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\author{R. de la Llave}
\address{
Dept. of Mathematics, Univ. of Texas, Austin TX 78712-1802}
\email{llave@math.utexas.edu}
\title{Bootstrap of regularity for integrable solutions of cohomology equations}
\begin{document}
\begin{abstract}
We present an elementary argument to bootstrap the regularity of
integrable solutions of cohomology equations.
if $f:M \to M$ is a smooth Anosov (partially hyperbolic with uniform
accessibility) we show that if $\varphi \in L^p$, $\eta \in C^\beta$,
$\beta > 0$
and
\[
\varphi =\eta \varphi \circ f,
\]
$p$
high enough, then $\varphi \in \alpha'$,
where $\beta'$ depends on the hyperbolicity properties
of $f$. ($\beta' = \beta$ if $f$ is Anosov.)
\end{abstract}
\maketitle
\markboth{R. de la Llave}{Bootstrap of regularity of cohomology equations}
\section{Introduction}
The goal of this note is to report a very simple trick to bootstrap
the regularity of measurable
solutions of cohomology equations over systems with
some hyperbolic behavior which preserve an absolutely
continuous invariant measure.
More precisely, given $M$ a $d$ dimensional
compact manifold, $G$ a Lie group
-- which we will assume to be a matrix group or, more generally
a subgroup of a Banach algebra $\cB$, possibly
infinite dimensional ---,
$f:M \to M$ a diffeomorphism preserving a measure $\mu$
equivalent to Lebesgue.
Given
$\eta:M \to G$ we consider the equation for $\vp:
M \to G$
\begin{equation}\label{cohomology}
\vp(x) = \eta(x) \vp\big(f(x)\big)\ .
\end{equation}
We will always assume that $\eta$ is rather regular, at least
$C^\beta$ for some $\beta > 0$.
Similarly we will always assume that $f$ is $C^r$,
$r > 1 + \beta $.
We will also assume that the density of $\mu$ is rather
smooth. At least continuous.
We note that, by the results of \cite{LlaveMM86},
when $f$ is Anosov,
the measure $\mu$ has a density which is $C^{r-1 - \delta}$.
The main goal is to see whether, just assuming
that $\vp \in L^p $ for some suitable $ 1 \le p \le \infty$,
high enough.
allows us to conclude that $\vp$ is actually rather
regular. Of course, \eqref{cohomology} is
supposed to hold almost everywhere $\mu$.
There is an extensive literature for these problems.
In particular, we note \cite{NicolP99}, which considers
the case $p = \infty$. We also note \cite{Llave01},
which uses similar techniques than the techniques developed
here.
In particular, we will make use of
Sobolev-like spaces (the potential spaces)
as intermediate steps.
(See Section \ref{sec:sobolev} for some notions
on these spaces.)
We note that both
\cite{NicolP99} and \cite{Llave01} -- as well as this paper --
use some remarkable cancellations that were discovered in
\cite{NiticaT98}.
As soon as we establish some regularity,
there
are other bootstrap arguments that can be employed -- at least
in the case that the diffeomorphism $f$ is Anosov.
Notably, when when we know that $\vp$ is continuous,
it is possible to use the results of
\cite{NiticaT98}. When $\vp \in W^{1,p}$, it is possible
to use the results in \cite{Llave01} to obtain
further regularity when the regularity of $\eta$ is
expressed in Sobolev classes.
Hence, in this paper we will only detail the bootstrap
till we obtain that $\eta$ is continuous or
$W^{1,p}$.
We also point out that the bootstrap regularities
is extremely easy when we assume that $\eta$ is
small or that the group $G$ has certain
growth properties. We will collect these
results in Section \ref{sec:small} but in the main part of the
paper, we will not make any assumptions either
on the fact that $\eta$ is small or in the
structure of the group.
We also make assumptions of hyperbolicity on
$f$ weaker than being Anosov.
See Theorem \ref{main} for a precise statement of
the results.
The argument presented here is extremely simple.
Hence, we decided to present it in its simplest
form and present some of the further developments
in a somewhat more informal way in Section~\ref{sec:final}.
If all the improvements were taken at the same time,
the paper -- even if conceptually simple -- would have
been hard to read.
\section{Preliminaries and notation}
\subsection{Some conventions that will be used
throughout the paper}
We will assume, that $G$ is a linear group.
(i.e. a subgroup of $GL(n,\real)$ or, more generally of
a Banach algebra $\cB$ of operators, possibly infinite dimensional.
This will be used in two ways in the paper.
First, it will allow to write to use the product
notation for the group operation and,
at the same time to use $\vp_1 - \vp_2$ and the like.
Of course, differences should be interpreted as elements of the Lie
algebra.
This is only a typographical convenience and can be eliminating
just by complicating the notation.
A more subtle point is that we will be able to
write the derivative of a product of
factors as a sum of terms. This, implicitly uses
that the group operation is linear.
