diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzznieb" "b/data_all_eng_slimpj/shuffled/split2/finalzznieb" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzznieb" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\nDecay estimate of solutions is a fundamental problem in the qualitative analysis of evolution equations. In most cases this problem can be reduced to differential or integral inequalities. For non-retarded evolution equations, numerous inequalities are available to make the performance of decay estimate fruitful (see e.g. \\cite{BS,LL,B.G.Pach,Qin}), among which is the remarkable Gronwall-Bellman inequality which was first proposed in Gronwall \\cite{Gron} and later extended to a more general form in Bellman \\cite{Bell1}. In contrast, the situation in the case of retarded equations seems to be more complicated. Although there have appeared many nice retarded differential and integral inequalities in the literature (see e.g. {\\cite{Dee,FT,Halanay,LL,Lip,WangGT,MP,B.G.Pach,YG}} and references cited therein), the existing ones are far from being adequate to provide easy-to-handle and efficient tools for studying the dynamics of this type of equations, and it is still a challenging task to derive decay estimates for their solutions, even if for the scalar functional differential equation $\\dot x=f(t,x,x_t)$. In fact, it is often the case that one has to fall his back on differential\/integral inequalities without delay when dealing with retarded differential or integral equations, which makes the calculations in the argument much involved and restrictive.\n\nIn this paper we investigate the following type of retarded integral inequalities:\n\\begin{equation}\\label{e1.1}\\begin{array}{ll}\ny(t)\\leq &E(t,\\tau)\\|y_\\tau\\|+\\int_\\tau^t K_1(t,s)\\|y_s\\|ds\\\\[2ex]\n&+\\int_t^\\infty K_2(t,s)\\|y_s\\|ds+\\rho,\\hspace{1cm}}\\def\\hs{\\hspace{0.5cm}\\forall\\,t\\geq\\tau\\geq 0,\n\\end{array}\n\\end{equation}\nwhere $E$, $K_1$ and $K_2$ are nonnegative measurable functions on $Q:=(\\mathbb{R}^+)^2$, $\\rho\\geq 0$ is a constant,\n$\\|\\.\\|$ denotes the usual sup-norm of the space ${{\\mathcal C}}:=C([-r,0])$ for some given $r\\geq0$, $y(t)$ is a nonnegative continuous function on $[-r,\\infty)$ (called a {\\em solution} of \\eqref{e1.1}), and $y_t$ ($t\\geq 0$) denotes the {\\em lift} of $y$ in ${{\\mathcal C}}$,\n\\begin{equation}\\label{e1.15}\ny_t(s)=y(t+s),\\hspace{1cm}}\\def\\hs{\\hspace{0.5cm} s\\in [-r,0].\n\\end{equation}\nOur main purpose is to establish some uniform decay estimates for its solutions.\nSpecifically, let $E$ be a function on $Q$ satisfying that\n\\begin{equation}\\label{e1.13}\\begin{array}{ll}\n\\lim_{t\\rightarrow \\infty}E(t+s,s)=0\\mbox{ uniformly w.r.t. $s\\in\\mathbb{R}^+$}, \\end{array}\n\\end{equation} and suppose\n\\begin{equation}\\label{e1.14E}\n\\vartheta(E):=\\sup_{t\\geq s\\geq0}E(t,s)\\leq \\vartheta<\\infty,\n\\end{equation}\n\\begin{equation}\\label{e1.14}\\begin{array}{ll}\nI(K_1,K_2):=\\sup_{t\\geq 0}\\(\\int_0^t K_1(t,s)ds+\\int_t^\\8K_2(t,s)ds\\)\\leq \\kappa<\\infty.\\end{array}\n\\end{equation}\nDenote ${\\mathscr L}_r(E;K_1,K_2;\\rho)$ the solution set of \\eqref{e1.1}, i.e.,\n\\begin{equation}\\label{e1.5}{\\mathscr L}_r(E;K_1,K_2;\\rho)=\\{y\\in C([-r,\\infty)):\\,\\,\\,y\\geq0\\mbox{ and satisfies } \\eqref{e1.1}\\}. \\end{equation}\nWe show that the following theorem holds true.\n\\begin{theorem}\\label{t:3.1}\nLet $\\vartheta$ and $\\kappa$ be the positive constants in \\eqref{e1.14E} and \\eqref{e1.14}.\n \\begin{enumerate}\n \\item[$(1)$] If $\\kappa<1$ then for any $R,\\varepsilon>0$, there exists $T>0$ such that\n \\begin{equation}\\label{e:t2.2}\n \\|y_t\\|<\\mu \\rho+\\varepsilon,\\hspace{1cm}}\\def\\hs{\\hspace{0.5cm} t>T\n \\end{equation}\n for all bounded functions $y\\in {\\mathscr L}_r(E;K_1,K_2;\\rho)$ with $\\|y_0\\|\\leq R$, where\n\\begin{equation}\\label{emu}\n \\mu =1\/(1-\\kappa).\\end{equation}\n\\vs\\item[$(2)$] If $\\kappa<1\/(1+\\vartheta)$ then there exist $M,\\lambda>0$ $($independent of $\\rho$$)$ such that\n\\begin{equation}\\label{e:gi}\n{\\|y_t\\|}\\leq M\\|y_0\\|e^{-\\lambda t}+\\gamma\\rho,\\hspace{1cm}}\\def\\hs{\\hspace{0.5cm} t\\geq0\n\\end{equation}\nfor all bounded functions $y\\in {\\mathscr L}_r(E;K_1,K_2;\\rho)$, where\n\\begin{equation}\\label{ec}\n\\gamma=({\\mu+1})\/{(1-\\kappa c)},\\hs c =\\max\\(\\vartheta \/(1-\\kappa),\\,1\\).\n\\end{equation}\n\\end{enumerate}\n\\end{theorem}\n\\begin{remark}\\label{r1.1}If $\\kappa<1\/(1+\\vartheta)$ then one trivially verifies that\n$\n\\kappa c <1.\n$\n\n\\end{remark}\n\n\nThe particular case where $K_2= 0$ is of crucial importance in applications. In such a case we show that if $I(K_1,0)\\leq\\kappa<1$ then any function $y\\in {\\mathscr L}_r(E;K_1,0;\\rho)$ is automatically bounded. Hence the boundedness requirement on $y$ in Theorem \\ref{t:3.1} can be removed. Consequently we have\n\n\\begin{theorem}\\label{t:3.2} Let $(K_1,K_2)=(K,0)$, and let $\\vartheta$, $\\kappa$, $\\mu$ and $\\gamma$ be the same constants as in Theorem \\ref{t:3.1}. Then the following assertions hold.\n\\begin{enumerate}\n\\item[$(1)$] If $\\kappa<1$ then for any $R,\\varepsilon>0$, there exists $T>0$ such that\n \\begin{equation}\\label{e:gia}\n \\|y_t\\|<\\mu \\rho+\\varepsilon,\\hspace{1cm}}\\def\\hs{\\hspace{0.5cm} t>T\n \\end{equation}\nfor all $y\\in {\\mathscr L}_r(E;K,0;\\rho)$ with $\\|y_0\\|\\leq R$.\n\\vs\n\\item[$(2)$] If $\\kappa<1\/(1+\\vartheta)$ then there exist $M,\\lambda>0$ such that for all $y\\in {\\mathscr L}_r(E;K,0;\\rho)$,\n\\begin{equation}\\label{e:gi1}\n{\\|y_t\\|}\\leq M\\|y_0\\|e^{-\\lambda t}+\\gamma\\rho,\\hspace{1cm}}\\def\\hs{\\hspace{0.5cm} t\\geq0.\n\\end{equation}\n\\end{enumerate}\n\\end{theorem}\n\nTheorem \\ref{t:3.1} can be seen as an extension of the following result in Hale \\cite{Hale1} (see \\cite[pp. 110, Lemma 6.2]{Hale1}) which plays a fundamental role in constructing invariant manifolds of differential equations.\n\\begin{proposition}\\label{p1.3}\\cite{Hale1}\n Suppose $\\alpha>0$, $\\gamma>0$, K, L, M are nonnegative constants and $u$ is a nonnegative bounded continuous solution of the inequality\n\\begin{equation}\\label{e1.8}\nu(t)\\le Ke^{-\\alpha t}+L\\int_{0}^{t}e^{-\\alpha(t-s)}u(s)ds+M\\int_{0}^{\\infty}e^{-\\gamma s}u(t+s)ds,\\hspace{1cm}}\\def\\hs{\\hspace{0.5cm} t\\ge0.\n\\end{equation}\nIf\n$\n\\beta:=L\/\\alpha+M\/\\gamma<1\n$\nthen\n\\begin{equation}\\label{e1.7}\nu(t)\\le(1-\\beta)^{-1}Ke^{-[\\alpha-(1-\\beta)^{-1}L]t},\\hspace{1cm}}\\def\\hs{\\hspace{0.5cm} t\\geq 0.\n\\end{equation}\n\\end{proposition}\nNote that there is in fact an additional requirement in \\eqref{e1.7} to guarantee the exponential decay of $u$, that is, $\\alpha-(1-\\beta)^{-1}L>0$, or equivalently,\n\\begin{equation}\\label{e1.9}\nL\/\\alpha+M\/\\gamma<1-L\/\\alpha.\n\\end{equation}\n\nLet us say a little more about the special case $M=0$, in which \\eqref{e1.9} reads as $L\/\\alpha<1\/2$. In such a case $L\/\\alpha$ coincides with the constant $\\kappa$ in Theorem \\ref{t:3.2}.\n Setting $K=u_0$ in \\eqref{e1.8}, we see that the upper bound $\\vartheta$ of the decay functor $E$ in \\eqref{e1.8} (corresponding to \\eqref{e1.1}) equals $1$. Consequently the smallness requirement on $\\kappa$ in assertion (2) of Theorem \\ref{t:3.2} reduces to that $\\kappa=L\/\\alpha<1\/2$.\n\n On the other hand, if $1\/2\\leq \\kappa=L\/\\alpha<1$ then we can only infer from \\eqref{e1.7} that $u$ has at most an exponential growth. However, Theorem \\ref{t:3.2} still assures that a function satisfying the corresponding integral inequality must approach $0$ in a uniform manner with respect to initial data in bounded sets.\n\nWe also mention that our proof for Theorem \\ref{t:3.1} is significantly different not only from the one for Proposition \\ref{p1.3} given in \\cite{Hale1}, but also from those in the literature for other types of differential or integral inequalities.\n\n\\begin{remark}\n The smallness requirement $\\kappa<1$ in the above theorems is optimal in some sense. This can be seen from the simple example of scalar equation:\n\\begin{equation}\\label{e1.6}\n\\dot x=-a x+bx(t-1),\n\\end{equation}\nwhere $a,b>0$ are constants, for which the assumption $\\kappa<1$ in Theorem \\ref{t:3.1} on the corresponding integral inequality to guarantee the global asymptotic stability of the $0$ solution of the equation amounts to require that $ba$ then simple calculations show that \\eqref{e1.6} has a positive eigenvalue and hence $0$ is unstable; see e.g. Kuang \\cite[Chap. 3, Sect. 2]{Kuang}.\\end{remark}\n\n\\begin{remark}\\label{r:1.6}\nIt remains open whether the assumption $\\kappa<1\/(1+\\vartheta)$ in Theorem \\ref{t:3.2} to guarantee global exponential decay for \\eqref{e1.1} can be further relaxed in the full generality of the theorem.\n\\end{remark}\n\nAs a simple example of applications, we consider the asymptotic stability of the scalar functional differential equation:\n\\begin{equation}\\label{e1.10}\n\\dot x=-a(t)x+B(t,x_t),\n\\end{equation}\nwhere $a\\in C(\\mathbb{R})$, and $B$ is a continuous function on $\\mathbb{R}\\times C([-r,0])$ for some fixed $r\\geq0$ with\n$\n|B(t,\\phi)|\\leq b(t)\\|\\phi\\|$.\nSpecial cases of the equation were studied in the literature by many authors. For instance, in an earlier work of Winston \\cite{Wins}, the author considered the case where $a(t)$ is nonnegative and $b(t)\\leq \\theta a(t)$ for some $\\theta<1$. Using Razumikhin's method\nthe author proved the exponential asymptotic stability and the asymptotic stability of the equation under the assumption $a(t)\\geq\\alpha>0$ and that $a(t)\\geq0$ with $\\int_0^\\8a(t)dt=\\infty$, respectively.\nHere we revisit this problem and allow $a(t)$ to be a function which may change sign. Assume\n $\\lim_{t\\rightarrow \\infty}\\int_s^{s+t} a(\\tau)d\\tau \\rightarrow \\infty\\,\\, \\mbox{uniformly w.r.t $s\\in\\mathbb{R}$}.$\nWe show that the null solution of \\eqref{e1.10} is globally asymptotically stable provided that\n$$\\begin{array}{ll}\n\\kappa_\\tau:= \\sup_{t\\geq \\tau}\\int_\\tau^tE(t,s)b(s)ds<1,\\hspace{1cm}}\\def\\hs{\\hspace{0.5cm} \\forall\\,\\tau\\in \\mathbb{R},\\end{array}\n$$\nwhere $\nE(t,s)=\\exp\\(-\\int_s^t a(\\sigma)d\\sigma\\).$ Some results on global exponential asymptotic stability will also be presented. It is not difficulty to check that if $a(t)$ is a nonnegative function with $\\int_0^\\8a(s)ds=\\infty$ and $b(t)\\leq \\theta a(t)$ ($t\\in\\mathbb{R}$) for some $\\theta<1$, then $\\kappa_\\tau\\leq \\theta<1$ for all $\\tau\\in\\mathbb{R}$.\n\nAs another example of applications for our integral inequalities, we discuss the existence of pullback attractor for ODE system\n\\begin{equation}\\label{e1.16}\n\\dot x=F_0(t,x)+\\sum_{i=1}^mF_i(t,x(t-r_i)),\\hs x=x(t)\\in\\mathbb{R}^n,\n\\end{equation}\nwhere $F_i(t,x)$ ($0\\leq i\\leq m$) are continuous mappings from $\\mathbb{R}\\times \\mathbb{R}^n$ to $\\mathbb{R}^n$ which are locally Lipschitz in $x$ in a uniform manner with respect to $t$ on bounded intervals, and $r_i:\\mathbb{R}\\rightarrow [0,r]$ ($1\\leq i\\leq m$) are measurable functions. The investigation of the dynamics of delayed differential equations in the framework of pullback attractor theory developed in \\cite{Crau, Kloeden2,Kloeden1} etc. was first initiated by Caraballo et al. \\cite{Carab}. In recent years there is an increasing interest on this topic for both retarded ODEs and PDEs; see e.g. \\cite{CMR,CMV, C,Chue, KL, MR,SC, WK, ZS}. However, we find that the existing works mainly focus on the case where the terms involving time lags have at most sublinear nonlinearities.\nHere we allow the nonlinearities $F_i(t,x)$ ($0\\leq i\\leq m$) in \\eqref{e1.16} to be superlinear in space variable $x$.\nSuppose\n\\begin{enumerate}\\item[{\\bf (F)}] there exist positive constants $p>q\\geq 1$, $\\alpha_i>0$ ($0\\leq i\\leq m$), and nonnegative measurable functions $\\beta_i(t)$ ($0\\leq i\\leq m$) on $\\mathbb{R}$ such that\n$$\n(F_0(t,x),x)\\leq -\\alpha_0 |x|^{p+1}+\\beta_0(t),\\hspace{1cm}}\\def\\hs{\\hspace{0.5cm} \\forall\\,x\\in\\mathbb{R}^n,\\,\\,t\\in\\mathbb{R},\n$$$$\n |F_i(t,x)|\\leq \\alpha_i |x|^{q}+\\beta_i(t),\\hspace{1cm}}\\def\\hs{\\hspace{0.5cm} \\forall\\,x\\in\\mathbb{R}^n,\\,\\,t\\in\\mathbb{R}.\n $$\n \\end{enumerate}\nWe show under some additional assumptions on $\\beta_i(t)$ ($0\\leq i\\leq m$) that system \\eqref{e1.16} is dissipative and has a global pullback attractor.\n\\vs\n\nAs our third example to illustrate applications of Theorems \\ref{t:3.1} and \\ref{t:3.2}, we finally consider the dynamics of retarded nonlinear evolution equations with sublinear nonlinearities in the general setting of the cocycle system:\n\\begin{equation}\\label{e1.18}\\frac{du}{dt}+Au=F(\\theta_tp,u_t),\\hspace{1cm}}\\def\\hs{\\hspace{0.5cm} p\\in {\\mathcal H}\\end{equation}\n in a Banach space $X$, where $A$ is a sectorial operator in $X$ with compact resolvent, ${\\mathcal H}$ is a compact metric space, and $\\theta_t$ is a dynamical system on ${\\mathcal H}$.\nWe will show under a hyperbolicity assumption on $A$ and some smallness requirements on the growth rate and the Lipschitz constant of $F(p,u)$ in $u$ that the system has a unique nonautonomous equilibrium solution $\\Gamma$. The global asymptotic stability and exponential stability of $\\Gamma$ will also be addressed.\n\n\n\\vs\nThis paper is organized as follows. Section 2 is devoted to the proofs of the main results, namely, Theorems \\ref{t:3.1} and \\ref{t:3.2}; and Section 3 consists of the two examples of ODE systems mentioned above. Section 4 is concerned with the dynamics of system \\eqref{e1.18}. We will also talk about in this section how to put a differential equation with multiple variable delays and external forces into the general setting of \\eqref{e1.18}.\n\n\n\n\n\\section{Proofs of Theorems \\ref{t:3.1} and \\ref{t:3.2}}\nFor convenience in statement, let us first introduce several classes of functions.\n\nDenote ${\\mathscr E}$ the family of {\\em bounded} nonnegative measurable functions on $Q:=(\\mathbb{R}^+)^2$ satisfying \\eqref{e1.13}, and let\n$$\\begin{array}{ll}\n{\\mathscr K}_1=\\{K\\in {\\mathscr M}^+(Q):\\,\\,\\,\\int_0^t K(t,s)ds<\\infty\\mbox{ for all }t\\geq 0\\},\n\\end{array}\n$$\n$$\\begin{array}{ll}\n{\\mathscr K}_2=\\{K\\in {\\mathscr M}^+(Q):\\,\\,\\,\\int_t^\\infty K(t,s)ds<\\infty\\mbox{ for all }t\\geq 0\\},\n\\end{array}\n$$\nwhere ${\\mathscr M}^+(Q)$ is the family of {nonnegative measurable functions} on $Q$.\nDenote $I(K_1,K_2)$ the constant defined in \\eqref{e1.14} for any $(K_1,K_2)\\in {\\mathscr K}_1\\times {\\mathscr K}_2$.\n\nLet ${\\mathcal C}$ be the space $C([-r,0])$ equipped with the usual sup-norm\n $$\n \\|\\phi\\|=\\sup_{s\\in[-r,0]}|\\phi(s)|,\\hspace{1cm}}\\def\\hs{\\hspace{0.5cm}\\phi\\in{\\mathcal C}.\n $$\nGiven $y\\in C([-r,T))$ ($T>0$), one can assign a function $y_t$ from $[0,T)$ to ${\\mathcal C}$ as\nfollows: for each $t\\in[0,T)$, $y_t$ is the {element} in ${{\\mathcal C}}$ defined by \\eqref{e1.15}. For convenience, $y_t$ will be referred to as the {\\em lift} of $y$ in ${\\mathcal C}$.\n\\subsection{Proof of Theorem \\ref{t:3.1}}\nWe begin with the following lemma:\n\\begin{lemma}\\label{l:2.1} Assume that $\\kappa<1$. Then for any bounded function $y\\in {\\mathscr L}_r(E;K_1,K_2;\\rho)$,\n\\begin{equation}\\label{e:t2.1}\n \\|y_{t}\\|\\leq c \\|y_0\\|+\\mu \\rho,\\hspace{1cm}}\\def\\hs{\\hspace{0.5cm} t\\geq 0,\n\\end{equation}\n where $c,\\mu$ are the constants defined in Theorem \\ref{t:3.1}.\n \\end{lemma}\n{\\bf Proof.} It can be assumed that there is $t>0$ such that $y(t)>\\|y_0\\|+\\mu \\rho$; otherwise \\eqref{e:t2.1} readily holds true.\nWrite $$\\sup_{t\\in{ \\mathbb{R}^+}}\\|y_t\\|=N_\\varepsilon(\\|y_0\\|+\\varepsilon)+\\mu \\rho$$ for $\\varepsilon>0$. We show that $N_\\varepsilon\\leq c $ for all $\\varepsilon>0$, and the conclusion follows.\n\n For each $\\delta>0$ sufficiently small, pick an $\\eta>0$ with\n$$y(\\eta)>\\sup_{t\\in{ \\mathbb{R}^+}}\\|y_t\\|-\\delta.$$\nThen by \\eqref{e1.1} we have\n$$\\begin{array}{ll}\nN_\\varepsilon (\\|y_0\\|+\\varepsilon)+\\mu \\rho-\\delta&=\\sup_{t\\in{ \\mathbb{R}^+}}\\|y_t\\|-\\delta0$, if we set $\\~y(t)=y(\\sigma+t)$ and define\n$$\\~E(t,s)=E(t+\\sigma,s+\\sigma),\\hs \\~K_i(t,s)=K_i(t+\\sigma,s+\\sigma)\\,\\,(i=1,2)\n $$\n for $t,s\\geq 0$, then one trivially checks that $\\~y\\in{\\mathscr L}_r(\\~E;\\~K_1,\\~K_2;\\rho)$ with\n $$I(\\~K_1,\\~K_2)\\leq I(K_1,K_2)\\leq\\kappa<1.$$\n Thus if $y$ is bounded, then by Lemma \\ref{l:2.1} one also concludes that\n \\begin{equation}\\label{e:3.4}\n \\|y_{t+\\sigma}\\|\\leq c \\|y_\\sigma\\|+\\mu \\rho,\\hspace{1cm}}\\def\\hs{\\hspace{0.5cm} t,\\sigma\\geq 0.\n \\end{equation}\n\\vs\n\n\\noindent{\\bf Proof of Theorem \\ref{t:3.1}.} (1) \\,Assume $\\kappa<1$. To verify assertion (1), we first show that if $y\\in {\\mathscr L}_r(E;K_1,K_2;\\rho)$ is a bounded function, then\n\\begin{equation}\\label{e:3.6}{\\limsup_{t\\rightarrow\\infty} \\|y_t\\|}\\leq \\mu\\rho.\n\\end{equation}\nLet us argue by contradiction and suppose\n$${\\limsup_{t\\rightarrow\\infty} \\|y_t\\|}=\\mu\\rho+\\delta\n$$\nfor some $\\delta>0$.\nTake a monotone sequence $\\tau_n\\rightarrow\\infty$ such that $\\lim_{n\\rightarrow\\infty}{ y({\\tau_n}})=\\mu\\rho+\\delta$.\nFor any $\\varepsilon>0$, take a $\\tau>0$ sufficiently large so that\n$$\n\\|y_t\\|<\\mu\\rho+\\delta+\\varepsilon,\\hspace{1cm}}\\def\\hs{\\hspace{0.5cm} t\\geq\\tau.\n$$\nThen for $\\tau_n>\\tau$, by \\eqref{e1.1} we deduce that\n$$\\begin{array}{ll}\ny(\\tau_n)&\\leq E(\\tau_n,\\tau)\\|y_{\\tau}\\|+\\int_{\\tau}^{\\tau_n} K_1(\\tau_n,s)\\|y_s\\|ds+\\int_{\\tau_n}^\\8K_2(\\tau_n,s)\\|y_{s}\\|ds+\\rho\\\\[2ex]\n&\\leq E(\\tau_n,\\tau)\\|y_{\\tau}\\|+ \\kappa\\(\\mu \\rho+\\delta+\\varepsilon\\)+\\rho.\n\\end{array}\n$$\nSetting $n\\rightarrow\\infty$ in the above inequality, it yields\n$$\n\\mu \\rho+\\delta\\leq \\kappa\\(\\mu \\rho+\\delta+\\varepsilon\\)+\\rho.\n$$\nSince $\\varepsilon$ is arbitrary, we conclude that\n$$\n\\mu \\rho+\\delta\\leq (\\kappa\\mu +1)\\rho+\\kappa\\delta.\n$$\nTherefore by \\eqref{e:2.3} one has $\\delta{ \\le}\\kappa\\delta$, which leads to a contradiction and verifies \\eqref{e:3.6}.\n\n\\vs\nNow we complete the proof of assertion (1).\nLet $R>0$. Denote\n$$\n{\\mathscr B}_R=\\{y\\in{\\mathscr L}_r(E;K_1,K_2;\\rho):\\,\\,y\\mbox{ is bounded with }\\|y_0\\|\\leq R\\}.\n$$\nBy \\eqref{e:t2.1} we see that ${\\mathscr B}_R$ is uniformly bounded. Hence the envelope\n$$\ny^*(t)=\\sup_{y\\in{\\mathscr B}_R}y(t)\n$$\nof the family ${\\mathscr B}_R$ is a bounded nonnegative measurable function on $[-r,\\infty)$.\n(The measurability of $y^*$ follows from the simple observation that\n$$\\begin{array}{ll}\\{t\\in(-r,\\infty):\\,\\,y^*(t)>a\\}=\\Cup_{y\\in {\\mathscr B}_R}\\{t\\in(-r,\\infty):\\,\\,y(t)>a\\}\\end{array}$$ is an open subset of $\\mathbb{R}$ for any $a\\in\\mathbb{R}$.) As in the case of a continuous function, we use the notation $y^*_t$ ($t\\geq0$) to denote the lift of $y^*$ in the space of measurable functions on $[-r,0]$ ($y^*_t(\\.)=y^*(t+\\.)$) and write $\\|y^*_t\\|=\\sup_{s\\in[-r,0]}y^*_t(s)$.\n(One should distinguish $\\|y^*_t\\|$ with the $L^\\infty$-norm $\\|y^*_t\\|_{L^\\infty(-r,0)}$ of $y^*_t$, although it can be shown by using the definition of $y^*$ and the continuity of the functions $y\\in{\\mathscr B}_R$ that the two quantities coincide for $y^*_t$.) We claim that $\\varphi(t):=\\|y^*_t\\|$ is a measurable function on $[0,\\infty)$. Indeed, one trivially verifies that\n$$\n\\|y^*_t\\|=\\sup_{y\\in{\\mathscr B}_R}\\|y_t\\|,\\hspace{1cm}}\\def\\hs{\\hspace{0.5cm} t\\geq 0.\n$$\nSince $\\|y_t\\|$ is continuous in $t$ for every $y$, the conclusion immediately follows.\n\nWe infer from \\eqref{e1.1} that\n$$\\begin{array}{ll}\ny(t)\\leq &E(t,\\tau)\\|y^*_\\tau\\|+\\int_\\tau^t K_1(t,s)\\|y^*_s\\|ds\\\\[2ex]\n&\\hspace{1cm}}\\def\\hs{\\hspace{0.5cm}\\hs+\\int_t^\\infty K_2(t,s)\\|y^*_s\\|ds+\\rho,\\hspace{1cm}}\\def\\hs{\\hspace{0.5cm}\\forall\\,t\\geq\\tau\\geq 0\n\\end{array}\n$$\nfor any $y\\in {\\mathscr B}_R$. Further taking supremum in the { lefthand} side of the above inequality with respect to $y\\in {\\mathscr B}_R$ it yields\n\\begin{equation}\\label{e:2.a1}\\begin{array}{ll}\ny^*(t)\\leq &E(t,\\tau)\\|y^*_\\tau\\|+\\int_\\tau^t K_1(t,s)\\|y^*_s\\|ds\\\\[2ex]\n&\\hspace{1cm}}\\def\\hs{\\hspace{0.5cm}\\hs+\\int_t^\\infty K_2(t,s)\\|y^*_s\\|ds+\\rho,\\hspace{1cm}}\\def\\hs{\\hspace{0.5cm}\\forall\\,t\\geq\\tau\\geq 0.\n\\end{array}\n\\end{equation}\nThe only difference between \\eqref{e1.1} and the above inequality \\eqref{e:2.a1} is that the function $y^*$ in \\eqref{e:2.a1} may not be continuous.\nNote that we do not make use of any continuity requirement on $y$ {in the proof of Lemma \\ref{l:2.1} and \\eqref{e:3.6}}. Therefore all the arguments {therein} can be directly carried over to $y^*$ without any modifications except that the function $y$ is replaced by $y^*$. As a result, we deduce that $\\limsup_{t\\rightarrow\\infty}\\|y^*_t\\|\\leq \\mu\\rho$. Hence for any $\\varepsilon>0$ there is a $T>0$ such that\n$$\n\\|y^*_t\\|<\\mu\\rho+\\varepsilon,\\hspace{1cm}}\\def\\hs{\\hspace{0.5cm} t>T,\n$$\nfrom which assertion (1) immediately follows.\n\n\n\\vskip8pt}\\def\\vs{\\vskip4pt\n(2) \\,Now we assume $\\kappa<1\/(1+\\vartheta)$. To obtain the exponential decay estimate in \\eqref{e:gi}, we first prove a temporary result:\n\n\\vs There exist $T,\\lambda>0$ such that if $\\|y_0\\|\\leq N_0+\\gamma\\rho$ with $N_0> 0$, then\n\\begin{equation}\\label{e:gi'}\n{\\|y_t\\|}\\leq N_0 e^{-\\lambda t}+\\gamma\\rho,\\hspace{1cm}}\\def\\hs{\\hspace{0.5cm} t\\geq T.\n\\end{equation}\n\n For this purpose, we take\n \\begin{equation}\\label{sig}\\sigma= (1+\\kappa c)\/2.\\end{equation} Since $\\kappa c<1$ (see Remark \\ref{r1.1}), it is clear that $\\sigma<1$. Define\n$$\\eta=\\min\\{s{ \\ge 0}:\\,\\,\\|y_t\\|\\leq \\sigma N_0+\\gamma\\rho\\mbox{ for all }t\\geq s\\}.$$\nThe key point is to estimate the upper bound of $\\eta$.\n\nBecause $\\gamma> \\mu$ and $N_0>0$, by \\eqref{e:3.6} it is clear that $\\eta<\\infty$. We may assume $\\eta> r$ (otherwise we are done). Then by continuity of $y$ one necessarily has $$\\|y_\\eta\\|=\\sigma N_0+\\gamma\\rho.$$\nFor simplicity, write $E(t,0):=b(t)$. Given $t\\in[\\eta-r,\\eta]$, by \\eqref{e1.1} we have\n$$\n\\begin{array}{ll}\ny(t)&\\leq { b(t)}\\|y_{0}\\|+\\int_0^{t} K_1(t,s)\\|y_s\\|ds +\\int_{t}^\\8K_2(t,s)\\|y_s\\|ds+\\rho\\\\[2ex]\n&\\leq (\\mbox{by }\\eqref{e:t2.1})\\leq\\|b_{\\eta}\\|\\|y_0\\|+\\kappa (c \\|y_0\\|+\\mu\\rho)+\\rho\\\\[2ex]\n&\\leq \\({ \\|b_{\\eta}\\|}+\\kappa c \\)\\|y_0\\|+(\\kappa\\mu+1)\\rho \\\\[2ex]\n&\\leq \\({ \\|b_{\\eta}\\|}+\\kappa c \\)(N_0+\\gamma\\rho)+\\mu\\rho.\n\\end{array}\n$$\nHere we have used the fact that $\\kappa\\mu+1=\\mu$ (see \\eqref{e:2.3}). Therefore\n\\begin{equation}\\label{e:3.25}\n\\begin{array}{ll}\n\\sigma N_0+\\gamma\\rho&=\\|y_\\eta\\|=\\max_{t\\in[\\eta-r,\\eta]}y(t)\\\\[2ex]\n&\\leq \\({ \\|b_{\\eta}\\|}+\\kappa c \\)N_0+\\(\\({ \\|b_{\\eta}\\|}+\\kappa c \\)\\gamma+\\mu\\)\\rho.\n\\end{array}\n\\end{equation}\n\nTake a number $t_0>0$ such that\n\\begin{equation}\\label{et0}\nE(t+s,s)\\gamma\\leq 1,\\hspace{1cm}}\\def\\hs{\\hspace{0.5cm} \\forall\\,t\\geq t_0,\\,\\,s\\in \\mathbb{R}^+.\n\\end{equation}\nIf $\\eta\\leq t_0+r$ then we are done. Thus we assume that $\\eta>t_0+r$. Then by the definition of $\\gamma$ and \\eqref{et0} one deduces that\n$$\n\\gamma=\\kappa c\\gamma+\\mu+1\\geq\\({ \\|b_{\\eta}\\|}+\\kappa c \\)\\gamma+\\mu.\n$$\nIt follows by \\eqref{e:3.25} that $\\sigma N_0\\leq \\({ \\|b_{\\eta}\\|}+\\kappa c \\)N_0$. Hence\n\\begin{equation}\\label{e:sig}\n\\|b_\\eta\\|\\geq \\sigma-\\kappa c =(1-\\kappa c)\/2>0.\\end{equation}\nTake a number $t_1>0$ such that\n\\begin{equation}\\label{e:3.27}\nE(t+s,s)< (1-\\kappa c )\/2,\\hspace{1cm}}\\def\\hs{\\hspace{0.5cm} t> t_1,\\,\\,s\\in\\mathbb{R}^+.\n\\end{equation}\n \\eqref{e:sig} then implies that\n$\\eta\\leq { t_1}+r.$\nHence we conclude that\n\\begin{equation}\\label{eT}\\eta\\leq T:=\\max\\(t_0,t_1\\)+r.\\end{equation}\n\n\\vs By far we have proved that if $\\|y_0\\|\\leq N_0+\\gamma\\rho$ ($N_0>0$) then\n$$\n \\|y_t\\|\\leq \\sigma N_0+\\gamma\\rho,\\hspace{1cm}}\\def\\hs{\\hspace{0.5cm} t\\geq T.\n $$\n\n\\vs Let $\\~y(t)=y(t+T)$, and set\n $$\n \\~E(t,s)=E(t+T,s+T),\\hs \\~K_i(t,s)=K_i(t+T,s+T)\n $$\nfor $t,s\\geq 0$, $i=1,2$. Then $\\~y\\in{\\mathscr L}_r(\\~E;\\~K_1,\\~K_2;\\rho)$ with $$I(\\~K_1,\\~K_2)\\leq I(K_1,K_2)\\leq\\kappa<1\/(1+\\vartheta).$$\n Since $\\|\\~y_0\\|\\leq \\sigma N_0+\\gamma\\rho$, the same argument as above applies to show that\n $$\n \\|\\~y_t\\|\\leq \\sigma(\\sigma N_0)+\\gamma\\rho,\\hspace{1cm}}\\def\\hs{\\hspace{0.5cm} t\\geq T,\n $$\n that is,\n $$\n \\|y_t\\|\\leq \\sigma^2 N_0+\\gamma\\rho,\\hspace{1cm}}\\def\\hs{\\hspace{0.5cm} t\\geq 2T.\n $$\n (We emphasize that the numbers $t_0$ and $t_1$ in \\eqref{et0} and \\eqref{e:3.27} can be chosen independent of $s\\in\\mathbb{R}^+$. This plays a crucial role in the above argument.)\n Repeating the above procedure we finally obtain that\n \\begin{equation}\\label{e:3.16}\n \\|y_t\\|\\leq \\sigma^n N_0+\\gamma\\rho,\\hspace{1cm}}\\def\\hs{\\hspace{0.5cm} t\\geq nT,\\,\\,n=1,2,\\cdots.\n \\end{equation}\n Setting $\\lambda=-(\\mbox{ln\\,} \\sigma)\/{2T}$, one trivially verifies that\n $$\\sigma^n\\leq e^{-\\lambda t},\\hspace{1cm}}\\def\\hs{\\hspace{0.5cm} t\\in[nT,(n+1)T]$$ for all $n\\geq1$. \\eqref{e:gi'} then follows from \\eqref{e:3.16}.\n\n\\vs\nWe are now in a position to complete the proof of the theorem.\\vs\n\nNote that \\eqref{e:t2.1} implies that if $\\|y_0\\|=0$ then\n$$\n\\|y_t\\|\\leq\\mu\\rho\\leq\\gamma\\rho,\\hspace{1cm}}\\def\\hs{\\hspace{0.5cm} t\\geq 0,\n$$\nand hence the conclusion readily holds true. Thus we assume that $\\|y_0\\|>0$.\nTake $N_0=\\|y_0\\|$. Clearly $\\|y_0\\|= N_0\\leq N_0+\\gamma\\rho$. Therefore by \\eqref{e:gi'} we have\n\\begin{equation}\\label{e:3.11b}\n\\|y_t\\|\\leq\\|y_0\\|e^{-\\lambda t}+\\gamma\\rho,\\hspace{1cm}}\\def\\hs{\\hspace{0.5cm} t\\geq T.\n\\end{equation}\nOn the other hand, by \\eqref{e:3.4} we deduce that\n$$\n\\|y_t\\|\\leq c \\|y_0\\|+\\mu\\rho\\leq c \\|y_0\\|+\\gamma\\rho,\\hspace{1cm}}\\def\\hs{\\hspace{0.5cm} t\\in[0,T].\n$$\nSet\n$M=c e^{\\lambda T}$. Then\n$$\n\\|y_t\\|\\leq c \\|y_0\\|+\\gamma\\rho\\leq Me^{-\\lambda t}\\|y_0\\|+\\gamma\\rho,\\hspace{1cm}}\\def\\hs{\\hspace{0.5cm} t\\in[0,T].\n$$\nCombining this with \\eqref{e:3.11b} we finally arrive at the estimate\n$$\n\\|y_t\\|\\leq M \\|y_0\\|e^{-\\lambda t}+\\gamma\\rho,\\hspace{1cm}}\\def\\hs{\\hspace{0.5cm} t\\geq 0.\n$$\nThe proof of the theorem is complete. $\\Box$\n\n\n \\begin{remark}\\label{r:2.2}\nIn many examples from applications, the function $E(t,s)$ in \\eqref{e1.1} takes the form:\n$$\nE(t,s)=M_0e^{-\\lambda_0(t-s)},\n$$\nwhere $M_0$ and $\\lambda_0$ are positive constants. In such a case one can write out the constants $M$ and $\\lambda$ in \\eqref{e:gi} and \\eqref{e:gi1} explicitly.\n\nIndeed, the number $t_0$ and $t_1$ in \\eqref{et0} and \\eqref{e:3.27} can be taken, respectively, as $$t_0=\\lambda_0^{-1}\\mbox{ln\\,}(M_0\\gamma),\\hs t_1=\\lambda_0^{-1}\\mbox{ln\\,}\\(\\frac{2M_0}{1-\\kappa c}\\).$$\nConsequently the number $T$ in \\eqref{eT} reads as\n$\nT=\\lambda_0^{-1}M_1+r,\n$\n{where } $$M_1=\\max\\(\\mbox{ln\\,}(M_0\\gamma),\\,\\mbox{ln\\,}\\(\\frac{2M_0}{1-\\kappa c}\\)\\).$$\nThus we infer from the proof of Theorem \\ref{t:3.1} that\n$$\n\\lambda=-\\frac{\\mbox{ln\\,}\\sigma}{2T}=\\frac{\\ln2-\\mbox{ln\\,}\\({1+\\kappa c}\\)}{2\\(M_1+r\\lambda_0\\)}\\,\\lambda_0,\n$$\n$$\nM=ce^{\\lambda T}=c\\sqrt{2\/(1+\\kappa c)}.\n$$\nIn particular, if $r=0$ then we have\n$$\\lambda=\\theta\\lambda_0,\\hs \\theta=\\frac{\\ln2-\\mbox{ln\\,}\\({1+\\kappa c}\\)}{2M_1}\\,.$$\n\\end{remark}\n\n\\begin{remark}\\label{r:2.3}In the general case, \\eqref{e1.13} implies that there is a bounded nonnegative function $e(t)$ on $\\mathbb{R}^+$ with $e(t)\\rightarrow 0$ as $t\\rightarrow\\infty$ such that\n\\begin{equation}\\label{e:2.E}E(t+s,s)\\leq e(t),\\hspace{1cm}}\\def\\hs{\\hspace{0.5cm} t,s\\geq 0.\\end{equation}\nOne can easily see that the numbers $t_0$ and $t_1$ in \\eqref{et0} and \\eqref{e:3.27} can be chosen in such a way that they only depend upon the constants $\\gamma,\\kappa,c$ and the function $e(t)$. Consequently the constants $M$ and $\\lambda$ in Theorem \\ref{t:3.1} $(2)$ (which are defined explicitly below \\eqref{e:3.16} in the proof of the theorem) only depend upon $\\gamma,\\kappa,c$, $\\sigma$ and $e(t)$. Since $\\gamma,c$ and $\\sigma$ are completely determined by $\\vartheta$ and $\\kappa$ (see Theorem \\ref{t:3.1} and \\eqref{sig} for the definitions of these constants), we finally conclude that $M$ and $\\lambda$ only depend upon $\\vartheta,\\kappa$ and $e(t)$.\\end{remark}\n\n\n\\subsection{Proof of Theorem \\ref{t:3.2}}\n\\noindent{\\bf Proof.} The conclusions of Theorem \\ref{t:3.2} immediately follow from Theorem \\ref{t:3.1} as long as Lemma \\ref{l:2.a1} below is proved. $\\Box$\n\n \\begin{lemma}\\label{l:2.a1} Let $E\\in{\\mathscr E}$, and $K_1=K\\in {\\mathscr K}_1$. Suppose $I(K,0)\\leq\\kappa<1$.\nLet $r,\\rho\\geq 0$, and let $y$ be a nonnegative continuous function on $[-r,T)$ $(0\\beta> 0$ are constants, then there exist $\\gamma>0$ and $k>0$ such that\n $$y(t)\\leq ke^{-\\gamma(t-t_0)},\\hspace{1cm}}\\def\\hs{\\hspace{0.5cm} t\\geq t_0;$$ see Halanay \\cite[pp. 378]{Halanay}. For simplicity we may put $t_0=0$.\n Using a similar argument as in the proof of Proposition \\ref{p:3.2} below, one can easily show that a function $y$ satisfying \\eqref{e:2.4} fulfills the integral inequality \\eqref{e1.1} with $K_2=0$ and\n $$\\begin{array}{ll}\nE(t,s)=e^{-\\alpha(t-s)},\\hs K_1(t,s)=\\beta E(t,s)\\end{array}\n$$\nfor $t,s\\geq 0$. Note that\n$$\n\\vartheta=\\sup_{t\\geq s\\geq 0}E(t,s)=1,\\hs \\kappa =\\sup_{t\\geq 0}\\int_0^tK_1(t,s)ds=\\beta\/\\alpha.\n$$\n Thus the assumption that $\\kappa<1$ in Theorem \\ref{t:3.2} amounts to say that $\\alpha>\\beta$. Hence Theorem \\ref{t:3.2} can be seen as a generalization of the Halanay's inequality.\n\n On the other hand, we emphasize that in the special case of \\eqref{e:2.4}, Halanay's result is stronger than Theorem \\ref{t:3.2} in the way that\n it guarantees the exponential convergence of $y(t)$ to $0$ under the assumption that $\\beta<\\alpha$, whereas under this weaker assumption Theorem \\ref{t:3.2} only gives convergence result.\nThis is also one of the reasons why we are interested in the question proposed in Remark \\ref{r:1.6}.\n\n An integral version of the Halanay's inequality can be found in a recent work of Chen \\cite[Lemma 3.2]{ChenH} along with a very simple proof: \\,\\,Let $y$ be a nonnegative continuous function on $[-r,\\infty)$. Suppose that for $\\alpha>0$, there exist two positive constants $M ,\\beta>0$ such that $y(t)\\leq M e^{-\\alpha t}$ $(t\\in[-r,0])$ and that\n \\begin{equation}\\label{e:2.5}\n y(t)\\leq M e^{-\\alpha t} +\\beta\\int_0^te^{-\\alpha(t-s)}\\|y_s\\|ds,\\hspace{1cm}}\\def\\hs{\\hspace{0.5cm} t\\geq 0.\n \\end{equation}\n If $\\beta<\\alpha$, then $y(t)\\leq M e^{-\\mu t}$ for $t\\geq -r$, where $\\mu\\in(0,\\alpha)$ is a constant satisfying that $\\frac{\\beta}{\\alpha-\\mu}e^{\\mu r}=1$. One advantage of this integral inequality is that it significantly reduces the smoothness requirement on the function $y$. This may greatly enlarge the applicability of the inequality.