Content-Type: multipart/mixed; boundary="-------------0505011705610" This is a multi-part message in MIME format. ---------------0505011705610 Content-Type: text/plain; name="05-157.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="05-157.keywords" Transfer operator, essential spectral radius, Banach space, Sobolev space, hyperbolic set ---------------0505011705610 Content-Type: application/x-tex; name="anisomay05.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="anisomay05.tex" \documentclass[10pt]{amsart} \usepackage{amsmath, latexsym} \usepackage{amssymb,amsthm} \usepackage{amscd, %showkeys } %\usepackage[dvips]{graphicx} %%%%%%% New theomrems %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \theoremstyle{definition} \newtheorem*{definition}{Definition} %%%%%%% Symbols %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \def\C{\mathbb{C}}% Complex number \def\real{\mathbb{R}}% Real number \def\integer{\mathbb{Z}}% Integer \def\disk{\mathbf{D}}% Disk \def\ann{\mathbf{A}}% Annulus \def\supp{\mathrm{supp}}% Support \def\sphere{\mathbf{S}^{d-1}}% Sphere \def\cone{\mathbf{C}} % Cone \def\BB{\mathcal{B}} \def\QQ{\mathcal{Q}} \def\WW{\mathcal{W}} \newcommand{\sob}[1]{_{W_*^{#1}}}% \newcommand{\hol}[1]{_{C_*^{#1}}}% \def\L{\mathcal{L}} \def\P{\mathcal{P}} \def\g{g} %%%%%%% \newcommand{\cout}[1]{} % Just cut. %%%%%%% Comment \newcommand{\comment}[1]{\marginpar{\footnotesize #1}} %Comment %\newcommand{\comment}[1]{} \begin{document} \title[Anisotropic H\"older and Sobolev spaces] {Anisotropic H\"older and Sobolev spaces for hyperbolic diffeomorphisms} \author{Viviane Baladi and Masato Tsujii} \address{CNRS-UMR 7586, Institut de Math\'e\-ma\-ti\-ques Jussieu, Paris, France} \email{baladi@math.jussieu.fr} \address{Department of Mathematics, Hokkaido University, Sapporo, Hokkaido, Japan} \email{tsujii@math.sci.hokudai.ac.jp} \date{May 2005} \begin{abstract} We study spectral properties of transfer operators for diffeomorphisms $T:X\to X$ on a Riemannian manifold $X$: Suppose that $\Omega$ is an isolated hyperbolic subset for $T$, with a compact isolating neighborhood $V\subset X$. We first introduce Banach spaces of distributions supported on $V$, which are anisotropic versions of the usual space of $C^p$ functions $C^p(V)$ and of the generalized Sobolev spaces $W^{p,t}(V)$, respectively. Then we show that the transfer operators associated to $T$ and a smooth weight $g$ extend boundedly to these spaces, and we give bounds on the essential spectral radii of such extensions in terms of hyperbolicity exponents. These bounds shed some light on those obtained by Kitaev for the radius of convergence of dynamical determinants. \end{abstract} \thanks{VB thanks Sculoa Normale Superiore Pisa for hospitality and Artur Avila for useful comments. MT thanks NCTS(Taiwan) for hospitality during his stay.} \maketitle %%%%%%%%%%%%%%%%%%%%% \section{Introduction} \label{intro} Let $X$ be a $d$-dimensional $C^\infty$ Riemannian manifold and let $T:X \to X$ be a diffeomorphism which is of class $C^1$ at least. For a given complex-valued continuous function $g$ on $X$, we define the Ruelle transfer operator $\L_{T,g}$ by \[ \L_{T,g}:C^0(X)\to C^0(X),\quad \L_{T,g} u(x)=g(x) \cdot u\circ T(x). \] Such operators appear naturally in the study of fine statistical properties of dynamical systems and provide efficient methods, for instance, to estimate of decay of correlations. (We refer e.g. to \cite{Ba0}.) Typical examples are the pull-back operator \begin{equation}\label{pullback} T^* u:=\L_{T,1}u=u\circ T , \end{equation} and the Perron-Frobenius operator \begin{equation}\label{PFoperator} \P u:=\L_{T^{-1},|\det DT^{-1}|}u=|\det DT^{-1}|\cdot u\circ T^{-1}. \end{equation} This paper is about spectral properties of the operator $\L_{T,g}$. We shall require a hyperbolicity assumption on the mapping~$T$: Let $\Omega\subset X$ be a compact isolated invariant subset for $T$, with a compact isolating neighborhood $V$, that