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\begin{document}
\date{\today}
\author{Rafael de la Llave}
\address{Department of Mathematics,
1 University Station C1200,
University of Texas, Austin,
TX 78712, USA}
\email{llave@math.utexas.edu}
\author{Victoria Sadovskaya}
\address{Department of Mathematics and Statistics,
ILB 325, University of South Alabama,
Mobile, AL 36688, USA}
\email{sadovska@jaguar1.usouthal.edu}
\title{On regularity of integrable conformal structures Invariant under
Anosov systems}
\begin{abstract} We consider conformal structures invariant under
a volume-preserving Anosov system. We show that if
such a structure is in $L^p$ for suffiently large $p$,
then it is continuous.
\end{abstract}
\maketitle
\markboth{R. de la Llave, V. Sadovskaya}{Regularity of invariant conformal structures}
\section{Introduction}
In this paper we consider integrable conformal structures which
are invariant under a volume preserving Anosov system.
Our goal is to show that such structures are actually continuous.
Our main results for Anosov diffeomorphisms and flows
are stated in Section~\ref{results}.
In Section~\ref{generalizations} we indicate some generalizations of these
results, in particular, an extension to partially hyperbolic accessible
diffeomorphisms.
This research was motivated by recent developments in the study of
rigidity properties of conformal Anosov systems.
The paper \cite{Llave02a} showed that conformal Anosov systems are
locally differentiably rigid. These results were extended \cite{Llave02c}.
More results on local and global differentiable rigidity of such systems
were obtained in \cite{Sadovskaya} and \cite{KalininS03}.
The above papers assumed that the conformal structures
were continuous or bounded. Various arguments were developed
there to show that continuous invariant structures are in fact differentiable.
Nevertheless, many of the tools from the theory of quasi-conformal
structures -- e.g. the measurable Riemann mapping theorem --
are designed to produce integrable conformal structures.
The goal of this paper is to bridge the gap
between the integrable theory and the continuous/differentiable one.
We will show that a conformal structure is continuous provided that
it belongs to a certain $L^p$ with $p$ high enough. Once
the conformal structure is known to be continuous, one can use the
results of \cite{Sadovskaya}, \cite{KalininS03} to obtain further regularity
for the conformal structure and differential rigidity for the system.
Bootstrap or regularity for measurable equations
has applications in ergodic theory. See for example,
\cite{ParryP97,NicolP99}.
The method of proof is based on the fact that the invariant
structures satisfy a functional equation very similar to
the equation for solutions of cohomology equations. Hence,
some of the techniques of \cite{Llave02b} apply. Moreover, taking
advantage of the form of the problem, the results
for conformal structures can be made somewhat sharper than
those for general solutions of cohomology equations.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Statement of results} \label{results}
Let $\M$ be a $m$-dimensional compact manifold, and $f:\M\to \M$ be
a $C^r$, $r\ge 2$, volume-preserving Anosov diffeomorphism.
We will assume that $\M$ is endowed with a $C^\infty$ ``background"
metric which is adapted to the Anosov diffeomorphism. More precisely,
there exists a continuous decomposition of the tangent bundle
$T\M$ into two invariant subbundles $E^u$, $E^s$ and
numbers $0<\lambda_s,\lambda_u <1$ such that for all $n\ge 0$,
\begin{equation} \label{1.1}
\begin{split}
|| df^n(v) || \leq \lambda_s^n || v ||&
\; \iff v \in E^s, \\
|| df^{-n}(v) || \leq \lambda_u^n || v ||&
\; \iff v \in E^u.
\end{split}
\end{equation}
Let $E\subset T\M$ be a subbundle invariant under $Df$,
$\dim E =d \geq 2$. A conformal structure $C_x$ on $E_x \subset T_x\M$
is a class of proportional positive definite quadratic forms on $E_x$.
Using the background metric, we can identify a quadratic form with
a symmetric linear operator. Hence, with this identification,
a conformal structure is an equivalence class of linear operators.
In this class, we can choose a unique representative
which has determinant $1$ with respect to the background metric.
{From} now on, we understand the conformal structure $C_x$ as
this linear operator on $E_x$.