\n Other types of extensions of the Halanay's inequality can be found in \\cite{HPT,WLL} etc. and references therein.\n\n\n\\end{remark}\n\n\n\n\n\\section{Asymptotic Behavior of ODE Systems}\n This section consists of two examples of ODE systems illustrating possible applications of the integral inequalities given here. For the general theory of delay differential equations, one may consult the excellent books \\cite{Hale2,Kuang,S,Wu}.\n\n\n\n\n \\subsection{Asymptotic stability of a scalar functional ODE}\\label{s:3.1}\nOur first example concerns the asymptotic stability of the scalar functional differential equation:\n\\begin{equation}\\label{e:5.1}\n\\dot x=-a(t)x+B(t,x_t),\\hs\n\\end{equation}\nwhere $x_t$ is the lift of $x=x(t)$ in ${\\mathcal C}:=C([-r,0])$ ($r\\geq 0$ is fixed), $a\\in C(\\mathbb{R})$, and $B$ is a continuous function on $\\mathbb{R}\\times{\\mathcal C}$. We always assume that $B$ satisfies the following local Lipschitz condition in the second variable: For any compact interval $J\\subset\\mathbb{R}$ and $R>0$, there exists $L>0$ such that\n$$\n|B(t,\\phi)-B(t,\\phi')|\\leq L\\|\\phi-\\phi'\\|,\\hspace{1cm}}\\def\\hs{\\hspace{0.5cm}\\forall\\,\\phi,\\phi'\\in\\overline{\\mathcal B}_R,\\,\\,t\\in J.\n$$\nHere and below ${\\mathcal B}_R$ denotes the ball in ${\\mathcal C}$ centered at $0$ with radius $R>0$.\n\n Given $(\\tau,\\phi)\\in\\mathbb{R}\\times{\\mathcal C}$, the above smoothness requirements on $a$ and $B$ are sufficient to guarantee the existence and uniqueness of a local solution $x(t)=x(t;\\tau,\\phi)$ ($t\\geq \\tau$) of \\eqref{e:5.1} with initial value $x_\\tau=\\phi\\in{\\mathcal C}$; see \\cite[Chap. 2, Theorems 2.1, 2.3]{Hale2}. We also assume that\n\\begin{equation}\\label{e:5.2}\n|B(t,\\phi)|\\leq b(t)\\|\\phi\\|, \\hspace{1cm}}\\def\\hs{\\hspace{0.5cm} (t,\\phi)\\in \\mathbb{R}\\times{\\mathcal C}\n\\end{equation}\nfor some nonnegative function $b\\in C(\\mathbb{R})$, so that $x(t;\\tau,\\phi)$ globally exists for each $(\\tau,\\phi)\\in\\mathbb{R}\\times{\\mathcal C}$. Furthermore, \\eqref{e:5.2} implies that $0$ is a solution of \\eqref{e:5.1}.\n\\vs\n\n\\begin{definition}\\label{d:3.1}The null solution $0$ of \\eqref{e:5.1} is said to be\n\\begin{enumerate}\n \\item[$(1)$] {globally asymptotically stable} (GAS in short), if \\,{\\em (i)}\\, it is {stable}, i.e, for every $\\tau\\in\\mathbb{R}$ and $\\varepsilon>0$, there exists $\\delta>0$ such that $x(t;\\tau,\\phi)\\in {\\mathcal B}_\\varepsilon$ for all $t\\geq \\tau$ and $\\phi\\in {\\mathcal B}_\\delta$, and {\\em (ii)}\\,\n it is globally attracting, meaning that $x(t;\\tau,\\phi)\\rightarrow 0$ as $t\\rightarrow \\infty$ for every $(\\tau,\\phi)\\in\\mathbb{R}\\times {\\mathcal C}$.\n \\vs\n\\item[$(2)$] { globally exponentially asymptotically stable} (GEAS in short), if for every $\\tau\\in \\mathbb{R}$, there exist positive constants $M,\\lambda>0$ such that\n\\begin{equation}\\label{e:5.3}\n|x(t;\\tau,\\phi)|\\leq M\\|\\phi\\|e^{-\\lambda (t-\\tau)},\\hspace{1cm}}\\def\\hs{\\hspace{0.5cm} \\forall\\,t\\geq \\tau,\\,\\,\\phi\\in {\\mathcal C}.\n\\end{equation}\n\\end{enumerate}\n\\end{definition}\n\\begin{remark}\nThe notions given in the above definition are the global versions of some corresponding local ones for functional differential equations in \\cite[Chap. 5, Def. 1.1]{Hale2} and \\cite[Def. 2.1-2.3]{Wins}, etc.\n\\end{remark}\n\n\nWe now assume that $a$ satisfies the following hypothesis:\n\\vs\\begin{enumerate}\n\\item[(A1)] \\,$\\int_s^{s+t} a(\\sigma)d\\sigma \\rightarrow \\infty$ as $t\\rightarrow \\infty$ uniformly with respect to $s\\in \\mathbb{R}$.\n \\end{enumerate}\n\\vs\\noindent\nDefine two functions $E(t,s)$ and $K(t,s)$ on $\\mathbb{R}^2$ as below: $\\forall\\,(t,s)\\in\\mathbb{R}^2$,\n$$\\begin{array}{ll}\nE(t,s)=\\exp\\(-\\int_s^ta(\\sigma)d\\sigma\\),\\hs K(t,s)=E(t,s)b(s).\\end{array}\n$$\nBy (A1) one trivially verifies that\n\\begin{equation}\\label{e:3E}\\begin{array}{ll}\n\\lim_{t\\rightarrow \\infty}E(t+s,s)=0\\mbox{ uniformly w.r.t. $s\\in\\mathbb{R}$}. \\end{array}\n\\end{equation}\nFor each $\\tau\\in\\mathbb{R}$, set\n$$\n\\vartheta_\\tau=\\sup_{t\\geq s\\geq \\tau}E(t,s),\\hs \\kappa_\\tau =\\sup_{t\\geq \\tau}\\int_\\tau^tK(t,s)ds.\n$$\n\n\n\\begin{proposition}\\label{p:3.2} The null solution of \\eqref{e:5.1} is GAS if $\\kappa_\\tau<1$ for all $\\tau\\in\\mathbb{R}$. If we further assume that $\\kappa_\\tau <1\/(1+\\vartheta_\\tau)$ for $\\tau\\in\\mathbb{R}$, then it is GEAS.\n\\end{proposition}\n\n\\noindent{\\bf Proof.} Let $\\tau\\in\\mathbb{R}$. Write $x(t)=x(t;\\tau,\\phi)$. For any $t\\geq\\eta\\geq\\tau$, multiplying \\eqref{e:5.1} with $E(t,\\eta)^{-1}=\\exp\\(\\int_\\eta^t a(\\sigma)d\\sigma\\)$, we obtain that\n\\begin{equation}\\label{e:5.0}\n\\frac{d}{dt}\\(E(t,\\eta)^{-1}x\\)= E(t,\\eta)^{-1}B(t,x_t).\n\\end{equation}\nIntegrating \\eqref{e:5.0} in $t$ between $\\eta$ and $t$, it yields\n\\begin{equation}\\label{e:5.5}\nx(t)=E(t,\\eta)x(\\eta)+\\int_\\eta^t E(t,s)B(s,x_s)ds.\n\\end{equation}\n(Here we have used the simple observation that $E(t,\\eta)E(s,\\eta)^{-1}=E(t,s)$.)\nHence\n\\begin{equation}\\label{e:3.7}\n|x(t)|\\leq E(t,\\eta)\\|x_\\eta\\|+\\int_\\eta^t K(t,s)\\|x_s\\|ds,\\hspace{1cm}}\\def\\hs{\\hspace{0.5cm} \\forall\\,t\\geq\\eta\\geq\\tau.\n\\end{equation}\nRewriting $t,s$ and $\\eta$ in \\eqref{e:3.7} as $t+\\tau$, $s+\\tau$ and $\\eta+\\tau$, respectively, i.e., performing a $\\tau$-translation on the variables in \\eqref{e:3.7}, we obtain that\n\\begin{equation}\\label{e:3.8a}\ny(t)\\leq E_\\tau(t,\\eta)\\|y_\\eta\\|+\\int_\\eta^t K_\\tau(t,s)\\|y_s\\|ds,\\hspace{1cm}}\\def\\hs{\\hspace{0.5cm} \\forall\\,t\\geq\\eta\\geq 0,\n\\end{equation}\nwhere $y(t)=|x(t+\\tau)|$, and\n\\begin{equation}\\label{e:3.8b}\nE_\\tau(t,s)=E(t+\\tau,s+\\tau),\\hs K_\\tau(t,s)=K(t+\\tau,s+\\tau)\n\\end{equation}\nfor $t,s\\geq 0.$ Note that\n$$\n\\vartheta_\\tau=\\sup_{t\\geq s\\geq 0}E_\\tau(t,s),\\hs \\kappa_\\tau =\\sup_{t\\geq 0}\\int_0^tK_\\tau(t,s)ds.\n$$\n\n\nAssume that $\\kappa_\\tau<1$. Then by Theorem \\ref{t:3.2} one deduces that for any $R,\\varepsilon>0$, there exists $T>0$ such that\n\\begin{equation}\\label{e:3.3}|x(t;\\tau,\\phi)|<\\varepsilon,\\hspace{1cm}}\\def\\hs{\\hspace{0.5cm} \\forall\\,t>\\tau+T,\\,\\,\\phi\\in{\\mathcal B}_R.\\end{equation}\nOn the other hand, we infer from Lemma \\ref{l:2.1} that\n$\n |x(t;\\tau,\\phi)|\\leq c_\\tau \\|\\phi\\|$ for all $t\\geq \\tau$ and $\\phi\\in {\\mathcal C}$,\n where $c_\\tau=\\max\\(\\vartheta_\\tau\/(1-\\kappa_\\tau),\\,1\\)$, from which it follows that the $0$ solution is stable at $\\tau$.\nThus we see that $0$ is GAS. (We mention that the stability of the null solution can be also deduced by using \\eqref{e:3.3} and the continuity property of $x(t;\\tau,\\phi)$ in $\\phi$. We omit the details.)\n\nThe second conclusion is a direct consequence of Theorem \\ref{t:3.2} (2). $\\Box$\n\n\\begin{remark}\\label{r:3.4} If $a$ is a bounded function on $\\mathbb{R}$ and $\\kappa_\\tau$ fulfills a stronger uniform smallness requirement:\n\\begin{equation}\\label{eu}\n\\kappa:=\\sup_{\\tau\\in\\mathbb{R}}\\kappa_\\tau<1\/(1+\\vartheta),\n\\end{equation}\nwhere $\\vartheta=\\sup_{\\tau\\in\\mathbb{R}}\\vartheta_\\tau$, then it can be shown that there exist positive constants $M,\\lambda>0$ independent of $\\tau\\in\\mathbb{R}$ such that \\eqref{e:5.3} holds true. In such a case we simply say that the solution $0$ of \\eqref{e:5.1} is uniformly GEAS.\n\nTo see this, we define for each $(\\tau,s)\\in\\mathbb{R}\\times\\mathbb{R}^+$ a function $e_{\\tau,s}$ on $\\mathbb{R}^+$:\n$$e_{\\tau,s}(t)=E_\\tau (s+t,s),\\hspace{1cm}}\\def\\hs{\\hspace{0.5cm} t\\in\\mathbb{R}^+.$$ By {\\em (A1)} we see that $\\lim_{t\\rightarrow\\infty}e_{\\tau,s}(t)=0$ uniformly with respect to $(\\tau,s)\\in\\mathbb{R}\\times\\mathbb{R}^+$. Using this simple fact and the boundedness of $a$ one easily examines that the family $\\{e_{\\tau,s}\\}_{(\\tau,s)\\in\\mathbb{R}\\times\\mathbb{R}^+}$ is uniformly bounded on $\\mathbb{R}^+$. Define\n$$\ne(t)=\\sup_{(\\tau,s)\\in\\mathbb{R}\\times\\mathbb{R}^+}e_{\\tau,s}(t),\\hspace{1cm}}\\def\\hs{\\hspace{0.5cm} t\\in\\mathbb{R}^+.\n$$\nThen $e(t)\\ra0$ as $t\\rightarrow\\infty$. Since for every $\\tau\\in\\mathbb{R}$, we have $$E_\\tau (s+t,s)\\leq e(t),\\hspace{1cm}}\\def\\hs{\\hspace{0.5cm} t,s\\geq 0,$$\ninvoking Remark \\ref{r:2.3} we deduce by \\eqref{e:3.8a} and \\eqref{eu} that there exist $M,\\lambda>0$ independent of $\\tau\\in\\mathbb{R}$ such that \\eqref{e:5.3} holds for all solutions of \\eqref{e:5.1}.\n\\end{remark}\n\n\n\n\\begin{remark} If $a(t)\\geq 0$ for $t\\in\\mathbb{R}$, then $\\vartheta_\\tau=1$ for all $\\tau\\in \\mathbb{R}$, and the hypothesis on $\\kappa_\\tau$ to guarantee GEAS of the null solution {reduces} to that $\\kappa_\\tau<1\/2$.\n\nIn such a case one can also easily verify that $\\kappa_\\tau\\leq\\theta<1$ for all $\\tau\\in \\mathbb{R}$ if the following hypotheses in Winston \\cite{Wins} are fulfilled:\n\\vs\n$(A2)$ $b(t)\\leq \\theta a(t)$ \\,$(t\\in\\mathbb{R})$ for some $\\theta<1$;\\, and $(A3)$ $\\int_0^\\8a(t)dt=\\infty$.\n\\vs\n\\noindent It follows that the null solution $0$ of \\eqref{e:5.1} is GAS. If $a$ is bounded and $\\theta<1\/2$, then we also infer from Remark \\ref{r:3.4} that $0$ is uniformly GEAS.\n\\end{remark}\n\n\n\n\n\\noindent{\\em Example} 3.1. Let $a(t)$ be a continuous $\\omega$-periodic ($\\omega>0$) function.\nDenote $a^+(t)$ ($a^-(t)$) the positive (negative) part of $a(t)$ (hence $a(t)=a^+(t)-a^-(t)$).\nLet\n$$I=\\int_0^\\omega a(t)dt,\\hs I^\\pm=\\int_0^\\omega a^\\pm(t)dt.$$ Clearly $I=I^+-I^-$.\nFor $s\\in\\mathbb{R}$ and $t\\geq 0$, we observe that\n\\begin{equation}\\label{e:3.8}\\begin{array}{ll}\n\\int_s^{s+t} a(\\sigma)d\\sigma&=\\int_s^{s+m_t\\omega} a(\\sigma)d\\sigma+\\int_{s+m_t\\omega}^{s+t} a(\\sigma)d\\sigma\\\\[2ex]\n&= m_t I+\\int_{s+m_t\\omega}^{s+t} a(\\sigma)d\\sigma\\\\[2ex]\n&\\geq m_t I-\\int_{s+m_t\\omega}^{s+t} a^-(\\sigma)d\\sigma\\geq m_t I-I^-,\n\\end{array}\\end{equation}\nwhere $m_t=[t\/\\omega]$ is the integer part of $t\/\\omega$.\n\nNow suppose that $I>0$. Then by \\eqref{e:3.8} we have that\n\\begin{equation}\\label{e:3.9}\n\\int_s^{s+t} a(\\sigma)d\\sigma\\geq m_t I-I^-\\geq \\(\\frac{t}{\\omega}-1\\)I-I^-=\\Lambda t-I^+,\n\\end{equation}\nwhere $\\Lambda=\\frac{I}{\\omega}$, and\n\\begin{equation}\\label{e:3.9b}\n\\int_s^{s+t} a(\\sigma)d\\sigma\\geq m_t I-I^-\\geq -I^-.\n\\end{equation}\nBy \\eqref{e:3.9} it is obviously that $a$ fulfills hypothesis (A1).\n\n\nWe infer from \\eqref{e:3.9b} that for any $\\tau\\in \\mathbb{R}$,\n\\begin{equation}\\label{e:5.9}\nE_\\tau(t,s)=\\exp\\(-\\int_s^ta(\\sigma+\\tau)d\\sigma\\)\\leq e^{I^-}:=\\vartheta,\\hspace{1cm}}\\def\\hs{\\hspace{0.5cm} t\\geq s\\geq 0.\n\\end{equation}\nAssume that the function $b$ in \\eqref{e:5.2} is bounded. Set $\\beta=\\sup_{t\\geq 0}b(t)$. Then\n\\begin{equation}\\label{e:5.11}\n\\int_0^tK_\\tau(t,s)ds=\\int_0^t E_\\tau(t,s)b(s+\\tau)ds\\leq (\\mbox{by }\\eqref{e:3.9})\\leq {\\beta\\omega e^{I^{+}}\/I}:=\\kappa\n\\end{equation}\nfor all $t\\geq 0$.\nThus in the case where $a$ is periodic and $b$ is bounded, we have\n\n\n\n\\begin{proposition}\\label{t:5.2}If $\\beta<\\beta_1:={I}\/({\\omega e^{I^{+}}})$, the null solution of \\eqref{e:5.1} is GAS; and if $\\beta<\\beta_2:={I\/(\\omega e^{I^+}(1+e^{I^{-}})})$, then it is GEAS.\n\\end{proposition}\n{\\bf Proof.} Assume $\\beta<\\beta_1$. Then by \\eqref{e:5.11} we see that\n $$\\begin{array}{ll}\\kappa_\\tau:=\\sup_{t\\geq 0}\\int_0^tK_\\tau(t,s)ds<1,\\hspace{1cm}}\\def\\hs{\\hspace{0.5cm} \\forall\\,\\tau\\in\\mathbb{R}.\\end{array} $$\n\nWe infer from \\eqref{e:5.9} that $\n\\vartheta_\\tau:=\\sup_{t\\geq s\\geq 0}E_\\tau(t,s)\\leq e^{I^-}:=\\vartheta$.\n {Thus if we assume $\\beta<\\beta_2$}, then one trivially verifies that $$\\kappa_\\tau\\leq(\\mbox{by }\\eqref{e:5.11})\\leq \\beta\\omega e^{I^{+}}\/I<1\/(1+\\vartheta),\\hspace{1cm}}\\def\\hs{\\hspace{0.5cm} \\tau\\in \\mathbb{R}.$$\n\nNow the conclusion directly follows from Propositions \\ref{p:3.2}. $\\Box$\n\\vskip8pt}\\def\\vs{\\vskip4pt\nA concrete example as in Example 3.1 is the linear equation:\n\\begin{equation}\\label{e:5.1c}\n\\dot x=-(\\sin t+\\varepsilon)x+\\beta\\, x(t-1),\\hs t>0,\n\\end{equation}\nwhere $0<\\varepsilon,\\beta<1$ are constants.\nSimple calculations show that $$\nI^+< 2+2\\pi\\varepsilon,\\hs I^-<2.\n$$\nIt is easy to check that if $\\beta<\\varepsilon e^{-(2+2\\pi\\varepsilon)}$, then the first hypothesis in Proposition \\ref{t:5.2} is fulfilled, and hence the null solution $0$ of the equation is GAS. If we further assume that $\\beta<\\varepsilon e^{-(2+2\\pi\\varepsilon)}\/\\(1+e^{2}\\)$, then it is GEAS.\n\n\n\\subsection{Pullback attractors of an ODE system with delays}\nAs a second example, we consider in this part the existence of pullback attractors of the ODE system:\n\\begin{equation}\\label{ode1}\n\\dot x=F_0(t,x)+\\sum_{i=1}^mF_i(t,x(t-r_i)),\\hs x=x(t)\\in\\mathbb{R}^n\n\\end{equation}\nwith superlinear nonlinearities $F_i$ ($0\\leq i\\leq m$).\n\nAssume that $F_i$ ($0\\leq i\\leq m$) are continuous mappings from $\\mathbb{R}\\times \\mathbb{R}^n\\rightarrow \\mathbb{R}^n$ which are locally Lipschitz in the space variable $x$ in a uniform manner with respect to $t$ on bounded intervals and satisfy the structure condition {\\bf (F)} given in Section 1, and $r_i:\\mathbb{R}\\rightarrow [0,r]$ ($1\\leq i\\leq m$) are measurable functions.\n\nDenote ${\\mathcal C}$ the space $C([-r,0],\\mathbb{R}^n)$ equipped with the usual norm $\\|\\.\\|$.\nBy the hypotheses on $F_i$ and the delay functions $r_i$, it can be easily shown that the initial value problem of \\eqref{ode1} is well-posed. Specifically, for each $\\tau\\in\\mathbb{R}$ and $\\phi\\in{\\mathcal C}$ the system has a unique solution $x(t;\\tau,\\phi):=x(t)$ on a maximal existence interval $[\\tau-r,T_\\phi)$ ($T_\\phi>\\tau$) with\n$$\nx(\\tau+s)=\\phi(s),\\hspace{1cm}}\\def\\hs{\\hspace{0.5cm} s\\in[-r,0].\n$$\nFor convenience, we call the lift $x_t$ of $x(t)$ the {\\em solution curve} of \\eqref{ode1} in ${\\mathcal C}$ with initial value $x_\\tau=\\phi$, denoted hereafter by $x_t(\\tau,\\phi)$.\n\n\n \\begin{lemma}\\label{l:ode1}Suppose that there exist $M,N>0$ such that\n \\begin{equation}\\label{ode3}\\begin{array}{ll}\n\\sum_{i=0}^m\\int_s^t\\beta_i(\\mu)d\\mu\\leq M(t-s)+N,\\hspace{1cm}}\\def\\hs{\\hspace{0.5cm} -\\infty0$ independent of $\\tau\\in\\mathbb{R}$ such that\n\\begin{equation}\\label{ode6}\n|x(t;\\tau,\\phi)|\\leq C\\|\\phi\\|e^{-\\lambda (t-\\tau)}+\\rho,\\hspace{1cm}}\\def\\hs{\\hspace{0.5cm} \\forall\\,t\\geq\\tau,\\,\\,(\\tau,\\phi)\\in\\mathbb{R}\\times {\\mathcal C}.\n\\end{equation}\n \\end{lemma}\n{\\bf Proof.} Let $x=x(t):=x(t;\\tau,\\phi)$ be a solution of \\eqref{ode1} with maximal existence interval $[\\tau-r,T_\\phi)$. Set $\\gamma:=p(q-1)\/(p-q)+1$.\nTaking the inner product of both sides of \\eqref{ode1} with $|x|^{\\gamma-1}x$, we find that\n$$\\begin{array}{lll}\n\\frac{1}{\\gamma+1}\\frac{d}{dt}|x|^{\\gamma+1}&=|x|^{\\gamma-1}(F_0(t,x),x)+|x|^{\\gamma-1}\\sum_{i=1}^m(F_i(t,x(t-r_i)),x)\\\\[2ex]\n&\\leq \\(-\\alpha_0|x|^{\\gamma+p}+\\beta_0(t)|x|^{\\gamma-1}\\)+\\sum_{i=1}^m\\(\\alpha_i |x|^{\\gamma}\\|x_t\\|^{q}+\\beta_i(t)|x|^{\\gamma}\\).\n\\end{array}\n$$\nThe classical Young's inequality implies that\n$$\n|x|^{\\gamma}\\|x_t\\|^{q}\\leq \\varepsilon\\|x_t\\|^{\\gamma+1}+C_\\varepsilon|x|^{\\gamma(\\gamma+1)\/((\\gamma+1)-q)}\n$$\nfor any $\\varepsilon>0$. Here and below $C_\\varepsilon$ denotes a general constant depending upon $\\varepsilon$. By the choice of $\\gamma$ one easily verify that $\\gamma(\\gamma+1)\/((\\gamma+1)-q)<\\gamma+p$. Hence using the Young's inequality once again we deduce that\n$$\n|x|^{\\gamma}\\|x_t\\|^{q}\\leq \\varepsilon\\|x_t\\|^{\\gamma+1}+\\varepsilon|x|^{\\gamma+p}+C_\\varepsilon.\n$$\nWe also have\n$$\n|x|^{\\gamma-1},|x|^{\\gamma}\\leq \\varepsilon |x|^{\\gamma+1}+C_\\varepsilon.\n$$\nCombining the above estimates together it gives\n \\begin{equation}\\label{ode2}\\begin{array}{lll}\n \\frac{1}{\\gamma+1}\\frac{d}{dt}|x|^{\\gamma+1}&\\leq -\\(\\alpha_0-\\varepsilon{\\alpha}\\)|x|^{\\gamma+p}+{\\varepsilon \\alpha}\\|x_t\\|^{\\gamma+1}\\\\[2ex]\n &\\hs+\\varepsilon \\beta(t)|x|^{\\gamma+1}+C_\\varepsilon(\\beta(t)+1),\\\\[2ex]\n \\end{array}\n \\end{equation}\nwhere $$\\begin{array}{ll}\\alpha=\\sum_{i=1}^m \\alpha_i,\\hs \\beta(t)=\\sum_{i=0}^m\\beta_i(t).\\end{array}$$\nIt can be assumed that $\\varepsilon\\alpha<\\alpha_0$. Noticing that $s^{\\gamma+1}\\leq s^{\\gamma+p}+1$ for all $s\\geq 0$, by \\eqref{ode2} we find that\n\\begin{equation}\\label{ode20}\\begin{array}{lll}\n\\frac{d}{dt}|x|^{\\gamma+1}&\\leq -a_\\varepsilon(t)|x|^{\\gamma+1}+{\\varepsilon (\\gamma+1)\\alpha}\\|x_t\\|^{\\gamma+1}+C_\\varepsilon(\\beta(t)+1),\n \\end{array}\n \\end{equation}\nwhere\n$\na_\\varepsilon(t)=(\\gamma+1)\\(\\alpha_0-{\\varepsilon\\alpha}-\\varepsilon \\beta(t)\\).\n$\n\\vs\nLet $E_\\varepsilon(t,s)=\\exp\\(-\\int_s^ta_\\varepsilon(\\mu)d\\mu\\)$ ($t\\geq s\\geq\\tau$). In what follows we always assume $\\varepsilon<1$ and that $\\varepsilon (\\gamma+1)(\\alpha+M)0$ sufficiently small so that\n$$\\begin{array}{ll}\n\\kappa:=\\varepsilon \\kappa_0<1\/(1+\\vartheta).\n\\end{array}\n$$\nThen the requirement in Theorem \\ref{t:3.2} is fulfilled.\nThus by virtue of Lemma \\ref{l:2.a1} we first deduce that $x(t)$ is bounded on $[\\tau-r,T_\\phi)$. It follows that $T_\\phi=\\infty$.\nFurther since $y\\in {\\mathscr L}_r(E;\\varepsilon K,0;C_\\varepsilon')$ for all $\\tau\\in\\mathbb{R}$, where ${\\mathscr L}_r(E;\\varepsilon K,0;C_\\varepsilon')$ denotes the family of nonnegative continuous functions on $\\mathbb{R}^+$ satisfying \\eqref{e:3.8a} (see also \\eqref{e1.5}), invoking Theorem \\ref{t:3.2}\none immediate concludes that there exist $C,\\lambda,\\rho>0$ independent of $\\tau$ such that \\eqref{ode6} holds. $\\Box$\n\n\n\\vskip8pt}\\def\\vs{\\vskip4pt\nLemma \\ref{l:ode1} enables us to define a {\\em process}\\, $\\Phi(t,\\tau)$ on ${\\mathcal C}$:\n\\begin{equation}\\label{e:3.20}\n\\Phi(t,\\tau)\\phi=x_t(\\tau,\\phi),\\hspace{1cm}}\\def\\hs{\\hspace{0.5cm} t\\geq\\tau>-\\infty,\\,\\,\\phi\\in{\\mathcal C},\n\\end{equation}\nwhere $x_t(\\tau,\\phi)$ is the { solution curve} of \\eqref{ode1} in ${\\mathcal C}$ with $x_\\tau(\\tau,\\phi)=\\phi$ defined as above. $\\Phi$ possesses the following basic properties:\n\\vs\n$\\bullet$ $\\Phi(t,\\tau):{\\mathcal C}\\rightarrow{\\mathcal C}$ is a continuous mapping for each fixed $(t,\\tau)\\in\\mathbb{R}^2$, $t\\geq\\tau$;\n\n$\\bullet$ $\\Phi(\\tau,\\tau)=\\mbox{id}_{\\mathcal C}$ for all $\\tau\\in\\mathbb{R}$, where $\\mbox{id}_{\\mathcal C}$ is the identity mapping on ${\\mathcal C}$;\n\n$\\bullet$ $\\Phi(t,\\tau)=\\Phi(t,s)\\Phi(s,\\tau)$ for all $t\\geq s\\geq\\tau$.\n\\vs\n\\noindent For system \\eqref{ode1}, the estimate given in Lemma \\ref{l:ode1} is sufficient to guarantee the existence of a global pullback attractor; see \\cite{Carab2,Carab} etc. (The interested reader is referred to \\cite{CLR} etc. for the general theory of pullback attractors.) Hence we have\n\n\\begin{theorem}\\label{t:ode1} Assume the hypotheses in Lemma \\ref{l:ode1}. Then $\\Phi$ has a (unique) global pullback attractor in ${\\mathcal C}$. Specifically, there is a unique family ${\\mathcal A}=\\{A(t)\\}_{t\\in\\mathbb{R}}$ of compact sets contained in the ball $\\overline{\\mathcal B}_\\rho$ in ${\\mathcal C}$ centered at $0$ with radius $\\rho$ such that\n\\begin{enumerate}\n\\item[$(1)$] $\\Phi(t,\\tau)A(\\tau)=A(t)$ for all $t\\geq\\tau$;\n\\item[$(2)$] for any bounded set $B\\subset{\\mathcal C}$,\n$$\\lim_{\\tau\\rightarrow-\\infty}d_H\\(\\Phi(t,\\tau)B,\\,A(t)\\)=0$$ for all $t\\in\\mathbb{R}$, where $d_H(\\.,\\.)$ denotes the Hausdorff semi-distance in ${\\mathcal C}$,\n$$\nd_H(M,N)=\\sup_{\\phi\\in M}\\inf\\{\\|\\phi-\\psi\\|:\\,\\,\\psi\\in N\\},\\hspace{1cm}}\\def\\hs{\\hspace{0.5cm} \\forall\\,M,N\\subset {\\mathcal C}.\n$$\n \\end{enumerate}\n\\end{theorem}\n\n\n\n\n\n\n\\section{On the Dynamics of Retarded Evolution Equations with Sublinear Nonlinearities}\nAs our third example to illustrate the application of Theorems \\ref{t:3.1} and \\ref{t:3.2}, we investigate the dynamics of abstract retarded functional differential equations with sublinear nonlinearities in the general setting of cocycle systems.\n\nLet ${\\mathcal H}$ be a compact metric space with metric $d(\\.,\\.)$.\nAssume that there has been given a dynamical system $\\theta$ on ${\\mathcal H}$, i.e., a continuous mapping $\\theta:\\mathbb{R}\\times{\\mathcal H}\\rightarrow{\\mathcal H} $ satisfying the group property: for all $p\\in{\\mathcal H}$ and $s,t\\in\\mathbb{R}$,\n$$\n{ \\theta(0,p)}=p,\\hspace{1cm}}\\def\\hs{\\hspace{0.5cm} \\theta({s+t},p)=\\theta(s,\\theta(t,p)).\n$$\n\n\nAs usual, we will rewrite $\\theta(t,p)=\\theta_tp$.\n\nIn what follows we always assume that ${\\mathcal H}$ is {\\em minimal} (with respect to $\\theta$). This means that $\\theta$ has no proper nonempty compact invariant subsets in ${\\mathcal H}$.\n\nLet $X$ be a real Banach space with norm $\\|\\.\\|_0$, and let $A$ be a sectorial operator on $X$ with compact resolvent. Denote $X^s$ ($s\\geq0$) the fractional power of $X$ generated by $A$ with norm $\\|\\.\\|_s$; see \\cite[Chap.\\,1]{Henry} for details.\n\nLet $0\\leq r<\\infty,$ and $\\alpha\\in [0,1)$. Denote ${{\\mathcal C}_\\alpha}=C([-r,0],X^\\alpha)$. ${{\\mathcal C}_\\alpha}$ is equipped with the norm {$\\|\\.\\|_{{\\mathcal C}_\\alpha}$} defined by\n$$\n\\|\\phi\\|_{{\\mathcal C}_\\alpha}=\\max_{[-r,0]}\\|\\phi(s)\\|_\\alpha,\\hspace{1cm}}\\def\\hs{\\hspace{0.5cm} \\phi\\in{{\\mathcal C}_\\alpha}.\n$$\nGiven a continuous function $u:[t_0-r,T)\\rightarrow X^\\alpha$, denote by $u_t$ the lift of $u$ in ${{\\mathcal C}_\\alpha}$,\n$$\nu_t(s)=u(t+s),\\hspace{1cm}}\\def\\hs{\\hspace{0.5cm} s\\in[-r,0],\\,\\,t\\geq t_0.\n$$\n\nThe retarded functional cocycle system we are concerned with is as follows:\n \\begin{equation}\\label{e:4.1}\\frac{du}{dt}+Au=F(\\theta_tp,u_t),\\hspace{1cm}}\\def\\hs{\\hspace{0.5cm} t\\geq 0,\\,\\, p\\in {\\mathcal H},\\end{equation}\nwhere $F$ is a continuous mapping from ${\\mathcal H}\\times {{\\mathcal C}_\\alpha}$ to $X$.\nLater we will show how to put a nonlinear evolution equation like\n $$\\frac{du}{dt}+Au={ f(u(t-r_1),\\cdots,u(t-r_m))}+h(t)$$\n into the abstract form of \\eqref{e:4.1}.\n For convenience in statement, ${\\mathcal H}$ and $\\theta$ are usually called the {\\em base space} and the {\\em driving system} of \\eqref{e:4.1}, respectively.\n\nDenote by ${\\mathcal B}_R$ the ball in ${{\\mathcal C}_\\alpha}$ centered at $0$ with radius $R$.\n\n Assume that $F$ satisfies the following conditions:\n \\begin{enumerate}\\item[(F1)] $F(p,\\phi)$ is {\\em locally Lipschitz} in $\\phi$ uniformly w.r.t $p\\in{\\mathcal H}$, namely, for any $R>0$, there exists $L_R>0$ such that\n$$\n\\|F(p,\\phi)-F(p,\\phi')\\|_0\\leq L_R\\|\\phi-\\phi'\\|_{{\\mathcal C}_\\alpha},\\hspace{1cm}}\\def\\hs{\\hspace{0.5cm} \\forall\\,\\phi,\\phi'\\in { \\overline{\\mathcal B}_R},\\,\\,p\\in{\\mathcal H}.\n$$\n\\item[(F2)] There exist $C_0,C_1>0$ such that\n $$\n \\|F(p,\\phi)\\|_{ 0}\\leq C_0\\|\\phi\\|_{{\\mathcal C}_\\alpha}+C_1,\\hspace{1cm}}\\def\\hs{\\hspace{0.5cm} \\forall\\,(p,\\phi)\\in {\\mathcal H}\\times{{\\mathcal C}_\\alpha}.\n $$\n \\end{enumerate}\n Under the above assumptions, the same argument as in the proof of \\cite[Proposition 3.1]{TW1} with minor modifications applies to show the existence and uniqueness of global mild solutions for \\eqref{e:4.1}: \\,For each initial data $\\phi\\in{{\\mathcal C}_\\alpha}:=C([-r,0],X^\\alpha)$ and $p\\in {\\mathcal H}$, there is a unique continuous function $u:[-r,\\infty)\\rightarrow X^{\\alpha}$ with $u(t)=\\phi(t)$ ($-r\\le t\\le0$) satisfying the integral equation\n$$\nu(t)=e^{-At}\\phi(0)+\\int_{0}^{t}e^{-A(t-s)}F(\\theta_sp,u_s)ds,\\hspace{1cm}}\\def\\hs{\\hspace{0.5cm} t\\geq0.\n$$\n\nA solution of \\eqref{e:4.1} clearly depends on $p$. For convenience, given $p\\in {\\mathcal H}$, we call a solution $u$ of \\eqref{e:4.1} a {\\em solution pertaining to $p$}.\nWe will use the notation $u(t;p,\\phi)$ to denote the solution of \\eqref{e:4.1} on $[-r,\\infty)$ pertaining to $p$ with initial value $\\phi\\in{{\\mathcal C}_\\alpha}$.\nThe solutions of \\eqref{e:4.1} generates a {\\em cocycle} $\\Phi$ on ${{\\mathcal C}_\\alpha}$,\n$$\n\\Phi(t,p)\\phi=u_t,\\hspace{1cm}}\\def\\hs{\\hspace{0.5cm} t\\geq 0,\\,\\,(p,\\phi)\\in{\\mathcal H}\\times{{\\mathcal C}_\\alpha},\n$$\nwhere $u_t$ is the lift of the solution $u(t)=u(t;p,\\phi)$ in ${{\\mathcal C}_\\alpha}$.\n\nSince ${\\mathcal H}$ is compact and $A$ has compact resolvent, using a similar argument as in the proof of \\cite[Proposition 4.1]{TW2}, it can be shown that for each fixed $t>r$, $\\Phi(t,p)\\phi$ is compact as a mapping from ${\\mathcal H}\\times{\\mathcal C}_\\alpha$ to ${\\mathcal C}_\\alpha$. Making use of this basic fact one can easily verify that $\\Phi$ is {\\em asymptotically compact}, that is, $\\Phi$ enjoys the following property:\n\n\\begin{enumerate}\\item[{\\bf(AC)}] For any sequences $t_n\\rightarrow \\infty$ and $(p_n,\\phi_n)\\in{\\mathcal H}\\times {{\\mathcal C}_\\alpha}$, if $\\Cup_{n\\geq 1}\\Phi([0,t_n],p_n)\\phi_n$ is bounded in ${{\\mathcal C}_\\alpha}$ then the sequence $\\Phi(t_n,p_n)\\phi_n$ has a convergent subsequence.\n\\end{enumerate}\n\n\\subsection{Basic integral formulas on bounded solutions}\n{\nSuppose $A$ has a spectral decomposition $\\sigma(A)=\\sigma^-\\cup\\sigma^+$, where\n \\begin{equation}\\label{e:4.3}\n \\mbox{Re}\\,z\\leq -\\beta<0\\,\\,(z\\in\\sigma^-),\\hs \\mbox{Re}\\,z\\geq \\beta>0\\,\\,(z\\in\\sigma^+)\n \\end{equation}\n for some $\\beta>0$. Let $X=X_1\\oplus X_2$ be the corresponding direct sum decomposition of $X$ with $X_1$\n and $X_2$ being invariant subspaces of $A$.\nDenote $P_i:X\\rightarrow X_i$ ($i=1,2$) the projection from $X$ to $X_i$, and write $A_i=A|_{X_i}$. By the basic knowledge on sectorial operators (see Henry \\cite[Chap.\\,1]{Henry}), there exists $M\\geq1$ such that\n\\begin{equation}\\label{e:4.6a}\\|\\Lambda^\\alpha e^{-A_1t}\\|\\leq{ Me^{\\beta t}},\\hs \\|e^{-A_1t}\\|\\leq{ Me^{\\beta t}},\\hspace{1cm}}\\def\\hs{\\hspace{0.5cm} t\\leq0,\\end{equation}\n\\begin{equation}\\label{e:4.6b}\\|\\Lambda^\\alpha e^{-A_2t}P_2\\Lambda^{-\\alpha}\\|\\leq Me^{-\\beta t},\\hs\\|\\Lambda^\\alpha e^{-A_2t}\\|\\leq Mt^{-\\alpha}e^{-\\beta t},\\hspace{1cm}}\\def\\hs{\\hspace{0.5cm} t>0.\\end{equation}\n\nThe verification of the following basic integral formulas on bounded solutions are just slight modifications of the corresponding ones for that of equations without delays (see e.g. \\cite[pp. 180, Lemma A.1]{Hale0} and \\cite{Ju}), and hence is omitted.\n\n\n\\begin{lemma}\\label{l:3.3}\n Let $u:[-r,+\\infty)\\to X^{\\alpha}$ be a bounded continuous function. Then $u$ is a solution of (\\ref{e:4.1}) on $[-r,\\infty)$ pertaining to $p\\in{\\mathcal H}$ if and only if $u$ solves the integral equation\n$$\\begin{array}{ll}\nu(t)=&\\,\\,e^{-A_2t}P_2 u(0)+\\int_{0}^{t}e^{-A_2(t-s)}P_2 F(\\theta_{s}p, u_s)ds\\\\[2ex]\n&-\\int_{t}^{\\infty}e^{-A_1(t-s)}P_1 F(\\theta_{s}p, u_s)ds,\\hs t\\ge0.\n\\end{array}\n$$\n\n\\end{lemma}\n\n}\n\n\n\\subsection{Existence of bounded complete solutions}\nFor nonlinear evolution equations, bounded complete solutions are of equal importance as equilibrium ones. This is because that the long-term dynamics of an equation\nis determined not only by the distribution of its equilibrium solutions, but also by that of all its bounded complete trajectories. In fact, for a nonautonomous evolution equation it may be of little sense to talk about equilibrium solutions in the usual terminology.\n\nIn this subsection we establish an existence result for bounded complete solutions of equation \\eqref{e:4.1}. For this purpose we need first to give some a priori estimates.\n\nLet $C_0,C_1$ be the constants in (F2), and set\n\\begin{equation}\\label{e:kap}\\begin{array}{ll}\\kappa_0=\\sup_{t\\ge0}\\(\\int_{0}^{t}(t-s)^{-\\alpha}e^{-\\beta (t-s)}ds+\\int_{t}^{\\infty}e^{\\beta(t-s)}ds\\).\\end{array}\\end{equation}\n\n\\begin{lemma}\\label{l:4.1} Suppose $A$ has a spectral decomposition as in \\eqref{e:4.3}, and that\n$C_0<{1}\/{(\\kappa_0 M)}.$\nThen for any $R,\\varepsilon>0$, there exists $T>0$ such that for all bounded solutions $u(t)=u(t;p,\\phi)$ of \\eqref{e:4.1} with $\\phi\\in\\overline{\\mathcal B}_R$,\n\\begin{equation}\\label{e:4.5}\n\\|u(t)\\|_\\alpha<\\rho+\\varepsilon,\\hspace{1cm}}\\def\\hs{\\hspace{0.5cm} t{ >} T,\n\\end{equation}\n where $\\rho={ C_1M(1-\\kappa_0 C_0M)^{-1}\\int_{0}^{\\infty}(1+s^{-\\alpha})e^{-\\beta s}ds}$. Consequently\n\\begin{equation}\\label{e:4.5b}\\sup_{t\\in\\mathbb{R}}\\|\\gamma(t)\\|_{\\alpha}\\leq \\rho\\end{equation} for all bounded complete solutions $\\gamma(t)$ of \\eqref{e:4.1}.\n\\end{lemma}\n\n\\noindent{\\bf Proof.} (1) {Let $u(t)=u(t;p,\\phi)$ be a bounded solution of \\eqref{e:4.1} on $[-r,\\infty)$. For any $\\tau\\geq 0$, set $v(t)=u(t+\\tau)$ ($t\\geq 0$). Then $v$ is a bounded solution of \\eqref{e:4.1} pertaining to $q=\\theta_\\tau p$.\nHence we infer from Lemma \\ref{l:3.3} that\n\\begin{equation}\\label{e:4.8}\n\\begin{array}{ll}\nv(t)=&e^{-A_2t}P_2 v(0)+\\int_{0}^{t}e^{-A_2(t-s)}P_2 F(\\theta_{s}{ q},v_s)ds\\\\[2ex]\n&-\\int_{t}^{\\infty}e^{-A_1(t-s)}P_1 F(\\theta_{s}q,v_s)ds,\\hs t\\ge0.\n\\end{array}\n\\end{equation}\nTherefore by \\eqref{e:4.6a}, \\eqref{e:4.6b} and (F2), we deduce that\n$$\n\\begin{array}{ll}\n\\|v(t)\\|_\\alpha\n\\leq&Me^{-\\beta t}{ \\|v_0\\|_{{\\mathcal C}_\\alpha}}+\\int_{0}^{t} K_1(t,s)\\|v_s\\|_{{\\mathcal C}_\\alpha} ds\\\\[2ex]\n&+\\int_{t}^{\\infty}K_2(t,s)\\|v_s\\|_{{\\mathcal C}_\\alpha} ds+C_2,\\hs t\\ge0,\n\\end{array}\n$$\nwhere\n\\begin{equation}\\label{ek1}K_1(t,s)={ C_0M(t-s)^{-\\alpha}e^{-\\beta(t-s)}},\\hs K_2(t,s)={ C_0Me^{\\beta(t-s)}},\\end{equation}\nand $C_2=C_1M{ \\int_{0}^{\\infty}(1+s^{-\\alpha})e^{-\\beta s}ds}$. That is, $u$ satisfies\n\\begin{equation}\\label{e:4.11}\n\\begin{array}{ll}\n{ \\|u(t)\\|_\\alpha}\n\\leq& Me^{-\\beta (t-\\tau)}{ \\|u_\\tau\\|_{{\\mathcal C}_\\alpha}}+\\int_{\\tau}^{t} K_1(t,s)\\|u_s\\|_{{\\mathcal C}_\\alpha} ds\\\\[2ex]\n&+\\int_{t}^{\\infty}K_2(t,s)\\|u_s\\|_{{\\mathcal C}_\\alpha} ds+C_2,\\hs t\\ge\\tau\\geq 0.\n\\end{array}\n\\end{equation}\n Applying Theorem \\ref{t:3.1} one deduces that if $C_0<1\/(\\kappa_0 M)$ then\n for any $R, \\varepsilon >0$, there exists { $T>0$} such that\n\\eqref{e:4.5} holds true for all $p\\in{\\mathcal H}$ and $\\phi\\in\\overline{\\mathcal B}_R$.\n\\vs\n(2) Let $\\gamma(t)$ be a bounded complete solution of \\eqref{e:4.1} pertaining to some $q\\in{\\mathcal H}$. Pick an $R>0$ such that $\\|\\gamma(t)\\|_\\alpha< R$ for all $t\\in\\mathbb{R}$. Then for any $\\varepsilon>0$, there is $T>0$ such that \\eqref{e:4.5} holds for all $p\\in{\\mathcal H}$ and $\\phi\\in\\overline{\\mathcal B}_R$. Taking $p=\\theta_{-T} q$ and $\\phi=\\gamma(-T)$, one finds that\n$$\n\\|\\gamma(0)\\|_\\alpha=\\|u(T;p,\\phi)\\|_\\alpha<\\rho+\\varepsilon.\n$$\nSince $\\varepsilon$ is arbitrary, we conclude that $\\|\\gamma(0)\\|_\\alpha\\leq\\rho$.\n\nIn a similar fashion it can be shown that $\\|\\gamma(t)\\|_\\alpha\\leq\\rho$ for all $t\\in\\mathbb{R}$. $\\Box$\n\n}\n\\vskip8pt}\\def\\vs{\\vskip4pt\nThanks to Lemma \\ref{l:4.1}, one can now show by very standard argument via the Conley index theory that equation \\eqref{e:4.1} has a bounded complete solution $u$. Specifically, we have the following existence result.