is, $\Omega=\cap_{m\in \mathbb{Z}}T^{m}(V)$. We assume that $\Omega$ is a hyperbolic subset, that is, there exists an invariant decomposition $T_{\Omega}M=E^u\oplus E^s$ of the tangent bundle over $\Omega$, satisfying $\|DT^m|_{E^s}\|\le C\lambda^{m}$ and $\|DT^{-m}|_{E^u}\|\le C\lambda^m$, for all $m\ge 0$ and $x\in \Omega$, with constants $C>0$ and $0<\lambda<1$. Up to decomposing $\Omega$, we may suppose that the dimensions of $E^u(x)$ and $E^s(x)$ are constant. We define two local hyperbolicity exponents for each $x\in \Omega$ and each $m\ge 1$ by \begin{equation}\label{hypexp} \begin{aligned} \lambda_{x}(T^{m})&=\sup_{v\in E^s(x)\setminus\{0\}} \frac{\|DT^{m}(v)\|}{\|v\|}\le C\lambda^m \quad \mbox{and}\\ \nu_x(T^{m}) &=\inf_{v\in E^u(x)\setminus\{0\}} \frac{\|DT^m (v)\|}{\|v\|}\ge C^{-1}\lambda^{-m}. \end{aligned} \end{equation} Let $\omega$ be the Riemannian volume form on $X$, and let $|\det DT|$ be the Jacobian of $T$, that is, the function given by $T^*\omega=|\det DT|\cdot \omega$. Put, for $m\ge 1$, \[ \g^{(m)}(x)= \prod_{k=0}^{m-1} g(T^k(x)). \] For real numbers $q\le 0\le p$, and for $1\le t\le \infty$, we set \[ R^{p,q,t}(T, \g, \Omega, m)= \sup_{\Omega} |\det DT^{m}_x|^{-1/t} |\g^{(m)}(x)| \max\bigl \{ (\lambda_{x} (T^m))^{p}, (\nu_{x} (T^m))^{q} \bigr \}, \] where we read $(\cdot)^{1/\infty}=1$ for $t=\infty$. As $\log R^{p,q,t}(T, \g, \Omega, m)$ is sub-additive with respect to $m$, we have \[ R^{p,q,t}(T,\g,\Omega):=\lim_{m\to \infty}\root{m}\of{R^{p,q,t}(T, \g,\Omega,m)} =\inf_{m\ge 1}\root{m}\of{R^{p,q,t}(T, \g,\Omega,m)}. \] In this paper, we introduce Banach spaces of distributions supported on~$V$, show that the transfer operators $\L_{T,g}$ extend boundedly to those spaces and then give bounds for the essential spectral radii of these transfer operators, using the quantities $R^{p,q,t}(T,\g,\Omega)$ introduced above. The main feature in our approach is that we work in Fourier coordinates. The definition and basic properties of the Banach spaces will be given in later sections. Here we state the main theorem as follows. Let $r_{ess} (L|_{\BB})$ be the essential spectral radius of a bounded linear operator $L:\BB\to \BB$. For non-integer $s>0$, a mapping is of class $C^s$ if all its partial derivatives of order $[s]$ are $(s-[s])$-H\"older. \begin{theorem}\label{main} Suppose that $T$ is $C^r$ for a real number $r>1$, and let $\Omega$ be a hyperbolic invariant set with compact isolating neighborhood $V$, as described above. Then, for any real numbers $q<00$ so that if $\widetilde T$ and $\widetilde g$, respectively, are $\epsilon$-close to $T$ and $g$, respectively, in the $C^r$, resp. $C^{r-1}$, topology, then the associated operator $\L_{\widetilde T,\widetilde g}$ has same spectral properties than $\L_{T,g}$ on {\it the same Banach spaces.} Spectral stability can then be proved, as it has been done \cite{BKL} or \cite{GL} for the norms defined there (see also the historical comments below). Furthermore, we note that controlling the essential spectral radius on a scale of Sobolev spaces may be useful in view of the kneading approach relating transfer operators and dynamical determinants (see \cite{BB}). \smallskip {\bf Organization of the paper.} After defining a version of our norms in $\real^d$ in Section \ref{S1}, we proceed in the usual way: prove compact embeddings in Section \ref{S2} and a Lasota-Yorke type estimate in Section \ref{S5}. In Section \ref{S6}, we prove Theorem ~\ref{main} by reducing to the model from Sections \ref{S1}--\ref{S5} starting from a $C^r$ diffeomorphism on a manifold, and applying Hennion's \cite{He} theorem. For the H\"older spaces, our proof is elementary: it only uses integration by parts. For the Sobolev spaces, we require in addition a standard $L^t$ estimate (Theorem \ref{th:Ta}) for (operator-valued) pseudodifferential operators with $C^\infty$ symbols $P(\xi)$ depending only on $\xi$. \smallskip {\bf Historical Remarks.