end{proposition}\n{\\bf Remark:} \n\\begin{itemize}\n\\item Note that this result, while clearly inferior to the \nCarleson-Hunt \ntheorem still implies a.e. convergence for any Cesaro summability method, when\napplied to $L^p(\\mathbf R^1)$ functions, and in fact for the larger class of $F^0_{p, \\infty}$ functions.  \n\\item Using the method of proof here, one may actually prove $L^2$ estimates\\footnote{And in fact\n$L^p\\to L^p$ estimates for $p=p(\\delta)$ close to $2$.} for the {\\it maximal\nBochner-Riesz operator} \n$$\nBR_\\delta f(x)=\\sup_{u>0} \\int_{{\\mathbf R}^n} (1-|\\xi|^2\/u^2)^{\\delta}_+  \ne^{2\\pi i \\xi x} \\hat{f}(\\xi) d\\xi.\n$$\n{\\it in any dimension}. \n\\end{itemize}\n\\begin{proof}\nIt clearly suffices to show the pointiwise estimate \n$|T_m f(x)|\\leq C M (\\sup_k P_k f)(x)$ for any $k$.\nThe  statements about $B^0_{p,1}\\to L^p$ bounds  follow by elementary Littlewood-Paley theory and\nthe $l^p$ bounds for the Hardy-Littlewood maximal function. The restricted-to-weak estimate \n$F^0_{1,\\infty}\\to L^{1,\\infty}$ for $C_\\delta$ follows by summing an \n{\\it exponentially decaying} series in\nthe quasi-Banach space $L^{1, \\infty}$. \n\nBy support considerations, it is clear that \n\\begin{eqnarray*}\nT_m f(x) & = &\\sum\\limits_k \\sup_{u>0} \\int_{\\mathbf R^1} \\varphi(2^m(1-\\xi^2\/u^2))  \ne^{2\\pi i \\xi x}\\varphi(2^{-k}\\xi) \\hat{f}(\\xi) d\\xi= \\\\\n&=&\n\\sum\\limits_k \\sup_{u\\in (2^{k-2}, 2^{k+2})} \\int_{\\mathbf R^1} \\varphi(2^m(1-\\xi^2\/u^2))  \ne^{2\\pi i \\xi x}\\varphi(2^{-k}\\xi) \\hat{f}(\\xi) d\\xi= \\\\\n&=& \\sum\\limits_k T_{m, u(\\cdot)\\in (2^{k-2}, 2^{k+2})} f_k.\n\\end{eqnarray*}\nClearly, the requirement $u\\in (2^{k-2}, 2^{k+2})$ creates (almost) \ndisjointness in the $x$ support, whence \n\\begin{equation}\n\\label{eq:819}\n|T_m f(x)|\\leq C \\sup\\limits_k |T_{m, u(\\cdot)\\in (2^{k-2}, 2^{k+2})} f_k(x)|.\n\\end{equation}\nOur basic claim is that \n\\begin{equation}\n\\label{eq:820}\n|T_{m, u(\\cdot)\\in (2^{k-2}, 2^{k+2})} f_k(x)|\\leq C M(f_k).\n\\end{equation}\nClearly \\eqref{eq:819} and \\eqref{eq:820} imply  $\\sup_m |T_m f(x)|\\leq C M(\\sup\\limits_k |f_k|)$, \nwhence the Proposition \\ref{prop:Carl}.  \\\\\nBy scale invariance, \\eqref{eq:820} reduces to the case $k=0$, that is we need to show \n$$\n|T_{m, u(x)\\in (1\/4, 4)} P_0 f(x)|\\leq C M (P_0 f)(x).\n$$\nfor any Schwartz function $f$ and any \n$m>>1$. By \\eqref{eq:32} (in the proof of Theorem \\ref{theo:3} below), it will suffice to\nshow \n\\begin{equation}\n\\label{eq:920}\n\\sum\\limits_l 2^l \\sup\\limits_x \\|P_l^\\xi [\\varphi(2^m(1-\\xi^2\/u(x)^2)) \\varphi(\\xi)]\\|_{L^1(\\mathbf R^1)}\\lesssim 1.\n\\end{equation}\nfor any measurable function $u$, which takes its values in $(1\/4, 4)$.  \\\\\nFor \\eqref{eq:920}, we have \n\\begin{eqnarray*}\n& & \\sum\\limits_{l<m}  2^l \\sup\\limits_x \\|P_l^\\xi [\\varphi(2^m(1-\\xi^2\/u(x)^2)) \\varphi(\\xi)]\\|_{L^1_\\xi(\\mathbf R^1)} \\\\ \n& & \\leq \\sum\\limits_{l<m} 2^l \\sup\\limits_{u\\in (1\/4, 4)} \\int|\\varphi(2^m(1-\\xi^2\/u^2)) \\varphi(\\xi)| d\\xi\\lesssim \n\\sum\\limits_{l<m} 2^{l-m}\\lesssim 1, \n\\end{eqnarray*}\nwhile \n\\begin{eqnarray*}\n& & \\sum\\limits_{l\\geq m}  2^l \\sup\\limits_x \n\\|P_l^\\xi [\\varphi(2^m(1-\\xi^2\/u(x)^2)) \\varphi(\\xi)]\\|_{L^1_\\xi(\\mathbf R^1)} \\\\\n& &\\leq \\sum\\limits_{l\\geq m}  2^{-l} \n\\sup\\limits_{u\\in (1\/4, 4)} \\int|\\f{d^2}{d \\xi^2} \\varphi(2^m(1-\\xi^2\/u^2)) \\varphi(\\xi)|d\\xi\\leq C \n\\sum\\limits_{l\\geq m} 2^{-l+m}\\lesssim 1.\n\\end{eqnarray*}\n\\end{proof}\n\\subsection{Maximal directional Hilbert transforms and the Kakeya  maximal\nfunction.}\nAnother interesting application is provided by the \ndirectional Hilbert transform\nin dimensions  $n\\geq 2$. Namely, take \n$$\nH^*_\\delta=\\sup_{u\\in \\mathbf S^{n-1}} \\int (\\dpr{u}{\\xi\/|\\xi|})^{\\delta}_+ \ne^{2\\pi i \\xi x} \\hat{f}(\\xi)\\varphi(\\xi) d\\xi,\n$$\nwhere $supp \\varphi\\subset \\{1\/2<|\\xi|<2\\}$. \\\\\nAs $\\delta\\to 0$, we obtain the operator $f\\to \n\\sup_{u\\in \\mathbf S^{n-1}} \\int_{\\{\\dpr{u}{\\xi}>0\\}}\ne^{2\\pi i \\xi x} \\hat{f}(\\xi) d\\xi$, which is closely related to the maximal\ndirectional Hilbert transform  \n$$\nH_* f(x)=\\sup_{u}|H_u f(x)|=\\sup_u |\\int sgn(u, \\xi) \ne^{2\\pi i \\xi x} \\hat{f}(\\xi) \\varphi(\\xi)d\\xi|.\n$$\n$H_*$ was of course shown to be $L^p(\\mathbf R^2), p>2$ bounded by \nLacey and Li, \\cite{LL} by very sophisticated time-frequency analysis methods. \n\\begin{proposition}\n\\label{prop:mdh}\nFor the ``thin big circle'' multiplier \n$$\nT_m f(x)= \\sup_{u\\in \\mathbf S^{n-1}} |\\int_{{\\mathbf R}^n} \\varphi(2^m\\dpr{u}{\\xi\/|\\xi|}) \ne^{2\\pi i \\xi x} \\hat{f}(\\xi) \\varphi(\\xi)d\\xi|.\n$$\nwe have \n\\begin{equation}\n\\label{eq:kak}\n\\|T_m f\\|_{L^2\\to L^2}\\leq C_\\varepsilon  2^{m(n\/2-1)}\n\\end{equation}\nIn particular \n$$\n\\|H^*_\\delta\\|_{L^2({\\mathbf R}^n)\\to L^2({\\mathbf R}^n)}\\leq C_{p, \\varepsilon, \\delta} 2^{n\/2-1}.\n$$\n\\end{proposition}\n{\\bf Remark:} \n\\begin{itemize}\n\\item We believe that  the operator $T_m$ ($m>>1$) \nhas a particular connection to the \nKakeya maximal function and the corresponding Kakeya problem. \nIndeed, the kernel\nof the corresponding singular integral behaves like  a \n($L^1$ normalized) characteristic function of a rectangle with long side along $u$ of length $2^m$\nand $(n-1)$ short sides of length $1$ in  the transverse directions! \n\\item In relation to that, one expects the conjectured Kakeya bounds \n$$\n\\|T_m f\\|_{L^p\\to L^p}\\leq C_\\varepsilon   2^{m(n\/p-1)}\n$$\nfor $p\\leq n$ \nto hold, while one only gets \n$$\n\\|T_m f\\|_{L^p\\to L^p}\\leq C_\\varepsilon   2^{m(n\/p-2\/p)}\n$$\nas a consequence of Theorem \\ref{theo:4}. Nevertheless, the two match when \n$p=2$. So it seems that  \\eqref{eq:kak}, at\nleast in principle,  captures the Kakeya conjecture for \n$p=2$ in general and in\nparticular the full Kakeya conjecture in two dimensions. \n\nSince our estimates do not seem to contribute much toward the resolution of any\nnew Kakeya estimates, we do not pursue here the exact relationship between $T_m$\nand the Kakeya maximal operator, although from our heuristic arguments above it\nshould be clear that it is a close one. \n\\end{itemize}\n\\begin{proof}\nWe proceed as in the proof of Proposition \\ref{prop:Carl}. We need only show \n\\begin{equation}\n\\label{eq:990}\n\\sum\\limits_{l} 2^{l(n-1)\/2} \\sup\\limits_x \\|P_l^{\\xi\/|\\xi|} \n\\varphi(2^m\\dpr{u(x)}{\\xi\/|\\xi|})\\|_{L^2(\\mathbf S^{n-1})}\\lesssim 1.\n\\end{equation}\nWe have \n\\begin{eqnarray*}\n& & \\sum\\limits_{l<m} 2^{l(n-1)\/2} \\sup\\limits_x \\|P_l^{\\xi\/|\\xi|} \n\\varphi(2^m\\dpr{u(x)}{\\xi\/|\\xi|})\\|_{L^2(\\mathbf S^{n-1})}\\\\ \n& &  \\leq C \\sum\\limits_{l<m} 2^{l(n-1)\/2}\\sup_u \n\\| \\varphi(2^m\\dpr{u(x)}{\\xi\/|\\xi|})\\|_{L^2(\\mathbf S^{n-1})}\\\\\n& &\\leq  C \\sum\\limits_{l<m} 2^{l(n-1)\/2} 2^{-m\/2}\n\\lesssim 2^{m(n\/2-1)}.\n\\end{eqnarray*}\n\\begin{eqnarray*}\n& & \\sum\\limits_{l\\geq m} 2^{l(n-1)\/2} \\sup\\limits_x \\|P_l^{\\xi\/|\\xi|} \n\\varphi(2^m\\dpr{u(x)}{\\xi\/|\\xi|})\\|_{L^2(\\mathbf S^{n-1})}\\\\ \n& & C \\leq \n\\sum\\limits_{l\\geq m} 2^{-l(n-1)\/2} \\sup\\limits_u \\|\\Omega^{(n-1)}\n\\varphi(2^m\\dpr{u(x)}{\\xi\/|\\xi|})\\|_{L^2(\\mathbf S^{n-1})} \\\\\n& & \\leq C  \\sum\\limits_{l\\geq m} 2^{-l(n-1)\/2} 2^{m(n-1)} 2^{-m\/2}\\lesssim 2^{m(n\/2-1)}. \n\\end{eqnarray*}\n\\end{proof}\n\n\n\n\\subsection{Estimates on $T_{\\sigma_1 \\sigma_2}$, $T_{e^{ \\sigma}}$ etc.}\nWe now present a result, which allows us to treat pseudodifferential \n operators, whose symbols are products, exponentials (or more generally entire functions) of symbols, which satisfy the requirements in \nTheorems \\ref{theo:1}, \\ref{theo:3}, \\ref{theo:4}. We would like to point out that similar in \nspirit (by essentially requiring $n\/2+\\varepsilon$ derivatives in $L^2$, but in a more general setting) \nfunctional calculus type result was obtained in \\cite{SeegerSogge}. \n\\begin{proposition}\n\\label{prop:7}\nLet $\\sigma, \n\\sigma_1, \\sigma_2:{\\mathbf R}^n\\times {\\mathbf R}^n\\to \\mathcal C$, so that $T_{\\sigma_j}, j=1,2$ define \n$L^p$ bounded operators, as in \\eqref{eq:7}, \\eqref{eq:8}.  \nThen for every $1<p<\\infty$, $T_{\\sigma_1\\sigma_2}$ is \n $L^p$ bounded and \n\\begin{eqnarray}\n\\label{eq:814}\n& & \\norm{T_{\\sigma_1 \\sigma_2}}{L^2({\\mathbf R}^n)\\to L^2({\\mathbf R}^n)}\\leq C  \\prod_{j=1}^2 \n(\\sum\\limits_l 2^{l n\/2} \\sup\\limits_x \\|P_l^\\xi \\sigma_j(x, \\cdot)\\|_{L^2({\\mathbf R}^n)}) , \\\\\n\\label{eq:815}\n& & \\norm{T_{\\sigma_1 \\sigma_2}}{L^p({\\mathbf R}^n)\\to L^p({\\mathbf R}^n)}\\leq C \n\\prod_{j=1}^2 \n(\\sum\\limits_l 2^{l n} \\sup\\limits_x \\|P_l^\\xi \\sigma_j(x, \\cdot)\\|_{L^1({\\mathbf R}^n)}).\n\\end{eqnarray}\nIn the same spirit, $T_{e^{\\sigma}}$ is also $L^p$ bounded and there is \n\\begin{equation}\n\\label{eq:816}\n\\norm{T_{e^\\sigma}}{L^p({\\mathbf R}^n)\\to L^p({\\mathbf R}^n)}\\leq C  \nexp(\\sum\\limits_l 2^{l n} \\sup\\limits_x \\|P_l^\\xi \\sigma(x, \\cdot)\\|_{L^1({\\mathbf R}^n)}) \n\\end{equation}\nSimilar statements can be made for homogeneous of \ndegree zero symbols \\\\ $\\mu_1(x, \\xi\/|\\xi|), \\mu_1(x, \\xi\/|\\xi|)$,  \n\\begin{eqnarray}\n\\label{eq:817}\n& & \n\\norm{T_{\\mu_1\\mu_2}}{L^2({\\mathbf R}^n)\\to L^2({\\mathbf R}^n)}\\leq C  \n\\prod_{j=1}^2 (\\sum\\limits_l 2^{l (n-1)\/2} \\sup\\limits_x \\|P_l^{\\xi\/|\\xi|} \\mu_j(x, \n\\cdot)\\|_{L^1({\\mathbf R}^n)}), \\\\\n& & \n\\label{eq:818}\n\\norm{T_{e^\\mu}}{L^2({\\mathbf R}^n)\\to L^2({\\mathbf R}^n)}\\leq C  \nexp(\\sum\\limits_l 2^{l (n-1)} \\sup\\limits_x \\|P_l^{\\xi\/|\\xi|} \\mu(x, \n\\cdot)\\|_{L^1({\\mathbf R}^n)}).\n\\end{eqnarray}\n\n\\end{proposition}\nThe proof of Proposition \\ref{prop:7} is based on the corresponding \nTheorem for $L^p$ boundedness, \ncombined with the fact that our requirements form a Banach \nalgebra under the multiplication. Take for example \\eqref{eq:814}. \nBy Theorem \\ref{theo:1}, we have \n$$\n\\norm{T_{\\sigma_1 \\sigma_2}}{L^2({\\mathbf R}^n)\\to L^2({\\mathbf R}^n)}\\leq C   \n(\\sum\\limits_l 2^{l n\/2} \\sup\\limits_x \\|P_l^\\xi (\\sigma_1\\sigma_2)(x, \\cdot)\\|_{L^2({\\mathbf R}^n)})\n$$\nWe finish by invoking the estimate\n\\begin{equation}\n\\label{eq:900}\n\\sum\\limits_l 2^{l n\/2} \\sup\\limits_x \\|P_l^\\xi (\\sigma_1\\sigma_2)(x, \\cdot)\\|_{L^2({\\mathbf R}^n)}\\leq \nC \\prod_{j=1}^2 \n(\\sum\\limits_l 2^{l n\/2} \\sup\\limits_x \\|P_l^\\xi \\sigma_j(x, \\cdot)\\|_{L^2({\\mathbf R}^n)}).\n\\end{equation}\nwhere this last inequality essentially means that $B^{n\/2}_{2,1}$ \nis a Banach algebra of functions\\footnote{This is well-known, but \ncan be verified easily by \nmeans of the Kato-Ponce estimate \n$\\|\\partial^{n\/2} (u v)\\|_{L^2}\\leq C (\\|\\partial^{n\/2} u\\|_{L^2}\\norm{v}{L^\\infty}+ \n\\|\\partial^{n\/2} v\\|_{L^2}\\norm{u}{L^\\infty})$, the embedding \n$B^{n\/2}_{2,1}\\hookrightarrow L^\\infty$ \nand  some Littlewood-Paley theory.}.\n\nThe argument above can be performed for the proof  of \n\\eqref{eq:815}. For  \\eqref{eq:816} (and more generally for \nany symbols of the form $g(\\sigma)$, where $g$ is entire function), \none  iterates  the product estimate \n\\eqref{eq:900} to \n\\begin{equation}\n\\label{eq:901}\n\\sum\\limits_l 2^{l n\/2} \\sup\\limits_x \\|P_l^\\xi (e^\\sigma)(x, \\cdot)\\|_{L^2({\\mathbf R}^n)}\\leq \nC exp(\\sum\\limits_l 2^{l n\/2} \\sup\\limits_x \\|P_l^\\xi \\sigma(x, \\cdot)\\|_{L^2({\\mathbf R}^n)}).\n\\end{equation}\nFor the proof of \\eqref{eq:817}, \\eqref{eq:818}, one has to use \n the fact that  $B^{(n-1)\/2}_{2, 1}(\\mathbf S^{n-1})$ is a \nBanach algebra as well, whence one gets an estimate similar to  \n\\eqref{eq:900} and \\eqref{eq:901}. \n\n\n\n\n\n\n\n\n\n\n\n\\section{Preliminaries}\n\\label{sec:prelim}\n\\noindent We start by introducing some basic concepts in Fourier analysis. \n\\subsection{Fourier analysis on ${\\mathbf R}^n$}\nFirst, define the Fourier transform and its inverse  \n\\begin{eqnarray*}\n& &\n\\hat{f}(\\xi)=\\int\\limits_{{\\mathbf R}^n} f(x)e^{-2\\pi i x\\cdot  \\xi} dx, \\\\\n& &\n f(x)=\\int\\limits_{{\\mathbf R}^n} \\hat{f}(\\xi)e^{2\\pi i x\\cdot  \\xi} d\\xi. \n\\end{eqnarray*}\nFor a \npositive, smooth and  even function $\\chi:{\\mathbf R}^n\\to \\mathbf R^1_+$, supported in \n $\\{\\xi:|\\xi|\\leq 2\\}$ and so that $\\chi(\\xi)=1$ for all\n$|\\xi|\\leq 1$.  \n Define $\\varphi(\\xi)=\\chi(\\xi)-\\chi(2\\xi)$, \nwhich is supported in the annulus $1\/2\\leq |\\xi|\\leq 2$. Clearly \n$\\sum_{k\\in \\mathcal Z} \\varphi(2^{-k} \\xi)=1$ for \nall\\footnote{The discussion henceforth will be \nfor ${\\mathbf R}^n$, unless explicitely specified otherwise.}  \n$\\xi\\neq 0$. \n\nThe $k^{th}$ Littlewood-Paley projection is given by \n$\\widehat{P_k f}(\\xi)=\\varphi(2^{-k}\\xi) \\hat{f}(\\xi)$.  \nNote that the kernel of $P_k$ is  integrable uniformly  in $k$  and thus \n$P_k :L^p\\to L^p$ for $1\\leq p\\leq \\infty$ and  \n$\\norm{P_k}{L^p\\to L^p}\\leq \nC_n \\norm{\\hat{\\chi}}{L^1}$. \nIn particular, the  bounds are independent of $k$. \n\nIt is a standard observation that $\\nabla P_k = P_k \\nabla = \n2^{k}\\tilde{P_k}$, where $\\tilde{P_k}$ is a multiplier type \noperator similar to $P_k$ and thus \n$\\norm{P_k \\nabla  \\psi}{L^p}\\sim 2^k \\norm{P_k \\psi}{L^p}$.  \nIn what follows, we will use the notation \n$P_l^x f(x, \\xi)$ to denote Littlewood-Paley operator acting on \nthe variable $x$, and $P_l^\\xi f(x, \\xi)$ will be a Littlewood-Paley operator in the variable $\\xi$. That is \n\\begin{eqnarray*}\n& & \nP_l^x f(x, \\xi)=2^{l n} \\int \\hat{\\varphi}(2^l(x-y)) f(y, \\xi) dy, \\\\\n& & P_l^\\xi f(x, \\xi)=2^{l n} \n\\int \\hat{\\varphi}(2^l(\\xi-\\eta)) f(x, \\eta) d\\eta\n\\end{eqnarray*}\nThe  Bernstein inequality takes the form\n$$\n\\norm{P_l f}{L^q}\\leq C_n 2^{l n (1\/p-1\/q)}\\norm{P_l f}{L^p}.\n$$\nfor $1\\leq p<q\\leq \\infty$. \\\\\nThe (uncentered) Hardy-Littlewood maximal function is\n$$\nMf(x)=\\sup_{Q\\supset x} \\f{1}{|Q|} \\int_Q |f(y)| dy.\n$$\nIt is well-known that $M:L^p\\to L^p$ for all $1<p\\leq \\infty$ and is of \nweak type $(1,1)$. It is also convenient to use the pointwise bound \n\\begin{equation}\n\\label{eq:30}\n\\sup_{t>0} t^{-n} |f*\\Phi(t^{-1} \\cdot)(x)|\\leq C \\|\\Phi\\|_{L^1} Mf(x),\n\\end{equation}\nfor a radially dominated function $\\Phi$. \nFor integer values of $s$, we may define $W^{p,s}$ to be \nthe Sobolev space with $s$ derivatives in $L^p$, $1\\leq p\\leq \\infty$, with the corresponding norm  \n$$\n\\norm{f}{W^{p,s}}:=\\sum\\limits_{|\\alpha|\\leq s} \\norm{D^\\alpha_x f}{L^p}. \n$$\nEquivalently, and for noninteger values of $s$, define \n$$\n\\norm{f}{W^{p,s}}:=\\norm{f}{L^p}+ \n\\|(\\sum\\limits_{l=0}^\\infty  2^{2l s}|P_l f|^2)^{1\/2}\\|_{L^p}\n$$\nand its  homogeneous analogue \n$$\n\\norm{f}{\\dot{W}^{p,s}}:=\n\\|(\\sum\\limits_{l=-\\infty}^\\infty  2^{2l s}|P_l f|^2)^{1\/2}\\|_{L^p}.\n$$\nNote $W^{p,s}=\\dot{W}^{p,s}\\cap L^p$.  \\\\\nThe  (homogeneous) Besov spaces $\\dot{B}_{p,q}^s$, \nwhich scale like $\\dot{W}^{p,s}$, are  defined as follows \n$$\n\\norm{f}{\\dot{B}_{p,q}^s}:= \n(\\sum\\limits_{l\\in \\mathcal Z}  2^{l sq }\\|P_l f\\|_{L^p}^q)^{1\/q}.\n$$\nThe Triebel-Lizorkin spaces are defined via \n$$\n\\norm{f}{\\dot{F}_{p,q}^s}:= \n\\|(\\sum\\limits_{l\\in \\mathcal Z} 2^{l sq }|P_l f|^q)^{1\/q}\\|_{L^p}.\n$$\n\n\\subsection{Fourier analysis on $\\mathbf S^{n-1}$} In this section, we define the Sobolev and Besov spaces for functions $q$ defined on $\\mathbf S^{n-1}$.  For that, the standard approach is to fix the basis of the spherical harmonics and define the Littlewood-Paley operators by projecting over the corresponding set of the harmonics within the fixed frequency.  \n\nIntroduce the angular differentiation operators $\\Omega_{i j}=x_j \\partial_i - x_i \\partial_j$.  It is well-known  that $\\{\\Omega_{ i j}\\}_{i\\neq j}$ generate the algebra of all differential operators, acting on $C^\\infty(\\mathbf S^{n-1})$. The spherical Laplacian is defined via \n$$\n\\Delta_{sph}=\\sum\\limits_{i< j} \\Omega_{i j}^2.\n$$\nThe spherical harmonics $\\{Y^n_{l,k}\\}_{k\\in A^n_l}$ \n are eigenfunctions of $\\Delta_{sph}$,  so that $\\Delta_{sph} Y^n_{l,k}=-l(n-2+l) Y^n_{l,k}$, where $l\\geq 0$, $k$ varies in a finite set $A_l^n$.  \n  An equivalent way to define them is to take all the \nhomogeneous of degree $l$ polynomials that are solutions to \n \\begin{equation}\n \\label{eq:45}\n (\\partial_r^2+ r^{-1} \\partial_r + r^{-2} \\Delta_{sph})Y^l=0\n \\end{equation}\n Iy turns out that \\eqref{eq:45} \nhas $\\left(\\begin{array}{c} n+l-1 \\\\ l\\end{array}\\right)$ \nlinearly independent solutions \n $\\{Y^n_{l,k}\\}_{k\\in A^n_l}$.  \nAnother  important property of the family  $\\{Y^n_{l,k}\\}$ \n is that it forms \n  an orthonormal  basis for $L^2(\\mathbf S^{n-1})$.  \n  \n Let $f:\\mathbf S^{n-1}\\to \\mathcal C$ be a smooth function.  One can then define \nthe expansion in spherical harmonics in the usual way \n$$\nf(\\theta)=\\sum\\limits_{l,k\\in A_l^n} c_{l,k}^n Y^n_{l,k}(\\theta), \n$$\nwhere $c_{l,k}^n=\\dpr{f}{Y^n_{l,k}}_{L^2(\\mathbf S^{n-1})}$. \n The Littlewood-Paley operators may be  defined via \n $$\n P_m^{\\xi\/|\\xi|} f = \\sum\\limits_{l,k\\in A_l^n} c_{l,k}^n\\varphi(2^{-m} l)  \nY^n_{l,k}(\\theta), \n $$\n and there is the equivalence for all\\footnote{The constant \nof equivalence here depends only on $p$ and the cutoff \nfunction $\\varphi$.}  $1<p<\\infty$, due to Strichartz \\cite{Str1972}\n $$\n \\norm{f}{L^p(\\mathbf S^{n-1})}\\sim \n\\|(\\sum_{k=-\\infty}^\\infty |P^{\\xi\/|\\xi|}_k f|^2 )^{1\/2}\\|_{L^p(\\mathbf S^{n-1})}.\n $$\n As a simple consequence, one has for all $1<p<\\infty$, \n $\\|P_k^{\\xi\/|\\xi|}\\|_{L^p\\to L^p}\\leq C_{n,p}$. \nSuch an inequality actually extends \n(as in the case of ${\\mathbf R}^n$) to the endpoint cases $p=1, \np=\\infty$, see \\cite{SteinWeiss}. \n One can also define the Sobolev spaces $W^{p,s}(\\mathbf S^{n-1}):1<p<\\infty$ via \n \\begin{eqnarray*}\n & & \\norm{f}{W^{p,s}(\\mathbf S^{n-1})}=\\sum\\limits_{|\\alpha|\\leq s}\\norm{ \\Omega^\\alpha f}{L^p(\\mathbf S^{n-1})} \\\\\n & & \\norm{f}{\\dot{W}^{p,s}(\\mathbf S^{n-1})}= \n\\|(\\sum\\limits_{k=-\\infty}^\\infty2^{2ks} |P^{\\xi\/|\\xi|}_k f|^2 )^{1\/2}\\|_{L^p(\\mathbf S^{n-1})}.\n \\end{eqnarray*}\n where $\\Omega=\\sqrt{-\\Delta_{sph}}$. These last two formulas \n  give equivalent definitions for the case of integer $s$.  The (homogeneous) \n  Besov spaces are defined in the usual manner as follows \n \\begin{eqnarray*}\n& &  \\norm{f}{\\dot{B}^s_{p,q}(\\mathbf S^{n-1})}=\n(\\sum\\limits_{k=-\\infty}^\\infty 2^{qks} \\|P^{\\xi\/|\\xi|}_k f\\|_{L^p(\\mathbf S^{n-1})}^q)^{1\/q}.\n\\end{eqnarray*}\nIt is worth mentioning at this point that a variant of Bernstein \ninequality holds\\footnote{The proof is simply \n that there are  $\\sim N^{n-1}$ spherical harmonics at frequency \n$N$, just as for the Bernstein inequality one uses that the volume of \n$\\{\\xi\\in{\\mathbf R}^n:|\\xi|\\sim N\\}$ is $\\sim N^n$.} in this context, \nsee \\cite{Sterbenz}, p. 201. This together with the Littlewood-Paley theory outlined above implies Sobolev embedding for $L^p(\\mathbf S^{n-1})$ spaces. \nFor future reference,  let us record  \nthis estimate \n\\begin{equation}\n\\label{eq:bern}\n\\|f\\|_{L^q(\\mathbf S^{n-1})}\\leq C_{p,q,n} \n\\norm{\\Omega^{(n-1)(1\/p-1\/q)}f}{L^p(\\mathbf S^{n-1})}, \n\\end{equation}\nwhich holds whenever for $1<p< q< \\infty$. \n\n  \n   \nIf $q$ is a finite sum of harmonics, it is actually an analytic function \n  (it is a fact a restriction of a polynomial to the unit sphere) and one may write \n\\begin{equation}\n\\label{eq:36}\nq(\\theta)=\\sum\\limits_{\\alpha>0} \\f{D_\\xi^\\alpha q(\\theta_0)}{\\alpha!}(\\theta-\\theta_0)^\\alpha.\n\\end{equation}\nHere, $D_\\xi^\\alpha q(\\theta_0)$ should be understood as taking $\\alpha$ derivatives of \nthe corresponding homogeneous polynomial and evaluating at $\\theta_0$. \nThe following lemma is standard, but since we need a specific dependence of our \nestimates  upon the parameter $\\alpha$, we state it here for completeness. \n\\begin{lemma}\n\\label{le:os}\nLet $q:\\mathbf S^{n-1}\\to \\mathcal C$ and $q_m=P^{\\xi\/|\\xi|}_m q$. Then, there is a constant $C_n$, so that \nfor every $1\\leq p\\leq \\infty$, there is a the estimate \n$$\n\\|D^\\alpha_\\xi q_m\\|_{L^p(\\mathbf S^{n-1})}\\leq C_n^{|\\alpha|} 2^{m |\\alpha|} \\norm{q_m}{L^p(\\mathbf S^{n-1})}.\n$$\n\\end{lemma}\nThe proof of Lemma \\ref{le:os} is standard. One way to proceed is to note that if we\nextend the function $q_m$ off $\\mathbf S^{n-1}$ to some annulus, say  via \n$Q_m(\\xi)=\\varphi(|\\xi|) q_m(\\xi\/|\\xi|)$, and \nthen \n$$\n \\|D^\\alpha_\\xi q_m\\|_{L^p(\\mathbf S^{n-1})}\\lesssim \\|D^\\alpha_\\xi Q_m \\|_{L^p({\\mathbf R}^n)}\n$$\n\n\n\n\n\\section{$L^p$ estimates for PDO with rough symbols}\nWe start with the $L^2$ estimate to illustrate the main ideas in the proof. \n\\subsection{$L^2$ estimates: Proof of Theorem \\ref{theo:1}}\nOur first remark is that we will for convenience consider only \nreal-valued symbols $\\sigma$, since of course the general \ncase follows from splitting into a real and imaginary part. \n\nTo show $L^2$ estimates for $T_\\sigma$, it is equivalent to show $L^2$ \nestimates for the adjoint operator, which takes the \nform\\footnote{ There is the small technical problem \nthat the $\\xi$ integral does not converge \nabsolutely. \nThis can be resolved by judicious placement of cutoffs \n$\\chi(\\xi\/N)$, after which, one may subsume that \npart in $\\hat{f}(\\xi)$. In  the end \nwe let $N\\to \\infty$ and all the estimates will be \nindependent of the cutoff constant $N$.} \n$$\nT_\\sigma^* g(x)=\\int e^{2\\pi i \\xi\\cdot x} (\\int e^{-2\\pi i \\xi\\cdot y} \n[g(y) \\sigma(y, \\xi)] dy )d\\xi.