\n\n\\begin{theorem}\\label{t:4.1b} Assume the hypotheses in Lemma \\ref{l:4.1}. Then for any $p\\in{\\mathcal H}$, \\eqref{e:4.1} has at least one bounded complete solution $u$ pertaining to $p$.\n\\end{theorem}\n{\\bf Proof.} The estimate \\eqref{e:4.5b} allows us to prove by using the Conley index theory and some standard argument that \\eqref{e:4.1} has at least one bounded complete solution $\\gamma=\\gamma(t)$ pertaining to some $p_0\\in{\\mathcal H}$. The interested reader is referred to \\cite[Sect. 7]{Wang} and \\cite{LLZ} for details.\n\nTo show that for any $p\\in{\\mathcal H}$, equation \\eqref{e:4.1} has at least one bounded complete solution $u$ pertaining to $p$, we consider the skew-product flow $\\Pi$ on ${\\mathscr X}={\\mathcal H}\\times {{\\mathcal C}_\\alpha}$ defined as below:\n\\begin{equation}\\label{e:sk}\n\\Pi(t)(p,\\phi)=\\(\\theta_tp,\\Phi(t,p)\\phi\\),\\hspace{1cm}}\\def\\hs{\\hspace{0.5cm} (p,\\phi)\\in {\\mathscr X},\\,\\,t\\geq 0.\n\\end{equation}\nThe asymptotic compactness of $\\Phi$ imply that $\\Pi$ is asymptotically compact.\nLet $\\varphi(t)=(\\theta_tp_0,\\,\\gamma_t)$. Then $\\varphi=\\varphi(t)$ is a bounded complete trajectory of $\\Pi$.\n\nLet ${\\mathcal S}=\\omega(\\varphi)$ be the $\\omega$-limit set of $\\varphi$,\n\\begin{equation}\\label{e:os}\\begin{array}{ll}\n\\omega(\\varphi)=\\bigcap}\\def\\Cup{\\bigcup_{\\tau\\geq0}\\,\\overline{\\{\\varphi(t):\\,\\,t\\geq\\tau\\}}.\\end{array}\n\\end{equation}\nBy the basic knowledge in the dynamical systems theory we know that ${\\mathcal S}$ is a nonempty compact invariant set of $\\Pi$.\nSet $K=P_{\\mathcal H}{\\mathcal S}$, where $P_{\\mathcal H}:{\\mathscr X}\\rightarrow {\\mathcal H}$ is the projection. One can easily verify that $K$ is a nonempty compact invariant set of the driving system $\\theta$. Hence due to the minimality hypothesis on ${\\mathcal H}$ we deduce that $K={\\mathcal H}$. Consequently for each $p\\in {\\mathcal H}$, there is a $\\phi\\in {{\\mathcal C}_\\alpha}$ such that $(p,\\phi)\\in{\\mathcal S}$. Let $(\\theta_tp,\\,u_t)$ be a bounded complete trajectory of $\\Pi$ in ${\\mathcal S}$ through $(p,\\phi)$. Set\n$$\nu(t)=u_t(0),\\hspace{1cm}}\\def\\hs{\\hspace{0.5cm} t\\in\\mathbb{R}.\n$$\nThen $u(t)$ is bounded complete solution of \\eqref{e:4.1} pertaining to $p$. $\\Box$\n\n\n\n\n\\subsection{Existence of a nonautonomous equilibrium solution}\\label{s:4.3}\nFor the sake of simplicity in statement, instead of (F1) and (F2), in this section we assume that $F(p,\\phi)$ is {\\em globally Lipschitz } in $\\phi$ uniformly w.r.t $p\\in {\\mathcal H}$, i.e.,\n \\begin{enumerate}\\item[(F3)] there exist $L>0$ such that\n$$\n\\|F(p,\\phi)-F(p,\\phi')\\|_0\\leq L\\|\\phi-\\phi'\\|_{{\\mathcal C}_\\alpha},\\hspace{1cm}}\\def\\hs{\\hspace{0.5cm} \\forall\\,\\phi,\\phi'\\in {\\mathcal C}_\\alpha,\\,\\,p\\in{\\mathcal H}.\n$$\n\\end{enumerate}\nIn such a case, since\n$$\n\\|F(p,\\phi)\\|_0= \\|F(p,\\phi)-F(p,0)\\|_0+\\|F(p,0)\\|_0\\leq L\\|\\phi\\|_{{\\mathcal C}_\\alpha}+\\|F(p,0)\\|_0,\n$$\nwe see that hypothesis (F2) is automatically fulfilled with\n\\begin{equation}\\label{e:4.31}C_0=L,\\hs C_1=\\max_{p\\in {\\mathcal H}}\\|F(p,0)\\|_0.\\end{equation}\n\n\n\n\\begin{definition} A nonautonomous equilibrium solution of \\eqref{e:4.1} is a continuous mapping $\\Gamma\\in C({\\mathcal H},X^\\alpha)$ such that\n$\\gamma_p(t):=\\Gamma(\\theta_tp)$ is a bounded complete solution of \\eqref{e:4.1} pertaining to $p$ for each $p\\in {\\mathcal H}$.\n \\end{definition}\n\\begin{theorem}\\label{t:4.1}Suppose $A$ has a spectral decomposition as in \\eqref{e:4.3}, and that\n$L<{1}\/{(\\kappa_0 M)}$.\n Then the following assertions hold:\n \\begin{enumerate}\\item[$(1)$] Equation $(\\ref{e:4.1})$ has a nonautonomous equilibrium solution $\\Gamma\\in C({\\mathcal H},{ X^{\\alpha}})$.\\vs\n \\item[$(2)$] For any $R,\\varepsilon>0$, there exists $T>0$ such that for any bounded solution $u(t)=u(t;p,\\phi)$ with $\\phi\\in\\overline{\\mathcal B}_R$,\n$$\n\\|u(t)-\\Gamma(\\theta_tp)\\|_\\alpha<\\varepsilon,\\hspace{1cm}}\\def\\hs{\\hspace{0.5cm} t> T.\n$$\n\\item[$(3)$] There exists $c>0$ such that for any bounded solution $u(t)=u(t;p,\\phi)$,\n$$\n\\|u(t)-\\Gamma(\\theta_tp)\\|_\\alpha\\leq c\\max_{s\\in[-r,0]}\\|\\phi(s)-\\Gamma(\\theta_s p)\\|_\\alpha,\\hspace{1cm}}\\def\\hs{\\hspace{0.5cm} t\\geq0.\n$$\n\\end{enumerate}\n\\end{theorem}\n{\\bf Proof.} (1)\\, We continue the argument in the proof of Theorem \\ref{t:4.1b}.\nSet $${\\mathcal S}[p]=\\{\\phi:\\,\\,(p,\\phi)\\in {\\mathcal S}\\},\\hspace{1cm}}\\def\\hs{\\hspace{0.5cm} p\\in{\\mathcal H},$$\n where ${\\mathcal S}$ is the $\\omega$-limit set of $\\varphi$ given by \\eqref{e:os}.\n Using the compactness of ${\\mathcal S}$ one easily checks that ${\\mathcal S}[p]$ is upper semicontinuous, i.e., given $p\\in{\\mathcal H}$, for any $\\varepsilon>0$, there is a $\\delta>0$ such that ${\\mathcal S}{ [q]}$ is contained in the $\\varepsilon$-neighborhood of ${\\mathcal S}{ [p]}$ for all $q$ with $d(q,p)<\\delta$.\nIn what follows we show that ${\\mathcal S}[p]$ is a singleton. Consequently the upper semicontinuity of ${\\mathcal S}[p]$ reduces to the continuity of ${\\mathcal S}[p]$ in $p$.\n\nLet $\\phi_1,\\phi_2\\in {\\mathcal S}[p]$. As in the proof of { Theorem \\ref{t:4.1b}} we know that $\\Phi$ has two bounded complete trajectories $\\gamma^i_t$ ($i=1,2$) in ${{\\mathcal C}_\\alpha}$ pertaining to $p$ with $\\gamma^i_0=\\phi_i$. We check that\n$\\gamma_t:=\\gamma^1_t-\\gamma^2_t\\equiv 0$ for $t\\in\\mathbb{R}$, or equivalently,\n\\begin{equation}\\label{e:4.8}\\gamma(t):=\\gamma^1(t)-\\gamma^2(t)\\equiv0,\\hs\\mbox{ where $\\gamma^i(t)=\\gamma^i_t(0)$}.\\end{equation} It then follows that $\\phi_1=\\phi_2$, hence ${\\mathcal S}[p]$ is a singleton.\n\n\nFor $\\eta\\in\\mathbb{R}$, we write $\\varphi^i(t)=\\gamma^i(t+\\eta)$. Then $\\varphi^i(t)$ is a solution of \\eqref{e:4.1} pertaining to $q=\\theta_\\eta p$. By Lemma \\ref{l:3.3} we have\n\n{\\begin{equation*}\n\\begin{array}{ll}\n\\varphi^{i}(t)=&e^{-A_2t}P_2 \\varphi^{i}(0)+\\int_{0}^{t}e^{-A_2(t-s)}P_2 F\\(\\theta_s q,\\,\\varphi^{i}_s\\)ds\\\\[2ex]\n&-\\int_{t}^{\\infty}e^{-A_1(t-s)}P_1 F\\(\\theta_s q,\\,\\varphi^{i}_{s}\\)ds,\\hs t\\ge0.\n\\end{array}\n\\end{equation*}\n Let $\\varphi(t):=\\varphi^1(t)-\\varphi^2(t)$. Then\n\\begin{equation*}\n\\begin{array}{ll}\n\\varphi(t)=\\,&\\,e^{-A_2t}P_2 \\varphi(0)+\\int_{0}^{t}e^{-A_2(t-s)}P_2 \\(F\\(\\theta_s q,\\,\\varphi^1_s\\)- F\\(\\theta_s q,\\,\\varphi^2_s\\)\\)ds\\\\[2ex]\n&-\\int_{t}^{\\infty}e^{-A_1(t-s)}P_1 \\(F\\(\\theta_s q,\\,\\varphi^1_s\\)-F\\(\\theta_s q,\\,\\varphi^2_s\\)\\)ds,\\hs t\\ge0.\n\\end{array}\n\\end{equation*}\nThus by ({ F3}) we deduce that\n\\begin{equation}\\label{eq2}\n\\begin{array}{ll}\n\\|\\varphi(t)\\|_\\alpha&\\leq M e^{-\\beta t}{ \\|\\varphi_0\\|_{{\\mathcal C}_\\alpha}}+{ L}M\\int_0^t(t-s)^{-\\alpha}e^{-\\beta(t-s)}\\|\\varphi_s\\|_{{\\mathcal C}_\\alpha} ds\\\\[2ex]\n&\\hs+{ L }M\\int_t^{\\infty} e^{-\\beta(s-t)}\\|\\varphi_s\\|_{{\\mathcal C}_\\alpha} ds,\\hspace{1cm}}\\def\\hs{\\hspace{0.5cm} \\forall\\,t\\geq 0.\n\\end{array}\n\\end{equation}\nSince $\\varphi(t)=\\gamma{ ^1}(t+\\eta)-\\gamma{ ^2}(t+\\eta)$ and $\\eta\\in\\mathbb{R}$ can be taken arbitrary, it can be easily seen that \\eqref{eq2} is readily satisfied by all the translations $\\varphi(\\.+\\tau)$ of $\\varphi$, i.e.,\n$$\n\\begin{array}{ll}\n\\|\\varphi(t+\\tau)\\|_\\alpha&\\leq M e^{-\\beta t}{ \\|\\varphi_\\tau\\|_{{\\mathcal C}_\\alpha}}+{ L}M\\int_0^t(t-s)^{-\\alpha}e^{-\\beta(t-s)}\\|\\varphi_{s+\\tau}\\|_{{\\mathcal C}_\\alpha} ds\\\\[2ex]\n&\\hs+{ L }M\\int_t^{\\infty} e^{-\\beta(s-t)}\\|\\varphi_{s+\\tau}\\|_{{\\mathcal C}_\\alpha} ds,\\hspace{1cm}}\\def\\hs{\\hspace{0.5cm} \\forall\\,t\\geq 0.\n\\end{array}\n$$\nRewriting $t+\\tau$ as $t$, the above inequality can be put into the following one:\n\\begin{equation}\\label{e:4.4}\n\\begin{array}{ll}\n\\|\\varphi(t)\\|_\\alpha&\\leq E(t,\\tau){ \\|\\varphi_\\tau\\|_{{\\mathcal C}_\\alpha}}+\\int_\\tau^t K_1(t,s)\\|\\varphi_{s}\\|_{{\\mathcal C}_\\alpha} ds\\\\[2ex]\n&\\hs+\\int_t^{\\infty} K_2(t,s)\\|\\varphi_{s}\\|_{{\\mathcal C}_\\alpha} ds,\\hspace{1cm}}\\def\\hs{\\hspace{0.5cm} \\forall\\,t\\geq \\tau.\n\\end{array}\n\\end{equation}\nwhere $E(t,s)=M e^{-\\beta (t-s)}$, and\n\\begin{equation}\\label{eEK}\\mbox{$K_1=LM(t-s)^{-\\alpha}e^{-\\beta(t-s)}$,\\hs $K_2=LMe^{-\\beta(s-t)}$.}\\end{equation}\n\nApplying Theorem \\ref{t:3.1} (1) to $y(t)=\\|\\varphi(t)\\|_\\alpha$, we deduce by \\eqref{e:4.4} that if $L<1\/(\\kappa_0 M)$, then for any $\\varepsilon>0$ there exists $T>0$ (independent of $\\eta$) such that\n$\n\\|\\varphi(t)\\|_\\alpha<\\varepsilon$ for all $t> T$, that is,\n\\begin{equation}\\label{e:4.17}\\|\\gamma({t+{\\eta}})\\|_\\alpha<\\varepsilon,\\hspace{1cm}}\\def\\hs{\\hspace{0.5cm} t> T,\\,\\,{\\eta}\\in \\mathbb{R}.\n\\end{equation}\nNow for any $\\tau\\in\\mathbb{R}$, setting $t=T+1$ and $\\eta=\\tau-(T+1)$ in \\eqref{e:4.17} we obtain that $\\|\\gamma({\\tau})\\|_\\alpha<\\varepsilon$. Since $\\varepsilon$ is arbitrary, one immediately concludes that $\\gamma(\\tau)=0$, which justifies the validity of \\eqref{e:4.8}.\n\n}\n\n\\vskip8pt}\\def\\vs{\\vskip4pt\nNow we write ${\\mathcal S}[p]=\\{\\phi_p\\}$. Then $\\phi_p$ is continuous in $p$, and the invariance property of ${\\mathcal S}$ implies that $\\gamma_{t}:=\\phi_{\\theta_tp}$ is a complete trajectory of the cocycle $\\Phi$ in ${{\\mathcal C}_\\alpha}$ for each $p\\in {\\mathcal H}$. Define\n$$\\Gamma(p)=\\phi_p(0),\\hspace{1cm}}\\def\\hs{\\hspace{0.5cm} p\\in{\\mathcal H}.$$ Clearly $\\Gamma\\in C({\\mathcal H},X^\\alpha)$. It is easy to see that for each $p\\in{\\mathcal H}$, $\\gamma_p(t):=\\Gamma(\\theta_tp)=\\phi_{\\theta_tp}(0)$ is a complete solution of \\eqref{e:4.1} pertaining to $p$. Hence $\\Gamma$ is a nonautonomous equilibrium solution of equation \\eqref{e:4.1}.\n\\vs\n\n(2)-(3)\\, Let $p\\in{\\mathcal H}, $ and let $u(t)=u(t;p,\\phi)$ be a bounded solution of \\eqref{e:4.1}. Then the same argument as above with minor modifications applies to show that \\eqref{e:4.4} is fulfilled by $\\varphi(t):=u(t)-\\Gamma(\\theta_tp)$ for all $t\\geq\\tau\\geq0$.\nAssertions (2) and (3) then immediately follows from Theorem \\ref{t:3.1} and Lemma \\ref{l:2.1}. $\\Box$\n\n\n\\subsection{Global asymptotic stability of the equilibrium}\\label{s:4.4}\n\nNow we pay some attention to the particular case where $\\sigma(A)$ lies in the right half plane.\nWe continue the argument in Section \\ref{s:4.3} and assume that $F$ satisfies the global Lipschitz condition (F3).\n\nGiven $(p,\\phi)\\in{\\mathcal H}\\times{\\mathcal C}_\\alpha$, we write $u(t)=u(t;p,\\phi)$. Since the spectral set $\\sigma^-=\\emptyset$, using the constant variation formula it can be shown that\n\\begin{equation}\\label{e:4.36}\n\\|{ u(t)}\\|_\\alpha\n\\leq Me^{-\\beta (t-\\tau)}{ \\|u_\\tau\\|_{{\\mathcal C}_\\alpha}}+\\int_{\\tau}^{t} K_1(t,s)\\|u_s\\|_{{\\mathcal C}_\\alpha} ds{ +\\rho_0},\\hs\\,\\, t\\ge\\tau\\geq 0\n\\end{equation}\n where $\\rho_0=C_1M\\int_{0}^{\\infty} s^{-\\alpha}e^{-\\beta s}ds$ ($C_1$ is the constant given in \\eqref{e:4.31}), and $\n K_1(t,s)$ is the function given in \\eqref{eEK}.\n The calculations involved here are similar to those as in the proof of Lemma \\ref{l:4.1}. We omit the details.\n Let $$\\begin{array}{ll}\\kappa_0=\\sup_{t\\ge0}\\(\\int_{0}^{t}(t-s)^{-\\alpha}e^{-\\beta (t-s)}ds \\),\\hs \\rho=(1-\\kappa_0 L M)^{-1}\\rho_0,\\end{array}$$\n where $L$ is the constant in (F3). Applying Theorem \\ref{t:3.2} (1) one deduces that if $L<{{1}\/{(\\kappa_0 M)}}$, then for any $R, \\varepsilon>0$, there exists $T>0$ such that\n\\begin{equation}\\label{e:4.20}\n\\|u(t)\\|_\\alpha<\\rho+\\varepsilon,\\hspace{1cm}}\\def\\hs{\\hspace{0.5cm}\\forall\\, t{ >} T,\\,\\,(p,\\phi)\\in{\\mathcal H}\\times{ \\overline{\\mathcal B}_R}.\n\\end{equation}\n\nLet $\\Gamma$ be the nonautonomous equilibrium solution given by Theorem \\ref{t:4.1}.\nAs a direct consequence of \\eqref{e:4.20} and Theorem \\ref{t:4.1}, we have\n\\begin{theorem}\\label{l:4.1c} Suppose $L<{{1}\/{(\\kappa_0 M)}}$, and that\n\\begin{equation}\\label{e:sa}\\mbox{\\em Re}\\,z\\geq \\beta>0,\\hspace{1cm}}\\def\\hs{\\hspace{0.5cm}\\forall\\,z\\in\\sigma(A).\n\\end{equation}\nThen $\\Gamma$ is uniformly globally asymptotically stable in the following sense:\n\\begin{enumerate}\n\\item[$(1)$] $\\Gamma$ is uniformly table, i.e., for any $\\varepsilon>0$, there exists $\\delta>0$ such that for all $(p,\\phi)\\in{\\mathcal H}\\times{\\mathcal C}_\\alpha$ with $\\max_{s\\in[-r,0]}\\|\\phi(s)-\\Gamma(\\theta_sp)\\|_\\alpha<\\delta$,\n\\begin{equation}\\label{e:4.34}\n\\|u(t)-\\Gamma(\\theta_tp)\\|_\\alpha<\\varepsilon,\\hspace{1cm}}\\def\\hs{\\hspace{0.5cm} t\\geq0.\n\\end{equation}\n\\item[$(2)$] $\\Gamma$ is uniformly globally attracting, i.e.,\nfor any $R,\\varepsilon>0$, there exists $T>0$ such that for all $p\\in{\\mathcal H}$ and $\\phi\\in\\overline{\\mathcal B}_R$,\n\\end{enumerate}\n\\begin{equation}\\label{e:4.35}\n\\|u(t)-\\Gamma(\\theta_tp)\\|_\\alpha<\\varepsilon,\\hspace{1cm}}\\def\\hs{\\hspace{0.5cm} t> T.\n\\end{equation}\n\\end{theorem}\n{\\bf Proof.} The uniform stability of $\\Gamma$ follows from Theorem \\ref{t:4.1} (3), and the\nuniform global attraction of $\\Gamma$ is a consequence of \\eqref{e:4.20} and some general results on the uniform forward attraction properties of pullback attractors; see e.g. \\cite[Theorem 3.3]{WLK}. $\\Box$\n\n\n\\vskip8pt}\\def\\vs{\\vskip4pt\nIf we impose on $L$ a stronger smallness requirement, then it can be shown that $\\Gamma$ is uniformly globally exponentially asymptotically stable.\n\\begin{theorem}\\label{l:4.1d} Assume that $A$ satisfies \\eqref{e:sa}. If $L<{1}\/{(\\kappa_0 M(1+M))}$, then there exist $C,\\lambda>0$ such that for all $(p,\\phi)\\in{\\mathcal H}\\times{\\mathcal C}_\\alpha$,\n$$\n\\|u(t)-\\Gamma(\\theta_tp)\\|_{\\alpha}\\leq { C}e^{-\\lambda t}\\max_{s\\in[-r,0]}\\|\\phi(s)-\\Gamma(\\theta_sp)\\|_\\alpha\\,,\\hspace{1cm}}\\def\\hs{\\hspace{0.5cm} t\\geq0.\n$$\n\\end{theorem}\n{\\bf Proof.}\nLet $\\varphi(t)=u(t)-\\Gamma(\\theta_t p)$. Using a parallel argument as in the proof of Lemma \\ref{l:4.1} (1), we can obtain that\n $$\n\\|\\varphi(t)\\|_\\alpha\n\\leq Me^{-\\beta (t-\\tau)}{\\|\\varphi_\\tau\\|_{{\\mathcal C}_\\alpha}}+\\int_{\\tau}^{t} K_1(t,s)\\|\\varphi_s\\|_{{\\mathcal C}_\\alpha} ds.\\hspace{1cm}}\\def\\hs{\\hspace{0.5cm} t\\ge\\tau\\geq 0,\n $$\n If $L<{1}\/{(\\kappa_0 M(1+M))}$ then the functions $E(t,s):=Me^{-\\beta (t-s)}$ and $K_1(t,s)$ fulfill the requirements in Theorem \\ref{t:3.2}. Thus there exist constants $C,\\lambda>0$ independent of $\\varphi$ such that\n$$\n\\|\\varphi(t)\\|_{\\alpha}\\leq { C}e^{-\\lambda t}\\|\\varphi_0\\|_{{\\mathcal C}_\\alpha},\\hspace{1cm}}\\def\\hs{\\hspace{0.5cm} t\\geq 0.\n$$\nThe conclusion of the theorem then immediately follows. $\\Box$\n\n\\subsection{Nonlinear evolution equations with multiple delays}\nLet us now consider the nonlinear evolution equation\n \\begin{equation}\\label{e:4.23}\\frac{du}{dt}+Au=f(u(t-r_1),\\cdots,u(t-r_m))+h(t)\\end{equation}\nwith multiple delays, where $X$ and $A$ are the same as in Subsection 4.1, $f$ is a continuous mapping from $(X^\\alpha)^{m}$ to $X$ for some $\\alpha\\in[0,1)$, $h\\in C(\\mathbb{R},X)$, $r_i\\in C(\\mathbb{R},\\mathbb{R}^+)$, and $$0\\leq r_i(t) \\leq r<\\infty,\\hs 1\\leq i\\leq m.$$ It is well known that \\eqref{e:4.23} covers a large number of concrete examples from applications. Our main goal here is to demonstrate how to put such an equation into the abstract form of \\eqref{e:4.1}.\n\nThe initial value problem of \\eqref{e:4.23} reads as follows:\n \\begin{equation}\\label{e:4.23b}\\left\\{\\begin{array}{ll}\\frac{du}{dt}+Au=f(u(t-r_1),\\cdots,u(t-r_m))+h(t),\\hs t\\geq\\tau,\\\\[1ex]\n u(\\tau+s)=\\phi(s),\\hs s\\in[-r,0],\\end{array}\\right.\\end{equation}\n where $\\phi\\in{{\\mathcal C}_\\alpha}=C([-r,0],X^\\alpha)$, and $\\tau\\in\\mathbb{R}$ is given arbitrary. Rewriting $t-\\tau$ as $t$, one obtains an equivalent form of \\eqref{e:4.23b}:\n{ \\begin{equation}\\label{e:4.23c}\\left\\{\\begin{array}{ll}\\begin{array}{ll}\\frac{dv}{dt}+Av=f(v(t-\\~r_1),\\cdots,v(t-\\~r_m))+\\~h(t),\\hs t\\geq 0,\\end{array}\\\\[1ex]\n v(s)=\\phi(s),\\hs s\\in[-r,0],\\end{array}\\right.\\end{equation}\n}\nwhere $v(t)=u(t+\\tau)$, and\n$$\n\\~r_i(t)=r_i(t+\\tau),\\hs \\~h(t)=h(t+\\tau).\n$$\n\nDenote ${\\mathcal Y}$ the space $C(\\mathbb{R})^m\\times C(\\mathbb{R},X)$ equipped with the {\\em compact-open topology} (under which a sequence $p_n(t)$ in ${\\mathcal Y}$ is convergent {\\em iff\\,} it is uniformly convergent on any compact interval $I\\subset \\mathbb{R}$).\nLet $\\theta$ be the translation operator on ${\\mathcal Y}$,\n$$\n\\theta_\\tau p=p(\\.+\\tau),\\hspace{1cm}}\\def\\hs{\\hspace{0.5cm} \\forall p\\in{\\mathcal Y},\\,\\,\\tau\\in\\mathbb{R}.\n$$ Set\n\\begin{equation}\\label{e:hp}\np^*(t)=(r_1(t),\\cdots,r_m(t),\\,h(t)),\n\\end{equation}\nand assume that $p^*(t)$ is {\\em translation compact } in ${\\mathcal Y}$, i.e., the hull\n$${\\mathcal H}={\\mathcal H}[p^*]:=\\overline{\\{\\theta_\\tau p^*:\\,\\,\\tau\\in\\mathbb{R}\\}}$$ of $p^*$ in ${\\mathcal Y}$ is a compact subset of ${\\mathcal Y}$.\n\nWe also assume that ${\\mathcal H}$ is minimal w.r.t $\\theta$. This requirement is naturally fulfilled when $p^*$ is, say for instance, periodic, pseudo-periodic, or almost periodic.\n\nDefine a function $F:{\\mathcal H}\\times {{\\mathcal C}_\\alpha}\\rightarrow X$ as\n\\begin{equation}\\label{e:F}\nF(p, \\phi)=f(\\phi(-p_1(0)),\\cdots,\\phi(-p_m(0)))+p_{m+1}(0)\n\\end{equation}\nfor any $p=(p_1,\\cdots,p_{m+1})\\in {\\mathcal H}$. Observing that\n$$\\(r_1(t+\\tau),\\cdots,r_m(t+\\tau),\\,h(t+\\tau)\\)=p^*(t+\\tau)=(\\theta_{t+\\tau}p^*)(0),$$\nwe can rewrite the righthand side of the equation in \\eqref{e:4.23c} as follows:\n{\n$$\\begin{array}{ll}\n&f(v(t-\\~r_1),\\cdots,v(t-\\~r_m))+\\~h(t)\\\\[1ex]\n=& F(\\theta_{t+\\tau}p^*,v_{t})= F(\\theta_{t}p,v_t),\\hs p=\\theta_\\tau p^*.\n\\end{array}\n$$\n}\nConsequently \\eqref{e:4.23c} can be reformulated as\n \\begin{equation}\\label{e:4.24}\\left\\{\\begin{array}{ll}\\frac{dv}{dt}+Av=F(\\theta_tp,v_t),\\hspace{1cm}}\\def\\hs{\\hspace{0.5cm} t\\geq 0,\\,\\,p\\in\\{\\theta_\\tau p^*:\\,\\,\\tau\\in\\mathbb{R}\\},\\\\\n v_0=\\phi.\\end{array}\\right.\\end{equation}\n Since $\\{p=\\theta_\\tau p^*:\\,\\,\\tau\\in\\mathbb{R}\\}$ is dense in ${\\mathcal H}$, for theoretical completeness we usually embed \\eqref{e:4.24} into the following cocycle system:\n \\begin{equation}\\label{e:4.25}\\left\\{\\begin{array}{ll}\\frac{dv}{dt}+Av=F(\\theta_tp,v_t),\\hspace{1cm}}\\def\\hs{\\hspace{0.5cm} t\\geq 0,\\,\\,p\\in {\\mathcal H},\\\\\n v_0=\\phi.\\end{array}\\right.\\end{equation}\n\n\\vs\nNow assume $f$ satisfies the following conditions:\n \\begin{enumerate}\\item[(f1)] $f$ is {\\em locally Lipschitz}, namely, for any $R>0$, there exists $L_f=L_f(R)>0$ such that for all $u_i,u_i'\\in X^\\alpha$ ($1\\leq i\\leq m$) with $\\|u_i\\|_\\alpha,\\|u_i'\\|_\\alpha\\leq R$,\n$$\n\\|f(u_1,\\cdots,u_m)-f(u_1',\\cdots,u_m')\\|_0\\leq L_f(\\|u_1-u_1'\\|_\\alpha+\\cdots+\\|u_m-u_m'\\|_\\alpha).\n$$\n\n\\vs \\item[(f2)] There exist $C_0,C_1>0$ such that\n $$\n \\|f(u_{ 1},\\cdots,u_m)\\|_{ 0}\\leq C_0(\\|u_1\\|_\\alpha+\\cdots+\\|u_m\\|_\\alpha)+C_1,\\hspace{1cm}}\\def\\hs{\\hspace{0.5cm} \\forall\\,u_i\\in X^\\alpha.\n $$\n \\end{enumerate}\nThen one can trivially verify that the mapping $F$ defined by \\eqref{e:F} satisfies hypotheses (F1) and (F2).\n\n\n\\begin{remark}\\label{r:4.6} Note that if the function $p^*$ in \\eqref{e:hp} is periodic (resp. quasi-periodic, almost periodic), then $\\theta_tp$ is periodic (resp. quasi-periodic, almost periodic) for any fixed $p\\in {\\mathcal H}:={\\mathcal H}[p^*]$. Let $\\Gamma$ be the equilibrium solution of \\eqref{e:4.25} given in Theorems \\ref{l:4.1c} and \\ref{l:4.1d}. Then since $\\Gamma(q)$ is continuous in $q$, we deduce that $\\gamma_p:=\\Gamma(\\theta_tp)$ is periodic (resp. quasi-periodic, almost periodic) as well. Therefore these two theorems give the existence of asymptotically stable periodic (resp. pseudo periodic, almost periodic) solutions for equation \\eqref{e:4.23}.\n\nThe interested reader is referred to \\cite{Hale2,Jones,Kaplan,Ken,LiYX,MN,MSS,NT,Nus2,Ou,Walt3,Walt2} etc. for some classical results and new trends on periodic solutions of delay differential equations,\nand to \\cite{Hino1,Layt,Naito,Seif,Wu,Yoshi,Yuan0,YuanR} and references therein for typical results on almost periodic solutions.\n\\end{remark}\n\\begin{remark}\\label{r:4.7}In the case where the functions $h$ and $r_i$ $(1\\leq i\\leq m)$ in the equation \\eqref{e:4.23} are not translation compact (or, the righthand side of the equation takes a more general form like $g(t,u(t-r_1),\\cdots,u(t-r_m))$\\,),\nthe framework of cocycle systems does not seem to be quite suitable to handle the problem, because the base space ${\\mathcal H}$ of the cocycle system corresponding to the equation may not be compact.\nInstead, the processes one may be more appropriate.\n\nSet\n$\nF(t,\\phi)=f(\\phi(-r_1),\\cdots,\\phi(-r_m))+h(t)$ \\,$(t\\in\\mathbb{R},\\,\\,\\phi\\in{\\mathcal C}_\\alpha).$ Then \\eqref{e:4.23} can be put into a functional one:\n\\begin{equation}\\label{e:4.2}\n\\frac{du}{dt}+Au=F(t,u_t).\\end{equation}\nSuppose \\eqref{e:4.2} has a unique global solution $x(t;\\tau,\\phi)$ $(t\\in[\\tau-r,\\infty))$ for each initial data $(\\tau,\\phi)\\in \\mathbb{R}\\times{\\mathcal C}_\\alpha$. Denote by $x_t(\\tau,\\phi)$ the lift of $x(t):=x(t;\\tau,\\phi)$ in ${\\mathcal C}_\\alpha$. Then as in \\eqref{e:3.20}, we can define a process $P(t,\\tau)$ on ${\\mathcal C}_\\alpha$.\nThis allows us to take some steps in the investigation of the dynamics of the equation. For instance, under similar hypotheses as in Section \\ref{s:4.4}, it is desirable to prove that the equation has a unique bounded complete (entire) solution $\\gamma(t)$ $(t\\in\\mathbb{R})$ which is uniformly globally (exponentially) asymptotically stable by employing the pullback attractor theory for processes.\n\n\nThe situation becomes quite complicated if the operator $A$ fails to be a dissipative one, i.e., the spectral set\n$\\sigma^-$ in \\eqref{e:4.3} is non-void. One drawback is that both the pullback attractor theory and the Conley index theory are not applicable in proving the existence of bounded complete solutions of the equation. If the delay functions $r_i(t)$ are constants, then since the external force $h(t)$ and the nonlinear term in the righthand side of \\eqref{e:4.23} are separate, one may try to get a bounded complete solution $\\gamma(t)$ of the equation by considering periodic approximations of $h(t)$.\nHowever, if the functions $r_i(t)$ also depend on $t$, we are not sure whether such a strategy still works. There are also many other interesting questions such as the synchronizing property of the bounded complete solution $\\gamma(t)$ with the external force (in case $r_i(t)$ are constant functions) and a more detailed description of the dynamics of the equation. (Note that even if in the case where $h(t)$ and $r_i(t)$ are translation compact, Theorem \\ref{t:4.1} only gives us some information on the asymptotic behavior of bounded solutions of the equation. A natural question is to ask: What can we say about those unbounded solutions\\,?) All these questions deserve to be clarified, and a further study on the geometric theory of functional differential equations in a processes fashion may be helpful for us to take some steps, in which the integral inequality \\eqref{e1.1} may once again play a fundamental role.\n\n\\end{remark}\n\n\\subsection{Neural networks with multiple delays}\n{As an concrete example, we consider the following reaction diffusion neural network system with multiple delays:\n\\begin{equation}\\label{e:4.27}\n\\left\\{\\begin{array}{ll}\\frac{\\partial{u_i}}{\\partial{t}}=\\mbox{div}\\(a_i(x)\\nabla u_i\\)\n+\\sum_{j=1}^{n}b_{ij}u_j+\\\\[1ex]\n\\hspace{1cm}}\\def\\hs{\\hspace{0.5cm}+\\sum_{j=1}^{n}T_{ij}g_j(x,u_j,u_j(x,t-r_{ij}))+J_{i}(x,t),\\\\[1ex]\nu_i(x,t)=0,\\hs t\\ge0,\\,\\,x\\in\\partial{\\Omega}, \\hs i=1,2,...,n.\n\\end{array}\\right.\n\\end{equation}\nHere $\\Omega\\subset \\mathbb{R}^m$ is a bounded domain with a smooth boundary $\\partial\\Omega$, $a_i\\in C^1(\\overline\\Omega)$ and is positive everywhere on $\\overline\\Omega$, $b_{ij}$ and $T_{ij}$\nare constant coefficients,\n$$0\\leq r_{ij}\\leq r<\\infty,\\hspace{1cm}}\\def\\hs{\\hspace{0.5cm} 1\\leq i,j\\leq n,$$ and $J_i(x,t)$\nare bounded inputs. We refer the interested reader to \\cite{He,Z} etc. for a physical background of this type of systems.\n\n Let $A_i$ be the elliptic operator given by\n$$\nA_iu=-\\sum_{k=1}^{m}\\frac{\\partial{}}{\\partial{x_k}}\\(a_i(x)\\frac{\\partial{u}}{\\partial{x_k}}\\)$$\nassociated with the corresponding boundary condition. It is a basic knowledge (see e.g. Henry \\cite[Chap.7]{Henry}) that $A_i$ is a sectorial operator in $L^2(\\Omega)$ with compact resolvent.\n\nFor notational simplicity, we use the same notation $g_j$ to denote the Nemytskii operator generated by the function $g_j(x, u,v)$, i.e.,\n$$g_j(u,v)(x)=g_j(x,u,v)\\,\\,(x\\in\\Omega),\\hspace{1cm}}\\def\\hs{\\hspace{0.5cm} u,v\\in L^2(\\Omega).$$\nLet $J_i(t)=J_i(\\.,t)$. Then \\eqref{e:4.27} takes a slightly abstract form:\n\\begin{equation}\\label{e:4.28}\n\\frac{d{u_i}}{d{t}}+{ A_i}u_i=\\sum_{j=1}^{n}b_{ij}u_j+\\sum_{j=1}^{n}T_{ij}g_j(u_j,u_j(t-r_{ij}))+J_{i}(t),\\hs 1\\leq i\\leq n.\n\\end{equation}\nSet $H=\\(L^2(\\Omega)\\)^n$, and let $u=(u_1,\\cdots,u_n)'$. Denote\n$$\nAu=(A_1u_1,...,A_nu_n)',\\hs u\\in D(A)\\subset H.\n$$ (It is clear that $A$ is a sectorial operator in $H$.)\nLet ${{\\mathcal C}_0}=C([-r,0],H)$, and define an operator $G:{{\\mathcal C}_0}\\rightarrow H^n=(L^2(\\Omega))^{n\\times n}$ as follows:\\,\\, $\\forall\\,\\phi=(\\phi_1,\\cdots,\\phi_n)'\\in{{\\mathcal C}_0}$,\n$$\nG(\\phi)=(\\psi_{ji})_{n\\times n},\\hs\\mbox{where } \\psi_{ij}=g_j(\\phi_j(0),\\phi_j(-r_{ij})).\n$$\n Let $T=\\(T_{ij}\\)_{n\\times n}$. Write $TG(\\phi)=\\([TG(\\phi)]_{ij}\\)_{n\\times n}$, and\n define\n $$\nF(\\phi)=(F_1(\\phi),F_2(\\phi),\\cdots,F_n(\\phi))',\\hs F_i(\\phi)=[TG(\\phi)]_{ii}.\n$$\n Then \\eqref{e:4.28} can be reformulated as\n\\begin{equation}\\label{e:4.29}\n\\frac{du}{dt}+Au=Bu+F(u_t)+J(t),\n\\end{equation}\nwhere $B=\\(b_{ij}\\)_{n\\times n}$, and $J=(J_1,\\cdots,J_n)'$.\n\n\nSince \\eqref{e:4.29} is nonautonomous,\ngenerally the initial value problem reads\n\\begin{equation}\\label{e:4.29b}\\left\\{\\begin{array}{ll}\n\\frac{dv}{dt}+Av=Bv+F({ v_t})+J(t+\\tau),\\hs t\\geq 0,\\\\\nv_0=\\phi\\in {{\\mathcal C}_0},\\end{array}\\right.\n\\end{equation}\nwhere $v(t)=u(t+\\tau)$, and $\\tau\\in\\mathbb{R}$ denotes the initial time.\nWe assume that $J$ is translation compact in ${\\mathcal Y}$. Denote ${\\mathcal H}$ the hull ${\\mathcal H}[J]$ of the function $J$ in ${\\mathcal Y}$. Then as in the previous subsection one can embed \\eqref{e:4.29b} into the cocycle system:\n \\begin{equation}\\label{e:4.30}\\left\\{\\begin{array}{ll}\\frac{dv}{dt}+(A-B)v=F(\\theta_tp,v_t),\\hspace{1cm}}\\def\\hs{\\hspace{0.5cm} t\\geq 0,\\,\\,p\\in {\\mathcal H},\\\\\nv_0=\\phi\\in {{\\mathcal C}_0},\\end{array}\\right.\\end{equation}\nwhere\n$$\nF(p,\\phi)=F(\\phi)+p(0),\\hspace{1cm}}\\def\\hs{\\hspace{0.5cm} p\\in{\\mathcal H},\\,\\,\\phi\\in{{\\mathcal C}_0}.\n$$\n\n\nFor simplicity, we always assume that $g_j(x,u,v)$ are continuous and {\\em globally Lipschitz} in $(u,v)$ uniformly for $x\\in\\Omega$, that is, there exists $L_{j}>0$ such that\n$$\n|g_j(x,t_1,s_1)-g_j(x,t_2,s_2)|\\leq L_{j}(|t_1-{ t_2}|+|s_1-s_2|)\n$$\nfor all $t_i,s_i\\in\\mathbb{R}$ and $x\\in\\Omega$. Then for the Nemytskii operator $g_j$ of the function $g_j(x,u,v)$, we have\n\\begin{align*}\n\\|g_j(u_1,v_1)-g_j(u_2,v_2)\\|_{L^{2}(\\Omega)}\\le& L_{j}(\\|u_1-u_2\\|_{L^{2}(\\Omega)}+\\|v_1-v_2\\|_{L^{2}(\\Omega)}).\n\\end{align*}\nFurther by some simple calculations it can be shown that\n\\begin{align*}\n\\|F(p,\\phi)-F(p,\\phi')\\|_{H}\\le L\\|\\phi-\\phi'\\|_{{\\mathcal C}_0}\n\\end{align*}\nwith $L=2\\(\\sum_{i=1}^{n}\\big(\\sum_{j=1}^{n}|T_{ij}|L_j\\big)^{2}\\)^{1\/2}$.\nThis allows us to carry over all the results on the abstract evolution equation \\eqref{e:4.1} to system \\eqref{e:4.30}. In particular, by Remark \\ref{r:4.6} we have the following theorem.\n\n\n \\begin{theorem}\\label{t:4.9} Suppose $\\mbox{Re}\\,(\\sigma(A-B))\\geq \\beta>0$, and that $L<1\/\\(M I\\)$, where $M$ is the constant appearing in \\eqref{e:4.6a} corresponding to operator $A-B$, and $$I=\\sup_{t\\geq0}\\int_{0}^te^{-\\beta(t-s)}ds.$$ Let $J(t)=(J_1(t),\\cdots,J_n(t))'$ be a periodic (resp. quasi-periodic, almost periodic) function. Then system \\eqref{e:4.27} has a unique periodic (resp. quasi-periodic, almost periodic) solution $\\gamma$ which is globally uniformly asymptotically stable.\n\n If we further assume $L<1\/\\(MI(1+M)\\)$, then $\\gamma$ is globally exponentially asymptotically stable.\n\\end{theorem}\n\n\\noindent{\\bf Acknowledgement.} Our sincere thanks go to the referees for their valuable comments and suggestions which helped us greatly improve the quality of the paper.\n}\n\\section*{References}\n\n\n\n\\begin {thebibliography}{44}\n\\addcontentsline{toc}{section}{References}\n\\bibitem{BS}D. Ba\\v{\\i}nov, P. Simeonov, Integral Inequalities and Applications, Kluwer Academic Publishers Group, Dordrecht, 1992.\n\n\\bibitem{Bell1}R. Bellman, The stability of solutions of linear differential equations, Duke Math. J. 10 (1943) 643-647.\n \\newblock \\href {https:\/\/doi.org\/10.1215\/S0012-7094-43-01059-2}\n {\\path{doi: 10.1215\/S0012-7094-43-01059-2}}.\n\n\\bibitem{Carab2} T. Caraballo, G. 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Representation learning in spaces of constant negative curvature, i.e. hyperbolic, has shown to outperform Euclidean embeddings significantly on data with latent hierarchical, taxonomic or entailment structures. Such data naturally arises in language modeling \\cite{nickel2017poincare, nickel2018learning, chamberlain2017neural, sala2018representation, ganea2018hyperbolic}, recommendation systems \\cite{chamberlain2019scalable}, image classification, and few-shot learning tasks \\cite{khrulkov2020hyperbolic}, to name a few. Grassmannian manifolds find applications in computer vision to perform video-based face recognition and image set classification \\cite{huang2018building, li2015face}. The Lie groups of transformations $\\mathrm{SO}(3)$ and $\\mathrm{SE}(3)$ appear naturally when dealing with problems like pose estimation and skeleton-based action recognition \\cite{hou2018computing, huang2017deep}. Stiefel manifolds are used for emotion recognition and action recognition from video data \\cite{chakraborty2019statistics, huang2017riemannian}. The space of symmetric positive definite (SPD) matrices characterize data from diffusion tensor imaging \\cite{pennec2006riemannian}, and functional magnetic resonance imaging \\cite{sporns2005human}. Among those advances is a family of neural network methods, which are parameterized by weights constrained to a particular non-Euclidean space \\cite{nickel2017poincare, chamberlain2017neural, huang2017deep, huang2018building, ganea2018hyperbolic, chamberlain2019scalable, khrulkov2020hyperbolic}.