Similarly, given a vector field $\delta$ on the manifold we will write
$x + \delta(x)$ to denote $\exp_x\delta(x)$. This is only for
typographical convenience. Of course, one has to be careful
because this addition is not associative.
We emphasize that for functions taking values in groups we understand
that $\vp \in L^p(M,G)$ means that $\vp, \vp^{-1} \in
L^p\big(M, \cB )$ when the later is understood in the usual
sense of Banach valued $L^p$ spaces.
We note that the fact that $\eta \in C^\beta$ and that the manifold
is compact implies that $\eta$ remains at a finite distance
of non-invertible elements in the algebra. Hence, $\eta^{-1}$ is
$C^\beta$.
\subsection{Some notions of hyperbolicity}
We recall a very weak notion of
hyperbolicity.
\begin{defin}
We say that a diffeomorphism $f$ of a manifold
$M$ preserves a dominated splitting
when there is a decomposition of the tangent bundle
$T_xM = E^{<} \oplus E^{>}$
and numbers $C> 0$, $\lambda_< < \lambda_> $ in such
a way that
\begin{equation} \label{characterization}
\begin{split}
& v \in E^{<}_x \iff \abs{Df^n(x) v } \le C (\lambda_<)^n \abs{v} \quad n \ge 0 \\
& v \in E^{>}_x \iff \abs{Df^n(x) v } \le C (\lambda_>)^n \abs{v} \quad n \le 0 \\
\end{split}
\end{equation}
\end{defin}
As it is well known, it is always possible to introduce
an analytic metric in $M$ in such a way that $C = 1$
in \eqref{characterization} at the only price of redefining
slightly $\lambda_<, \lambda_>$. We will henceforth assume that
this redefinition has taken place.
In case that $\lambda_< < 1$, $\lambda_> < 1$
the system is Anosov.
In the case that $\lambda_< < 1$, we
say that the system has a dominated splitting with
a stable component. When $\lambda_> > 1$, we say that it
has a dominated splitting with an unstable component.
When the component is stable, we will use
the notation $E^s, \lambda_s$ to denote $E^<, \lambda_<$.
Analogously for the unstable case.
A very important property of systems
with a dominated splitting with
a stable component is the fact that the foliation $W^s$
associated to the stable component of the
splitting is absolutely continuous.
Analogously, for the unstable foliation.
See \cite{Anosov69} Theorem 9.
\begin{lem}\label{absolutecontinuity}
If the mapping $f$ is $C^{1 + \alpha}$, $\alpha > 0$,
and preserves a stable distribution
around every point, it is possible to find
a neighborhood $U_x$ and
a mapping $\Lambda_{U_x}: U_x \to \real^d$ which is
a local homeomorphism and which transforms the
stable foliation into a product foliation in $\real^d$.
Moreover, it can be arranged that
the mapping $\Lambda_{U_x}$ is continuous and absolutely
continuous. The Jacobian of $\Lambda_{U_x}$ is $C^\alpha$
along the leaves of the stable foliation,
continuous over $U_x$ and bounded away from zero.
\end{lem}
The Lemma \ref{absolutecontinuity} can also be
formulated by saying that the holonomy maps of the stable
and unstable foliations are absolutely continuous.
See \cite{PughS72}.
It is possible to show that if the map $f$ is $C^r$,
the Jacobian of $\Lambda_{U_x}$ can be made to $C^{r-1}$
along the leaves of the stable foliation and H\"older
with its derivatives across the manifold.
See \cite{LlaveMM86}.
\begin{defin}\label{displacements}
Let $W^s$ be the stable foliation of a $C^{1+\alpha}$,
$\alpha > 0$ dynamical system, (in particular absolutely continuous).
We say that a continuous vector field $\delta$ is adapted to the
foliation $W^s$ when:
\begin{itemize}
\item[i)] $x + \delta(x) \in W^s_x$
\item[ii)] Denoting by $h_\delta(x) = x + \delta(x) $,
$h_\delta$ is absolutely continuous.
\end{itemize}
\end{defin}
A system is called partially hyperbolic when it has two
dominated splittings, one with an stable component,
another one with an unstable component.
The following definition is somewhat
standard.
\begin{defin}\label{uniformlyaccessible}
We say that a partially hyperbolic system is uniformly accessible when
$\exists\quad \ep_0 > 0$, $N \in \nat$, $K \in [1,\infty)$, $C > 0$
such that:
Given $x, y \in M$, $d(x,y)\leq \ep$,
we can find
\[
u_1 \cdots u_N, v_1\cdots v_N,\in M
\]
in such a way that
\begin{itemize}
\item[a)] $u_1 = x$, $v_N = y$
\item[b)] $v_i \in W^s_{u_i}$, $u_{i+1} \in W^u_{v_i}$
\item[c)] $d(u_i, v_i) \leq C \ep^{1/K}$,
$d({v}_i, {u}_{i+1}) \leq C \ep^{1/K}$.