\n$$\nOur next task is to decompose $T_\\sigma^* g$ and we start by \ntaking a Littlewood-Paley partition of unity in the $\\xi$ variable for $g$. We have \n$$\nT_\\sigma^* g (x)=\\sum\\limits_{l\\in \\mathcal Z} \\int e^{2\\pi i \\xi\\cdot x} \n(\\int e^{-2\\pi i \\xi\\cdot y} \n[ g(y) P_l^\\xi \\sigma(y, \\xi)] dy )d\\xi\n$$ \nNow that the function $g$ is frequency localized at frequency $2^l$, \nwe  introduce further decomposition in the  \n$\\xi$ integration. \n\nFor  the $L^2$ estimates, because of the orthogonality, \nwe only need  rough partitions, so for each fixed $l$, \ntake a tiling of ${\\mathbf R}^n$ composed of cubes $\\{Q\\}$ \nwith diameter $2^{-l}$. Denote the  characteristic functions of $Q$ by $\\chi_Q$. We have \n$$\nT_\\sigma^* g (x)=\\sum\\limits_{l\\in \\mathcal Z} \\sum\\limits_{Q:d(Q)=2^{-l}} \n\\int e^{2\\pi i \\xi\\cdot x} \\chi_Q(\\xi)\n(\\int e^{-2\\pi i \\xi\\cdot y} \n[ g(y) P_l^\\xi \\sigma(y, \\xi)] dy )d\\xi\n$$\nThe main point of our next decompositions is that the \nfunction $P_l^\\xi \\sigma$ is essentially constant in $\\xi$ \nover any fixed cube $Q$. We exploit that by observing  that \n$\\xi\\to P_l^\\xi\\sigma(x, \\xi)$ is an entire  function and there is \n the expansion \n$$\nP_l^\\xi \\sigma(y, \\xi) \\chi_Q(\\xi)=[P_l^\\xi \\sigma(y, \\xi_Q)+\\sum\\limits_{\\alpha: |\\alpha|>0}^\\infty\n\\f{D_\\xi^\\alpha P_l^\\xi \\sigma(y, \\xi_Q)}{\\alpha!} (\\xi-\\xi_Q)^\\alpha]\\chi_Q(\\xi)\n$$\nfor any fixed $y$ and for any $\\xi_Q\\in Q$. Note that \n$D_\\xi^\\alpha P_l^\\xi \\sigma(y, \\xi_Q)\\sim 2^{l|\\alpha|} P_l^\\xi \\sigma(y, \\xi_Q)$ and \n$ |(\\xi-\\xi_Q)^\\alpha|\\lesssim  2^{-l|\\alpha|}$, by support consideration \n(recall $d(Q)=2^{-l}$). On a heuristic level, by the presence of  $\\alpha!$, \none should think that  the series \nabove behave like $P_l^\\xi \\sigma(y, \\xi_Q)$ plus exponential tail. \n\nGoing back to $D_\\xi^\\alpha P_l^\\xi$, as  we have mentioned in Section \\ref{sec:prelim}, we can write \n$D_\\xi^\\alpha P_l^\\xi=2^{l|\\alpha|} P_{l, \\alpha}^\\xi$, where \n$P_{l, \\alpha}^\\xi$ is given by the multiplier $\\varphi(2^{-l}\\xi) (2^{-l} \\xi)^\\alpha$. \nIt is clear that \n$\\|P_{l, \\alpha}^\\xi f\\|_{L^2({\\mathbf R}^n)}\\leq C_n^{|\\alpha|}\\|P_l^\\xi f\\|_{L^2({\\mathbf R}^n)}$. \n\nThus, we have arrived at \n\\begin{eqnarray*}\n& & T_\\sigma^* g (x)=\\sum\\limits_{|\\alpha|\\geq 0} \\sum\\limits_{l\\in \\mathcal Z} \\sum\\limits_{Q:d(Q)=2^{-l}} \n\\int e^{2\\pi i \\xi\\cdot x} \\chi_Q(\\xi) \n\\f{2^{l|\\alpha|}(\\xi-\\xi_Q)^\\alpha}{\\alpha!} \\times \\\\\n& & \\times \n(\\int e^{-2\\pi i \\xi\\cdot y} \n[ g(y) P_{l,\\alpha}^\\xi \\sigma(y, \\xi_Q)] dy )d\\xi= \n\\sum\\limits_{l,\\alpha}(\\alpha!)^{-1} \\sum\\limits_{l, Q:d(Q)=2^{-l}}  P_{Q, l, \\alpha} [ g(\\cdot) \nP_{l,\\alpha}^\\xi \\sigma(\\cdot, \\xi_Q)], \n\\end{eqnarray*}\nwhere $P_{Q, l, \\alpha}$ acts via $\\widehat{P_{Q, l, \\alpha} f}(\\xi)= \\chi_Q(\\xi) \n2^{l|\\alpha|}(\\xi-\\xi_Q)^\\alpha \\hat{f}(\\xi)$. Note \n$$\n\\|P_{Q, l, \\alpha}\\|_{L^2\\to L^2}=\\sup_\\xi | \\chi_Q(\\xi) \n2^{l|\\alpha|}(\\xi-\\xi_Q)^\\alpha|\\leq 1. \n$$\n\nFor fixed $l, \\alpha$, take $L^2$ norm. Using the orthogonality of \n$P_{Q, l, \\alpha}$ and its boundedness on $L^2$, we obtain \n\\begin{eqnarray*}\n& & \\|\\sum\\limits_{Q:d(Q)=2^{-l}}  P_{Q, l, \\alpha} [ g(\\cdot) \nP_{l,\\alpha}^\\xi \\sigma(\\cdot, \\xi_Q)]\\|_{L^2}^2= \n\\sum\\limits_{Q:d(Q)=2^{-l}} \\|P_{Q, l, \\alpha} [ g(\\cdot) \nP_{l,\\alpha}^\\xi \\sigma(\\cdot, \\xi_Q)]\\|_{L^2}^2\\leq  \\\\\n& & \\leq \\sum\\limits_{Q:d(Q)=2^{-l}} \\|g(\\cdot) \nP_{l,\\alpha}^\\xi \\sigma(\\cdot, \\xi_Q)]\\|_{L^2}^2=\\int |g(y)|^2 \n(\\sum\\limits_{Q} |P_{l,\\alpha}^\\xi \\sigma(y, \\xi_Q)|^2) dy . \n\\end{eqnarray*}\nWe now again use   $P_{l,\\alpha}^\\xi \\sigma(y, \\xi_Q)\\sim P_{l,\\alpha}^\\xi \\sigma(y, \\eta)$ for any\n$\\eta\\in Q$, this time to estimate the contribution of $\\sum_{Q} |P_{l,\\alpha}^\\xi \\sigma(y, \\xi_Q)|^2$. \nThis is done as follows. Expand \n\\begin{equation}\n\\label{eq:859}\nP_{l, \\alpha} ^\\xi \\sigma(y, \\xi_Q)= \\sum\\limits_{\\beta: |\\beta|\\geq 0}^\\infty\n\\f{D_\\eta^\\beta P_{l, \\alpha}^\\eta \\sigma(y, \\eta)}{\\beta!} (\\xi_Q-\\eta)^\\beta,\n\\end{equation}\nto be used for  $\\eta\\in Q$. Thus, if we average over $Q$, \n\\begin{eqnarray*}\n& & |P_{l, \\alpha} ^\\xi \\sigma(y, \\xi_Q)|=\\left(|Q|^{-1} \\int_Q |\\sum\\limits_{\\beta: |\\beta|\\geq 0}^\\infty\n\\f{D_\\xi^\\beta P_{l, \\alpha}^\\xi \\sigma(y, \\eta)}{\\beta!} (\\xi_Q-\\eta)^\\beta|^2 d\\eta\n\\right)^{1\/2}\\leq  \\\\\n& &\\leq |Q|^{-1\/2} \\sum\\limits_{\\beta: |\\beta|\\geq 0}^\\infty \\f{C_n^{|\\beta|} \n2^{-l|\\beta|}}{\\beta!}  \\left(\\int\\limits_Q | D_\\xi^\\beta P_{l, \\alpha}^\\xi \\sigma(y, \\eta)|^2 d\\eta\\right)^{1\/2}.\n\\end{eqnarray*}\nand so (recalling $|Q|\\sim 2^{-l n}$)\n\\begin{eqnarray*}\n& &(\\sum\\limits_{Q} |P_{l,\\alpha}^\\xi \\sigma(y, \\xi_Q)|^2)^{1\/2}\\leq C 2^{l n\/2} \\sum\\limits_{\\beta} \\f{C_n^{|\\beta|} \n2^{-l|\\beta|}}{\\beta!} \\|D_\\xi^\\beta P_{l, \\alpha}^\\xi \\sigma(y, \\cdot)\\|_{L^2} \\leq \\\\\n& & \\leq \nC 2^{l n\/2} \\|P_{l, \\alpha}^\\xi \\sigma(y, \\cdot)\\|_{L^2}.\n\\end{eqnarray*}\nThus, \n\\begin{eqnarray*}\n& & \\|T_\\sigma^* g\\|_{L^2} \\lesssim  \\sum\\limits_{l, \\alpha} 2^{l n\/2} (\\alpha!)^{-1} \\left(\\int |g(y)|^2   \n \\|P_{l, \\alpha}^\\xi \\sigma(y, \\cdot)\\|_{L^2}^2 dy\\right)^{1\/2}\\lesssim \\\\\n & &\\lesssim \n \\sum\\limits_{l, \\alpha} 2^{l n\/2} (\\alpha!)^{-1} \\|g\\|_{L^2} \\sup\\limits_y \\|P_{l, \\alpha}^\\xi \\sigma(y, \\cdot)\\|_{L^2}. \n\\end{eqnarray*}\nFurthermore, \n$$\n\\sup_y \\|P_{l,\\alpha}^\\xi \\sigma(y,\\cdot)\\|_{L^2}\\leq C_n^{|\\alpha|} \n\\sup_y \\|P_{l}^\\xi \\sigma(y,\\cdot)\\|_{L^2}.\n$$\nPut everything together \n\\begin{eqnarray*}\n& & \\|T_\\sigma^* g\\|_{L^2}\\leq  C_n \n\\norm{g}{L^2}\\sum\\limits_{\\alpha} (\\alpha!)^{-1} C_n^{|\\alpha|} \n\\sum_l 2^{ln\/2} \n \\sup_y \\|P_{l}^\\xi \\sigma(y,\\cdot)\\|_{L^2} \n\\leq \\\\\n& & \\leq D_n \\norm{g}{L^2} \\sum_l 2^{ln\/2} \n \\sup_y \\|P_{l}^\\xi \\sigma(y,\\cdot)\\|_{L^2}, \n\\end{eqnarray*}\nas desired. \n\\subsection{$L^p$ estimates: $2<p\\leq \\infty$}\nThe result anounced in Theorem \\ref{theo:3} \nfollows by interpolation between the $L^2$ estimate just \nproved and the boundedness of $T_\\sigma:L^\\infty\\to L^\\infty$, \nwhich we need to show next. \n\nWe do  that by showing that the adjoint $T_\\sigma^*:L^1\\to L^1$.  \nThis is relatively  easy, since one can reduce to showing that \n\\begin{equation}\n\\label{eq:31}\n\\sup_{a\\in{\\mathbf R}^n} \\norm{T_\\sigma^* \\delta(a)}{L^1({\\mathbf R}^n)}<C\\sum_l 2^{ln\/2} \n \\sup_a \\|P_{l}^\\xi \\sigma(a,\\cdot)\\|_{L^2}.\n\\end{equation}\n\nThis is standard,  since one can embed $L^1$  into the space \nof all Borel measures $M({\\mathbf R}^n)$. The next observation is that \nby Krein-Milman's theorem, the convex combinations of \nthe set of Dirac masses $\\{\\delta_a: a\\in {\\mathbf R}^n\\}$  are weak* dense \nin the unit ball of $M({\\mathbf R}^n)$. \n\nFix $a\\in {\\mathbf R}^n$. We have \n\\begin{eqnarray*}\n& & T_\\sigma^* \\delta_a(x)= \\int e^{2\\pi i \\xi\\cdot(x-a)} \\sigma(a, \\xi) d\\xi=\\mathcal F_\\xi(\\sigma(a, \\cdot))(a-x).\n\\end{eqnarray*}\nwhere $\\mathcal F_\\xi$ signifies the Fourier transform in the $\\xi$ \nvariable. Denote $g_a(z)=\\mathcal F_\\xi(\\sigma(a, \\cdot))(z)$. \nWe have by Cauchy-Schwartz and the Plancherel's theorem \n\\begin{eqnarray*}\n& & \\|T_\\sigma^* \\delta_a\\|_{L^1}=\\int |g_a(a-x)| dx=\n\\sum\\limits_{l\\in \\mathcal Z} \\int_{|x-a|\\sim 2^l} |g_a(a-x)| dx\\leq  \\\\\n& &\\leq C_n \n\\sum\\limits_{l\\in \\mathcal Z} 2^{ln\/2} \n(\\int_{z: |z|\\sim 2^l} |g(z)|^2 dz)^{1\/2}\\leq  C_n \n\\sum\\limits_{l\\in \\mathcal Z} 2^{ln\/2} \\|P_l^\\xi \\sigma(a, \\cdot)\\|_{L^2}\n\\end{eqnarray*}\nThis is \\eqref{eq:31}, whence \\eqref{eq:22}.  \n\n\\subsection{$L^p$ estimates: $1<p\\leq 2$}\nWe take slightly different approach than in the case of $L^2$ estimates. \nNamely, we will show  that $T_\\sigma$ \nis of weak type $(1,1)$ operator, whence, by interpolation with the \n$L^2$ estimate, one gets the full range $1<p\\leq 2$. \nNote that the $L^2$ estimate comes with \n$$\n\\norm{T_\\sigma}{L^2\\to L^2}\\leq C_n \\sum\\limits_l \n2^{ln\/2} \\sup_y \\|P_l^\\xi \\sigma(y, \\cdot)\\|_{L^2}\n\\leq C_n \\sum\\limits_l  2^{l n} \\sup_y \\|P_l^\\xi \\sigma(y, \\cdot)\\|_{L^1}, \n$$\nwhere in the last inequality, we have used the Bernstein inequality \\\\\n$\\|P_l^\\xi \\sigma(a, \\cdot)\\|_{L^2}\\leq C_n 2^{l n\/2} \n\\|P_l^\\xi \\sigma(a, \\cdot)\\|_{L^1}$. Thus, it remains to show weak type $(1,1)$ bounds for $T_\\sigma$. \nWe proceed by performing a decomposition for $T_\\sigma$, \n inspired by the $L^2$ bounds. Our goal is to show the pointwise estimate \n\\begin{equation}\n\\label{eq:32}\n|T_\\sigma f(x)|\\leq C_n (\\sum\\limits_{l\\in \\mathcal Z} \n2^{l n}\\sup_y  \\|P_l^\\xi \\sigma(y, \\cdot)\\|_{L^1}) Mf(x),\n\\end{equation}\nwhich implies the desired weak type bounds since $M:L^1\\to L^{1, \\infty}$. \n\nTo achieve that, we have to be a bit more careful than in the $L^2$ case, \nsince the rough cutoffs in the $\\xi$ variable will be insuficient to \nshow \\eqref{eq:32}. \n\nFor any integer $l$, introduce smooth partition of unity, which is \nadapted to \nthe cover ${\\mathbf R}^n=\\cup_{Q: diam(Q)= 2^{-l}}Q$, that is a family of \nfunctions $\\{h_{l,Q}\\}$, with \n$supp\\  h_{l, Q}\\subset Q^*$ and \n$ |D^\\alpha h_{l, Q}(\\xi)|\\leq C_\\alpha 2^{l|\\alpha|}$ for \nevery multiindex $\\alpha$.  Choose and fix a family of arbitrary points $\\xi_Q\\in Q$. \nBy rescaling, \none can choose $h_{l, Q}:=\\psi_{l,Q}(2^{l}(\\xi-\\xi_Q))$, \nwhere $supp\\ \\psi_{l, Q}\\subset \\{|\\xi|<2\\}$ and  $ |D^\\beta_{\\eta}\n\\psi_{l, Q}(\\eta)|\\leq C_\\beta$ and \n$$\n\\sum\\limits_{Q}\\psi_{l, Q}(2^l(\\xi-\\xi_Q))=1\n$$ \nWrite \n\\begin{eqnarray*}\n& & T_\\sigma f(x)=\\sum\\limits_{l\\in \\mathcal Z} \n\\int\\limits_{{\\mathbf R}^n} P_l^\\xi \\sigma(x,\\xi) e^{2\\pi i \\xi x} \\hat{f}(\\xi) d\\xi= \\\\\n& &= \n\\sum\\limits_{l\\in \\mathcal Z}\\sum\\limits_Q \\int\\limits_{{\\mathbf R}^n} P_l^\\xi \\sigma(x,\\xi)\n\\psi_{l,Q}(2^l(\\xi-\\xi_Q)) \n e^{2\\pi i \\xi x} \\hat{f}(\\xi) d\\xi\n\\end{eqnarray*}\nWe now expand the $P_l^\\xi \\sigma(x,\\xi)$ around $\\xi_Q$. We have \n$$\nP_l^\\xi \\sigma(x, \\xi) \\psi_{l,Q}(2^l(\\xi-\\xi_Q))= \n(\\sum_{\\alpha: |\\alpha|\\geq 0}^\\infty\n\\f{D_\\xi^\\alpha P_l^\\xi \\sigma(x, \\xi_Q)}{\\alpha!} (\\xi-\\xi_Q)^\\alpha) \n\\psi_{l,Q}(2^l(\\xi-\\xi_Q))\n$$\nPlugging that in the formula for $T_\\sigma f$ yields \n\\begin{eqnarray*}\n& & T_\\sigma f(x)=\\sum\\limits_{l, \\alpha} (\\alpha!)^{-1}   \\sum\\limits_Q 2^{-l|\\alpha|}\nD_\\xi^\\alpha P_l^\\xi \\sigma(x, \\xi_Q)  \nZ_{l, Q}^{\\alpha} f(x). \n\\end{eqnarray*}\nwhere $\\widehat{Z_{l, Q}^{\\alpha} f}(\\xi)=\n\\psi_{l,Q} (2^l(\\xi-\\xi_Q)) (2^l(\\xi-\\xi_Q))^{\\alpha}\n \\hat{f}(\\xi)=\\psi^\\alpha_{l,Q} (2^l(\\xi-\\xi_Q))\\hat{f}(\\xi)$, i.e. \\\\\n$\\psi^\\alpha_{l,Q}(z)=\\psi_{l,Q}(z) z^\\alpha$. \nBy \\eqref{eq:30}, we get \n$$\n|Z_{l, Q}^{\\alpha} f(x)|\\leq C_n \\|\\widehat{\\psi^\\alpha_{l,Q}}\\|_{L^1} M f(x).\n$$\nBy the elementary properties of the Fourier transform \n\\begin{eqnarray*}\n& &\\|\\widehat{\\psi^\\alpha_{l,Q}}\\|_{L^1} = C_n^{|\\alpha|}  \\int |D^\\alpha_\\eta[\n \\widehat{\\psi_{l,Q}}(\\eta)]| d\\eta\\leq C_n^{|\\alpha|}  \\sum_{k=-\\infty}^\\infty  2^{k |\\alpha|} \n\\int |P_k^\\eta[ \\widehat{\\psi_{l,Q}}(\\eta)]| d\\eta\n\\end{eqnarray*}\nBut by support considerations, \n $P_k^\\eta [\\widehat{\\psi_{l,Q}}]=0$ if $k>3$. Also  since  $P_k:L^1\\to L^1$, we get \n$$\n\\|\\widehat{\\psi^\\alpha_{l,Q}}\\|_{L^1}\\leq C_n^{|\\alpha|} \n\\|P_k[\\widehat{\\psi_{l,Q}}]\\|_{L^1}\\leq  C_n^{|\\alpha|} \n\\|\\widehat{\\psi_{l,Q}}\\|_{L^1}\\leq \nC_n^{|\\alpha|}.\n$$\nThus, it remains to show for every $x$  and for {\\it any}\n$\\{\\xi_Q\\}, \\xi_Q\\in Q$\n\\begin{equation}\n\\label{eq:35} \n\\sum\\limits_\\alpha \\f{2^{-l|\\alpha|}}{\\alpha!} \\sum\\limits_Q \n|D_\\xi^\\alpha P_l^\\xi \\sigma(x, \\xi_Q)|  \\leq C_n \n2^{ln} \\sup_y \\|P_l^\\xi \\sigma(y, \\cdot)\\|_{L^1({\\mathbf R}^n)}. \n\\end{equation}\nThis is done similar to the $L^2$ case. By  \\eqref{eq:859} and by averaging over \nthe corresponding $Q$\n\\begin{eqnarray*}\n& & \\sum\\limits_\\alpha \\f{2^{-l|\\alpha|}}{\\alpha!} \\sum\\limits_Q \n|D_\\eta^\\alpha P_l^\\eta \\sigma(x, \\xi_Q)|\\leq \\sum\\limits_{\\alpha, \\beta} \\f{2^{-l|\\alpha|}}{\\alpha!\\beta!}  \n \\sum\\limits_Q |Q|^{-1} \\int_Q  |D_\\eta^{\\alpha+\\beta} P_l^\\eta \\sigma(x, \\eta)(\\eta-\\xi_Q)^\\beta|d\\eta\\\\\n & & \\lesssim 2^{ln}  \\sum\\limits_{\\alpha, \\beta} \\f{2^{-l|\\alpha|}}{\\alpha!\\beta!} \n \\int |D_\\eta^{\\alpha+\\beta} P_l^\\eta \\sigma(x, \\eta)(\\eta-\\xi_Q)^\\beta|d\\eta\\lesssim 2^{ln} \\|P_l^\\eta \\sigma(x,\n \\cdot)\\|_{L^1} \n\\end{eqnarray*}\n\n\n\n\n\\section{$L^p$ estimates for homogeneous of degree zero symbols}\nWe start with the $L^2$ estimate, since it is very similar to the corresponding \nestimate \\eqref{eq:7} and contains the main ideas for the $L^p$ estimate. \n\n\\subsection{$L^2$ estimates for homogeneous of degree zero symbols}\nConsider $T_\\sigma^*$ and introduce the Littlewood-Paley partition of unity $P_l^{\\xi\/|\\xi|}$. We have \n$$\nT_\\sigma^* g (x)=\\sum\\limits_{l=0}^\\infty \\int e^{2\\pi i \\xi\\cdot x} \n(\\int e^{-2\\pi i \\xi\\cdot y} \n[ g(y) P_l^{\\xi\/|\\xi|}  q(y, \\xi\/|\\xi|)] dy )d\\xi\n$$ \nFor every $l\\geq 0$, introduce a partition of unity on $\\mathbf S^{n-1}$, say $\\{K\\}$, which consists of disjoint \nsets of diameter comparable to $2^{-l}$. One may form $\\{ K\\}$ by introducing a $2^{-l}$ net on $\\mathbf S^{n-1}$, say $\\xi^m_{l}$,  form  the conic sets  $H_m^l=\\{\\xi\\in{\\mathbf R}^n:  \\: |\\xi\/|\\xi|-\\xi^m_l|\\leq 2^{-l}\\}$ and construct \\\\ $K^l_m=H_m^l\\setminus \\cup_{j=0}^{m-1} H_{j}^l$.   We have \n\\begin{equation}\n\\label{eq:400}\nT_\\sigma^* g (x)=\\sum\\limits_{l=0}^\\infty  \\sum\\limits_m \\int e^{2\\pi i \\xi\\cdot x} \\chi_{K^l_m}(\\xi)\n(\\int e^{-2\\pi i \\xi\\cdot y} \n[ g(y) P_l^{\\xi\/|\\xi|} q (y, \\xi\/|\\xi|)] dy )d\\xi\n\\end{equation}\nNow, that the symbol is frequency localized around frequencies $\\sim 2^l$ \n and the sets $K^l_m\\cap \\mathbf S^{n-1}$ have diameters less \nthan $2^{-l}$,  we expand $q(y, \\xi\/|\\xi|)$ around an {\\it arbitrary point} $\\theta_l^m\\in K_l^m$.\nAccording to \\eqref{eq:36}, we have for all $\\xi\\in K^l_m$, \n$$\nq(y, \\xi\/|\\xi|) \n=\\sum\\limits_{\\alpha\\geq 0} \\f{D_\\xi^\\alpha q(y,\\theta_l^m)}{\\alpha!}(\\xi\/|\\xi|-\\theta_l^m)^\\alpha.\n$$\nEntering this new expression in \\eqref{eq:400} yields \n\\begin{eqnarray*}\n& & \nT_\\sigma^* g (x)=\\sum\\limits_{l=0}^\\infty  \\sum\\limits_m \\sum\\limits_{\\alpha} (\\alpha!)^{-1} \n\\int e^{2\\pi i \\xi\\cdot x} \\chi_{K^l_m}(\\xi) (\\xi\/|\\xi|-\\theta_l^m)^\\alpha \\times \\\\\n& & \\times \n(\\int e^{-2\\pi i \\xi\\cdot y} \n[ g(y)  D_\\xi^\\alpha P_l^{\\xi\/|\\xi|} q (y, \\theta_l^m)] dy )d\\xi= \\\\\n& & =\n\\sum\\limits_{l=0}^\\infty  \\sum\\limits_m \\sum\\limits_{\\alpha} (\\alpha!)^{-1}  Z_{l,m}^\\alpha [g(\\cdot) 2^{-l|\\alpha|} \n D_\\xi^\\alpha P_l^{\\xi\/|\\xi|} q (\\cdot , \\theta_l^m)] ,\n\\end{eqnarray*}\nwhere $Z_{l,m}^\\alpha$ is given by the multiplier $\\chi_{K^l_m}(\\xi) 2^{l |\\alpha|} \n(\\xi\/|\\xi|-\\theta_l^m)^\\alpha$.  Note the disjoint support of the multipliers \n$\\{Z_{l,m}^\\alpha\\}_m$ and $\\|Z_{l, m}^\\alpha\\|_{L^2\\to L^2}=\\sup_\\xi|\\chi_{K^l_m}(\\xi) 2^{l |\\alpha|} \n(\\xi\/|\\xi|-\\theta_l^m)^\\alpha| \\leq 4^{|\\alpha|}$. \\\\\nTake $L^2$ norm of $T_\\sigma^* g$.   \n\\begin{eqnarray*}\n& & \\norm{T_\\sigma^* g}{L^2({\\mathbf R}^n)}\\lesssim \n\\sum\\limits_{l=0}^\\infty  \\sum\\limits_{\\alpha} (\\alpha!)^{-1}\\left(\\sum\\limits_m \n\\norm{Z_{l, m}^\\alpha   [g(\\cdot) 2^{-l|\\alpha|} \n D_\\xi^\\alpha P_l^{\\xi\/|\\xi|} q (\\cdot , \\theta_l^m) ]}{L^2}^2 \\right)^{1\/2}\\leq \\\\\n& &\\leq  4^{|\\alpha|} \\sum\\limits_{l=0}^\\infty \\sum\\limits_{\\alpha} (\\alpha!)^{-1} \\left(\\sum\\limits_m \\norm{g(\\cdot) 2^{-l|\\alpha|} \n D_\\xi^\\alpha P_l^{\\xi\/|\\xi|} q (\\cdot , \\theta_l^m) }{L^2}^2\\right)^{1\/2}.\n\\end{eqnarray*}\nWe proceed to further bound the expression in the  $m$ sum. Since\n$$\n\\sum\\limits_m \\norm{g(\\cdot) 2^{- l|\\alpha|} \n D_\\xi^\\alpha P_l^{\\xi\/|\\xi|} q (\\cdot , \\theta_l^m)}{L^2}^2=2^{-2 l|\\alpha|} \\int |g(y)|^2 \\left(\\sum\\limits_m \n| D_\\xi^\\alpha P_l^{\\xi\/|\\xi|} q (y , \\theta_l^m)|^2\\right) dy, \n$$\nmatters reduce to a good estimate for $\\sum\\limits_m \n| D_\\xi^\\alpha P_l^{\\xi\/|\\xi|} q (y , \\theta_l^m)|^2$. \nWe proceed as before. By \\eqref{eq:36}, we get for all $\\eta\\in K^l_m\\cap \\mathbf S^{n-1}$, \n\\begin{equation}\n\\label{eq:n3}\n  D_\\xi^\\alpha P_l^{\\xi\/|\\xi|} q(y, \\theta_l^m) \n=\\sum\\limits_{\\beta\\geq 0} \\f{D_\\xi^{\\alpha+\\beta} P_l^{\\eta\/|\\eta|}q(y,\\eta)}{\\beta!}(\\theta_l^m-\\eta)^\\beta.\n\\end{equation}\nAveraging over $K^l_m\\cap \\mathbf S^{n-1}$ and taking into account $|K^l_m\\cap \\mathbf S^{n-1}|\\sim 2^{l(n-1)}$  yields \n\\begin{eqnarray*}\n& & (\\sum\\limits_m \n| D_\\xi^\\alpha P_l^{\\xi\/|\\xi|} q (y , \\theta_l^m)|^2)^{1\/2} \\lesssim \\\\\n& & \\lesssim \\sum\\limits_{\\beta} \\f{2^{-l |\\beta|}}{\\beta!}\n(\\sum\\limits_m \n|K^l_m\\cap \\mathbf S^{n-1}|^{-1} \\int\\limits_{K^l_m\\cap \\mathbf S^{n-1}} |D_\\xi^{\\alpha+\\beta} P_l^{\\eta\/|\\eta|}q(y,\\eta)|^2\nd\\eta)^{1\/2}\\\\\n& &\\lesssim 2^{l(n-1)\/2} \\sum\\limits_{\\beta} \\f{2^{-l |\\beta|}}{\\beta!} \n\\|D_\\xi^{\\alpha+\\beta} P_l^{\\eta\/|\\eta|}q(y,\\cdot)\\|_{L^2} \\\\\n& &\\lesssim 2^{l[(n-1)\/2+|\\alpha|]} \n\\sum\\limits_{\\beta} \\f{C_n^{|\\alpha|+|\\beta|}}{\\beta!} \n\\|P_l^{\\eta\/|\\eta|}q(y,\\cdot)\\|_{L^2(\\mathbf S^{n-1})} \\\\ \n& &\\leq C_n^{|\\alpha|}2^{l[|\\alpha|+(n-1)\/2]}\n\\|P_l^{\\eta\/|\\eta|}q(y,\\cdot)\\|_{L^2(\\mathbf S^{n-1})}.\n\\end{eqnarray*}\nPutting this back into the estimate for $\\|T_\\sigma^* g\\|_{L^2({\\mathbf R}^n)}$ implies \n$$\n\\norm{T_\\sigma^* g}{L^2({\\mathbf R}^n)}\\lesssim \\|g\\|_{L^2} \\sum\\limits_l 2^{l(n-1)\/2}  \\sup\\limits_y \n\\|P_l^{\\eta\/|\\eta|}q(y,\\cdot)\\|_{L^2(\\mathbf S^{n-1})}\n$$\nas desired. \n\\subsection{$L^p$ estimates for homogeneous of degree zero multipliers}\nFix $p: 2\\leq p<\\leq \\infty$. \nTo verify the estimate $\\|T\\|_{B^0_{p,1}\\to L^p}$, it will suffice to fix $k$ and show \n\\begin{equation}\n\\label{eq:01}\n\\|T(P_k f)\\|_{L^p}\\leq C \\|f\\|_{L^p}.\n\\end{equation}\nFurthermore, by the scale invariance of the quantity  $\\sum_l 2^{l(n-1)} \n\\sup_y\\|P_l^{\\xi\/|\\xi|} q(y, \\cdot)\\|_{L^1(\\mathbf S^{n-1})}$  this is equivalent to  verifying \\eqref{eq:01} \nonly for  $k=0$. \nThat is, it suffices to establish \nthe $L^p, p\\geq 2$ boundedness of the  operator   \n$$\nG f(x)=\\int\\limits_{{\\mathbf R}^n} q(x,\\xi\/|\\xi|) e^{2\\pi i \\xi x} \\varphi(|\\xi|) \\hat{f}(\\xi) d\\xi.\n$$\nprovided the multiplier  $m$ satisfies \n$\\sum\\limits_l 2^{l(n-1)} \\sup_y\\|P_l^{\\xi\/|\\xi|} q(y, \\cdot)\\|_{L^1(\\mathbf S^{n-1})}<\\infty$. \n\nNext, we make the angular decomposition as in the case of the $L^2$ estimates for the adjoint \noperator $G^*$.However, this time we will have to be more careful and \ninstead of the rough cutoffs $\\chi_{K^l_m}$, we shall use a \nsmoothed out versions of them. Fix $l$.  Choose and fix a \n $2^{-l}$ net  $\\theta_m^l\\in K_m^l\\cap \\mathbf S^{n-1}$, so that the family \n$\\{\\theta\\in \\mathbf S^{n-1}: \n|\\theta_m^l-\\theta|\\leq 2^{-l}\\}_m$ has the finite intersection\nproperty. \nIntroduce a family of functions $\\varphi_{l,m}:{\\mathbf R}^n\\to [0,1]$, so that for every $\\xi\\in {\\mathbf R}^n$, \n\\begin{eqnarray}\n\\label{eq:fun}\n& & \\sum\\limits_m \\varphi_{l,m} (2^l(\\xi\/|\\xi|-\\theta_m^l))=1 \\\\ \n\\nonumber\n& & \\sup_\\eta |D^\\beta_\\eta \\varphi_{l,m} (\\eta)|\\leq C_\\beta.\n\\end{eqnarray}\nIn other words, the family of functions $\\{\\varphi_{l,m}\\}$ provides a \nsmooth partition of unity, subordinated to the cover $\\{K_m^l\\}$. \n\nAs before, write \n$$\nG^* g(x)=\\sum\\limits_{l\\geq 0} \\int\\limits_{{\\mathbf R}^n} e^{2\\pi i \\xi x}  \\varphi(|\\xi|)  \\int  e^{-2\\pi i \\xi y} [g(y) \nP_l^{\\xi\/|\\xi|} q(y, \\xi\/|\\xi|)]dy  d\\xi.\n$$\nInserting the partition of unity discussed above   into the ($l^{th}$ term of the) \nlast formula for $G^*$ yields \n$$\nG^* g (x)=\\sum\\limits_{l\\geq 0} \\sum\\limits_m  \\int  e^{2\\pi i \\xi (x-y)} \n\\varphi_{l,m} (2^l(\\xi\/|\\xi|-\\theta_m^l)) \\varphi(|\\xi|)  [g(y) \nP_l^{\\xi\/|\\xi|} q(y, \\xi\/|\\xi|)]dy d\\xi.  \n$$\nFollowing the same strategy as before, we expand $q(y, \\xi\/|\\xi|)$ around  $\\theta_m^l\\in K_m^l$.\nAccording to \\eqref{eq:36}, we have \n$$\nP_l^{\\xi\/|\\xi|} q(y, \\xi\/|\\xi|) \n=\\sum\\limits_{\\alpha\\geq 0} \\f{D_\\xi^\\alpha P_l^{\\xi\/|\\xi|} \nq(y, \\theta_m^l)  }{\\alpha!}(\\xi\/|\\xi|-\\theta_m^l)^\\alpha.\n$$\nOf course,  the last formula is useful only when \n$|\\xi\/|\\xi|-\\theta_m^l|\\lesssim  2^{-l}$, in particular on the \nsupport of $\\varphi_{l,m} (2^l(\\xi\/|\\xi|-\\xi_m^l)) $. \nThis gives us the representation \n\\begin{eqnarray*}\n& & G^* g =\\sum\\limits_{l\\geq 0}\\sum\\limits_m   \\sum\\limits_{|\\alpha|\\geq 0} (\\alpha!)^{-1}\n\\int   e^{2\\pi i \\xi x} \n\\varphi_{l,m} (2^l(\\xi\/|\\xi|-\\theta_m^l)) (\\xi\/|\\xi|-\\theta_m^l)^\\alpha \\varphi(|\\xi|) \\times \\\\\n& &\\times \\int e^{2\\pi i \\xi y} g(y) \nP_l^{\\xi\/|\\xi|} D_\\xi^\\alpha P_l^{\\xi\/|\\xi|}  q(y, \\theta_m^l)dy d\\xi= \\\\\n& & =\n\\sum\\limits_{l\\geq 0}\\sum\\limits_m   \\sum\\limits_{|\\alpha|\\geq 0}(\\alpha!)^{-1} Z_{l,m}^\\alpha \n[g(\\cdot)  2^{-l|\\alpha|} D_\\xi^\\alpha  P_l^{\\xi\/|\\xi|}  q(\\cdot, \\theta_m^l)]\n\\end{eqnarray*}\nwhere \n$$\n\\widehat{Z_{l,m}^\\alpha f}(\\xi)=  \\varphi_{l,m} (2^l(\\xi\/|\\xi|-\\theta_m^l))  2^{l|\\alpha|} \n(\\xi\/|\\xi|-\\theta_m^l)^\\alpha \n\\varphi(|\\xi|) \\hat{f}(\\xi)=\\varphi_{l,m}^\\alpha (\\xi\/|\\xi|-\\theta_m^l)\\varphi(|\\xi|) \\hat{f}(\\xi).\n$$\nTaking $L^p$ norm of $G^* g$, we get \n\\begin{eqnarray*}\n& & \\|G^* g\\|_{L^p}\\leq \\sum\\limits_{l\\geq 0} \\sum\\limits_{|\\alpha|\\geq 0}(\\alpha!)^{-1}\\|\\sum\\limits_m Z_{l,m}^\\alpha \n[g(\\cdot)  2^{-l|\\alpha|} D_\\xi^\\alpha  P_l^{\\xi\/|\\xi|}  q(\\cdot, \\theta_m^l)]\\|_{L^p}\n\\end{eqnarray*}\nLemma \\ref{le:sum} in the Appendix allows us to treat   expressions  of the type \n$\\|\\sum\\limits_m  Z_{l,m}^\\alpha g_m^\\alpha\\|_{L^p}$. Indeed, according to \\eqref{eq:n1}, we have \n\\begin{eqnarray*}\n& & \\|\\sum\\limits_m Z_{l,m}^\\alpha \n[g(\\cdot)  2^{-l|\\alpha|} D_\\xi^\\alpha  P_l^{\\xi\/|\\xi|} q(\\cdot, \\theta_m^l)]\\|_{L^p}\\lesssim  (\\sum\\limits_m  \n\\|g(\\cdot)  2^{-l|\\alpha|} D_\\xi^\\alpha  P_l^{\\xi\/|\\xi|}  q(\\cdot, \\theta_m^l)\\|_{L^p}^p)^{1\/p} \\\\\n& &= 2^{-l|\\alpha|} (\\int |g(y)|^p (\\sum\\limits_m  \n   |D_\\xi^\\alpha  P_l^{\\xi\/|\\xi|}  q(y, \\theta_m^l)|^p) dy)^{1\/p} \n\\end{eqnarray*}\nBy virtue of  \\eqref{eq:n3}, we get \n\\begin{eqnarray*}\n& & \nD_\\xi^\\alpha P_l^{\\xi\/|\\xi|} q(y, \\theta_l^m) \n=\\sum\\limits_{\\beta\\geq 0} \\f{D_\\xi^{\\alpha+\\beta} P_l^{\\eta\/|\\eta|}q(y,\\eta)}{\\beta!