For each $x\in \M$, we denote the space of conformal structures on $E_x$ by $\C_x=\C_x^E$. Thus we obtain a bundle $\C=\C^E$ over $\M$
whose fiber over $x$ is $\C_x$.
A section $C$ of the bundle $\C$ is called a conformal structure on $E$.
The diffeomorphism $f$ induces a natural pull-back action $F$ on
conformal structures as follows. For a conformal structure
$C_{fx}\in \C_{fx},\;$
$F_x(C_{fx})\in \C_{x}$ is given by
\begin{equation}\label{1.2}
F_x(C_{fx})=\frac{1}{\det \, ((Df_x)^* \circ Df_x) }
(Df_x)^* \circ C_{fx} \circ Df_x.
\end{equation}
Here $C_{fx}$ is the linear operator on $E_{fx}$, and
$(Df_x)^*:\; T_{fx}\M \to T_{x} \M$
denotes the conjugate operator of $Df_x$.
Clearly, $F_x: \C_{fx}\to \C_{x}$ is a linear operator.
We say that a conformal structure $C$ is $f$-invariant
if for all $x\in \M$,
\begin{equation}\label{1.3}
F_x(C_{fx})=C_x.
\end{equation}
Note that the subbundle $E$ can carry an invariant conformal
structure only if $E\subset E^s$ or $E\subset E^u$.
We define the norm of a conformal structure $C_x$ as the norm
of the quadratic form with respect to the background metric:
\begin{equation} \label{1.4}
|| C_x ||= \underset{0\ne v\in E_x}{\sup} \frac{}{||v||^2}.
\end{equation}
Since the operator $C_x$ is symmetric and positive definite, $||C_x||$
is equal to its largest eigenvalue.
We also define
\begin{equation} \label{1.5}
|| C_x^{-1} ||= \underset{0\ne v\in E_x}{\sup} \frac{||v||^2}{},
\end{equation}
which is the inverse of the smallest eigenvalue of $C_x$.
Since the product of all the eigenvalues equals 1, it is easy
to see that
\begin{equation}\label{Cm1}
\begin{split}
||C_x^{-1}||^{d-1} &\ge || C_x || \ge ||C_x^{-1}||^{\frac{1}{d-1}}, \\
||C_x||^{d-1} &\ge || C_x^{-1} || \ge ||C_x ||^{\frac{1}{d-1}}.
\end{split}
\end{equation}
With respect to these norms, we can define $L^p$ spaces of conformal
structures. We say that a conformal structure $C$ belongs to $L^p$
if it is measurable and
its norm is an $L^p$ function with respect to the invariant volume
on $\M$. It is clear that the property that $C \in L^p$ does not depend
on the choice of the background metric.
The inequalities \eqref{Cm1} show that $||C|| \in L^p$ implies
$||C^{-1}|| \in L^{p/(d-1)}$ and, similarly, $||C^{-1}|| \in L^p$ implies
$||C|| \in L^{p/(d-1)}$.
Conformal structures at two nearby points can be identified as follows.
When points $x$ and $y$ are close enough, they
can be joined by a unique shortest geodesic.
We transport $E_x$ along this geodesic using the Levi-Civita
connection, and then project it to $E_y$.
We denote by $S_x^y : \C^E_x \to \C^E_y$ the corresponding
identification of conformal structures on $E_x$ and $E_y$.
If $y$ is in a small neighborhood of $x$, then the dependence of $S_x^y$
on $y$ is as smooth as the dependence of $E_y$ on the base point.
In particular, when $E$ is the stable distribution, $S^x_y$ depends
smoothly on $y$ when $y$ moves along the stable manifold.
\smallskip
We now define the $\L_\alpha^{p,(s)}$ spaces for
conformal structures. They are a natural extension to manifolds
endowed with an Anosov system of
similar spaces which are standard in
harmonic analysis. We
refer to \cite{Stein70} for the results from harmonic
analysis that we will need.
In this paper we consider dynamical applications of these spaces
similar to those in \cite{Llave02b}.
\begin{definition}
Let $W^s$ be the stable foliation of $f$. We say that a
homeomorphism $h:\M\to \M$ is adapted to the stable foliation
if for any $x\in \M$ we have $h(x) \in W^s_x$.