\end{itemize}
\end{defin}
The notion of uniformly accessible systems means that all sufficiently
close points can be connected by a sequence of steps along stable and
unstable manifolds. The intuition is that, if we
want to connect two points, at the beginning we may have to go
rather far away, but then, we come back.
The number of steps is uniformly bounded and the length
of each step is bounded is a root of
the length of distance among the original points.
The importance of conditions similar to this for cohomology equations
was emphasized in \cite{KatokK96}.
The amount to the stable and unstable foliations being quite
non-integrable and they have played a role in many studies of
partially hyperbolic systems, starting with \cite{BrinP73}.
We will need the following property which is a
version of Definition \ref{uniformlyaccessible}
including measurability assumptions.
\begin{defin}\label{uniformmeasure}
We say that a partially hyperbolic system is measurably uniformly
accessible when for some
$\ep > 0$, $n \in \nat$, $K \in [1, \infty)$, $L \in \real^+$
we have the following property:
Given a $C^1$ measure preserving vector field
with $\norm{ h - \Id}_{C^1} \le \ep_0$,
we can find vector fields $u_1,\ldots u_N$,
$v_1, \ldots, v_N$.
\begin{itemize}
\item[a)] $u_1,\ldots u_N$ adapted to the stable foliation,
$v_1, \ldots, v_N$ adapted to the unstable foliation.
\item[b)] $h = h_{v_N} \circ h_{u_N} \circ \ldots h_{v_1}\circ h_{u_1}$.
\item[c)] $\norm{v_i}_{L^\infty}, \norm{u_i}_{L^\infty} \le
L \norm{ h - \Id}_{L^\infty}^{1/K}$.
\item[d)] $\norm{ J[h_{u_i}]}_{L^\infty}, \norm{ J[h_{v_i}]}_{L^\infty} \le L$.
Where $J[h]$ denotes the Jacobian of the function $h$.
\end{itemize}
\end{defin}
Note that Definition \ref{uniformmeasure}
is a version of Definition \ref{uniformlyaccessible}
in which
we consider the motions connecting $x$ to $h(x)$
but we require that the decompositions are not only small
-- which more or less follows from Definition \ref{uniformlyaccessible}--
but also we have condition $d)$ which requires that the maps
$h_u$ are
absolutely continuous.
The absolute continuity of the maps and the fact that they
have bounded Jacobians comes into play when we consider
their action on $L^p$ functions under composition.
Note that $\int \abs{\vp\circ h(x)}^p \, dx =
\int \abs{\vp(y)}^p J_h(y) \, dy$
where $J_h$ is the Jacobian of $h^{-1}$.
Therefore:
\begin{equation} \label{normtransform}
\norm{\vp \circ h}_{L^p} \le
\norm{\vp }_{L^p} \norm{J_h}_{L^\infty}^{1/p}
\end{equation}
We do not know how to characterize
systems satisfying Definition \ref{uniformmeasure}.
We will just give two examples of systems that satisfy
Definition \ref{uniformmeasure}.
\begin{prop} \label{anosov}
An Anosov system satisfies Definition \ref{uniformmeasure}.
The algebraic examples in \cite{BurnsPSW01} p. 330
satisfy Definition \ref{uniformmeasure}.
\end{prop}
{\bf Proof.}
Given a manifold $W^s_x$, we consider the manifold
$h(W^s_x)$. Both of them are $C^1$ manifolds.
Restricted to $W^s_x$, the map $h$ is absolutely
continuous and the Jacobian is bounded uniformly
when $h$ ranges in a $C^1$ neighborhood of the identity.
Moreover, $W^s_x$, $h( W^s_x)$ are $C^1$ close.
We can, therefore define a holonomy map $H_x$ along the
unstable foliation from $h(W^s_x) $ to $W^s_x$.
Because of Lemma \ref{absolutecontinuity}, the holonomy
map will be absolutely continuous
when the manifolds are endowed with the
Riemannian measures and the Jacobian of
the holonomy map is bounded. We note that the bound
is uniform over all the points in the manifold and over
all the maps $h$ which are in a $C^1$ neighborhood of the identity.
We denote by $\tilde h = H_x \circ h$. The map $\tilde h$ is
adapted to the stable foliations and absolutely continuous
when restricted to each leaf. Moreover,
we have bounds on the Jacobian which are uniform over
the point in the manifold and over the maps $h$ in
a $C^1$ neighborhood.
Given the absolute continuity of the stable foliation,
it is clear that a map which preserves the leaves and
is uniformly absolutely continuous over all of them, is
absolutely continuous and we can bound the Jacobian.
The fact that the algebraic examples satisfy te definition is
clear because the stable and unstable foliations are
smooth.
\qed
\subsection{The potential spaces}\label{sec:sobolev}
In this subsection, we briefly recall the
definition and a few facts about
the so-called potential spaces and adapt them to
an slightly more geometric meaning.
The results we will present here are quite well known and
can all be found in \cite{Stein70} p. 134 -- 165.
\cite{Taylor97}, \cite{Nikolskii61}.