} The first estimates on the essential spectral radius of transfer operators were obtained for one-sided subshifts of finite type (Sinai, Ruelle, Bowen, ...) and expanding (or piecewise expanding) endomorphisms (Lasota-Yorke, Ruelle, Hofbauer-Keller, ...). In order to study Anosov diffeomorphisms, a reduction to the expanding case was used in the eighties (Pollicott, Haydn, Ruelle, ...). Since this essentially involves quotienting along a dynamical foliation, and since these foliations are in general only H\"older (even if $r=\infty$), this severely limited the sharpness of the bounds. In the early nineties, Rugh, and then Fried, introduced some ideas which allow to bypass this reduction to the expanding case, but only for analytic Anosov (or Axiom ~A) diffeomorphisms, and for a transfer operator associated to a``model" for the dynamics. Since there are no partitions of unity in the analytic category, translating the results for the model back into the manifold is not trivial. Precise statements and references for the, by now ``classical," results mentioned in this paragraph may be found in the book \cite{Ba0}. Under the assumptions of Theorem~\ref{main} Kitaev \cite{Ki} proved that the following ``dynamical Fredholm determinant'' \begin{equation} d(z)=\exp -\sum_{n=1}^\infty \frac {z^n} {n} \sum_{T^n(x)=x} \frac {1}{ |\det (DT^n(x)-\mathrm{Id})|} \end{equation} extends to a holomorphic function in the disc $\{ z \, : \, |z| \cdot R^{p,q,\infty}(T,1,\Omega) < 1\}$ for all $q<0From now on, we fix a compact subset $K\subset \real^d$ with non-empty interior. Let $C^{\infty}(K)$ be the space of complex-valued $C^\infty$ functions on $\real^d$ supported on $K$. For a real number $p$ and $1< t < \infty$, we define on $C^{\infty}(K)$ the norms \[ \|u\|\hol{p}=\sup_{n\ge 0} \; 2^{pn}\|u_n\|_{L^\infty} \quad \mbox{and}\quad \|u\|\sob{p,t}=\left\| \left(\sum_{n\ge 0} 4^{pn} |u_n |^2\right) ^{1/2}\right\|_{L^t}. \] It is known that the norm $\|u\|\hol{p}$ is equivalent to the H\"older norm \[ \|u\|_{C^p}=\max\left\{ \max_{|\alpha|\le [p]} \sup_{x\in \real^d} |\partial^\alpha u(x)|, \max_{|\alpha|= [p]} \sup_{x\in \real^d}\sup_{y\in \real^d/\{0\}} \frac{|\partial^\alpha u(x+y)-\partial^\alpha u(x)|}{|y|^{p-[p]}} \right\} \] provided that $p>0$ is not an integer, and $\|u\|\sob{p,t}$ is equivalent to the generalized Sobolev norm \[ \|u\|_{W^{p,t}}=\left\|(1+\Delta)^{p/2} u\right\|_{L^t} \] for any $p\in \real$ and $10$ is not an integer \footnote{If $p$ is an integer, we get a Zygmund class: our ``H\"older" terminology is slightly unprecise, but convenient.}). The generalized Sobolev space $W^{p,t}_*(K)$ for $10$;}\\ \chi_n(\xi)/2,\quad&\mbox{ if $n=0$.} \end{cases} \] Note that the $\psi_{\Theta, n, \sigma}(\xi)$ enjoy similar properties as those of the $\psi_n$, in particular the $L^1$-norm of the rapidly decaying function $\widehat \psi_{\Theta, n,\sigma}$ is bounded uniformly in $n$. For a $C^\infty$ function $u:\real^d\to \C$ with compact support, an integer $n\in \mathbb Z_+$, $\sigma\in \{+,-\}$, and a combination $\Theta=(\cone_+,\cone_-,\varphi_+,\varphi_-)$, we define \[ u_{\Theta,n,\sigma}=\psi_{\Theta,n,\sigma}(D) u =\widehat \psi_{\Theta,n,\sigma} * u. \] Since $1=\sum_{n= 0}^{\infty}\sum_{\sigma=\pm}\psi_{\Theta, n, \sigma}(\xi)$ by definition, we have $ u=\sum_{n\ge 0}\sum_{\sigma=\pm}u_{\Theta,n,\sigma}$. Let $p$ and $q$ be real numbers. For $u\in C^\infty(K)$, we define the anisotropic H\"older norm $\|u \|\hol{\Theta,p,q}$ by \begin{equation} \|u\|\hol{\Theta,p,q}=\max\left\{\; \sup_{n\ge 0}\; 2^{pn}\| u_{\Theta,n,+}\|_{L^\infty}, \; \;\sup_{n\ge 0} \; 2^{qn}\|u_{\Theta,n,-}\|_{L^\infty}\;\right\}, \end{equation} and the anisotropic Sobolev norm $\|u \|\sob{\Theta,p,q,t}$ for $10$ such that \[ |u_{\Theta,n,\sigma}(x)| \le \frac{C\sum_{\tau=\pm}\sum_{\ell\ge 0} 2^{-c\max\{n,\ell\}}\|u_{\Theta,\ell,\tau}\|_{L^t}} {d(x,\supp(u))^{b}} , \] for all $n\ge1$, all $u\in C^\infty(K)$, and all $x\in \real^d$ satisfying $d(x,\supp(u))>\epsilon$. \end{lemma} Note that the numerator of the right hand side above is bounded by $C\|u\|\hol{\Theta,p,q}$ in the case $t=\infty$, and by $C\|u\|\sob{\Theta,p,q,t}$ in the case $1p$. \begin{proof} Choose a $C^\infty$ function $\rho:\real^d\to[0,1]$ supported in the disk of radius $\epsilon/4$ centered at the origin and so that $\int \rho(x) dx =1$. Fix $u\in C^\infty(K)$. Let $U(\epsilon)$ be the $\epsilon$-neighborhood of $\supp(u)$. Put $\chi_0(x)=\int \mathbf{1}_{U(\epsilon/4)}(y) \cdot \rho(x-y) dy$, where $\mathbf{1}_{Z}$ denotes the indicator function of a subset $Z\subset \real^d$. Then $\chi_0$ is supported in $U(\epsilon/2)$, with $0\le \chi_0(x)\le 1$ for any $x\in \real^d$, and $\chi_0(x)= 1$ for $x\in \supp(u)$. Since $\|\chi_0\|_{C^c_*}$ is bounded by a constant depending only on $c$ and $\epsilon$, we have \begin{equation}\label{est1} \|\psi_j(D)\chi_0\|_{L^{\infty}}\le C(c, \epsilon) 2^{-c j}. \end{equation} Furthermore, integrating several times by parts on $\xi$ in \[ \psi_j(D)\chi_0(y)=(2\pi)^{-d} \int e^{i(y-w)\xi}\psi_j(\xi)\chi_0(w) d\xi dw, \] we can see that for any $y\in \real^d$ satisfying $d(y,\supp(\chi_0))\ge \epsilon/4$ \begin{equation}\label{est2} |\psi_j(D)\chi_0(y)|\le C(b,c,\epsilon)\cdot 2^{-c j} d(y,\supp(\chi_0))^{-b}. \end{equation} We assume $d(x,\supp(u))>\epsilon$ henceforth and estimate \[ \psi_{\Theta,n,\sigma}(D) u(x)=\psi_{\Theta, n, \sigma}(D)(\chi_0 u)(x) = \sum_{(\ell,\tau)\in \Gamma} \widehat\psi_{\Theta, n, \sigma}* (\chi_0 u_{\Theta, \ell, \tau})(x). \] By the H\"older inequality, we have \begin{align}\label{hpe} |\widehat\psi_{\Theta, n, \sigma}*(\chi_0 u_{\Theta, \ell, \tau})(x)|& \le \|\mathbf{1}_{U(\epsilon/2)}\cdot \widehat\psi_{\Theta, n, \sigma}(x-\cdot)\|_{L^{t'}} \|\chi_0 u_{\Theta, \ell, \tau}\|_{L^{t}}\\ &\le C(b,c,\epsilon, t')\cdot 2^{-c n} \cdot d(x,\supp(u))^{-b}\cdot \|u_{\Theta, \ell, \tau}\|_{L^{t}}\notag \end{align} for any $n$ and $\ell$, where $t'$ is the conjugate exponent of $t$, i.e. $t^{-1}+(t')^{-1}=1$. Suppose that $\ell \ge n+3$. Then we have \[ \psi_{\Theta, n, \sigma}(D) ((\psi_j(D)\chi_0)\cdot u_{\Theta, \ell, \tau})=0\quad \mbox{for $j<\ell-2$,} \] because $\supp(\psi_{\Theta, n, \sigma})$ does not meet $\supp(\psi_j)+\supp(\psi_{\Theta, \ell, \tau})$ which supports the Fourier transform of $(\psi_j(D)\chi_0)\cdot u_{\Theta, \ell, \tau}$. Thus \[ \psi_{\Theta, n, \sigma}(D) (\chi_0 u_{\Theta, \ell, \tau})= \sum_{j\ge \ell-2}\widehat\psi_{\Theta, n, \sigma}*((\psi_j(D)\chi_0)\cdot u_{\Theta, \ell, \tau}). \] For each $j\ge \ell-2$ with $\ell\ge n+3$, we can see from (\ref{est1}-\ref{est2}) that \begin{align*} &|\widehat\psi_{\Theta, n, \sigma}*( (\psi_j(D)\chi_0)\cdot u_{\Theta, \ell, \tau})(x)|\\ &\qquad\qquad \le \|\widehat \psi_{\Theta, n, \sigma}\|_{L^{\infty}}\cdot \|\mathbf{1}_{ \real^d \setminus U(\delta)} \cdot \psi_j(D)\chi_0\|_{L^{t'}}\cdot \|u_{\Theta, \ell, \tau}\|_{L^t}\\ &\qquad\qquad \qquad + \|\mathbf{1}_{U(\delta)}\cdot\widehat \psi_{\Theta, n, \sigma}(x-\cdot)\|_{L^{t'}} \cdot \|\psi_j(D)\chi_0\|_{L^{\infty}}\cdot \|u_{\Theta, \ell, \tau}\|_{L^t} \\ &\qquad \qquad\le C(b,c,\epsilon,t)\cdot 2^{-c j}\cdot d(x,\supp(u))^{-b} \cdot \|u_{\Theta, \ell, \tau}\|_{L^{t}}, \end{align*} where $\delta=\epsilon/2+d(x,\supp(u))/4$. (We decomposed the domain of integration in the convolution into $U(\delta)$ and its complement.) Hence, if $\ell\ge n+3$, we have \[ |\psi_{\Theta, n, \sigma}(D) (\chi_0 u_{\Theta, \ell, \tau})(x)| \le C(b,c,\epsilon,t)\cdot 2^{-c \ell}d(x,\supp(u))^{-b} \cdot \|u_{\Theta, \ell, \tau}\|_{L^{t}}. \] With this and (\ref{hpe}) we conclude the proof of the lemma. \end{proof} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Compact embeddings} \label{S2} Recall that $K\subset \real^d$ is a compact subset with non-empty interior. If $p'\le p$ and $q'\le q$, we have the obvious continuous inclusions \begin{equation}\label{eqn:inc} C_*^{\Theta,p,q}(K)\subset