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Transfer operator, essential spectral radius, Banach space, 
Sobolev space, hyperbolic set
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\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 $q<0<p$ with $p-q<r-1$, there exist Banach spaces
$C^{p,q}_*(T,V)$ and $W^{p,q,t}_*(T,V)$, for $1<t<\infty$, of  distributions supported on $V$, 
such that, for any $C^{r-1}$ function $g:X\to \C$ supported on $V$, 
\begin{enumerate}
\item (H\"older spaces)
 $\L_{T,g}$ extends boundedly  to $C^{p,q}_*(T,V)$ and
\[
r_{ess}(\L_{T,g}|_{C^{p,q}_*(T,V)})\le 
R^{p,q,\infty}(T,\g,\Omega).
\]
\item (Sobolev spaces)
 $\L_{T,g}$ extends boundedly to 
$W^{p,q,t}_*(T, V)$ and 
\[
r_{ess}(\L_{T,g}|_{W^{p,q,t}(T,V)}) 
\le  R^{p,q,t}(T,\g,\Omega) \, , \forall \, 1<t<\infty.
\]
\end{enumerate}
The Banach spaces $C^{p,q}_*(T,V)$ and $W^{p,q,t}_*(T,V)$ contain the set $C^p(V)$ of $C^p$ 
functions supported on $V$. 
\end{theorem}

\smallskip

{\bf Hyperbolic attractors and SRB measures.}
Let us see how to apply Theorem \ref{main} to a hyperbolic attractor:
Assume in addition to the above that $\Omega$ is an attracting hyperbolic
set and take the isolating neighborhood $V$ so that 
$T(V)\subset \mbox{interior}(V)$. Consider the pull-back operator $T^*$ defined by (\ref{pullback}) and the Perron-Frobenius operator $\P$ defined by (\ref{PFoperator}).
Note that these operators 
are adjoint to each other:
\begin{equation}\label{adjoint}
\int_{X} T^* u\cdot v \; d\omega=\int_{X} u \cdot \P v \;d\omega.
\end{equation}
Let $h:X\to [0,1]$ be a $C^\infty$ function supported on $V$ and satisfying $h\equiv 1$ 
on $T(V)$. 
Then the action of the 
operator $\mathcal{L}_{T^{-1},g}$ with $g(x)=|\det DT^{-1}(x)| \cdot h(x)$ coincides with that 
of the Perron-Frobenius operator $\P$ on $C^{p,q}_*(T^{-1},V)$ and 
$W^{p,q,t}_*(T^{-1},V)$, $1<t<\infty$. Therefore Theorem \ref{main} easily implies:

\begin{theorem}\label{PF} Let $\Omega$ be a hyperbolic attractor for a $C^r$diffeomorphism 
$T:X\to X$, and let $V$ be a compact neighborhood of $\Omega$ such that 
$T(V)\subset \mbox{{\rm interior}} (V)$ and  $\cap_{m\ge 0}T^m(V)=\Omega$. Then, 
for real numbers $q<0<p$ with $p-q<r-1$, the Perron-Frobenius operator 
$\P$ extends boundedly to $\P:C^{p,q}_*(T^{-1},V)\to C^{p,q}_*(T^{-1},V)$ and also to 
$\P:W^{p,q,t}_*(T^{-1},V)\to W^{p,q,t}_*(T^{-1},V)$, and it holds
\[
r_{ess}(\P|_{C^{p,q}_*(T^{-1},V)})\le R^{-q,-p,1}(T,1,\Omega)<1
\]
and
\[
r_{ess}(\P|_{W^{p,q,t}_*(T^{-1},V)})\le R^{-q,-p,t/(t-1)}(T,1,\Omega)<1
\quad \mbox{for $1<t<\infty$.}
\]
If we assume further that $T:X\to X$ is an Anosov diffeomorphism on a closed manifold $X$, then,  for real numbers $q<0<p$ with $p-q<r-1$, the pull-back operator $T^*$ extends boundedly to 
$T^*:C^{p,q}_*(T,X)\to C^{p,q}_*(T,X)$ and also to $T^*:W^{p,q,t}_*(T,X)\to W^{p,q,t}_*(T,X)$, 
and it holds
\[
r_{ess}(T^*|_{C^{p,q}_*(T,X)})\le R^{p,q,\infty}(T,1,X)<1 ,
\]
and
\[
r_{ess}(T^*|_{W^{p,q,t}_*(T,X)})\le R^{p,q,t}(T,1,X)<1, 
\quad \mbox{for $1<t<\infty$.}
\]
\end{theorem}


\begin{remark}
From (\ref{adjoint}) and Theorem \ref{PF}, if $T$ is Anosov, the 
Perron-Frobenius operator 
$\P$ acts naturally on the (strong) dual spaces of $C^{p,q}_*(T,X)$ and $W^{p,q,t}_*(T,X)$, 
and we have for instance
\[
r_{ess}(\P|_{(C^{p,q}_*(T,X))^*})\le R^{p,q,\infty}(T,1,X)<1 ,
\]
for real numbers $q<0<p$ with $p-q<r-1$.
\end{remark}

Once we have the estimates in Theorem \ref{PF}, 
it is not difficult to see that the spectral radii of the Perron-Frobenius operator 
$\P$ on $C^{p,q}_*(T^{-1},V)$ and $W^{p,q,t}_*(T^{-1},V)$ are equal to one.
If $(T,\Omega)$ is topologically mixing in addition, then $1$ is the unique eigenvalue on the unit circle, it is a simple eigenvalue, and
the fixed vector gives rise to  the SRB measure $\mu$ on $\Omega$: This 
corresponds to exponential decay of correlations for $C^p$ observables and $\mu$. (See Blank--Keller--Liverani \cite[\S3.2]{BKL} for example.)


\smallskip

{\bf Spectral stability and kneading theory.}
We  point out that, in the setting of Theorem \ref{main}, 
there is $\epsilon >0$  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 $q<0<p$ with $p-q<r-1$.
Kitaev's bounds are in fact slightly more general 
(see Remark \ref{MTO}),
but
no spectral interpretation of his result  is given in \cite{Ki}.


The first article which analyses the  essential spectrum 
of the transfer operator $\P$ associated to an Anosov diffeomorphism
and  acting on
a space of distributions on the manifold  is due to
Blank, Keller, and Liverani \cite{BKL}. 
Another breakthrough of that paper is
that perturbation theory was made possible, because the same
Banach space could be used for nearby Anosov diffeomorphisms.
The methods in \cite{BKL}, however,
only allowed to exploit very limited smoothness of the diffeomorphism
in the Lasota-Yorke  
bounds, so that the estimates there are far from being sharp.