\end{definition}
Recall that $C$ is a conformal structure on an invariant subbundle $E$.
\begin{definition}
We say that $C\in \L_\alpha^{p, (s)}$,
$1\le p < \infty$, $0<\alpha \le 1$, if
\begin{equation}\label{1.6}
\left( \int_\M ||C_x-S_{h(x)}^x (C_{h(x)}) ||^p \right)^{1/p}
\le K \cdot ||h-\Id||_{L^\infty}^{\alpha}
\end{equation}
for any absolutely continuous homeomorphism $h$ adapted to the stable foliation
with $|| h - \Id||_{L^\infty}$ sufficiently small and
$|| J_h||_{L^\infty}, ||J_{h^{-1}} ||_{L^\infty}\le L$.
Analogously, we define the space $\L_\alpha^{p,(u)}$ for the
unstable foliation.
We say that $C\in \L_\alpha^{p}$ if the condition \eqref{1.6} is satisfied
for any diffeomorphism $h$ sufficiently close to the identity.
\end{definition}
The spaces $\L_\alpha^{p, (s)}$, $\L_\alpha^{p, (u)}$, and
$\L_\alpha^{p}$ are Banach spaces with respect to the norm given
by the best possible $K$ in \eqref{1.6}.
\smallskip
Our main results are the following two statements.
\begin{theorem} \label{main} Let $f$ be a $C^r$, $r\ge 2$,
volume-preserving Anosov diffeomorphism of a compact manifold $\M$.
Let $E\subset T\M$ be an invariant distribution of dimension $d \ge 2$
which is H\"older continuous with exponent $\alpha>0$.
\smallskip
Let $C$ be an invariant conformal structure on $E$
such that $||C||\in L^p$, $p\ge d +1$.
Then
$C\in \L^{p/(d+1)}_\alpha$.
\end{theorem}
As a consequence, we obtain the following.
\begin{corollary} \label{continuous} In addition to the assumptions
of Theorem~\ref{main}, suppose that $p > \frac{d \cdot m}{\alpha}$,
where $m=\dim\,\M$. Then the conformal structure $C$ is continuous.
\end{corollary}
\medskip
For the continuos-time case, we obtain the following analogs of
the above statements.
\begin{theorem} \label{main_flow} Let $\v^t$ be a $C^r$, $r\ge 2$,
volume-preserving Anosov flow on a compact manifold $\M$.
Let $E\subset T\M$ be an invariant distribution of dimension $d \ge 2$
which is H\"older continuous with exponent $\alpha>0$.
\smallskip
Let $C$ be an invariant conformal structure on $E$
such that $||C||\in L^p$, $p\ge d +1$. Then
$C\in \L^{p/(d+1)}_\alpha$.
\end{theorem}
\begin{corollary} \label{continuous_flow} In addition to the assumptions
of Theorem~\ref{main_flow}, suppose that $p > \frac{d \cdot m}{\alpha}$,
where $m=\dim\,\M$.
Then the conformal structure $C$ is continuous.
\end{corollary}
Once it is known that the conformal structure $C$ is continuous,
its regularity can be improved as follows.
Suppose that the diffeomorphism $f$ (the flow $\v^t$) is $C^\infty$,
and $E$ is tangential to a continuous
foliation $W$ with $C^\infty$ leaves, for example,
$E$ is the (strong) stable or unstable distribution.
Then, the continuity of $C$
implies that $C$ is actually $C^\infty$ along the leaves of
the foliation $W$ (\cite{Sadovskaya}).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Generalizations} \label{generalizations}
The main results can be generalized in some respects.
The modifications required for the proof can be found
in Section 5 of \cite{Llave02b}.
\subsection{} We can replace the assumption of $f$ being Anosov
by $f$ being partially hyperbolic with uniform
accesibility in a measure theoretic sense.
\subsection{} The assumption that $f$ is measure preserving can
be weakened to $|| J f - 1||_{L^\infty} $ sufficiently
small depending on the assumptions.