\n\nStochastic Gradient Descent (SGD) \\cite{robbins1951stochastic} algorithm has been used for the backpropagation learning in connectionist neural network models. The SGD is known to generate undesirable oscillations around optimal values of model parameters \\cite{qian1999momentum}. Using a momentum term \\cite{rumelhart1986learning} has been shown to improve the optimization convergence. More recently, adaptive learning rates \\cite{hinton2012neural, kingma2014adam, zeiler2012adadelta} have been proposed, and are popular optimization algorithms used for iterative optimization while training neural networks.\n\nThese adaptations of the SGD methods have been studies extensively under the lenses of Riemannian geometry and translated to non-Euclidean settings \\cite{bonnabel2013stochastic, roy2018geometry, becigneul2018riemannian, kasai2019riemannian}. Remannian stochastic optimization algorithms have been robustly integrated with many popular machine learning toolboxes. However, previous work in this space was mainly motivated by research use cases \\cite{townsend2016pymanopt, meghwanshi2018mctorch, miolane2020geomstats, kochurov2020geoopt}. Whereas practical aspects, such as deploying and maintaining machine learning models, were often overlooked.\n\nWe present TensorFlow ManOpt, a Python library for optimization on Riemannian manifolds in TensorFlow \\cite{abadi2016tensorflow}, to help bridge the aforementioned gap. The library is designed with the aim for a seamless integration with the TensorFlow ecosystem, targeting not only research, but also end-to-end machine learning pipelines with the recently proposed TensorFlow Extended platform \\cite{baylor2017tfx}.\n\n\\section{Riemannian optimization}\n\nIn this section we provide a brief overview of Riemannian optimization with a focus on stochastic methods. We refer to \\cite{edelman1998geometry, smith1994optimization, absil2009optimization} for more rigorous theoretical treatment.\n\nOptimization on Riemannian manifolds, or Riemannian optimization, is a class of methods for optimization of the form\n\\begin{equation}\n \\min_{x \\in \\mathcal{M}} f\n \\label{eq:optim}\n\\end{equation}\nwhere $f$ is an objective function, subject to constraints which are smooth, in the sense that the search space $\\mathcal{M}$ admits the structure of a differentiable manifold, equipped with a positive-definite inner product on the space of tangent vectors $\\operatorname{T}_{x} \\mathcal{M}$ at each point $x \\in \\mathcal{M}$. Conceptually, the approach of Riemannian optimization translates the constrained optimization problem (\\ref{eq:optim}) into performing optimization in the Euclidean space $\\mathbb{R}^n$ and projecting the parameters back onto the search space $\\mathcal{M}$ after each iteration \\cite{absil2009optimization}.\n\nIn particular, several first order stochastic optimization methods commonly used in the Euclidean domain, such as the SGD algorithm and related adaptive methods, have already been adapted to Riemannian settings.\n\nThe update equation of the (Euclidean) SGD for differentiable $f$ takes the form\n\\begin{equation}\n x_{t+1} = x_t - \\alpha \\nabla f_t\n \\label{eq:sgd}\n\\end{equation}\nwhere $\\nabla f_t$ denotes the gradient of objective $f$ evaluated at the minibatch taken at step $t$, and $\\alpha > 0$ is the learning rate. For a Riemannian manifold $\\mathcal{M}$, Riemannian SGD \\cite{bonnabel2013stochastic} performs the update\n\\begin{equation}\n x_{t+1} = \\operatorname{Exp}_{x_t}(- \\alpha \\operatorname{\\pi}_{x_t} \\nabla f_t)\n \\label{eq:rsgd}\n\\end{equation}\nwhere $\\operatorname{\\pi}_{x} : \\mathbb{R}^n \\to \\operatorname{T}_{x} \\mathcal{M}$ denotes the projection operator from the ambient Euclidean space on the tangent space $\\operatorname{T}_{x} \\mathcal{M}$. And $\\operatorname{Exp}_x(u) : \\operatorname{T}_{x} \\mathcal{M} \\to \\mathcal{M}$ is the exponential map operator, which moves a vector $u \\in \\operatorname{T}_{x} \\mathcal{M}$ back to the manifold $\\mathcal{M}$ from the tangent space at $x$.\n\nIn a neighborhood of $x$, the exponential map identifies a point on the geodesic, thus guarantees decreasing the objective function. Intuitively, the exponential map enables to perform an update along the shortest path in the relevant direction in unit time, while remaining in the manifold. In practice, when $\\operatorname{Exp}_x(u)$ is not known in closed-form or it is computationally expensive to evaluate, it is common to replace it by a retraction map $\\operatorname{R}_x(u)$, which is a first-order approximation of the exponential.\n\nAdaptive optimization techniques compute smoother estimates of the gradient vector using its first or second order moments. \\mbox{RMSProp} \\cite{hinton2012neural} is an example of a gradient based optimization algorithm which does this by maintaining an exponentially moving average of the squared gradient, which is an estimate of the second raw moment of the gradient. The update equations for RMSProp can be expressed as follows\n\\begin{equation}\n \\begin{aligned}\n m_{t+1} = \\rho m_{t} + (1 - \\rho) (\\nabla f_t \\odot \\nabla f_t) \\\\\n x_{t+1} = x_t - \\alpha \\frac{\\nabla f_t}{\\sqrt{m_{t+1}} + \\epsilon}\n \\end{aligned}\n \\label{eq:rmsprop}\n\\end{equation}\nwhere $\\rho$ is a hyperparameter, $\\alpha$ is the learning rate, and $\\odot$ denotes the \\textit{Hadamard} product.\n\nIn the Euclidean settings, these estimates can be obtained by linearly combining previous moment vectors due to the inherent ``flat'' nature of the underlying space. However, since general Riemannian manifolds can be curved, it is not possible to simply add moment vectors at different points on the manifold, as the resulting vectors may not lie in the tangent spaces at either points.\n\nRiemannian versions of adaptive SGD algorithms use the parallel transport to circumvent this issue \\cite{roy2018geometry, becigneul2018riemannian}. The parallel transport operator $\\operatorname{P}_{x \\to y}(v) : \\operatorname{T}_x \\mathcal{M} \\to \\operatorname{T}_y \\mathcal{M}$ takes $v \\in \\operatorname{T}_x \\mathcal{M}$ and outputs $v' \\in \\operatorname{T}_x \\mathcal{M}$. Informally, $v'$ is obtained by moving $v$ in a ``parallel'' fashion along the geodesic curve connecting $x$ and $y$, where the intermediate vectors obtained through this process have constant norm and always belong to tangent spaces. Such transformation enables combining moment vectors computed at different points of the optimization trajectory.\n\nGeometry-aware constrained RMSProp (cRMSProp) \\cite{roy2018geometry} translates the Equation (\\ref{eq:rmsprop}) to Riemannian settings as follows\n\\begin{equation}\n \\begin{aligned}\n m_{t+1} = \\rho \\operatorname{P}_{x_{t-1} \\to x_{t}} m_{t} + (1 - \\rho) \\operatorname{\\pi}_{x_t} (\\nabla f_t \\odot \\nabla f_t) \\\\\n x_{t+1} = \\operatorname{Exp}_{x_t}(- \\alpha \\frac{\\nabla f_t}{\\sqrt{m_{t+1}} + \\epsilon})\n \\end{aligned}\n \\label{eq:crmsprop}\n\\end{equation}\n\nIn practice, if $\\operatorname{P}_{x\\to y}(v)$ is not known for a given Riemannian manifold or it is computationally expensive to evaluate, it is replaced by its first-order approximation $\\mathfrak{T}_{x\\to y}(v)$.\n\nWhile many optimization problems are of the described form, technicalities of differential geometry and implementation of corresponding operators often pose a significant barrier for experimenting with these methods.\n\u2039\n\\section{Design goals}\n\nWorking out and implementing gradients is a laborious and error prone task, particularly when the objective function acts on higher rank tensors. TensorFlow ManOpt builds upon the TensorFlow framework \\cite{abadi2016tensorflow}, and leverages its automatic differentiation capabilities for computing gradients of composite functions. The design of TensorFlow ManOpt was informed by three main objectives:\n\n\\begin{enumerate}\n \\item \\textbf{Interoperability with the TensorFlow ecosystem.} All components, including optimization algorithms and manifold-constrained variables, can be used as drop-in replacements with the native TensorFlow API. This ensures transparent integration with the rich ecosystem of tools and extensions in both research and production settings.\n Specifically, TensorFlow ManOpt was tested to be compatible with the TensorFlow Extended platform in \\textit{graph} and \\textit{eager} execution modes.\n \\item \\textbf{Computational efficiency.} TensorFlow ManOpt aims for providing closed-form expressions for manifold operators. The library also implements numerical approximation as a fallback option, when such solutions are not available. TensorFlow ManOpt solvers can perform updates on dense and sparse tensors efficiently.\n \\item \\textbf{Robustness and numerical stability.} The library makes use of half-, single-, and double- precision arithmetic where appropriate.\n\\end{enumerate}\n\n\\section{Implementation overview}\n\nThe package implements concepts in differential geometry, such as manifolds and Riemannian metrics with the associated exponential and logarithmic maps, geodesics, retractions, and transports. For manifolds, where closed-form expressions are not known, the library provides numerical approximations. The core module also exposes functions for assigning TensorFlow variables to manifold instances.\n\n\\subsection{Manifolds}\n\nThe module \\texttt{manifolds} is modeled after the Manopt \\cite{boumal2014manopt} API, and provides implementations of the following manifolds:\n\n\\begin{itemize}\n \\item \\texttt{Cholesky} -- manifold of lower triangular matrices with positive diagonal elements \\cite{lin2019riemannian}\n \\item \\texttt{Euclidian} -- an unconstrained manifold with the Euclidean metric\n \\item \\texttt{Grassmannian} -- manifold of $p$-dimensional linear subspaces of the $n$-dimensional space \\cite{edelman1998geometry}\n \\item \\texttt{Hyperboloid} -- manifold of $n$-dimensional hyperbolic space embedded in the $n+1$-dimensional Minkowski space\n \\item \\texttt{Poincare} -- the Poincar{\\'e} ball model of the hyperbolic space\\cite{nickel2017poincare}\n \\item \\texttt{Product} -- Cartesian product of manifolds \\cite{gu2018learning}\n \\item \\texttt{SPDAffineInvariant} -- manifold of symmetric positive definite (SPD) matrices endowed with the affine-invariant metric \\cite{pennec2006riemannian}\n \\item \\texttt{SPDLogCholesky} -- SPD manifold with the Log-Cholesky metric \\cite{lin2019riemannian}\n \\item \\texttt{SPDLogEuclidean} -- SPD manifold with the Log-Euclidean metric \\cite{arsigny2007geometric}\n \\item \\texttt{SpecialOrthogonal} -- manifold of rotation matrices\n \\item \\texttt{Sphere} -- manifold of unit-normalized points\n \\item \\texttt{StiefelEuclidean} -- manifold of orthonormal $p$-frames in the $n$-dimensional space endowed with the Euclidean metric\n \\item \\texttt{StiefelCanonical} -- Stiefel manifold with the canonical metric \\cite{zimmermann2017matrix}\n \\item \\texttt{StiefelCayley} -- Stiefel manifold the retraction map via an iterative Cayley transform \\cite{li2019efficient}\n\\end{itemize}\n\nEach manifold is implemented as a Python class, which inherits the abstract base class \\texttt{Manifold}. The minimal set of methods required for optimization includes:\n\n\\begin{itemize}\n \\item \\texttt{inner(x, u, v)} -- inner product (i.e., the Riemannian metric) between two tangent vectors $u$ and $v$ in the tangent space at $x$\n \\item \\texttt{proju(x, u)} -- projection of a tangent vector $u$ in the ambient space on the tangent space at point $x$\n \\item \\texttt{retr(x, u)} -- retraction $\\operatorname{R}_x(u)$ from point $x$ with given direction $u$. Retraction is a first-order approximation of the exponential map introduced in \\cite{bonnabel2013stochastic}\n \\item \\texttt{transp(x, y, v)} -- vector transport $\\mathfrak{T}_{x\\to y}(v)$ of a tangent vector $v$ at point $x$ to the tangent space at point $y$. Vector transport is the first-order approximation of parallel transport\n\\end{itemize}\n\nSelected manifolds also support exact computations for additional operators:\n\n\\begin{itemize}\n \\item \\texttt{exp(x, u)} -- exponential map $\\operatorname{Exp}_x(u)$ of a tangent vector $u$ at point $x$ to the manifold\n \\item \\texttt{log(x, y)} -- logarithmic map $\\operatorname{Log}_{x}(y)$ of a point $y$ to the tangent space at $x$\n \\item \\texttt{ptransp(x, y, v)} -- parallel transport $\\operatorname{P}_{x\\to y}(v)$ of a vector $v$ from the tangent space at $x$ to the tangent space at $y$\n\\end{itemize}\n\nAll methods support broadcasting of tensors with different shapes to compatible shapes for arithmetic operations.\n\n\\subsection{Optimizers}\n\nThe module \\texttt{optimizers} provides TensorFlow 2 API for optimization algorithms on Riemannian surfaces, including recently proposed adaptive methods:\n\n \\begin{itemize}\n \\item \\texttt{RiemannianSGD} -- stochastic Riemannian gradient descent \\cite{bonnabel2013stochastic}\n \\item \\texttt{ConstrainedRMSprop} -- constrained RMSprop learning method \\cite{roy2018geometry}\n \\item \\texttt{RiemannianAdam} -- Riemannian Adam and AMSGrad algorithms \\cite{becigneul2018riemannian}\n \\end{itemize}\n\n Algorithms are implemented to support dense and sparse updates, as well as serialization, which is crucial for compatibility with TensorFlow functions.\n\n\\subsection{Layers}\n\nFinally, the module \\texttt{layers} exemplify building blocks of neural networks, which can be used alongside with the native Keras \\cite{chollet2015keras} layers.\n\n\\section{Usage}\n\nA simple illustrative example of using low-level API is depicted in Listing~\\ref{lst:low_level_api}. There, TensorFlow Manopt closely follows the Manopt semantics and naming convention. Geometric meaning of those operations is visualized on Figure~\\ref{fig:low_level_api}.\n\n\\begin{minipage}{\\linewidth}\n\\begin{lstlisting}[language=Python,caption={Low-level API usage example},label={lst:low_level_api}]\nimport tensorflow_manopt as manopt\n\n# Instantiate a manifold\nS = manopt.manifolds.Sphere()\nx = S.projx(tf.constant([0.1, -0.1, 0.1]))\nu = S.proju(x, tf.constant([1., 1., 1.]))\nv = S.proju(x, tf.constant([-0.7, -1.4, 1.4]))\n\n# Compute the exponential map and vector transports\ny = S.exp(x, v)\nu_ = S.transp(x, y, u)\nv_ = S.transp(x, y, v)\n\\end{lstlisting}\n\\end{minipage}\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.5\\linewidth]{usage.png}\n \\caption{Geometric operations on the $\\mathbb{S}^2$ manifold}\n \\label{fig:low_level_api}\n\\end{figure}\n\nConstructing an optimization problem is demonstrated on Listing~\\ref{lst:optimization_api}. Manifold-valued variables can be transparently passed to standard TensorFlow functions. And, conversely, native TensorFlow tensors are treated by TensorFlow ManOpt algorithms as data in the Euclidean space.\n\n\\begin{minipage}{\\linewidth}\n\\begin{lstlisting}[language=Python,caption={Optimization API usage example},label={lst:optimization_api}]\nimport tensorflow as tf\nimport tensorflow_manopt as manopt\n\n# Number of training steps\nSTEPS = 10\n\n# Instantiate a manifold\nsphere = manopt.manifolds.Sphere()\n\n# Instantiate and assign a variable to the manifold\nvar = tf.Variable(sphere.random(shape=(N, 2)))\nmanopt.variable.assign_to_manifold(var, sphere)\n\n# Instantiate an optimizer algorithm\nopt = manopt.optimizers.RiemannianAdam(learning_rate=0.2)\npole = tf.constant([0., 1.])\n\n# Compute the objective function and apply gradients\nfor _ in range(STEPS):\n with tf.GradientTape() as tape:\n loss = tf.linalg.norm(var - pole)\n grad = tape.gradient(loss, [var])\n opt.apply_gradients(zip(grad, [var]))\n\\end{lstlisting}\n\\end{minipage}\n\n\\section{Advanced Usage}\n\nThe folder \\texttt{examples} contains reference implementations of several neural network architectures with manifold-valued parameters. For example, the Rotation Mapping layer of the LieNet \\cite{huang2017deep} with weights constrained to the $\\mathrm{SO}(3)$ manifold is shown on Listing~\\ref{lst:layer_api}.\n\n\\begin{minipage}{\\linewidth}\n\\begin{lstlisting}[language=Python,caption={Layers API usage example},label={lst:layer_api}]\n@tf.keras.utils.register_keras_serializable(name=\"RotMap\")\nclass RotMap(tf.keras.layers.Layer):\n \"\"\"Rotation Mapping layer.\"\"\"\n\n def build(self, input_shape):\n \"\"\"Create weights depending on the shape of the inputs.\n\n Expected `input_shape`:\n `[batch_size, spatial_dim, temp_dim, num_rows, num_cols]`, where\n `num_rows` = 3, `num_cols` = 3, `temp_dim` is the number of frames,\n and `spatial_dim` is the number of edges.\n \"\"\"\n input_shape = input_shape.as_list()\n so = SpecialOrthogonal()\n self.w = self.add_weight(\n \"w\",\n shape=[input_shape[-4], input_shape[-2], input_shape[-1]],\n initializer=so.random,\n )\n assign_to_manifold(self.w, so)\n\n def call(self, inputs):\n return tf.einsum(\"sij,...stjk->...stik\", self.w, inputs)\n\\end{lstlisting}\n\\end{minipage}\n\n\\section{Related work}\nThis section compares TensorFlow ManOpt with related implementations on differential geometry and learning. While inspired by related work, the main difference of our library lies in the choice of the underlying framework and design objectives.\n\nThe library Geoopt \\cite{kochurov2020geoopt} is the most closely related to TensorFlow ManOpt. Geoopt is a research-oriented toolbox, which builds upon the PyTorch \\cite{paszke2019pytorch} library for tensor computation and GPU acceleration. Geoopt supports the Riemannian SGD, as well as adaptive optimization algorithms on Riemannian manifolds, and Markov chain Monte Carlo methods for sampling. However, PyTorch, being a research-first framework, lacks tooling for bringing machine learning pipelines to production, which limits Geoopt applicability in industrial settings.\n\nMcTorch \\cite{meghwanshi2018mctorch} is a manifold optimization library for deep learning, that extends the PyTorch C++ code base to closely follow its architecture. McTorch supports Riemannian variants of stochastic optimization algorithms, and also implements a collection of neural network layers with manifold-constrained parameters. McTorch shares the same limitations as Geoopt due to its dependency on PyTorch.\n\nPymanopt \\cite{townsend2016pymanopt} is a Python package that computes gradients and Hessian-vector products on Riemannian manifolds, and provides the following solvers: steepest descent, conjugate gradients, the Nelder-Mead algorithm, and the Riemannian trust regions. Pymanopt leverages on Theano \\cite{team2016theano} symbolic differentiation and on TensorFlow automatic differentiation for computing derivatives. Pymanopt is a great general-purpose tools for Riemannian optimization. However, it is not well-suited for neural network applications due to lack of supports for SGD algorithms. It is also not intended for production use.\n\nLastly but not least, there is Geomstats \\cite{miolane2020geomstats} Python package for computations and statistics on nonlinear manifolds. Geomstats supports a broad list of manifolds such as hyperspheres, hyperbolic spaces, spaces of symmetric positive definite matrices and Lie groups of transformations. It also provides a modular library of differential geometry concepts, which includes the parallel transports, exponential and logarithmic maps, Levi-Civita connections, and Christoffel symbols. Geomstats focuses on research in differential geometry and education use cases, by providing low-level code that follows the Scikit-Learn \\cite{pedregosa2011scikit} semantics. Geomstats examples also include a modified version of the Keras framework with support of the Riemannian SGD algorithm. However, it lacks engineering maintenance.\n\n\\section{Conclusion}\nWe propose TensorFlow ManOpt, a library that combines optimization on Riemannian manifolds with automated differentiation capabilities of the TensorFlow framework. The library enables researchers to experiment with different state of the art solvers for optimization problems in non-Euclidean domains, while also allowing practitioners to efficiently pursue large-scale machine learning. The benefits of TensorFlow ManOpt are most noticeable when it comes to taking Riemannian machine learning models from research to production, where it unlocks advantages of TensorFlow tooling, such as continuous training and model validation.\n\n\\bibliographystyle{abbrv}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Centrality determination in NA61\/SHINE}\nCentrality is a key parameter in heavy ion collisions as it allows to restrict the volume of the created system. The classical definition of centrality is based on impact parameter which is not directly measurable. Therefore experiments use different techniques to define centrality indirectly. The most common method is to apply restrictions on the produced particle multiplicity in some pseudorapidity intervals \\cite{ALICE centrality}. This definition possibly biases multiplicity fluctuation measurements. Therefore NA61\/SHINE uses for centrality determination information from a forward hadron colorimeter PSD \\cite{Abgrall:2014fa}, which measures the energy of all non-interacted forward nucleons. \n\\section{Fluctuation measures}\nFluctuation quantities are studied to probe the critical point CP of strongly interacting matter. The most widely used is the scaled variance of the multiplicity distribution:\n\\begin{eqnarray}\n &\\omega[N] = (\\langle N^{2}\\rangle -{\\langle N\\rangle}^{2})\/\\langle N\\rangle,\n\\end{eqnarray}\nwhere $\\langle\\cdots\\rangle$ stands for averaging over all events. Within the Wounded Nucleon Model (WNM) or the ideal Boltzmann multi-component gas in the grand canonical ensemble (IB-GCE) one can write \\cite{Gorenstein:2011vq}:\n\\begin{eqnarray}\n &\\omega[N] = \\omega[n]+\\omega[W]\\langle N\\rangle\/\\langle W\\rangle,\n\\end{eqnarray}\nwhere $n$ is the multiplicity produced from one wounded nucleon or from a fixed volume (IB-GCE) and $W$ is the number of wounded nucleons. To probe the CP it is important to suppress the volume fluctuation ($\\omega[W]$). Therefore the scaled variance should be measured for the most central collisions or strongly intensive quantities should be used. In Refs.~\\cite{Gorenstein:2011vq,Gazdzicki:2013ana} the following measures were proposed:\n\\begin{eqnarray}\n &\\Delta[A,B] = \\frac{1}{C_{\\Delta}} \\biggl[ \\langle B \\rangle \\omega[A] -\n \\langle A \\rangle \\omega[B] \\biggr] \\\\\n &\\Sigma[A,B] = \\frac{1}{C_{\\Sigma}} \\biggl[ \\langle B \\rangle \\omega[A] +\n \\langle A \\rangle \\omega[B] - 2 \\bigl( \\langle AB \\rangle -\n \\langle A \\rangle \\langle B \\rangle \\bigr) \\biggr],\n\\end{eqnarray}\nwhere $A$ and $B$ are extensive quantities and $C_{\\Delta}$ and $C_{\\Sigma}$ are normalization coefficients, which are chosen so that $\\Delta=\\Sigma=1$ for the models of independent particle production.\nA new quantity $\\Omega$~\\cite{Poberezhnyuk:2015Omega} can be constructed from $\\Delta$ and $\\Sigma$ by setting $C_{\\Delta}=C_{\\Sigma}=\\langle B \\rangle$:\n\\begin{eqnarray}\n&\\Omega[A,B]=\\frac{1}{2}(\\Delta[A,B]+\\Sigma[A,B]),\n\\end{eqnarray}\nIf $A$ and $B$ are uncorrelated from a fixed volume within the IB-GCE or from a single source then $\\Omega[A,B]=\\omega[a]$, where $\\omega[a]$ is $\\omega[A]$ in the fixed volume or the scaled variance of $A$ from a single source. Therefore for the most central collisions one expects that $\\Omega[A,B]\\approx\\omega[A]$.\n\\section{Analysis procedure}\nPreliminary results were obtained from forward energy selected Ar+Sc collisions at 19, 30, 40, 75 and 150{\\it A} GeV\/c respectively. The analysis was performed in the NA61\/SHINE acceptance with the restriction $00\n\\]\n\\[\n\\|u\\|_{X_k}=2^{\\frac{k}2} \\|u\\|_{L^2_{t,x}(A_{<-k})} + \\sup_{j\\ge -k} \\|(|x|+2^{-k})^{-1\/2} u\\|_{L^2_{t,x}(A_j)},\n\\quad k\\le 0.\n\\]\nThe local smoothing space ${X}$ is the completion of the Schwartz space\nwith respect to the norm\n\\[\n\\|u\\|_{{X}}^2 = \\sum_{k=-\\infty}^\\infty 2^k \\|S_k u\\|^2_{X_k}.\n\\]\nIts dual ${X}'$ has norm \n\\[\n\\|f\\|_{{X}'}^2=\\sum_{k=-\\infty}^\\infty 2^{-k} \\|S_k f\\|^2_{X_k'}.\n\\]\n\nIn dimension $n \\geq 3$ the space ${X}$ is a space of distributions,\nand we have the Hardy type inequality\n\\begin{equation}\n\\| \\langle x\\rangle^{-1} u\\|_{L^2_{t,x}} \\lesssim \\| u\\|_{{X}}.\n\\label{Hardy}\\end{equation}\nOn the other hand in dimensions $n=1,2$, the space ${X}$ is a space of\ndistributions modulo constants, and we have the BMO type inequality\n\\begin{equation}\n\\sum_{j\\ge 0}\\| \\langle x\\rangle^{-1} (u-u_{D_j})\\|^2_{L^2_{t,x}(A_j)} \\lesssim \\| u\\|^2_{{X}}\n\\label{Hardylow}\\end{equation}\nwhere $u_{D_j}$ represents the (time dependent) average \nof $u$ in $D_j$.\n At the same time ${X}'$ contains only\nfunctions with integral zero. We refer the reader to \\cite{T} for\nmore details. \n\n\nIn \\cite{T} the case of a small perturbation\nof the Laplacian is considered, and it is proved that\n\\begin{theorem}~\\cite{T}. Assume that either\n\n(i) $n \\geq 3$ and \\eqref{coeff}, \\eqref{coeffb},\\eqref{coeffc} hold\nwith a sufficiently small $\\kappa$ or \n\n(ii) $n = 1,2$, $b^i=0$, $c=0$ and \\eqref{coeff} holds\nwith a sufficiently small $\\kappa$.\n\nThen the local smoothing estimate \n\\begin{equation}\n \\label{lsesmall}\n\\| u\\|_{{X} \\cap L^\\infty_t L^2_x} \\lesssim \\|u_0\\|_{L^2}\n+ \\|f\\|_{{X}'+L^1_tL^2_x} \n\\end{equation}\nholds for all solutions $u$ to \\eqref{main.equation}.\n\\end{theorem}\n\nAs one can see, the assumptions are more restrictive in low\ndimensions. This is related to the spectral structure of the operator\n$A$, precisely to the presence of a resonance at zero. This is the case\nif $A = -\\Delta$ or, more generally, if $b^i=0$ and $c=0$. However\nthe zero resonance is unstable with respect to lower order perturbations.\nTo account for non-resonant situations, it is convenient to \nintroduce a stronger norm which removes the quotient structure,\n\\[\n\\begin{split}\n\\|u\\|_{{{\\tilde{X}}}}^2 = &\\ \\| \\langle x\\rangle^{-1} u\\|_{L^2_{t,x}}^2 + \n \\sum_{k=-\\infty}^\\infty 2^k \\|S_k u\\|^2_{X_k}, \\qquad n \\neq 2\n\\\\\n\\|u\\|_{{{\\tilde{X}}}}^2 = &\\ \\| \\langle x\\rangle^{-1} (\\ln (2+|x|))^{-1} u\\|_{L^2_{t,x}}^2 + \n \\sum_{k=-\\infty}^\\infty 2^k \\|S_k u\\|^2_{X_k}, \\qquad n = 2.\n\\end{split}\n\\]\nIts dual is \n\\[\n{{\\tilde{X}}}' = {X}' + \\langle x \\rangle L^2_{t,x}, \\quad n \\neq 2,\\qquad {{\\tilde{X}}}' = {X}' + \\langle x\n \\rangle (\\ln(2+|x|)) L^2_{t,x}, \\quad n = 2.\n\\]\nDue to the Hardy inequality above, if $n \\geq 3$ we have ${{\\tilde{X}}} = {X}$. \nOn the other hand in low dimension the ${{\\tilde{X}}}$\n norm adds some local\nsquare integrability to the ${X}$ norm. Precisely, we have \n\\begin{lemma}\nLet $n=1,2$. Then\n\\begin{equation}\n\\| u\\|_{{{\\tilde{X}}}} \\lesssim \\|u\\|_{{X}} + \\|u\\|_{L^2_{t,x}(\\{|x| \\leq 1\\})}.\n\\end{equation}\n\\label{txdx}\n\\end{lemma}\n\n\nThe first goal of this article is to show, without any trapping\nassumption, that loss-less (with respect to regularity), global-in-time local smoothing and\nStrichartz estimates hold exterior to a sufficiently large ball,\nmodulo a localized error term. It is hoped that this error term can\nbe separately estimated for applications of interest. Moreover, in\nthe case of finite times, this error term can be trivially estimated\nby the energy inequality and immediately yields a $C^2$, long range,\ntime dependent analog of the result of \\cite{BT}.\n\nFor $M$ fixed and sufficiently large so that\n\\eqref{coeff.M}, \\eqref{coeffb.M} and \\eqref{coeffc.M} hold,\nwe consider a smooth, radial, nondecreasing cutoff function $\\rho$ which is supported in\n$\\{|x|\\ge 2^M \\}$ with $\\rho(|x|)\\equiv 1$ for $|x|\\ge 2^{M+1}$. \nThen we define the exterior local smoothing space ${{\\tilde{X}}}_e$\nwith norm\n\\[\n\\| u\\|_{{{\\tilde{X}}}_e} = \\| \\rho u\\|_{{{\\tilde{X}}}} + \\| (1-\\rho) u\\|_{L^2_{t,x}}\n\\]\nand the dual space ${{\\tilde{X}}}'_e$ with norm \n\\[\n\\|f\\|_{{{\\tilde{X}}}'_e} = \\inf_{f = \\rho f_1 +\n (1-\\rho) f_2} \\| f_1\\|_{{{\\tilde{X}}}'} + \\|f_2\\|_{L^2_{t,x}}. \n\\]\nNow we can state our exterior local smoothing estimates.\n\\begin{theorem}\n \\label{main.ls.theorem}\n Let $n \\geq 1$. Assume that the coefficients $a^{ij}$, $b^i$ and\n $c$ are real and satisfy \\eqref{coeff}, \\eqref{coeffb},\n \\eqref{coeffc}. Then the solution $u$ to \\eqref{main.equation}\n satisfies\n\\begin{equation}\n \\label{lsext}\n\\| u\\|_{{{\\tilde{X}}}_e \\cap L^\\infty_t L^2_x} \\lesssim \\|u_0\\|_{L^2}\n+ \\|f\\|_{{{\\tilde{X}}}'_e+L^1_tL^2_x} + \n \\|u\\|_{L^2_{t,x}(\\{|x|\\le 2^{M+1}\\})}.\n\\end{equation}\n\\end{theorem}\n\nIn the low dimensional resonant case the situation is a bit more\ndelicate. First of all, the above theorem does not give a meaningful estimate in the \n$n=1,2$ resonant case as the last term in the right of \\eqref{lsext} blows up for\nconstant functions, which correspond to the zero resonance.\n Since we do not control the local $L^2$ norm for ${X}$\nfunctions, truncation by the cutoff function $\\rho$ does not preserve\nthe ${X}$ space. To remedy this we define a time dependent local\naverage for $u$, namely\n\\[\nu_\\rho = \\left( \\int_{{\\mathbb R}^n} (1-\\rho) \\ dx \\right)^{-1} \\int_{{\\mathbb R}^n} (1-\\rho) u\\ dx,\n\\]\nand define a modified truncation by the self-adjoint operator\n\\[\nT_\\rho u = \\rho u + (1-\\rho) u_\\rho.\n\\]\nWe note that $T_\\rho$ leaves constant functions unchanged, as well as the\nintegral of $u$ (if finite).\n\nThen we set\n\\[\n\\| u\\|_{{X}_e} = \\| T_\\rho u\\|_{{X}} + \\| u - T_\\rho u \\|_{L^2_{t,x}}\n\\]\nand have the dual space ${X}'_e$ with norm \n\\[\n\\|f\\|_{{X}'_e} = \\inf_{f = T_\\rho f_1 + (1-T_\\rho) f_2} \n\\| f_1\\|_{{X}'} + \\|f_2\\|_{L^2_{t,x}}. \n\\]\nWe now have the following alternative to\nTheorem~\\ref{main.ls.theorem} which is consistent with operators with\na constant zero resonance:\n\n\n\\begin{theorem}\n \\label{main.ls.theorem.res}\n Let $n = 1,2$. Assume that \n\n(i) the coefficients $a^{ij}$ are real and satisfy \\eqref{coeff};\n\n(ii) the coefficients $b^i$ are real, satisfy \\eqref{coeffb},\nand $\\partial_i b^i = 0$;\n\n(iii) there are no zero order terms, $c = 0$.\n\n Then the solution $u$ to \\eqref{main.equation}\n satisfies\n\\begin{equation}\n \\label{lsext.res}\n\\| u\\|_{{X}_e \\cap L^\\infty_t L^2_x} \\lesssim \\|u_0\\|_{L^2}\n+ \\|f\\|_{{X}'_e+L^1_tL^2_x} + \n \\|u - u_\\rho\\|_{L^2_{t,x}(\\{|x|\\le 2^{M+1}\\})}.\n\\end{equation}\n\\end{theorem}\n\nOnce we have the local smoothing estimates, the parametrix\nconstruction in \\cite{T} allows us to obtain corresponding Strichartz\nestimates. If $(p,q)$ is a Strichartz pair we define the exterior\nspace ${{\\tilde{X}}}_e(p,q)$ with norm\n\\[\n\\| u\\|_{{{\\tilde{X}}}_e(p,q)} = \\| u\\|_{{{\\tilde{X}}}_e} + \\| \\rho u\\|_{L^p_t L^q_x}\n\\]\nand the dual space ${{\\tilde{X}}}'(p,q)$ with norm \n\\[\n\\|f\\|_{{{\\tilde{X}}}'_e(p,q)} = \\inf_{f = f_1 + \\rho f_2} \\| f_1\\|_{{{\\tilde{X}}}'_e} + \n\\|f_2\\|_{L^{p'}_t L^{q'}_x}. \n\\]\n\n\\begin{theorem}\n \\label{main.est.theorem}\n Let $n \\geq 1$. Assume that the coefficients $a^{ij}$, $b^i$ and\n $c$ are real and satisfy \\eqref{coeff}, \\eqref{coeffb},\n \\eqref{coeffc}. Then for any two Strichartz pairs\n $(p_1,q_1)$ and $(p_2, q_2)$, the solution $u$ to\n \\eqref{main.equation} satisfies\n\\begin{equation}\n \\label{fse}\n\\| u\\|_{{{\\tilde{X}}}_e(p_1,q_1) \\cap L^\\infty_t L^2_x} \\lesssim \\|u_0\\|_{L^2}\n+ \\|f\\|_{{{\\tilde{X}}}'_e(p_2,q_2)+L^1_tL^2_x} + \n \\|u\\|_{L^2_{t,x}(\\{|x|\\le 2^{M+1}\\})}.