}(\\theta_l^m-\\eta)^\\beta.\n\\end{eqnarray*}\nwhence by averaging\\footnote{this step is identical to the one performed earlier for the \n$L^2$ bounds, except that now the $l^2$ sums are replaced by $l^p$ sums.} over $K^l_m\\cap \\mathbf S^{n-1}$, \n\\begin{eqnarray*}\n & & (\\sum\\limits_m \n| D_\\xi^\\alpha P_l^{\\xi\/|\\xi|} q (y , \\theta_l^m)|^p)^{1\/p} \\lesssim \\\\\n& & \\lesssim \\sum\\limits_{\\beta} \\f{2^{-l |\\beta|}}{\\beta!}\n(\\sum\\limits_m \n|K^l_m\\cap \\mathbf S^{n-1}|^{-1} \\int\\limits_{K^l_m\\cap \\mathbf S^{n-1}} |D_\\xi^{\\alpha+\\beta} P_l^{\\eta\/|\\eta|}q(y,\\eta)|^p\nd\\eta)^{1\/p}\\\\\n& &\\lesssim 2^{l(n-1)\/p} \\sum\\limits_{\\beta} \\f{2^{-l |\\beta|}}{\\beta!} \n\\|D_\\xi^{\\alpha+\\beta} P_l^{\\eta\/|\\eta|}q(y,\\cdot)\\|_{L^p(\\mathbf S^{n-1})} \\\\\n& &\\lesssim 2^{l[(n-1)\/p+|\\alpha|]} \n\\sum\\limits_{\\beta} \\f{C_n^{|\\alpha|+|\\beta|}}{\\beta!} \n\\|P_l^{\\eta\/|\\eta|}q(y,\\cdot)\\|_{L^p(\\mathbf S^{n-1})} \\\\ \n& &\\leq C_n^{|\\alpha|}2^{l[|\\alpha|+(n-1)\/p]}\n\\|P_l^{\\eta\/|\\eta|}q(y,\\cdot)\\|_{L^p(\\mathbf S^{n-1})}.\n\\end{eqnarray*}\nAll in all, \n\\begin{eqnarray*}\n& & \\|G^* g\\|_{L^p}\\leq C_n \\|g\\|_{L^p}\\sum\\limits_{l\\geq 0} \\sum\\limits_{|\\alpha|\\geq 0}(\\alpha!)^{-1} \n C_n^{|\\alpha|}2^{l(n-1)\/p}\n\\sup\\limits_y \\|P_l^{\\eta\/|\\eta|}q(y,\\cdot)\\|_{L^p(\\mathbf S^{n-1})}\\\\ \n& &\\leq C_n \\|g\\|_{L^p} \\sum\\limits_l \n2^{l(n-1)\/p}\n\\sup\\limits_y \\|P_l^{\\eta\/|\\eta|}q(y,\\cdot)\\|_{L^p(\\mathbf S^{n-1})}.\n\\end{eqnarray*}\nas desired. \n\n\\section{Counterexamples}\n\\label{sec:counter}\n\\subsection{Theorem \\ref{theo:1} is sharp.}\nGiven $p>2$, we will construct \n an explicit symbol $\\sigma(x, \\xi)$, \nso that the corresponding PDO $T_\\sigma$  is not bounded on $L^2({\\mathbf R}^n)$, \nbut which satisfies \\\\\n$\\sup_x |D_\\xi^\\alpha \\sigma(x, \\xi)|\\leq C_\\alpha |\\xi|^{-|\\alpha|}$ and \n$\\sup_x \\norm{\\sigma(x, \\cdot)}{W^{p,n\/p}}<\\infty$. \nThe construction is a minor modification of the \nstandard example of a symbol in $S^0_{1, 1}$, which \nis not bounded on $L^2$, see for example \\cite{Stein}, page 272. \nWe carry out the construction in $n=1$, \nbut this can be easily generalized to higher dimensions.\n\nFor the given $p>2$,  fix small $0<\\delta<1\/2$, \nso that\\footnote{The reason for this choice of $\\delta$ will become apparent in the proof below.}\n $2+4\\delta\/(1-2\\delta)<p$. \nDefine \n$$\n\\sigma(x, \\xi):=\\sum\\limits_{j=8}^\\infty \\f{e^{-2\\pi i 2^j x}}{j^{1\/2-\\delta}}\n \\varphi(2^{-j} \\xi),\n$$\nwhere the function $\\varphi$ is $C^\\infty$, -supported \n in  $1\/2\\leq |\\xi|\\leq 3\/2$, and $\\varphi(\\eta)=1$ for all $3\/4\\leq|\\eta|\\leq 5\/4$.  \\\\\nTo show the unboundedness of $T_\\sigma$ on $L^2$, \nlet us test it against the function \n$$\nf_N(x)=\\sum\\limits_{j=8}^N \\f{e^{-2\\pi i 2^j x}}{j^{1\/2+\\delta}} f_0(x),\n$$\nwhere $f_0$ is a Schwartz function, whose Fourier transform is \nsupported in $\\{\\xi: |\\xi|\\leq 1\/10\\}$. Clearly \n$\\norm{f_N}{L^2}=(\\sum_{j=8}^N j^{1+2\\delta})^{1\/2}\\norm{f_0}{L^2}\n\\leq (\\sum_{j=1}^\\infty j^{1+2\\delta})^{1\/2}\\norm{f_0}{L^2}=C_\\delta\\norm{f_0}{L^2}$, while  \n\\begin{eqnarray*}\n& & T_\\sigma f_N(x)=\\sum_{j_1\\geq 8, N\\geq j_2\\geq 8} \n\\int \\f{e^{-2\\pi i 2^{j_1} x}}{j_1^{1\/2-\\delta}} \\f{\\varphi(2^{-j_1} \n\\xi)}{j_2^{1\/2+\\delta}} \\hat{f_0}(\\xi-2^{j_2}) e^{2\\pi i \\xi x} d\\xi.\n\\end{eqnarray*}\nClearly, by Fourier support considerations the \nterms $j_1\\neq j_2$ disappear and we get \n$$\nT_\\sigma f_N(x)=(\\sum_{N\\geq j\\geq 8} j^{-1}) f_0(x),\n$$\nwhence $\\norm{T_\\sigma}{L^2\\to L^2}\\gtrsim \\ln(N)$, whence $T_\\sigma$ \n{\\it is not bounded} on $L^2$. \\\\ On the other hand, it is clear that for $|\\xi|>1$, \n$$\n\\sup_x |D_\\xi^\\alpha \\sigma(x, \\xi)|\\sim |\\xi|^{-|\\alpha|} \\ln^{\\delta-1\/2}(|\\xi|) \n\\leq |\\xi|^{-|\\alpha|}.\n$$\nFinally, to \nestimate $\\sup_x \\norm{\\sigma(x, \\cdot)}{W^{p,1\/p}}$, write \n$$\n\\sigma=\\sum\\limits_{s=3}^\\infty \\sum\\limits_{j=2^s}^{2^{s+1}} \\f{e^{-2\\pi i 2^j x}}{j^{1\/2-\\delta}}\n \\varphi(2^{-j} \\xi)=:\\sum\\limits_{s=3}^\\infty \\sigma^s,\n$$\nBy the convexity of the norms, we have with $\\theta: 1\/p=\\theta\/2$, \n$$\n\\norm{\\sigma^s(x, \\cdot)}{W^{p,1\/p}}\\leq \n\\norm{\\sigma^s(x, \\cdot)}{H^{1\/2}}^\\theta\n\\norm{\\sigma^s(x, \\cdot)}{L^\\infty}^{(1-\\theta)}\n$$\nIt is now easy to compute the norms on the right hand side. We have \n$$\n\\sup_x \\norm{\\sigma^s(x, \\cdot)}{H^{1\/2}} \n\\sim (\\sum_{j=2^{s}}^{2^{s+1}} \\f{1}{j^{1-2\\delta}})^{1\/2}\n\\sim 2^{\\delta s}.\n$$\nOn the other hand, \n$$\n\\norm{\\sigma^s(x, \\cdot)}{L^\\infty}\\sim  2^{-s(1\/2-\\delta)},\n$$\nwhence $\\sup_x \\norm{\\sigma^s(x, \\cdot)}{W^{p,1\/p}}\\leq \n2^{s(\\delta\\theta-(1\/2-\\delta)(1-\\theta))}$. Clearly, \nsuch an expression dyadically sums in $s\\geq 3$, \nprovided $\\delta\\theta<(1\/2-\\delta)(1-\\theta)$ or equivalently $p>2+4\\delta\/(1-2\\delta)$.\n\\subsection{Proposition \\ref{prop:5}: Theorem \\ref{theo:4} is sharp}\n\\begin{proof}(Proposition \\ref{prop:5}) \nWe construct a sequence of symbols $\\sigma_\\delta:\\mathbf R^2\\times \\mathbf R^2\\to\\mathbf R^1$, \nso that for a fixed Schwartz function $f$\n$$\n\\lim_{\\delta\\to 0+} |T_{\\sigma_\\delta} f|=|H_* f(x)|=\\sup_{u\\in \\mathbf S^1} |H_u f(x)|\n$$\nSince we already know, \\cite{LL}, that \n$H_*$ is {\\it unbounded} on $L^2(\\mathbf R^2)$, \nwe should have\n\\begin{equation}\n\\label{eq:715}\n\\limsup_{\\delta\\to 0+} \\|T_{\\sigma_\\delta}\\|_{L^2\\to L^2}=\\infty. \n\\end{equation}\nIn our construction \n$\\sigma_\\delta$ will depend on $f$, but it is still clear \nthat one can achieve \\eqref{eq:715}. \nNamely, take a sequence $f_N: \\norm{f_N}{L^2(\\mathbf R^2)}=1$, \nso that $\\|H_* (f_N)\\|_{L^2(\\mathbf R^2)}\\geq N$. \nThen construct $\\sigma_{N, \\delta}$, so that $\\lim_{\\delta\\to 0+} \n|T_{\\sigma_{\\delta, N}} f_N|=\nH_* f_N$. Then clearly, \\\\\n$\\limsup_{ N\\to \\infty, \\delta\\to 0+} \n\\|T_{\\sigma_{N,\\delta}}\\|_{L^2\\to L^2}=\\infty$. \n\nNow, from the $L^2$ boundedness results of \nTheorem \\ref{theo:4} (or rather the lack thereof), we must have \n\\begin{equation}\n\\label{eq:713}\n\\limsup_{\\delta\\to 0} \\sum\\limits_l 2^{l\/2} \n\\sup_x \\|P_l^{\\xi\/|\\xi|} \\sigma_\\delta(x, \\cdot)\\|_{L^2(\\mathbf S^1)}=\\infty.\n\\end{equation}\nOn the other hand, we will see that \n$\\sup_{x, \\xi, \\delta}|\\sigma_\\delta(x, \\xi)|\\leq 1$ and \n\\begin{equation}\n\\label{eq:714}\n\\sup_{\\delta, x} \\|\\sigma_\\delta(x, \\cdot)\\|_{W^{1, 1}(\\mathbf S^1)}<\\infty. \n\\end{equation}\nNote in contrast that (at least heuristically)  \\eqref{eq:713} states \n$$\n\\limsup_{\\delta\\to 0} \\sup_x \\|\\sigma_\\delta(x, \\cdot)\\|_{B^{1\/2}_{2, 1}}=\\infty\n$$\nand by the Sobolev embedding estimate on the sphere \n\\eqref{eq:bern} (and up  to the usual  Besov spaces \nadjustments at the endpoints), one should have that the quantity \nin \\eqref{eq:714} (at least in principle) controls \\eqref{eq:713}. Having both \n\\eqref{eq:713} and \\eqref{eq:714} for a concrete example suggests that \nthe conditions imposed in Theorem \\ref{theo:4} \n are extremely tight. \n\nLet us now describe the construction of $\\sigma_\\delta$. First of all, \n\\begin{eqnarray*}\nH_*f(x) = \\sup_{u\\in\\mathbf S^1} |H_u f(x)| &=& \\sup_{u\\in\\mathbf S^1}|\\int \nsgn(u\\cdot \\xi\/|\\xi|) \\hat{f}(\\xi) e^{2\\pi i \\xi\\cdot x} d\\xi|=\\\\\n&=&\n|\\int \nsgn(u(x)\\cdot \\xi\/|\\xi|) \\hat{f}(\\xi) e^{2\\pi i \\xi\\cdot x} d\\xi|,\n\\end{eqnarray*}\nfor some measurable function $u(x):\\mathbf R^1\\to \\mathbf S^1$. \nClearly $u(x)$ will depend on the function $f$, see the remarks above after \\eqref{eq:715}. \\\\\nIntroduce a function $\\psi:\\psi\\in C^\\infty, -1\\leq \\psi(x)\\leq 1$, and \nso that $\\psi(z)=-1: z\\in (-\\infty, -1]$, $\\psi(z)=1: \nz\\in [1, \\infty)$. Clearly \n$$\nH_* f(x)=\\lim_{\\delta\\to 0+} T_{\\sigma_\\delta} f(x)= \\lim_{\\delta\\to 0+}|\\int \n\\psi\\left(\\f{u(x)\\cdot \\xi\/|\\xi|}{\\delta}\\right) \n\\hat{f}(\\xi) e^{2\\pi i \\xi\\cdot x} d\\xi|, \n$$\nthat is $\\sigma_\\delta(x, \\xi\/|\\xi|)=\\psi(\\f{u(x)\\cdot \\xi\/|\\xi|}{\\delta})$, \nfor which we will verify \\eqref{eq:714}, while it \nis clearly bounded in absolute value by one. \n\nWe pause \nfor a second to comment on the particular form of $T_{\\sigma_\\delta}$. Note that the function $u(x)$ \nin general will not be smooth\\footnote{Note that under some extra smoothness assumptions on $u$, \nLacey and Li have managed to prove $L^2$ boundedness!} and therefore will not fall under \nthe scope of any standard boundedness theory for PDO. Also, note that while the map \n $\\xi\\to \\sigma_\\delta(x, \\xi)$ is definitely smooth, its  derivatives are quite large and \nblow up at the important limit $\\delta\\to 0$.  This shows that in order to \ntreat maximal operators, build upon singular multipliers (as is the case here), \none needs the full strength  of Theorems \\ref{theo:1}, \\ref{theo:4} and beyond.   \n\nGoing back to the proof of \\eqref{eq:714}, compute \n\\begin{eqnarray*}\n& & \\f{\\partial \\sigma}{\\partial_{\\xi_1}}= \n\\f{\\psi'(\\f{u(x)\\cdot \\xi\/|\\xi|}{\\delta})}{\\delta|\\xi|^3}\\left(u_1(x)\\xi_2^2- u_2(x)\\xi_1\\xi_2\\right)= \n\\f{\\psi'(\\f{u(x)\\cdot \\xi\/|\\xi|}{\\delta})\\xi_2}{\\delta|\\xi|^2}u(x)\\cdot \n(\\xi\/|\\xi|)^{\\perp} \\\\\n& & \\f{\\partial \\sigma}{\\partial_{\\xi_2}}=\n\\f{\\psi'(\\f{u(x)\\cdot \\xi\/|\\xi|}{\\delta})}{\\delta|\\xi|^3}\n\\left(u_1(x)\\xi_1^2- u_2(x)\\xi_1\\xi_2\\right)= \\f{\\psi'(\\f{u(x)\\cdot \n\\xi\/|\\xi|}{\\delta})\\xi_1}{\\delta|\\xi|^2} u(x)\\cdot \n(\\xi\/|\\xi|)^{\\perp} \n\\end{eqnarray*}\nClearly, the supports of both derivatives are in \n$\\xi: |u(x)\\cdot \\xi\/|\\xi||\\leq \\delta<<1$. Also, on their support, \n$|\\nabla \\sigma(x, \\xi\/|\\xi|)|\\sim C \\delta^{-1}$. It follows \n$$\n\\|\\sigma_\\delta(x, \\cdot)\\|_{W^{1, 1}(\\mathbf S^1)}\\leq \n\\int_{\\xi\\in \\mathbf S^1: |u(x)\\cdot \\xi\/|\\xi|\\leq \\delta}\n|\\nabla \\sigma_\\delta(x, \\xi)|d\\xi\\leq C,\n$$\nwhere $C$ is independent of $\\delta$. This was the claim  in \\eqref{eq:714}. \n\\end{proof}\n\n\n\n\n\n\\section{Appendix}\n\n\\subsection{Estimates for Fourier transforms of functions supported on small spherical caps.}\nIn this section, we present a pointwise estimate for the  kernels of \nmultipliers that   restrict the Fourier transform to a small spherical cap. \n\\begin{lemma}\n\\label{le:900}\nLet $\\theta\\in \\mathbf S^{n-1}$  and \n$\\varphi$ is a $C^\\infty$ function with $supp \\ \\varphi\\subset \\{\\xi: 1\/2\\leq |\\xi|\\leq 2\\}$. \nLet also $l>0$ be any integer. Define $K_{l, \\theta}$ to be the inverse \nFourier transform of $\\varphi(2^l(\\xi\/|\\xi|-\\theta))\\varphi(|\\xi|)$, that is \n$$\nK_{l, \\theta} (x)=\\int \\varphi(2^l(\\xi\/|\\xi|-\\theta))\\varphi(|\\xi|) e^{2\\pi i x\\cdot \\xi} d\\xi.\n$$\nThen, for every $N>0$, there exists $C_N$, so that \n\\begin{equation}\n\\label{eq:201}\n|K_{l, \\theta}(x)|\\leq C_N 2^{-l(n-1)}(1+|\\dpr{x}{\\theta}|)^{-N} \n(1+2^{-l}|x-\\dpr{x}{\\theta}\\theta|)^{-N}.\n\\end{equation}\nThat is, in the direction of $\\theta$, the function has any polynomial decay, while in the directions transversal to $\\theta$, one has decay like $(2^{-l}<x'>)^{-N}$, where $x=\\dpr{x}{\\theta}\\theta+x'$. \nIn particular, \n\\begin{equation}\n\\label{eq:202}\n\\sup_{\\theta, l} \\int |K_{l, \\theta}(x)| dx\\leq C_n<\\infty,\n\\end{equation}\nwhere the constant $C_n$ depends on $\\|\\varphi\\|_{L^\\infty}$ and \nthe smoothness properties of $\\varphi$. \n\\end{lemma} \n\\begin{proof}\nBy rotation symmetry, we can assume without loss of generality that $\\theta=e_1=(1, 0, \\ldots, 0)$. \nFix  $l$ and drop the subindices for notational convenience. \nWe will need to show that for every \n$x=x_1 e_1+x'$, \n\\begin{equation}\n\\label{eq:203}\n|K(x)|\\leq C_N 2^{-l(n-1)}  <x_1>^{-N} <2^{-l} x'>^{-N}\n\\end{equation}\nFirst of all,  by support considerations, one has \n$|K(x)|\\leq C_n 2^{-l(n-1)}.$ \\\\\nNext, we will show that integration by parts in the variable $\\xi_1$ \nyields\n\\begin{equation}\n\\label{eq:700} \nK(x)= x_1^{-1} \\tilde{K}(x),\n\\end{equation}\nwhereas  integration by parts in each of the variables  $\\xi_j: j=2, \\ldots, n$ yield \n\\begin{equation}\n\\label{eq:701} \nK(x)=(2^{-l} x_j)^{-1} \\tilde{K}(x),\n\\end{equation}\nwhere $\\tilde{K}(x)$ is different in each instance, but it has the form \n$$\n\\tilde{K}(x)=\\int \\varphi_1(2^l(\\xi\/|\\xi|-e_1))\\varphi_2(|\\xi|) e^{2\\pi i x\\cdot \\xi} d\\xi.\n$$\nfor some $C^\\infty$ \nfunctions\\footnote{As we shall see the functions $\\varphi_1, \\varphi_2$ are obtained in a specific way \nfrom $\\varphi$ via the operations \ndifferentiation and  multiplication by monomial.}  \n$\\varphi_1, \\varphi_2$ with $supp \\ \\varphi_{k}\\subset \\{\\xi: 1\/2\\leq |\\xi|\\leq 2\\}, k=1,2$. \n\nThat is enough to deduce \\eqref{eq:203} and thus Lemma \\ref{le:900}. \nIndeed, by iterating  \\eqref{eq:700} and  \\eqref{eq:701}, one  gets the formula \n$$\nK(x)= x_1^{-N_1} (\\prod_{j=2}^n (2^{-l} x_j)^{N_j})^{-1} \\tilde{K}_{N_1, \\ldots, N_n}(x),\n$$ \nfor any $n$ tuple of integers $(N_1, N_2, \\ldots, N_n)$. Combining this representation with the \nestimate $|\\tilde{K}_{N_1, \\ldots, N_n}(x)|\\leq C_{n, N_1, \\ldots, N_n} 2^{-l(n-1)}$, one deduces \\eqref{eq:203}. \n\nFor \\eqref{eq:700}, integration by parts yields \n\\begin{eqnarray*}\n& & K(x)=-\\f{1}{2\\pi i x_1} \\int \\partial_{\\xi_1} [\\varphi(2^l(\\xi\/|\\xi|-e_1))\\varphi(|\\xi|)] \ne^{2\\pi i x\\cdot \\xi} d\\xi = \\\\\n& & =\n-\\f{1}{2\\pi i x_1}\\int  \\sum_{j=2}^n 2^l \\f{\\xi_j^2}{|\\xi|^2}\n \\partial_1\\varphi(2^l(\\xi\/|\\xi|-e_1)) \n\\varphi(|\\xi|)|\\xi|^{-1} e^{2\\pi i x\\cdot \\xi} d\\xi+\\\\\n& & \n+\\f{1}{2\\pi i x_1} \\int \\sum_{j=2}^n 2^l \\f{\\xi_j\\xi_1}{|\\xi|^2} \n\\partial_j \\varphi(2^l(\\xi\/|\\xi|-e_1))\\varphi(|\\xi|)|\\xi|^{-1} e^{2\\pi i x\\cdot \\xi} d\\xi+\\\\\n& &- \\f{1}{2\\pi i x_1}\\int \\varphi(2^l(\\xi\/|\\xi|-e_1)) \\varphi'(|\\xi|)\\f{\\xi_1}{|\\xi|} \ne^{2\\pi i x\\cdot \\xi} d\\xi\n\\end{eqnarray*}\nThe third term is clearly in the form $x_1^{-1} \\tilde{K}(x)$, by taking into account that $supp \\varphi\\subset \\{\\xi: 1\/2\\leq |\\xi|\\leq 2\\}$. \\\\\nThe second term above can be rewritten in the form \n\\begin{eqnarray*}\n& &\n\\f{1}{2\\pi i x_1} \\int \n\\varphi_1(2^l(\\xi\/|\\xi|-e_1))\\f{\\xi_1}{|\\xi|}\\varphi(|\\xi|)|\\xi|^{-1} e^{2\\pi i x\\cdot \\xi} d\\xi=\\\\\n& & = \\f{1}{2\\pi i x_1}\\int \n[\\varphi_1(2^l(\\xi\/|\\xi|-e_1))+2^{-l} \n\\tilde{\\varphi}_{1}(2^l(\\xi\/|\\xi|-e_1))]\\varphi(|\\xi|)|\\xi|^{-1} e^{2\\pi i x\\cdot \\xi} d\\xi=\\\\\n& & = x_1^{-1} \\tilde{K}(x),\n\\end{eqnarray*}\nwhere $\\varphi_1(\\eta)=\\sum_{j=2}^n \\eta_j \\partial_{\\eta_j}\\varphi(\\eta)$ and \n$\\tilde{\\varphi}_1(\\eta)=\\eta_1\\varphi_1(\\eta)$. \\\\\nAnalogously, one can rewrite the first term of $K(x)$ in the form\n$2^{-l} x_1^{-1} \\tilde{K}(x)$, i.e. it has an extra decay factor of $2^{-l}$. \nThis establishes \\eqref{eq:700}. \n\nFor \\eqref{eq:701}, we obtain by integration by parts in $\\xi_j, 2\\leq j\\leq n$, \n\\begin{eqnarray*}\n& & K(x)=\\f{1}{2\\pi i x_j} \\int \\partial_j\\varphi(2^l(\\xi\/|\\xi|-e_1))\\f{2^l \\xi_1 \\xi_j}{|\\xi|^3} \\varphi(|\\xi|) \ne^{2\\pi i x\\cdot \\xi} d\\xi + \\\\\n& &+ \\f{1}{2\\pi i x_j} \\sum\\limits_{k\\neq j, k=2}^n \n\\int \\partial_k\\varphi(2^l(\\xi\/|\\xi|-e_1))\\f{2^l \\xi_k \\xi_j}{|\\xi|^3} \\varphi(|\\xi|) \ne^{2\\pi i x\\cdot \\xi} d\\xi+\\\\\n& &- \\f{1}{2\\pi i x_j} \\int \\partial_j \\varphi(2^l(\\xi\/|\\xi|-e_1)) 2^l(\\sum_{k\\neq j, k=1}^n \n\\f{\\xi_k^2}{|\\xi|^2}) \\varphi(|\\xi|) e^{2\\pi i x\\cdot \\xi} d\\xi + \\\\\n& &- \\f{1}{2\\pi i x_j}\\int \\varphi(2^l(\\xi\/|\\xi|-e_1)) \\varphi'(|\\xi|)\\f{\\xi_j}{|\\xi|} \ne^{2\\pi i x\\cdot \\xi} d\\xi\n\\end{eqnarray*}\nBy performing similar analysis as in the proof of \\eqref{eq:700}, we easily see that the first term above \nis in the form $x_j^{-1} \\tilde{K}(x)$, the second and the fourth terms are  in fact even better, since \nthey are in the form $2^{-l}x_j^{-1} \\tilde{K}(x)$. The third term has two types of terms. Clearly, \n$$\n\\f{1}{2\\pi i x_j} \\int \\partial_j \\varphi(2^l(\\xi\/|\\xi|-e_1)) 2^l(\\sum_{k\\neq j, k=2}^n \n\\f{\\xi_k^2}{|\\xi|^2}) \\varphi(|\\xi|) e^{2\\pi i x\\cdot \\xi} d\\xi +\n$$\nis of the form $2^{-l}x_j^{-1} \\tilde{K}(x)$, while lastly, \n$$\n\\f{1}{2\\pi i x_j} \\int \\partial_j \\varphi(2^l(\\xi\/|\\xi|-e_1)) \n2^l \\f{\\xi_1^2}{|\\xi|^2} \\varphi(|\\xi|) e^{2\\pi i x\\cdot \\xi} d\\xi\n$$\nis of the form $2^l x_j^{-1} \\tilde{K}(x)$, as is the statement of \\eqref{eq:701}. \n\\end{proof}\n\\subsection{$l^p$ functions of cone multipliers}\nIn this section, we discuss a simple  extension of Lemma \\ref{le:900}, \nwhich is concerned with appropriate \n $L^p$ bounds \nfor $l^p$ functions of such cone multipliers. \n\\begin{lemma}\n\\label{le:sum}\nLet $l>>1$ and $\\{\\theta_m^l\\}_{m}$ be a $2^{-l}$ net in $\\mathbf S^{n-1}$, so that the family \n$\\{\\theta\\in \\mathbf S^{n-1}: \n|\\theta_m^l-\\theta|\\leq 2^{-l}\\}_m$ has the finite intersection property. Define \n$$\n\\widehat{P_m f}(\\xi)=\\varphi_{l, m}(2^l(\\xi\/|\\xi|-\\theta_m^l)) \\varphi(|\\xi|) \\hat{f}(\\xi).\n$$\nwhere $\\varphi_{l, m}$ are as in \\eqref{eq:fun}.\nThen one has   \n\\begin{eqnarray}\n\\label{eq:n1}\n& & \\|\\sum\\limits_m P_m g_m\\|_{L^p({\\mathbf R}^n)}\\leq C \\left(\\sum\\limits_m \n\\|g_m\\|_{L^p}^p\\right)^{1\/p}\\quad\\quad \\textup{if}\\quad 1\\leq p\\leq 2 \\\\\n\\label{eq:n2}\n& & (\\sum\\limits_m \\|P_m g\\|_{L^q({\\mathbf R}^n)}^q )^{1\/q}\\leq  C \\|f\\|_{L^q}\\quad\\quad \\textup{if}\\quad 2\\leq\nq\\leq \\infty.\n\\end{eqnarray}\n\\end{lemma}\n\\begin{proof} \nSince \\eqref{eq:n1} and \\eqref{eq:n2} are dual, it will suffice to check \\eqref{eq:n2}. Next, the\n$L^2$ estimate is trivial by the Plancherel's theorem and the finite intersection property of the\nsupports of $\\varphi_{l, m}(2^l(\\xi\/|\\xi|-\\theta_m^l))$. Thus, by interpolation it suffices to check \n$$\n\\sup\\limits_m \\|P_m g\\|_{L^\\infty}\\leq C\\|g\\|_{L^\\infty}.\n$$\nBut $P_m g(x)=K_{l, \\theta_m^l}*g(x)$ and so \n$$\n\\|P_m g\\|_{L^\\infty}\\leq \\|K_{l, \\theta_m^l}\\|_{L^1}\\|g\\|_{L^\\infty}\\leq C \\|g\\|_{L^\\infty}.\n$$\nwhere the last inequality follows from \\eqref{eq:202}.\n\\end{proof}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nWe study equivalence classes of decompositions of $S^3$ and also decompositions of other manifolds.\nThese decompositions are given by toroidal defining sequences \n(we use the term {toroidal} for a subspace of an $n$-dimensional manifold \nbeing homeomorphic to the disjoint union of finitely many copies of $S^{n-2} \\times D^2$) \nalthough more generally\nit would be possible to get similar results by considering  handlebodies instead of solid tori in the \ndefining sequences. \nThe problem of classifying decompositions was studied by many authors.\nBy \\cite{Sh68b} so-called Antoine decompositions in $\\mathbb R^3$ are equivalently embedded   if and only if\ntheir toroidal defining sequences can be mapped into each other by homeomorphisms of the stages.\nMore generally \\cite{ALM68}   \nfor a  decomposition $G$ of $\\mathbb R^3$ given by an arbitrary  \ndefining sequence made of handlebodies\nthe homeomorphism type of the pair  $(\\mathbb R^3 \/ G, \\mathrm {cl} \\thinspace \\pi_G ( H_G))$, where\n$\\pi_G$ is the decomposition map and \n$H_G$ is the union of the non-degenerate elements,\nis determined by the homeomorphism types of the consecutive stages of the defining sequence of $G$.\nBy \\cite{GRWZ11} two Bing-Whitehead decompositions of $S^3$ are equivalently embedded if and only if\nthe stages of the  toroidal defining sequences\nare homeomorphic to each other after some number of iterations (counting only the Bing stages).\nDecompositions given by defining sequences are upper semi-continuous\nand many shrinkability conditions are known about them.\nFor example, Bing-Whitehead decompositions are shrinkable under some conditions \\cite{AS89, KP14}\njust like Antoine's necklaces, which are wild Cantor sets.\nIn \\cite{Ze05} the maximal  genus of handlebodies being associated with a  defining sequence \nis used to study  Cantor sets. \n\nIn the present paper we define the  concordance of decompositions (see Section~\\ref{concdef}) \nwhich come with \ntoroidal defining sequences. As for knots, slice decompositions \nplay an important role in the classification: a decomposition is slice if \neach  component of a defining sequence is slice in a way that the $D^{n-1} \\times D^2$ thickened slice disk stages are nested into each other.\nBeing concordant means the analogous concordance of the solid tori in the defining sequence\nand this makes the well-known knot and link concordance invariants possible to apply\nin order to distinguish between the concordance classes of such decompositions.\nFor example, we show that \nthe concordance group of decompositions of $S^3$, where the defining sequences have some intrinsic properties, \nhas at least uncountably many elements, see Theorem~\\ref{uncount1}.\nThe uncountably many elements that we find are represented by Antoine's necklaces. \n\n\nDecompositions appear in studying  manifolds, where cell-like \nresolutions of homology manifolds \\cite{Qu82, Qu83, Qu87, Th84, Th04} provide a tool of \nobtaining topological manifolds. \nDecompositions  also appear\nin the proof of the Poincar\\'e conjecture in dimension four, see \\cite{Fr82, FQ90, BKKPR21}, where\n a cell-like decomposition of a $4$-dimensional manifold yields \n a decomposition space which is a topological manifold. \nIn higher dimensions the decomposition space given by a cell-like decomposition of a compact topological manifold\n is a homology manifold being also a topological manifold if it satisfies the disjoint disk property \\cite{Ed16}. \n A particular result \\cite{Ca78, Ca79, Ed80, Ed06} is that the double suspension of every integral homology $3$-sphere\nis homeomorphic to $S^5$,  \nthat is  for every homology $3$-sphere $H$ \nthere is a cell-like decomposition $G$ of $S^5$ such that \nthe decomposition space is the  homology manifold $\\Sigma^2 H$\n and since $\\Sigma^2 H$ satisfies  the disjoint disk property,\n the decomposition $G$ is shrinkable (and this implies that the decomposition space is $S^5$).  \n\nBeside concordance, \n we also define and study another equivalence relation, which is \n the cobordism of decompositions, see Definition~\\ref{borddecomp}\nand Section~\\ref{computebord}. This yields a cobordism group, which has\na natural homomorphism into \nthe  cobordism group  of homology manifolds \\cite{Mi90, Jo99, JR00}. We study  how homological manifolds  are related to \nthe cobordism group of cell-like decompositions via taking the decomposition space. \nIt turns out that \nevery such decomposition space is cobordant to a topological manifold in the cobordism group of homology manifolds and\nthey generate a subgroup isomorphic to the cobordism group of topological manifolds, \n see Proposition~\\ref{bordkep}. \n Often we state and prove our results only for unoriented cobordisms but all the arguments\n obviously work for the oriented cobordisms as well giving the corresponding results. \n\nThe paper is organized as follows.\nIn Section~\\ref{prelim}\nwe give some basic lemmas and the definitions of the most important notions and \nin Section~\\ref{results}\nwe state and prove our main results.\n\nThe author  would like to thank the referee for the helpful comments, which improved the paper. \n\n\n\\section{Preliminaries}\\label{prelim}\n\n\n\n\\subsection{Cell-like decompositions}\n\nThroughout the paper we suppose that \nif $X$ is a compact manifold with boundary and \n$Y$ is a compact manifold with corners, then \nan embedding $e \\co Y \\to X$ is such that \nthe corners of $Y$ are mapped into $\\partial X$ and the pairs of boundary components near  the corners  of $Y$ are \nmapped into $\\int X$ and into $\\partial X$, respectively.\nWe also suppose that $e(\\int Y) \\subset \\int X$. \n If $Y$ has no corners, then \n$\\partial X \\cap e(Y) = \\emptyset$. \n We generalize the notions of defining sequence, cellular set and cell-like set  in the obvious way  \nfor manifolds with boundary as follows. Recall that a decomposition of a topological space  $X$ is \n a collection of pairwise disjoint subsets of $X$ whose union is equal to $X$. \n\n\\begin{defn}[Defining sequence for a subset]\\label{defseq}\nLet $X$ be an  $n$-dimensional manifold with possibly non-empty boundary.\n A \\emph{defining sequence}  for a subset $C \\subset X$ \n is a sequence \n $$c \\co \\mathbb N \\to \\mathcal P(X)$$\n $$C_0, C_1, C_2, \\ldots, C_n, \\ldots$$ of compact  $n$-dimensional \n submanifolds-with-boundary  possibly with corners \n in $X$ such\n that \n \\begin{enumerate}\n \\item\n  every\n$C_{n+1}$ has a neighbourhood $U$ such that  \n$U \\subset C_n$,  \n\\item\nin every component of $C_n$ there is a component of $C_{n+1}$,\n\\item\n$\\cap_{n=0}^{\\infty} C_n = C$ and\n\\item\nif $\\partial X \\neq \\emptyset$, then \nthere is an $\\varepsilon > 0$ such that \n$\\partial X \\times [0, \\varepsilon)$ is  a collar neighbourhood \nof $\\partial X$ and for every $C_n$ such that \n $C_n \\cap \\partial X \\neq \\emptyset$\n we have $C_n \\cap (\\partial X \\times [0, \\varepsilon)) = (C_n \\cap \\partial X) \\times [0, \\varepsilon)$. \n\\end{enumerate}\n A decomposition of $X$ defined by the  defining sequence $c$  is the triple \n $(X, \\mathcal D, C)$, where \n $C = \\cap_{n=0}^{\\infty} C_n$ and \n the elements of $\\mathcal D \\subset \\mathcal P(X)$ \n are \n\\begin{enumerate}\n\\item\nthe  connected components of $C$ and \n\\item\nthe points in $X -C$.\n\\end{enumerate}\nWe denote the decomposition map by $\\pi$. \n\\end{defn}\n\nObserve that for a decomposition $(X, \\mathcal D, C)$ the set $C$ is non-empty and each of the non-degenerate elements is a subset of $C$. \nThere could be singletons in $C$ as well. \nFor example in the case of an Antoine's necklace there are no non-degenerate elements, we  choose\n $C$ to be the Cantor set Antoine's necklace itself and so  $C$ consists of singletons. \nEvery decomposition  defined by some defining sequence is upper semi-continuous. \nA decomposition $\\mathcal D$  of a manifold induces a decomposition on its boundary by intersecting the decomposition elements\nwith the boundary. \nThe decomposition of the boundary $\\partial X$  induced  by a defining sequence in $X$ is upper semi-continuous.\n This induced decomposition is given by an induced  defining sequence $C_{n} \\cap \\partial X$\n  if $D_{n, k} \\cap \\partial X \\neq \\emptyset$ for  every component $D_{n, k}$ of each $C_n$. \n If all $C_n$ in a defining sequence are connected, then  $\\cap_{n=0}^{\\infty} C_n$ is connected.\n\n\n\n\\begin{defn}[Cell-like set]\nA compact subset $C$ of a metric space $X$  is \\emph{cell-like} if \nfor every neighbourhood $U$ of $C$ there is a neighbourhood $V$ of $C$ in $U$ such that\nthe inclusion map $V \\to U$ is homotopic in $U$ to a constant map.\nA decomposition is called cell-like if\neach of its decomposition elements is  cell-like.\n\\end{defn}\n\nCell-like sets given by defining sequences are connected \nbecause if the connected components could be separated by open neighbourhoods, then   \n a  homotopy could not deform the set into one single point in the neighbourhoods.\n\n\nA  space $X$ is \\emph{finite dimensional} if for every open cover $\\mathcal U$ of $X$ there exists\na refinement $\\mathcal V$ of $\\mathcal U$ such that no points of $X$ lies in more \nthan $K_X$ of the elements of $\\mathcal V$, where $K_X$ is a constant depending only on $X$.\n\n\n\\begin{lem}\\label{dimdecomp}\nLet $\\mathcal D$ be a decomposition of a manifold  $X$ possibly with non-empty boundary\ngiven by a defining sequence.\n Then the  decomposition space $X \/ \\mathcal D$ \n is finite dimensional.\n\\end{lem}\n\\begin{proof}\nIf $X$ has no boundary, then the statement follows from \n Theorem~2 and Proposition~3 in \\cite[Chapter~34]{Da86}.\n If $X$ has non-empty boundary, then \n the argument is also similar. \n\\end{proof}\n\n\n\\subsection{Homology manifolds}\n\n\n\nRecall that \na metric space $Y$ is an \\emph{absolute neighbourhood retract}  (or \\emph{ANR} for short)\nif \nfor every metric space $Z$ and embedding $i \\co Y \\to Z$  such that \n$i(Y)$ is closed there is a neighbourhood $U$ of $i(Y)$ in $Z$ which retracts onto $i(Y)$, that is\n$r|_{i(Y)} = \\mathrm {id}_{i(Y)}$ for some map $r \\co U \\to i(Y)$.\nIt is a fact that every manifold is an ANR.\nA space is called a  \\emph{Euclidean neighbourhood retract}  (or \\emph{ENR} for short)\nif it can be embedded into a Euclidean space as a closed subset  so that it is a retract of \nsome of its  neighbourhoods. \nIt is well-known that \na space is an ENR if and only if it is a locally compact, finite dimensional, separable  ANR. \n\n\n\n\n\n\n\n\n\\begin{defn}[Homology manifold]\nLet $n \\geq 0$ and let $X$ and $Y$ be finite dimensional  ANR spaces, where\n$Y$ is a closed subset of $X$. \nSuppose that for every $x \\in X$ we have\n\\begin{enumerate}\n\\item\n$H_k ( X, X - \\{ x \\} ) = 0$ for $k \\neq n$ and \n\\item\n$H_n ( X, X - \\{ x \\} )$ is isomorphic to  $\\mathbb Z$ if $x \\in X - Y$ and  it is isomorphic to  $0$ if $x \\in Y$.\n\\end{enumerate}\nThen $X$ is an $n$-dimensional  \\emph{homology manifold}. \nThe set of points $x \\in Y$  \nare the \\emph{boundary points of $X$} and the set $Y$  is denoted by $\\partial X$.\nA homology manifold is called \\emph{closed} if it is compact and has no boundary.\n\\end{defn}\n\nSince  locally compact and separable homology manifolds  are ENR spaces, a locally compact and separable\n homology manifold is called an \\emph{ENR homology manifold}.\nIn \\cite{Mi90} it is proved that\nfor $n \\geq 1$ and for  every compact and locally compact $n$-dimensional homology manifold  $X$\n the set of boundary points $\\partial X$ \n is an $(n-1)$-dimensional  homology manifold.\n\n\n\nSometimes a space $X$ without the ANR property but having \n$H_k ( X, X - \\{ x \\} ) = 0$ for $k \\neq n$ and \n$H_n ( X, X - \\{ x \\} ) = \\mathbb Z$ in the sense of \\v{C}ech homology is also called a homology manifold. These spaces arise as \nquotient spaces of acyclic decompositions of topological manifolds \n\\cite{DW83} while ANR homology manifolds are often homeomorphic to \nquotients of cell-like decompositions \\cite{Qu82, Qu83, Qu87}.\n\n\n\nIn the case of cell-like decompositions the decomposition spaces are homology manifolds if they are finite dimensional essentially because \nof the Vietoris-Begle theorem \\cite[Theorem~0.4.1]{DV09}. In more detail, we will use the following. \nLet $X'$ be a compact $n$-dimensional  manifold with possibly non-empty boundary, let\n$Y$ be  $\\partial X' \\times [0,1]$ and attach $Y$ to $X'$ as a collar  to get a manifold $X$.\n\n\\begin{lem}\\label{decomphomolmanif}\nLet $\\mathcal D'$ be a cell-like decomposition of $X'$ given by a defining sequence such that \n$X'$ contains a small open set (intersecting the possibly non-empty boundary) which consists of singletons. \n Suppose that \n the induced decomposition  on $\\partial X'$ is cell-like and it is given by the induced defining sequence. \n Suppose that \n in $Y$  a cell-like decomposition $\\mathcal E$ is given, where\n $\\mathcal E$ is the product of the  decomposition \n induced by $\\mathcal D'$ on $\\partial X'$ and the trivial decomposition of $[0,1]$. \n  Denote by $\\mathcal D$ the resulting decomposition on $X$. \nThen $X\/\\mathcal D$ \nis an $n$-dimensional ENR homology manifold with possibly non-empty boundary. \n The boundary points of $X\/\\mathcal D$ are exactly the points of the ENR homology manifold \n$\\pi (\\partial X)$.\n\\end{lem}\n\\begin{proof}\nWe have to show that \nthe quotient space \n$$X \/ \\mathcal D$$\nis an $n$-dimensional  homology manifold with boundary the homology manifold\n$\\pi (\\partial X)$.\nTake the closed manifold $$X  \\cup_{\\varphi} X,$$\nwhere\n$\\varphi \\co \\partial X  \\to \\partial X$\nis the identity map.\n\nThe decomposition space $X' \/ \\mathcal D'$\n(that is the part of the decomposition space $X \/ \\mathcal D$ which is obtained from \n$X'$)   \nis finite dimensional by Lemma~\\ref{dimdecomp}. \nThe doubling of the decomposition $\\mathcal D$ on \n$X  \\cup_{\\varphi} X$\nyields\na finite dimensional quotient space, we get this by using \nestimations for the covering dimension, see \\cite{HW41} and \\cite[Corollary~2.4A]{Da86}.\nSo the decomposition space $P$ obtained by factorizing \n$X  \\cup_{\\varphi} X$\nby the double of $\\mathcal D$ is a closed  finite dimensional homology manifold by  \\cite[Proposition~8.5.1]{DV09}. \nSince a small neighbourhood of a singleton results  an open set in $P$ homeomorphic to $\\mathbb R^{n}$, it is $n$-dimensional.\nWe obtain the space $X \/ \\mathcal D$\nby cutting $P$ into two pieces along  $\\pi(\\partial X)$. Because of a similar argument  \nthe space $\\pi(\\partial X)$ is a closed  $(n-1)$-dimensional homology manifold. \nThe set $\\pi(\\partial X)$ is closed in the decomposition space $X \/ \\mathcal D$ since\n$\\mathcal D$ is upper semi-continuous and $\\partial X$ is closed. \nAlso, the homology group  $H_n(X \/ \\mathcal D; X \/ \\mathcal D - \\{p\\} )$ is equal to \n$0$ for every $p \\in \\pi(\\partial X)$. So $\\pi(\\partial X)$ is the boundary of $X \/ \\mathcal D$. \n\nMoreover the space $X \/ \\mathcal D$ is \n a locally compact separable metric space because $X$ is so. By \\cite[Corollary~7.4.8]{DV09}\nthe space $X\/\\mathcal D$ is an ANR so it follows that it is an ENR. The same holds for $\\pi(\\partial X)$. \n\\end{proof}\n\n\n\n\\begin{defn}[Cobordism of homology manifolds]\nThe  closed $n$-dimensional  \nhomology manifolds $X_1$ and $X_2$ are \\emph{cobordant} \nif there exists  a compact $(n+1)$-dimensional homology manifold $W$ such that \n$\\partial W$ is homeomorphic to the disjoint union of $X_1$ and $X_2$.\nThe induced cobordism group (the group operation is the disjoint union) is \ndenoted by $\\mathfrak N_n^H$. In a similar way the induced oriented cobordism group is denoted by $\\Omega_n^H$.\n\\end{defn}\nNote that the connected sum of homology manifolds does not always exist. \nAnalogously let $\\mathfrak N_n^E$ and $\\Omega_n^E$ denote the cobordism group and\noriented cobordism group \nof ENR homology manifolds (the cobordisms are also ENR), respectively.\n\n\nAlmost all oriented cobordism groups $\\Omega_n^H$ are computed \\cite{BFMW96, Jo99, JR00}:\n\\[\n\\Omega_n^H = \n\\left \\{\n\\begin{array}{cc}\n\\mathbb Z  & \\mbox{if $n = 0$}   \\\\\n0  &   \\mbox{if $n = 1, 2$}   \\\\\n \\Omega_n^{TOP}[8\\mathbb Z  +1]   &     \\mbox{if $n \\geq 6$},\n\\end{array}\n\\right.\n\\]\nwhere $\\Omega_n^{TOP}$ denotes the cobordism group of topological manifolds\n and the group $$\\Omega_n^{TOP}[8\\mathbb Z  +1]$$ denotes \n the group of finite linear combinations \n $\\sum_{i \\in 8\\mathbb Z +1} \\omega_i i$ of cobordism classes of topological manifolds.\n By \\cite[Corollary~4.2]{Ma71} the oriented cobordism group of manifolds $\\Omega_n$ is always a subgroup\nof $\\Omega_n^H$.\n\n\nA \\emph{resolution} of a homology manifold  $N$ \nis a topological manifold $M$ and a cell-like decomposition of $M$ \nsuch that the decomposition space is homeomorphic to the homology manifold $N$,\nthe quotient map $\\pi$ is proper \n and\n$\\pi^{-1} (\\partial N) = \\partial M$. \nBy \\cite{Qu82, Qu83, Qu87} homology manifolds are resolvable if a local obstruction is equal to $1$, more precisely we have the following.\n\\begin{thm}[\\cite{Qu82, Qu83, Qu87}]\\label{resol}\nFor every $n \\geq 4$ and every non-empty connected $n$-dimensional ENR  homology manifold $N$ there is an integer local obstruction $i(N) \\in 8\\mathbb Z + 1$\nsuch that \n\\begin{enumerate}[\\rm (1)]\n\\item\nif $U \\subset N$ is open, then $i(U) = i(N)$,\n\\item\nif $\\partial N \\neq \\emptyset$, then \n$i(\\partial N) = i(N)$, \n\\item\n$i(N \\times N_1) = i(N) i(N_1)$ for any other homology manifold $N_1$,\n\\item\nif $\\dim N = 4$ and $\\partial N$ is a manifold, then there is a resolution if and only if $i(N)=1$ and \n\\item\nif $\\dim N \\geq 5$, then there is a resolution if and only if $i(N)=1$.\n\\end{enumerate}\n\\end{thm}\nBy \\cite{Th84, Th04} a closed  $3$-dimensional ENR homology manifold $N$ is resolvable if \nits singular set has general position dimension less than or equal to one, that is\nany map of a disk into $N$ can be approximated by one whose image meets the singular set \n (i.e.\\ the set of non-manifold points) \nof $N$ in a $0$-dimensional set.\n\n\n\\begin{lem}\\label{resolcob}\nLet $M_{1}$ and $M_{2}$ be two closed $n$-dimensional manifolds, where $n \\geq 4$.\nIf both of them are resolutions of the ENR homology manifold $N$, then \n$M_1$ and $M_2$ are cobordant as manifolds.\n\\end{lem}\n\\begin{proof}\nIf there are two resolutions $f_1 \\co M_1 \\to N$ and $f_2 \\co M_2 \\to N$ of \n a closed $n$-dimensional homology manifold $N$, then \n as in the proof of \\cite[Theorem~2.6.1]{Qu82}\n take a resolution \n $$Y \\to X_{f_1} \\cup X_{f_2}$$ \n of the double mapping cylinder $X_{f_1} \\cup X_{f_2}$  \n of the maps $f_1$ and $f_2$ \n  by applying \\cite[Theorem~1.1]{Qu83} and \\cite{Qu87}. \n This resolution exists because $X_{f_1} \\cup X_{f_2}$ is an $(n+1)$-dimensional ENR homology manifold and \n $i( X_{f_1} \\cup X_{f_2} )  =1$.\n Let \n $$X_{f_1} \\cup X_{f_2} \\to N \\times [-1,1]$$\nbe the natural map of the double mapping cylinder \nonto $N \\times [-1,1]$, where the target $N$ of the two mapping cylinders is mapped \nonto $N \\times \\{ 0 \\}$. \n \nIt follows that the composition \n $$Y \\to X_{f_1} \\cup X_{f_2} \\to N \\times [-1,1]\n$$ \n is a resolution, moreover by \\cite[Theorem~1.1]{Qu83}\n the cell-like map \n $Y \\to X_{f_1} \\cup X_{f_2}$ can be chosen \nso that \n it is a homeomorphism over the boundary hence\n $Y$ is a cobordism  between $M_1$ and $M_2$.\n\\end{proof}\n\n\n\n\n\n\n\\subsection{Concordance and cobordism of decompositions}\\label{concdef}\n\n\nWe will study decompositions given by defining sequences $C_0, C_1, C_2, \\ldots$ such that \neach $C_n$ is a disjoint union of solid tori. \n We remark  that more generally all the following notions work for  decompositions whose stages are handlebodies instead of just tori.\nIn a closed $n$-dimensional manifold $M$\n instead of decompositions $(M, \\mathcal D, A)$\nwe will consider decompositions with \nsome thickened link which contains the set $A$ so \n in the following\n a decomposition in $M$ is a quadruple $(M, \\mathcal D, A, L)$, where $L \\subset M$ is the thickened link and $A \\subset L$.\nFor example an Antoine's necklace is situated inside an unknotted \nsolid torus while it can be knotted in many different ways in the solid torus.\n \n\n\n\\begin{defn}[Concordance of decompositions]\\label{concordecomp}\nLet $M_1$ and $M_2$ be closed $n$-dimensional manifolds. \nThe  decompositions\n $(M_1, \\mathcal D_1, A, L_1)$ and $(M_2, \\mathcal D_2, B, L_2)$\nare \\emph{cylindrically related}\nif  \nthere exist\n toroidal  defining sequences\n$C_0, C_1, C_2, \\ldots$ for $A$ and $D_0, D_1, D_2, \\ldots$ for $B$ and \nthere exists \na defining sequence $E_0, E_1, E_2, \\ldots$ for a   decomposition $\\mathcal E$ \nof a compact $(n+1)$-dimensional manifold $W$ such that \n\\begin{enumerate}\n\\item\n$C_0 = L_1$ and $D_0 = L_2$, \n\\item\n$\\partial W = M_1 \\sqcup M_2$,\n\\item\neach $E_i$ is homeomorphic to  $C_i \\times [0,1]$ and \n\\item\neach $E_i$  bounds the components of $C_i \\subset M_1$ and $D_i \\subset M_2$ that is  \n $C_i \\times \\{ 0 \\}$ corresponds to $C_i$ and $C_i \\times \\{ 1 \\}$ corresponds to $D_i$. \n\\end{enumerate}\nTwo decompositions $(M_1, \\mathcal D_1, A, L_1)$ and $(M_2, \\mathcal D_2, B, L_2)$\n are \\emph{concordant} if \n there exist closed $n$-dimensional manifolds\n $M_1', \\ldots, M_k'$ and  decompositions $(M_i', \\mathcal D_i',  A_i', L_i')$ for every $i = 1, \\ldots, k$ \n  such that \n  \\begin{enumerate}\n  \\item\n  $(M_1, \\mathcal D_1, A, L_1)$ is cylindrically related to \n  $(M_1', \\mathcal D_1',  A_1', L_1')$, also for $i = 1, \\ldots, k-1$ every \n  $(M_i', \\mathcal D_i',  A_i',  L_i')$ is cylindrically related to \n  $(M_{i+1}', \\mathcal D_{i+1}',  A_{i+1}',  L_{i+1}')$\n and \n $(M_k', \\mathcal D_k',  A_k',  L_k')$ is cylindrically related to\n $(M_2, \\mathcal D_2, B, L_2)$ and \n \\item\nfor each $A_i' \\subset M_i'$, where $i = 1, \\ldots, k$,  the two toroidal defining sequences \n $C_{i, 0}', C_{i, 1}', \\ldots$ and  $C_{i, 0}'', C_{i, 1}'', \\ldots$  in $M_i'$ \n appearing in these successive cylindrically related decompositions\n are such that the $0$-th stages \n $C_{i, 0}'$ and $C_{i, 0}''$ are equal as subsets of $M_i'$.\n\\end{enumerate}\n Being concordant is an equivalence relation \n and the equivalence classes are called \\emph{concordance classes}.\n \\end{defn}\n \nHence being concordant implies that the two decompositions\n are in the same equivalence class of \nthe equivalence relation generated by being cylindrically related, that is \nthe two decompositions can be connected by a finite\nnumber of cylindrically related  decompositions. \n Being concordant also implies that \n the $0$-th stages of two toroidal defining sequences for the two decompositions\n are connected by a single concordance in the usual sense. \nClearly in the definition \neach $E_i$ intersects  some fixed collar of $\\partial W$ as the defining sequence in (4) of Definition~\\ref{defseq}.\n The  concordance classes form a \ncommutative semigroup under the operation ``disjoint union\". Moreover this \nsemigroup is a monoid because the neutral element \nis the ``empty manifold'', that is the empty set $\\emptyset$.\nTo have a more meaningful neutral element we define the following.\n\n\n\\begin{defn}[Slice decomposition]\\label{slicedecomp}\nLet $M$  be a closed $n$-dimensional manifold and \nlet $(M, \\mathcal D, A, L)$  \nbe a   decomposition of $M$ such that \nthere exists a toroidal  defining sequence $C_0, C_1, C_2, \\ldots$ with $C_0 = L$ for  $A$. \nThen $(M, \\mathcal D, A, L)$ is \\emph{slice} if \nit is concordant to a decomposition $(M', \\mathcal D', A', L')$\nwith defining sequence $C_0', C_1', C_2', \\ldots$ with $C_0' = L'$ \n such that\nthere exists\n a defining sequence $E_0, E_1, E_2, \\ldots$ for a   decomposition $\\mathcal E$ \nof the  $(n+1)$-dimensional manifold $M' \\times [0,1)$, where \neach $E_i$ consists of finitely many $D^{n-1} \\times D^2$ bounding the torus components $S^{n-2} \\times D^2$ of $C_i' \\subset M' \\times \\{0\\}$. \n \\end{defn}\n \n Analogously \nto Definitions~\\ref{concordecomp} and \\ref{slicedecomp}, we \ndefine the \\emph{oriented concordance} of decompositions by requiring \nall the manifolds  to be oriented in the usual consistent way, \nin this way we also get  a corresponding monoid.   \nObserve that the set of concordance classes of slice decompositions is a submonoid \nof the monoid of concordance classes of decompositions.  \n To obtain a group \nwe factor out the concordance classes by the classes represented by the slice decompositions\nand also by the classes of the form $$[(M, \\mathcal D, A, L)] + [(-M, \\mathcal D, A, L)],$$\nwhere $-M$ denotes the opposite orientation. Observe that all these classes form a submonoid. \n\n\\begin{defn}[Decomposition concordance group]\nDefine the relation $\\sim$ on the set of concordance classes of decompositions by the following rule:  \n$a \\sim b$ exactly if\nthere exist  slice decompositions   $s_1$ and $s_2$ \nand decompositions $(M, \\mathcal D, A, L)$  and \n$(M', \\mathcal D', A', L')$   \nsuch  that \n$\n a + [s_1] +  [( M, \\mathcal D, A, L )] + [(- M, \\mathcal D, A, L )] = \n  b + [s_2] + \n [( M', \\mathcal D', A', L')] + [(- M', \\mathcal D', A', L')].\n$\nThe relation $\\sim$ is a congruence and we obtain a commutative group by factoring out by this \ncongruence. We call this group the \\emph{oriented decomposition concordance group} \nand denote it by $\\Gamma_n$.\n\\end{defn}\n\n\nIf we confine the closed $n$-dimensional  manifolds to $S^n$ and the cobordisms to $S^n \\times [0,1]$, then we obtain something similar to the classical link concordance.\nFor the convenience of the reader we repeat the definitions. \n\n\\begin{defn}[Concordance group of decompositions in $S^n$]\\label{concordecompS}\nLet $(S^n, \\mathcal D_1, A, L_1 )$ and $(S^n, \\mathcal D_2, B, L_2 )$\nbe  decompositions of $S^n$ in the complement of $\\infty$.