\subsection{} In the main statements we assumed that
$||C|| \in L^p$, which automatically implies
that $||C^{-1}|| \in L^{p/(d-1)}$. If we assume in addition that
$||C^{-1}|| \in L^q$ with $q > \frac{p}{d-1}$, the conclusions
can be strengthened. For example, if $q=p$, then in
in the corollaries we obtain that $C$ is continuous if
$p>\frac{2m}{\alpha}$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Some results from Harmonic Analysis}
We will use the following results from harmonic
analysis, which already were a basic tool in \cite{Llave02b}.
\begin{proposition}\label{L^p}
For $1\le p\le \infty$, $0< \alpha \le 1$,
\[
\L_\alpha^{p, (s)} \cap \L^{p, (u)}_\alpha \subset\L^{p}_{\alpha}.
\]
\end{proposition}
We start by observing that any diffeomorphism $h$ close to
the identity can be written as $h = h^u\circ h^s$,
where $h^s, h^u$ are absolutely continuous homeomorphisms
adapted to the stable and unstable foliations respectively.
This follows from the absolute
continuity of stable and unstable foliations of Anosov
diffeomorphisms, and the implicit function theorem
(we refer for more details to \cite{Llave02b} Proposition 2.6).
Moreover, we have
\[
||h^{s,u} - \Id||_{L^\infty} \le K_1 \cdot || h - \Id ||_{L^\infty},
\]
and the Jacobians of $h^s$ and $h^u$ are uniformly bounded.
Then, for any $\Psi \in \L_\alpha^{p, (s)} \cap \L^{p, (u)}_\alpha $,
we can estimate
\[
\begin{split}
||\Psi\circ h - \Psi||_{L^p} &\le
||\Psi\circ h^u \circ h^s - \Psi \circ h^s ||_{L^p} +
||\Psi\circ h^s - \Psi ||_{L^p} \\
&\le K_2\cdot ||\Psi\circ h^u - \Psi||_{L^p} + ||\Psi\circ h^s - \Psi ||_{L^p} \\
&\le K_2 K_3\cdot || h^u - \Id ||_{L^\infty}^\alpha+
K_3\cdot || h^s - \Id ||_{L^\infty}^\alpha\\
&\le K\cdot || h - \Id ||_{L^\infty}.
\end{split}
\]
\qed
We will use the following results from harmonic analysis.
\begin{proposition}\label{L^q}
Assume that $0 < \alpha < 1$, $1 < p < \infty$. Then,
\smallskip
(a) $\;\L^{p}_\alpha \subset W^p_{\alpha'}$
for any $\alpha' < \alpha$;
\smallskip
(b) $\;\L^{p}_\alpha \subset L^{q-\epsilon}$ for any $\epsilon>0$,
where $\frac{1}{q}=\frac{1}{p}-\frac{\alpha}{d}$;
\smallskip
(c) $\;\L^{p}_\alpha \subset C^0$ for $p> \frac{m}{\alpha}$,
where $m=\dim \M.$
\smallskip
In all cases, the embeddings are continuous.
\end{proposition}
Using partitions of unity and coordinate patches,
we reduce the proof of Proposition \ref{L^q} to a proof in
Euclidean space.
Then, to prove (a), one can use the estimates in \cite{Stein70}
\S 3.3 and in \S 3.5.2 p. 141.
Note that the spaces that we are calling here $\L^{p}_\alpha$
are called in \cite{Stein70} $\Lambda^{p,\infty}_\alpha$.
The potential spaces $W^{p,\alpha}$ are denoted by another letter
in \cite{Stein70}.
In some cases, e.g. $p = 2$, the results can be improved
slightly, but we will not be concerned with this.
After that, (b) and (c) are consequences of
the standard Sobolev embedding theorem. A standard proof
for the fractional cases we need is in \cite{Taylor97}, p. 22 ff.
\qed
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Proofs}
\subsection {Proof of Theorem~\ref{main}.}
Using invariance of the conformal structure under the
powers of $F$, we rewrite the difference $C_x-S_{hx}^x (C_{hx})$
in a form suitable for making estimates.