\n\\end{equation}\n\\end{theorem}\n\n\n\nCorrespondingly, in the resonant case we define\n\\[\n\\| u\\|_{{X}_e(p,q)} = \\| u\\|_{{X}_e} + \\| \\rho u\\|_{L^p_t L^q_x}\n\\]\nand the dual space ${X}'_e(p,q)$ with norm \n\\[\n\\|f\\|_{{X}'_e(p,q)} = \\inf_{f = f_1 + \\rho f_2} \\| f_1\\|_{{X}'_e} + \n\\|f_2\\|_{L^{p'}_t L^{q'}_x}. \n\\]\nThen we have \n\n\\begin{theorem}\n \\label{main.est.theorem.res}\n Let $n = 1,2$. Assume that the coefficients of $P$ are as in\n Theorem~\\ref{main.ls.theorem.res}. Then for any two Strichartz\n pairs $(p_1,q_1)$ and $(p_2, q_2)$, the solution $u$ to\n \\eqref{main.equation} satisfies\n\\begin{equation}\n \\label{fselow}\\| u\\|_{{X}_e(p_1,q_1)\\cap L^\\infty_t L^2_x} \\lesssim \n\\|u_0\\|_{L^2} + \\|f\\|_{{X}'_e(p_2,q_2)+L^1_t L^2_x} \n + \\|u-u_\\rho\\|_{L^2_{t,x}(\\{|x|\\le 2^{M+1}\\})}.\n\\end{equation}\n\\end{theorem}\n\n\nIn both cases the space-time norms are over $[0,T]\\times {\\mathbb R}^n$ for any\ntime $T>0$ with constants independent of $T$.\nIf the time $T$ is finite, then we may use energy estimates to\ntrivially bound the error term. Doing so results in the following,\nwhich is a $C^2$-analog of the exterior Strichartz estimates of\n\\cite{BT}.\n\\begin{corr}\n \\label{corr.main.theorem}\n(a.) Assume that the coefficients $a^{ij}$, $b^i$, and $c$ are as in\n Theorem~\\ref{main.ls.theorem}. Then for any two Strichartz pairs\n $(p_1,q_1)$ and $(p_2, q_2)$, the solution $u$ to\n \\eqref{main.equation} satisfies\n\\begin{equation}\n \\label{fse2}\n\\| u\\|_{{{\\tilde{X}}}(p_1,q_1)\\cap L^\\infty_t L^2_x} \\lesssim_T \\|u_0\\|_{L^2}\n+ \\|f\\|_{{{\\tilde{X}}}'(p_2,q_2)+L^1_t L^2_x}.\n\\end{equation}\n\n(b.) Assume that the coefficients $a^{ij}$ and $b^i$ are as in\nTheorem~\\ref{main.ls.theorem.res}. Then for any two Strichartz pairs\n$(p_1,q_1)$ and $(p_2,q_2)$, the solution $u$ to \\eqref{main.equation}\nsatisfies\n\\begin{equation}\n \\label{fse2.res}\n\\|u\\|_{{X}(p_1,q_1)\\cap L^\\infty_t L^2_x}\\lesssim_T \\|u_0\\|_{L^2}\n+ \\|f\\|_{{X}'(p_2,q_2)+L^1_tL^2_x}.\n\\end{equation}\n\nIn both cases, the space-time norms are over $[0,T]\\times {\\mathbb R}^n$ and \n$T>0$ is finite.\n\\end{corr}\n\nWe conclude this subsection with a few remarks concerning \nseveral alternative set-ups for these results.\n\n\n\\subsubsection{ Boundary value problems }\n\nOur proof of Theorems~\\ref{main.ls.theorem},\\ref{main.ls.theorem.res},\n\\ref{main.est.theorem}~\\ref{main.est.theorem.res} treats the interior \nof the ball $B = \\{ |x| < 2^M \\}$ as a black box with the sole property\nthat the energy is conserved by the evolution.\nHence the results remain valid for exterior boundary problems.\nPrecisely, take a bounded domain $\\Omega \\subset B$\nand consider either the Dirichlet problem\n\\begin{equation}\n\\left\\{ \\begin{array}{lc} Pu = f & \\text{in } \\Omega^c \n\\cr u(0) = u_0 & \\cr \nu = 0& \\text{in } \\partial \\Omega \\end{array} \\right.\n\\label{D}\\end{equation}\nor the Neumann problem\n\\begin{equation}\n\\left\\{ \\begin{array}{lc} Pu = f & \\text{in } \\Omega^c \n\\cr u(0) = u_0 & \\cr \\displaystyle \n\\frac{\\partial u}{\\partial \\nu} = 0& \\text{in } \\partial \\Omega\n \\end{array} \\right.\n\\label{N}\\end{equation}\nwhere\n\\[\n\\frac{\\partial}{\\partial \\nu} = \\nu_i (a^{ij} D_j + b^i)\n\\]\nand $\\nu$ is the unit normal to $ \\partial \\Omega$.\n\n\nThen we have \n\\begin{corr}\n a) The results in Theorems~\\ref{main.ls.theorem} and\n \\ref{main.est.theorem} remain valid for both the Dirichlet\n problem \\eqref{D} and the Neumann problem \\eqref{N}.\n\n b) The results in Theorems~ \\ref{main.ls.theorem.res} and\n \\ref{main.est.theorem.res} remain valid for \nthe Neumann problem \\eqref{N} with the additional condition\n$b^i \\nu_i = 0$ on $ \\partial \\Omega$.\n\\end{corr}\n\nThe more restrictive hypothesis in part (b) is caused by the \nrequirement that constant functions solve the homogeneous\nproblem.\n\n\n\\subsubsection{Complex coefficients} \n\nThe only role played in our proofs by the assumption\nthat the coefficients $b^i$ and $c$ are real is to insure \nthe energy conservation in the interior region. Hence \nwe can allow complex coefficients in the region $\\{ |x| > 2^{M+1}\\}$\nwhere the coefficients satisfy the smallness condition.\n\nIn addition, allowing $c$ to be complex in the interior region does\nnot affect energy conservation either, since we are assuming \nan a priori control of the local $L^2$ space-time norm of the solution.\nHence we have\n\n\\begin{remark}\na) The results in Theorems~\\ref{main.ls.theorem} and \n\\ref{main.est.theorem} remain valid for complex coefficients\n $b^i$, $c$ with the restriction that $b^i$ are real \nin the region $\\{ |x| < 2^{M+1}\\}$.\n\nb) The results in Theorems~ \\ref{main.ls.theorem.res} and\n\\ref{main.est.theorem.res} remain valid for coefficients $b^i$ which\nare real in the region $\\{ |x| < 2^{M+1}\\}$.\n\\end{remark}\n\n\n\\subsection{ Non-trapping metrics}\nThe second goal of the article is to consider the previous setup but\nwith an additional non-trapping assumption. To state it we consider\nthe Hamilton flow $H_a$ for the principal symbol of the operator \n$A$, namely \n\\[\na(t,x,\\xi) = a^{ij}(t,x) \\xi_i \\xi_j.\n\\]\nThe spatial projections of the trajectories of the Hamilton flow $H_a$\nare the geodesics for the metric $a_{ij} d x^i dx^j$ where\n$(a_{ij})=(a^{ij})^{-1}$.\n\n\\begin{deff}\n We say that the metric $(a_{ij})$ is non-trapping if for each $R > 0$\n there exists $L > 0$ independent of $t$ so that any portion of a\n geodesic contained in $\\{|x| < R\\}$ has length at most $L$.\n\\end{deff}\n\nThe non-trapping condition allows us to use standard propagation of\nsingularities techniques to bound high frequencies inside a ball in\nterms of the high frequencies outside. Then the cutoff function\n$\\rho$ which was used before is no longer needed, and we obtain\n\\begin{theorem}\n \\label{main.ls.theoremnt}\n Let $R > 0$ be sufficiently large. Assume that the coefficients $a^{ij}$, $b^i$ and $c$ are real and\n satisfy \\eqref{coeff}, \\eqref{coeffb}, \\eqref{coeffc}. Assume also\n that the metric $a_{ij}$ is non-trapping. Then the solution $u$ to\n \\eqref{main.equation} satisfies\n\\begin{equation}\n \\label{lsnt}\n\\| u\\|_{{{\\tilde{X}}}} \\lesssim \\|u_0\\|_{L^2}\n+ \\|f\\|_{{{\\tilde{X}}}'} + \n \\|u\\|_{L^2_{t,x}(\\{|x|\\le 2R\\})},\n\\end{equation}\n\\end{theorem}\n\nrespectively \n\n\\begin{theorem}\n \\label{main.ls.theoremnt.res}\n Let $R > 0$ be sufficiently large, and let $n=1,2$. Assume that the coefficients of $P$ are as\n in Theorem~\\ref{main.ls.theorem.res}. Assume also that the metric\n $a_{ij}$ is non-trapping. Then the solution $u$ to\n \\eqref{main.equation} satisfies\n\\begin{equation}\n \\label{lsntres}\n\\| u\\|_{{X}} \\lesssim \\|u_0\\|_{L^2}\n+ \\|f\\|_{{X}'} + \n \\|u-u_\\rho\\|_{L^2_{t,x}(\\{|x|\\le 2R\\})}.\n\\end{equation}\n\\end{theorem}\n\nWe note that the high frequencies in the error term on the right are\ncontrolled by the ${X}$ norm on the left. Also the low frequencies ($\\ll\n1$) are controlled by the ${X}$ norm using the uncertainty\nprinciple. Hence the only nontrivial part of the error term corresponds\nto intermediate (i.e. $\\approx 1$ ) frequencies.\n\nThe proof combines the arguments used for the exterior estimates\nwith a standard multiplier construction from the theory of propagation\nof singularities. Adding to the above results the parametrix\nobtained in \\cite{T} we obtain\n\n\n\\begin{theorem}\n \\label{main.theoremnt}\n Let $R > 0$ be sufficiently large. Assume that the coefficients $a^{ij}$, $b^i$ and\n $c$ are real and satisfy \\eqref{coeff}, \\eqref{coeffb}, \\eqref{coeffc}. Assume\n also that the metric $a_{ij}$ is non-trapping. Then for any two\n Strichartz pairs $(p_1,q_1)$ and $(p_2, q_2)$, the solution $u$ to\n \\eqref{main.equation} satisfies\n\\begin{equation}\n \\label{fsent}\n\\| u\\|_{{{\\tilde{X}}} \\cap L^{p_1}_t L^{q_1}_x} \\lesssim \\|u_0\\|_{L^2}\n+ \\|f\\|_{{{\\tilde{X}}}'+L^{p'_2}_t L^{q'_2}_x} + \n \\|u\\|_{L^2_{t,x}(\\{|x|\\le 2R\\})},\n\\end{equation}\n\\end{theorem}\n\nrespectively \n\n\\begin{theorem}\n \\label{main.theoremnt.res}\n Let $n = 1,2$, and let $R > 0$ be sufficiently large. Assume that the coefficients of $P$ are as\n in Theorem~\\ref{main.ls.theorem.res}. Assume also that the metric\n$a_{ij}$ is non-trapping. Then for any two Strichartz pairs $(p_1,q_1)$\nand $(p_2, q_2)$, the solution $u$ to \\eqref{main.equation} satisfies\n\\begin{equation}\n \\label{fsentres}\\| u\\|_{{X}\\cap L^{p_1}_t L^{q_1}_x} \\lesssim \\|u_0\\|_{L^2}\n + \\|f\\|_{{X}'+L^{p'_2}_t L^{q'_2}_x} \n + \\| u - u_\\rho\\|_{L^2_{t,x}(\\{|x|\\le 2R\\})}.\n\\end{equation}\n\\end{theorem}\n\n\n\\subsubsection{ An improved result for trapped metrics}\n\nA variation on the above theme is obtained in the case when there are\ntrapped rays, but not too many. If they exist, they must be confined\nto the interior region $\\{|x| \\leq 2^M\\}$. Then we can define the\nconic set\n\\[\n\\begin{split}\n \\Omega_{trapped}^L = \\{&\\ (t,x,\\xi) \\in {\\mathbb R} \\times T^* B(0,2^M);\n \\text{ the $H_a$ bicharacteristic through } (t,x,\\xi)\n\\\\ & \\text{ has\n length at least } L \\text{ within } |x| \\leq 2^M\\}.\n\\end{split}\n\\]\nGiven a smooth zero homogeneous symbol $q(x,\\xi)$ which \nequals $1$ for $|x| > 2^M$, we define modified exterior spaces\nby\n\\[\n\\| u\\|_{{{\\tilde{X}}}_q} = \\| q(x,D) u\\|_{{{\\tilde{X}}}} + \\| u\\|_{L^2(\\{|x| \\leq 2^{M+1}\\})} \n\\]\nwith similar modifications for ${{\\tilde{X}}}'_q$, ${X}_q$ and ${X}'_q$.\n\nThen the same argument as in the proof of the above Theorems\ngives\n\n\\begin{corr} \n Assume that $q$ is supported outside $ \\Omega_{trapped}^L$ for some\n $L > 0$. Then the results in Theorems~\\ref{main.ls.theorem},\n \\ref{main.est.theorem}, \\ref{main.ls.theorem.res} and\n \\ref{main.est.theorem.res} remain valid with ${{\\tilde{X}}}_e$, ${{\\tilde{X}}}'_e$,\n ${X}_e$ and ${X}'_e$ replaced by ${{\\tilde{X}}}_q$, ${{\\tilde{X}}}'_q$, ${X}_q$ and\n ${X}'_q$.\n\\end{corr}\n\nWe also note that if $A$ has time independent coefficients then\n$\\Omega_{trapped}^L$ is translation invariant. Hence a compactness\nargument allows us to replace $\\Omega_{trapped}^L$ by\n$\\Omega_{trapped}^\\infty$, which contains all the trapped geodesics.\n\n\\subsubsection{Boundary value problems}\n\nConsider solutions $u$ for either the Dirichlet problem \\eqref{D} or\nthe Neumann problem \\eqref{N}. Then singularities will propagate along\ngeneralized broken bicharacteristics (see \\cite{MS1, MS2},\\cite{Hor},\\cite{BGT2}).\nHence the non-trapping condition needs to be modified accordingly.\n\n\\begin{deff}\n We say that the metric $(a_{ij})$ is non-trapping if for each $R > 0$\n there exists $L > 0$ independent of $t$ so that any portion of a\n generalized broken bicharacteristic is contained in $\\{|x| < R\\}$\n has length at most $L$.\n\\end{deff}\n\nWith this modification the results of\nTheorems~\\ref{main.ls.theoremnt}, \\ref{main.ls.theoremnt.res}, remain\nvalid. However, some care must be taken with the results on\npropagation of singularities near the boundary, as not all of them are\nknown to be valid for operators with only $C^2$ coefficients.\n\nOn the other hand we do not know whether the bounds in Theorems\n\\ref{main.theoremnt}, \\ref{main.theoremnt.res} are true or not. These\nhinge on the validity of local Strichartz estimates near the boundary.\nThis is currently an unsolved problem.\n\n\\subsubsection{ Complex coefficients}\n\nAgain, one may ask to what extent are our results in this section \nare valid if complex coefficients are allowed. We have \n\\begin{remark}\n The results in Theorems~\\ref{main.ls.theoremnt},\n \\ref{main.ls.theoremnt.res}, \\ref{main.theoremnt},\n \\ref{main.theoremnt.res} remain valid if the coefficients $b^i$ and\n $c$ are allowed to be complex.\n\n\\end{remark}\n\nThis result is obtained without making any changes to our proofs\nprovided that the constant $\\kappa$ in \\eqref{coeffb} is sufficiently\nsmall. Otherwise, the multiplier $q$ used in the proof has to change \ntoo much along bicharacteristics from entry to exit from \n$B(0,2^M)$; this in turn forces a modified multiplier for the exterior\nregion. See, e.g., \\cite{D1, Doi} and \\cite{ST}.\n\n\n\\subsection{ Time independent metrics}\n\nIt is natural to ask when can one eliminate the error term altogether.\nThis is a very delicate question, which hinges on the local in space\nevolution of low frequency solutions. For general operators $A$ with \ntime dependent coefficients this question seems out of reach for now.\n\n\nThis leads us to the third part of the paper where, in addition to the\nflatness assumption above and the non-trapping hypothesis on $a_{ij}$, we\ntake our coefficients $a^{ij},b^i,c$ to be time-independent. Then the natural \nobstruction to the dispersive estimates comes from possible\neigenvalues and zero resonances of the operator $A$.\n\nSince the operator $A$ is self-adjoint, it follows that its spectrum is\nreal. More precisely, $A$ has a continuous spectrum $\\sigma_c=\n[0,\\infty)$ and a point spectrum $\\sigma_p$ consisting of discrete\nfinite multiplicity eigenvalues in ${\\mathbb R}^-$, whose only possible\naccumulation point is $0$. \n\nFrom the point of view of dispersion there is nothing we can do about\neigenvalues. Consequently we introduce the spectral projector $P_c$\nonto the continuous spectrum, and obtain dispersive estimates only for\n$P_c u$ for solutions $u$ to \\eqref{main.equation}. \n\nThe resolvent \n\\[\nR_\\lambda = (\\lambda - A)^{-1}\n\\]\nis well defined in ${\\mathbb C} \\setminus (\\sigma_c \\cup \\sigma_p)$.\nOne may ask whether there is any meromorphic continuation \nof the resolvent $R_\\lambda$ across the positive real axis, starting\non either side. This is indeed possible. The poles of this\nmeromorphic continuation are called resonances.\nThis is of interest to us because the resonances which are close to\nthe real axis play an important role in the long time behavior\nof solutions to the Schr\\\"{o}dinger equation. \n\nIn the case which we consider here (asymptotically flat), there are no\nresonances nor eigenvalues inside the continuous spectrum i.e. in\n$(0,\\infty)$. However, the bottom of the continuous spectrum,\nnamely $0$, may be either an eigenfunction (if $n \\geq 5$) or \na resonance (if $n \\leq 4$). For zero resonances we use\na fairly restrictive definition:\n\n\n\\begin{deff}\nWe say that $0$ is a resonance for $A$ if there is a function \n$u \\in {{\\tilde{X}}}^0$ so that $Au = 0$. The function $u$ \nis called a zero resonant state of $A$.\n\\end{deff}\n\nHere ${{\\tilde{X}}}^0$ denotes the spatial part of the ${{\\tilde{X}}}$ norm. I.e. ${{\\tilde{X}}} = L^2_t {{\\tilde{X}}}^0$.\n\n\nThe main case we consider here is when $0$ is neither an eigenfunction\n(if $n \\geq 5$) nor a resonance (if $n \\leq 4$). This implies that\nthere are no eigenvalues close to $0$. Then $A$ has at most finitely\nmany negative eigenvalues, and the corresponding eigenfunctions decay\nexponentially at infinity.\n\n\n\\begin{theorem}\\label{theorem.PcSmoothing} \n Suppose that $a^{ij},b^i,c$ are real, time-independent, and satisfy the\n conditions \\eqref{coeff},\\eqref{coeffb}, and\n \\eqref{coeffc}. We also assume that the Hamiltonian vector field\n $H_a$ permits no trapped geodesics and that $0$ is not an\n eigenvalue or a resonance of $A$. Then for all solutions $u$ to\n \\eqref{main.equation} we have\n\\begin{equation}\\label{PcSmoothing}\n\\|P_c u\\|_{{\\tilde{X}}}\\lesssim \\|u_0\\|_2 + \\| f\\|_{{{\\tilde{X}}}'}.\n\\end{equation}\n\\end{theorem}\nFrom this, using the parametrix of \\cite{T}, we immediately obtain the corresponding global-in-time\nStrichartz estimates:\n\\begin{theorem}\\label{corr.nontrap.Strichartz}\n \n Suppose that $a^{ij},b^i,c$ are real, time-independent, and\n satisfy the conditions \\eqref{coeff},\\eqref{coeffb},\n and \\eqref{coeffc}. Moreover, assume that the Hamiltonian vector\n field $H_a$ permits no trapped geodesics. Assume, also, that $0$ is\n not an eigenvalue or a resonance of $A$. Then for all solutions $u$\n to \\eqref{main.equation}, we have\n\\begin{equation}\\label{PcStrichartz}\n\\|P_c u\\|_{L^{p_1}_t L^{q_1}_x \\cap {{\\tilde{X}}}} \\lesssim \\|u_0\\|_2 + \\|f\\|_{L^{p'_2}_t\n L^{q'_2}_x+{{\\tilde{X}}}'} ,\n\\end{equation}\nfor any Strichartz pairs $(p_1,q_1)$ and $(p_2,q_2)$.\n\\end{theorem}\n\nOne can compare this with the result of \\cite{RT}, where the authors\nconsider a smooth compactly supported perturbation of the metric in\n$3+1$ dimensions where no eigenvalues are present. Estimates in the\nspirit of \\eqref{PcStrichartz} have also recently be shown by\n\\cite{BT2}, though only for smooth coefficients and with a more\nrestrictive spectral projection. We also note the related work\n\\cite{EGS} on Schr\\\"odinger equations with magnetic potentials. In\ntheir work, the second order operator is taken to be $-\\Delta$.\nTheorem \\ref{corr.nontrap.Strichartz} is a more general version of\nthe main theorem in \\cite{EGS} in the sense that it allows a more\ngeneral leading order operator and that it assumes less flatness on\nthe coefficients.\n\n\nIn dimension $n \\geq 3$ zero is not an eigenvalue or a resonance \nfor $-\\Delta$, nor for small perturbations of it. However, in\ndimension $n=1,2$, zero is a resonance and the corresponding\nresonant states are the constant functions. This spectral picture\nis not stable with respect to lower order perturbations, but it does\nremain stable with respect to perturbations of the metric $a^{ij}$.\nHence there is some motivation to also investigate this case \nin more detail. We prove the following result.\n\n\n\n\\begin{theorem}\\label{theorem.PcSmoothing.res} \n Assume that the coefficients of $P$ are time-independent, but otherwise as in\n Theorem~\\ref{main.ls.theorem.res}. Assume also that the Hamiltonian\n vector field $H_a$ permits no trapped geodesics, and that there are\n no nonconstant zero resonant states of $A$. Then for all\n solutions $u$ to \\eqref{main.equation}, we have\n\\begin{equation}\\label{PcSmoothing.res}\n\\| u\\|_{X}\\lesssim \\|u_0\\|_2 + \\| f\\|_{{X}'}.\n\\end{equation}\n\\end{theorem}\n\nIn terms of Strichartz estimates, this has the following consequence:\n\n\\begin{theorem}\\label{corr.nontrap.Strichartz.res}\n Assume that the coefficients of $P$ are time-independent, but otherwise as in\n Theorem~\\ref{main.ls.theorem.res}. Assume also that the Hamiltonian\n vector field $H_a$ permits no trapped geodesics, and that there are\n no nonconstant zero resonant states of $A$. Then for all\n solutions $u$ to \\eqref{main.equation}, we have\n\\begin{equation}\\label{PcStrichartz.res}\n\\| u\\|_{L^{p_1}_t L^{q_1}_x \\cap {X}} \\lesssim \\|u_0\\|_2 + \\|f\\|_{L^{p'_2}_t\n L^{q'_2}_x+{X}'} \n\\end{equation}\nfor any Strichartz pairs $(p_1,q_1)$ and $(p_2,q_2)$.\n\\end{theorem}\n\nImplicit in the above theorems is the fact that there are, under their\nhypothesis, no eigenvalues for $A$. There is another\nsimplification if we make the additional assumption that $b = 0$.\n\n\\begin{remark}\\label{remark.nootherres}\nIf in addition $b = 0$, then there are no nonconstant generalized zero\neigenvalues of $A$.\n\\end{remark}\n\nIn order to prove Theorems \\ref{theorem.PcSmoothing} and\n\\ref{theorem.PcSmoothing.res}, we restate the bounds\n\\eqref{PcSmoothing} and \\eqref{PcSmoothing.res} in terms of estimates\non the resolvent using the Fourier transform in $t$. We then argue\nvia contradiction. Using the positive commutator method, we show an\noutgoing radiation condition (see Steps 8-10 of the proof), which allows us\nto pass to subsequences and claim that if \\eqref{PcSmoothing} were\nfalse, then there is a resonance or an eigenvalue $v$ within the\ncontinuous spectrum. By hypothesis this cannot occur at $0$. We use\nanother multiplier and the radiation condition to then show that $v\\in\nL^2$ and thus cannot be a resonance. As results of \\cite{KT2} show\nthat there are no eigenvalues embedded in the continuous spectrum, we\nreach a contradiction. If instead \\eqref{PcSmoothing.res} were false,\nthen the same argument produces a nonconstant zero resonance, again\nreaching a contradiction.\n \nThe paper is organized as follows. In the next section, we fix some\nfurther notations and our paradifferential setup. It is here that we\nshow that we may permit the lower order terms in the local smoothing\nestimates in a perturbative manner.\nIn the third section, we prove the local smoothing estimates using the\npositive commutator method, first in the exterior local smoothing\nspaces and then in the non-trapping case. The fourth section is\ndevoted to non-trapping, time-independent operators.\n In the final section, we review the parametrix of \\cite{T} \nand use it to show how the Strichartz estimates follow\nfrom the local smoothing estimates. \n\n{\\em Acknowledgements:} The authors thank W. Schlag and M. Zworski for helpful discussions\nregarding some of the spectral theory, and in particular the behavior of resonances, contained herein.\n\n\\bigskip\n\\newsection{Notations and the paradifferential setup}\\label{not_para}\n\n\\subsection{Notations}\nWe shall be using dyadic decompositions of both space and frequency.\nFor the spatial decomposition, we let $\\chi_k$ denote smooth functions\nsatisfying\n\\[\n1=\\sum_{j=0}^\\infty \\chi_j(x),\\quad \\text{supp } \\chi_0\\subset\\{|x|\\le 2\\},\n\\quad \\text{supp }\\chi_j\\subset \\{2^{j-1}<|x|<2^{j+1}\\}\\text{ for } j\\ge 1.\n\\]\nWe also set\n\\[\n\\chi_{k}$. In frequency, we use a\nsmooth Littlewood-Paley decomposition\n\\[\n1=\\sum_{j=-\\infty}^\\infty S_j(D), \\quad \\text{supp }\ns_j\\subset\\{2^{j-1}<|\\xi|<2^{j+1}\\}\n\\]\nand similar notations for $S_{k}$ are applied.\n\nWe say that a function is frequency localized at frequency $2^k$ if its Fourier transform is supported\nin the annulus $\\{2^{k-1}<|\\xi|<2^{k+1}\\}$. An operator $K$ is said to be frequency localized if\n$Kf$ is supported in $\\{2^{k-10}<|\\xi|<2^{k+10}\\}$ for any function $f$ which is frequency localized at\n$2^k$.\n\nFor $\\kappa$ as in \\eqref{coeff}, we may choose a positive, slowly varying sequence $\\kappa_j\\in \\ell^1$ satisfying\n\\begin{equation}\n\\label{kappa_j}\n\\sup_{A_j} \\langle x\\rangle^2 |\\partial_x^2 a(t,x)| + \\langle x\\rangle |\\partial_x a(t,x)| + |a(t,x)-I_n|\\le \\kappa_j,\n\\end{equation}\n$$\\sum \\kappa_j \\lesssim \\kappa,$$\nand\n$$|\\ln \\kappa_j - \\ln \\kappa_{j-1}|\\le 2^{-10}.$$\nWhen the lower order terms are present, we may choose $\\kappa_j$ so that each dyadic piece of\n\\eqref{coeffb} is also controlled similarly.\nWe may also assume that $M$ in \\eqref{coeff.M} is chosen sufficiently large that\n$$\\sum_{j\\ge M} \\kappa_j \\lesssim \\varepsilon.$$\nAssociated to this slowly varying sequence, we may choose functions $\\kappa_k(s)$ with\n$$\\kappa_0<\\kappa_k(s)<2\\kappa_0,\\quad 0\\le s < 2,$$\n$$\\kappa_j<\\kappa_k(s)<2\\kappa_j,\\quad 2^j 0$, we\nhave the dyadic bound\n\\[\n\\| \\langle x \\rangle^{-1} S_k u\\|_{L^2} \\lesssim \\|S_k u\\|_{X_k} \n\\]\nwhich we can easily sum over $k$ to obtain\n\\[\n\\| \\langle x \\rangle^{-1} S_{>0} u\\|_{L^2} \\lesssim \\|u\\|_{{X}}.\n\\]\n\n\nFor frequencies $k < 0$ it is easy to see that \n\\begin{equation}\\label{Tkbdd}\n\\|(1-T_k) S_k u \\|_{X_k} \\lesssim \\|S_k u \\|_{X_k}\n\\end{equation}\nfollows from the bound\n\\begin{equation}\\label{bernstein}\n \\|\\chi_{<-k} S_k u\\|_{L^2_t L^\\infty_x}\\lesssim 2^{\\frac{n-1}{2}k}\\|S_k u\\|_{X_k},\\quad k\\le 0.\n\\end{equation}\nwhich is a consequence of Bernstein's inequality.\n\nThe gain is that $(1-T_k) S_k u(t,0)=0$. This leads to the improved\npointwise bound\n\\[\n |x|^{-1}|(1-T_k) S_k u| \\lesssim 2^{\\frac{n+1}2 k} \\|S_k u \\|_{X_k},\\quad |x|<2^{-k}\n\\]\nand further to the improved $L^2$ bound\n\\begin{equation}\n\\sup_{j} \\| (2^k|x|+2^{-k}|x|^{-1})^\\frac12 |x|^{-1} (1-T_k) S_k\nu\\|_{L^2(A_j)} \\lesssim 2^{\\frac{k}2} \\|S_k u \\|_{X_k}.\n\\label{impl2}\\end{equation}\nThen, by orthogonality with respect \nto spatial dyadic regions, we can sum up\n\\[\n\\| \\langle x\\rangle^{-1} \\sum_{k < 0} (1-T_k) S_k u \\|_{L^2}^2 \\lesssim \\|u\\|_{{X}}^2 \n\\]\nwhich combined with the previous high frequency bound yields\n\\begin{equation}\n\\|\\langle x \\rangle^{-1} u^{out}\\|_{L^2} \\lesssim \\|u\\|_{{X}}.\n\\label{l2uout}\\end{equation}\n\n\nFor the terms in $u^{in}$, differentiation yields a $2^k$ factor, and\ntherefore we can estimate\n\\begin{equation}\n\\| u^{in}\\|_{\\dot H^1} \\lesssim \\|u\\|_{{X}}.\n\\label{l2uin}\\end{equation}\nIt remains to prove the bounds\n\\begin{equation}\n\\| \\langle x \\rangle^{-1} v \\|_{L^2} \\lesssim \\|v\\|_{L^2(B(0,1))} + \\|\nv\\|_{\\dot H^1},\n\\qquad n=1\n\\label{n=1emb} \n\\end{equation}\nrespectively\n\\begin{equation}\n\\| \\langle x \\rangle^{-1} (\\ln (1+ \\langle x \\rangle))^{-1}\n v \\|_{L^2} \\lesssim \\|v\\|_{L^2(B(0,1))} + \\|\nv\\|_{\\dot H^1},\n\\qquad n=2.\n\\label{n=2emb} \n\\end{equation}\n\nDue to the first factor in the right of both estimates, we may without loss of generality take\n$v$ to vanish in $B(0,1\/2)$.\nFor \\eqref{n=1emb} we integrate\n\\[\n2 \\int_{1\/2}^R x^{-1} v v_x dx = \\int_{1\/2}^R x^{-2} v^2 dx + R^{-1}\nv^2(R).\n\\]\nUsing Cauchy-Schwarz the conclusion follows.\n\nFor \\eqref{n=2emb} we argue in a similar fashion.\nWe have\n\\begin{multline*}\n2 \\int_{B_R \\setminus B_{1\/2}} |x|^{-2} (\\ln (2+|x|^2))^{-1} v x \\nabla v dx = \n\\int_{B_R \\setminus B_{1\/2}} (2+|x|^2)^{-1} (\\ln (2+|x|^2))^{-2} v^2 dx \n\\\\ + \\int_{\\partial B_R}|x|^{-1} (\\ln (2+|x|^2))^{-1} v^2 d\\sigma\n\\end{multline*}\nand conclude again by Cauchy-Schwarz. The lemma is proved. \\qed\n\nOn a related note, we include here another result which\nsimplifies the type of local error terms we allow in the non-trapping\ncase.\n\n\\begin{lemma}\nLet $n \\geq 1$ and $R > 0$. Then for each $\\epsilon > 0$ there is \n$m_\\epsilon > 0$ and $c_\\epsilon > 0$\nso that\n\\begin{equation}\\label{dXerror}\n\\| \\langle x \\rangle^{-\\frac32} u\\|_{L^2} \\leq \\epsilon \\|u\\|_{{X}} + c_\\epsilon\n\\|S_{k} S_{k} S_{k} u\\|_{L^2} < 1, \\qquad \n\\|u\\|_{L^2(\\{|x| < R\\})} = 0.\n\\]\nBut $u$ is also frequency localized in $|\\xi| < 2^{m+1}$ and\nis therefore analytic. Then the last condition above implies $u = 0$ which is a \ncontradiction.\n\\end{proof}\n\n\n\n\n\\subsection{Paradifferential calculus}\n\nHere, we seek to frequency localize the coefficients of $P$.\nA similar argument is present in \\cite{T}, where for solutions at \nfrequency $2^k$ the coefficients are localized at frequency\n\\[\n|\\xi| \\ll 2^{k\/2} \\langle x\\rangle^{-1\/2}.\n\\]\nSuch a strong localization was essential there in order to carry out\nthe parametrix construction. Here we are able to keep the setup\nsimpler and use a classical paradifferential construction, where for\nsolutions at frequency $2^k$ the coefficients are localized at\nfrequency below $ 2^k$. For a fixed frequency scale $2^k$, we set\n\\begin{align*}\na^{ij}_{(k)} = S_{0\n\\\\\n|\\partial^\\alpha (a^{ij}_{(k)}-I_n)|&\\lesssim \\kappa_k(|x|)\n{2^{|\\alpha| k}}{\\langle 2^k x \\rangle^{-|\\alpha|}},\\quad \n|\\alpha|\\le 2, \\quad k\\leq 0.\n\\end{split}\n\\end{equation}\n\n\nThe next proposition will be used to pass back and forth between $A_{(k)}$\nand $A$. We first define\n\\[\n\\tilde{A}=\\sum_k A_{(k)}S_k.\n\\]\n\n\\begin{prop}\n\\label{lemma.A.to.Ak}\nAssume that the coefficients $a^{ij}$ satisfy \\eqref{coeff}, and that\n$b=0$, $c=0$. Then\n\\begin{equation}\n \\label{aak}\n \\sum_{k} 2^{-k} \\|S_k (A-A_{(k)}) u\\|_{X_k'}^2 \\lesssim \n\\kappa^2 \\|u\\|_{{X}}^2, \n\\end{equation}\n\\begin{equation}\\label{amta}\n\\|(A-\\tilde{A})u\\|_{{X}'}\\lesssim \\kappa \\|u\\|_{{X}},\n\\end{equation}\n\\begin{equation}\n \\label{comak}\n2^{-k} \\|[A_{(k)},S_k] u\\|_{X'_k}\\lesssim \\kappa \\|u\\|_{X_k}.\n\\end{equation}\n\\end{prop}\n\\begin{proof}[ Proof of Lemma \\ref{lemma.A.to.Ak}:]\nWe begin by writing\n\\[\nS_k(A- A_{(k)})=A_k^{med}+A_k^{high}\n\\]\nwith\n\\begin{align*}\nA_k^{med}&= \n \\sum_{l=k-4}^{k+4} \\sum_{m=-\\infty}^{k+8} S_k D_i(S_l a^{ij})\nD_j S_m \\\\\nA_k^{high}&= \\sum_{l>k+4} \\sum_{m=l-4}^{l+4} \nS_k D_i (S_la^{ij})D_j S_m.\n\\end{align*}\n\n\nFor $A_k^{med}$ we take $l=k \\geq m$ for simplicity; then it suffices \n to establish the off-diagonal decay \n\\begin{equation}\n \\label{dplow3.small}\n\\|S_k D_i(S_k a^{ij}D_j S_m v)\\|_{X'_k}\\lesssim \\kappa\n2^m \\|S_m v\\|_{X_m}.\n\\end{equation}\n\nIf $k \\geq m \\geq 0$ then we have\n\\[\n\\begin{split}\n\\|S_k D_i(S_k a^{ij} D_j S_m v)\\|_{X'_k}\n&\\ \\lesssim 2^k \\|S_k a^{ij} D_j S_m v \\|_{X'_k}\n\\\\ &\\ \\lesssim \\kappa 2^{-k} \\| \\langle x\\rangle^{-2} D_j S_m v \\|_{X'_k}\n\\\\ &\\ \\lesssim \\kappa 2^{-k} \\| D_j S_m v \\|_{X_m}\n\\\\ &\\ \\lesssim \\kappa 2^{m-k} \\|S_m v \\|_{X_m}.\n\\end{split}\n\\]\n\nIf $k \\geq 0 > m$ then we have two spatial scales to deal with, namely\n$1$ and $2^{-m}$. To separate them we use the cutoff function\n$\\chi_{<-m}$. For contributions corresponding to large $x$ we estimate\n\\[\n\\begin{split}\n\\|S_k D_i(S_k a^{ij} \\chi_{\\ge -m} D_j S_m v)\\|_{X'_k}\n&\\ \\lesssim 2^k \\|S_k a^{ij} \\chi_{\\ge -m} D_j S_m v \\|_{X'_k}\n\\\\ &\\ \\lesssim \\kappa 2^{-k} \\| |x|^{-2} \\chi_{\\ge -m} D_j S_m v \\|_{X'_k}\n\\\\ &\\ \\lesssim \\kappa 2^{m-k} \\| D_j S_m v \\|_{X_m}\n\\\\ &\\ \\lesssim \\kappa 2^{2m-k} \\| S_m v \\|_{X_m}.\n\\end{split}\n\\]\nFor contributions corresponding to small $x$, we first note that by \nBernstein's inequality, see \\eqref{bernstein}, we have \n\\begin{equation}\\label{bern.2}\n\\| D_j S_m v\\|_{L^2_t L_x^\\infty(A_{\\leq -m})} \\leq 2^{\\frac{n+1}2 m} \\|S_m v\\|_{X_m}.\n\\end{equation}\nThen\n\\[\n\\begin{split}\n\\|S_k D_i(S_k a^{ij} \\chi_{<-m} D_j S_m v)\\|_{X'_k}\n&\\ \\lesssim 2^k \\|S_k a^{ij} \\chi_{<-m} D_j S_m v\\|_{X'_k}\n\\\\\n&\\ \\lesssim 2^{-k} 2^{\\frac{n+1}2 m} \\|\\langle x \\rangle^{-2} \\chi_{<-m}\n\\kappa(|x|)\\|_{(X_k^0)'} \\|S_m v\\|_{X_m}\n\\\\\n&\\ \\lesssim \\kappa 2^{-k}2^{\\frac{n+1}2 m} \\max\\{ 1, 2^{\\frac{3-n}{2} m} \\} \\|S_m v\\|_{X_m}\n\\\\\n&\\ \\lesssim \\kappa 2^{-k} \\max\\{ 2^{\\frac{n+1}2 m}, 2^{2 m} \\} \\|S_m v\\|_{X_m}\n\\end{split}\n\\]\nwhere $(X_k^0)'$ is the spatial part of the $X_k'$ norm, i.e. $X_k' = L^2_t (X_k^0)'$.\n\nFinally if $ 0 >k \\geq m$ then the spatial scales are $2^{-k}$ and\n$2^{-m}$, and we separate them using the cutoff function\n$\\chi_{<-m}$. The exterior part is exactly as in the previous case. For\nthe interior part we use again \\eqref{bern.2} \nto compute\n\\[\n\\begin{split}\n\\|S_k D_i(S_k a^{ij} \\chi_{<-m} D_j S_m v)\\|_{X'_k}\n&\\ \\lesssim 2^k \\|S_k a^{ij} \\chi_{<-m} D_j S_m v\\|_{X'_k}\n\\\\\n&\\ \\lesssim 2^{k} 2^{\\frac{n+1}2 m} \\|\\langle 2^k x \\rangle^{-2} \\chi_{<-m}\n\\kappa(|x|)\\|_{(X_k^0)'} \\|S_mv\\|_{X_m}\n\\\\\n&\\ \\lesssim \\kappa 2^{k}2^{\\frac{n+1}2 m} \n\\max\\{ 2^{-\\frac{n+1}2 k}, 2^{-2k} \n2^{\\frac{3-n}{2} m} \\} \\|S_m v\\|_{X_m}\n\\\\\n&\\ \\lesssim \\max\\{ 2^{\\frac{1-n}2 k} 2^{\\frac{n+1}2 m}, 2^{-k} 2^{2 m} \\} \\|S_m v\\|_{X_m}.\n\\end{split}\n\\]\nHence \\eqref{dplow3.small} is proved, which by summation yields the bound\n\\eqref{aak} for $A_k^{med}$. The bound for $A_k^{high}$ follows from\n summation of \\eqref{dplow3.small} in a duality argument.