\n They \nare \\emph{cylindrically related}\nif  \n there exist\ntoroidal  defining sequences\n$C_0, C_1, C_2, \\ldots$  for $A$ and $D_0, D_1, D_2, \\ldots$   for $B$ and \nthere exists \na defining sequence $E_0, E_1, E_2, \\ldots$ for a  decomposition $\\mathcal E$ \nof the compact $(n+1)$-dimensional manifold $S^n \\times [0,1]$ in the complement of $\\{ \\infty\\} \\times [0,1]$ such that \n\\begin{enumerate}\n\\item\n$C_0 = L_1$ and $D_0 = L_2$, \n\\item\neach $E_i$ is homeomorphic to  $C_i \\times [0,1]$ and \n\\item\neach $E_i$  bounds the components of $C_i \\subset S^n \\times \\{0\\}$ and $D_i \\subset S^n \\times \\{1\\}$. \n\\end{enumerate}\nTwo decompositions are \\emph{concordant} if \n\\begin{enumerate}\n\\item\nthey are in the same equivalence class of \nthe equivalence relation generated by being cylindrically related \nso  the two decompositions can be connected by a finite\nnumber of cylindrically related  decompositions\nand \n\\item\nthe $0$-th stages of the defining sequences \nappearing in this sequence of  cylindrically related  decompositions\n are concordant as thickened links in the usual sense.\n\\end{enumerate}\nThe obtained  equivalence classes are called \\emph{concordance classes}. \n If two decompositions of $S^n$ are given by defining sequences, \nthen in the connected sum  (at  $\\infty$)  of the two $n$-spheres the ``disjoint union''   \ninduces \n a commutative semigroup operation on the set of concordance classes. \n Then \n by factoring out  by the \nsubmonoid of classes of slice decompositions and \nclasses of the form $[(S^n, \\mathcal D, A, L)] + [(-S^n, \\mathcal D, A, L)]$\nwe\n get a group called the \\emph{decomposition concordance group in $S^n$}.\n  We denote this group by $\\Delta_n$.\n\\end{defn}\n\n For example, the Whitehead decomposition in $S^3$ \nis slice \\cite{Fr82} and the Bing decomposition in $S^3$ is also slice\nbecause the Bing double of the unknot is slice. Observe that \nthe Bing decomposition $(S^3, \\mathcal B, C)$ has only singletons, where $C$ is a wild Cantor set.\nAs another example, a defining sequence in $S^3$ given by the replicating pattern of a solid torus and inside of it \n a  link made of a sequence of ribbon knots linked with each other circularly \n can yield a  slice decomposition.\n\n\n\nSince being concordant  implies  that the two decompositions can be connected by a finite\nnumber of cylindrically related  decompositions, all \ninvariants \n of concordance classes defined through defining sequences \n  are invariant under choosing another defining sequence for the same decomposition (while leaving the $0$-th stage unchanged). \nFor $n = 3$ in the following we restrict ourselves only to such toroidal \ndefining sequences $C_0, C_1, C_2, \\ldots$ of decompositions of the closed $n$-dimensional manifolds  \nin Definitions~\\ref{concordecomp}-\\ref{concordecompS}\nwhich \nsatisfy \nthe following  conditions:\n\\begin{defn}[Admissible defining sequences and decompositions]\\label{adm}\nSuppose \n\\begin{enumerate}\n\\item\nfor $m \\geq 1$ each $C_m$ has at least four components in a component of $C_{m-1}$ and \neach component $T$ of  $C_m$ \nis linked to exactly  two other components of $C_m$ in the ambient space $S^3$ with algebraic linking number non-zero\nand the splitting number of $T$ and each of  the other components is equal to $0$, \n\\item\nfor $m \\geq 1$ the components $A_1, \\ldots, A_k$ of  $C_m$ which are in a component $D$ of $C_{m-1}$ \nare linked in such a way that \nif a component $A_i$  is null-homotopic in a solid torus $T$ whose boundary is disjoint from all $A_i$, then \nall $A_i$  are in this solid torus $T$,\n\\item\n$\\cap_{m=0}^{\\infty} C_m$ is not separated by and not contained in any \n $2$-dimensional sphere \n$S$ for which \n$S \\subset C_m$ for some $m$,\n\\item\n every embedded circle in  the boundary of a component of $C_m$\nwhich bounds no $2$-dimensional disk in this boundary\ncannot be shrunk to a point in the complement of $\\cap_{m=0}^{\\infty} C_m$. \n\\end{enumerate}\nWe call such defining sequences and decompositions \\emph{admissible}. \n\\end{defn}\n\n\\begin{prop}\nIn the connected sum  (at  $\\infty$)  of two $3$-spheres the ``disjoint union''   as in Definition~\\ref{concordecompS} \nof two admissible toroidal decompositions  is an  admissible toroidal decomposition.\n\\end{prop}\n\\begin{proof}\nChecking the conditions (1)-(4) in Definition~\\ref{adm} is obvious, details are left to the reader. \n\\end{proof}\n\n\n\n\nThen we denote the arising concordance  group in $S^3$ by  \n$\\Delta_3^a$. \nFor example, Antoine's necklaces (or Antoine's decompositions) for $n = 3$ \nhave defining sequences satisfying these conditions \\cite{Sh68b}. \nWe note that by \\cite{Sh68b} their defining sequences also have  the property  of \nsimple chain type, which means that \nthe torus components are unknotted and they are linked like the Hopf link. \n We have \nthe natural group homomorphisms\n$$\n\\Delta_3^a \\to \\Delta_3\\mbox{\\ \\ \\ \\ \\ and\\ \\  \\ \\ }\\Delta_3 \\to \\Gamma_3$$\nand also for arbitrary $n$ the group homomorphism\n$$\n\\Delta_n \\to \\Gamma_n.$$\nWe will show that \nthe number of elements of the group $\\Delta_3^a$ is at least uncountable. \n\n\n\nNow we define \\emph{cobordism} of decompositions, where \nwe restrict ourselves to \\emph{cell-like} decompositions (not necessarily admissible)  at the cobordisms \nand at the representatives as well.\n\n\n\\begin{defn}[Cobordism of decompositions]\\label{borddecomp}\nLet $M_1$ and $M_2$ be closed $n$-dimensional manifolds and \nlet $(M_1, \\mathcal D_1, A)$ and $(M_2, \\mathcal D_2, B)$\nbe cell-like decompositions  such that \nthere exist\n toroidal  defining sequences\n$C_0, C_1, C_2, \\ldots$ for $A$ and $D_0, D_1, D_2, \\ldots$ for $B$. \nThen $(M_1, \\mathcal D_1, A)$ and $(M_2, \\mathcal D_2, B)$\nare \\emph{coupled}\nif  there exists \na defining sequence $E_0, E_1, E_2, \\ldots$ for a cell-like  decomposition $\\mathcal E$ \nof a compact $(n+1)$-dimensional manifold $W$ such that \n\\begin{enumerate}\n\\item\n$\\partial W = M_1 \\sqcup M_2$,\n\\item\neach $E_i$ is homeomorphic to  the disjoint union \nof finitely many manifolds $P_j^{n-1} \\times D^2$, $j = 1, \\ldots m_i$, where all $P_j^{n-1}$ are  \ncompact $(n-1)$-dimensional manifolds and \n\\item\neach $E_i$  bounds the components of $C_i$ and $D_i$.\n\\end{enumerate}\nWe attach a collar $\\partial W \\times [0,1]$ to $W$ along its boundary and extend the decomposition \n$\\mathcal E$ to the collar by taking the product  of $\\mathcal D_1$ and \n$\\mathcal D_2$  with the trivial decomposition on $[0,1]$, respectively. We say that this extended manifold  \n$W \\cup (\\partial W \\times [0,1])$ and its decomposition is a \\emph{coupling} between $(M_1, \\mathcal D_1, A)$ and $(M_2, \\mathcal D_2, B)$.\n Finally, two decompositions are \\emph{cobordant} if they are in the same equivalence class of \nthe equivalence relation generated by being coupled. \nThe generated equivalence classes are called \\emph{cobordism classes}. \n\\end{defn}\n\nClearly\neach $E_i$ intersects  some fixed collar of $\\partial W$ as the defining sequence in (4) of Definition~\\ref{defseq}.\n The cobordism classes  form a commutative group \nunder the operation ``disjoint union\". \nDenote this group by $\\mathcal B_n$.\n\nWe will show that \nfor a cobordism between arbitrary given cell-like decompositions $\\mathcal D_{1, 2}$ as in Definition~\\ref{borddecomp} if we \n take the decomposition  space, then we get\n a group homomorphism \ninto the cobordism group of homology manifolds. \n\n\n\n\\section{Results}\\label{results}\n\n\n\\subsection{Computations in the concordance groups}\n\n\nWe are going  to define invariants  of  elements of\nthe group $\\Delta_3^a$. \n With the help of these invariants, we will show that \n the group $\\Delta_3^a$  has at least uncountably many elements.\n\n\\begin{defn}\nFor a given defining sequence \n$C_0, C_1, C_2, \\ldots, C_n, \\ldots$ in $S^3$  let $$n_{C_0, C_1, C_2, \\ldots } = (n_0, n_1, n_2, \\ldots )$$ be \nthe sequence of the numbers of components of the manifolds $C_0, C_1, C_2, \\ldots$.\n\\end{defn}\n\nIf two decompositions of $S^3$ as in Definition~\\ref{concordecompS} are cylindrically related, then \nthey have defining sequences \n$C_0, C_1, C_2, \\ldots$ and $D_0, D_1, D_2, \\ldots$   such that \n$$\nn_{C_0, C_1, C_2, \\ldots }  = n_{D_0, D_1, D_2, \\ldots }. \n$$\nBy \\cite[Theorem~3]{Sh68b}   for \ncanonical defining sequences of an Antoine's necklace (or an  Antoine decomposition) \nthe sequence $n_{C_0, C_1, C_2, \\ldots }$ \n uniquely exists (note that $C_0$ is only an unknotted solid torus which is not \n appearing in \\cite{Sh68b}).\n\n\\begin{prop}\\label{compnumbsame}\n Let  $(S^3, \\mathcal D, A, C_0)$ be an admissible\ndecomposition  and  let\n  $C_0, C_1, C_2, \\ldots$ and $D_0, D_1, D_2, \\ldots$  be\n  admissible defining sequences \n  for $(S^3, \\mathcal D, A, C_0 )$, where we suppose that   $C_0 = D_0$. \nThen we have\n$$n_{C_0, C_1, C_2, \\ldots }  = n_{D_0, D_1, D_2, \\ldots}.$$\n\\end{prop}\n\\begin{proof}\nSuppose that \n$C_0, C_1, C_2, \\ldots$ and $D_0, D_1, D_2, \\ldots$ are admissible defining sequences for \na decomposition $(S^3, \\mathcal D, A, C_0)$ such that $C_0 = D_0$. \nOf course \n$$\\cap_{n = 0}^{\\infty} C_n = A = \\cap_{n = 0}^{\\infty} D_n.$$\nWe use an algorithm applied in \\cite[Proof of Theorem~2]{Sh68b}. \nWe restrict ourselves to \none component of $C_0$ and to the components of the  defining sequences in it, the following argument\n works the same way for the other components. \nWe can suppose that \n$\\partial C_1 \\cap \\partial D_1$ is a closed $1$-dimensional submanifold of $S^3$.\n Suppose some  component $P$ of $\\partial C_1 \\cap \\partial D_1$\nbounds a  $2$-dimensional disk $Q \\subset \\partial D_1$.\nAlso suppose that $P$ is an innermost component of $\\partial C_1 \\cap \\partial D_1$ in \n$\\partial D_1$ so $\\int Q \\cap \\partial C_1 = \\emptyset$. \nBy (4) in Definition~\\ref{adm} if $P$ does not bound a disk $Q'$ in $\\partial C_1$, then \n$P$ is not homotopic to constant in \nthe complement of $A$ but then $P$ cannot bound the disk $Q \\subset \\partial D_1$. \nHence \n$P$ bounds a disk $Q' \\subset \\partial C_1$ as well. \nThen the interior of the sphere $Q \\cup Q'$ \ndoes not intersect $A$ \nbecause of (3) in Definition~\\ref{adm}.\nSo we can modify $C_1$ by pushing $Q'$  through the sphere $Q \\cup Q'$ by a self-homeomorphism of the complement \nof $A$ \nand hence we obtain  fewer circles in the new $\\partial C_1 \\cap \\partial D_1$.\nAfter repeating these steps finitely many times we obtain \na new $C_1$ such that $\\partial C_1 \\cap \\partial D_1$ contains no circles which bound disks\n on $\\partial C_1 \\cup \\partial D_1$. \n Similarly, by further adjusting $C_1$ in the complement of $A$ as written on  \\cite[page~1198]{Sh68b}\n  in order to eliminate the circles in $\\partial C_1 \\cap \\partial D_1$ which bound annuli\n we finally obtain a $C_1$ such that \n\n\\begin{itemize}\n\\item\nthe intersection $\\partial C_1 \\cap \\partial D_1$ is empty,\n\\item\nno component of $C_1$ is disjoint from all the components of $D_1$ and vice versa,\n\\item\neach component of $C_1$ is inside \na component of $D_1$ or it contains some components of $D_1$.\n\\end{itemize} \n\nThen we can see that there is a bijection between the \nnumber of components of $C_1$ and $D_1$ because of the following.\n\nIf a component of $C_1$ is in $\\int D_1$ and it is homotopic to constant \nin $\\int D_1$, then all the other components of $C_1$ are in the same component of $\\int D_1$ by (2) in Definition~\\ref{adm}.\nThis would result that no part of $A$ is in other components of $D_1$, which would  contradict to \n(1) in Definition~\\ref{adm}\n so \nno \ncomponent of $C_1$ in $\\int D_1$  is homotopic to constant \nin $\\int D_1$. The same holds if we switch the roles of $C_1$ and $D_1$. \nThis means that \n\\begin{itemize}\n\\item\nthe winding number\nof a component $T$ of $C_1$ in the component of $D_1$ which contains $T$ is not equal to $0$ and the same holds\nfor $D_1$ and $C_1$ with opposite roles. \n\\end{itemize} \n\n\nFurthermore \nsuppose that $T$ is some component of $D_1$ and\n $T$  contains at least two  components $T_1$ and $T_2$ of $C_1$.\n Then \n   $T$ is linking with other component $T'$ of $D_1$ by (1) in Definition~\\ref{adm} with algebraic \n   linking number non-zero.  \nLet $T_3$ be a component of $C_1$ such that \n $T_3 \\subset T'$ or $T' \\subset T_3$. \nIf $T_3 \\subset T'$, then \n$T_1$ and $T_2$ are linking with $T_3$ with algebraic linking number non-zero.\nIf $T' \\subset T_3$, then \n$T$ is not in $T_3$ because for example \n$T_1$ cannot be in $T_3$. \nBut then $T$ is linking with $T_3$ with algebraic linking number non-zero since the same holds for $T$ and $T'$. \n So again we obtained that $T_1$ and $T_2$ are linking with $T_3$ with linking number non-zero.\nNow, there is a $T''$ component of $D_1$\n which is linking with $T'$ with linking number non-zero and which is disjoint from all the previously mentioned tori \n ($T', T'' \\subset T_3$ is impossible because then both of $T', T''$ are linking with $T$ and also with each other \n and this contradicts to (1) in Definition~\\ref{adm}).\n  Let $T_4$ be a component of $C_1$  such that \n $T_4 \\subset T''$ or $T'' \\subset T_4$. There are a number of cases to check. \n If $T_3 \\subset T'$\n  and  $T_4 \\subset T''$, then \n  $T_3$ is linking with $T_4$. \n  If $T_3 \\subset T'$ but $T'' \\subset T_4$, then \n  since $T_4$ cannot contain \n  $T$ or $T'$, we have again that \n  $T_3$ is linking with $T_4$. \n  Finally, if $T' \\subset T_3$, then since $T''$ cannot be in $T_3$, we have that \n  $T_4 \\subset T''$ implies that $T_3$ and $T_4$ are linking and\n  $T'' \\subset T_4$ implies that \n since  $T_4$ is disjoint from all the other tori, again $T_4$ is linking with $T_3$.\nSo we obtain that \n$T_1$ and $T_2$ are linking with $T_3$\n and $T_3$ is linking with $T_4$\n resulting that $T_3$ is linking with three other components of $C_1$ which contradicts to \n(1) in Definition~\\ref{adm}.\nSummarizing, we obtained the following.\n\\begin{itemize}\n\\item\nThe intersection $\\partial C_1 \\cap \\partial D_1$ is empty,\n\\item\nno component of $C_1$ is disjoint from all the components of $D_1$ and vice versa,\n\\item\nevery component of $C_1$ contains one component of $D_1$ or is contained in one component of $D_1$, \n\\item\nno component of $C_1$ contains more than one component of $D_1$ and vice versa.\n\\end{itemize}\nAll of these imply that \nthe number of components of $C_1$ is equal to the number of components of $D_1$. \nWe repeat the same line of arguments for the components of $C_2$ and $D_2$ lying in each component of $C_1$ or $D_1$ separately, \nwhere we perform the previous algorithm in the larger component which contains the smaller one, \n and so on, in this way we get the result.\n\\end{proof}\n\n\\begin{rem}\nIf in (1) in Definition~\\ref{adm}\nwe require having splitting number greater than $0$ instead of having \nalgebraic linking number non-zero, then \nthe previous arguments could be repeated  \nto get a similar result \nif we could prove that \nhaving two solid tori with splitting number greater than $0$ \nand embedding one circle into each of these tori \nwith non-zero winding numbers \nresults that the splitting number of these two knots is greater than $0$. \nFor similar results about knots and their unknotting numbers, see \\cite{ST88, HLP22}. \n\\end{rem}\n\n\nIt follows that if two admissible decompositions of $S^3$ are\nin the same equivalence class \nof \nthe equivalence relation generated by being cylindrically related, \nthen they determine the  same sequence of numbers of components. \nSo if we define the operation \n$$\n(n_0, n_1, \\ldots )  +  (m_0, m_1, \\ldots ) = (n_0 + m_0, n_1 + m_1, \\ldots )\n$$\non the set of sequences, then \nthe induced map \n$$\n[ ( S^3,  \\mathcal D, A, C_0 ) ] \\mapsto n_{C_0, C_1, C_2, \\ldots},$$\nwhere $C_0, C_1, C_2, \\ldots$ is some admissible defining sequence, \nis a monoid homomorphism. \n\n\\begin{defn}\nFor an equivalence class $x$ represented by \n the admissible decomposition $( S^3,  \\mathcal D, A, C_0 )$ and for its admissible defining sequence\n $C_0, C_1, \\ldots$ \nlet \n$$\nL(x) = (l_1, l_2, \\ldots)$$\nbe the sequence of numbers mod $2$ \nof the components of $C_m$ which have non-zero algebraic linking number with some other component of $C_m$.\n\\end{defn}\n\n\\begin{lem}\nThe map $L$ is well-defined i.e.\\ \nadmissible decompositions being concordant through finitely many\ncylindrically related admissible decompositions have the same value of $L$.\n\\end{lem}\n\\begin{proof}\nIf decompositions with defining sequences $C_0, C_1, \\ldots$ and \n$D_0, D_1, \\ldots$ \nare cylindrically related, then for every $m \\geq 0$ \n the pairs of components of $C_m$ and the pairs of corresponding components of $D_m$ \n have the same algebraic linking numbers. Suppose for a decomposition there are two \n admissible defining sequences \n$C_0, C_1, \\ldots$ and \n$D_0, D_1, \\ldots$ \n such that $C_0 = D_0$, we have to show that the linking numbers are equal to $0$ simultanously for both of them (for the components of\n $C_0$ and $D_0$ this is obviously true). Of course we know\n that the components are in bijection with each other \n  by the proof of Proposition~\\ref{compnumbsame}\n and in every component of $C_0$\n  after some deformation\n  we have that \n \\begin{itemize}\n\\item\nthe intersection $\\partial C_1 \\cap \\partial D_1$ is empty,\n\\item\nno component of $C_1$ is disjoint from all the components of $D_1$ and vice versa,\n\\item\nevery component of $C_1$ contains one component of $D_1$ or is contained in one component of $D_1$, \n\\item\nno component of $C_1$ contains more than one component of $D_1$ and vice versa.\n\\end{itemize}\nIf a component $T$ of $C_1$ is linked with a component $T'$ of $C_1$\nwith linking number $0$, then \nany knot in $T'$ is linked with $T$ with linking number $0$.\nAlso, if a knot in $T'$ is linked with $T$ with linking number $0$, then \n$T$ and $T'$ are linked with linking number $0$.\nFor every $m \\geq 1$ \nafter a finite number of iterations\nof the algorithm in the proof of Proposition~\\ref{compnumbsame} we get the result.\n\\end{proof}\n\n\n\nOf course \nthe map $L$\nis a monoid homomorphism moreover \nfor a class $x$ represented by a slice decomposition \nwe have $L(x) = (0, 0, \\ldots)$.\n Also, for a class $x$ of the form $[(S^n, \\mathcal D, A)] + [(-S^n, \\mathcal D, A)]$\nwe have $L(x) = (0, 0, \\ldots)$ since all the linking components appear twice.\n\n\n\n\\begin{defn}\nWe call the function \n$$\n\\nu \\co \\Delta_3^a \\to \\mathbb Z_2^{\\mathbb N}$$\nobtained by \n$\\nu ( [x] ) = L(x)$ \n the \\emph{mod $2$ component number sequence} of the elements of $\\Delta_3^a$. \n\\end{defn}\n\n\n\\begin{thm}\\label{uncount1}\nThere are at least uncountably many different elements in the concordance group $\\Delta^a_3$.\nThese can be represented by Antoine decompositions.\n\n\\end{thm}\n\\begin{proof}\nFor every element $(l_0, l_1, \\ldots) \\in \\mathbb Z_2^{\\mathbb N}$, where\n $l_0 = 0$,\n we have an Antoine decomposition representing a class $x$ such that \n $\\nu([x]) = (l_0, l_1, \\ldots)$.\n Hence \nwe get uncountably many different classes in the concordance group.\n\\end{proof}\n\n\n\n\n\n\n\n\\subsection{Computations in the cobordism group}\\label{computebord}\n\n\n\n\n\n\n\n\\begin{prop}\\label{manifcob}\nSuppose that  $n \\geq 0$ and  $M$ is a closed manifold. \nA closed  $n$-dimensional homology manifold $N$\nhaving a resolution   $M \\to N$  \nis cobordant in $\\mathfrak N_n^E$ to $M$.\n\\end{prop}\n\\begin{proof}\nTake $M \\times [0,1]$ and consider the cell-like decomposition $\\mathcal D$ of $M$ which results the homology manifold $N$.\n If $\\mathcal S(X)$ denotes the collection of singletons in a space $X$, \nthen\n$\\mathcal D \\times \\mathcal S([0, 1\/2])$ union $\\mathcal S( M \\times (1\/2, 1] )$ \n is a cell-like decomposition of $M \\times [0,1]$, denote it by $\\mathcal E$.\nWe have to show that \nthe quotient space \n$$M \\times [0,1] \/ \\mathcal E$$\nis an $(n+1)$-dimensional  homology manifold with boundary homology manifolds\n$N$ and $M$.\nTake the closed manifold $$M \\times [0,1]  \\cup_{\\varphi} M \\times [0,1],$$\nwhere\n$\\varphi \\co \\partial (  M \\times [0,1])  \\to \\partial (M \\times [0,1])$\nis the identity map.\nSince $M\/ \\mathcal D$ is $n$-dimensional, \nthe doubling of the decomposition $\\mathcal E$ on \n$M \\times [0,1]  \\cup_{\\varphi} M \\times [0,1]$\nyields\na finite dimensional quotient space, we get this by using \nestimations for the covering dimension, see \\cite{HW41} and \\cite[Corollary~2.4A]{Da86}.\nSo the decomposition space $P$ obtained by factorizing \n$M \\times [0,1]  \\cup_{\\varphi} M \\times [0,1]$\nby the double of $\\mathcal E$ is a closed  finite dimensional homology manifold by  \\cite[Proposition~8.5.1]{DV09}. \nSince this space \nhas an open set homeomorphic to $\\mathbb R^{n+1}$, it is $(n+1)$-dimensional.\nWe obtain the space $M \\times [0,1] \/ \\mathcal E$\nby cutting $P$ into two pieces along two subsets homeomorphic to \n$M$ and $N$. This means that $M$ and $N$ are cobordant in $\\mathfrak N_n^E$.\n\\end{proof}\n\n\nSo if every $3$-dimensional  homology manifold is resolvable, then \n$\\mathfrak N_3^E = 0$.\nAlso note that  the decomposition space $S^3 \/ \\mathcal W$ of the Whitehead decomposition $\\mathcal W$\nis a null-cobordant $3$-dimensional homology manifold, because $[S^3] = 0$.\n\n\n\n\\begin{prop}\\label{cobsubgroup}\nFor $n \\geq 4$ the cobordism  group $\\mathfrak N_n$ is a subgroup of $\\mathfrak N_n^E$.\n\\end{prop}\n\\begin{proof}\nLet $M_{1}$ and $M_2$ be closed manifolds. \nIf the two cobordism classes $[M_1]$  and $[M_2]$ in \n$\\mathfrak N_n^E$ coincide, then \nsince $M_i$ are manifolds, we have $i(M_i)=1$ hence \na cobordism in $\\mathfrak N_n^E$ between $M_1$ and $M_2$ also has index $1$ so\nthis cobordism is resolvable. \nBy \\cite[Theorem~1.1]{Qu83} and Lemma~\\ref{resolcob} there is a manifold cobordism between \n$M_1$ and $M_2$.\n\\end{proof}\n\n\n\n\n\n  In Definition~\\ref{borddecomp} for $i = 1, 2$ the space $M_i\/\\mathcal D_i$ is an $n$-dimensional ENR homology manifold and \n$W \/ \\mathcal E$ is an $(n+1)$-dimensional ENR homology manifold if we add the appropriate collars by Lemma~\\ref{decomphomolmanif}.  \n If $(M, \\mathcal D)$ is such a cell-like decomposition, then \nwe can assign the cobordism class of the decomposition space $M \/  \\mathcal D$\nto the cobordism class of $(M, \\mathcal D)$. \nThis map \n$$\\beta_n \\co \\mathcal B_n \\to \\mathfrak N_n^E$$\n$$ [(M, \\mathcal D) ] \\mapsto [M \/\\mathcal D]$$\nis a group homomorphism.\nThe image of $\\beta_n$\ncontains the classes represented by \ntopological manifolds since trivial decompositions always exist\nand it contains also the classes represented by homology manifolds having appropriate resolutions.\nFor $n = 1, 2$  all the homology manifolds are topological manifolds \\cite{Wi79}\n so the homomorphism $\\beta_n$ is \nsurjective. \nTake the natural forgetting homomorphism \n$$\nF_n \\co  \\mathcal B_n  \\to \\mathfrak N_n$$\n$$\n[(M, \\mathcal D) ] \\to [M].$$\nFor every $n \\geq 0$ the diagram \n\\begin{center}\n\\begin{graph}(6,2)\n\\graphlinecolour{1}\\grapharrowtype{2}\n\\textnode {A}(0.5,1.5){$\\mathcal B_n$}\n\\textnode {X}(5.5, 1.5){$\\mathfrak N_n^E$}\n\\textnode {U}(3, 0){$\\mathfrak N_n$}\n\\diredge {A}{U}[\\graphlinecolour{0}]\n\\diredge {U}{X}[\\graphlinecolour{0}]\n\\diredge {A}{X}[\\graphlinecolour{0}]\n\\freetext (3,1.2){$\\beta_n$}\n\\freetext (1.2, 0.6){$F_n$}\n\\freetext (4.6, 0.6){$\\varphi_n$}\n\\end{graph}\n\\end{center}\nis commutative by Proposition~\\ref{manifcob}, where $\\varphi_n$ is the natural map assigning the cobordism class \n$[M] \\in \\mathfrak N_n^E$ to the cobordism class $[M] \\in \\mathfrak N_n$.\n\n\\begin{prop}\\label{deltaimage}\nFor every $n \\geq 0$ \n the image of $\\beta_n$ is equal to the \nsubgroup of $\\mathfrak N_n^E$ generated by the cobordism classes of topological  \nmanifolds.\n\\end{prop}\n\\begin{proof}\nThe statement follows from the fact that\n$F_n$ is surjective. \n\\end{proof}\n\n\n\\begin{prop}\\label{bordkep}\nFor $n \\geq 1$, we have $\\beta_n(\\mathcal B_n ) = \\mathfrak N_n$ in $\\mathfrak N_n^E$.\n\\end{prop}\n\\begin{proof}\nBy  Proposition~\\ref{cobsubgroup} and Proposition~\\ref{deltaimage} \n we have $\\beta_n(\\mathcal B_n ) = \\mathfrak N_n$ for $n \\geq 4$.\nFor $n = 3$, since $\\mathfrak N_3 = 0$, \nthe statement also holds. For $n = 2$,\nThe group $\\mathfrak N_2$ is isomorphic to  $\\mathbb Z_2$\nso by Proposition~\\ref{deltaimage} it is enough to show that \n$\\beta_2(\\mathcal B_2 ) = \\mathbb Z_2$. But\n$[{\\mathbb R}{P}^2]$ is not null-cobordant in $\\mathfrak N_2^E$\nbecause ${\\mathbb R}{P}^2$ has a non-zero characteristic number as a smooth or topological manifold and then \nby \\cite{BH91} it cannot be null-cobordant.\nFor $n = 1$, of course $\\mathfrak N_1^E = \\mathfrak N_1 = 0$.\n\\end{proof}\n\n\n\\begin{rem}\nInstead of cell-like decompositions, which result homology manifolds, it would be possible to study \ndecompositions which are just homologically acyclic and nearly $1$-movable, see \\cite{DW83}. These result  homology manifolds as well.\n Without being nearly $1$-movable, these can result non-ANR homology manifolds. \n\\end{rem}\n\n\n\n\nAs we could see, the class $\\beta_n([(M, \\mathcal D)])  = [M\/\\mathcal D] \\in \\mathfrak N_n$ \ncould  not expose a lot of things about the decomposition $\\mathcal D$.\nIf we add more details to the homology manifolds and their cobordisms,\n then we could obtain a finer invariant of the cobordism group of decompositions. \n Recall that \n the singular set \n of a homology manifold is the set of non-manifold points, which is a closed set.\n \n\n\\begin{defn}[$0$- and $1$-singular homology manifolds]\\label{0-1-homolmanif}\nA  homology manifold is \\emph{$0$-singular} if its singular set  is a \n$0$-dimensional set.\nA compact homology manifold with collared boundary is \\emph{$1$-singular}  if its singular set $S$\n consists of properly embedded arcs such that $S$ is a direct product \n in the  collar.  \nThe  closed $n$-dimensional  $0$-singular \nhomology manifolds $X_1$ and $X_2$ are \\emph{cobordant} \nif there exists  a compact $(n+1)$-dimensional \\emph{$1$-singular} homology manifold $W$ such that \n$\\partial W$ is homeomorphic to the disjoint union of $X_1$ and $X_2$ and\n $\\partial W \\cap S$ coincides with  the singular set of $X_1 \\sqcup X_2$ under this homeomorphism.\n The set of (oriented) cobordism classes is denoted by \n$\\mathfrak N_n^S$ (and $\\Omega_n^S$).\n\\end{defn}\n\nThe set of cobordism classes\n$\\mathfrak N_n^S$ and $\\Omega_n^S$ are groups with the disjoint union as group operation. \nDenote by $\\mathfrak M_n^0$ the cobordism group of $0$-singular manifolds where the cobordisms are arbitrary but \nthe singular set of the cobordisms is not the entire manifold.\n\n\n\nNote that the representatives of the classes in $\\beta_n(\\mathcal B_n)$ are $0$-singular \nand the cobordisms between them \nhave not only singular points because the boundary has not only singular points since the singular set is a compact $0$-dimensional set. \nThere are  natural homomorphisms\n$$\n i_n' \\co \\mathcal B_n' \\to  \\mathfrak N_n^S \\mbox{, \\ \\ \\ \\ \\ }i_n \\co \\mathcal B_n \\to  \\mathfrak M_n^0 \\mbox{\\ \\ \\ \\ \\ and\\ \\ \\ \\ \\ } \\mathcal B'_n \\to \\mathcal B_n\n $$\n where $ \\mathcal B_n'$ is the version of $\\mathcal B_n$ yielding $0$-singular spaces\n and $1$-singular cobordisms, there is \n  the forgetful map \n$$\n\\varphi_n \\co \\mathfrak N_n^S \\to \\mathfrak M_n^0$$\nand then the diagram \n\\begin{equation*}\n\\begin{CD}\n \\mathcal B'_n @> i_n' >>  \\mathfrak N_n^S    \\\\\n @VVV @VV \\varphi_n V  \\\\\n\\mathcal B_n @> i_n >> \\mathfrak M_n^0  @>  \\psi_n >> \\mathfrak N_n^E \n\\end{CD}\n\\end{equation*}\ncommutes.\nObserve that $\\psi_n$ is injective, $\\varphi_n$ is surjective  and since $\\beta_n ( \\mathcal B_n ) = \\psi_n \\circ i_n ( \\mathcal B_n )= \\mathfrak N_n$, \nthe image $ i_n'(  \\mathcal B_n' )$ is in $\\varphi^{-1}_n \\circ \\psi^{-1}_n(  \\mathfrak N_n )$, which could be a larger group\nthan $\\mathfrak N_n$. \n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\n\n\n\n\n\n\n\n\\section*{Acknowledgements}\nThis work is partially supported by NSF CREST Grant HRD-1736209, NSF grant CNS-1423481, and DoD ARL Grant W911NF-15-1-0518.\n\n\\section*{Acknowledgements}\nThis research is partially supported by NSF Grants CNS-1423481, CNS-1538418, DoD ARL Grant W911NF-15-1-0518.\n\\clearpage \\fi\n\\bibliographystyle{IEEEtran}\n\n\\section{Implementation in AWS}\\label{sec_imp}\nIn this section we present a proof of concept implementation of AB-ITS model in Amazon Web Services (AWS) \\cite{aws}.  We use AWS IoT service along with AWS Greengrass \\cite{grass} (to provide edge functionality) to setup a realistic environment where vehicles are simulated as AWS IoT things. In particular, these stand alone services are implemented as a Lambda function \\cite{lambda} using Boto \\cite{boto} which is AWS SDK for Python. It should be noted that in this implementation no long term GPS data coordinates of vehicles are collected in cloudlets. This reduces privacy concerns of end users and encourages adoption of the proposed model.\n\n\\subsection{Use Cases Overview}\nUS-DOT has proposed an extensive list of ITS applications \\cite{its-app} which we have used to create our real life connect use-cases. Our implementation addresses trust, security and privacy issues concerning end users which must be satisfied before bringing ITS technology in practice. As most applications are safety related, we have considered accident and ice-on-road (tire slip) alerts as our running use-case along with real-time detection and prevention of rogue (or malicious) vehicles on road. In the use-cases, we have also shown how different entities (S, TC, {$\\mathrm{V_T}$}~etc.) fit in the formal model definitions.\n\n\\textbf{Accidental Safety and Ice-Threat :} Moving vehicles (S) can generate warnings for other vehicles ({$\\mathrm{V_T}$}) in their surrounding based on an event which they sensed or encountered. In our use-case, we consider `ice-threat' alerts based on a tire slip wherein vehicles are notified a warning, if any nearby vehicle `feels' it and broadcasts, after satisfying security policies implemented at the edge infrastructures (TC). These policies take into account: who is the source of alert, location of vehicle (ATT) and how many other vehicles encountered similar event, before forwarding (OP) these alerts to other nearby approaching vehicles ({$\\mathrm{V_T}$}). It is possible that a single vehicle (S) sends an ice-threat alert to associated cloudlet (TC), while other vehicles in the area sense no such movement.  Therefore the edge will be able to filter such malfunctioning or deliberate malicious attempt from the vehicle and also notify law enforcement and put that vehicle in  rogue vehicles list. Further, in case of an accident, alert messages will be generated and sent only to police or medical vehicles in the area. Based on the type of alerts and who generates it, policies are defined in the system to ensure trusted, anonymized and relevant notifications.\n\\input{Figures\/comp}\n\\input{Tables\/table_conf1}\n\n\\textbf{Compromised Rogue Vehicles :} Rogue vehicle either intentionally or due to sensor failure can send fake messages to other vehicles. Misbehaving and compromised vehicles must be detected in smart transportation and alerts must be issued immediately to discard the information sent by them. In our use-case, central cloud authority (S) informs edge infrastructures (TC) with a list of detected rogue vehicles and when any message is received by an edge from these vehicles, it is not forwarded to other vehicles. Further, law enforcement is informed about the location (ATT) of a rogue vehicle to prevent fake message dissemination. This approach prevents the need to update and publish revocation list to all vehicles eliminating the bandwidth and connectivity issues.\n\n\\input{Figures\/Policy}\n\n\n\\subsection{Proof of Concept}\nWe will first go over the system configuration along with implemented security policies defined in the cloudlet before we delve into more details of our developed prototype.\n\n\\textbf{System Architecture :} Figure \\ref{fig-comp} represents system architecture along with different components implemented for our prototype. All the vehicles and static smart entities including edge infrastructures must be registered with a central cloud controller to ensure trusted authorized participating entities. Further, the controller also helps in the administrative phase (discussed later) which includes providing a list of edge infrastructures on designated path of moving vehicle. Once the registration is done and vehicles are sent a list of edge infrastructures, the vehicles publish and subscribe to secure (and reserved) MQTT topics created in each of cloudlets which get dynamically assigned based on vehicle current GPS coordinates.\nIt is also possible that the moving vehicle keeps on sending coordinates to the cloud and the controller lets them know the IP address of the nearby edge infrastructures to which the vehicle has to associate.\nThese cloudlets (represented as AWS Greengrass) hold the implemented security policies, a lambda function (similar to policy decision point - PDP \\cite{hu2013guide}) for policy evaluation and the policy enforcement point (PEP) to check messages received, anonymize and filter them and based on the type of alert send them to relevant entities. It should be noted that only alert messages go through the enforcement point, whereas no alerts messages are discarded after logging. Table \\ref{tab-2} lists different AWS system parameters to provide a better understanding of performance metrics shown later in this section.\n\n\\textbf{Security Policies :} We defined attributes based policies which are enforced at the edge, to check who is allowed to send messages, conditions when the message is forwarded to other vehicles and who are authorized recipients for different types of alerts in the system.\n\\input{Figures\/gps}\n\\input{Figures\/seq}\nVarious attributes can be included in policy but for the sake of simplicity we used only vehicle type to determine the source and destination of messages. As shown in Figure \\ref{fig-policy}, security policies are listed in JSON format, where three types of alerts are being generated, `TireSlip', `Accident' and `Rogue' vehicle updates, as denoted by red rectangular boxes.\nWe defined separate set of conditions for each alert type. For example, in `TireSlip' alerts, it is first checked if it is generated (`Source' attribute) by a regular vehicle (specified by attribute value `Vehicle') or by law enforcement (`Police' or `Medical'). Policy then checks number of vehicles which created similar alerts (specified by \"Number\" attribute). Notification to other vehicles depends on how many alerts were generated or who is the source of alert. If the number of alerts are greater than or equal to 2 from regular vehicles, or even a single alert from police or medical vehicle, \"Ice-threat High\" notifications are sent to other associated vehicles of the cloudlet. However, if an alert is generated by one regular vehicle, \"Ice Threat - Low\" is sent for all member vehicles.\nIt must be noted that the sender vehicles and the receiving vehicle must be associated with the same cloudlet to exchange notifications, which also ensure relevance of alerts being received. Similarly, for accident use case, notification is only sent to nearby police vehicles and medical with assistance message. Here the source is not defined, since any smart entity including vehicle, or nearby smart road side sensor or a pedestrian can send message to police or medical vehicles. It is also possible that information about the vehicle including color, license plate number or other identifying information can be sent to law enforcement.\n Another important use case is to enable a central law enforcement that can regularly publish and update the list of rogue vehicles. This list for example, could help locate vehicles that have been stolen or implicated in amber alert\n In the last part of our policy for `Rogue', vehicle IDs \\texttt{Car-X, Car-Y, Vehicle-Z} are stated as rogue and any message from these vehicles is not forwarded. This is a dynamic policy as the list is periodically updated by a central authority. Also to extend the use-case, it is possible when an edge receives a message from a rogue vehicle, it can forward that information to nearby police along with vehicle information like license number and color. The defined policies are only for alert messages, and other `no alerts' messages are just checked by the policy and are logged and dropped without forwarding to any vehicle. Note that policies can also be implemented inside the smart vehicle as well to provide user privacy preference aware notifications, but are not implemented in our prototype\n\n\\textbf{Implementation Details :} The implementation of our proposed solution involves two steps: the administrative phase and the operational phase. Administrative phase includes setup of cloudlets by city administration, setting up the boundaries for each cloudlet, dynamic assignment of moving vehicles to edge infrastructures, and attributes and alerts inheritance from edges to the member vehicles. To be part of ITS, vehicles and smart infrastructures need to have one time registration with central cloud which ensures that smart entities are trusted and benign.\nOnce registered, the moving vehicles can be provided with a mapped list of edges which will arrive in their designated route to which they are allowed to connect. As the vehicles get dynamically associated to different cloudlets, they are able to publish and subscribe to the reserved topics on each edge infrastructures. The operational phase consists of how these attributes and assignment to cloudlets ensure the relevance of alerts to the vehicles and how the edge deployed security policies are used to mitigate security and privacy concerns of users who are using AB-ITS system.\n\n\n\nIn our prototype, we demarcated a big geographic location area into several smaller regions and each region has a trusted cloudlet (TC) which serves all the smart entities in the region as shown in Figure \\ref{fig gps}. We used a python script to simulate the movement of vehicles in the system, shown as green dots, which sends MQTT messages containing GPS coordinates to a central cloud. Service in cloud determines which edge cloudlets are in the surrounding area of the vehicle and then assigns the vehicle to the nearby cloudlets. Following is the sample MQTT payload sent by a moving vehicle to its shadow reserved topic \\texttt{\\$aws\/things\/`Vehicle-Name'\/shadow\/update} in the cloud for dynamic cloudlet assignment:\n\n \\begin{itemize}[leftmargin=*]\n  \\item[] \\texttt{\\{\"state\": \\{\"reported\": }\n      \\item[] \\texttt{\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\; \\{\"Latitude\": \"28.1452683\",}\n  \\item[] \\texttt{ \\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\; \"Longitude\":\"-97.567259\"\\}\\}\\}}\n\\end{itemize}\n\\iffalse\n\\begin{itemize}[leftmargin=*]\n  \\item[] \\texttt{\\{\"state\": \\{\"reported\": }\n      \\item[] \\texttt{\\{\"Latitude\": \"28.14526\",\"Longitude\":\"-97.5672\"\\}\\}\\}}\n\n\\end{itemize}\n\\fi\nAs the path of vehicle is mostly known, these edge assignments can be pro-active in nature as well, mitigating the concern of cloud latency. In such a case, the cloud controller can send a list of edge infrastructures which will be on the designated path of the vehicle to get them associated when vehicles come in their range. It is also possible that these cloudlets have a wireless range and the vehicles which are in the range get automatically assigned to these cloudlets. A vehicle can associate to multiple cloudlets at a time based on their overlapping location. In Figure \\ref{fig gps}, static smart objects like stop warning signs, road work ahead or other  infrastructures have fixed allocation to cloudlets, and the dotted lines represent predicted future cloudlets of vehicle along with current cloudlets by solid pink lines.\n\n\nOnce vehicles get assigned to nearby cloudlets, operational phase starts where the vehicles send messages to its shadow reserved topic (which gets created when the vehicle becomes member of a cloudlet) in their associated edges, which enforce security policies to ensure trusted and authorized alerts to nearby vehicles in near real time manner. In all the policies defined, privacy of the sender is well preserved as the messages do not contain any personal identifiable information and are anonymous. Following is a sample MQTT message sent by vehicle:\n \\begin{itemize}[leftmargin=*]\n  \\item[] \\texttt{\\{\"state\":\\{\"reported\": }\n  \\item[] \\texttt{\\;\\; \\{\"Longitude\": \"29.472741982\", }\n  \\item[] \\texttt{\\;\\;\\;\\; \"Latitude\": \"-98.50038363\", }\n  \\item[] \\texttt{\\;\\;\\;\\; \"Time\": \"2019-03-19 11:27:40.237734\", }\n  \\item[] \\texttt{\\;\\;\\;\\; \"Velocity\": \"30\", \"Direction\": \"north\", }\n  \\item[] \\texttt{\\;\\;\\;\\;\\;\\;\"Elevation\": \"650\", \"Posit. Accuracy\":}\n  \\item[] \\texttt{\\;\\;\\;\\;  \"5\", \"Steering Wheel Angle\": \"0\",}\n  \\item[] \\texttt{\\;\\;\\;\\; \"Alert\": myAlert\\}\\}\\}}\n\\end{itemize}\nIn this message, beside BSM \\cite{bsm} attributes, an attribute \"Alert\" also exists, which defines what kind of alert has been sent from the vehicle to cloudlets. For our use-cases, it can be an \"Accident\", \"Tireslip\", or \"Null\" value where Null signifies no alert. Once the message is received by cloudlet, and is checked against the policies, the edge infrastructure forwards the following Tireslip alert message to a generic topic \\texttt{test\/devices} to which the vehicles subscribe when they become member of the edge.\n \\begin{itemize}[leftmargin=*]\n\\item[] \\texttt{\\{\"message\": \"Ice Threat - Low', }\n\\item[] \\texttt{ 'myEvent': '2019-03-19 10:56:15.921834'\"\\}}\n\\end{itemize}\nIn case of accident alert following message:\n\\begin{itemize}[leftmargin=*]\n\\item[] \\texttt{\\{\"message\":\"Accident- Require Assistance',}\n\\item[] \\texttt{ 'myEvent': '2019-03-19 11:27:40.237734'\"\\}}\n\\end{itemize}\nis sent to topic \\texttt{test\/medical} and \\texttt{test\/police} to which nearby medical and police vehicles are subscribed respectively. Note that event time has also been added to messages, to ensure when the message  is not obsolete. Similarly, for updating the rogue vehicle list from the transportation authority via central cloud to the edge infrastructures, message\n\\begin{itemize}[leftmargin=*]\n\\item[] \\texttt{\\{\"Alert\": myAlert, \"myVehicle\": myVehicle\\}}\n\\end{itemize}\nis sent to \\texttt{test\/Rogue-Vehicle} topic. In this message, 'myAlert' variable can be \\texttt{ADD}, \\texttt{DELETE} or \\texttt{LIST} operation, and 'myVehicle' can hold the vehicles to be added or deleted. In case of list operation, 'myVehicle' attribute value is NULL.\nThe complete sequence of events for the administrative and operational phase in cloudlet supported ITS is shown in Figure \\ref{fig_seq}.\n\\input{Figures\/perform}\n\\iffalse\n\\paragraph{Edge Cloudlet Allocation}\n\n\\paragraph{Dynamic Policy Update}\n\n\\paragraph{Secure and Trusted Communication}\n\\fi\n\\subsection{Performance Metrics and Discussion}\n\nWe evaluated the performance of our proposed AB-ITS model in AWS and provide metrics for the use-cases in proof of concept. We first calculate the execution time for the proposed policy enforcer to evaluate the attribute based security polices (shown in Figure \\ref{fig-policy}) against the number of vehicles associated with a cloudlet and scaling the number of messages sent per vehicle per second. In Figure \\ref{fig-perform} (a) and (b), as the number of vehicles increase (along x axis) with more messages being sent, the enforcer takes more time to evaluate the polices and impact performance. This enforced policy engine in cloudlet has the worse case execution time less than 200 microseconds, for any number of messages sent per second (from 1 to 20) by vehicles which could range from 1 to 50. In case of no-alerts, this execution time will be zero as the policies will not be evaluated.