In 2004, 
Gou\"ezel and Liverani \cite[v1]{GL}   obtained
bounds for the essential spectral
radius of $\P$ for an Anosov diffeomorphism $T$, acting on 
Banach spaces $\BB^{p,q}$,  for
{\it integers} $q<0<p$ with $p-q<r-1$, using methods different from ours.
The space $\BB^{p,q}$ is similar in spirit to (but not quite as simple as)  
the dual of our H\"older space $C_*^{p,q}(T,V)$. 
Although the upper bounds for $r_{ess}(\P|_{\BB^{p,q}})$
in \cite[v1]{GL} are not
sharp, they tend to zero as $(p-q) \to \infty$ 
if $r=\infty$, and this
is one of the important contributions of \cite[v1]{GL}.


Also in 2004, if $r=\infty$ and {\it under the very strong additional
assumption} that either the stable or the unstable foliation of the Anosov
diffeomorphism $T$
is $C^\infty$, the first named author
of the present paper introduced tangential pseudodifferential
methods \cite{Ba} to  define ``foliated''
Sobolev spaces $\widetilde W^{p,q,t}$
for $1<t<\infty$,
and to prove e.g. the  upper bounds 
$R^{-q,-p,t/(1-t)}(T, 1, X)$  
on the essential spectral radius of $\P$ 
acting on  $\widetilde W^{p,q,t}( T^{-1},X)$.
(The proof in \cite{Ba} requires  pseudodifferential
operators with symbols $\Psi(x,\xi)$ which are not
$C^\infty$ in $x$.)  As mentioned in \cite{Ba}, this foliated approach
should extend to the case when either the stable or the unstable foliation
is $C^{1+\epsilon}$. However this generalization requires
paradifferential techniques.


As we were finishing to write the present
paper, Gou\" ezel and Liverani substantially
improved their bounds in a second
version \cite[v2]{GL} of their article, still in the
Anosov case. 
Their bounds now look similar to ours:
In our notation, they  prove that
if $p-q<r-1$, where 
$p\ge 1$ is an integer, the essential
spectral radius of $\P$ on their Banach space $\BB^{p,q}$ 
is bounded by $R^{-q,-p,\infty}(T^{-1},1,X)=R^{p,q,\infty}(T,1,X)$.
They do not have an analogue of our Sobolev spaces. 

%%%%%%%%%%%%%%%%%%%%%
\section{Definition of the anisotropic norms.}
\label{S1}

We recall a few facts on Sobolev  and H\"older norms, which motivate our definition 
of anisotropic norms.

Fix an integer $d \ge 1$ and a $C^\infty$ function $\chi:\real\to [0,1]$ with
\[
\chi(s)=
1, \quad \mbox{for $s\le 1$,}\qquad
\chi(s)=0, \quad \mbox{for $s\ge 2$.}
\] 
For $n\in \mathbb Z_+$, define 
$\chi_n:\real^d\to [0,1]$ as $\chi_n(x)=\chi(2^{-n}|\xi|)$
and, setting $\chi_{-1}\equiv 0$,
\begin{align*}
&\psi_n:\real^d\to [0,1],\qquad 
\psi_n(\xi)=\chi_n(\xi)-\chi_{n-1}(\xi) .
\end{align*}
We have  
$1=\sum_{n= 0}^{\infty}\psi_{n}(\xi)$, and   $\supp(\psi_n)\subset 
\{\xi\mid 2^{n-1}\le |\xi|\le  2^{n+1}\}$. 
Also $\psi_n(\xi)=\psi_1(2^{-n+1}\xi)$ for $n\ge 1$. Thus,
for every multi-index $\alpha$, there exists a constant $C_\alpha$ such that  
$\|\partial^\alpha \psi_n\|_{L^\infty} \le C_\alpha 2^{-n|\alpha|}$ for all $n\ge 0$, 
and the inverse 
Fourier transform  of $\psi_n$,
$$\widehat \psi_n(x)=
(2\pi)^{-d} \int  e^{ix\xi} \psi_n(\xi)  d\xi,
$$  
decays rapidly, satisfies $\widehat \psi_n(x)=2^{d(n-1)} \widehat \psi_1(2^{n-1}x)$
for $n\ge1$ and all $x$,
and is bounded uniformly in $n$ with 
respect to the $L^1$-norm.

We decompose each $C^\infty$ function $u:\real^d\to \C$ with compact support as
$u=\sum_{n\ge 0}u_n$ by defining for
integer 
$n\in \mathbb Z_+$, 
\begin{equation}\label{decomp}
u_n(x)=\psi_n(D) u(x):=(2\pi)^{-d}\int e^{i(x-y)\xi}\psi_n(\xi)u(y) dy d\xi
=\widehat \psi_n * u  .
\end{equation}
\begin{remark}
The operator $\psi_n(D)$ in (\ref{decomp}) is 
the ``pseudodifferential operator with symbol $\psi_n$."
We refer 
to the books \cite{Ho} and \cite{Ta2} for more about pseudodifferential 
operators, although our text is self-contained,
except for Theorem~\ref{th:Ta}.   
\end{remark}

>From 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 $1<t<\infty$. 
(See \cite[Appendix A]{Ta} for a brief account.) 


The H\"older space  $C^{p}_*(K)$ is the completion of $C^{\infty}(K)$ with respect 
to 
the norm $\|\cdot \|\hol{p}$ (by the above, it  coincides with the space of 
$C^p$ functions supported on $K$, if $p>0$ 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 $1<t<\infty$ is the completion of $C^{\infty}(K)$ 
with respect to the norm $\|\cdot \|\sob{p,t}$. 