\n\nWe note that in all cases there is some room to spare in the\nestimates. This shows that our hypothesis is too strong for this\nlemma. Indeed, one could prove it without using at all the bound on\nthe second derivatives of the coefficients.\n\n\nThe bound \\eqref{amta} follows by duality from \\eqref{aak}. The proof\nof \\eqref{comak}, as in \\cite{T}, follows from the $|\\alpha|=1$ case\nof \\eqref{coeffak.greater}. \n\\end{proof}\n\n\n\nThe next proposition allows us to treat lower order terms perturbatively \nin most of our results.\n\n\n\\begin{prop}\\label{lemma.lower.order}\n a) Assume that $b,c$ satisfy \\eqref{coeffb} and \\eqref{coeffc}.\nThen\n \\begin{equation}\\label{bc}\n \\|(b^i D_i + D_i b^i +c)u\\|_{{{\\tilde{X}}}'}\\lesssim \\kappa \\|u\\|_{{{\\tilde{X}}}}.\n \\end{equation}\n \nb) Assume that $b$ satisfies \\eqref{coeffb} and $\\div \\ b=0$.\nThen\n \\begin{equation}\\label{bnoc}\n \\|(b^i D_i + D_i b^i)u\\|_{{X}'}\\lesssim \\kappa \\|u\\|_{{X}}.\n\\end{equation}\n\n\\end{prop}\n\n\\begin{proof}\nThis proof parallels a similar argument in \\cite{T}. However in there \nonly dimensions $n \\geq 3$ are considered, and the bound\n\\eqref{coeffc} is stronger to include the full gradient of $b$.\nThus we provide a complete proof here. We consider two cases,\nthe first of which is similar to \\cite{T}, while the second\nrequires a new argument.\n\n\n{\\bf Case 1: The estimate \\eqref{bc} for $n \\geq 3$ and \\eqref{bnoc}\n for $n=1,2$.}\nThe estimate for the $c$ term is straightforward since, by \\eqref{coeffc}, \n\\[ \\langle cu, v\\rangle \\lesssim \\kappa \\|\\langle x\\rangle^{-1} u\\|_{L^2_{t,x}} \\|\\langle x\\rangle^{-1} v\\|_{L^2_{t,x}}\n\\lesssim \\kappa \\|u\\|_{{{\\tilde{X}}}} \\|v\\|_{{{\\tilde{X}}}}.\\]\n \nFor the $b$ term, we consider a\nparadifferential decomposition,\n\\begin{multline}\\label{trichotomy}\n(b^i D_i + D_i b^i)u=\\ \\sum_{k} (S_{k}b^i D_i + D_i S_{>k} b^i)S_k u.\n\\end{multline}\n\nThe frequency localization is preserved in the first term; therefore \nit suffices to verify that\n\\[\n\\| (S_{k}b^i D_i S_k u -i S_{>k} \\div\\ b \\ S_k u.\n\\label{hl}\n\\end{equation}\nWe consider the two terms separately. The second one occurs only in \nthe case of \\eqref{bc} but the first one occurs also in \\eqref{bnoc}.\nSo we need to show that\n\\[\n\\| \\sum_{k} S_{>k}b^i D_i S_k u \\|_{{X}'} \\lesssim \\kappa \\| u\\|_{{X}}.\n\\]\nThis will follow from the dyadic estimates\n\\[\n\\| S_{m}b^i S_k u \\|_{X'_m} \\lesssim \\kappa \\| S_k\nu\\|_{X_k}, \\qquad m > k.\n\\]\nGiven the pointwise bound on $S_m b^i$, this reduces to\n\\[\n\\| S_k u \\|_{X_m} \\lesssim \\| S_k\nu\\|_{X_k}.\n\\]\nFor $|x| > \\max\\{2^{-k},1\\}$ this is trivial. For smaller $x$\nwe use \\eqref{bernstein},\nand the conclusion is obtained by a direct computation.\n\n\nIt remains to consider the second term in \\eqref{hl}, for which we \nwant to show that in dimension $n \\geq 3$\n\\begin{equation}\n\\| \\sum_{k} S_{>k} \\div\\ b \\ S_k u \\|_{{{\\tilde{X}}}'} \\lesssim \\kappa \\| u\\|_{{{\\tilde{X}}}}.\n\\label{hhll}\\end{equation}\n\nFor this we establish again off-diagonal decay,\n\\begin{equation}\n\\| S_{m} \\div\\ b \\ S_k u \\|_{X'_m} \\lesssim \\kappa (m-k) 2^k \\| S_k u\\|_{X_k},\n\\qquad m > k.\n\\label{odhhll}\\end{equation}\nThis follows from the pointwise bounds\n\\[\n| S_{m} \\div\\ b | \\leq \\kappa 2^{2m}\\langle 2^m x\\rangle^{-2}, \\qquad m < 0\n\\]\n\\[\n| S_{m} \\div\\ b | \\leq \\kappa \\langle x\\rangle^{-2}, \\qquad m \\geq 0.\n\\]\nWe consider the worst case $0 > m > k$ and leave the rest for the\nreader. We use $\\chi_{<-k}$ to separate small and large values of $x$.\nFor large $x$ we have\n\\[\n\\| \\chi_{>-k} S_{m} \\div\\ b \\ S_k u \\|_{X'_m} \\lesssim \n\\kappa \\| |x|^{-2} \\chi_{>-k} \\ S_k u \\|_{X'_k} \\lesssim \n\\kappa 2^k \\| S_k u\\|_{X_k}.\n\\]\nFor small $x$ we use \\eqref{bernstein} instead,\n\\[\n\\| \\chi_{<-k} S_{m} \\div\\ b \\ S_k u \\|_{X'_m} \\lesssim \n \\kappa 2^{2m} 2^{\\frac{n-1}2 k} \\| \\chi_{<-k} \\langle 2^m x\\rangle^{-2}\\|_{(X_m^0)'} \\|S_k u\\|_{X_k}\n\\lesssim \\kappa 2^{ k} \\|S_k u\\|_{X_k}.\n\\]\nThe last computation above is accurate if $n \\geq 4$. In dimension\n$n=3$ we encounter a harmless additional logarithmic factor $|m-k|$.\nHowever if $n=1,2$ then the above off-diagonal decay can no longer \nbe obtained.\n\n\n\n{\\bf Case 2: The estimate \\eqref{bc} in dimension $n=1,2$.}\nThe $c$ term is again easy to deal with. We write the \nestimate for $b$ in a symmetric way,\n\\[\n|\\langle (b^i D_i + D_i b^i) u,v \\rangle | \\lesssim \\kappa \\|u\\|_{{{\\tilde{X}}}} \\|v\\|_{{{\\tilde{X}}}}.\n\\]\nWe use the decomposition in Section~\\ref{embxs},\n\\[\nu = u^{in} + u^{out}, \\qquad v = v^{in} + v^{out}.\n\\]\nWe consider first the expression \n\\[\n\\langle (b^i D_i + D_i b^i) u^{out},v^{out} \\rangle.\n\\]\nFor this we can take advantage of the improved $L^2$ bound\n\\eqref{impl2} to carry out the same computation as in dimension \n$n \\geq 3$, establishing off-diagonal decay. Precisely,\nthe difference arises in the proof of \\eqref{odhhll}, whose\nreplacement is \n\\begin{equation}\n\\| S_{m} \\div\\ b \\ (1-T_k) S_k u \\|_{X'_m} \\lesssim \\kappa (m-k)2^k \\| S_k u\\|_{X_k},\n\\qquad m > k.\n\\label{odhhll.new}\\end{equation}\n\nConsider now one of the cross terms, \n\\[\n\\langle (b^i D_i + D_i b^i) u^{in},v^{out} \\rangle = \\langle (2 b^i D_i -i \\div b) u^{in},v^{out} \\rangle.\n\\]\nThe proof for the other cross term will follow similarly.\nFor the $\\div \\ b$ term we use the $L^2$ bound for both $u^{in}$ and\n$v^{out}$, as in the case of $c$. For the rest we use \\eqref{l2uin}\nand \\eqref{l2uout} to estimate\n\\[\n|\\langle b^i D_i u^{in},v^{out} \\rangle| \\lesssim \\|u^{in}\\|_{\\dot H^1} \\| b\nv^{out}\\|_{L^2} \\lesssim \\|u\\|_{{X}} \\|v\\|_{{X}}.\n\\]\n\n Finally, consider the last term\n \\[\n\\langle (b^i D_i + D_i b^i) u^{in},v^{in} \\rangle.\n\\]\nIn dimension $n=1$, we can easily estimate it by\n\\[\n|\\langle (b^i D_i + D_i b^i) u^{in},v^{in} \\rangle| \\lesssim \\|u^{in}\\|_{\\dot\n H^1} \\| \\langle x \\rangle^{-1} v^{in}\\|_{L^2} + \\|v^{in}\\|_{\\dot\n H^1} \\| \\langle x\\rangle^{-1} u^{in}\\|_{L^2} \\lesssim \\|u\\|_{{{\\tilde{X}}}} \\|v\\|_{{{\\tilde{X}}}}.\n\\]\nThis argument fails for $n=2$ due to the logarithmic factor in the\n$L^2$ weights. Instead we will take advantage of the\nspherical symmetry of both $u^{in}$ and $v^{in}$.\n\nIn polar coordinates we write\n\\[\nb^i D_i = b^r D_r + r^{-1} b^\\theta D_\\theta \n\\] \nand\n\\[\n\\div \\ b = \\partial_r b^r + r^{-1} b^r + r^{-1} \\partial_\\theta b^\\theta.\n\\]\nFor a function $b(r,\\theta)$, we denote $\\bar b(r)$ its spherical\naverage. By spherical symmetry, we compute\n\\[\n\\langle b^i D_i u^{in},v^{in} \\rangle = \\langle (b^r D_r + r^{-1} b^\\theta D_\\theta ) u^{in},v^{in} \\rangle\n= \\langle D_r u^{in}, \\bar{b^r} v^{in} \\rangle.\n\\]\nThen we can estimate\n\\[\n|\\langle (b^i D_i + D_i b^i) u^{in},v^{in} \\rangle| \\lesssim \\|u^{in}\\|_{\\dot\n H^1} \\| \\bar{b^r} v^{in}\\|_{L^2} + \\|v^{in}\\|_{\\dot H^1} \\|\\bar{b^r}u^{in}\\|_{L^2}\n\\lesssim \\|u\\|_{{{\\tilde{X}}}} \\|v\\|_{{{\\tilde{X}}}}\n\\]\nprovided we are able to establish the improved bound\n\\begin{equation}\n| \\bar{b^r}(r)| \\lesssim \\langle r \\rangle^{-1}(\\ln (2+r))^{-1}.\n\\label{bbr}\\end{equation}\n\nFor this we take spherical averages in the divergence equation\nto obtain\n\\[\n \\partial_r \\bar b^r + r^{-1} \\bar b^r = \\overline{\\div \\ b}.\n\\]\nAt infinity we have $b(r) = o(r^{-1})$. Integrating from infinity\nwe obtain\n\\[\n \\bar b^r(r) = \\int_{r}^\\infty \\frac{s}{r}\\ \\overline{\\div \\ b}(s) ds.\n\\]\nHence\n\\[\n| \\bar b^r(r) | \\lesssim \\int_{r}^\\infty \\frac{s}{r} (1+s)^{-2}\n(\\ln(2+s))^{-2} ds\n\\]\nand \\eqref{bbr} follows.\n\\end{proof}\n\n\n\\bigskip\n\\newsection{Local smoothing estimates}\\label{smoothing.section}\n\nIn this section we prove our main local smoothing estimates, \nfirst in the exterior region and then in the non-trapping case.\n\n\n\\subsection{The high dimensional case $n \\geq 3$: Proof of Theorem~\\ref{main.ls.theorem}}\nThe proof uses energy estimates and the positive commutator\nmethod. This turns out to be rather delicate. The difficulty \nis that the trapping region acts essentially as a black box,\nwhere the energy is conserved but little else is known. \nHence all the local smoothing information has to be estimated \nstarting from infinity along rays of the Hamilton flow which are \n incoming either forward or backward in time.\n\n\n\nWe begin with the energy estimate. This is standard\nif the right hand side is in $L^1_t L^2_x$, but we would like to allow \nthe right hand side to be in the dual smoothing space as well.\n\n\\begin{proposition}\nLet $u$ solve the equation\n\\begin{equation}\nD_t +A u = f_1 + f_2, \\qquad u(0) = u_0\n\\end{equation}\nin the time interval $[0,T]$. Then we have \n\\begin{equation}\n\\| u\\|_{L^\\infty_t L^2_x}^2 \\lesssim \\| u_0\\|_{L^2}^2 + \\| f_1\\|_{L^1_t L^2_x}^2\n+ \\| u\\|_{{{\\tilde{X}}}_e} \\|f_2\\|_{{{\\tilde{X}}}'_e}. \n\\label{eest}\\end{equation}\n\\end{proposition}\n\\begin{proof}\nThe proof is straightforward. We compute\n\\[\n\\frac{d}{dt} \\frac12 \\|u(t)\\|_{L^2}^2 = \\Im \\langle u, f_1+f_2 \\rangle.\n\\]\nHence for each $t \\in [0,T]$ we have\n\\[\n\\| u(t)\\|_{L^2}^2 \\lesssim \\|u(0)\\|_{L^2}^2 + \\|u\\|_{L^\\infty_t L^2_x}\n\\| f_1\\|_{L^1_t L^2_x} + \\| u\\|_{{{\\tilde{X}}}_e} \\|f_2\\|_{{{\\tilde{X}}}'_e}.\n\\]\nWe take the supremum over $t$ on the left and use bootstrapping\nfor the second term on the right. The conclusion follows.\n\\end{proof}\n\nTo prove \\eqref{lsext} we need a complementary estimate,\nnamely \n\\begin{equation}\n \\| \\rho u\\|_{{{\\tilde{X}}}}^2 \\lesssim \\| u\\|_{L^\\infty_t L^2_x}^2 + \n \\| f_1\\|_{L^1_t L^2_x}^2 \n + \\|\\rho f_2\\|_{{{\\tilde{X}}}'}^2 + \\| \\langle x \\rangle^{-2} u\\|_{L^2_{t,x}}^2. \n\\label{lsest}\\end{equation}\nGiven \\eqref{eest} and \\eqref{lsest}, the bound \\eqref{lsext}\nis obtained by bootstrapping, with some careful balancing \nof constants. \n\nIt remains to prove \\eqref{lsest}. \nWe will use a positive commutator method. We shall assume that $b=0$ and $c=0$.\nFor a self-adjoint operator\n$Q$, we have\n\\[\n2\\Im\\langle A u, Qu\\rangle = \\langle Cu,u\\rangle\n\\]\nwhere\n\\[\nC=i[A,Q].\n\\]\nAs a consequence of this, we see that\n\\[\n\\frac{d}{dt}\\langle u, Qu\\rangle = -2\\Im\\langle (D_t+A)u, Qu\\rangle + \\langle\nCu,u\\rangle.\n\\]\nTaking this into account, the estimate \\eqref{lsest} is an immediate\nconsequence of the following lemma.\n\n\\begin{proposition} \\label{Qprop}\n There is a family $\\mathcal Q$ of bounded self-adjoint operators $Q_\\rho$ with \nthe following properties:\n\n(i) $L^2$ boundedness,\n\\[\n\\|Q_\\rho\\|_{L^2 \\to L^2} \\lesssim 1\n\\]\n\n(ii) ${{\\tilde{X}}}$ boundedness,\n\\[\n|\\langle Q_\\rho u, f \\rangle | \\lesssim \\| \\rho f\\|_{{{\\tilde{X}}}'} \\| \\rho u\\|_{{{\\tilde{X}}}}\n\\]\n\n(iii) Positive commutator,\n\\[\n\\sup_{Q_\\rho \\in \\mathcal Q} \\langle Cu,u \\rangle \\geq c_1 \\|\\rho u\\|_{{{\\tilde{X}}}}^2 -\nc_2 \\|\\langle x\\rangle^{-2} u\\|_{L^2_{t,x}}^2.\n\\]\n\\end{proposition}\n\n\n We first note that the condition (ii) shows that $Q_\\rho u$ is supported\n in $\\{|x| > 2^M\\}$ and depends only on the values of $u$ in the same\n region. Hence for the purpose of this proof we can modify the\n operator $A$ arbitrarily in the inner region $\\{|x| < 2^M\\}$. In\n particular we can improve the constant $\\kappa$ in \\eqref{coeff} to\n the extent that \\eqref{coeff.M} holds globally. Similarly, we can\n assume without any restriction in generality that $u = 0$ in $\\{|x|\n < 2^M\\}$. \n\nUsing (ii), we may argue similarly and assume that \\eqref{coeffb.M} and\n\\eqref{coeffc.M} hold globally if lower order terms are present. \nThe estimate \\eqref{bc} then justifies\n neglecting the \nlower order terms in $A$. I.e., we may assume that $b=0$, $c=0$.\n\n\n\\begin{proof}\nThe main step in the proof of the proposition is to construct\nsome frequency localized versions of the operator $Q_\\rho $. Precisely,\nfor each $k \\in {\\mathbb Z}$ we produce a family ${\\mathcal Q}_k$ \nof operators $Q_k$, which we later use to construct $Q_\\rho $.\nWe consider two cases, depending on whether $k$ is positive or\nnegative.\n\n\nWe first introduce some variants of the spaces $X_k$. Let $k \\in {\\mathbb Z}$ and\n$k^-=\\frac{|k|-k}{2}$ be its negative part. For any positive, slowly varying\nsequence $(\\alpha_m)|_{m\\geq k^-}$ with\n\\[\n\\sum_{k \\ge k^-}\\alpha_j =1, \\qquad \\alpha_{k^-} \\approx 1,\n\\]\nwe define the space $X_{k,\\alpha}$ with norm\n\\begin{align*}\n \\|u\\|_{X_{k,\\alpha}}^2 &= 2^{-k^-} \\|u\\|^2_{L^2(A_{\\leq k^-})} +\n \\sum_{j>k^-} \\alpha_j\\| |x|^{-1\/2} u\\|^2_{L^2(A_j)}.\n\\end{align*}\nThen our low frequency result has the form\n\\begin{lemma} \\label{lemma.low.freq} \nLet $n \\geq 1$ and $k<0$. Then for any slowly varying sequence $(\\alpha_m)$ with\n$\\alpha_{-k} \\approx 1$ and $\\sum_{m\\ge -k}\\alpha_m=1$, there\n is a self-adjoint operator $Q_k$ so that\n\\begin{align}\n\\|Q_k u\\|_{L^2}&\\lesssim \\|u\\|_{L^2},\\label{Q.L2.low}\\\\\n\\|Q_k u\\|_{X_{k,\\alpha}}&\\lesssim \\|u\\|_{X_{k,\\alpha}},\\label{Q.X.low}\\\\\n\\langle C_k u, u\\rangle &\\gtrsim 2^{k} \\|u\\|^2_{X_{k,\\alpha}}, \\qquad \nC_k = i [ A_{(k)}, Q_k] \n\\label{C.low}\n\\end{align}\nfor all functions $u$ frequency localized at frequency $2^k$.\n\\end{lemma}\n\n\\begin{proof}\n We argue exactly as in \\cite[Lemma 9]{T}. The only difference is\n that here we work with the operator $A_{(k)}$ whose coefficients have\n less regularity, but this turns out to be nonessential.\n \n We first increase the sequence $(\\alpha_m)$ so that\n\\begin{equation}\n\\label{new.alpha}\n\\begin{cases}\n(\\alpha_m)\\text{ remains slowly varying,}\\\\\n\\alpha_m = 1\\text{ for } m\\le -k \\\\\n\\displaystyle \\sum_{m>-k} \\alpha_m \\approx 1,\\\\\n\\kappa_m \\le \\epsilon \\alpha_{m}\\text{ for } m > -k.\n\\end{cases}\n\\end{equation}\nTo this slowly varying sequence we may associate a slowly varying\nfunction $\\alpha(s)$ with\n\\[\n\\alpha(s)\\approx \\alpha_m,\\quad s\\approx 2^{m+k}.\n\\]\n\nWe construct an even smooth symbol $\\phi$ of order $-1$ satisfying\n\\begin{align}\n\\phi(s)&\\approx \\langle s\\rangle^{-1},\\quad s>0 \\label{phi1}\\\\\n\\phi(s)+s\\phi'(s)&\\approx \\frac{\\alpha(s)}{\\langle s\\rangle},\\quad\ns>0.\\label{phi2} \t\n\\end{align}\nWe notice that the radial function $ S_{<10}(D)\\phi(|x|)$ satisfies\nthe same estimates; therefore without any restriction in generality we\nassume that $\\phi(|x|)$ is frequency localized in $|\\xi| < 2^{10}$.\n\nWe now define the self-adjoint multiplier\n\\[\nQ_k(x,D)= \\delta(Dx\\phi( 2^k \\delta |x|)+\\phi(2^k \\delta |x|)xD).\n\\]\nFor small $\\delta$ this takes frequency $2^k$ functions to frequency\n$2^k$ functions. The first property \\eqref{Q.L2.low} follows immediately.\n\\begin{comment}\n\\begin{com}\n Indeed, this is easy since we are in $L^2$ and $\\phi_k$ preserves\n the frequency support. Thus, the $D$ introduces the $2^k$, and the\n $\\delta |x| \\phi_k \\le 1$ by the first property of $\\phi$ above.\n\\end{com}\n\\end{comment}\nThe estimate \\eqref{Q.X.low} is also straightforward as the weight in\nthe $X_{k,\\alpha}$ norm is slowly varying on the dyadic scale. It\nremains to prove \\eqref{C.low} for which we begin by computing the\ncommutator\n\\begin{equation}\n\\label{C}\n\\begin{split}\nC_k =&\\ 4\\delta D_i \\phi(2^k\\delta |x|)a^{ij}_{(k)}D_j \n\\\\ &\\ + 2^{k+1}\n\\delta^2 \\Bigl(Dx |x|^{-1}\\phi'(2^k \\delta |x|)\nx_i a^{ij}_{(k)} D_j + D_i a^{ij}_{(k)} x_j |x|^{-1}\\phi'(2^k \\delta |x|)xD\\Bigr)\n\\\\ &\\ -2\\delta D_i\\phi(2^k \\delta |x|)(x_l\\partial_l a^{ij}_{(k)})D_j + \\partial_i(a^{ij}_{(k)}(\\partial_j \\partial(\n\\delta x\\phi(2^k \\delta |x|)))).\n\\end{split}\n\\end{equation}\nThe positive contribution comes from the first two terms. Replacing\n$a^{ij}_{(k)}$ by the identity leaves us with the principal part\n\\[\nC_k^0=4\\delta D \\phi(2^k\\delta |x|)D + 4\\delta D \\frac{x}{|x|} 2^k \n\\delta |x|\\phi'(2^k\\delta |x|)\\frac{x}{|x|}D\n\\]\nwhich by \\eqref{phi2} satisfies\n\\[\n\\langle C_k^0 u,u\\rangle \\ge 4\\delta \\langle (\\phi(2^k \\delta |x|)+2^k \\delta\n|x|\\phi'(2^k \\delta |x|))\\nabla u,\\nabla u\\rangle\n\\gtrsim \\delta 2^{2k}\\Bigl\\langle \\frac{\\alpha(2^k \\delta |x|)}{\\langle 2^k \\delta\n x\\rangle}u,u\\Bigr\\rangle.\n\\]\nSince $a^{ij}_{(k)}(x)-\\delta^{ij} = O(\\kappa_k(|x|))$, the error we produce\nby substituting $a^{ij}_{(k)}$ by the identity has size\n\\[\n\\delta 2^{2k} \\left\\langle \\frac{\\kappa_k(|x|)}{\\langle 2^k \\delta x\\rangle}u,u\\right\\rangle.\n\\]\nIt remains to examine the last two terms in $C_k$. Using\n\\eqref{coeffak.greater}, we see that\n\\[\n|\\delta \\phi(2^k \\delta |x|)(x_l\\partial_l a^{ij}_{(k)})|\\lesssim\n\\frac{\\delta\\kappa_k(|x|)}{\\langle 2^k \\delta x\\rangle}.\n\\]\nSo, the third term yields an error similar to the above one.\n\nFinally, \n\\[\n|\\partial_i(a^{ij}_{(k)}(\\partial_j\\partial(\\delta x\\phi(2^k \\delta |x|))))|\\lesssim\n\\frac{\\delta^3 2^{2k}}{\\langle 2^k\\delta x\\rangle^3}\\lesssim\n\\frac{\\delta^3 2^{2k} \\alpha(2^k \\delta |x|)}{\\langle 2^k \\delta x\\rangle},\n\\]\nwhich yields\n\\[\n\\langle\\partial_i(a^{ij}_{(k)}(\\partial_j\\partial\\delta x\\phi(2^k \\delta |x|)))u,u\\rangle\n\\lesssim \\delta^3 2^{2k} \\Bigl\\langle \\frac{\\alpha(2^k \\delta |x|)}{\\langle 2^k \\delta\n x\\rangle} u,u\\Bigr\\rangle.\n\\]\n\nSumming up, we have proved that\n\\begin{equation}\n\\langle C_ku,u\\rangle \\geq c_1 \\delta 2^{2k}\\Bigl\\langle \\frac{\\alpha(2^k \\delta |x|)}{\\langle 2^k \\delta\n x\\rangle}u,u\\Bigr\\rangle\\! - c_2 \\delta^3 2^{2k}\n\\Bigl\\langle \\frac{\\alpha(2^k \\delta |x|)}{\\langle 2^k \\delta\n x\\rangle} u,u\\Bigr\\rangle \\!- c_3 \\delta 2^{2k} \\left\\langle \\frac{\\kappa_k(|x|)}{\\langle 2^k \\delta x\\rangle}u,u\\right\\rangle.\n\\label{lowcom}\\end{equation}\nIn order to absorb the second term into the first we need to know that\n$\\delta$ is sufficiently small. This determines the choice of $\\delta$\nas a small universal constant. In order to absorb the third term into\nthe first we use the last part of \\eqref{new.alpha} and the fact that \n$\\alpha$ is slowly varying on the dyadic scale to estimate\n\\[\n\\kappa(|x|) \\lesssim \\epsilon \\alpha(2^k |x|) \\lesssim \\delta^{-1}\n\\epsilon \\alpha(2^k \\delta |x|).\n\\]\nThus the third term is negligible if $\\epsilon \\ll \\delta$. This\ndetermines the choice of $\\epsilon$ in \\eqref{coeff.M},\n\\eqref{coeffb.M} and \\eqref{coeffc.M}.\n\\end{proof}\n\n\nWe continue with the result for high frequencies.\n\n\n\\begin{lemma}\n\\label{lemma.high.freq}\nLet $n \\geq 1$ and $k \\geq 0$. Then for any sequence $(\\alpha_m)$ with\n$\\alpha_0=1$ and $\\sum_{m\\ge 0} \\alpha_m=1$ there\nis a self-adjoint operator $Q_k$ so that\n\\begin{align}\n\\|Q_k u\\|_{L^2} &\\lesssim \\|u\\|_{L^2},\\label{Q.L2.high}\\\\\n\\|Q_k u\\|_{X_{k,\\alpha}} \n&\\lesssim \\|u\\|_{X_{k,\\alpha}},\\label{Q.X.high}\\\\\n\\langle C_k u,u\\rangle &\\gtrsim 2^{k}\\| u\\|^2_{X_{k,\\alpha}}\n, \\qquad \nC_k = i [ A_{(k)}, Q_k],\n\\label{C.high}\\\\\n2\\Im\\langle[A_{(k)},\\rho_{ 0$ and by $1$ for $k < 0$. This generates error terms\nwhich we need to estimate. \n\n\nIf $k < 0$ then these error terms are estimated\nas follows. We first want to substitute $A$ by $A_{(k)}$, and as such, we see errors of the\nform\n\\begin{equation}\\label{error1}\n |\\langle [A,\\rho]u,\\sum_{k<0} S_k Q_k S_k \\rho u\\rangle|\n+ |\\sum_{k<0} \\langle (A-A_{(k)})\\rho u, S_k Q_k S_k \\rho u\\rangle|.\n\\end{equation}\nFor the first term, we use \\eqref{Q.X.low} (after optimizing in $\\alpha$)\n\\[\n\\begin{split}\n|\\langle [A,\\rho]u, \\sum_{k<0}S_k Q_k S_k \\rho u\\rangle|&\\lesssim\n|\\langle -2iD_i a^{ij}(\\partial_j \\rho) u + \\partial_j((\\partial_i \\rho)a^{ij})u,\\sum_{k<0}S_k Q_k S_k \\rho u\\rangle|\\\\\n&\\lesssim \\|u\\|_{L^2_{t,x}(2^{M}< |x|< 2^{M+1})} \\|\\sum_{k<0}S_k Q_k S_k \\rho u\\|_{L^2_{t,x}(\\{\n2^{M}<|x|<2^{M+1}\\})}\\\\\n&\\lesssim \\|\\langle x\\rangle^{-2} u\\|_{L^2_{t,x}} \\|\\sum_{k<0} S_k Q_k S_k \\rho u\\|_{{X}}\\\\\n&\\lesssim \\|\\langle x\\rangle^{-2} u\\|_{L^2_{t,x}} \\|\\rho u\\|_{{{\\tilde{X}}}}.\n\\end{split}\n\\]\nFor the second term in \\eqref{error1}, we use \\eqref{aak} and \\eqref{Q.X.low} to see that\n\\[\n\\begin{split}\n |\\sum_{k<0} \\langle (A-A_{(k)})\\rho u, S_k Q_k S_k \\rho u\\rangle|&\\lesssim\n\\Bigl(\\sum 2^{-k} \\|S_k(A-A_{(k)})\\rho u\\|^2_{X'_k}\\Bigr)^{\\frac12} \\|\\rho u\\|_{{X}}\\\\\n&\\lesssim \\epsilon \\|\\rho u\\|^2_{{{\\tilde{X}}}}. \n\\end{split}\n\\]\nFor the remaining errors, we use the fact that $A_{(k)}$ preserves\nlocalizations at frequency $2^k$ combined with\n\\eqref{coeffak.greater}, and \\eqref{Q.L2.low} to see that\n\\[\n\\begin{split}\n |\\sum_{k<0} \\langle A_{(k)} (1-\\rho) u, S_k Q_k S_k \\rho u\\rangle|\n&\\lesssim \\|\\langle x\\rangle^{-2} u\\|_{L^2_{t,x}} \\|\\langle x\\rangle^2 \\sum_{k<0} S_k Q_k S_k A_{(k)} (1-\\rho)u\\|_{L^2_{t,x}}\\\\\n&\\lesssim \\|\\langle x\\rangle^{-2} u\\|_{L^2_{t,x}} \\|(1-\\rho)u\\|_{L^2_{t,x}}\n\\end{split}\n\\]\nand respectively,\n\\[\n\\begin{split}\n |\\sum_{k<0} \\langle A_{(k)} u, S_k Q_k S_k (1-\\rho)u\\rangle|&\\lesssim\n\\|\\langle x\\rangle^{-2} u\\|_{L^2_{t,x}}\\|\\langle x\\rangle^2 A_{(k)} \\sum_{k<0}S_k Q_k S_k (1-\\rho)u\\|_{L^2_{t,x}}\\\\\n&\\lesssim \\|\\langle x\\rangle^{-2} u\\|_{L^2_{t,x}} \\|(1-\\rho)u\\|_{L^2_{t,x}}.\n\\end{split}\n\\]\nIn both formulas above the last step is achieved by commuting the\n$x^2$ factor to the right, where it is absorbed by the $(1-\\rho)$\nfactor. The two possible commutators may yield an extra $2^{-2k}$\nfactor, which is compensated for by the two derivatives in $A_{(k)}$.\n\n\nOn the other hand if $k \\geq 0$ then we have the bound\n\\[\n| \\rho - \\rho_{ 0} 2^k \\| S_k \\rho_{ 0} 2^k \\| S_k \\rho_{ < k}\n u\\|_{X_{k}}^2 \\right)\n\\\\ &\\\n- c_2 \\left( \\|\\langle x \\rangle^{-2}u\\|_{L^2_{t,x}}^2 + \n\\epsilon\n\\|\\rho u\\|_{{{\\tilde{X}}}}^2 \\right)\n\\end{split}\n\\]\nwhich for $\\epsilon$ sufficiently small yields part (iii) of the proposition. \n\\end{proof}\n\n\\subsection{The non-resonant low dimensional case $n=1,2$: Proof of\n Theorem~\\ref{main.ls.theorem}}\n\n Almost all the arguments\nin the high dimensional case apply also in low dimension. The only\ndifference arises in part (ii) of Proposition \\ref{Qprop}. Since\nthe multiplication by $\\rho$ is bounded in both ${{\\tilde{X}}}$ and ${{\\tilde{X}}}'$,\nthe property (ii) reduces to proving that\n\\[\n\\sum_{k=-\\infty}^\\infty S_k Q_k S_k : {{\\tilde{X}}} \\to {{\\tilde{X}}}.\n\\]\nIn dimension $n \\geq 3$ the ${{\\tilde{X}}}$ norm is described in terms of the\n$X_k$ norms of its dyadic pieces, and the above property follows\nfrom the $X_k$ boundedness of $Q_k$ at frequency $2^k$. \n\nHowever, in dimension $n=1,2$ the ${{\\tilde{X}}}$ norm also has a weighted $L^2$\ncomponent. The high frequency part $k \\geq 0$ of the above sum causes\nno difficulty, but the low frequency part does. We do know that\n\\[\n\\sum_{k=-\\infty}^0 S_k Q_k S_k : {X} \\to {X}.\n\\]\nTherefore, due to Lemma~\\ref{txdx}, it would remain to prove that\n\\[\n\\Bigl\\| \\sum_{k=-\\infty}^0 S_k Q_k S_k u\\Bigr\\|_{L^2_{t,x}(\\{|x| \\leq 1\\})} \\lesssim\n\\|u\\|_{{{\\tilde{X}}}}.\n\\]\nUnfortunately, the operators $S_k Q_k S_k$ act on the $2^{-k}$ spatial\nscale; therefore without any additional cancellation there is no\nreason to expect a good control of the output in a bounded region. The\naim of the next few paragraphs is to replace the above low frequency\nsum by a closely related expression which exhibits the desired\ncancellation property.\n\nFirst of all, it is convenient to replace the discrete parameter $k$\nby a continuous one $\\sigma$. The operators $S_\\sigma$ are defined in\nthe same way as $S_k$ by scaling. Let $\\phi_k$ be the functions in\nLemma~\\ref{lemma.low.freq}. The functions $\\phi_\\sigma$ are defined\nfrom $\\phi_k$ using a partition of unity on the unit scale in $\\sigma$.\nThe normalization we need is very simple, namely $\\phi_k(0) = 1$,\nwhich leads to $\\phi_{\\sigma}(0) = 1$. The operators $Q_\\sigma$ are\ndefined in a similar way. Then it is natural to substitute\n\\[\n \\sum_{k=-\\infty}^0 S_k Q_k S_k \\to \\int_{-\\infty}^0 S_\\sigma\n Q_\\sigma S_\\sigma d\\sigma\n\\]\nand all the estimates for the second sum carry over identically\nfrom the discrete sum.\n\nHowever, the desired cancellation is still not present in the second\nsum. To obtain that we consider a spherically symmetric Schwartz function \n$\\phi^0$ localized at frequency $\\ll 1$ with $\\phi^0(0) = 1$. \nThen we write $\\phi_\\sigma$ in the form\n\\[\n\\phi_\\sigma(x) = \\phi^0(x) + x^2 \\psi_\\sigma(x).\n\\]\n\nThe modified self-adjoint operators ${{\\tilde{Q}}}_\\sigma$ are defined as \n\\[\n{{\\tilde{Q}}}_\\sigma = S_\\sigma Q_{\\sigma,\\phi^0} S_\\sigma + 2^{2\\sigma} \\delta^2 x S_\\sigma\nQ_{\\sigma,\\psi_\\sigma} S_\\sigma x\n\\]\nwhere, as in Lemma~\\ref{lemma.low.freq}, we set\n\\[\nQ_{\\sigma,\\phi} = \\delta(Dx\\phi( 2^\\sigma \\delta |x|)+\\phi(2^\\sigma \\delta |x|)xD).\n\\]\n\nWe claim that the conclusion of Proposition~\\ref{Qprop} is valid with\nthe operator $Q$ defined as \n\\begin{equation}\nQ_\\rho = \\rho Q \\rho, \\qquad Q = \\int_{-\\infty}^0 {{\\tilde{Q}}}_\\sigma d\\sigma + \\sum_{k=0}^\\infty S_k Q_k S_k. \n\\label{qlowd}\n\\end{equation}\nThe family $\\mathcal Q$ is obtained as before by allowing the\nchoice of the functions $\\phi_k$ to depend on the slowly varying\nsequences $(\\alpha^\\sigma_{j})_{j \\in {\\mathbb N}}$ which are chosen\nindependently\\footnote{In effect, without any restriction in generality,\n one may also assume that $\\alpha^\\sigma_{j}$ is also slowly varying\n with respect to $\\sigma$} for different $k$.\n\nThere is no change in part (i) of Proposition~\\ref{Qprop}. For part\n(ii) we need to prove that\n\\begin{equation} \\label{qtx}\n\\left\\| Q u \\right \\|_{{{\\tilde{X}}}} \\lesssim \\|u\\|_{{{\\tilde{X}}}}.\n\\end{equation}\nThe high frequencies are estimated directly from the ${X}$ norm;\ntherefore we have to consider the integral term in $Q$ and show that\n\\[\n\\left\\| \\int_{-\\infty}^0 {{\\tilde{Q}}}_\\sigma u\\, d\\sigma\\right \\|_{{{\\tilde{X}}}} \\lesssim \\|u\\|_{{{\\tilde{X}}}}.\n\\]\n\nThe ${X}$ component of the ${{\\tilde{X}}}$ norm is easily estimated by Littlewood-Paley\ntheory, so due to Lemma~\\ref{txdx}, it would remain to prove the local\n$L^2$ bound\n\\begin{equation}\\label{quo}\n\\left\\| \\int_{-\\infty}^0 {{\\tilde{Q}}}_\\sigma u d\\sigma\\right \\|_{L^2_{t,x}(\\{|x| \\leq 1\\})} \\lesssim\n\\|u\\|_{{{\\tilde{X}}}}.\n\\end{equation}\n\nWe can neglect the time variable in the sequel.\nWe have the $L^2$ bound\n\\[\n\\| {{\\tilde{Q}}}_{\\sigma} u\\|_{X_\\sigma} \\lesssim\n \\| S_\\sigma u\\|_{X_\\sigma}\n\\]\nwhich leads to \n\\[\n\\| \\nabla {{\\tilde{Q}}}_{\\sigma} u\\|_{X_\\sigma} \\lesssim\n2^{\\sigma} \\| S_\\sigma u\\|_{X_\\sigma}\n\\]\nand the corresponding pointwise bound\n\\[\n\\| \\nabla {{\\tilde{Q}}}_{\\sigma} u\\|_{L^\\infty (A_{<-\\sigma})} \\lesssim\n2^{\\frac{n+1}2 \\sigma } \\| S_\\sigma u\\|_{X_\\sigma}\n\\]\nwhich establishes the convergence and the bound for the corresponding integral\n\\[\n\\left\\| \\int_{-\\infty}^0 \\nabla {{\\tilde{Q}}}_{\\sigma} u\\,d\\sigma \\right\\|_{L^\\infty(A_{<0})} \\lesssim\n\\| S_{\\leq 0} u\\|_{{X}}.\n\\]\nHence in order to prove \\eqref{quo} it remains to establish a similar\nbound for the integral at $x=0$. Assume first that $u \\in L^2$, which\narguing as above guarantees the uniform convergence of the integral.\nDenoting by $K_\\sigma$ the kernel of $S_\\sigma$ we have\n\\[\n\\begin{split}\n({{\\tilde{Q}}}_{\\sigma} u)(0) = &\\ (S_\\sigma Q_{\\sigma,\\phi^0} S_\\sigma) u(0) \n\\\\\n= &\\ \\langle K_\\sigma, Q_{\\sigma,\\phi^0} S_\\sigma u \\rangle\n= \\langle Q_{\\sigma,\\phi^0} K_\\sigma, S_\\sigma u \\rangle \n\\\\ \n= & \\ \\int Q_{\\sigma,\\phi^0}(x,D_x) K_\\sigma(x) \\int K_\\sigma(x-y) u(y) dy\ndx\n\\\\ =&\\ (S_\\sigma^1 u)(0)\n\\end{split}\n\\]\nwhere $S_\\sigma^1$ is the frequency localized multiplier with\nspherically symmetric Schwartz kernel\n\\[\nK_\\sigma^1 = Q_{\\sigma,\\phi^0}(x,D_x) K_\\sigma * K_\\sigma.\n\\]\nDue to the frequency localization we can define\n\\[\nS_{<0}^1 = \\int_{-\\infty}^{0} S_\\sigma^1 d \\sigma.\n\\]\nThe punch line is that by construction the operators $S_\\sigma^1$ have\nthe same kernel up to the appropriate rescaling. This implies that the\nsymbols of $S_{<0}^1$ are constant for $|\\xi| \\leq\n2^{-4}$. Hence both the symbols and the kernels\n$K_{<0}^1$ of $S_{<0}^1$ are Schwartz functions which\ncoincide modulo rescaling. Hence for all functions $u \\in L^2$ we have\n\\[\n\\int_{-\\infty}^0 {{\\tilde{Q}}}_{\\sigma} u (0) d\\sigma = \\langle K_{<0}^1, u\\rangle\n\\]\nwhich leads to the estimate\n\\[\n\\left | \\int_{-\\infty}^0 ({{\\tilde{Q}}}_\\sigma u)(0) d\\sigma\\right| \\lesssim\n\\|u\\|_{{{\\tilde{X}}}}.\n\\]\nThis completes the proof of the estimate \\eqref{quo} for all $u \\in\nL^2$, and, by density, shows that the integral\n\\[\n\\int_{-\\infty}^0 {{\\tilde{Q}}}_\\sigma d\\sigma\n\\]\nhas a unique bounded extension to ${{\\tilde{X}}}$.\n\nIt remains to prove part (iii) of Proposition~\\ref{Qprop}. If\n${{\\tilde{Q}}}_\\sigma$ is replaced by $S_\\sigma Q_\\sigma S_\\sigma$ then the high\ndimensional argument applies by simply replacing sums with integrals.\nHence it remains to estimate the difference. Commuting we obtain\n\\[\n{{\\tilde{Q}}}_\\sigma - S_\\sigma Q_\\sigma S_\\sigma = i \\delta^2 2^{2\\sigma} \\left( S'_\\sigma\nQ_{\\sigma,\\psi} x S_\\sigma - S_\\sigma x Q_{\\sigma,\\psi} \nS'_\\sigma - S'_\\sigma Q_{\\sigma,\\psi} S'_\\sigma (D) \\right).\n\\]\nCommuting again to take advantage of the cancellation between the\nfirst two terms, by semiclassical pdo calculus we can write\n\\[\n{{\\tilde{Q}}}_\\sigma - S_\\sigma Q_\\sigma S_\\sigma = \\delta^2 R_\\sigma (2^\\sigma\n\\delta x, 2^{-\\sigma} D)\n\\]\nwhere the symbol $r_\\sigma(y,\\eta)$ is localized in $\\{ |\\eta|\n\\approx 1\\}$ and satisfies\n\\[\n|\\partial_y^\\alpha \\partial_\\eta^\\beta r_\\sigma(y,\\eta)| \\leq\nc_{\\alpha \\beta} \\langle y \\rangle^{-2}.\n\\]\nThis implies the bound\n\\[\n\\| ({{\\tilde{Q}}}_\\sigma - S_\\sigma Q_\\sigma S_\\sigma) u \\|_{X'_\\sigma} \\lesssim\n\\delta^2 2^{-\\sigma} \\|S_\\sigma u\\|_{X_\\sigma}. \n\\]\nTherefore without any commuting we obtain\n\\[\n| \\langle [{{\\tilde{Q}}}_\\sigma - S_\\sigma Q_\\sigma S_\\sigma,A_{(\\sigma)}] u,u\\rangle |\n\\lesssim \\delta^2 \\| u \\|_{{X}}^2.\n\\]\nThis error is negligible since, as one can note in the proofs of\nLemmas~\\ref{lemma.low.freq}, \\ref{lemma.high.freq},\n the constant $c_1$ in (iii) has size $c_1 = O(\\delta)$.\n\n\n\n\n\n\n\n \\subsection{The resonant low dimensional case $n=1,2$: Proof of\n \\ref{main.ls.theorem.res}}\n\nThe proof follows the same outline as in the non-resonant case, with minor\nmodifications. The energy estimate \\eqref{eest} is now replaced\nby \n\\begin{equation}\n\\| u\\|_{L^\\infty_t L^2_x}^2 \\lesssim \\| u_0\\|_{L^2}^2 + \\| f_1\\|_{L^1_t L^2_x}^2\n+ \\| u\\|_{{X}_e} \\|f_2\\|_{{X}'_e}. \n\\label{eest.res}\\end{equation}\nInstead of the exterior smoothing estimate \\eqref{lsest},\nwe need to prove\n\\begin{equation}\n \\| T_\\rho u\\|_{{X}}^2 \\lesssim \\| u\\|_{L^\\infty_t L^2_x}^2 + \n \\| f_1\\|_{L^1_t L^2_x}^2 \n + \\|T_\\rho f_2\\|_{{X}'}^2 + \\| \\langle x \\rangle^{-2} (u-u_\\rho) \\|_{L^2_{t,x}}^2.