\nTotal trip time performance of our model includes time at which vehicle generates an alert till it is received by target vehicles which includes the policy evaluation time. As shown in Table \\ref{tab-acc} and \\ref{tab-tire}, the total trip time is within the permissible limits ($\\sim$100 ms \\cite{Xu:2004:VSM:1023875.1023879}) for most of the case scenarios. However, the trip time goes beyond the limits when 50 vehicles get associated to single edge cloudlet at one time. The variation in total trip time is due to network traffic and latency, but the average and standard deviation infer that the performance is very comparable to peer to peer ITS. It should be noted that the extra overhead induced by policy execution (in microseconds) is very negligible as compared to the total trip time (in milliseconds). In our approach MQTT protocol has been used, therefore, if some one does not want use DSRC due to cost of transmitter and receiver, our approach can still work with the traditional IoT MQTT based communication based on LTE, 5G or WiFi connectivity.\n\\input{Tables\/perform}\nWe understand that there may be hundreds of vehicles during heavy traffic time, therefore, to scale the system and accommodate all vehicles we can install more cloudlets and infrastructure devices in busy areas that will reduce the number of vehicles which will get associated with single cloudlet at a time. This implementation in AWS showcases the practical viability and use of fine grained polices in context of intelligent transportation system, without the need to capture data points from real world traffic. It must be also noted that, AWS Greengrass has limit of 200 devices per Greengrasss group, which means maximum number of vehicles which can be associated can not be more than 200. We can add more cloudlets in the system which can cater to higher population of vehicles and smart entities.\nAs mentioned earlier, this proposed cloudlet supported V2V and V2I complements the current DSRC approach and is not considered a replacement.\n\n\n\n\n\n\n\n\n\n\n\\section{Introduction and Motivation}\\label{sec_intro}}\n\\IEEEPARstart{F}{uture} smart world will be equipped with technologies and autonomous devices which collaborate among themselves with minimal human interference. Automotive industry is one of the front runners that has quickly embraced this technological change. Connected vehicles (CVs) and smart cars have been introduced, with a plethora of on-board sensors and applications with internet connectivity to offer safety and comfort services to users. Intelligent transportation for smart cities envision moving entities interacting and exchanging information with other vehicles, infrastructures or on-road pedestrians. Federal and private agencies are defining communication standards and technologies for Intelligent Transportation System (ITS) to ensure safety, and address security and privacy concerns of end users.\n\nVehicle to Vehicle (V2V) and Vehicle to Infrastructure (V2I) are two proposed technological innovations which can change current transportation. V2V will enable vehicles to exchange information about speed, location, position, direction, or brake status with other surrounding vehicles where receiving vehicles will aggregate these messages and make smart decisions using on-board applications which will warn drivers about accidents, over-speed, slow traffic ahead, aggressive driver, blind spot or a road hazard. V2I will enable road side units (RSUs) or traffic infrastructures to transmit information about bridge permissible height, merging traffic, work zone warning or road hazard detection to complement V2V applications. Vehicle to pedestrian (V2P) is also envisioned to cater to pedestrians, such as with visual or physical impairments, and send corresponding alerts to approaching vehicles. These communication technologies will use Dedicated Short Range Communications (DSRC) \\cite{dsrc} to exchange data packets, called Basic Safety Messages (BSMs) \\cite{bsm}, with nearby vehicles and entities between 300-500 meters range. Messages will be sent up to 10 times per second providing a 360-degree view of proximity, with on-board applications using the information for triggering alerts and warnings. US Department of Transportation (DOT) and National Highway Traffic Safety Administration (NHTSA) estimate around 80\\% of non-impaired collisions \\cite{us-dot,us-dot-1} and 6.9 billions traffic hours can be reduced by using V2V, V2I and V2P communications.\n\nVehicles in ITS are communicating and exchanging information with external entities including toll booths, gas stations, parking lots, and other vehicles, which raises security and privacy issues. Incidents on Jeep and Tesla \\cite{jeep,tesla} have been demonstrated where car engine was shut and steering wheel controlled remotely by adversaries. These smart cars are equipped with 100's of electronic control units (ECUs) and more than 100 million lines of code, thereby, exposing broad attack surface for critical car systems including transmission control, air-bag, telematics, engine or infotainment systems. In-vehicle controller area network (CAN) bus also needs security to prevent unauthorized data exchange and tampering among ECUs and software manipulation. Cyber attacks on smart connected vehicles \\cite{gao,nhtsa-1,nhtsa-2,elmaghraby2014cyber} include: unauthorized over the air updates (OTA) for firmware, stealing user private data, spoofing sensors, coordinated attacks on road side infrastructure or malware injection. Dynamic and mobile nature of V2X (Vehicle to everything) communication makes it additionally difficult to secure the distributed system where vehicles will be exchanging data with random unknown entities on road. Impersonation and fake message from malicious compromised vehicles is also a grave concern as the information exchanged is used by other vehicles to make alerts and notifications. Vehicle users also have privacy concerns where every movement can be tracked continuously by agencies or data collected from vehicles can be used to extrapolate personal identifiable information (PII).  These concerns lead to reluctance in embracing these future transportation technologies.\n\nAttribute-based access control (ABAC) \\cite{jin2012unified,hu2013guide,gupta2016mathrm} provides fine grained authorization capabilities for resources in a system. This mechanism offers flexibility in a distributed multi-entity dynamic environment where the attributes of entities along with contextual information are used to make access and communication authorization decisions. Intelligent transportation system involves interaction and messages exchange among entities with no prior association. Attributes of vehicles or transportation infrastructure can be used to authorize communication decision based on their current location, ownership or degree of trust. Such security mechanisms can help to prevent fake messages, stop rogue vehicles and ensure privacy aware message communication besides ensuring location and time sensitive relevance of exchanged information.\n\nIn this work, we present a privacy-aware secure attribute-based V2V and V2I communication architecture and model using trusted cloudlets. These cloudlets are setup in wide geographic locations with defined coverage area.  Each cloudlet will receive messages from vehicles in its range and forward it to all other vehicles associated with that cloudlet. Vehicles are dynamically assigned to these cloudlets as they move along geographic boundaries based on their GPS coordinates and predicted path. An important benefit of this indirect V2V and V2I communication is the deployment of security policies at edge cloudlets which can restrict or block fake messages, and ensure trustworthiness in communication. Moreover edge cloudlets also enable message anonymization and user privacy, as the receiver cannot detect who is the sender as all messages come through edge infrastructures. These cloudlets can also be used to forward certificate revocation lists (CRLs) to vehicles in the range beside blocking the vehicles themselves. Rogue vehicle list can be dynamically updated at the edges, and messages from a vehicle in the rogue list can be blocked. The proposed architecture and attribute-based policies ensure the important security properties of message integrity, originator authenticity and user privacy concerns in V2V and V2I communication. This MQTT \\cite{mqtt} based approach for messages exchange can be used in addition to DSRC to enable use cases with acceptable latency (discussed in implementation section) without the need for additional hardware cost\\footnote{NHTSA proposed V2V equipment and communication is between \\$341 to \\$350 per vehicle in 2020 \\cite{faq}} and work with familiar technologies such as WiFi, LTE or 5G. This work proposes a formalized communication security model for V2V and V2I called attribute-based intelligent transportation system (AB-ITS). We have implemented our proposed architecture and model using AWS and collected several performance metrics, which reflect the plausibility and efficiency of our proposal.\n\n\nRest of the paper is as follows: Section \\ref{sec_related} discusses related work along with USDOT proposed Security Credential Management System (SCMS). Security and privacy requirements along with the proposed cloudlet supported ITS architecture is given in Section \\ref{sec_arch}. Section \\ref{sec_model} presents formal attribute-based V2V and V2I communication model (AB-ITS). Section \\ref{sec_imp} describes our implementation with real-world use cases using AWS, and discusses performance parameters. Section \\ref{sec_summary} concludes the paper.\n\n\\iffalse\nDepartment's National Highway Traffic Safety Administration (NHTSA) estimates that safety applications enabled V2V could eliminate or mitigate the severity of up to 80\\% of non-impaired crashes, including crashes at intersections or while changing lanes.\nOn the other hand, when such a vehicle connects with road infrastructure, the exchange of information such as information of road signals, weather condition, and nearby traffic condition takes place.\n\nIt would enable vehicles to transmit their location, speed, direction and other information 10 times per second. That lets cars detect, for example, when another vehicle is about to run a red light or coming around a blind turn in time to prevent a crash.\n\\fi\n\n\n\n\n\n\\section{Discussion and Limitations}\\label{sec-limit}\n\\section{ Cloudlets Enabled Attribute Based V2V and V2I Communication}\\label{sec_model}\nEdge cloudlets supported V2V and V2I communication has many advantages, as discussed in previous section. These cloudlets can support attributes based fine-grained policies based on which communication decisions can be made. These attributes offer flexibility and take into account different environmental factors along with dynamic policies based on administrator needs. Further, individual users are also allowed to set their own privacy preferences, to decide on what and from whom messages are allowed to receive. In this section, we formally define our proposed cloudlets supported attributes based intelligent transportation system model, which we refer to as AB-ITS.\n\n\\subsection{AB-ITS Communication Model}\n\\input{Figures\/model}\nThe conceptual AB-ITS communication model is shown in Figure \\ref{fig_model} and formal definitions elaborated in Table \\ref{tab 1}. The model has following components: Vehicles (V), Transportation Infrastructure Devices (I), Users (U), Sources (S), Trusted Cloudlets (TC), Target Vehicles ({$\\mathrm{V_T}$}), Operations (OP), Authorization Policies (POL), and Attributes (ATT).\n\n\\noindent\n\\textbf{Sources (S) :} A source initiates operations on cloudlets (discussed below) in the system. A source can be from a set of vehicles (V), transportation infrastructure (I) or an administrator user (U). For instance, in case of V2V communication, a source is a vehicle which wants to send messages to other vehicles in its vicinity. Similarly, law enforcement and city administration can initiate theft and accident alerts in a particular area via cloudlets, which are forwarded to all vehicles associated with cloudlet.\n\n\\noindent\n\\textbf{Trusted Cloudlets (TC) :} Cloudlets are introduced, which are trusted edge infrastructures set up across locations and facilitate secure V2V and V2I communication. These cloudlets have a limited geographic range and all vehicles in that range get associated with one or more TCs automatically based on their moving location coordinates. Any communication between vehicles and other entities including transportation infrastructures (or RSUs) is done via TC, which checks security policies to forward or block the messages sent by different sources. Also TCs have attributes which are propagated to associated vehicles and can also help setting alerts and warnings based on attribute values. For instance, when a vehicle enters forest and gets associated with the cloudlet, it can automatically inherit a wildlife area attribute ON from TC.\n\n\\noindent\n\\textbf{Target Vehicles ($\\mathbf{V_T}$) :} These vehicles are subset of total vehicles (V) in the transportation system and are potential receiver of messages sent by a source. Both target vehicles and source must be associated with same TC to enable V2V and V2I communication.\n\n\\noindent\n\\textbf{Operations (OP) :} Operations are actions which are performed by source on TC. TC also execute operations against associated vehicles and infrastructures. For example, a source initiating a join operation to get associated with a TC, or trying to send a message to vehicles via TC. Also, TC forwarding a message sent by sources to its member vehicles is another example of operations in ITS. These also include administrative actions performed by a user including updating, deleting or adding attribute values for an attribute or rogue vehicles list in TC.\n\n\\noindent\n\\textbf{Authorization Policies (POL) and Attributes (ATT) :} Sources, TCs, vehicles and other relevant ITS entities can have personal defined individual policies along with system wide authorization policies needed for the overall secure functioning of the ecosystem. Vehicles owners can set individual privacy preferences which enable them to allow or disallow any particular private information from being shared with a third party remotely. Similarly, city traffic department may set its own rules when to trigger an alert or warnings to vehicles in a sensitive or accident prone area. Administrative policies are also needed to authorize a legitimate user to change attributes, send notifications to TCs or update rogue vehicles list. Entities like vehicles and sources also have individual characteristics, called attributes, which are used to make authorization and communication decisions in ITS. For a vehicle, sample attributes can be: vehicle ID, speed, heading angle, brake, vehicle size, vehicle type or preferred notifications. Vehicles and infrastructure can also inherit attributes from their associated TCs, which can have common location wide attributes like speed limit, road work ahead or blind turn.\n\nBoth attributes and policies are dynamic which can be changed by administrators or vehicle owners based on system needs and personal preferences. The attributes of vehicles like location, speed or heading angle are continuously changing, but other attributes like vehicle size remain static. Policies are also dynamic in nature, as reflected in use-case implementation in the next section, where we defined a security policy with a list of black-listed rogue vehicles which are notified to law enforcement when detected by TCs. This list is dynamic in nature and is continuously updated by administrators, demonstrating how dynamic policies are used and enforced in ITS.\nIt must be noted that in a session the proposed model assumes a static set of policies and attributes which are used to make V2V and V2I communication decision. All relevant polices including system defined and user preferences are evaluated to make the final communication decision.\n\\input{Tables\/table_model-new}\n\nIn our proposed model, TCs evaluate security policies and ensure that un-trusted or fake messages are not forwarded to associated vehicles in its geographic coverage boundary. These connected vehicles must initiate association with TCs pro-actively based on their predicted path, and once they get into the range of the TCs, vehicles become the member of TC. Such communication with TCs can be done using encrypted and secure cellular or WiFi technologies with no added equipment cost. It should be noted that our model complements the proposed DSRC based direct V2V and V2I communication, and can be used to assist in situations where the authenticity and integrity of messages is much needed. Our use-cases in the next section will highlight the real world enforcement and use of AB-ITS model.\n\n\n\\subsection{Formal Definitions}\n\nTable \\ref{tab 1} elaborates the formal AB-ITS communication model definitions, which comprise of vehicles (V), transportation infrastructure devices (I), administrative users (U) and edge cloudlets (TC). A source in S initiating an operation op $\\in$ OP can be from a set of vehicles, transportation infrastructures or users, whereas target vehicles {$\\mathrm{V_T}$}~is a subset of total vehicles in the entire transportation system ({$\\mathrm{V_T}$}~$\\subseteq$ V). Attributes are functions defined for source and edge cloudlets, where functions can be set or atomic valued (stated by attType) and are assigned values from Range(att) for each att $\\in$ ATT. The atomic valued attributes are assigned single value including null (denoted as $\\bot$) whereas set valued attribute can have a subset of values assigned from power set of the range of attribute function. Some attributes are also defined system wide, which reflect the state of entire transportation system (like level of threat or city traffic) and are set by administrators. Authorization security policies are defined for individual sources and TCs, which are either stated based on personal privacy preferences or are enforced system wide as defined by central administrators. For example, a driver may not want to receive marketing commercials on dashboard, so she can set such personal preference as choosing the desired policy, whereas police can define a policy with a list of black-listed cars and blocking communication from them.\n\nSource and target vehicles are dynamically assigned to one or many trusted edge cloudlets based on their current GPS coordinates and predicted path as defined by associated\\_cloudlets function. The association with edge cloudlets is fixed for transportation infrastructures or administrators which are assigned at the time of system deployment whereas for vehicles it keeps on changing as the vehicles move. Each cloudlet has defined geographic coverage area and when vehicles enter the area, they get associated with the cloudlet. A vehicle may be associated with multiple cloudlets in areas where coverage areas are overlapping, thereby, a vehicle is always associated to at least one cloudlet at all times. These cloudlets mediate the V2V and V2I communication by enforcing security policies, stop fake messages and ensure privacy, as discussed later in the model definitions. Further, sources (including vehicles) inherit attributes from their associated cloudlets, which helps in administration and propagation of common attributes to all associated entities with single administrative action. For instance, at a location where flash flood warning is issued, the edge cloudlet installed there will set attribute flash-flood = ON for all its associated vehicles when they become members of that cloudlet. In case of set valued attribute function, the effective attribute values for att $\\in$ ATT of source (defined as $\\mathrm{\\ensuremath{\\mathrm{effS}}_{att}}$), including target vehicles, is the union of direct values assigned to the source for attribute att and the values assigned to att for each associated cloudlets. However, in case of atomic valued attribute, it is necessary to define which attribute values take precedence when multiple edge clouds are associated. In our model, we propose that most recently connected cloudlet with non null value for the attribute will be inherited by the associated source or vehicles.\\footnote{There are other approaches also to deal with atomic value inheritance, but for moving vehicles which are dynamically assigned to new cloudlets, we believe this approach is the most appropriate and relevant.} For example, the speed-limit attribute of most recently associated cloudlet will be populated for all member vehicles, and as the vehicle moves, this value is inherited from next associated edge cloudlet and so on. This inheritance in atomic values attribute only takes place when edge cloudlets have non null values, whereby with all associated cloudlets having null values, the direct attribute value of the source holds as its effective value also.\n\nAuthorization functions are parameterized propositional logic formulae defined to represent access control security policies stated in the policy language defined in Table \\ref{tab 1}. The function $\\mathrm{Auth_{op}}$(s:S, tc:TC) specify conditions under which source s (including vehicles) can perform an operation op $\\in$ via cloudlet tc $\\in$ TC. These boolean authorization functions are evaluated substituting actual arguments for formal parameters along with direct and effective attributes values of actual arguments. Similar syntax and policy language can be defined for other set of policies including personal vehicle specific policies or system wide policies with attributes of relevant entities substituted in authorization requests evaluation.\nAuthorization decision to allow {$\\mathrm{s^\\prime}$}~$\\in$ S to perform an operation op $\\in$ OP on {$\\mathrm{tc^\\prime}$}~$\\in$ TC is determined when the authorization function is evaluated with the actual arguments ({$\\mathrm{s^\\prime}$}~$\\in$ S, {$\\mathrm{tc^\\prime}$}~$\\in$ TC) to be True.\nSimilarly, the decision for operation op from {$\\mathrm{tc^\\prime}$}~$\\in$ TC to {$\\mathrm{v^\\prime}$} $\\in$ {$\\mathrm{V_T}$}~is made by calling the relevant authorization function with actual parameters.