Iterating the invariance equation \eqref{1.3} we have
\begin{equation}\label{3.1}
C_{x} = F_x \circ \dots \circ F_{f^{n-1}x} (C_{f^n x}).
\end{equation}
Note that, by \eqref{1.2}, the mapping
\[
F^n_x \equiv F_x \circ \dots \circ F_{f^{n-1}x}:\;
\C_{f^n x} \to \C_{x}
\]
is given by
\[
F^n_x (C_{f^n x}) = (A^n_x)^* \circ C_{f^n x} \circ A^n_x,
\quad \text{ where } \quad
A^n_x=\frac{1}{ \det (Df^n_x)} Df^n_x.
\]
When $E_x$ is equipped with
the metric given by $C_x$ and
$E_{f^n x}$ is equipped with the metric given by $C_{f^n x}$,
$A_n:\; E_x \to E_{f^n x}$ is an isometry. It follows from
\eqref{1.4} and \eqref{1.5}
that
\[
||(A^n_x)^*||= ||A^n_x|| \le \sqrt{||C_{f^{n}x}^{-1}||}\cdot \sqrt{||C_x||}.
\]
and hence
\begin{equation} \label{3.2}
||F_x^n|| \le ||(A^n_x)^*||\cdot ||A^n_x||
\le ||C_{f^{n}x}^{-1}||\cdot ||C_x||.
\end{equation}
Considering \eqref{3.1}
at the point $hx$, we have
\[
C_{hx} = F_{hx} \circ \dots \circ F_{f^{n-1}(hx)} (C_{f^n(hx)}).
\label{3.3a}
\]
Now we assume that $h$ is adapted to the stable foliation of $f$
so that $f^{i}(hx)$ is sufficiently close to $f^{i}(x)$,
and thus the space $E_{f^{i}(hx)}$ can be identified with $E_{f^{i}x}$.
We have
\begin{equation}\label{3.3b}
S_{hx}^{x}(C_{hx}) =
S_{hx}^{x} \circ F_{hx} \circ \left( S_{f(hx)}^{fx} \right)^{-1}
\circ S_{f(hx)}^{fx}
\circ \dots
\end{equation}
\[
\circ \left( S_{f^{n-1}(hx)}^{f^{n-1}x}\right)^{-1}
\circ S_{f^{n-1}(hx)}^{f^{n-1}x} \circ
F_{f^{n-1}(hx)} \circ \left(S_{f^n(hx)}^{f^nx} \right)^{-1} \circ
S_{f^n(hx)}^{f^nx} \left(C_{f^n(hx)} \right) .
\]
We denote
\[
\F_{f^i x}= S_{f^i(hx)}^{f^i x} \circ F_{f^i(hx)}
\circ \left(S_{f^{i+1}(hx)}^{f^{i+1}x}\right)^{-1}
: \; \C_{f^{i+1}x} \to \C_{f^{i}x}.
\]
and
\begin{equation}\label{tildeF2}
\F_{x}^i=\F_x \circ \dots\circ \F_{f^{i-1}x}.
\end{equation}
Now, we can write \eqref{3.3b} as
\begin{equation} \label{3.4}
S_{hx}^{x}(C_{hx}) =
\F_x\circ \dots \circ \F_{f^{n-1} x}
\circ S_{f^n(hx)}^{f^n x} \left(C_{f^n(hx)} \right).
\end{equation}
Using \eqref{3.1} and \eqref{3.4}, we rewrite the difference
$C_x-S_{hx}^x (C_{hx}) $
by adding and substracting appropriate terms.