We are going to introduce  anisotropic versions of the norms and spaces above.   
Let  $\cone_+$ and $\cone_-$ be
closed cones in $\real^d$  with nonempty interiors 
such that $\cone_+\cap\cone_-=\{0\}$. 
Let then
$\varphi_+, \varphi_-:\sphere\to [0,1]$ be
$C^{\infty}$ functions on the unit sphere $\sphere$ in $\real^d$ satisfying
\begin{equation}\label{vp}
\varphi_+(\xi)=
\begin{cases}
1, &\mbox{if $\xi\in \sphere\cap \cone_{+}$;}\\
0, &\mbox{if $\xi\in \sphere\cap \cone_{-}$,}
\end{cases} \qquad 
\varphi_-(\xi)=1-\varphi_+(\xi).
\end{equation}
We shall work with combinations
$\Theta=(\cone_+,\cone_-,\varphi_+,\varphi_-)$ as above.
For another such combination $\Theta'=(\cone'_+, \cone'_-, \varphi'_+, \varphi'_-)$,
we write $\Theta'<\Theta$ if
\[
\mbox{closure} (\real^d\setminus \cone_+)\subset\mbox{interior}
(\cone'_-) \cup \{0\} 
\]
(This implies $\cone'_+\subset \cone_+$ and $\cone'_-\supset \cone_-$ in particular.) 
For $n\in \mathbb{Z_+}$ and $\sigma\in\{+,-\}$,  we define
\[
\psi_{\Theta, n, \sigma}(\xi)=
\begin{cases}
\psi_n(\xi)\varphi_{\sigma}(\xi/|\xi|),\quad&\mbox{ if $n>0$;}\\
\chi_n(\xi)/2,\quad&\mbox{ if $n=0$.}
\end{cases}
\]
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 $1<t<\infty$ by
\begin{equation}
\|u\|\sob{\Theta,p,q,t}=\; 
\left\| \left(\sum_{n\ge 0} 4^{pn}|  u_{\Theta,n,+}|^2
+ 4^{qn}|u_{\Theta,n,-}|^2 \right)^{1/2}\right\|_{L^t}.
\end{equation}
Let $C^{\Theta,p,q}_*(K)$ be the completion of $C^{\infty}(K)$ with respect to the norm 
$\|\cdot \|\hol{\Theta, p,q}$. 
Likewise, for  $1<t<\infty$, let $W^{\Theta,p,q,t}_*(K)$  be the completion of 
$C^{\infty}(K)$ with respect to the norm $\|\cdot \|\sob{\Theta, p, q,t}$. 
We will call these spaces $C^{\Theta,p,q}_*(K)$ and $W^{\Theta,p,q,t}_*(K)$ of 
distributions the anisotropic H\"older and Sobolev space respectively. In Section~\ref{S6}, we will construct the Banach spaces in Theorem \ref{main} by patching these
H\"older and Sobolev spaces 
using local coordinates. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Preliminaries}\label{prelim}
In studying the anisotropic H\"older and Sobolev norms, it is convenient to work 
in different ``coordinates" that we introduce next.
Let $\Gamma=\{ (n,\sigma)\mid n\in \integer_+ ,\sigma\in\{+,-\}\}$ and put
\[
\C^{\Gamma}=\{(f_{n,\sigma})_{(n,\sigma)\in \Gamma}\mid f_{n,\sigma}\in \C\}.
\]
For real numbers $p$ and $q$, and for $\mathbf{f}=(f_{n,\sigma})_{(n,\sigma)\in \Gamma}$ 
and 
$\mathbf{g}=(g_{n,\sigma})_{(n,\sigma)\in \Gamma}$ in
$\C^\Gamma$, we define a norm associated to
a scalar product
\[
|\mathbf{f}|_{\mathcal{W}^{p,q}}=\sqrt
{(\mathbf{f}, \mathbf{f})_{\mathcal{W}^{p,q}}},\qquad
(\mathbf{f}, \mathbf{g})_{\mathcal{W}^{p,q}}=
\sum_{n=0}^{\infty}
\biggl ( 4^{pn}f_{n,+}\cdot \overline{g_{n,+}}+4^{qn}f_{n,-}\cdot \overline{g_{n,-}} \biggr ),
\]
and a norm
$
|\mathbf{f}|_{\mathcal{C}^{p,q}}=
\max\left\{\sup_{n\ge0} 2^{pn}|f_{n,+}|, \sup_{n\ge 0} 2^{qn}|f_{n,-}|\right\}
$.
We then set
\[
\mathcal{W}^{p,q}=\{\mathbf{f}\in \C^{\Gamma}\mid |\mathbf{f}|_{\mathcal{W}^{p,q}}<\infty\}\quad
\mbox{and}\quad
\mathcal{C}^{p,q}=\{\mathbf{f}\in \C^{\Gamma}\mid |\mathbf{f}|_{\mathcal{C}^{p,q}}<\infty\}.
\]
Recall that $K\subset \real^d$ is a fixed compact set.
Consider  the operator $\QQ_{\Theta}:C^{\infty}(K)
\to C^{\infty}(\real^d,\C^\Gamma)$ defined  by
\[
\QQ_{\Theta}u=(\psi_{\Theta, n,\sigma}(D)u)_{(n,\sigma)\in \Gamma}.
\]
By definition, this operator extends to
\[
\QQ_{\Theta}:C^{\Theta,p,q}_*(K)\to L^{\infty}(\real^d, \mathcal{C}^{p,q}),\quad 
\QQ_{\Theta}:W^{\Theta,p,q,t}_*(K)\to L^{t}(\real^d, \mathcal{W}^{p,q}).
\]
If we define norms 
\[
\|\mathbf{u}\|_{p,q,\infty}=\| |\mathbf{u}|_{\mathcal{C}^{p,q}}(x)\|_{L^{\infty}}
\quad\mbox{for $\mathbf{u}\in L^{\infty}(\real^d, \mathcal{C}^{p,q})$}
\]
and
\[
\|\mathbf{u}\|_{p,q,t}=\| |\mathbf{u}|_{\mathcal{W}^{p,q}}(x)\|_{L^{t}}
\quad\mbox{for $\mathbf{u}\in L^{t}(\real^d, \mathcal{W}^{p,q})$},
\]
respectively,
the anisotropic H\"older norm and the Sobolev norms coincide with their 
respective pull-backs  
by $\QQ_{\Theta}$:
\begin{equation}\label{pqt}
\|u\|\hol{\Theta, p,q}=\|\QQ_{\Theta}u\|_{p,q,\infty}
\quad\mbox{and}\quad
\|u\|\sob{\Theta, p,q,t}=\|\QQ_{\Theta}u\|_{p,q,t}.
\end{equation}


\smallskip

The pseudodifferential operator $\psi(D)$ with symbol $\psi\in C^{\infty}_0(\real^d)$ extends
to a continuous operator $\psi(D):L^t(\real^d)\to L^t(\real^d)$ for $1\le t\le \infty$ whose 
operator norm is bounded by $\|\widehat\psi\|_{L^1}$, because 
\begin{equation}\label{Young}
\|\psi(D)u\|_{L^t}=\|\widehat\psi*u\|_{L^t}\le \|\widehat\psi\|_{L^1}\|u\|_{L^t}
\end{equation}
by Young's inequality. We will use the following more general result on operator-valued 
pseudodifferential operators:



\begin{theorem}[{\cite[{Theorem 0.11.F}]{Ta}}]\label{th:Ta}
Let $\mathcal{H}_1$ and $\mathcal{H}_2$ be Hilbert spaces and let  $\mathcal{L}(\mathcal{H}_1,
\mathcal{H}_2)$ be the space of bounded linear operators from $\mathcal{H}_1$ 
to $\mathcal{H}_2$ equipped with the operator norm. 
If $P(\cdot)\in C^{\infty}(\real^d, \mathcal{L}(\mathcal{H}_1,
\mathcal{H}_2))$ satisfies
\begin{equation}\label{eqn:szero}
\|D^\alpha_\xi P(\xi)\|_{\mathcal{L}(\mathcal{H}_1,
\mathcal{H}_2)}\le
C_{\alpha}(1+|\xi|^2)^{-|\alpha|/2}
\end{equation}
for each multi-index $\alpha$, then for each  $1<t<\infty$ the operator
\[
P(D):L^t(\real^d,\mathcal{H}_1)\to L^t(\real^d,\mathcal{H}_2) 
\]
is bounded.
\end{theorem}

\begin{remark}
The operator-valued pseudodifferential operator $P(D)$  is defined by 
\[
P(D)u=(2\pi)^{-d}\int e^{i(x-y)\xi}P(\xi) u(y) d\xi dy. 
\]
\end{remark}
\begin{remark}
The proof of Theorem \ref{th:Ta} does not need much knowledge on the theory of
pseudodifferential operators and, in fact,  is rather simple. 
Since the case $t=2$ is proved by using Parseval's identity, one only has to 
check that the arguments in Sections 0.2 and 0.11 of  \cite{Ta} 
extend straightforwardly to the operator-valued case.  
\end{remark}

By ``integration by parts on $w$", we will mean application, 
for $f\in C^2(\real^d)$ and  $g\in C^1_0(\real^d)$,
of the formula
\begin{align*}
 \int e^{if(w)}g(w) dw
&=
-\sum_{k=1}^d 
\int i(\partial_k f(w))e^{if(w)}\cdot 
\frac{i(\partial_k f(w))\cdot g(w)}
{\sum_{j=1}^{d}(\partial_j f(w))^2} dw
\\
&=
i\cdot \int e^{if(w)}\cdot \sum_{k=1}^d \partial_k\left(\frac
{\partial_k f(w)\cdot  g(w)}
{\sum_{j=1}^{d}(\partial_j f(w))^2}\right) dw ,
\end{align*}
where $w=(w_k)_{k=1}^{d}\in \real^d$, and $\partial_k$ denotes partial differentiation 
with respect to $w_k$. 

%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{A pseudolocal property}

Although the pseudodifferential operators $\psi_{\Theta,n,\sigma}(D)$ are not local operators,
i.e., $\psi_{\Theta,n,\sigma}(D)u$  does not necessarily
vanish  outside of the support of~$u\in C^{\infty}(K)$, we have the following rapid decay
property, which
will be used in Sections \ref{S2} and \ref{Spartition}: 

\begin{lemma}\label{lm:plp}
For all positive real numbers $b$, $c$,  $\epsilon$ and each
$1< t\le \infty$, there exists a constant $C=C(b,c,\epsilon,t)>0$ 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 $1<t<\infty$ provided  $c>p$. 

\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 $1<t< \infty$.}
\end{equation}
Here we prove:

\begin{proposition}\label{prop:cpt}  
If $p'< p$ and $q'< q$, the inclusions (\ref{eqn:inc}) are compact. 
\end{proposition}
\begin{proof}
Take any sequence $u^{(k)}$, $k\ge 1$, in $C^{\infty}(K)$  such that $
\|u^{(k)}\|\hol{\Theta, p,q}<E$ (respectively $\|u^{(k)}\|\sob{\Theta, p,q,t}<E$)
for some positive constant $E>0$. 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 $q<0<p$ such that $p-q<r-1$, 
the operator $L$ extends to continuous operators
\[
L: C_*^{\Theta, p,q}(K')\to C_*^{\Theta',p,q}(K),\qquad
L: W_*^{\Theta, p,q,t}(K')\to W_*^{\Theta',p,q,t}(K)
\]
for $1<t<\infty$. 
Furthermore, for any $0\le p'< p$ and $q'< q$ such that $p-q'<r-1$, we have the following 
Lasota-Yorke type inequalities:\\
{\bf H\"older case:} There exist a constant $C$, that does not depend on $T$
or $\g$, 
and a constant $C(T,\g)$, that may depend on $T$ and
$\g$, such that for any $u\in C_*^{\Theta,p,q}(K)$
\begin{equation*}
\|L u\|\hol{\Theta',p,q}\le C\|\g\|_\infty\cdot 
 \max\{ 
 \|T\|_+^{p},\|T\|_-^{q}
\}\|u\|\hol{\Theta,p,q}+C(T,\g)\|u\|\hol{\Theta,p',q'}.
\end{equation*}
\\
{\bf Sobolev case:} For each $1<t<\infty$, there exist a constant $C(t)$, that 
does not depend on 
$T$ or $\g$, 
and a constant $C(T,\g,t)$, that may depend on $T$
and $\g$, such that for any 
$u\in W_*^{\Theta,p,q,t}(K)$
\begin{equation*}
\|L u\|\sob{\Theta',p,q,t}\le C(t)\|\g\|_\infty\cdot 
\frac{
 \max\{ \|T\|_+^{p},
 \|T\|_-^{q}
\}}
{\inf |\det DT|^ {1/t}}
\|u\|\sob{\Theta,p,q,t}+C(T,\g,t)\|u\|\sob{\Theta,p',q',t} .
\end{equation*}
\end{theorem}


For the proof of Theorem \ref{th:main2}, we need more notation. 
By (\ref{conehyp}) there exist  closed cones
$\widetilde \cone_+$, $\widetilde \cone_-$  such that
\[
\widetilde \cone_+ \subset \mbox{interior} \, (\cone_+) \, \mbox{ and }
\;\widetilde \cone_-\subset \mbox{interior} \, (\cone_-),  
\]
and that
\begin{equation}\label{conehyp2}
DT^{tr}_x(\real^d\setminus \mbox{interior}(\widetilde \cone_{+})) \subset  \, \mbox{interior}(\cone'_{-})\cup \{0\} \quad
\mbox{for all $x\in \supp(g)$.}
\end{equation}
Let 
$\tilde \varphi_+$, $\tilde\varphi_-:\sphere\to [0,1]$ be
$C^{\infty}$ functions  satisfying
\[
\tilde \varphi_+(\xi)=
\begin{cases}
1, &\mbox{if $\xi\notin \sphere\cap \cone_{-}$;}\\
0, &\mbox{if $\xi\in \sphere\cap \widetilde \cone_{-}$.}
\end{cases}, \qquad 
\tilde\varphi_-(\xi)=
\begin{cases}
0, &\mbox{if $\xi\in \sphere\cap \widetilde\cone_{+}$;}\\
1, &\mbox{if $\xi\notin \sphere\cap \cone_{+}$.}
\end{cases}
\]
Recall the function $\chi$ we fixed in the beginning. Put, for $\ell\ge 0$,  
\[
\tilde\psi_\ell(\xi)=
\begin{cases}
\chi(2^{-\ell-1}|\xi|)-\chi(2^{-\ell+2}|\xi|),&\mbox{if $\ell\ge 1$;}\\
\chi(2^{-1}|\xi|),&\mbox{if $\ell=0$;}
\end{cases}
\]
and then define, for $(\ell,\tau)\in\Gamma$, 
\[
\tilde \psi_{\Theta, \ell, \tau}(\xi)=
\tilde{\psi}_\ell(\xi)\tilde \varphi_{\tau}(\xi/|\xi|).
\]
Note that $\tilde \psi_{\Theta,\ell,\tau}(\xi)=1$ 
if $\xi \in \supp (\psi_{\Theta,\ell,\tau})$.