More in depth treatments of Sobolev spaces can be found in
\cite{Adams75}, \cite{Nikolskii75}.
In this paper
we will only be concerned with those spaces for
which there exist characterizations in terms of modulus of
continuity. The reason is that the modulus of continuity
of solutions of cohomology equations can be studied using
dynamical arguments.
\begin{defin} \label{ourspaces}
For $1 \le p \le \infty$,
given a $\psi \in L^p(\real^d)$,
we say that $\psi \in \ccL^p_\alpha(\real^d)$
when $\norm{ \psi( \cdot) - \psi( \cdot + t)}_{L^p} \le C |t|^\alpha$
for all $t$, $|t| \le \gamma_0$ for some $\gamma_0$.
\end{defin}
These spaces can be endowed the norm
\[
\max( \norm{\psi}_{L^p},
\sup_{0 < |t| \le \gamma_0}
\norm{ \psi( \cdot) - \psi( \cdot + t)}_{L^p} |t|^{-\alpha}
\]
which makes them Banach spaces.
We recall that the potential spaces, defined in
\cite{Stein70} p. 134 are the image of $L^p$ under
$(- \Delta + \Id)^{-\alpha/2}$.
(i.e. $W^{\alpha, p} = \{ f | (-\Delta + \Id)^{\alpha/2} \in L^p\}$
endowed with the norm
$\norm{f}_{W^{\alpha, p}} = \norm{(-\Delta + \Id)^{\alpha/2}f}_{ L^p}$.
Hence, it is natural
to consider them as Sobolev spaces for fractional regularities.
Hence, we use the notation $W^{\alpha, p}$.
\begin{remark}\label{notation}
In \cite{Stein70},
the spaces that we call $W^p_\alpha$ are called $\cL^p_\alpha$ and, in p. 135, it is shown
that, for $\alpha \in \nat$, this definition agrees with
a more customary definition in terms of $L^p$ norms of
derivatives. We will follow the more customary notation
$W^{\alpha,p}$ to avoid having to resort to different
fonts for $L$ as in \cite{Stein70}.
The notation $\ccL^p_\alpha$ does not occur in
\cite{Stein70}.
\end{remark}
Some relations between these spaces
are summarized in Lemma \ref{inmersions} below.
\begin{lem}\label{inmersions}
With the above definitions:
\begin{itemize}
\item[a)]
When $\alpha = 1, 1 < p < \infty$ or when
$0 < \alpha < 1$ and $p = 2$, then
\[
\ccL^p_\alpha = W^{\alpha,p}
\]
\item[b)]
When $0 \le \alpha < 1$ we have
\[
\ccL^p_\alpha \subset W^p_{\alpha'}
\]
for every $\alpha' < \alpha < 1$.
\end{itemize}
\end{lem}
Part a) of Lemma \ref{inmersions} can be found in
\cite{Stein70} \S 3. 3 p. 138. Part b) in
and in \S 3.5.2 p. 141. (It suffices to work out the
integrals in the sufficient conditions stated there.)
Two important results for $\cL^p_\alpha$ spaces
that we will need are:
\begin{lem} \label{results}
We have:
\begin{itemize}
\item[a)]
$\ccL^p_\alpha \subset L^q$
where $ \frac{1}{q} = \frac{1}{p} - \frac{\alpha}{d}$.
\item[b)]
If $p > d/\alpha$, then $W^p_\alpha \subset C^0$.
\end{itemize}
In both cases, the embeddings are continuous.
\end{lem}
Both results are versions of the Sobolev embedding theorem
which can be found in \cite{Stein70} p. 124 -- for integer $\alpha$.
The standard result for $0 < \alpha < 1$ can be found
e.g. in \cite{Taylor97} p. 22 ff.
It is also true that in case b) of Lemma~\ref{results} one
can conclude that the function is H\"older. For us, this is not
so important, since, once we know that the functions we
are interested in are continuous, we can use dynamic
arguments to obtain results which are sharper than those
obtained by applying embedding theorems.
The definition of spaces above generalize straightforwardly to manifolds.
The results mentioned above remain valid. All of them are local
(notice that we need only to consider only displacements that
are small enough. Hence we can use partition of unity arguments).
To apply the results to dynamical systems, we will find it useful
to introduce also analogues of the $\ccL^p_\alpha$ associated
to the foliations that appear in dynamical systems.
\begin{defin}
Let $0 < \alpha \le 1$, $1 \le p \le \infty$.
Given a function $\psi \in L^p$, we say that
$\psi \in \ccL^{p, (s)}_{\alpha} $
when
\[
\norm{\psi - \psi\circ h_\delta}_{L^p} \le C \norm{\delta}_{L^\infty}^\alpha
\]
for all the vector fields $\delta$ that are adapted to the
foliation $W^s$ and $\norm{\delta}_{L^\infty} \le \gamma_0$ for
some $\gamma_0 > 0$.
Analogous definitions, of course, are true for the
unstable foliations.