\[
\begin{split}
C_x- & S_{hx}^x (C_{hx})
= F_x \circ \dots \circ F_{f^{n-1}x} (C_{f^n x})-
\F_x\circ \dots \circ \F_{f^{n-1} x}
\circ S_{f^n(hx)}^{f^n x} (C_{f^n(hx)}) \\
= & F_x \circ \dots \circ F_{f^{n-1}x} (C_{f^n x}) -
\F_x \circ F_{fx} \circ \dots \circ F_{f^{n-1} x} (C_{f^n x}) \\
& +
\F_x \circ F_{fx} \circ \dots \circ F_{f^{n-1} x} (C_{f^n x}) -
\F_x\circ \dots \circ \F_{f^{n-1} x}
\circ S_{f^n(hx)}^{f^n x} (C_{f^n(hx)}) \\
= &
(F_x-\F_x)\circ F_{fx} \circ \dots \circ F_{f^{n-1} x}
(C_{f^n x}) \\
& +
\F_x \circ \left( F_{fx}\circ \dots \circ F_{f^{n-1}x} (C_{f^n x}) -
\F_{fx} \circ \dots \circ \F_{f^{n-1} x}
\circ S_{f^n(hx)}^{f^n x} C_{f^n(hx)}) \right) \\
= & \cdots \\
= &
\sum_{i=0}^{n-1} \F_x \circ \dots\circ \F_{f^{i-1}x} \circ
\left( F_{f^i x}-\F_{f^i x} \right) \circ F_{f^{i+1}x} \circ \dots \circ F_{f^{n-1}x}
(C_{f^n x}) \\
& +
\F_{x}\circ \dots \circ\F_{f^{n-1}x} \left( C_{f^n x}-
S_{f^n(hx)}^{f^n x} (C_{f^n(hx)}) \right) \\
= &
\sum_{i=0}^{n-1} \F_{x}^i \circ \left( F_{f^i x}-\F_{f^i x} \right)
(C_{f^{i+1} x}) + \F^n_{x} \left( C_{f^n x}-
S_{f^n(hx)}^{f^n x}(C_{f^n(hx)}) \right).
\end{split}
\]
Thus,
\begin{equation}\label{3.5}
\begin{split}
C_x-S_{hx}^x (C_{hx})=&
\sum_{i=0}^{n-1} \F_{x}^i \circ \left( F_{f^i x}-\F_{f^i x} \right)
(C_{f^{i+1} x}) \\
& + \F^n_{x} \left(C_{f^n x}-
S_{f^n(hx)}^{f^n x} (C_{f^n(hx)}) \right).
\end{split}
\end{equation}
Now we proceed as in \cite{Llave02b}.
First we estimate the general term of the sum in \eqref{3.5}.
Using \eqref{3.2} we obtain
\[
|| F^i_{hx} || \le ||C_{f^i(hx)}^{-1}|| \cdot ||C_{hx}||.
\]
This can be viewed as an analog of cancellations in \cite{NiticaT98}.
Note that $\F^i_x =S_{hx}^{x} \circ F_{hx}^i \circ
\left(S_{f^{i}(hx)}^{f^{i} x}\right)^{-1}$. Since $h$ is
$C^0$-close to the identity,
\begin{equation} \label{3.6}
|| \F^i_{x} || \le K_1\cdot ||C_{f^i(hx)}^{-1}|| \cdot ||C_{hx}||.
\end{equation}
Since the restriction of the derivative of $f$ to $E$ is H\"older
continuous, $F$ is also H\"older continuous. Hence,
\[
\begin{split}
||F_{f^i x}-\F_{f^i x}|| & =
||F_{f^i x}- S_{f^i(hx)}^{f^i x} \circ F_{f^i(hx)}
\circ \left(S_{f^{i+1}(hx)}^{f^{i+1}x}\right)^{-1}|| \\
& \le
K_2\cdot \dist \left( f^i x, f^i(hx) \right)^\alpha \le
K_2 \cdot \left(\lambda_s^i \cdot \dist(h, hx) \right)^\alpha,
\end{split}
\]
where $\lambda_s$ is the contraction coefficient in \eqref{1.1} and $\alpha>0$
is the H\"older exponent.
Therefore, we can estimate the general term in \eqref{3.5} as follows
\[
\begin{split}
||\F_{x}^i & \circ \left( F_{f^i x}-\F_{f^i x} \right)
(C_{f^{i+1} x})|| \\
& \le
K_3\cdot ||C_{f^i(hx)}^{-1}|| \cdot ||C_{hx}|| \cdot
||C_{f^{i+1}x}||
\cdot \lambda_s^{i\alpha} \cdot \dist(h, hx) ^\alpha.