C_*^{\Theta,p',q'}(K),\quad W^{\Theta, p,q,t}_*(K)\subset W^{\Theta, p',q',t}_*(K)\quad\mbox{for $10$. We show that there exists a subsequence $\{k(j)\}$ such that $\{u^{(k(j))}\}$ is a Cauchy sequence in the norm $\|\cdot \|\hol{\Theta, p', q'}$(respectively $\|\cdot \|\sob{\Theta, p', q',t}$). For each $(n,\sigma)\in \Gamma$, the Fourier transform $\hat{u}^{(k)}_{\Theta,n,\sigma}$ of $u^{(k)}_{\Theta,n,\sigma}$ is a $C^{\infty}$ function supported on $\{\xi\mid 2^{n-1}\le |\xi|\le 2^{n+1}\}$, and its first order derivatives are bounded uniformly for $k\ge 1$ and $\xi\in \real^d$ since $(1+|x|)u^{(k)}_{\Theta,n,\sigma}(x)$ are uniformly bounded in $L^1$norm from Lemma \ref{lm:plp}. Hence, by Ascoli-Arzel\'a's theorem and by the diagonal argument, we can choose a subsequence $\{k(j)\}$ such that the sequences $\{\hat{u}^{(k(j))}_{\Theta,n,\sigma}\}_{j=0}^{\infty}$ are all Cauchy sequences with respect to the $L^\infty$-norm and so is the sequence $\{u^{(k(j))}_{\Theta,n,\sigma}\}_{j=0}^{\infty}$. This is the subsequence with the required property. Indeed, for given $\epsilon>0$, we can choose an integer $N>0$ so that $\sum_{n>N} (2^{(q'-q)n}+2^{(p'-p)n})E<\epsilon/2$, and then we have \begin{align*} &\|u^{(k(j))}-u^{(k(j'))}\|\hol{\Theta,p',q'}\\ &\qquad\le \epsilon/2+ \sum_{n\le N}\left( 2^{p'n} \left\|{u}^{(k(j))}_{\Theta,n,+}-{u}^{(k(j'))}_{\Theta,n,+}\right\|_{L^\infty}+2^{q'n} \left\|{u}^{(k(j))}_{\Theta,n,-}-{u}^{(k(j'))}_{\Theta,n,-}\right\|_{L^\infty}\right), \end{align*} (respectively the same inequality with the norms $\|\cdot \|\hol{\Theta,p',q'}$ and $\|\cdot \|_{L^\infty}$ replaced by $\|\cdot \|\sob{\Theta,p',q',t}$ and $\|\cdot\|_{L^t}$). The right hand side is $<\epsilon$ for large enough $j$, $j'$. \end{proof} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{A Lasota-Yorke type inequality} \label{S5} Let $r >1$. Let $K, K'\subset \real^d$ be compact subsets with non-empty interiors, and take a compact neighborhood $W$ of $K$. Let $T:W\to K'$ be a $C^{r}$ diffeomorphism onto its image. Let $\g:\real^d\to \C$ be a $C^{r-1}$ function such that $\supp(\g)\subset K$. In this section we study the transfer operator on $\real^d$: \[ L: C^{r-1}(K')\to C^{r-1}(K), \qquad L u(x)= \g(x)\cdot u\circ T(x). \] For two fixed combinations $\Theta=(\cone_+,\cone_-,\varphi_+, \varphi_-)$ and $\Theta'=(\cone'_+,\cone'_-,\varphi'_+, \varphi'_-)$ as in Section \ref{S1}, we make the following {\it cone-hyperbolicity} assumption on $T$: \begin{equation}\label{conehyp} DT_x^{tr}(\real^d \setminus \mbox{interior}\,(\cone_{+})) \subset \mbox{interior}\,(\cone'_{-})\cup\{0\}\quad \mbox{for all $x\in W$,} \end{equation} where $DT_x^{tr}$ denotes the transpose of the derivative of $T$ at $x$. We put \begin{align*} \|T\|_+&=\sup_{x\in \supp(g)} \sup_{0\neq DT_x^{tr}(\xi)\notin \cone'_{-}} \frac{\|DT_x^{tr}(\xi)\|} {\|\xi\|}, \quad \|T\|_-&=\inf_{x\in W} \inf_{0\neq \xi \notin \cone_{+}} \frac{\|DT_x^{tr}(\xi)\|}{\|\xi\|}. \end{align*} \begin{theorem}\label{th:main2} Fix $\Theta$ and $\Theta'$ and assume (\ref{conehyp}). For any $q<0 \|T\|_{-}-5,\qquad 2^{h_{\max}^{+}}< \|T\|_{+}+5. \] We write $(\ell, \tau) \hookrightarrow (n,\sigma)$ if either \begin{itemize} \item $(\tau,\sigma)=(+,+)$ and $n\le \ell+h_{\max}^+$, or \item $(\tau,\sigma)=(-,-)$ and $\ell+h_{\min}^{-}\le n$, or \item $(\tau,\sigma)=(+,-)$ and $\ell+h_{\min}\le n\le \ell+h_{\max}$. \end{itemize} We write $(\ell, \tau) \not\hookrightarrow (n,\sigma)$ otherwise. By the definition of $\not\hookrightarrow$, and by (\ref{conehyp2}), there exists an integer $N(T)>0$ such that for all $x\in \supp(g)$ \begin{equation}\label{lowerbd} d(\supp(\psi_{\Theta',n,\sigma}), DT_{x}^{tr}(\supp(\tilde \psi_{\Theta,\ell,\tau}))) \ge 2^{\max\{n,\ell\}-N(T)} \quad \mbox{if $(\ell, \tau) \not\hookrightarrow (n,\sigma)$.