\end{defin}
The following proposition is a very simple argument.
\begin{prop} \label{decomposition}
Assume that the system satisfies the Definition \ref{uniformmeasure}
Then, for $1 \le p \le \infty$, $0 < \alpha \le 1$,
\begin{equation} \label{intersection}
\ccL^{p, (s)}_\alpha \cap \ccL^{p, (u)}_\alpha \subset \ccL^p_{\alpha/K}
\end{equation}
\end{prop}
The Proposition \ref{decomposition} is very elementary.
The only thing that is used is adding and subtracting increments along
the stable and unstable directions.
\[
\psi \circ h - \psi =
\big( \psi \circ h_{\delta^u_N} - \psi \big) \circ h_{\delta^s_N} \circ
\ldots \circ h_{\delta^u_1} \circ h_{\delta^s_1}
+ \cdots +
\psi \circ h_{\delta^s_1} - \psi
\]
For other higher regularity Sobolev spaces, when $\alpha \in \nat$,
there are results similar to \eqref{intersection} for Anosov systems.
We refer for more details to \cite{LlaveMM86} and
\cite{Llave01}. The proos use methods from elliptic regularity theory.
\section{Statement of results}
We recall that our standing assumptions are
that $M$ is a compact manifold, $f$ is a $C^{1 + \alpha}$
diffeomorphism and $\eta$ is $C^\beta$ $ 0 < \beta \le \Lip$.
The main technical result of this note
is:
\begin{thm}\label{main}
Assume that $f$ admits a dominated splitting
with a stable component $E^s$.
Then, if $\vp \in L^p(M,G)$, $1\le p < \infty$, then,
\[
\vp \in \ccL^{p/3, (s) }_{\beta}
\]
\end{thm}
Of course, we have similar results for $\vp^{-1}$ in place
of $\vp$ and for the unstable foliation.
Once we have Theorem~\ref{main},
using the known properties of the potential spaces
summarized in Section \ref{sec:sobolev},
we obtain in the conditions of Theorem \ref{main}
\begin{cor} \label{bootstrapfinal}
Assume that $f$ is uniformly
accessible as in Definition \ref{uniformlyaccessible}.
Assume that
\begin{equation}\label{condition}
p \ge 2 \frac{d K}{\alpha}
\end{equation}
where $d$ is the dimension of the manifold.
Then $\vp$ is continuous.
(Therefore
we have $\vp \in C^{\beta/K}$. )
\medskip
Moreover, if $f$ is Anosov and $\beta = \Lip$, then
$\vp \in W^{1,p}$.
\end{cor}
The proof of the Corollary is quite simple given the material
mentioned in Section \ref{sec:sobolev}.
We note that, by Theorem \ref{main}
if $\vp, \vp^{-1} \in \ccL^{p/3,{s}}_\beta \cap \ccL^{p/3,{u}}_\beta
\subset \ccL^{p}_{\beta/K}$.
By Sobolev embedding theorem, we conclude
$\vp, \vp^{-1} \in L^q$ where
$\frac{1}{q} = \frac{3}{p} - \frac{d K }{\beta} - \ep$.
We note that, if $p$ satisfies $\eqref{condition}$, we
have that $q > p$. We can keep repeating the process
till we obtain that
$\vp, \vp^{-1} \in \ccL^{q}_{\beta/K}$
with $q \ge \frac{d K}{\beta}$.
Then, we can conclude that $\vp, \vp^{-1}$ are continuous
by Sobolev embedding theorem.
Once that they are continuous, there are arguments in
\cite{NiticaT98}
to conclude that $\vp, \vp^{-1}$ are H\"older when
restricted to the stable and unstable manifolds.
Then, we obtain that $\vp, \vp^{-1}$ are H\"older.
Since we will use variants of the method of
\cite{NiticaT98}, we
we postpone till Remark \ref{holderbootstrap}
the discussion of the bootstrap from continuous to
$C^{\beta/K}$.
\section{Proof of Theorem \ref{main}}
We will use a method which is similar to the method used in
\cite{Llave01} in the commutative case -- iterate
the equation \eqref{cohomology} and derived formulas
for the increments.
for the non-commutative
case, we use several cancellations that were discovered in
\cite{NiticaT98}.
\subsection{Some representation formulas}
We note the equation \eqref{cohomology} implies that for every, almost every
$x$ and for every $n \in \nat$ we have
\begin{equation} \label{iterate}
\vp(x) = \eta(x) \cdots \eta \circ f^{n}(x)
\vp\circ f^{n+1}(x)
\end{equation}
for $\mu$-almost all $x$.
Let $\delta(x)$ be a vector field on the manifold $M$.
For a function $\Psi: M \to G$ we define
$h_\delta(x) = x + \delta(x) $
and
\begin{equation} \label{difference}
\big[\Delta \vp\big](x) = \vp\big(x + \delta(x)\big) - \vp(x)\ .