\n\\label{lsest.res}\\end{equation}\nThe estimate \\eqref{lsext.res} then follows from the previous two estimates as\nwell as \\eqref{dXerror}.\n\nThe lower order terms will still be negligible. Indeed, letting\n$B=2b^i D_i$, we have\n\\[\n T_\\rho Bu = B T_\\rho u - (B \\rho)(u-u_\\rho) + (1-\\rho) \\left( \\int\n (1-\\rho) dx\\right)^{-1} \\int (B\\rho)(u-u_\\rho) dx.\n\\]\nTherefore by \\eqref{bnoc}, we obtain\n\\[\n\\| T_\\rho Bu\\|_{{X}'} \\lesssim \\epsilon \\|u\\|_{{X}_e},\n\\]\nwhich combined with the ${X}$ boundedness of our multiplier below shows that the lower order\nterms can be neglected.\n\nThe estimate \\eqref{lsest.res} follows from \n\n\\begin{proposition} \\label{Qprop.res}\nThere is a family $\\mathcal Q_{res}$ of bounded self-adjoint operators $Q_{res}$ with \nthe following properties:\n\n(i) $L^2$ boundedness,\n\\[\n\\|Q_{res}\\|_{L^2 \\to L^2} \\lesssim 1,\n\\]\n\n(ii) ${X}$ boundedness,\n\\[\n|\\langle Q_{res} u, f \\rangle | \\lesssim \\| T_\\rho f\\|_{{X}'} \\| T_\\rho u \\|_{{X}},\n\\]\n\n(iii) Positive commutator,\n\\[\n\\sup_{Q_{res} \\in \\mathcal Q_{res}} \\langle Cu,u \\rangle \\geq c_1 \\|T_\\rho u\\|_{{X}}^2 -\nc_2 \\|\\langle x\\rangle^{-2} (u-u_\\rho) \\|_{L^2_{t,x}}^2.\n\\]\n\\end{proposition}\n\n\\begin{proof}\n We construct $Q_{res}$ as in the non-resonant case but with the\n modified truncation operator\n\\[\nQ_{res} u = T_\\rho Q T_\\rho.\n\\]\nwith $Q$ given by \\eqref{qlowd}.\n\nThe properties (i) and (ii) are straightforward.\nFor (iii) we note that \n\\[\nS_k T_\\rho u = S_k \\rho(u-u_\\rho)\n\\]\nwhile\n\\[\n T_\\rho A u = \\rho Au + c (1-\\rho) \\int (1-\\rho) A (u-u_\\rho) dx = \n \\rho A(u-u_\\rho) - c (1-\\rho) \\int (u-u_\\rho) A \\rho dx.\n\\]\nHence we can express the bilinear form $\\langle Au, Q_{res} u \\rangle$ in\nterms of the operator $Q_\\rho$ in the nonresonant case\n\\[\n\\langle Au, Q_{res} u \\rangle = \\langle A(u-u_\\rho), Q_\\rho (u-u_\\rho) \\rangle - c \\int\n(u-u_\\rho) A \\rho dx \\ \\langle (1-\\rho), Q T_\\rho u \\rangle\n\\]\nwhich implies that\n\\[\n\\langle C_{res} u,u\\rangle = \\langle C (u-u_\\rho),u-u_\\rho\\rangle + c \\Im \\int\n(u-u_\\rho) A \\rho dx\\ \\langle (1-\\rho), Q T_\\rho u \\rangle.\n\\]\nHence we can apply part (iii) of Proposition~\\ref{Qprop} and\n\\eqref{qtx} to obtain the desired conclusion.\n\\end{proof}\n\n\n\n\\subsection{Non-trapping metrics: Proof of Theorem~\\ref{main.ls.theoremnt}.}\nThis requires some modifications of the previous argument. First of\nall, instead of the energy estimate \\eqref{eest}, we need a\nstraightforward modification of it, namely\n\\begin{equation}\n\\| u\\|_{L^\\infty_t L^2_x}^2 \\lesssim \\| u_0\\|_{L^2}^2 + \\| f_1\\|_{L^1_t L^2_x}^2\n+ \\| u\\|_{{{\\tilde{X}}}} \\|f_2\\|_{{{\\tilde{X}}}'}.\n\\label{eestnt}\\end{equation}\nWe still need the exterior local smoothing estimate \\eqref{lsest}.\nHowever, now we can complement it with an interior estimate,\nnamely\n\\begin{equation}\n\\| (1-\\rho) u\\|_{{{\\tilde{X}}}}^2 \\lesssim \\| u\\|_{L^\\infty_t L^2_x}^2 + \n\\| f_1\\|_{L^1_t L^2_x}^2 + \\| \\rho u\\|_{{{\\tilde{X}}}}^2\n+ \\|(1-\\rho) f_2\\|_{{{\\tilde{X}}}'}^2 + \\| (1-\\rho) u\\|_{L^2_{t,x}}^2. \n\\label{lsestnt}\\end{equation}\nThe conclusion of Theorem~\\ref{main.ls.theoremnt} is obtained by combining \nthe three estimates \\eqref{eestnt}, \\eqref{lsest} and \\eqref{lsestnt}.\n\nIt remains to prove \\eqref{lsestnt}. This is obtained by applying \nto the function $v = (1-\\rho) u$ the local bound\n\n\\begin{proposition}\n Assume that the coefficients $a^{ij}$, $b^i$, $c$ are real and satisfy \\eqref{coeff}, \\eqref{coeffb},\n and \\eqref{coeffc}. Moreover, assume that\n the metric $a^{ij}$ is non-trapping. Let $v$ be a\n function supported in $\\{|x| \\leq 2^{M+1}\\}$ which solves the\nequation\n\\begin{equation}\n(D_t +A) v = g_1 + g_2, \\qquad v(0) = v_0\n\\end{equation}\nin the time interval $[0,T]$. \nThen we have \n\\begin{equation}\n\\| v \\|_{L^2_t H^{\\frac12}_x}^2 \\lesssim \\| v\\|_{L^\\infty_t L^2_x}^2 + \n\\| g_1\\|_{L^1_t L^2_x}^2 + \\|g_2\\|_{L^2_t H^{-\\frac12}_x}^2 + \\| v\\|_{L^2_{t,x}}^2. \n\\label{leint}\\end{equation}\n\\end{proposition}\n\n\\begin{proof}\n We use again the multiplier method. The following lemma tells us\n how to choose an appropriate multiplier.\n\\begin{proposition}\\label{Doi_construction}\n Assume that the coefficients $a^{ij}$ satisfy \\eqref{coeff}.\n Moreover, we assume that the Hamiltonian vector field $H_a$ permits\n no trapped geodesics. Then there exists a smooth, time-independent, real-valued symbol $q\\in S^0_{hom}$\n so that\n\\[\nH_a q\\gtrsim |\\xi|,\\quad \\text{ in } \\{|x| \\leq 2^{M+1}\\}.\n\\]\n\\end{proposition}\n\nThis proposition is essentially from \\cite{D1}, if $a^{ij}$ were smooth. See also \nLemma 1 of \\cite{ST}, which includes some discussion of the limited regularity. \n\nWorking in the Weyl calculus and using this multiplier $Q$, we compute\n\\[\n\\frac{d}{dt}\\langle v,Qv\\rangle = -2\\Im\\langle (D_t+A)v,Qv\\rangle\n+ i\\langle [A,Q]v,v\\rangle \n\\]\nwhich after time integration yields\n\\[\n\\langle i [A,Q]v,v\\rangle = \\langle v,Qv \\rangle |^T_0 + 2\\Im\\langle g_1+g_2,Qv\\rangle. \n\\]\nFor the second term on the right, we apply Cauchy-Schwarz and use the\n$L^2$ and $H^\\frac12$ boundedness of $Q$ to obtain\n\\[\n|\\langle (D_t+A)v,Qv\\rangle| \\lesssim \\|v\\|_{L^\\infty_t L^2_x}^2 + \\|g_1\\|_{L^1_t\n L^2_x}^2 + \\|g_2\\|_{L^2_t H_x^{-\\frac12}} \\| v\\|_{L^2_t H_x^\\frac12}.\n\\]\nHence\n\\[\n\\langle i [A,Q]v,v\\rangle \\lesssim \n\\|v\\|_{L^\\infty_t L^2_x}^2 + \\|g_1\\|_{L^1_t\n L^2_x}^2 + \\|g_2\\|_{L^2_t H^{-\\frac12}_x} \\| v\\|_{L^2_t H^\\frac12_x}.\n\\]\nThen it remains to prove the positive commutator bound\n\\begin{equation}\n\\langle i [A,Q]v,v\\rangle \\geq c_1 \\|v\\|_{L^2_t H^\\frac12_x}^2 - c_2 \\|v\\|_{L^2_{t,x}}^2.\n\\end{equation}\nThe positive contribution comes from the second order terms in $P$.\nPrecisely, we have\n\\[\ni[ D_i a^{ij} D_j,Q(x,D)] = Op(H_a q) +O(1)_{L^2 \\to L^2}.\n\\]\nThe first symbol is positive, and we can obtain a bound from below by\nG\\aa rding's inequality. The first order term yields an $L^2$ bounded commutator, and the zero\norder term is $L^2$ bounded by itself.\n \nHere, we remind the reader that\nwe are not working with classical smooth symbols but instead with\nsymbols of limited regularity, and we refer the interested reader to\nthe discussion in Taylor \\cite[p. 45]{TaylorIII} for further details\non these otherwise classical results.\n\\end{proof}\n\n\\subsection{Non-trapping metrics: Proof of Theorem~\\ref{main.ls.theoremnt.res}.}\n\nThe argument is similar to the above one, with some obvious\nmodifications. Instead of \\eqref{eestnt} we have\n\\begin{equation}\n\\| u\\|_{L^\\infty_t L^2_x}^2 \\lesssim \\| u_0\\|_{L^2}^2 + \\| f_1\\|_{L^1_t L^2_x}^2\n+ \\| u\\|_{{X}} \\|f_2\\|_{{X}'} \n\\label{eestnt.res}\\end{equation}\nwhile \\eqref{lsestnt} is replaced by \n\\begin{multline}\n\\| (1-\\rho) (u-u_\\rho)\\|_{{X}}^2 \\lesssim \\| u\\|_{L^\\infty_t L^2_x}^2 + \n\\| f_1\\|_{L^1_t L^2_x}^2 + \\| \\rho (u-u_\\rho)\\|_{{X}}^2\n+ \\|f_2\\|_{{X}'}^2 \\\\+ \\| \\langle x\\rangle^{-2} (u-u_\\rho)\\|_{L^2_{t,x}}^2. \n\\label{lsestnt.res}\\end{multline}\n\nThe conclusion of Theorem~\\ref{main.ls.theoremnt.res} is obtained by combining \nthe estimates \\eqref{eestnt.res}, \\eqref{lsest.res} and \\eqref{lsestnt.res} and\napplying \\eqref{dXerror} to reduce the error terms to the form presented in \\eqref{lsntres}.\n\nIt remains to prove \\eqref{lsestnt.res}. We first compute\n\\[\\begin{split}\nD_t u_\\rho &= \\Bigl(\\int (1-\\rho)\\:dx\\Bigr)^{-1}\\Bigl[\\langle (D_t+A)u, (1-\\rho)\\rangle\n- \\langle Au, (1-\\rho)\\rangle\\Bigr]\\\\\n&= \\Bigl(\\int(1-\\rho)\\:dx\\Bigr)^{-1}\\Bigl[\\langle f_1+f_2,(1-\\rho)\\rangle - \\langle u-u_\\rho,A(1-\\rho)\\rangle\\Bigr].\n\\end{split}\n\\]\nThe function $v = (1-\\rho) (u-u_\\rho)$ solves\n\\begin{multline*}\nP v = (1-\\rho)(f_1+f_2) -(1-\\rho)\\Bigl(\\int(1-\\rho)\\:dx\\Bigr)^{-1}\\Bigl[\\langle f_1+f_2,(1-\\rho)\\rangle\n-\\langle u-u_\\rho, A(1-\\rho)\\rangle\\Bigr]\n\\\\+[A,(1-\\rho)](u-u_\\rho).\n\\end{multline*}\nThen we apply \\eqref{leint} to $v$ to obtain\n\\[\n\\begin{split}\n \\| v\\|_{L^2_t H^\\frac12_x}^2 &\\lesssim \\|v\\|_{L^\\infty_t L^2_x}^2 + \\|\n (1-\\rho) f_1 \\|_{L^1_t L^2_x}^2 +\\| [A,(1-\\rho)] (u-u_\\rho)\\|_{L^2_t\n H^{-\\frac12}_x} \\\\ &\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad + \\| (1-\\rho) f_2 \\|_{L^2_t H^{-\\frac12}_x}^2 \n + \\| \\langle x\\rangle^{-2}(u- u_\\rho)\\|_{L^2_{t,x}}^2 \\\\ & \\\n \\lesssim \\|u\\|_{L^\\infty_t L^2_x}^2 + \\| f_1 \\|_{L^1_t L^2_x}^2 +\n \\|\\rho(u-u_\\rho)\\|_{{X}}^2 + \\| f_2 \\|_{{X}'}^2 \n+ \\| \\langle x\\rangle^{-2}(u- u_\\rho)\\|_{L^2_{t,x}}^2\n\\end{split}\n\\]\nand \\eqref{lsestnt.res} follows.\n\n\n\n\\bigskip\n\\newsection{Time independent nontrapping metrics}\n\n\nThe aim of this section is to prove\nTheorems~\\ref{theorem.PcSmoothing},\\ref{theorem.PcSmoothing.res}.\nThus we work with a nontrapping, self-adjoint operator $A$ whose\ncoefficients are time independent. We prove\nTheorem~\\ref{theorem.PcSmoothing} in detail, and then outline the\nmodifications which are needed for\nTheorem~\\ref{theorem.PcSmoothing.res}.\n\n\n\\subsection{Proof of Theorem~\\ref{theorem.PcSmoothing}} Here we shall provide the\ndetails for the $n\\neq 2$ case. The general case follows with the obvious logarithmic adjustments\nto the ${{\\tilde{X}}}$ spaces in $n=2$.\n\nWe break the proof into steps.\n\n \\par{\\bf Step 1:} Without any restriction, we assume that $u_0=0$\n and that $u$ is the forward solution to \\eqref{main.equation}.\n Nonzero initial data $u_0$ can be easily added in via a $TT^*$\n argument.\n\n \\par{\\bf Step 2:} We add a damping term to the equation\n\\[\n(D_t+A-i\\varepsilon)u_\\varepsilon = f\n\\]\nin order to insure global square integrability of the solution\n$u_\\epsilon$. Applying our nontrapping estimate \\eqref{lsnt} we have\n\\begin{equation}\n\\|u_\\varepsilon\\|_{{\\tilde{X}}}\\lesssim\n\\|f\\|_{{{\\tilde{X}}}'}+\\|u_\\varepsilon\\|_{L^2_{t,x}({\\mathbb R}\\times B(0,2R))}.\n\\label{veps}\\end{equation}\nWe want to eliminate the second term on the right (when we add $P_c$\non the left).\n\n \\par{\\bf Step 3:} \n We want to take a Fourier transform in time and use Plancherel's\n theorem. For this we need to work with Hilbert spaces. These are\n defined using the structure introduced in the previous section. We\n denote by $\\alpha$ a family of positive sequences $(\\alpha(k)_j)_{j\n \\geq k^-}$ which have sum $1$ for each $k$ and by $\\mathcal\n A$ the collection of such sequences. For $\\alpha \\in \\mathcal A$ we\n define the Hilbert space ${{\\tilde{X}}}_\\alpha$ with norm\n\\[\n\\| u\\|_{{{\\tilde{X}}}_\\alpha}^2 = \\sum_{k} 2^k\\|S_k u\\|_{X_{k,\\alpha(k)}}^2 + \\|\\langle\nx\\rangle^{-1} u\\|_{L^2_{t,x}}^2\n\\]\nas well as its dual ${{\\tilde{X}}}_\\alpha'$. \nSince\n\\[\n\\| u\\|_{{{\\tilde{X}}}} \\approx \\sup_{\\alpha \\in \\mathcal A} \\| u\\|_{{{\\tilde{X}}}_\\alpha},\n\\qquad\n\\| u\\|_{{{\\tilde{X}}}'} \\approx \\inf_{\\alpha \\in \\mathcal A} \\| u\\|_{{{\\tilde{X}}}'_\\alpha}\n\\]\nwe can rewrite \\eqref{veps} in the equivalent form\n\\[\n\\|u_{\\varepsilon}\\|_{{{\\tilde{X}}}_\\alpha}\\lesssim \\|f\\|_{{{\\tilde{X}}}'_\\beta}+\n\\|u_\\varepsilon\\|_{L^2_{t,x}({\\mathbb R}\\times B(0,2R))}, \\qquad \\alpha, \\beta\n\\in \\mathcal A.\n\\]\nWe denote by $X^0_\\alpha$ the spatial version of $X_\\alpha$, i.e.\n$X_\\alpha = L^2_t X_\\alpha^0$. Then we take a time Fourier transform,\nand by Plunderer this is equivalent to\n\\[\n\\|\\hat{u}_\\varepsilon\\|_{L^2_\\tau {{\\tilde{X}}}_\\alpha^0}\\lesssim \n\\|\\hat{f}\\|_{L^2_\\tau ({{\\tilde{X}}}^0_\\beta)'} + \\|\\hat{u}_\\varepsilon\n\\|_{L^2_{\\tau,x}({\\mathbb R}\\times B(0,2R))}.\n\\]\nThis is in turn equivalent to the fixed $\\tau$ bound\n\\[\n\\|\\hat{u}_\\varepsilon(\\tau)\\|_{{{\\tilde{X}}}_\\alpha^0}\\lesssim\n\\|\\hat{f}(\\tau)\\|_{({{\\tilde{X}}}_\\beta^0)'}\n+\\|\\hat{u}_\\varepsilon(\\tau)\\|_{L^2(B(0,2R))},\n\\]\nwhich we rewrite in the form\n\\[\n\\|v \\|_{{{\\tilde{X}}}^0_\\alpha}\\lesssim\n\\|(A-\\tau-i\\varepsilon) v \\|_{({{\\tilde{X}}}_\\beta^0)'}+\\| v \\|_{L^2(B(0,2R))},\n\\]\nor, optimizing with respect to $\\alpha,\\beta \\in \\mathcal A$,\n\\begin{equation}\\label{have}\n\\|v \\|_{{{\\tilde{X}}}^0}\\lesssim\n\\| (A-\\tau-i\\varepsilon)v \\|_{({{\\tilde{X}}}^0)'}+\\| v \\|_{L^2(B(0,2R))}.\n\\end{equation}\nA similar\ncomputation shows that the estimate that we want to prove, namely \n\\eqref{PcSmoothing} with $u_0=0$, can be rewritten in the equivalent\nform \n\\begin{equation}\\label{want}\n \\|P_c v \\|_{{{\\tilde{X}}}^0}\\lesssim \\| (A-\\tau-i\\epsilon) v \\|_{({{\\tilde{X}}}^0)'}\n\\end{equation}\nuniformly with respect to $\\tau \\in {\\mathbb R}$, $\\epsilon > 0$.\n\n\n \\par{\\bf Step 4:} When $|\\tau|$ is large, \\eqref{want}\nfollows from \\eqref{have} combined with the elliptic bound\n\\begin{equation}\\label{ellip}\n\\tau^{1\/4} \\|v\\|_{L^2(B(0,2R))}\\lesssim \\|v\\|_{{{\\tilde{X}}}^0}+\\|(A-\\tau-i\\epsilon)v\\|_{({{\\tilde{X}}}^0)'}.\\end{equation}\nTo prove this we replace $v$ by $w = (1-\\rho) v$ and rewrite \nit in the form\n\\[\n\\tau^{1\/4} \\|w\\|_{L^2}\\lesssim \n\\|w\\|_{H^\\frac12}+\\|(A-\\tau-i\\epsilon)w\\|_{H^{-\\frac12}}\n\\]\nfor $w$ with compact support. Since\n\\[\n\\tau \\| w \\|_{H^{-\\frac32}} \\lesssim \\|(A-\\tau-i\\epsilon)w\\|_{H^{-\\frac32}}\n+ \\| A w\\|_{H^{-\\frac32}} \\lesssim \\|(A-\\tau-i\\epsilon)w\\|_{H^{-\\frac12}}\n+ \\| w\\|_{H^{\\frac12}},\n\\]\nthe bound \\eqref{ellip} follows by interpolation.\n\\par{\\bf Step 5:}\nFor $\\tau$ in a bounded set we argue by contradiction.\nIf \\eqref{want} does not hold uniformly then we find sequences \n\\[\n\\varepsilon_n\\to 0, \\qquad \\tau_n\\to \\tau, \n\\]\nand $ v_n \\in {{\\tilde{X}}}^0$ with $P_c v_n=v_n$ and\n\\[\n\\|(A-\\tau_n-i\\varepsilon_n)v_n\\|_{({{\\tilde{X}}}^0)'}\\to 0,\\quad\n\\|v_n\\|_{L^2(B(0,2R))}=1.\n\\]\nOn a subsequence we have\n\\[\nv_n\\to v\\quad \\text{weakly* in} \\quad {{\\tilde{X}}}^0.\n\\]\nSince ${{\\tilde{X}}}^0 \\subset H^{\\frac12}_{loc}$, on a subsequence\nwe have the strong convergence\n\\[\nv_n \\to v \\qquad \\text{in } L^2_{loc}.\n\\]\nHence we have produced a function $v $ with \n\\begin{equation}\n v \\in {{\\tilde{X}}}^0, \\qquad P_c v = v, \\qquad (A-\\tau) v = 0, \n\\qquad \\| v\\|_{L^2(B(0,2R))}=1.\n\\label{vprop}\\end{equation}\nDepending on the sign of $\\tau$ we consider three cases.\n\n\\par{\\bf Step 6:}\nIf $\\tau < 0$ then, using the bound \\eqref{bc} for the lower order terms\nin $A$, we obtain\n\\[\n\\| D_i a^{ij} D_j v -\\tau v\\|_{({{\\tilde{X}}}^0)'} \\lesssim \\| v\\|_{{{\\tilde{X}}}^0}.\n\\]\nThen\n\\[\n\\| v\\|_{{{\\tilde{X}}}^0}^2 \\gtrsim \\langle v, D_i a^{ij} D_j v -\\tau v \\rangle \n\\gtrsim \\|v\\|_{H^1}^2 ,\n\\]\nand therefore $v \\in L^2$ is an eigenfunction. This contradicts \nthe relation $P_c v = v$.\n\n\\par{\\bf Step 7:} \nIf $\\tau = 0$ then there is either a zero eigenvalue or a zero\nresonance, both of which are excluded by hypothesis.\n\n\\par{\\bf Step 8:} \nIt remains to consider the most difficult case $\\tau > 0$. Here \nthe properties \\eqref{vprop} of $v$ are no longer sufficient \nto obtain a contradiction. Instead we will establish an additional\nproperty of $v$, namely that $v$ satisfies an outgoing radiation\ncondition. In order to state this, we need an additional \nregularity property for $v$. We define the space ${{\\tilde{X}}}^{0}_{med}$\nwith norm\n\\[\n\\| v \\|_{{{\\tilde{X}}}^{0}_{med}} = \\| v\\|_{L^2(D_{0})} + \\|\\nabla\nv\\|_{L^2(D_{ 0})} + \\sup_{j > 0} \\| |x|^{-\\frac12} v\\|_{L^2(D_j)} +\n \\| |x|^{-\\frac12} \\nabla v\\|_{L^2(D_j)}\n\\]\nwhich coincides with the ${{\\tilde{X}}}^0$ norm for intermediate frequencies but\nimproves it at both low and high frequencies. Then we claim that $v\n\\in {{\\tilde{X}}}^0_{med}$. More precisely, we will prove the elliptic bound\n\\begin{equation}\n\\| v \\|_{{{\\tilde{X}}}^{0}_{med}} \\lesssim \\|v\\|_{{{\\tilde{X}}}^0} +\n \\|(A-\\tau-i\\epsilon)v\\|_{({{\\tilde{X}}}^0)'}, \\qquad 0 < \\tau_0 < \\tau < \\tau_1 \n\\label{vregb} \\end{equation}\nwith implicit constants which may depend on the thresholds\n$\\tau_0$, $\\tau_1$. \n\nNow we define the closed subspace ${{\\tilde{X}}}^0_{out}$ of ${{\\tilde{X}}}^0$,\n\\[\n{{\\tilde{X}}}^0_{out}=\\{v\\in {{\\tilde{X}}}^0_{med}\\,:\\, \\lim_{j\\to\\infty} \\|r^{-1\/2}(\\partial_r\n-i\\tau^{1\/2})v\\|_{L^2(D_j)}=0\\},\n\\]\nand also claim that $v$ has the additional property\n\\begin{equation}\nv \\in {{\\tilde{X}}}^0_{out}.\n\\label{vout} \\end{equation}\nIn other words this implies that $v$ is a resonance contained \ninside the continuous spectrum.\n\nWe postpone the proof of \\eqref{vregb} and \\eqref{vout} \nand conclude first our proof by contradiction, by showing\nthat there are no resonances inside the continuous spectrum.\nSuch results are known, see for instance \\cite{Agmon}, but perhaps\nnot in the degree of generality we need here. In any case,\nfor the sake of completeness, we provide a full proof. \n\nLet $\\chi$ be a smooth spherically symmetric increasing bump function\n$\\chi$ with $\\chi(r)\\equiv 0$ for $r<1\/2$ and $\\chi(r) \\equiv 1$\nfor $r>2$. Since $A$ is self-adjoint, for large $j$ we commute\n\\[\n\\begin{split}\n0=&\\ \\frac{i}{2}\\langle [A,\\chi(2^{-j}r)]v,v\\rangle \\\\\n = &\\ \\Im\\left\\langle 2^{-j} \\chi'(2^{-j} r)\n\\Bigl(\\frac{x_ia^{ij}}{r}\\partial_j -i\\tau^{1\/2}\\Bigr)v,v\\right\\rangle\n+2^{-j} \\tau^{1\/2}\\langle \\chi'(2^{-j}r) v,v\\rangle\n\\\\&\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\n+2^{-j}\\Bigl\\langle b^i\\frac{x_i}{r} \\chi'(2^{-j}r) v, v\\Bigr\\rangle .\n\\end{split}\n\\]\nUsing the Schwarz inequality, \\eqref{coeffb.M}, and the outgoing radiation condition, we\nconclude that\n\\begin{equation}\\label{initial.decay}\n\\lim_{j\\to\\infty} \\|r^{-1\/2} v\\|_{L^2(D_j)}=0\n\\end{equation}\nwhich shows that $v$ has better decay at infinity. We note that \nthis is the only use we make of the radiation condition. From this, by\nelliptic theory, we also obtain a similar decay for the gradient,\n\\begin{equation}\\label{dinitial.decay}\n\\lim_{j\\to\\infty} \\|r^{-1\/2} \\nabla v\\|_{L^2(D_j)}=0.\n\\end{equation}\n\nTo conclude we use \\eqref{initial.decay} and \\eqref{dinitial.decay} to\nshow that in effect $v \\in L^2$; i.e. $v$ is an eigenvalue. Then by\nthe results of \\cite{KT2} $v$ must be $0$. Here, we shall again use a\npositive commutator argument. The multiplier we use is the operator\n$Q_k$, for some $k\\le 0$, in Lemma~\\ref{lemma.low.freq} but where for simplicity we set\n$\\delta = 1$. We have\n\\[\n0=-2\\Im \\langle Q_k v, (A-\\tau)v\\rangle = \\langle C_k v,v\\rangle - 2 \n\\Im \\langle Q_k v, (b^j D_j + D_j b^j + c)v\\rangle\n\\]\nwhere\n\\[\n C_k = i [D_l a^{lm} D_m,Q_k].\n\\]\nThe expression of the operator $C_k$ is exactly as in the \nformula \\eqref{C} but with unmollified coefficients $a^{ij}$.\nThe main contribution $C_k^0$ is estimated as there\nby\n\\[\n\\langle C_k^0 v,v\\rangle \\gtrsim \\left\\langle \\frac{\\alpha(2^k |x|)}{\\langle 2^k\n x\\rangle} \\nabla v,\\nabla v \\right\\rangle, \n\\]\nwhile the error terms are bounded by\n\\[\n\\left\\langle \\frac{\\kappa ( |x|)}{\\langle 2^k x\\rangle} \\nabla v,\\nabla v \\right\\rangle \n\\]\nrespectively \n\\[\n\\left\\langle \\langle x \\rangle^{-2} v, v \\right\\rangle. \n\\]\nThe expression $\\Im \\langle Q_k v, (b^j D_j + D_j b^j + c)v\\rangle$ can\nalso be included in the two error terms.\nThus we obtain\n\\[\n\\left\\langle \\frac{\\alpha(2^k |x|)}{\\langle 2^k x\\rangle} \\nabla v,\\nabla v \\right\\rangle \n\\lesssim \\left\\langle \\frac{\\kappa ( |x|)}{\\langle 2^k x\\rangle} \\nabla v,\\nabla v \\right\\rangle \n+ \\left\\langle \\langle x \\rangle^{-2} v, v \\right\\rangle. \n\\]\nFor $|x| > 2^{M}$ we have, by \\eqref{new.alpha},\n\\[\n\\kappa(x) \\lesssim \\epsilon \\alpha(2^k x);\n\\]\ntherefore the first term on the right is essentially negligible. We obtain\n\\[\n\\int \\frac{\\alpha(2^k |x|)}{\\langle 2^k x\\rangle} |\\nabla v|^2 dx \n\\lesssim \\int_{D_{ < M}} |\\nabla v|^2 dx \n+ \\int \\langle x \\rangle^{-2} |v|^2 dx .\n\\]\nAt the same time we have\n\\[\n0 = \\left\\langle \\frac{\\alpha(2^k |x|)}{\\langle 2^k x\\rangle} v, (A-\\tau) v \\right\\rangle, \n\\]\nwhich after an integration by parts yields\n\\[\n\\tau \\int \\frac{\\alpha(2^k |x|)}{\\langle 2^k x\\rangle} |v|^2 dx \n\\lesssim \\int \\frac{\\alpha(2^k |x|)}{\\langle 2^k x\\rangle} |\\nabla v|^2 dx \n+ \\int \\langle x \\rangle^{-2} |v|^2 dx. \n\\]\nCombining the two relations we obtain\n\\[\n\\int \\frac{\\alpha(2^k |x|)}{\\langle 2^k x\\rangle} ( |\\nabla v|^2 +|v|^2 ) dx \n\\lesssim \\int_{D_{ < M}} |\\nabla v|^2 dx \n+ \\int \\langle x \\rangle^{-2} |v|^2 dx. \n\\]\nFinally we let $k \\to -\\infty$ to obtain\n\\[\n\\int |\\nabla v|^2 +|v|^2 dx \n\\lesssim \\int_{D_{ < M}} |\\nabla v|^2 dx \n+ \\int \\langle x \\rangle^{-2} |v|^2 dx < \\infty \n\\]\nwhich shows that $v \\in L^2$. \n\nWe note that \\eqref{initial.decay} and\n\\eqref{dinitial.decay} are not used in any quantitative way but serve\nonly to justify the previous computations. More precisely, one can \nintroduce in the computation a cutoff outside a large enough ball\nand then pass to the limit. \n\nIt remains to prove \\eqref{vregb} and \\eqref{vout}.\n\n\n{\\bf Step 9:} Here we prove \\eqref{vregb}. We begin with the \nbounds on $v$. This is trivial for the high frequencies of $v$,\n\\[\n \\| S_{>0} v\\|_{X^0_0} \\lesssim \\| v\\|_{{{\\tilde{X}}}^0}.\n\\]\nTo estimate the low frequencies, we compute\n\\[\n(\\tau+i\\epsilon) S_{<0} v = S_{< 0} A v - S_{<0}(A-\\tau-i\\epsilon)v.\n\\]\nWriting $A$ in the generic form\n\\[\nA = D^2 a + D b + c, \n\\]\nwe have\n\\[\n\\begin{split}\n\\| S_{<0} v\\|_{X^0_{0}} \\lesssim &\\ \\| S_{< 0} D^2 av\\|_{X^0_0} +\n\\| S_{<0} bv \\|_{X^0_0} + \\| S_{<0} cv\\|_{X^0_0} + \\|S_{<0} (A-\\tau-i\\epsilon) v\\|_{X^0_0}\n\\\\\n \\lesssim &\\ \\| av\\|_{{X}^0} +\n\\| bv \\|_{X^0_0} + \\| cv\\|_{X^0_0} + \\|(A-\\tau-i\\epsilon)v\\|_{({{\\tilde{X}}}^0)'}\n\\\\ \\lesssim &\\ \\| v\\|_{{X}^0} +\n\\| \\langle x\\rangle^{-1} v \\|_{L^2} + \\|(A-\\tau-i\\epsilon)v\\|_{({{\\tilde{X}}}^0)'}.\n\\end{split}\n\\]\nOnce we control $\\| v\\|_{X^0_0}$, we can also obtain control of $\\| \\nabla\nv\\|_{X^0_0}$ by a straightforward elliptic estimate.\n\n\n\n{\\bf Step 10:} Here we prove the outgoing radiation condition\n\\eqref{vout} for $v$. This is obtained from similar outgoing radiation\nconditions for the functions $v_n$. However, $v_n$ only converges to\n$v$ in a weak sense. Hence we need to produce some uniform estimates\nfor $v_n$ which will survive in the limit.\n\\begin{multline}\\label{out}\n\\| r^{-\\frac12} (D_r - \\tau^\\frac12) u\\|_{L^2(D_j)}^2 \\\\\\lesssim \n\\sum_{k=0}^\\infty \n2^{-\\delta(k-j)^-} \n\\left( \\|\\langle r\\rangle^\\frac12 (A-\\tau-i\\epsilon) u\\|_{L^2(D_k)} \\| \\langle r\\rangle^{-\\frac12}\n (u,\\nabla u)\\|_{L^2(D_k)} \\right. \\\\ \\left. + \n\\kappa_k \\| r^{-\\frac12} (u,\\nabla u)\\|_{L^2(D_k)}^2 \\right).\n\\end{multline}\nIn other words, there is decay when $k2^{j+2} \\cr\n (2^{-j}R)^\\delta, & 1 < R< 2^{j+2}\n\\end{array} \\right.\n\\]\nwith $\\delta$ a small parameter.\nWe write\n\\begin{equation}\\label{commute.eqn}\n-2\\Im \\langle Qu, (A-\\tau-i\\varepsilon)u\\rangle = \\langle i [A,Q]u,u\\rangle - 2\\varepsilon \\langle Qu,u\\rangle.\n\\end{equation}\nWe expect to get the main positive contribution from the first term on\nthe right. The second term on the right on the other hand is\nessentially negative definite due to the fact that its symbol is\nnegative on the characteristic set of $A - \\tau$. Finally, the term on\nthe left is bounded simply by Cauchy-Schwarz.\n\n\nTo shorten the notations, in the sequel we denote by $E$ error terms\nof the form\n\\[\nE = D O( b(R) r^{-1} \\kappa(|x|)) D + O(b(R) r^{-1} \\kappa(|x|)).\n\\]\nSuch terms occur whenever $a^{ij}$ is either differentiated or\nreplaced by the identity and are easily estimated in terms of the \nright hand side of \\eqref{out}.\n\nWe evaluate the commutator $ i [A,Q]$. A similar computation\nwas already carried out in \\eqref{C.high.calc}, which we reuse with\n$k = 0$, $\\delta = 1$ and $\\phi(r) = b(R)\/R$. We obtain\n\\[\n\\begin{split}\n i [A,Q]= &\\ 4 D \\frac{b(R)}{R} D + 4 D x \\left( \\frac{b'(R)}{R^2} -\n \\frac{b(R)}{R^3}\\right) xD - 2\\tau^\\frac12 \\left(\\frac{b'(R)}{R} x D + D x \\frac{b'(R)}{R}\\right) +E\n\\\\\n= &\\ 2D \\left( 2 \\frac{b(R)}{R} - b'(R)\\right) D - 2 Dx \\left(\n 2 \\frac{b(R)}{R^3} - \\frac{b'(R)}{R^2}\\right) xD \n\\\\&\\qquad\n+ \n {b'(R)} (A -\\tau) + (A -\\tau) {b'(R)}\n+ 2 \\left( Dx - \\tau^\\frac12 r\\right ) \\frac{b'(R)}{r R} \\left(\n xD - r \\tau^\\frac12 \\right ) + E.\n\\end{split}\n\\]\nOur choice of $b$ insures that the coefficient in the first two terms\nis positive,\n\\[\n2 \\frac{b(R)}{R} - {b'(R)} \\geq 0 \\qquad R > 1.\n\\]\nHence we obtain\n\\[\n\\langle i [A,Q] u, u\\rangle \\gtrsim 2 \\langle b'(R) (D_r - \\tau^\\frac12) u, (D_r -\n\\tau^\\frac12) u\\rangle + 2 \\Re \\langle (A -\\tau-i\\epsilon) u, b'(R) u \\rangle +\n\\langle Eu,u\\rangle\n\\]\nwhere we have inserted a harmless $\\epsilon$ term.\n\n\nIt remains to evaluate the second term on the right in\n\\eqref{commute.eqn}. We have \n\\[\n\\begin{split}\n\\tau^\\frac12 Q = &\\ - \\Bigl(D_k \\frac{x_l a^{kl}}{R}-\\tau^{1\/2}\\Bigr)\nb(R) \\Bigl(\\frac{x_i a^{ij}}{R}D_j -\\tau^{1\/2}\\Bigr)\n + \\frac{b(R)}2 (A-\\tau) + (A-\\tau) \\frac{b(R)}2\n\\\\\n&\\ - \\Bigl(D_i -D_l\\frac{a^{lk}x_k x_i}{R^2}\\Bigr) a^{ij} b(R)\n\\Bigl(D_j -\\frac{x_j x_m a^{mn}}{R^2}D_n\\Bigr) - \\frac12 (A b(R)).\n\\end{split}\n\\]\nThe first and third terms are negative while the last term can be\nincluded in $E$. Hence we obtain\n\\[\n\\tau^{\\frac12} \\langle Q u,u\\rangle \\leq \\Re \\langle b(R) u, (A-\\tau-i\\epsilon) u \\rangle +\n\\langle Eu,u\\rangle .\n\\]\n\nReturning to \\eqref{commute.eqn}, we insert the bounds \nfor the two terms on the right to obtain\n\\[\n\\langle b'(R) (D_r - \\tau^\\frac12) u, (D_r -\\tau^\\frac12) u\\rangle \n\\lesssim \\Re \\langle (A -\\tau-i\\epsilon) u, (2 b'(R)+\\epsilon \\tau^{-\\frac12} b(R)\n+i Q)\nu \\rangle + \\langle Eu,u \\rangle.\n\\]\nIn the region $D_j$, we have $b' \\approx 2^{-j} \\approx r^{-1}$;\ntherefore \\eqref{out} follows.\n\n\n\n\n\\subsection{Proof of Theorem~\\ref{theorem.PcSmoothing.res}}\n\nWe proceed as in the nonresonant case. The bound \\eqref{veps}\nis replaced by\n\\begin{equation}\n\\|u_\\varepsilon\\|_{X}\\lesssim\n\\|f\\|_{{X}'}+\\|u_\\varepsilon-u_{\\varepsilon \\rho}\n\\|_{L^2_{t,x}({\\mathbb R}\\times B(0,2R))}.\n\\label{veps.res}\\end{equation}\nUsing Plancherel as in Step 3, this is equivalent to the spatial\nbound\n\\begin{equation}\\label{have.res}\n\\|v \\|_{{X}^0}\\lesssim\n\\| (A-\\tau-i\\varepsilon)v \\|_{({X}^0)'}+\\| v-v_\\rho \\|_{L^2(B(0,2R))}\n\\end{equation}\nwhere ${X}^0$ is the fixed time counterpart of ${X}$. On the other\nhand the estimate that we want to prove, namely\n\\eqref{PcSmoothing.res} with $u_0=0$, has the\nequivalent form\n\\begin{equation}\\label{want.res}\n \\|v \\|_{{X}^0}\\lesssim \\| (A-\\tau-i\\epsilon) v \\|_{({X}^0)'}\n\\end{equation}\nuniformly with respect to $\\tau \\in {\\mathbb R}$, $\\epsilon > 0$.\n\nFor $\\tau$ away from $0$ we can easily bound the local average\nof $v$. We have\n\\[\n(\\tau+i\\epsilon) v_\\rho = (Av)_\\rho - ((A-\\tau-i\\varepsilon)v )_\\rho.\n\\]\nTherefore, by Cauchy-Schwarz,\n\\[\n\\tau |v_\\rho| \\lesssim \\| (A-\\tau-i\\varepsilon)v \\|_{({X}^0)'} + \\|\nv \\|_{L^2(B(0,2R))}.\n\\]\nHence we are able to bound $v$ in ${{\\tilde{X}}}^0$ as well,\n\\begin{equation}\\label{have.resa}\n\\|v \\|_{{{\\tilde{X}}}^0}\\lesssim\n\\| (A-\\tau-i\\varepsilon)v \\|_{({X}^0)'}+\\| v \\|_{L^2(B(0,2R))},\n\\qquad |\\tau| > \\tau_0.\n\\end{equation}\nConsequently, the argument for large $\\tau$ rests unchanged.\n\nConsider now the proof by contradiction. \n\nIn the case $\\tau < 0$,\nwe use the bound \\eqref{bnoc} instead of \\eqref{bc} for the \nlower order terms and show that $v$ is an eigenvalue.\nHowever, by the maximum principle, there can be no \nnegative eigenvalue for $A$.\n\n\nThe case $\\tau = 0$ is the interesting one. Then $v$ satisfies\n\\[\nv \\in {X}, \\qquad A v = 0, \\qquad \\| v-v_\\rho\\|_{L^2(B(0,2R))}=1. \n\\]\nHence $v$ is a zero generalized eigenvalue; therefore it must be\nconstant. But this contradicts the last relation.\n\nFinally, due to \\eqref{have.resa}, the case $\\tau > 0$ is identical to\nthe nonresonant case. \n\n\\subsection{ Proof of Remark~\\ref{remark.nootherres}}\n\nIf $Av = 0$ then from\n\\[\n0 = \\langle A(v-v_{D_j}),\\chi_{0}\nf(t,0) -\\sum_{j=k}^{-1} (\\phi_{j+1}(x)-\\phi_j(x))D_t^{-1} S^t_{>2j} f(t,0) \n\\]\nwith $\\phi_k(x) = \\phi(2^{k} x)$ and\n\\[\n\\phi(0) = 1, \\qquad \\text{supp } \\hat{\\phi} \\subset \\{|\\xi| \\in [1\/2,2]\\}.\n\\]\n\nNotice that $Tu=u^{in}$ with $u^{in}$ as in Section \\ref{embxs}. As such, the bound\n\\eqref{umtu} follows directly from \\eqref{Tkbdd} and \\eqref{l2uout}.\nThe bound \\eqref{lplqtu} follows similarly using a Bernstein bound, Littlewood-Paley theory, and\n\\eqref{kkf}.\nFor \\eqref{atu} we use Proposition~\\ref{lemma.A.to.Ak} to replace \n$A$ by $\\sum A_{(k)} S_k$. Then we use the spatial localization\ncoming from $T$, \\eqref{bernstein}, and the two derivatives gain from $ A_{(k)}$.\n\n\nWe consider now the ${X}$ bounds in \\eqref{arf}. For the second term in the left of \\eqref{arf}, \nusing Bernstein's inequality twice yields\n\\begin{align*}\n\\left\\| (\\phi_{j+1}(x)-\\phi_j(x))D_t^{-1}S_{>2j}^t\n(S_k f)(t,0)\\right\\|_{X_j}\n&\\lesssim 2^{\\frac{2-n}2 j} 2^{2j(-1+\\frac{1}{p_2'}-\\frac{1}{2})}\n\\|S_k f(t,0)\\|_{L^{p_2'}_t} \\\\\n&\\lesssim 2^{\\frac{2-n}2j} 2^{2j(-1+\\frac{1}{p_2'}-\\frac{1}{2})}\n2^{\\frac{n}{q_2'}k} \\|S_k f\\|_{L^{p_2'}_tL^{q_2'}_x} \\\\\n&= \n2^{\\frac{n}{q_2'}(k-j)} \\|S_k f\\|_{L^{p_2'}_tL^{q_2'}_x}.\n\\end{align*}\nThe $j=0$ term in $R_k$ is estimated in a similar fashion. Summing\nwith respect to $k \\leq j \\leq 0$ we use the off-diagonal decay to\nobtain\n\\begin{align*}\n\\|Rf\\|_X &\\lesssim \\left(\\sum_{j=-\\infty}^0 \\left(\\sum_{k=-\\infty}^j \n2^{\\frac{n}{q_2'}(k-j)} \\|S_k f\\|_{L^{p_2'}_tL^{q_2'}_x}\\right)^2\\right)^\\frac12\\\\\n&\\lesssim \\left(\\sum_{k=-\\infty}^0 \\|S_k f\\|^2_{L^{p_2'}_tL^{q_2'}_x}\\right)^\\frac12.\n\\end{align*}\n The bound $X$ bound for the second term in the left of \\eqref{arf} then follows from Littlewood-Paley theory.\nThe $L^{p_1}_tL^{q_1}_x$ estimate follows from similar applications of Bernstein estimates and\nLittlewood-Paley theory.\n \nFor the first term in the left of \\eqref{arf}, we may apply Proposition \\ref{lemma.A.to.Ak} to again\nreplace $A$ by $\\sum A_{(k)}S_k$. As the derivatives in $A_{(k)}$ yield a $2^{2k}$ factor, the estimate\nfor the first term in \\eqref{arf} follows from a very similar argument.\n\n\n\n\nIn order to complete the proof of \\eqref{arf}, we examine the $L^2$\npart of the ${{\\tilde{X}}}$ norm. We may first apply \\eqref{Hardy} and\n\\eqref{txdx} to reduce the problem to the bound\n\\[\n\\|\\sum_{k<0} R_k S_k f\\|_{L^2_{t,x}(\\{|x|\\le 1\\})}\\lesssim\n\\|f\\|_{L^{p_2'}_tL^{q_2'}_x}\n\\]\nin dimensions $n=1,2$.