} \end{equation} \begin{proof}[Proof of Theorem \ref{th:main2}] For $v:=L u$, we have \[ v_{\Theta',n,\sigma}=\sum_{(\ell,\tau)\in \Gamma} \psi_{\Theta',n,\sigma}(D) L (u_{\Theta, \ell, \tau}). \] We define $\mathbf{S}$ as the formal matrix of operators \[ S_{n,\sigma}^{\ell,\tau}u = \begin{cases} \psi_{\Theta',n,\sigma}(D) L \, u,& \mbox{ if $(\ell,\tau)\hookrightarrow (n,\sigma)$},\\ \psi_{\Theta',n,\sigma}(D) L\,\tilde{\psi}_{\Theta,\ell,\tau}(D) u &\mbox{ if $(\ell,\tau)\not\hookrightarrow (n,\sigma)$}, \end{cases} \] for $((\ell,\tau), (n,\sigma))\in \Gamma\times \Gamma$. That is, we set \[ \mathbf{S}\left((u_{\Theta,\ell,\tau})_{(\ell,\tau)\in \Gamma}\right) =\left(\sum_{(\ell,\tau)\in \Gamma}S_{n,\sigma}^{\ell,\tau} u_{\Theta,\ell,\tau}\right)_{(n,\sigma)\in \Gamma}. \] Since $\tilde{\psi}_{\Theta,\ell,\tau}(D)u_{\Theta,\ell,\tau}=u_{\Theta,\ell,\tau}$, we have the commutative relation $\mathbf{S}\circ \QQ_\Theta=\QQ_{\Theta'}\circ L$. For the proof of Theorem \ref{th:main2}, it is enough to show \[ \|\mathbf{S}\mathbf{u}\|_{p,q,\infty} 1$.} \end{equation} The required estimate on $H^{\ell,\tau}_{n,\sigma}$ follows if we show \begin{equation}\label{eqn:Kernelest} |V_{n,\sigma}^{\ell,\tau}(x,y)|\le C(T,g) 2^{-(r-1)\max\{n,\ell\}}\cdot 2^{d\min\{n,\ell\}}b(2^{\min\{n,\ell\}}(x-y)) \end{equation} for some $C(T,g)>0$ and all $(\ell,\tau)\not\hookrightarrow (n,\sigma)$. Indeed, as the right hand side of (\ref{eqn:Kernelest}) is written as a function of $x-y$, say $B(x-y)$, we have, by Young's inequality, \[ \|H^{\ell,\tau}_{n,\sigma}v\|_{L^t}\le \|B * v\|_{L^t}\le \|B\|_{L^1} \|v\|_{L^t} \le C(T) 2^{-(r-1)\max\{n,\ell\}}\cdot \|b\|_{L^1}\cdot \|v\|_{L^t}. \] Below we prove the estimate (\ref{eqn:Kernelest}). We may assume that $r$ is not an integer, up to replacing $r$ by $r' \in (p-q+1, r]$. Integrating (\ref{Vkernel}) by parts $[r]-1$ times on $w$, we obtain \[ V_{n,\sigma}^{\ell,\tau}(x,y)= \int e^{i(x-w)\xi+i(T(w)-T(y))\eta} F(\xi,\eta,w)\psi_{\Theta',n,\sigma}(\xi) \tilde{\psi}_{\Theta,\ell,\tau}(\eta)dwd\xi d\eta, \] where $F(\xi,\eta,w)$ is a $C^{r-[r]}$ function in $w$ which is $C^\infty$ in the variables $\xi$ and $\eta$. Using (\ref{lowerbd}), we can see that for all $\alpha$, $\beta$ \[ \|\partial_\xi^\alpha\partial_\eta^\beta F\|_{C^{r-[r]}}\le C_{\alpha,\beta}(T,g) 2^{-n|\alpha|-\ell|\beta|-([r]-1)\max\{n,\ell\}}. \] Put $F_{k}(\xi,\eta,w) =\psi_{k}(D)F (\xi,\eta,w)$, where the pseudodifferential operator $\psi_{k}(D)$ acts on $F (\xi,\eta,w)$ as a function of $w$. Then each $F_{k}(\xi,\eta,w)$ is $C^\infty$ and satisfies \begin{equation}\label{diff} \|\partial_\xi^\alpha\partial_\eta^\beta\partial_w^{\gamma} F_{k}\|_{L^\infty} \le C_{\alpha,\beta,\gamma}(T,g)\cdot 2^{-n|\alpha|-\ell|\beta|-([r]-1)\max\{n,\ell\}-(r-[r]-|\gamma|)k} \end{equation} for any $\alpha$, $\beta$, and $\gamma$. (Note that $\partial_\xi^\alpha\partial_\eta^\beta\partial_w^{\gamma}F$, as a function of $w$, lies to $C^{r-[r]-|\gamma|}_*(\real^{d})$ with $r-[r]-|\gamma|\in \real$.) Correspondingly, we decompose $V_{n,\sigma}^{\ell,\tau}(x,y)$ into \begin{equation}\label{Wk} W^{(k)}(x,y)=\int e^{i(x-w)\xi+i(T(w)-T(y))\eta} G_{k}(\xi,\eta,w) dwd\xi d\eta \quad \mbox{for $k\ge 0$,} \end{equation} where $G_{k}(\xi,\eta,w)=F_{k}(\xi,\eta,w)\psi_{\Theta',n,\sigma}(\xi) \tilde{\psi}_{\Theta,\ell,\tau}(\eta)$. Recall the choice of $N(T)$ from (\ref{lowerbd}). We first estimate $W^{(k)}(x,y)$ for $k>\max\{n,\ell\}-N$. Decompose $W^{(k)}(x,y)$ into \begin{align*} \int e^{i(x-w)\xi+i(T(w)-T(y))\eta} G_{k}(\xi,\eta,w) \psi_i(2^{n}(x-w))\psi_j(2^{\ell}(T(w)-T(y))) dwd\xi d\eta \end{align*} for $i\ge 0$ and $j\ge 0$, each of which is denoted by $W^{(k)}_{i,j}(x,y)$. Integrating by parts on $\xi$ for $d+1$ times if $i>0$, and integrating by parts on $\eta$ for $d+1$ times if $j>0$, we can see \begin{align}\label{Wregij} |W_{i,j}^{(k)}(x,y)| %&\le C(T) 2^{-n-\ell -d\max\{n-i,\ell-j\}+(d+1)(n-i)+(d+1)(\ell-j) %-([r]-1)\max\{n,\ell\}-(r-[r])k}\\ &\le C(T,g) 2^{d\min\{n-i,\ell-j\}-([r]-1)\max\{n,\ell\}-(r-[r])k-i-j} . \end{align} In fact the case $i>0$ and $j>0$ can be shown as follows