\end{equation}
Adding and subtracting extra terms in \eqref{difference}
we have:
\begin{equation}\label{productformula}
\big[\Delta (\Psi\vp)\big](x) = \Psi\big(x +
\delta(x)\big)\big[\Delta \vp\big](x) +
\big[\Delta\Psi\big](x)\vp(x)\ .
\end{equation}
Note that this is one of the places where we use the fact that
we are considering a group which is a Banach algebra. For a general
group, the derivative of the composition operation would appear.
Substituting \eqref{productformula} into \eqref{iterate}, we have
\begin{equation}\label{cohomologyn}
\begin{split}
\Delta \vp(x) & = \sum_{i=0}^n \eta\circ h_\delta(x) \cdots
\eta \circ f^{i-1} \circ h_\delta(x) \\
&\quad \quad \quad \quad \quad
\cdot \big[\Delta(\eta\circ f^i)\big](x)
\eta\circ f^{i+1}(x) \cdots \eta\circ f^n(x) \vp\circ
f^{n+1}(x) \\
& \quad + \eta\circ h_\delta(x) \cdots
\eta \circ f^n\circ h_\delta(x) \big[\Delta(\vp\circ f^{n+1})\big](x)
\end{split}
\end{equation}
Using again \eqref{iterate} we have:
\begin{equation} \label{cancellation}
\begin{split}
\Delta \vp(x) & = \sum_{i=0}^n \vp\circ h_{\delta}(x)
\vp^{-1}\circ f^i\circ h_\delta (x)
\big[\Delta(\eta\circ f^i)\big](x)
\vp\circ f^{i+1}(x) \\
& \quad + \vp\circ h_\delta(x)\vp^{-1}\circ f^{n+1}\circ h_{\delta}(x)
\big[\Delta(\vp \circ f^{n+1})\big](x)\ .
\end{split}
\end{equation}
Even if formula \eqref{iterate} is quite general,
we restrict our attention to the case that $\delta$ is
adapted to the stable foliation.
In such a case, we estimate the general term of
the sum in
\eqref{cancellation}
as:
\begin{equation}\label{generalterm}
\begin{split}
\norm{
\vp\circ h_{\delta}
\vp^{-1}\circ & f^i\circ h_\delta
\Delta(\eta\circ f^i)
\vp\circ f^{i+1}
}_{L^{p/3}} \\
&\le C \norm{\vp}_{L^p}^2\norm{\vp^{-1}}_{L^p}
\norm{\Delta(\eta\circ f^i)}_{L^\infty} \\
&\le C \norm{\vp}_{L^p}^2\norm{\vp^{-1}}_{L^p}
\norm{\eta}_{C^\beta}
\norm{\delta}_{L^\infty}^\beta \lambda_s^{i \beta}
\end{split}
\end{equation}
Hence we see that
the $L^{p/3}$ limit as $n \to \infty$ in the sum in
\eqref{cancellation} exists.
Now we turn to study the last term in \eqref{cancellation}.
We note that
\[
\norm{ \vp\circ h_\delta(x)\vp^{-1}\circ f^{n+1}\circ h_{\delta}
}_{L^{p/2}}
\]
is bounded uniformly in $n$.
We also note that, when $\vp$ is continuous,
\[
\norm{ \Delta(\vp \circ f^{n+1})}_{L^\infty} \to 0.
\]
Since the space of continuous functions is $L^p$-dense on
$L^p$, an $\epsilon/3$ argument shows that
\begin{equation}\label{lasterm}
\norm{ \vp\circ h_\delta(x)\vp^{-1}\circ f^{n+1}\circ h_{\delta}(x)
\big[\Delta(\vp \circ f^{n+1})\big] }_{L^{p/3}} \to 0
\end{equation}
Hence, if we take limits in \eqref{cancellation},
we obtain that
\begin{equation} \label{incrementformula}
\Delta \vp(x) = \sum_{i=0}^\infty \vp\circ h_{\delta}(x)
\vp^{-1}\circ f^i\circ h_\delta (x)
\big[\Delta(\eta\circ f^i)\big](x)
\vp\circ f^{i+1}(x)
\end{equation}
{From} \eqref{incrementformula}
and summing the estimates in \eqref{generalterm}, we obtain
\[
\norm{\Delta \vp}_{L^{p/3}} \le C \norm{\delta}_{L^\infty}^\beta
\]
where $C$ may depend on $\vp, \vp^{-1} $ and, of course, of
$\eta$, $f$ but is independent on
$\delta$.
That is, we conclude that $\vp \in \ccL^{p/3}_{\beta, (s)}$.
A similar argument -- or just use this result for
replacing $f$, $\eta$, $\vp$ by $f^{-1}$, $(\eta^{-1})^t$
$(\vp^{-1})^t$ -- shows that
$\vp \in \ccL^{p/3}_{\beta, (u)}$.
An appeal to Lemma \ref{decomposition}
ends the proof of Theorem \ref{main}.
\qed
\begin{remark}\label{holderbootstrap}
We also note that the formula
\eqref{incrementformula}
also shows that if $\vp, \vp^{-1}$ are continuous and
$\eta$ is $C^\beta$, then, we can bound
$|| \Delta \vp||_{L^\infty} \le C || \delta ||_{L^\infty} $.