\end{split}
\]
Now we estimate the $L^{p/3}$ norm of the general term in \eqref{3.5}.
Let
\[
T_i(x)=\F_{x}^i \circ \left( F_{f^i x}-\F_{f^i x} \right)
(C_{f^{i+1} x}).
\]
Since $||C|| \in L^p$, the equations \eqref{Cm1} imply that
$||C^{-1}|| \in L^{p/(d-1)}$.
Thus, using H\"older inequality, we obtain
\begin{equation} \label{Holder}
\begin{split}
& || T_i ||_{L^{p/(d+1)}} \le
K_3 \cdot \lambda_s^{i\alpha} \cdot || h-\Id||_{L^\infty}^\alpha \\
&
\phantom{ || T_i ||_{L^{p/3}} \le } \quad
\times
\left( \int_\M (||C_{f^i(hx)}^{-1}|| \cdot ||C_{hx}|| \cdot
||C_{f^{i+1}x}||)^{p/(d+1)}
\right)^{(d+1)/p}
\\
& \le
K_3\lambda_s^{i\alpha} \cdot || h-\Id||_{L^\infty}^\alpha
\cdot ||C_{f^i(h(\cdot))}^{-1}||_{L^{p/(d-1)}} \cdot
||C_{h(\cdot)}||_{L^p} \cdot ||C_{f^{i+1}(\cdot)}||_{L^p}
\end{split}
\end{equation}
Since $f$ preserves the volume,
\[
||C_{f^{i+1}(\cdot)}||_{L^p}= ||C||_{L^p},
\]
and since $h$ is a $C^1$-close to the identity diffeomorphism,
\begin{equation} \label{K_4}
||C_{h(\cdot)}||_{L^p}\le K_4 \cdot ||C||_{L^p},
\end{equation}
\[
||C_{f^i(h(\cdot))}^{-1}||_{L^{p/(d-1)}}= ||C_{h(\cdot)}^{-1}||_{L^{p/(d-1)}}\le
K_4\cdot ||C^{-1}||_{L^{p/(d-1)}}.
\]
Hence we obtain that if $C\in L^p,$ the general term in \eqref{3.5}
is bounded in $L^{p/(d+1)}$ by
\[
||T_i||_{L^{p/(d+1)}} \le K_5 \cdot || h-\Id||_{L^\infty}^\alpha
\cdot || C||^2_{L^p} \cdot || C^{-1}||_{L^{p/(d-1)}} \cdot \lambda_s^{i\alpha}.
\]
Thus,
\[
\begin{split}
||C_x- & S_{hx}^x (C_{hx})||_{L^{p/(d+1)}} \\
&\le
\sum_{i=0}^{n-1} ||T_i||_{L^{p/(d+1)}}+
||\F^n_{x}(C_{f^n x}-
S_{f^n(hx)}^{f^n x} (C_{f^n(hx)}))||_{L^{p/(d+1)}} \\
&\le
K\cdot || h-\Id||_{L^\infty}^\alpha +
||\F^n_{x}(C_{f^n x}-
S_{f^n(hx)}^{f^n x} (C_{f^n(hx)}))||_{L^{p/(d+1)}}. \label{3.7}
\end{split}
\]
Now we show that the last term in \eqref{3.5} tends to 0 as
$n\to \infty$. Let us define linear operators $R_n$ by
\[
(R_n (C))_x=
\F^n_{x} \left(C_{f^n x}- S_{f^n(hx)}^{f^n x} (C_{f^n(hx)})\right).
\]
Then by \eqref{3.6},
\[
||(R_n (C))_x|| \le
K_1\cdot ||C_{f^{n}(hx)}^{-1}||\cdot ||C_{hx}|| \cdot
||C_{f^n x}- S_{f^n(hx)}^{f^n x} (C_{f^n(hx)})||.