and the other cases are similar: The result of the integration by parts is a sum of terms of the form \[ \pm \int e^{i(x-w)\xi+i(T(w)-T(y))\eta} \partial_{\xi}^\alpha\partial_{\eta}^\beta G_k(\xi,\eta,w) G_{\alpha,\beta}(x,w,y) dwd\xi d\eta , \] where $|\alpha|=|\beta|=d+1$, and \[ G_{\alpha,\beta}(x,w,y) =\frac{(x-w)^{\alpha} (T(w)-T(y))^{\beta} \cdot \psi_i(2^{n}(x-w))\psi_j(2^{\ell}(T(w)-T(y)))} {|x-w|^{2|\alpha|}|T(w)-T(y)|^{2|\beta|}}. \] Note that $|G_{\alpha,\beta}(x,w,y)|\le C(T)\cdot 2^{(d+1)(n-i)+(d+1)(\ell-j)}$ and that $G_{\alpha,\beta}(x,w,y)\neq 0$ only if $|x-w|<2^{-n+i+1}$ and $|T(w)-T(y)|<2^{-\ell+j+1}$. Together with (\ref{diff}), we see \begin{align*} &\int |\partial_{\xi}^\alpha\partial_{\eta}^\beta G_k(\xi,\eta,w) G_{\alpha,\beta}(x,w,y)| dw \le \\ &C(T,g) 2^{-n(d+1)-\ell(d+1) -([r]-1)\max\{n,\ell\} -d\max\{n-i,\ell-j\} +(d+1)(n-i)+(d+1)(\ell-j)-(r-[r])k}\\ &= C(T,g) 2^{-nd-\ell d -([r]-1)\max\{n,\ell\} +d\min\{n-i,\ell-j\} -(r-[r])k-i-j}. \end{align*} Since $\partial_{\xi}^\alpha\partial_{\eta}^\beta G_k(\xi,\eta,w)\neq 0$ only if $\xi\in \supp (\psi_{\Theta',n,\sigma})$ and $\eta\in \supp (\tilde{\psi}_{\Theta,\ell,\tau})$, we get (\ref{Wregij}). Since $r>[r]$, for all $x$, $y$ and $i$, $j$, (\ref{Wregij}) implies \begin{equation}\label{zero} \sum_{k>\max\{n,\ell\}-N}|W_{i,j}^{(k)}(x,y)| \le C(T,g) 2^{d\min\{n-i,\ell-j\}-(r-1)\max\{n,\ell\}-i-j}. \end{equation} Thus, for all $x$ and $y$, \begin{equation}\label{one} \sum_{k>\max\{n,\ell\}-N}\sum_{i\ge 0}\sum_{j\ge 0} |W_{i,j}^{(k)}(x,y)| \le C(T,g) 2^{-(r-1)\max\{n,\ell\}+d\min\{n,\ell\}} . \end{equation} If $|x-y| \ge 2^{-\min\{n,\ell\}}$ we have better estimates: Let $M=M(T)$ be so that $|T(x)-T(y)|\ge 2^{-M} |x-y|$ for all $x, y\in W$. Let $q_0=q_0(x,y)\le \min\{n,\ell\}$ be the smallest integer satisfying $|x-y|> 2^{-q_0}$. If $\min\{n-i,\ell-j\}\ge q_0+M+4$, we have $W^{(k)}_{i,j}(x,y)=0$ since $\psi_i(2^{n}(x-z))\psi_j(2^{\ell}(T(z)-T(y)))\equiv 0$. Therefore, we get from (\ref{zero}) that \begin{equation}\label{two} \sum_{k>\max\{n,\ell\}-N}\sum_{i\ge 0}\sum_{j\ge 0} |W_{i,j}^{(k)}(x,y)| \le C(T,g) 2^{-(r-1)\max\{n,\ell\}-\min\{n,\ell\}+(d+1)q_0}. \end{equation} To finish, we consider the case $k\le\max\{n,\ell\}-N$. Integrate (\ref{Wk}) by parts on $w$ once, obtaining \[ W^{(k)}(x,y)=\int e^{i(x-w)\xi+i(T(w)-T(y))\eta} \widetilde G_k(\xi,\eta,w)dwd\xi d\eta \quad \mbox{for $k\ge 0$.} \] Decompose $W^{(k)}(x,y)$ into the sum of $\widetilde W^{(k)}_{i,j}(x,y)$ for $i\ge 0$ and $j\ge 0$, each of which is defined in the same manner as $W^{(k)}_{i,j}(x,y)$, but with $G_k(\xi,\eta,w)$ replaced by $\widetilde G_k(\xi,\eta,w)$. Integrate $\widetilde W^{(k)}_{i,j}(x,y)$ by parts on $\xi$ for $d+1$ times if $i>0$, on $\eta$ for $d+1$ times if $j>0$. Then we obtain, by (\ref{diff}) for $|\gamma|=1$, \begin{align*} |\widetilde W^{(k)}_{i,j}(x,y)| %&\le C(T) 2^{-[r]\max\{n,\ell\}-n-\ell-(r-[r]-1)k-d\max\{n-i,\ell-j\}+(d+1)(n-i+\ell-j)}\\ &\le C(T,g) 2^{d\min\{n-i,\ell-j\}-[r]\max\{n,\ell\}+([r]+1-r)k -i-j}, \end{align*} and hence, since $[r]+1 > r$, we find for any $x$, $y$, $i$ and $j$, \begin{equation*} \sum_{k\le \max\{n,\ell\}-N}|\widetilde W^{(k)}_{i,j}(x,y)| \le C(T,g) 2^{d\min\{n-i,\ell-j\}-(r-1)\max\{n,\ell\}-i-j} . \end{equation*} Therefore \begin{equation}\label{three} \sum_{k\le \max\{n,\ell\}-N}\sum_{i\ge 0}\sum_{j\ge 0}|\widetilde W_{i,j}^{(k)}(x,y)| \le C(T,g) 2^{-(r-1)\max\{n,\ell\}+d\min\{n,\ell\}}. \end{equation} By the same argument as above, we see that, for $|x-y|> 2^{-q_0}\ge 2^{-\min\{n,\ell\}}$, \begin{equation}\label{four} \sum_{k\le \max\{n,\ell\}-N}\sum_{i\ge 0}\sum_{j\ge 0}|\widetilde W_{i,j}^{(k)}(x,y)| \le C(T,g) 2^{-(r-1)\max\{n,\ell\}-\min\{n,\ell\}+(d+1)q_0 }. \end{equation} The inequalities (\ref{one},\ref{two},\ref{three},\ref{four}) imply (\ref{eqn:Kernelest}) for noninteger $r$. \end{proof} %%%%%%%%%%%% \section{Partitions of unity} \label{Spartition} Let $r>0$ and recall $K\subset \real^d$ is