Next, take integers $h_{\max}$, $h_{\min}$,
$h_{\min}^{-}$, and 
$h_{\max}^{+}$  such that for all $x\in W$
\begin{align*}
&2^{h_{\min}+4}\|\xi\|\le \|DT^{tr}_{x}(\xi)\|\le 2^{h_{\max}-4}\|\xi\|
\qquad\mbox{for any $\xi\in \real^d$,}\\
&\phantom{2^{h_{\min}+4}\|\xi\|\le} \|DT^{tr}_{x}(\xi)\|\le 2^{h_{\max}^{+}-4}\|\xi\|
\qquad\mbox{if $DT^{tr}_{x}(\xi)\notin \cone'_{-}$,}\\
&2^{h_{\min}^{-}+4}\|\xi\|\le \|DT^{tr}_{x}(\xi)\|
\phantom{\le 2^{h_{\max}-4}\|\xi\|}
\qquad\mbox{if $\xi\notin \widetilde \cone_{+}$.}
\end{align*}
By modifying the cones $\widetilde \cone_{\pm}$ if necessary, we may and do assume 
\[
2^{h_{\min}^{-}}> \|T\|_{-}-5,\qquad 
2^{h_{\max}^{+}}< \|T\|_{+}+5.
\]

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  
\[
\|\mathbf{S}\mathbf{u}\|_{p,q,\infty}<C
2^{\max\{p h_{\max}^+, q h_{\min}^-\}}\|g\|_{L^\infty} \|\mathbf{u}\|_{p,q,\infty}
+C(T,g)\|\mathbf{u}\|_{p',q',\infty}
\]
and that, for $1<t<\infty$,
\[
\|\mathbf{S}\mathbf{u}\|_{p,q,t}
<\frac{C(t)2^{\max\{p h_{\max}^+, q h_{\min}^-\}}\|g\|_{L^\infty}}
{\inf |\det DT|^{1/t}}\|\mathbf{u}\|_{p,q,t}
+C(T,g,t)\|\mathbf{u}\|_{p',q',t} .
\]
To prove the above inequalities, we split the matrix of operator $\mathbf{S}$ into two parts:
\[
\mathbf{S}_0=(\widetilde S_{n,\sigma}^{\ell,\tau}),
\quad
\widetilde S_{n,\sigma}^{\ell,\tau}
=
\begin{cases}
S_{n,\sigma}^{\ell,\tau},&\quad\mbox{if $(\ell,\tau)\hookrightarrow(n,\sigma)$};\\
0&\quad\mbox{if $(\ell,\tau)\not\hookrightarrow (n,\sigma)$},
\end{cases}
\]
and   
$\mathbf{S}_1=\mathbf{S}-\mathbf{S}_0$. 
We first consider $\mathbf{S}_0$. 
This is the composition $\Phi(D)\circ \Psi\circ \mathbf{L}$ of  
\begin{itemize}
\item the operator 
$\mathbf{L}$ defined by $
\mathbf{L}(\mathbf{u})(x)=\g(x)\cdot \mathbf{u}\circ T(x)$,
\item the operator $\Psi$ defined by
\[
\Psi\left((f_{\ell,\tau}(x))_{(\ell,\tau)\in \Gamma}\right)_{(n,\sigma)}=
\sum_{(\ell,\tau)\hookrightarrow(n,\sigma)}f_{\ell,\tau}(x),
\]
where $\sum_{(\ell,\tau)\hookrightarrow(n,\sigma)}$ is the sum over $(\ell,\tau)\in \Gamma$ 
such that $(\ell,\tau)\hookrightarrow(n,\sigma)$, 
\item the pseudodifferential operator $\Phi(D)$ with  symbol 
$\Phi:\real^d\to \mathcal{L}(\C^\Gamma, \C^\Gamma)$,
\[
\Phi(\xi)\left((f_{\ell,\tau})_{(\ell,\tau)\in \Gamma}\right)_{(n,\sigma)}=
\psi_{\Theta',n,\sigma}(\xi) f_{n,\sigma}.
\]
\end{itemize}
Clearly $\|\mathbf{L}\|_{p,q,\infty}\le \|\g\|_\infty$ and $\|\mathbf{L}\|_{p,q,t}\le 
\|\g\|_\infty\sup |\det DT|^{-1/t}$ for $1<t<\infty$. 
Also we can prove 
\begin{equation}\label{b4}
\|\Psi\|_{p,q,t}\le C2^{\max\{p h_{\max}^+, q h_{\min}^-\}}\quad\mbox{ for $1<t\le \infty$}
\end{equation}
as follows. Set  $c(+)=p$, $c(-)=q$ and observe that there is $C$ so that 
\begin{equation}\label{cnorm}
\sum_{(\ell,\tau)\hookrightarrow (n,\sigma)} 2^{2c(\sigma)n-2c(\tau)\ell}
\le C2^{2\max\{p h_{\max}^+, q h_{\min}^-\}},\, \forall (n,\sigma),
\end{equation}
For $\mathbf{f}(x)=(f_{n,\sigma}(x))_{(n,\sigma)\in \Gamma}$, we have, at each point $x\in \real^d$, 
\begin{align}
\label{b2}
&|\Psi(\mathbf{f})|_{C^{p,q}}(x)\le 
C2^{\max\{p h_{\max}^+, q h_{\min}^-\}}
|\mathbf{f}|_{C^{p,q}}(x) 
\intertext
{
and also, by Cauchy-Schwartz and (\ref{cnorm}), 
}
\label{b3}
&|\Psi(\mathbf{f})|_{\mathcal{W}^{p,q}}(x)\le 
C2^{\max\{p h_{\max}^+, q h_{\min}^-\}}
  |\mathbf{f}|_{\mathcal{W}^{p,q}} (x).
\end{align}
Taking the supremum and $L^t$ norm of the both sides respectively, we obtain (\ref{b4}). 
The operator $\Phi(D)$ is bounded with respect to the norm $\|\cdot\|_{p,q,t}$ for 
$1<t\le \infty$: 
If $t=\infty$, this follows from (\ref{Young}) since 
$\widehat\psi_{\Theta',n,\sigma}$ is bounded uniformly for $(n,\sigma)\in \Gamma$ in $L^1$-norm, and
the case $1<t<\infty$ follows from Theorem \ref{th:Ta}. 
Thus we conclude 
\[
\|\mathbf{S}_0\|_{p,q,\infty}\le 
C\|\g\|_\infty\cdot 2^{\max\{p h_{\max}^+, q h_{\min}^-\}},
\]
and
\[
\|\mathbf{S}_0\|_{p,q,t}\le 
\frac{C(t)\|\g\|_\infty\cdot 2^{\max\{p h_{\max}^+, q h_{\min}^-\}}}
{\inf |\det DT|^{1/t}}\quad \mbox{for $1<t<\infty$. }
\]