\nHere we use the fact that\n$\\phi_{j+1}(0)-\\phi_j(0)=0$. \nUsing this gain in a fashion similar to that from Section \\ref{embxs},\nwe have \n\\[\n\\|\\phi_{j+1}-\\phi_j\\|_{L^2(\\{|x|\\le 1\\})}\\lesssim 2^j.\n\\]\nThus, arguing as above,\n\\begin{align*}\n\\|R_k S_k f\\|_{L^2(\\{|x|\\le 1\\})}&\\lesssim \\sum_{j\\ge k} 2^j 2^{2j(-1+\\frac{1}{p_2'}-\\frac{1}{2})}\n2^{\\frac{n}{q_2'}k}\\|S_k f\\|_{L^{p_2'}_tL^{q_2'}_x}\\\\\n&\\lesssim 2^{\\frac{n}{2}k}\\|S_k f\\|_{L^{p_2'}_t L^{q_2'}_x}.\n\\end{align*}\nThis can clearly be summed to yield the desired bound.\n\n\nIt remains to prove \\eqref{tmdtr}. For this we will show the bound\n\\begin{equation}\n\\|\\langle x \\rangle (T-D_t R)f\\|_{L^2} \\lesssim \\| f\\|_{L^{p'}_tL^{q'}_x}.\n\\label{fdsa}\\end{equation}\nWe have \n\\[\n(T-D_t R)f = - \\sum_{k < 0} \\left(\\phi_0S^t_{\\le 0} (S_k f)(t,0) \n+\\sum_{j= k}^{-1} (\\phi_{j+1}-\\phi_j)S^t_{\\le 2j} (S_k f)(t,0)\\right).\n\\]\nArguing as above we obtain\n\\[\n \\| (\\phi_{j+1}-\\phi_j)S^t_{\\le 2j} (S_k f)(t,0)\\|_{L^2} \n\\lesssim 2^j 2^{\\frac{n}{q_2'}(k-j)} \\|S_k f\\|_{L^{p_2'}_tL^{q_2'}_x}\n\\]\nrespectively \n\\[\n\\| x (\\phi_{j+1}-\\phi_j)S^t_{\\le 2j} (S_k f)(t,0)\\|_{L^2}\n\\lesssim 2^{\\frac{n}{q_2'}(k-j)} \\|S_k f\\|_{L^{p_2'}_tL^{q_2'}_x}\n\\]\nand similarly for the $j=0$ term. Then \\eqref{fdsa} is obtained by\nsummation using the off-diagonal decay and Littlewood-Paley theory.\n\\end{proof}\n\nTheorems~\\ref{prop.Tataru.parametrix.nr}, \\ref{prop.Tataru.xtolp.nr}\nwill allow us to derive Theorems \\ref{main.est.theorem},\n\\ref{main.theoremnt}, \\ref{corr.nontrap.Strichartz} from\nTheorems~\\ref{main.ls.theorem}, \\ref{main.ls.theoremnt},\n\\ref{theorem.PcSmoothing}. Similarly,\nTheorems~\\ref{prop.Tataru.parametrix}, \\ref{prop.Tataru.xtolp}\nwill allow us to derive Theorems \\ref{main.est.theorem.res},\n\\ref{main.theoremnt.res}, \\ref{corr.nontrap.Strichartz.res} from\nTheorems~\\ref{main.ls.theorem.res}, \\ref{main.ls.theoremnt.res},\n\\ref{theorem.PcSmoothing.res}.\n\n\\subsection{Proof of Theorems~\\ref{main.theoremnt},\n \\ref{corr.nontrap.Strichartz},~\\ref{main.theoremnt.res},\n \\ref{corr.nontrap.Strichartz.res}}\n\nThe four proofs are almost identical, so we discuss only the first\ntheorem. Suppose the function $u$ solves\n\\[\nPu = f +g, \\qquad f \\in {{\\tilde{X}}}', \\quad g \\in L^{p'_2}_t L^{q'_2}_x\n\\]\nwith initial data \n\\[\nu(0) = u_0.\n\\]\nWe let $K$ be the parametrix of\nTheorem~\\ref{prop.Tataru.parametrix.nr} and denote\n\\[\nv = u - K g.\n\\]\nThen\n\\[\nPv = f + g -PKg, \\qquad v(0) = u(0) - Kg(0).\n\\]\nUsing the bounds \\eqref{bc}, \\eqref{kft}, and \\eqref{lperror2bt}, we obtain\n\\[\n\\| v(0)\\|_{L^2} + \\| Pv\\|_{{{\\tilde{X}}}'} \\lesssim \\|u(0)\\|_{L^2} + \n\\| f\\|_{{{\\tilde{X}}}'} + \\|g\\|_{L^{p'_2}_t L^{q'_2}_x}.\n\\]\nThen Theorem~\\ref{main.ls.theoremnt} gives\n\\[\n\\| v\\|_{L^\\infty_t L^2_x \\cap {{\\tilde{X}}}} + \\| Pv\\|_{{{\\tilde{X}}}'} \\lesssim \\|u(0)\\|_{L^2} + \n\\| f\\|_{{{\\tilde{X}}}'} + \\|g\\|_{L^{p'_2}_t L^{q'_2}_x} + \\|v\\|_{L^2_{t,x}(A_{<2R})}.\n\\]\nHence by \\eqref{bc} and Theorem~\\ref{prop.Tataru.xtolp.nr} it follows that\n\\[\n\\| v\\|_{L^\\infty_t L^2_x \\cap {{\\tilde{X}}}}+ \\|v\\|_{L^{p_1}_t L^{p_2}_x}\n \\lesssim \\|u(0)\\|_{L^2} + \n\\| f\\|_{{{\\tilde{X}}}'} + \\|g\\|_{L^{p'_2}_t L^{q'_2}_x} + \\|v\\|_{L^2_{t,x}(A_{<2R})}.\n\\]\nUsing again \\eqref{lperror2bt} we return to $u$ to obtain\n\\[\n\\| u\\|_{L^\\infty_t L^2_x \\cap {{\\tilde{X}}}}+ \\|u\\|_{L^{p_1}_t L^{p_2}_x}\n \\lesssim \\|u(0)\\|_{L^2} + \n\\| f\\|_{{{\\tilde{X}}}'} + \\|g\\|_{L^{p'_2}_t L^{q'_2}_x} + \\|u\\|_{L^2_{t,x}(A_{<2R})}\n\\]\nconcluding the proof of the Theorem.\n\n\n\n\\subsection{Proof of Theorem~\\ref{main.est.theorem}}\n Suppose the function $u$ solves\n\\[\nPu = f +\\rho g, \\qquad f \\in {{\\tilde{X}}}'_e, \\quad g \\in L^{p'_2}_t L^{q'_2}_x\n\\]\nwith initial data \n\\[\nu(0) = u_0.\n\\]\nWe consider two\nadditional spherically symmetric cutoff functions $\\rho_1$ and\n$\\rho_2$ supported in $\\{|x| > 2^M\\}$ so that $\\rho_2 =1$ in the\nsupport of $\\rho_1$ and $\\rho_1 =1$ in the\nsupport of $\\rho$.\n\n Let $K$ be the parametrix of\nTheorem~\\ref{prop.Tataru.parametrix.nr} and denote\n\\[\nv = u - \\rho_1 K \\rho g.\n\\]\nThen\n\\[\nPv = f +\\rho_2(\\rho_1( \\rho g - P K \\rho g) - [P,\\rho_1]K\\rho g)\n, \\qquad v(0) = u(0) - \\rho_1 K \\rho g(0).\n\\]\nUsing the bounds \\eqref{bc}, \\eqref{kft}, and \\eqref{lperror2bt}, we obtain\n\\[\n\\| v(0)\\|_{L^2} + \\| Pv\\|_{{{\\tilde{X}}}'_{e2}} \\lesssim \\|u(0)\\|_{L^2} + \n\\| f\\|_{{{\\tilde{X}}}'_e} + \\|g\\|_{L^{p'_2}_t L^{q'_2}_x}\n\\]\nwhere ${{\\tilde{X}}}'_{e2}$ is similar to ${{\\tilde{X}}}'_{e}$ but with $\\rho$ replaced\nby $\\rho_2$. Then we can apply Theorem~\\ref{main.ls.theorem}\nto $v$ to obtain\n\\[\n\\| v\\|_{L^\\infty_t L^2_x \\cap {{\\tilde{X}}}_{e}}\n + \\| Pv\\|_{{{\\tilde{X}}}'_{e2}} \\lesssim \\|u(0)\\|_{L^2} + \n\\| f\\|_{{{\\tilde{X}}}'_e} + \\|g\\|_{L^{p'_2}_t L^{q'_2}_x}+ \\|v\\|_{L^2_{t,x}(|x|\\le 2^{M+1})}.\n\\]\nWe truncate $v$ with $\\rho$ and compute\n\\[\nP \\rho v = [P,\\rho]v + \\rho P v .\n\\]\nThen we can estimate\n\\[\n\\| v\\|_{L^\\infty_t L^2_x} + \\|\\rho v\\|_{{{\\tilde{X}}}}\n + \\| P(\\rho v)\\|_{{{\\tilde{X}}}'} \\lesssim \\|u(0)\\|_{L^2} + \n\\| f\\|_{{{\\tilde{X}}}'_e} + \\|g\\|_{L^{p'_2}_t L^{q'_2}_x} + \\|v\\|_{L^2_{t,x}(|x|\\le 2^{M+1})}.\n\\]\nHence by \\eqref{bc} and Theorem~\\ref{prop.Tataru.xtolp.nr} applied to $\\rho v$,\nwe obtain\n\\[\n\\|v\\|_{L^\\infty_t L^2_x} + \\|\\rho v\\|_{{{\\tilde{X}}} \\cap L^{p_1}_t L^{q_1}_x}\n \\lesssim \\|u(0)\\|_{L^2} + \n\\| f\\|_{{{\\tilde{X}}}'_e} + \\|g\\|_{L^{p'_2}_t L^{q'_2}_x}+ \\|v\\|_{L^2_{t,x}(|x|\\le 2^{M+1})}.\n\\]\nFinally, we use \\eqref{kft} to return to $u$ and obtain\n\\[\n\\|u\\|_{L^\\infty_t L^2_x} + \\|\\rho u\\|_{{{\\tilde{X}}} \\cap L^{p_1}_t L^{q_1}_x}\n\\lesssim \\|u(0)\\|_{L^2} + \n\\| f\\|_{{{\\tilde{X}}}'_e} + \\|g\\|_{L^{p'_2}_t L^{q'_2}_x}+ \\|u\\|_{L^2_{t,x}(|x|\\le 2^{M+1})},\n\\]\nconcluding the proof of the Theorem.\n\n\n\n\\subsection{Proof of Theorem~\\ref{main.est.theorem.res}}\n\nThe argument is similar to the one above. The chief difference \nis that we can no longer use the truncations by $\\rho$, $\\rho_1$,\n$\\rho_2$ and instead we use the modified truncation operators\nsuch as $T_\\rho$.\n\n Suppose the function $u$ solves\n\\[\nPu = f +\\rho g, \\qquad f \\in {X}'_e, \\quad g \\in L^{p'_2}_t L^{q'_2}_x\n\\]\nwith initial data \n\\[\nu(0) = u_0.\n\\]\nWe let $K$ be the parametrix of\nTheorem~\\ref{prop.Tataru.parametrix} and denote\n\\[\nv = u - T_{\\rho_1} K \\rho g\n\\]\nThen we can write\n\\[\nPv = f +T_{\\rho_2}(T_{\\rho_1}( \\rho g - P K \\rho g) - \n[P,T_{\\rho_1}]K\\rho g)\n, \\qquad v(0) = u(0) - T_{\\rho_1}K\\rho g(0).\n\\]\nHere we compute the commutator\n\\[\n[A,T_{\\rho_1}] w = A \\rho_1 (w-w_{\\rho_1}) - \\rho_1 A (w-w_{\\rho_1}) -\n(1-\\rho)(Aw)_{\\rho_1}\n= [A,\\rho_1] (w -w_{\\rho_1}) - (1-\\rho)(Aw)_{\\rho_1}.\n\\]\nAlso we have \n\\[\n(Aw)_{\\rho_1} = c_\\rho \\int (1-\\rho_1)A(w-w_{\\rho_1}) dx\n = -c_\\rho \\int (w-w_{\\rho_1}) A\\rho_1 dx.\n\\]\nThen using the bounds \\eqref{bnoc}, \\eqref{kf}, and \\eqref{lperror2b}, we obtain\n\\[\n\\| v(0)\\|_{L^2} + \\| Pv\\|_{{X}'_{e2}} \\lesssim \\|u(0)\\|_{L^2} + \n\\| f\\|_{{X}'_e} + \\|g\\|_{L^{p'_2}_t L^{q'_2}_x}.\n\\]\nBy Theorem~\\ref{main.ls.theorem} for $v$ we get\n\\[\n\\| v\\|_{L^\\infty_t L^2_x \\cap {X}_{e}}\n + \\| Pv\\|_{{X}'_{e2}} \\lesssim \\|u(0)\\|_{L^2} + \n\\| f\\|_{{X}'_e} + \\|g\\|_{L^{p'_2}_t L^{q'_2}_x}+ \\|(1-\\rho)(v-v_\\rho)\\|_{L^2_{t,x}}.\n\\]\nWe truncate $v$ with $T_\\rho$ and compute as above the \ncommutator $[P,T_\\rho]$.\nThen we estimate\n\\[\n\\| v\\|_{L^\\infty_t L^2_x} + \\|T_\\rho v\\|_{{X}}\n + \\| P(T_\\rho v)\\|_{{X}'} \\lesssim \\|u(0)\\|_{L^2} + \n\\| f\\|_{{X}'_e} + \\|g\\|_{L^{p'_2}_t L^{q'_2}_x} + \\|(1-\\rho)(v-v_\\rho)\\|_{L^2_{t,x}}.\n\\]\nHence by \\eqref{bnoc} and Theorem~\\ref{prop.Tataru.xtolp} applied to $T_\\rho v$,\nwe obtain\n\\[\n\\|v\\|_{L^\\infty_t L^2_x} + \\|T_\\rho v\\|_{{X} \\cap L^{p_1}_t L^{q_1}_x}\n \\lesssim \\|u(0)\\|_{L^2} + \n\\| f\\|_{{X}'_e} + \\|g\\|_{L^{p'_2}_t L^{q'_2}_x}+ \\|(1-\\rho)(v-v_\\rho)\\|_{L^2_{t,x}}.\n\\]\nFinally, we use \\eqref{kf} to return to $u$ and obtain\n\\[\n\\|u\\|_{L^\\infty_t L^2_x} + \\|T_\\rho u\\|_{{X} \\cap L^{p_1}_t L^{q_1}_x}\n\\lesssim \\|u(0)\\|_{L^2} + \n\\| f\\|_{{X}'_e} + \\|g\\|_{L^{p'_2}_t L^{q'_2}_x}+ \\|(1-\\rho)(u-u_\\rho)\\|_{L^2_{t,x}},\n\\]\nconcluding the proof of the Theorem.\n\n\n\\begin{comment}\n\\newsection{Remarks}\n\nThis is a temporary section which contains some more detailed remarks and proofs. It will be removed\nfor the final version.\n\\subsection{Remarks from Section \\ref{not_para}:}\n\n\\begin{remark}\n\\label{remark.1}\nHere we shall show\n$$S_{8$.\n\nIndeed, the left side is bounded by\n\\begin{equation}\\label{remark.1.1}\n\\sum_{j\\ge 0} \\int_{y\\in D_j} \\frac{2^{kn\/2}}{(1+2^{k\/2}|x-y|)^N} (a^{ij}-I)(y)\\:dy,\\end{equation}\nfor any $N\\ge 0$. We examine the cases $0\\le jm+3$ separately.\n\\begin{itemize}\n\\item For $0\\le jm+3} 2^{-(j-m)(N-n)} (e^{2^{-10}})^{j-m}\\lesssim \\kappa_m.$$\n\\end{itemize}\n\\item Now, we look at the case $l+m<0$.\nHere, we have two cases to examine.\n\\begin{itemize}\n\\item For $0\\le j\\le -l+2$, we have that the above is\n$$\\lesssim \\sum_{0\\le j\\le -l+2} 2^{(l+j)n}\\kappa_j \\lesssim \\kappa_{-l} \\sum_{0\\le j\\le -l+2} 2^{(l+j)n}\n(e^{2^{-10}})^{|l+j|}\\lesssim \\kappa_{-l}.$$\n\\item For $j>-l+2$, we have that the above is\n$$\\lesssim \\sum_{j>-l+2} 2^{-N(l+j)}\\kappa_j \\lesssim \\kappa_{-l} \\sum_{j>-l+2} 2^{-N(l+j)}(e^{2^{-10}})^{l+j}\n\\lesssim \\kappa_{-l}.$$\n\\end{itemize}\n\\end{itemize}\n\\qed\n\\end{remark}\n\n\n\n\\end{comment}\n\n\n\n\n\n\n\n\\bigskip\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nThe occurrence of CO$_2$ within magmas and volcanic gases indicates a\nsignificant carbon presence within the Earth's lower mantle\n\\cite{Marty_1987,Halliday_2013}. Carbon has a low solubility in\nmantle silicates and the majority of the oxidized carbon in the\nEarth's mantle is believed to exist in the form of carbonates.\nCalcium and magnesium carbonate (CaCO$_3$ and MgCO$_3$) are the main\nsources and sinks of atmospheric CO$_2$ within the Earth's mantle.\nCarbonates are conveyed into the deep Earth by subduction, and carbon\nis recycled to the surface via volcanic processes in the form of\nCO$_2$-containing fluids and solids, and diamonds\n\\cite{Ghosh_2009,Frezzotti_2011}. However, the details of carbon\nstorage within the Earth's interior are unclear. The Deep Carbon\nObservatory \\cite{DCO} has been set up to investigate carbon within\nthe Earth's deep interior. CaCO$_3$ and MgCO$_3$ play fundamental\nroles in the global carbon cycle and influence the climate of our\nplanet \\cite{Dasgupta_2010,Hazen_2013_DCO}. Knowledge of the\nstructures, energetics and other properties of CaCO$_3$ and MgCO$_3$\nat high pressures is therefore important in understanding the Earth's\nmantle, and especially the carbon cycle.\n\nThe low-pressure calcite form \\cite{Bragg_structure_of_calcite_1914}\nof CaCO$_3$ is one of the most abundant minerals on the Earth's\nsurface and is the main constituent of metamorphic marbles. Several\nmetastable calcite-like phases have been observed\n\\cite{Bridgman_1939,Suito_2001,Merlini_2012}, and a calcite-related\nphase has been reported at around 25 GPa\n\\cite{Catalli_2005,Merlini_2012}. At pressures of about 2 GPa calcite\ntransforms to the aragonite structure\n\\cite{Bragg_structure_of_aragonite_1924} of $Pnma$ symmetry. At about\n40 GPa aragonite transforms into the ``post aragonite'' ($Pmmn$)\nstructure of CaCO$_3$, which is stable up to at least 86 GPa\n\\cite{Ono_2005,Oganov_2006}. The low pressure magnesite phase of\nMgCO$_3$ has the same structure as calcite. Experiments indicate that\nmagnesite is stable up to 80 GPa \\cite{Fiquet_2002}, and a phase\ntransition occurs above 100 GPa to an unknown magnesite II structure\n\\cite{Isshiki_2004_magnesite_and_high_pressure_form,Boulard_2011}.\n\n\\section{Structure searches}\n\nDensity functional theory (DFT) calculations for high pressure phases\nof CaCO$_3$ and MgCO$_3$ were performed by Oganov \\textit{et al.}\\\nusing an evolutionary structure searching algorithm\n\\cite{Oganov_2006,Oganov_2008}. These calculations predicted a\ntransition from the calcite to aragonite to ``post aragonite''\nstructures of CaCO$_3$, followed by a transition to a structure of\n$C222_1$ symmetry at pressures over 100 GPa. Similar calculations for\nMgCO$_3$ predicted transitions from magnesite to a structure of $C2\/m$\nsymmetry at 82 GPa, followed by a transition to a structure of $P2_1$\nsymmetry at 138 GPa, and a phase of $Pna2_1$ symmetry at 160 GPa\n\\cite{Oganov_2008}.\n\nCalculations using the \\textit{ab initio} random structure searching\n(AIRSS) technique \\cite{Airss_review} have led to the discovery of\nstructures that have subsequently been verified by experiment, for\nexample, in silane \\cite{pickard_silane}, aluminium hydride\n\\cite{pickard_aluminum_hydride}, ammonia monohydrate\n\\cite{fortes_ammonia_monohydrate_II} and ammonia dihydrate\n\\cite{griffiths_ammonia_dihydrate_II}. In the basic AIRSS approach a\ncell volume and shape is selected at random from within reasonable\nranges, the atoms are added at random positions, and the system is\nrelaxed until the forces on the atoms are negligible and the pressure\ntakes the required value. This procedure is repeated many times,\nleading to a reasonably unbiased scheme which allows a significant\nportion of the ``structure space'' to be investigated, although the\nsampling may be rather sparse. This approach is often successful for\nsmall systems, but it involves sampling a large portion of the\nhigh-energy structure space which is not normally of interest. We\ntherefore reduce the size of the structure space investigated by\nconstraining the searches.\n\nWe first perform searches in small cells, constraining the initial\nstructures so that all of the atoms are at least 1 \\AA\\ apart. The\nlow-enthalpy structures obtained from these calculations give us\ninformation about the favorable bonding configurations and likely\nnearest neighbor distances between the different atomic types. At low\npressures we find that the low-enthalpy structures contain\nwell-defined triangular CO$_3$ or ring C$_3$O$_9$ units, and therefore\nwe place these units and Ca or Mg atoms randomly within the cells of\nrandom shapes. We ensure that the atoms are not too close together by\nconstraining the initial values of the minimum distances between atoms\nfor each of the six possible pairs of atomic species. The six minimum\ndistances are obtained from low-enthalpy structures found in the\nsmall-cell searches. To construct the initial structures at higher\npressures we use minimum distances from low-enthalpy small-cell\nstructures to prepare new larger structures that approximately satisfy\nthe minimum distance constraints. This approach helps to space out\nthe different species appropriately, while retaining a high degree of\nrandomness. We perform searches at both low and high pressures, using\nstructures which are constrained to have a certain symmetry which is\nenforced during the relaxation, but are otherwise random\n\\cite{Airss_review}. This approach is useful because low energy\nstructures often possess symmetry \\cite{Pauling_1929,Wales_1998},\nalthough symmetry constraints break up the allowed structure space\ninto disconnected regions and can prevent some structures from\nrelaxing to lower energy ones \\cite{Airss_review}. We consider\nstructures containing up to eight formula units (f.u.) for CaCO$_3$\nand twelve f.u.\\ for MgCO$_3$.\n\nOur first-principles DFT calculations are performed using the\n\\textsc{Castep} plane-wave basis set pseudopotential code\n\\cite{ClarkSPHPRP05}. We use the Perdew-Burke-Ernzerhof (PBE)\ngeneralized gradient approximation (GGA) density functional\n\\cite{Perdew_1996_PBE}, default \\textsc{Castep} ultrasoft\npseudopotentials \\cite{Vanderbilt90}, and a plane-wave basis set\nenergy cutoff of 440 eV. We use a Brillouin zone sampling grid of\nspacing $2\\pi\\times$0.1~\\AA$^{-1}$ for the searches, and a finer\nspacing of $2\\pi\\times$0.05~\\AA$^{-1}$ for the final results reported\nin this paper.\n\n\\section{C\\lowercase{a}CO$_3$, pressure $\\leq$ 50 GPa}\n\nCalculated enthalpy-pressure curves for CaCO$_3$ phases are shown in\nFig.\\ \\ref{fig:enthalpy_CaCO3}, relative to the enthalpy of the ``post\naragonite'' phase. The transition from aragonite to ``post\naragonite'' becomes energetically favorable at about 42 GPa, in\nagreement with previous DFT results\n\\cite{Oganov_2006,Arapan_2007,Oganov_2008,Arapan_2010} and experiment\n\\cite{Ono_2005}. We performed calculations for the CaCO$_3$-VI\nstructure reported in Ref.\\ \\onlinecite{Merlini_2012}, which was\nsuggested as a possible high pressure phase of CaCO$_3$. However, we\nfound it to be very high in enthalpy, with a strongly anisotropic\nstress and large forces on the atoms. Relaxation of the CaCO$_3$-VI\nstructure at 40 GPa led to a reasonably stable structure with an\nenthalpy close to that of aragonite, but the relaxed structure does\nnot have a region of stability on our phase diagram (Fig.\\\n\\ref{fig:enthalpy_CaCO3}). We also found a structure of $Pnma$\nsymmetry (``{CaCO$_3$-$Pnma$-$h$}'', where $h$ denotes ``high\npressure'') that is predicted to be more stable than aragonite above\n40 GPa, and more stable than ``post aragonite'' below 47 GPa.\nHowever, {CaCO$_3$-$Pnma$-$h$} does not have a region of thermodynamic\nstability on our phase diagram because we find a previously unknown\nstructure of $P2_1\/c$ symmetry (``{CaCO$_3$-$P2_1\/c$-$l$}'', where $l$\ndenotes ``low pressure'') which is calculated to be the most stable\nphase in the pressure range 32--48 GPa, see Fig.\\\n\\ref{fig:enthalpy_CaCO3}.\n\nAt 42 GPa {CaCO$_3$-$P2_1\/c$-$l$} is calculated to be about 0.05 eV\nper f.u.\\ more stable than aragonite and ``post aragonite'' and,\nbecause these $sp^2$ bonded structures are similar, we expect that DFT\ncalculations should give rather accurate enthalpy differences between\nthem. However, our {CaCO$_3$-$P2_1\/c$-$l$} and {CaCO$_3$-$Pnma$-$h$}\nstructures do not provide as good a fit to the experimental X-ray\ndiffraction data as the ``post aragonite'' phase \\cite{Oganov_2006}.\nIt is possible that large energy barriers hinder formation of the\n{CaCO$_3$-$P2_1\/c$-$l$} structure. Another possibility is that the\nlaser-heated sample melts and the least stable polymorph crystallizes\nfrom the melt first, in analogy to ``Ostwald's rule''\n\\cite{Ostwald_1897}.\nIn any case, the conditions within the Earth's mantle are not the same\nas in diamond anvil cell experiments, and the timescales associated\nwith geological processes are enormously longer than those for\nlaboratory experiments.\n\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.4\\textwidth,clip]{Figure_1_Enthalpy.eps}\n \\caption{(Color online) Enthalpies per f.u.\\ of CaCO$_3$ phases\n relative to ``post aragonite'', with the number of f.u.\\ per\n primitive unit cell given within square brackets. The enthalpies\n of phases known prior to the current study are shown as dashed\n lines, while those found in the current study are shown as solid\n lines. The dotted red line shows the collapse of the\n {CaCO$_3$-$P2_1\/c$-$l$} structure into the more stable\n {CaCO$_3$-$P2_1\/c$-$h$} structure at 80--90 GPa.}\n \\label{fig:enthalpy_CaCO3}\n\\end{figure}\n\n\n\\section{C\\lowercase{a}CO$_3$, pressure $>$ 50 GPa}\n\nAt higher pressures we find another CaCO$_3$ structure of $P2_1\/c$\nsymmetry (``{CaCO$_3$-$P2_1\/c$-$h$}'') to be stable from 67 GPa to\nwell above 100 GPa. Our {CaCO$_3$-$P2_1\/c$-$h$} structure is about\n0.18 eV per f.u.\\ more stable than the $C222_1$ structure found by\nOganov \\textit{et al.}\\ \\cite{Oganov_2006}, see Fig.\\\n\\ref{fig:enthalpy_CaCO3}, and $C222_1$ does not have a region of\nthermodynamic stability. We also find that at about 80--90 GPa\n{CaCO$_3$-$P2_1\/c$-$l$} transforms into the more stable\n{CaCO$_3$-$P2_1\/c$-$h$} structure without any apparent energy barrier\n(dotted red line in Fig.\\ \\ref{fig:enthalpy_CaCO3}). Our calculations\nlead to the prediction of a new and more stable polymorph of CaCO$_3$\nat pressures $>67$ GPa.\n\n\n\\section{M\\lowercase{g}CO$_3$}\n\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.4\\textwidth,clip]{Figure_2_Enthalpy.eps}\n \\caption{(Color online) Enthalpies per f.u.\\ of MgCO$_3$ phases\n relative to the $C2\/m$ phase, with the number of f.u.\\ per\n primitive unit cell given within square brackets. Previously\n known phases are shown as dashed lines, and those found in the\n current study are shown as solid lines.}\n \\label{fig:Enthalpies_MgCO3_higher_pressure}\n\\end{figure}\n\n\nCalculated enthalpy-pressure curves for MgCO$_3$ phases in the\npressure range 50--200 GPa are shown in Fig.\\\n\\ref{fig:Enthalpies_MgCO3_higher_pressure}, relative to the $C2\/m$\nphase. We find a previously unreported structure of $P\\bar{1}$\nsymmetry to be the most stable in the range 85--101 GPa. We also find\na phase of $P2_12_12_1$ symmetry that is marginally the most stable at\npressures around 144 GPa, see Fig.\\\n\\ref{fig:Enthalpies_MgCO3_higher_pressure}.\n\n\n\\section{Structures and bonding}\n\nThe carbon atoms in the calcite, aragonite, ``post aragonite'', and\nour {CaCO$_3$-$P2_1\/c$-$l$} and {CaCO$_3$-$Pnma$-$h$} structures\ncontain threefold coordinated carbon atoms, as does the magnesite\nphase of MgCO$_3$. These structures contain triangular CO$_3^{2-}$\nions with $sp^2$ bonding. In aragonite and ``post aragonite'' the\nCO$_3^{2-}$ ions are coplanar, but in our {CaCO$_3$-$Pnma$-$h$}\nstructure they are somewhat tilted, while in {CaCO$_3$-$P2_1\/c$-$l$}\nthey are tilted at approximately 90$^\\circ$ to one another, see Fig.\\\n\\ref{fig:structures_CaCO3_lower_pressure}. More details of the\nstructures are given in the Supplemental Material \\cite{Supplemental}.\n\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.475\\textwidth,clip]{Figure_3_P21c-low.eps}\\\\\n \\includegraphics[width=0.37\\textwidth,clip]{Figure_3_Pnma.eps}\\\\ \\vspace{0.375cm}\n \\includegraphics[width=0.3\\textwidth,clip]{Figure_3_Pmmn.eps}\n \\caption{(Color online) The {CaCO$_3$-$P2_1\/c$-$l$} (top),\n {CaCO$_3$-$Pnma$-$h$} (middle), and ``post aragonite'' (bottom)\n structures of CaCO$_3$ at 40 GPa. The Ca atoms are in green, the\n carbon in grey, and the oxygen in red.}\n \\label{fig:structures_CaCO3_lower_pressure}\n\\end{figure}\n\n\nThe high-pressure {CaCO$_3$-$P2_1\/c$-$h$} and $C222_1$ structures\ncontain fourfold coordinated carbon atoms and are of the pyroxene\ntype. {CaCO$_3$-$P2_1\/c$-$h$} and $C222_1$ possess very similar\ncalcium lattices but the packing of the pyroxene chains is different,\nas can be seen in Fig.\\ \\ref{fig:structures_CaCO3_higher_pressure}.\nIn $C222_1$ each of the chains is orientated in the same manner, but\n{CaCO$_3$-$P2_1\/c$-$h$} alternate chains run in the reverse direction,\nsee Fig.\\ \\ref{fig:structures_CaCO3_higher_pressure}, and consequently\nthe unit cell of {CaCO$_3$-$P2_1\/c$-$h$} contains four f.u., whereas\n$C222_1$ contains two. When viewed along the axis of the chains, the\n{CaCO$_3$-$P2_1\/c$-$h$} and $C222_1$ structures appear almost\nidentical. {CaCO$_3$-$P2_1\/c$-$h$} and $C222_1$ have very similar\nvolumes at high pressures, with $C222_1$ being slightly denser, which\nleads to almost parallel enthalpy-pressure relations, see Fig.\\\n\\ref{fig:enthalpy_CaCO3}. The lower enthalpy of\n{CaCO$_3$-$P2_1\/c$-$h$} must therefore arise from more favorable\nelectrostatic interactions between the pyroxene chains.\n\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.35\\textwidth,clip]{Figure_4_C2221.eps}\n \\includegraphics[width=0.35\\textwidth,clip]{Figure_4_P21c.eps}\n \\caption{(Color online) The $C222_1$ (top) and {CaCO$_3$-$P2_1\/c$-$h$}\n pyroxene-type (bottom) structures of CaCO$_3$ at 60 GPa. The Ca\n atoms are in green, carbon in grey, and oxygen in red.}\n \\label{fig:structures_CaCO3_higher_pressure}\n\\end{figure}\n\n\n\\subsection{High-pressure X-ray data for C\\lowercase{a}CO$_3$}\n\nOno \\textit{et al.}\\ \\cite{Ono_2007} performed laser-heated diamond\nanvil cell experiments on CaCO$_3$ at 182 GPa. X-ray diffraction data\nfor the $C222_1$ \\cite{Oganov_2006} and {CaCO$_3$-$P2_1\/c$-$h$}\nstructures are compared in Fig.\\ \\ref{fig:XRD_CaCO3} with the\nexperimental data from Fig.\\ 1 of Ref.\\ \\onlinecite{Ono_2007}. Note\nthe appearance of three peaks marked with stars in the experimental\ndata that arise from the platinum used to enhance heat absorption\nduring the laser heating and as a pressure calibrant. The\nexperimental data is not of very high resolution. The diffraction\npatterns of the theoretical $C222_1$ and {CaCO$_3$-$P2_1\/c$-$h$}\nstructures share many common features. There are also clear\nsimilarities between the theoretical and experimental X-ray data, but\nthe experimental data is of insufficient resolution to allow the\nstructure to be determined unambiguously. We suggest that our\nCaCO$_3$-$P2_1\/c$-$h$ structure is the best available candidate for\nthe observed high pressure phase because it has a much lower enthalpy\nthan $C222_1$.\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.4\\textwidth,clip]{Figure_5_Xray.eps}\n \\caption{(Color online) X-ray diffraction patterns of the $C222_1$\n \\cite{Oganov_2006} and {CaCO$_3$-$P2_1\/c$-$h$} phases of CaCO$_3$,\n compared with experimental data from Fig.\\ 1(b) of Ref.\\\n \\onlinecite{Ono_2007}. Data at 182 GPa are reported, with an\n incident wavelength of 0.415 \\AA. The stars indicate that the peak\n immediately to the right arises from platinum.}\n \\label{fig:XRD_CaCO3}\n\\end{figure}\n\n\n\\section{Chemical reactions in Earth's mantle}\n\nWe have investigated possible chemical reactions involving the mantle\nmaterials CaCO$_3$, MgCO$_3$, CO$_2$, MgSiO$_3$, CaSiO$_3$, SiO$_2$,\nCaO and MgO, following the approach of Oganov \\textit{et al.}\\\n\\cite{Oganov_2008}. The most stable structures of each compound at\nthe relevant pressures are used, as provided by DFT studies. We use\nthe $Pa\\bar{3}$, $P4_2\/mnm$, and $I\\bar{4}2d$ structures of CO$_2$\n\\cite{Ma_CO2_2013}, the stishovite, CaCl$_2$ and pyrite structures of\nSiO$_2$ \\cite{Karki_silica_1997}, the rocksalt structure of MgO, the\northorhombic structure of perovskite CaSiO$_3$ and the perovskite and\npost-perovskite structures of MgSiO$_3$ \\cite{Murakami_2004,\n Oganov_2004, Tsuchiya_2004}.\n\nDecomposition of CaCO$_3$ and MgCO$_3$ into the alkaline earth oxides\nplus CO$_2$ is found to be unfavorable. Under conditions of excess\nSiO$_2$, the reaction\n\\begin{eqnarray}\n{\\rm MgCO}_3 + {\\rm SiO}_2 & \\rightarrow & {\\rm MgSiO}_3 + {\\rm CO}_2\n\\end{eqnarray}\nis found to be energetically unfavorable up to 138 GPa, which is just\nabove the pressure at the mantle-core boundary, see Fig.\\\n\\ref{fig:Enthalpy_MgCO3+SiO2-MgSiO3+CO2}. We find that the reaction\n\\begin{eqnarray}\n{\\rm CaCO}_3 + {\\rm SiO}_2 & \\rightarrow & {\\rm CaSiO}_3 + {\\rm CO}_2\n\\end{eqnarray}\ndoes not occur below 200 GPa, see Fig.\\\n\\ref{fig:Enthalpy_CaSiO3+CO2-CaCO3+SiO2}, which is much higher than\nthe value of 135 GPa reported in Ref.\\ \\onlinecite{Oganov_2008}. We\nconclude that both MgCO$_3$ and CaCO$_3$ are stable within the Earth's\nmantle under conditions of excess SiO$_2$. These results suggest that\nfree CO$_2$ does not occur as an equilibrium phase within the Earth's\nmantle.\n\nMgCO$_3$ has generally been believed to be the dominant carbonate\nthroughout the Earth's mantle. This assumption can be tested when\nexcess MgO is present by determining the relative stability of\nCaCO$_3$+MgO and MgCO$_3$+CaO. We find that CaCO$_3$+MgO is the more\nstable up to pressures of about 200 GPa, so that CaCO$_3$ is the\nstable carbonate under these conditions. In the case of excess\nMgSiO$_3$ we consider the reaction\n\\begin{eqnarray} \n{\\rm CaCO}_3 + {\\rm MgSiO}_3 & \\rightarrow & {\\rm CaSiO}_3 + {\\rm MgCO}_3,\n\\end{eqnarray}\nfinding that CaCO$_3$ is more stable than MgCO$_3$ from 100 GPa up to\npressures well above those of 136 GPa found at the mantle-core\nboundary, see Fig.\\ \\ref{fig:Enthalpy_CaCO3+MgSiO3-CaSiO3+MgCO3}.\n\n\n\\begin{figure}[h!]\n \\centering\n \\includegraphics[width=0.4\\textwidth,clip]{Figure6-Mg-CO2.eps}\n \\caption{(Color online). The relative stabilities per f.u.\\ as a\n function of pressure of MgCO$_3$+SiO$_2$ and MgSiO$_3$+CO$_2$.\n The vertical gray line indicates the pressure at the base of the\n mantle (136 GPa). In this and the following figures, the kinks\n arise from phase transitions. }\n \\label{fig:Enthalpy_MgCO3+SiO2-MgSiO3+CO2}\n\\end{figure}\n\n\n\\begin{figure}[h!]\n \\centering\n \\includegraphics[width=0.4\\textwidth,clip]{Figure7-Ca-CO2.eps}\n \\caption{(Color online). The relative stabilities per f.u.\\ as a\n function of pressure of CaSiO$_3$+CO$_2$ and CaCO$_3$+SiO$_2$.\n The vertical gray line indicates the pressure at the base of the\n mantle (136 GPa). }\n \\label{fig:Enthalpy_CaSiO3+CO2-CaCO3+SiO2}\n\\end{figure}\n\n\n\\begin{figure}[h!]\n \\centering\n \\includegraphics[width=0.45\\textwidth,clip]{Figure8-CaCO3vMgCO3.eps}\n \\caption{(Color online) Enthalpy per f.u.\\ of CaCO$_3$+MgSiO$_3$\n compared with that of CaSiO$_3$+MgCO$_3$. Below 100 GPa we find\n that CaSiO$_3$+MgCO$_3$ is the most stable, while above 100 GPa\n CaCO$_3$+MgSiO$_3$ is the most stable. }\n \\label{fig:Enthalpy_CaCO3+MgSiO3-CaSiO3+MgCO3}\n\\end{figure}\n\n\n\\section{Conclusions}\n\nIn conclusion, we have searched for structures of CaCO$_3$ and\nMgCO$_3$ at mantle pressures using AIRSS\n\\cite{Airss_review,pickard_silane}. We have found a\n{CaCO$_3$-$P2_1\/c$-$l$} structure with $sp^2$ bonded carbon atoms that\nis predicted to be stable within the range 32--48 GPa. We have also\nfound a high pressure {CaCO$_3$-$P2_1\/c$-$h$} structure with $sp^3$\nbonded carbon atoms that is about 0.18 eV per f.u.\\ more stable than\nthe $C222_1$ phase proposed by Oganov \\textit{et al.}\\\n\\cite{Oganov_2006}. Both the {CaCO$_3$-$P2_1\/c$-$h$} and $C222_1$\nstructures are compatible with the available X-ray diffraction data\n\\cite{Ono_2007}. However, {CaCO$_3$-$P2_1\/c$-$h$} is the most stable\nstructure from 67 GPa to pressures well above those encountered within\nthe Earth's lower mantle ($\\leq$ 136 GPa).\nOur AIRSS calculations suggest a previously unknown phase of MgCO$_3$\nof $P\\bar{1}$ symmetry that is predicted to be thermodynamically\nstable in the pressure range 85--101 GPa.\nOur results suggest that CO$_2$ is not a thermodynamically stable\ncompound under deep mantle conditions.\nUnder conditions of excess MgSiO$_3$ we find that CaCO$_3$ is more\nstable than MgCO$_3$ above 100 GPa. This result arises directly from\nour discovery of the highly stable {CaCO$_3$-$P2_1\/c$-$h$} phase. The\nresults of our study change our understanding of the carbon cycle in\nthe lower part of the mantle and may have important consequences for\ngeodynamics\n\\cite{Javoy_carbon_geodynamic_cycle,Marty_1987_geodynamics,Dobretsov_2012_geodynamics}\nand geochemistry\n\\cite{Schrag_2013_geochemistry,Deines_geochem_review}.\n\n\n\n\n\n\\begin{acknowledgments}\n\n We acknowledge financial support from the Engineering and Physical\n Sciences Research Council (EPSRC) of the United Kingdom.\n\n\\end{acknowledgments}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}