This means that $\vp$ is $C^\beta$ along the stable
leafs. A similar argument shows that
$\vp$ is $C^\beta$ along the unstable leaves.
It is then, elementary to show that $\vp \in C^{\beta/K} $
provided that the system satisfies Definition \ref{uniformlyaccessible}.
This is the argument promised in the
proof of Corollary\ref{bootstrapfinal}
\end{remark}
\section{Some generalizations} \label{sec:final}
We note that the present method can be generalized in several
respects. All of them are reasonably straightforward and
they can all be carried out simultaneously.
The formulation, however would obscure the simple idea.
\subsection{Different integrability conditions for $\vp$,
$\vp^{-1}$}
The main result, Theorem \ref{main} can be modified so
that we do not assume the same integrability for $\vp$ and
$\vp^{-1}$.
A more precise version of the argument, proved by the same method
is
\begin{thm}\label{main2}
Assume that $f$ admits a dominated splitting
with an stable component $E^s$.
Then, if $\vp \in L^p(M, \cB)$, $1\le p < \infty$,
$\vp^{-1} \in L^q(M, \cB)$, $1\le q < \infty$,
then,
\[
\vp \in \ccL^{p', (s) }_{\beta} \quad \quad
\vp^{-1} \in \ccL^{q', (s) }_{\beta}
\]
where
\[
\frac{1}{p'} = \frac{2}{p} + \frac{1}{q}; \quad \quad
\frac{1}{q'} = \frac{1}{p} + \frac{2}{q}.
\]
\end{thm}
If the system satisfies Definition \ref{uniformmeasure}, from
the assumptions of integrability in Theorem \ref{main2}
we can conclude that
\[
\begin{split}
& \vp \in \ccL^{p'}_{\beta/K} \subset L^{p'' - \ep} \\
& \vp^{-1} \in \ccL^{q'}_{\beta/K} \subset L^{q'' - \ep}
\end{split}
\]
where
\[
\frac{1}{p''} = \frac{2}{p} + \frac{1}{q} - \frac{\beta}{d K}; \quad \quad
\frac{1}{q''} = \frac{1}{p} + \frac{2}{q} - \frac{\beta}{d K}
\]
Of course, when the RHS of the above equation is negative, it means
that the corresponding function is continuous.
We can see that when $\frac{1}{p} + \frac{1}{q} < \frac{\beta}{d K} $
we can apply repeatedly the bootstrap argument and
get that the function is continuous.
\subsection{$\ccL^q_\beta$ regularity of $\eta$}
The following result is true:
\begin{thm}\label{main3}
Assume that $f$ admits a dominated splitting
with an stable component $E^s$.
Then, if $\vp \in L^p(M, \cB)$,
$\eta, \eta^{-1} \in \ccL^q_\beta$, $q,p \in [1,\infty]$
$0 \le \beta \le 1$.
then,
\[
\vp \in \ccL^{p', (s) }_{\beta} \quad \quad
\]
where $1/p' = 3/p + 1/q$.
\end{thm}
The proof of the result is the same as in the previous
cases.
We just bound the $L^{p'}$ norm
of the general term in \eqref{cancellation}
using the H\"older inequality.
The last term goes to zero.
If $q > p$, one can do better.
Using the embedding theorems, we
obtain that in the conditions of Theorem \ref{main3},
we have $\vp \in L^{p''}$
where $1/p'' = 3/p + 1/q - \beta/(K d) - \ep $.
Under the condition
$2/p + 1/q \le - \beta/(K d) $, $p'' > p$
we can repeat the process till we conclude
that $\vp \in \ccL^{q - \ep}_{\beta/K}$.
\subsection{Weakening the assumption that $f$
preserves a smooth measure. }
We point out that the fact that $f$ preserves a smooth measure
is not used much. The only place is in the estimates of
the iterated maps. Similar arguments to those outlined here
appear in \cite{Llave01}.
Hence, instead of \eqref{generalterm}, we
could bound:
\begin{equation}\label{generalterm2}
\begin{split}
\norm{
\vp\circ h_{\delta} &
\vp^{-1}\circ f^i\circ h_\delta
\Delta(\eta\circ f^i)
\vp\circ f^{i+1}
}_{L^{p/3}} \\
&
\le C \norm{\vp}_{L^p}^2\norm{\vp^{-1}}_{L^p}
\norm{\eta}_{C^\beta}
\norm{\delta}_{L^\infty}^\beta \lambda_s^{i \beta} \norm{J_f}_{L^\infty}^{i/p}
\end{split}
\end{equation}
We see that, provided that
\begin{equation}\label{condition1}
\lambda_s^\beta \norm{J_f}_{L^\infty}^{1/p} < 1
\end{equation}
the series in \eqref{cancellation} converges.
The analogue of \eqref{lasterm} will require more
analysis and more assumptions.