Next we consider  $\mathbf{S}_1$. 
It only remains to show the following two estimates:
\[
\|\mathbf{S}_1\mathbf{u}\|_{p,q,\infty}<C(T,g)\|\mathbf{u}\|_{p',q',\infty}
\,\, \mbox{
and}\,\, \,
\|\mathbf{S}_1\mathbf{u}\|_{p,q,t}<C(T,g,t)\|\mathbf{u}\|_{p',q',t}, 
\forall 1 < t < \infty .
\]
For this, it is enough to prove that for $1<t\le \infty$,
\begin{equation}\label{eqn:s}
\|S_{n,\sigma}^{\ell,\tau}u\|_{L^t}\le C(T,g)2^{-(r-1)\max\{n,\ell\}}\|u\|_{L^t}\quad 
\mbox{if $(\ell,\tau)\not\hookrightarrow(n,\sigma)$.}
\end{equation}
Indeed, (\ref{eqn:s}) with $t=\infty$ implies the first estimate:
\begin{align*}
\|\mathbf{S}_1\mathbf{u}\|_{p,q,\infty}&
\le \sup_{(n,\sigma)\in \Gamma}
\sum_{(\ell,\tau)\not\hookrightarrow(n,\sigma)}
2^{p n}\|S_{n,\sigma}^{\ell,\tau} u_{\Theta,\ell,\tau}\|_{L^\infty}
\\
&\le C(T,g) \cdot 
\sup_{(n,\sigma)\in \Gamma} \left(\sum_{(\ell,\tau)\not\hookrightarrow(n,\sigma)}
 2^{pn-q'\ell-(r-1)\max\{n,\ell\}}\right)\|\mathbf{u}\|_{p',q',\infty},
\end{align*}
and the sum in $(\cdot)$ above is finite from the assumption $p-q'<r-1$.  Next, 
the Cauchy-Schwartz inequality implies that 
\begin{align*}
(|\mathbf{S}_1\mathbf{u}|_{\mathcal{W}^{p,q}}(x))^2=&\sum_{(n,\sigma)\in \Gamma}2^{2c(\sigma) n}\,
\biggl |
\sum_{(\ell,\tau)\not\hookrightarrow(n,\sigma)}S_{n,\sigma}^{\ell,\tau}
u_{\Theta,\ell,\tau}(x)
\biggr|^2
\end{align*}
is bounded by
\begin{align*}
&\sup_{(n,\sigma)\in \Gamma}
\sum_{(\ell,\tau)\in \Gamma}\biggl(
2^{c(\sigma) n-c(\tau)\ell -(r-1)\max\{n,\ell\}}
\biggr )^2\\ 
&\qquad\qquad \cdot \sum_{(n,\sigma)\in \Gamma}
\sum_{(\ell,\tau)\not\hookrightarrow(n,\sigma)}\biggl(
 2^{c(\tau)\ell+(r-1)\max\{n,\ell\}}
|S_{n,\sigma}^{\ell,\tau}u_{\Theta,\ell,\tau}(x)| \biggr )^2 .
\end{align*}
The assumption $p-q<p-q'<r-1$  implies that the 
sum in the second line above is finite.
Setting $c'(+)=p'$, and $c'(-)=q'$,
we have 
\begin{align*}
\biggl \| \biggl (
\sum_{(n,\sigma)\in \Gamma} &
\sum_{(\ell,\tau)\not\hookrightarrow(n,\sigma)}\bigl(
 2^{c(\tau)\ell+(r-1)\max\{n,\ell\}}
|S_{n,\sigma}^{\ell,\tau}u_{\Theta,\ell,\tau}(x)| \bigr )^2
\biggr ) ^{1/2} \biggr \|_{L^t(\real^d)}\\
&\le  \sum_{(n,\sigma)\in \Gamma} 
\sum_{(\ell,\tau)\not\hookrightarrow(n,\sigma)}
2^{c(\tau)\ell+(r-1)\max\{n,\ell\}}
\|S_{n,\sigma}^{\ell,\tau}u_{\Theta,\ell,\tau}\|_{L^t}
\\
&\le C(T,g)\cdot  \sum_{(n,\sigma)\in \Gamma} 
\sum_{(\ell,\tau)\not\hookrightarrow(n,\sigma)}
2^{c(\tau)\ell}
\|u_{\Theta,\ell,\tau}\|_{L^t}\,\,\,\,\mbox{by (\ref{eqn:s}) for $1<t<\infty$}
\\
&\le C(T,g) \cdot \biggl(\sum_{(n,\sigma)\in \Gamma} 
\sum_{(\ell,\tau)\not\hookrightarrow(n,\sigma)}
2^{(c(\tau)-c'(\tau))\ell }\biggr)\| \mathbf{u}\|_{p'q',t}.
\end{align*}
Since the sum in $(\cdot)$ in the last line is finite, this implies the second estimate. 


To show (\ref{eqn:s}),
we rewrite the operator $S_{n,\sigma}^{\ell,\tau}$ as 
\[
(S_{n,\sigma}^{\ell,\tau}u)(x)
=(2\pi)^{-2d}\int V_{n,\sigma}^{\ell,\tau}(x,y) \cdot u\circ T(y) |\det DT(y)|
dy,
\]
where 
\begin{equation}\label{Vkernel}
V_{n,\sigma}^{\ell,\tau}(x,y)=\int e^{i(x-w)\xi+i(T(w)-T(y))\eta} 
\g(w)\psi_{\Theta',n,\sigma}(\xi)
\tilde{\psi}_{\Theta,\ell,\tau}(\eta)dw d\xi d\eta.
\end{equation}
Since $\| u\circ T \cdot |\det DT|\|_{L^t}\le C(T,g)\|u\|_{L^t}$, the inequality (\ref{eqn:s}) 
follows if we show that  there exists
$C(T)$ such that for all $(\ell,\tau)\not\hookrightarrow (n,\sigma)$
and all $1<t\le \infty$
the operator norm of the integral operator 
\[
H^{\ell,\tau}_{n,\sigma}:v\mapsto \int V_{n,\sigma}^{\ell,\tau}(x,y)v(y) dy
\]
acting on $L^t(\real^d)$ is 
bounded by $C(T,g)\cdot 2^{-(r-1)\max\{n,\ell\}}$. 