\n\n\nAs discussed in authorization property, the model has defined two primitive operations, `send' and `forward' relevant for V2V and V2I communication. A source uses `send' operation (defined as  $\\mathrm{Auth_{send}}$({$\\mathrm{s^\\prime}$} : S, {$\\mathrm{tc^\\prime}$} : TC)) to communicate a `send message' to trusted cloudlet, whereas `forward' operation (defined as $\\mathrm{Auth_{forward}}$({$\\mathrm{tc^\\prime}$} : TC, {$\\mathrm{v^\\prime}$} : {$\\mathrm{V_T}$})) is between trusted cloudlet and target vehicle defining a `forward message'. For allowing, communication from {$\\mathrm{s^\\prime}$} to {$\\mathrm{v^\\prime}$} requires a common {$\\mathrm{tc^\\prime}$}~to which both {$\\mathrm{s^\\prime}$} and {$\\mathrm{v^\\prime}$} are associated and the required authorization functions for send and forward messages i.e  $\\mathrm{Auth_{send}}$({$\\mathrm{s^\\prime}$} : S, {$\\mathrm{tc^\\prime}$} : TC) and $\\mathrm{Auth_{forward}}$({$\\mathrm{tc^\\prime}$} : TC, {$\\mathrm{v^\\prime}$} : {$\\mathrm{V_T}$}) as well as the system defined security policies evaluate to True. Additional relevant operations and messages can be similarly defined.\n\nThe proposed AB-ITS model leverages attributes and GPS coordinates of communicating entities to enable and secure V2V and V2I communication. The introduction of trusted cloudlets provide benefits of enforcing security policies at the edge to stop fake messages, enhance user privacy and integrity of messages before forwarded to other target vehicles. These edge cloudlets ensure low latency and near real time communication much needed in most ITS applications without bandwidth issues. It must be noted that the messages shared among source and vehicles are end to end encrypted and can still use the proposed DSRC wireless technology for communication with cloudlet and then to the vehicles. Our model complements the USDOT proposed V2V and V2I architecture functionalities and support applications which need additional message integrity and confidentiality, and can be used as an add on to current ITS peer to peer communication.\n\n\\iffalse\nAuthorization functions are parameterized propositional logic formulae defined to represent access control security policies stated in the policy language as shown in Table \\ref{tab 1}. We have stated syntax for two authorization functions $\\mathrm{Auth_{op}}$(s:S, tc:TC) and $\\mathrm{Auth_{op}}$(tc : TC, v : {$\\mathrm{V_T}$}), which specify conditions under which source s can perform operation op $\\in$ OP on tc, and policies when tc can perform operation op on vehicle v respectively. These boolean authorization functions take formal parameters which are passed when function call is made using the actual arguments. The values of direct and effective attributes of these formal parameters (s $\\in$ S, tc $\\in$ TC, v $\\in$ {$\\mathrm{V_T}$}) are used to evaluate the authorization policy. Similar syntax and policy language can be defined for other set of policies including personal vehicle specific policies or system wide policies with attributes of relevant entities included in authorization requests evaluation. Authorization decision for allowing {$\\mathrm{s^\\prime}$}~$\\in$ S to send or perform an operation op $\\in$ OP on {$\\mathrm{tc^\\prime}$}~$\\in$ TC is determined when the authorization function call is made using the actual parameters ({$\\mathrm{s^\\prime}$}~$\\in$ S, {$\\mathrm{tc^\\prime}$}~$\\in$ TC) and if function returns True, the operation op is allowed. Similarly, the decision for operation op from {$\\mathrm{tc^\\prime}$}~$\\in$ TC to {$\\mathrm{v^\\prime}$} $\\in$ {$\\mathrm{V_T}$}~is made by calling the relevant authorization function with actual parameters. The communication decision for a message from a source {$\\mathrm{s^\\prime}$}~to vehicle {$\\mathrm{v^\\prime}$}~requires a common trusted cloudlet {$\\mathrm{tc^\\prime}$}~to which both {$\\mathrm{s^\\prime}$}~and {$\\mathrm{v^\\prime}$}~are associated, and the required authorization policies including system defined policies are taken into account to make the final ITS V2V and V2I communication decision. The following authorization property must hold true in AB-ITS model:\n\n\nFor allowing, communication and message flow from {$\\mathrm{s^\\prime}$} to {$\\mathrm{v^\\prime}$} requires a common {$\\mathrm{tc^\\prime}$} to which both {$\\mathrm{s^\\prime}$} and {$\\mathrm{v^\\prime}$} are associated and the required atomic authorization functions i.e  $\\mathrm{Auth_{send}}$({$\\mathrm{s^\\prime}$} : S, {$\\mathrm{tc^\\prime}$} : TC) and $\\mathrm{Auth_{forward}}$({$\\mathrm{tc^\\prime}$} : TC, {$\\mathrm{v^\\prime}$} : {$\\mathrm{V_T}$}) as well as the system defined security policies must return True.\n\n\nThe proposed AB-ITS model leverages attributes and GPS coordinates of communicating entities to enable and secure V2V and V2I communication. The introduction of trusted cloudlets provide benefits of enforcing security policies at the edge to stop fake messages, enhance user privacy and integrity of messages before forwarded to other target vehicles. These edge cloudlets ensure low latency and near real time communication much needed in most ITS applications without bandwidth issues. It must be noted that the messages shared among source and vehicles are end to end encrypted and can still use the proposed DSRC wireless technology for communication with cloudlet and then to the vehicles. Our model complements the USDOT proposed V2V and V2I architecture functionalities and support applications which need additional message integrity and confidentiality, and can be used as an add on to current ITS peer to peer communication.\n\n\\fi\n\n\n\n\\section{Proposed Cloudlets Supported ITS Architecture}\\label{sec_arch}\nThe current peer to peer V2V and V2I communication as represented in Figure \\ref{fig_v2v} is proposed to use SCMS to ensure secure trusted basic safety messages exchange among entities. However, the vast and complex scale of this PKI based system has user privacy and security concerns which need to be addressed before its deployment. In this section, we will discuss security and privacy requirements of ITS and smart cars ecosystem and highlight how the proposed trusted cloudlets supported communication offers the required security and complements current solutions.\n\n\\subsection{Security and Privacy Requirements}\n\\input{Figures\/architecture}\nIntelligent Transportation System (ITS) involves real time sharing of location and sensitive information about vehicles and passengers, which pose a serious privacy threat and a strong deterrent for its adoption. Dynamic and distributed ITS will enable interaction with random entities on road with no prior trust established, and the information sent from these smart vehicles will be used by on-board applications to provide safety and warning signals, which itself has some inherent security risks. An adversary can compromise a road-side unit or vehicle to send fake information about traffic or accident, which can trigger unnecessary alerts and may distract drivers. Basic safety messages (BSMs) are designed to contain no personal identifiable information (PII) and are attached with a certificate issued by certificate authority in SCMS. However, limited number of certificates and number of messages sent per minute can reveal the identity of a targeted vehicle with advanced computer techniques. Untrackability of vehicles and users is paramount to ensure privacy in ITS. Also, the system must not save personal or individual information and use it as law enforcement or issuing speeding tickets. Anonymity of sender must always be maintained. Over the air messages exchanged among smart entities must have integrity, and authenticity. Security mechanisms to protect smart cars and their critical systems from unauthorized access, control and tampering are important to strengthen intelligent transportation. Integrated approach of DSRC and cellular technologies is needed based on different ITS applications. Cloud and cloudlets supported architectures will provide resiliency and reduce system stress. Encrypted and secure data transfer link is the backbone needed from DSRC, cellular LTE or any communication technologies involved in ITS. However, limited bandwidth and latency issues in cloud connectivity needed for certificate updates and revocation needs attention\n\nIn smart city, location based notifications for connected vehicles must allow user to have personal preferences where a user may want weather warning and not parking advertisements on board. Dynamic policies are required, for example, in case of a traffic jam in an area a policy may ask all drivers to follow route A but considering the heavy traffic on route A, the policy may be changed to move traffic to route B or C. This can be implemented at the edge level and triggered by central administrators. In such a case, whether the administrative subject is authorized to change the policy or trigger an alert, also needs security checks.\n\n\n\n\\subsection{How Cloudlets Can Provide Security?}\nFigure \\ref{fig_arch} shows the proposed edge supported architecture for V2V and V2I communication. Trusted edge infrastructures (setup by city administration) will work as a middle man and relay messages to vehicles and other entities inside its geographic range. Instead of peer to peer connection, all vehicles publish to edges, where security policies defined are checked to ensure validity and integrity of the communication, and relevance of messages, before forwarding to other vehicles. A vehicle can be in range of multiple infrastructures, depending on its location.  Each vehicle will be dynamically associated with edges as it moves. All participating vehicles and RSUs still need to enroll with a central authority to be part of the system, to ensure that only trusted vehicles are allowed to exchange messages among themselves. Communication technologies used for vehicles to cloudlets can be cellular LTE, WiFi or DSRC. MQTT messaging protocol can be used, as discussed in implementation which will obviate the cost of DSRC equipments needed in smart cars. The proposed architecture is implemented in addition to V2V and V2I direct communication and is supported in NPRM \\cite{nhtsa-6} documents which recommend both DSRC and secondary communication for ITS.\n\nTrusted cloudlets installed in wide geographic area offer the needed fog infrastructure functionality required in an IoT environment. They can address security concerns by deploying and enforcing security policies to ensure trusted communication among smart entities on the road.\nThis proposed architecture offers an alternate edge supported V2V and V2I communication with minimal message latency and in permissible time limits \\cite{Xu:2004:VSM:1023875.1023879,articleV2V}.\nA vehicle sending and receiving BSM or other messages, must be associated with an edge infrastructure, which will enforce policies, sanitize messages, prevent fake messages dissemination and offer administrative advantages. Each cloudlet will have a geographic range and all the vehicles within it will get associated with the edge automatically. Since the range of edge is within a restricted limited area, it also ensures location sensitivity of messages exchanged, as vehicles communicating messages must be associated to a common edge cloudlet. Message anonymity and sanitization can be done, since\nthe messages sent by a vehicle are relayed via the edge cloudlet without direct peer to peer communication, which will have less security and privacy implications.\n\nFurther, using cloudlets offers administrative benefits as single notification from edge infrastructure will trigger alerts for all the vehicles which are connected to it in a geographic range. If an agency or a police vehicle wants to send alerts, instead of sending to each individual vehicle, they can send it to a trusted cloudlet, which after checking the policies to ensure the sender is allowed to generate such requests, forwards or stops the message.\nAlso, entities present in a particular area have certain characteristics (for example, stop sign warning, speed limits, deer-threat, flash flood warnings etc.) in common, which can be inherited by getting dynamically associated to edge infrastructures, without the need to generate messages 10 times per second \\cite{nhtsa-3} to get this information from other vehicles or RSUs saving network bandwidth.\n\nIt is also possible to limit the messages to a specific set of vehicles, for example, in case of a kidnapped child warning, messages can be sent to nearby edge infrastructures and then to only police vehicles in the area, and not to the common public using security policies defined at the cloudlet. Edge infrastructure can also have the capacity to filter unwanted and incorrect messages from the vehicles and infrastructure using a majority rule policy.  For example, if an adversary is sending accident message (either deliberately or a malfunction sensor on vehicle) to subvert the traffic whereas other vehicles notify no accident and clear traffic messages, installed trusted edge will have the intelligence and policy to filter such fake messages and forward the correct information to its associated vehicles.\nThis will not be possible in peer to peer V2X (vehicle to anything) architecture immediately, until certificate revocations (by a central authority) are propagated to individual vehicle, which may take time and also require internet connectivity which cannot be guaranteed all times in terrains where the vehicle is moving. Also, instead of sending CRLs to each vehicle, only edge servers can be sent with list of revoked certificates and based on the information, edge can decide if the messages sent by vehicle should be forwarded or not.\n\nFurther, if an adversary is detected by an edge with fake or wrong messages, policies can be defined to inform appropriate agencies and law enforcement in the area where such malicious behaviour is detected. It is also possible to have different levels of alerts based on the degree of trust and who is the sender. Law enforcement initiating a bomb threat in the vicinity will be treated as major threat and edge infrastructure states it as code red alert, with immediate rerouting and emergency exit directions.\n\\iffalse\nV2V-- Forward Collision Warning, Emergency Electronic Brake Light, Blind Spot\/Lane Change Warning, Do Not Pass Warning Intersection, Movement Assist Left Turn Assist\n\nv2I-- Curve Speed Warning Red Light Violation Warning\n\\fi\n\n\n\n\n\n\n\n\\section{Related Work}\\label{sec_related}\nConnected and smart vehicle applications need wireless exchange of V2X messages among unknown moving vehicles, RSUs and pedestrians. The proposed intelligent transportation system (ITS) for future cities has underlying technologies, security concerns and proposed solutions, which we briefly review in this section.\n\n\\subsection{Security Credentials Management System}\nUnited States Department of Transportation (USDOT) has suggested a PKI-based security infrastructure system, called Security Credentials Management System or SCMS \\cite{its-scms2, its-scms1}, to ensure trusted V2V and V2I communication among random moving entities. Authorized participating vehicles use digital certificates issued by SCMS to validate and authenticate basic safety messages (BSMs), by attaching these certificates with each message to ensure integrity, confidentiality and privacy of the communication. Vehicles need initial enrollment into SCMS to obtain security certificates from trusted certificate authorities (CA). Each BSM will include vehicle related information digitally signed using private key corresponding to the digital certificate attached with BSM. Different certificate types are used including enrollment, pseudonym and identification for vehicle and enrollment applications for RSUs. Certificates can be cancelled for potential adversaries or reported misbehaving vehicles by CAs by disseminating certificate revocation lists (CRLs). USDOT and NHTSA claim \\cite{us-dot-1} that BSMs will exchange anonymized information and no personal identifiable data will be shared with other entities. SCMS is considered as a central system to be trusted by entities participating to revolutionize transportation\n\nHowever, there are some challenges \\cite{scms-issues,scms-issues1} that need to be addressed before the system is deployed. Each vehicle will receive 20 certificates weekly to sign the BSMs \\cite{nhtsa-5}, which will rotate every 5 minutes. Therefore, a vehicle will use a new set of 20 certificates every 100 minutes. In such a scenario a computer can analyse all the certificates a vehicle used in a day and then use these certificates to track it for a week. Although, PKI based SCMS system ensures who signed the certificate, it is difficult to prove how correct or true the information sent  from the vehicle is. A malfunctioning device in the vehicle can result in false BSMs exchanged even though the sender is trusted. Further, the proposed SCMS system will be largest and complex ever built producing 265B to 800B certs\/year depending on weekly rate supporting 17M vehicles\/year \\cite{scms-issues1}. The revocation of certificates for bad actors would result in pushing CRLs to all enrolled vehicles, which will be time and bandwidth consuming.\n\n\\subsection{Relevant Background and Technologies}\n\nSeveral general IoT architectures \\cite{atzori2010internet,gubbi2013internet,alshehri2016access} have been proposed with different middleware layers in multi-layer stack representing physical objects, communication or service layer, cloud and end-user applications. Gupta and Sandhu proposed \\cite{gupta2018authorization} enhanced access control oriented architecture (E-ACO) particularly relevant to smart cars and intelligent transportation. The work introduced clustered objects (smart objects with multiple sensors like cars) as component of object layer which interact with other objects similar to V2V and V2I communication. As shown in Figure \\ref{fig_eaco}, E-ACO architecture has four layers: \\textbf{Object Layer} at the bottom representing physical objects including connected cars, vehicles and RSUs. \\textbf{Virtual Object Layer} maintains cyber entity (like an AWS shadow stored as JSON) of each physical object which is imperative in a moving and dynamic ecosystem like smart cars, where the connectivity of a vehicle is not continuously guaranteed. With virtual objects, when direct communication with physical object is not possible, its virtual entity maintains last reported and desired state information. Further, it resolves the issues of heterogeneity as objects support different communication technologies. Using virtual objects, physical entities communicate with corresponding virtual objects where messages are exchanged with virtual entities of other object which is then passed to actual physical object. \\textbf{Cloud Services and Application Layer} together harness data sent by physical objects and use it to extrapolate value, analytics and provide end user cloud supported applications.\n\n\\input{Figures\/v2v}\n\nSmart cars security incidents including Jeep \\cite{jeep} and Tesla Model X \\cite{tesla} hacks have demonstrated how engine was stopped\nand steering remotely controlled exhibiting cyber threats. Security and privacy issues in smart cars and ITS are serious concerns where several federal agencies are working along with industry partners to ``fully'' proof the system before final deployment and use by common public. European Union Agency for Network and Information Security (ENISA) \\cite{enisa} has studied vulnerable assets in smart cars with related threat and risks, and proposed some prevention approaches with recommendations. Cooperative Intelligent Transport Systems (C-ITS) \\cite{c-its1,c-its2} also highlighted the need of data communication integrity and authenticity in V2V and V2I, and proposed PKI based trust model using pseudonym certificates. NHTSA report \\cite{nhtsa-3} has thoroughly explored the technical, legal and policy related issues pertinent to V2V communication and studied technological solutions for safety and privacy issues. US Government Accountability Office (GAO) \\cite{gao} has also discussed security risks and potential attack surfaces in smart vehicles, and  proposed solutions to prevent cyber threats.\n\nAttribute based access control \\cite{hu2015attribute,jin2012unified,gupta2018attribute} provides fine grained authorization capabilities most appropriate in dynamic and distributed systems similar to ITS. Recently dynamic groups and ABAC model \\cite{Gupta:2019:DGA:3292006.3300048,gupta2019secure} was proposed for smart cars ecosystem which caters to mobile needs of vehicles. However the model is more suitable to cloud assisted applications and a real time V2V and V2I edge supported model is still missing. Role based access controls \\cite{sandhu1996role,ferraiolo2001proposed} were designed particularly for enterprise applications with a limited set of roles and administrators assigning roles to users. Similar concept does not seem to fit dynamic and random unknown IoT smart cars setting where devices and vehicles are in different administrative domains spread across geographic area.\n\n\n\n\n\n\\section{Summary}\\label{sec_summary}\n\nThis research work proposes a cloudlet assisted secure V2V and V2I communication in intelligent transportation system, which ensures trusted and reliable messages exchange among moving entities on road. We introduce the novel notion of dynamic edge associations in which the smart entities get connected to different pre-installed cloudlets on road, which help them relay the basic safety messages and perform the needed filtering and reduces privacy concerns of the users. These cloudlets can anonymize the messages, ensure trustworthiness and ensure their relevance to entities which receive them. We also present the formal model which specifies attributes based polices for V2V and V2I communication. Several use-cases of ITS have been discussed along with implementation in Amazon Web Services (AWS). Performance has been evaluated against time taken to evaluate the polices in cloudlets and the total trip time from the moment message is generated till it gets received and relayed by the cloudlets. In future work we would incorporate additional privacy preserving approaches wherein the exact location GPS coordinates are not required to be shared with cloud. The work can be complemented using homomorphic encryption or other similar approaches which will further mitigate privacy concerns of the users.\n\\iffalse\n---To ADD---\n\n\n\nPERFORMANCE---\n\nAs more cars are in the GG, it do not impacr performace as all assigned to same topic (Need conforamtion)\n\\fi\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}