\]
Since $h$ is $C^1$-close to the identity, the norms of
the operators $S_{f^n(hx)}^{f^n x}$ are uniformly bounded in $x$
and $n$. Now, using \eqref{K_4} and the fact that $f$ is volume-preserving,
we obtain:
\[
\begin{split}
||C_{f^n x}- S_{f^n(hx)}^{f^n x} (C_{f^n(hx)})||_{L^p} &\le
||C_{f^n x}||+ K_6 \cdot ||C_{f^n(hx)}||_{L^p} \\
&\le (1+K_6 \cdot K_4) \cdot ||C||_{L^p}.
\end{split}
\]
Using H\"older inequality as in \eqref{Holder} it is easy to see
that the norms of the linear operators $R_n$
from $L^p$ to $L^{p/(d+1)}$ are bounded uniformly in $n$. Note that
$||(R_n (C))_x||$ tends to $0$ uniformly on $\M$ for any continuous
structure $C$. Since the continuous structures are dense in $L^{p}$,
we conclude that $||(R_n (C))||_{L^{p/(d+1)}} \to 0$ as $n \to \infty$
for any $L^{p}$ structure $C$.
\smallskip
Thus we conclude that
\[
||C_x-S_{hx}^x (C_{hx})||_{L^{p/(d+1)}} \le
K\cdot || h-\Id||_{L^\infty}^\alpha,
\]
and hence $C \in \L_\alpha^{\frac{p}{d+1}, (s)}.$
A similar argument shows that $C \in \L_\alpha^{\frac{p}{d+1}, (u)}.$
Then Proposition~\ref{L^p} implies that $C\in L_\alpha^{p/(d+1)}.$
This completes the proof of Theorem~\ref{main}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Proof of Theorem~\ref{main_flow}}
The argument here is similar to the proof of
Theorem~\ref{main}. We will indicate modifications
required for the flow case.
Instead of the stable and unstable foliations we consider
the strong stable and strong unstable foliations.
We say that a conformal structure $C$ is in $\L^{p,(s)}_\alpha$
(in $\L^{p,(u)}_\alpha$) if the condition \eqref{1.6} is satisfied
for any diffeomorphism $h$ adapted to the strong stable (unstable)
foliation. The argument in the proof of Theorem~\ref{main}
shows that
$C\in \L^{\frac{p}{d+1},(s)}_\alpha \cap \L^{\frac{p}{d+1},(u)}_\alpha$.
We can also define the space $\L^{p,(o)}_\alpha$ by considering the
diffeomorphisms adapted to the orbit foliation. For this case, there exists
a natural identification of conformal structures on $E_x$ and on $E_{h(x)}$
given by the flow. We use this identification in place of $S^x_{h(x)}$
in the condition \eqref{1.6}. With this definition, the difference in \eqref{1.6}
is identically 0 for any invariant conformal structure, and
the condition is trivially satisfied. Thus,
\[
C\in \L^{\frac{p}{d+1},(s)}_\alpha \cap \L^{\frac{p}{d+1},(u)}_\alpha
\cap \L^{\frac{p}{d+1},(o)}.
\]
Now, an analog of Proposition~\ref{L^p} shows that $C\in \L^{\frac{p}{d+1}}$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Proof of Corollaries~\ref{continuous} and \ref{continuous_flow}}
Since the conformal structure $C$ is in $L^p$, Theorem \ref{main}
(\ref{main_flow}) together with Proposition \ref{L^q}(b)
imply that
\[
C \in \L^{p/(d+1)}_\alpha\subset L^{q-\epsilon}
\]
for any $\epsilon >0$, where $\frac 1{q} = \frac {d+1}{p} - \frac{\alpha}{m}$.
Calculating $q$ we obtain
\[
q=p\cdot \frac{m}{m(d+1)-\alpha p}>p \quad\text{ for } \;
p >\frac{d\cdot m}{\alpha}
\]
Since the factor $\frac{m}{m(d+1)-\alpha p}$ increases with $p$,
we can apply the theorem repeatedly until we obtain that
$C\in \L^{q}_{\alpha}$ with $q\ge \frac{m}{\alpha}$.
Then it follows from Proposition ~\ref{L^q}(c) that
the conformal structure $C$ is continuous.
\qed
\section{Acknowledgments}
We thank B. Kalinin for discussions and encouragement.
The work of R.L. has been partially supported by NSF grants.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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