Define the positive-valued integrable function $b:\real^d\to \real$ by
\begin{equation}\label{convol}
b(x)=
1\quad\mbox{ if $|x|\le 1$}, \qquad
b(x)=|x|^{-d-1}\quad\mbox{ if $|x|> 1$.}
\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 $p-q<r$, and let $p'$ and $q'$ 
be real numbers with $p'<p$
and $q'<q$. 
For every $C^r$ partition of unity $\{g_i\}$ whose intersection multiplicity  is $\nu$, there are constants
$C$ and $C(t)$ (that may depend on the $g_i$'s)
so that for any $u\in C^\infty(K)$
\[
\|u\|\hol{\Theta,p,q}\le \nu\cdot  
\max_{1\le i\le I} \|u_i\|\hol{\Theta,p,q}+C \sum_{1\le i\le I} \|u_i\|\hol{\Theta,p',q'}
\]
and
\[
\|u\|\sob{\Theta,p,q,t}\le \nu \cdot
\left[ \sum_{1\le i\le I} \|u_i\|\sob{\Theta,p,q,t}^t\right]^{1/t}
+C(t) 
\sum_{1\le i\le I} \|u_i\|\sob{\Theta,p',q',t}, \, 
\forall 1<t<\infty.
\]
\end{lemma}

\begin{proof} Let $U(i,\epsilon)$ be the $\epsilon$-neighborhood of the support of $g_i$.
Take $\epsilon>0$ 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 $p'<p$, $q'<q$ and $p-q'<r$. 
If $\Theta'< \Theta$, there are constants $C_0$ and $C_0(t)$
so that   for every $C^r$ partition of unity $\{g_i\}$  there are
constants $C(\{g_i\})$ and $C(\{g_i\},t)$ (which may depend 
on the $g_i$'s) so that for all $u\in C^{\infty}(K)$
\[
\max_{1\le i\le I} \|u_i\|\hol{\Theta',p,q}\le
 C_0\|u\|\hol{\Theta,p,q} +C(\{g_i\})\|u\|\hol{\Theta,p',q'},
\]
and, for $1<t<\infty$, 
\[
\left[\sum_{1\le i\le I}\|u_i\|\sob{\Theta',p,q,t}^{t}\right]^{1/t}
\le 
C_0(t)\|u\|\sob{\Theta,p,q,t} +
C(\{g_i\},t)\|u\|\sob{\Theta,p',q',t} .
\]
\end{proposition}

\begin{proof}
We revisit the proof of Theorem~\ref{th:main2}, setting $T=id$. (Note that assumption (\ref{conehyp}) 
holds since we are assuming $\Theta'<\Theta$.)
Recall $\Phi(D)$ and $\Psi$  there, and let $\mathbf{S}^{(i)}$, $\mathbf{S}^{(i)}_0$, $\mathbf{S}^{(i)}_1$ and $\mathbf{L}^{(i)}$ be  the operators defined  in the same way as $\mathbf{S}$, $\mathbf{S}_0$, $\mathbf{S}_1$ and $\mathbf L$ respectively with $g$ replaced by $g_i$.  
Obviously $|\mathbf{L}_i(\mathbf {f})|_{C^{p,q}}(x)\le |\g_i(x)|
|\mathbf{f}|_{C^{p,q}}(x)$ and $ |\mathbf{L}_i(\mathbf{f})|_{\mathcal{W}^{p,q}}(x)\le 
|\g_i(x)|  |\mathbf{f}|_{\mathcal{W}^{p,q}} (x)$ at each point $x$.
These and (\ref{b2}-\ref{b3}) imply  
\[
\max_{i} \|\Psi\circ \mathbf{L}^{(i)}(\mathbf{u})
\|_{p,q,\infty}\le C_1 \|\mathbf{u}\|_{p,q}\]
and 
\[
\left[\sum_i \|\Psi\circ \mathbf{L}^{(i)}
(\mathbf{u})\|_{p,q,t}^t\right]^{1/t}\le C_1
\|\mathbf{u}\|_{p,q,t}, \forall 1<t<\infty
\]
for $\mathbf{u}=(u_{\Theta,n,\sigma})_{(n,\sigma)\in \Gamma}$. 
By boundedness of $\Phi(D)$, the same estimates holds with $\Psi\circ \mathbf{L}^{(i)}$ replaced by  $\mathbf{S}^{(i)}_0=\Phi(D)\circ \Psi\circ \mathbf{L}^{(i)}$. The conclusion of the proposition then follows from those estimates and the estimates on the operators $\mathbf{S}_1^{(i)}$ parallel to that on $\mathbf{S}_0$ 
in the proof of Theorem~\ref{th:main2}.
\end{proof}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Transfer operators for hyperbolic diffeomorphisms}
\label{S6}

In this section we prove
Theorem~\ref{main} by reducing  to the model of
Sections \ref{S1}--\ref{Spartition}.

\begin{proof}[Proof of  Theorem~\ref{main}]
We first define the  spaces  $C_*^{p,q}(T,V)$ and $W_*^{p,q,t}(T,V)$, 
by using local charts
to patch the  anisotropic H\"older and Sobolev spaces from
Section ~\ref{S1}. Fix 
a finite system of $C^{\infty}$ local charts $\{(V_j, \kappa_j)\}_{j=1}^{J}$ 
that cover the compact isolating neighborhood  $V$ of $\Omega$, 
and a finite system of
pairs of closed cones 
\footnote{We regard $\cone_{j,\pm}$ as constant cone fields in the {\em cotangent} bundle 
$T^*\real^d$.}  
$\{(\cone_{j,+}, \cone_{j,-})\}_{j=1}^{J}$ in $\real^d$
with the 
properties that for all  $1\le j,k\le J$:

\renewcommand{\labelenumi}{(\alph{enumi})}
\begin{enumerate}
\item $\mbox{Closure}(\kappa_j(V_j))$ is a compact subset of $\real^d$.
\item The cones $\cone_{j,\pm}$ are transversal to each other: 
$\cone_{j,+} \cap \cone_{j,-}=\{0\}$.
\item If $x\in V_j\cap \Omega$, the cones 
$(D\kappa_j)^{*}(\cone_{j,+})$ and 
$(D\kappa_j)^{*}(\cone_{j,-})$ in the cotangent space 
contain the normal subspaces of $E^s(x)$ and $E^u(x)$, respectively. 
\item  If 
$T^{-1}(V_k)\cap V_j \neq \emptyset$, setting 
$U_{jk}=\kappa_j(T^{-1}(V_k)\cap V_j)$,
the map in charts 
\[
T_{jk}:=\kappa_k\circ T\circ \kappa_j^{-1}:U_{jk} \to \real^d
\]
enjoys the cone-hyperbolicity condition:
\begin{equation}\label{charthyp}
DT_{jk,x}^{tr}(\real^d\setminus \mbox{interior } (\cone_{j,+}))
\subset \mbox{interior} (\cone_{k,-} )\cup\{0\}, \quad
\forall x \in U_{jk} .
\end{equation}
\end{enumerate}


Choose $C^\infty$ functions $\varphi_j^+, \varphi_j^-: \sphere\to [0,1]$ 
for $1\le j\le J$ 
which satisfy condition (\ref{vp}) with $\cone_\pm=\cone_{j,\pm}$, giving  combinations 
$\Theta_j=(\cone_{j,+}, \cone_{j,-}, \varphi_j^+,\varphi_j^-)$ 
as in Section~\ref{S1}. 
Choose finally a $C^{\infty}$ partition of the unity
$\{\phi_j\}$ on $V$ subordinate to the covering $\{V_j\}_{j=1}^J$, that is, the support of 
each $\phi_j:X\to [0,1]$ is contained in $V_j$ and 
we have $\sum_{j=1}^{J} \phi_j\equiv 1$ on $V$. 

We define the Banach spaces $C_*^{p,q}(T,V)$ and $W^{p,q,t}_*(T,V)$ 
for $1<t<\infty$, respectively, to be 
the completion of $C^{\infty}(V)$ for the norm
\[
\|u\|_{C_*^{p,q}(T,V)}:=\max_{1\le j\le J} 
\|(\phi_j\cdot u)\circ \kappa_j^{-1}\|\hol{\Theta_j, p,q}
\]  
and
\begin{equation}\label{normsob}
\|u\|_{W^{p,q,t}_*(T,V)}:= \max_{1\le j\le J}
\|(\phi_j \cdot u)\circ \kappa_j^{-1}\|\sob{\Theta_j, p,q,t} .
\end{equation}
By this definition, we have that $C_*^{p,q}(T,V)$ and $W^{p,q,t}_*(T, V)$,
respectively, contain 
$C^p(V)\subset C_*^p(V)$ and $W^{p,t}(V)$, respectively, as  dense subsets. 
Take and fix real numbers $0\le p'<p$ and $q'<q$ such that $p-q'<r-1$.  
By Lemma ~\ref{prop:cpt} and a finite
diagonal argument over $\{1, \ldots, J\}$, 
we can see that the inclusions 
$C^{p,q}_*(T, V) \subset C^{p',q'}_*(T, V)$ and 
$W^{p,q,t}_*(T,V) \subset W^{p',q',t}_*(T, V)$ are compact. 
  

For $m\ge 1$ and $j$, $k$ so that
\[
V_{m, jk}:=T^{-m}(V_k)\cap V_j\cap \left(\cap_{i=0} ^{m}T^{-i}(V)\right) \ne \emptyset,
\]
we may consider the map  in charts 
\[
T^m_{jk}=\kappa_k\circ T^m\circ \kappa_j^{-1}:
\kappa_j(V_{m,jk})\to \real^d.
\]
Note that (\ref{charthyp}) implies that
\begin{equation}\label{charthyp2}
(DT^m_{jk,x})^{tr}(\real^d\setminus \mbox{interior } (\cone_{j,+}))
\subset \mbox{interior} (\cone_{k,-} )\cup\{0\}, \quad
\forall x \in \kappa_j(V_{m,jk}) .
\end{equation}

For $1<t\le \infty$, we 
set
\[
 \Lambda_{m,t}= \max_j \max_k \sup_{x\in \kappa_j(V_{m,jk})}  
 \frac{|\g^{(m)}\circ \kappa_j^{-1}(x)|\cdot
 \max\{ (\|T^m_{jk}\|_+(x))^p, (\|T^m_{jk}\|_-(x))^{q} \}}{|\det DT^{m}_x|^{1/t}}
\]
where 
\[
\|T_{jk}^m\|_+(x)=
\sup\left\{ \frac{\|(DT_{jk}^m)_x^{tr}(\xi)\|}{\|\xi\|}\;;\;
 0\neq (DT_{jk}^m)_x^{tr}(\xi)\notin \cone_{k,-}\right\},
\]
and
\[
\|T_{jk}^m\|_-(x)=
\inf\left\{ \frac{\|(DT_{jk}^m)_x^{tr}(\xi)\|}{\|\xi\|}\;;\;
 0\neq \xi \notin \cone_{j,+}\right\}.
\]
Then a standard argument on hyperbolic sets gives a constant $C(t)>1$ 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 $1<t<\infty$
\begin{align*}
\|\mathcal{L}_{T,g}^m u\|_{W_*^{p,q,t}(T,V)}\le C_2(t) \cdot J\cdot \Lambda_{m,t}\cdot 
\|u\|_{W_*^{p,q,t}(T,V)} 
+C_2(m,t) \| u\|_{W_*^{p',q',t}(T,V)}.
\end{align*}
Finally Hennion's theorem \cite{He}
gives the claimed upper bounds 
\[
\liminf_{m \to \infty}
(C(t) \Lambda_{m,t})^{1/m}=
R^{p,q,t}(T,\g,\Omega)
\]
for the essential
spectral radius of $\mathcal{L}_{T,g}$. 
\end{proof}


\begin{remark}\label{MTO}
The proof above applies to  (hyperbolic) mixed transfer operators \cite{Ki}.
\end{remark}



\begin{remark}
Though it is not  explicit in our notation, the definition of the spaces 
$C_*^{p,q}(T,V)$ and $W_*^{p,q,t}(T,V)$ depends on the system of charts 
$\{(V_j,\kappa_j)\}_{j=1}^J$, the set of combinations  
$\{(\cone_{j,+}, \cone_{j,-}, \varphi_{j,+},\varphi_{j,-})\}_{j=1}^{J}$, 
and the partition of unity $\{\phi_j\}_{j=1}^J$. Choosing a different system of local charts, 
a different set of combinations, or a different partition of unity, does not a 
priori give rise to equivalent norms, though Theorem \ref{th:main2} gives relations.  
This is a little unpleasant, but does not
cause  problems. 
\end{remark}





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