compact with nonempty interior. A $C^r$ partition of unity on $K$ is by definition a finite family of $C^r$ functions $g_{i}:\real^d\to [0,1]$, $1\le i\le I$, such that $\sum_{i} g_i(x)= 1$ for $x\in K$ and $\sum_{i} g_i(x)\le 1$ for $x\in \real^d$. The intersection multiplicity of a partition of unity is $\nu:=\sup_x\#\{i \mid g_i(x)>0\}$. For $u\in C^\infty(K)$, we set $u_i:=g_i u$ so that $u=\sum_i u_i$. In this section, we compare the norms of $u$ and those of the $u_i$'s. (This will be useful to refine partitions in the proof of Theorem~\ref{main} in the next section.) \begin{lemma} \label{lm:pu} Let $q\le 0\le p$ satisfy $p-q0$ so small that the intersection multiplicity of the sets $U(i,\epsilon)$ is $\nu$. Decompose $\QQ_{\Theta} u_i$ (recall Section \ref{prelim}) into \[ \mathbf{u}^{\mathrm{body}}_i=\mathbf{1}_{U(i,\epsilon)}\cdot \QQ_{\Theta}u_i\quad \mbox{and}\quad \mathbf{u}^{\mathrm{tail}}_i=\QQ_{\Theta}u_i-\mathbf{u}^{\mathrm{body}}_i. \] On the one hand, Lemma \ref{lm:plp} implies \[ \|\mathbf{u}^{\mathrm{tail}}_i\|_{p,q,\infty}\le C \|u_i\|\hol{\Theta,p',q'}\quad \mbox{and}\quad \|\mathbf{u}^{\mathrm{tail}}_i\|_{p,q,t}\le C \|u_i\|\sob{\Theta,p',q',t}. \] On the other hand, since the bound on the intersection multiplicity is $\nu$, we have \[ \left\|\sum_{i}\mathbf{u}_i^{\mathrm{body}}\right\|_{p,q,\infty}\le \nu \cdot \max_i \left\|\mathbf{u}_i^{\mathrm{body}}\right\|_{p,q,\infty}, \] and, using the H\"older inequality, \[ \left\|\sum_{i}\mathbf{u}_i^{\mathrm{body}}\right\|_{p,q,t}\le \nu^{1/t'} \cdot \left[ \sum_i \left\|\mathbf{u}_i^{\mathrm{body}}\right\|_{p,q,t}^t\right]^{1/t}. \] Therefore we obtain the estimates in the lemma by using (\ref{pqt}). \end{proof} The next proposition gives bounds in the opposite direction. \begin{proposition}\label{prop:pu} Let $q\le 0\le p$, and let $p'$ and $q'$ be real numbers with $p'1$ that does not depend on $m>0$ such that \begin{equation}\label{eta} C(t)^{-1} R^{p,q,t}(T,\g,\Omega,m)\le \Lambda_{m,t}\le C(t) R^{p,q,t}(T,\g,\Omega,m). \end{equation} The definition of $\Lambda_{m,t}$ involves first taking a maximum and a product, and then taking the supremum over $x$. We shall apply Theorem~\ref{th:main2} in a moment: the upper bound there corresponds to taking a supremum first. Since different points in $\kappa_j(V_{m,jk})$ may have very different itineraries, it is necessary to refine our partition of unity, depending on $m$. This will not cause problems since we can take arbitrarily fine finite $C^\infty$ partitions of unity on $\real^d$, with intersection multiplicities bounded uniformly by a constant depending only on $d$. Using such a partition of unity, we decompose the function $u_{jk}=(\phi_k(\phi_{j} \circ T^{-m}) \cdot u )\circ \kappa_k^{-1} $ into $u_{jk,i}$ for $1\le i \le I_{jk}$. Take combinations $ \Theta'_{k}>\Theta_k$ (close to $\Theta_k$) so that the iterated cone-hyperbolicity condition (\ref{charthyp2}) holds with $\Theta_k$ replaced by $\Theta'_{k}$. For each $m$, by taking a sufficiently fine partition of unity, we can apply Theorem~\ref{th:main2} to obtain, for $1\le i\le I_{jk}$, \[ \|g^{(m)}\circ \kappa_j^{-1}\cdot u_{jk,i}\circ T^m_{jk}\|\hol{\Theta_j,p,q} \le 2\Lambda_{m,\infty}\cdot \|u_{jk, i}\|\hol{\Theta'_k,p,q} +C\|u_{jk,i}\|\hol{\Theta'_k,p',q'}. \] Then, using Lemma \ref{lm:pu} and Proposition \ref{prop:pu}, we get \begin{align*} \|g^{(m)}\circ \kappa_j^{-1}\cdot u_{jk}\circ T^m_{jk}\|\hol{\Theta_j,p,q}\le C_1\cdot \Lambda_{m,\infty}\cdot \|u_{jk}\|\hol{\Theta_k,p,q}+C_1(m)\cdot \|u_{jk}\|\hol{\Theta_k,p',q'}, \end{align*} where $C_1$ is a constant that does not depend on $m$. Thus, using again Lemma~ \ref{lm:pu} and Proposition \ref{prop:pu}, we obtain the following Lasota-Yorke type inequalities: \begin{align*} \|\mathcal{L}_{T,g}^m u\|_{C_*^{p,q}(T,V)}\le C_2 \cdot J\cdot \Lambda_{m,\infty}\cdot \|u\|_{C_*^{p,q}(T,V)} +C_2(m) \| u\|_{C_*^{p',q'}(T,V)} , \, m \ge 1. \end{align*} Likewise, we obtain for $1