We start by studying the effect on highly iterated
compositions on the differences.
Note that we have:
$\vp \circ f^n \circ h_\delta = \vp \circ h_{\tilde \delta_n} \circ f^n$
where $h_{\tilde \delta_n}$ is defined so that the equation is satisfied.
By the contractivity of the fibers we have
\[
\norm{\tilde \delta_n}_{L^\infty} \le \norm{\delta}_{L^\infty} \lambda_s^n
\]
We also have
\[
\norm{ J_{\tilde h_n} }_{L^\infty} \le C
(\norm{J_f}_{L^\infty} \norm{J_{f^{-1}} }_{L^\infty} )^n
\]
If we assume that $\vp \in \ccL^p_\nu$ for some
$\nu > 0$ we see
that
\[
\norm{ \Delta(\vp \circ f^{n+1}}_{L^p} \le
C \lambda_s^{\nu (n+1)} \norm{\delta}_{L^\infty}^\nu
(\norm{J_f}_{L^\infty} \norm{J_{f^{-1}} }_{L^\infty} )^n
\]
Since
\[
\norm{ \vp\circ h_\delta \vp^{-1}\circ f^{n+1}\circ h_{\delta}}_{L^{p/2}}
\le
\norm{J_f}_{L^\infty}^{n /p}
\]
We see that we recover \eqref{lasterm} provided
that
\begin{equation}\label{condition2}
\norm{J_f}_{L^\infty}^{1 + 1/p} \norm{J_{f^{-1}} }_{L^\infty}
\lambda_s^\nu < 1.
\end{equation}
The rest of the proof goes through.
Hence, we can see that the assumption that $f$ preserves a smooth
measure can be somewhat relaxed to
assuming that $\vp \in \ccL^p_\gamma$
where the parameters satisfy \eqref{condition1}, \eqref{condition2}.
\subsection{Exploiting the structure of the group and
smallness assumptions on $\eta$} \label{sec:small}
We assume that we have bounds
\begin{equation}\label{expbounds}
|\eta(x) \cdots \eta\circ f^i(x)| \le C \rho^n
\end{equation}
with $C \ge 0$, $ 1 \le \rho $ sufficiently small.
(of course, if $\rho < 1$, it is trivial to show
that there are only trivial solutions)
Such bounds happen for example when $\eta$ takes values in
unipotent groups or when $\eta$ is uniformly small.
We also assume that
$\vp \in \cL^p_\nu$ for some $\nu > 0$.
Then, we can proceed to
we could proceed to bound the general term in
\eqref{cohomologyn} rather than bounding the general
term in \eqref{cancellation}.
We see that the general term in the sum in
\eqref{cohomologyn} can be bounded
\begin{equation}\label{generalterm3}
\norm{ \eta\circ h_\delta\cdots
\eta \circ f^{i-1} \circ h_\delta
\big[\Delta(\eta\circ f^i)\big]
\eta\circ f^{i+1} \cdots \eta\circ f^n(x) \vp\circ
f^{n+1} }_{L^p} \le
C \rho^n \lambda_< ^{i \beta }
\norm{\delta}_{L^\infty}^{\beta}
\end{equation}
The last term in \eqref{cohomologyn} can be bounded
by $C \rho^n \lambda_<^{n \nu} \norm{\delta}_{L^\infty}^{\nu}$
Summing \eqref{generalterm3} from $i = 0 $ to $n$ and
adding the bound for the last term we
obtain that for every $n \in \nat$, we have:
\begin{equation} \label{intermediate}
\norm{\Delta \vp}_{L^p} \le C \rho^n( \norm{\delta}_{L^\infty}^{\beta}
+ \lambda_<^{n\nu} \norm{\delta}_{L^\infty}^{\nu} )
\end{equation}
Choosing $n = \log \delta^{\beta - \nu}/\log( \lambda_<^\nu)$,
we obtain that \eqref{intermediate}
implies
\[
\norm{\Delta \vp}_{L^p} \le \norm{\delta}_{L^\infty}^{\gamma}
\]
with
\[
\gamma = (\beta - \nu)\frac{\log \rho}{\log( \lambda_<^\nu) } + \beta
\]
In the nilpotent case, when we can take $\rho$ as close to $1$
as desired we see that we can make $\gamma$ as close to $\beta$
as desired.
This tells that, in the nilpotent case, from $\vp \in L^p$
we can conclude $\vp \in L^{p, (s)}_{\beta -\ep}$.
This is very similar to the result Theorem \ref{main}
except because we integrability exponent is not divided by $3$.
This makes much easier to bootstrap.
Using the Sobolev embedding Theorem, we conclude
that $\vp \in L^{p'}$
where \[
\frac{1}{p'} = \frac{1}{p} - \frac{\beta}{d K}.
\]
Since we can repeat the steps, we
end up concluding that $\vp$ is continuous.
Hence, we have established that in the case that $\eta$ is nilpotent,
in the same conditions of Corollary \ref{bootstrapfinal},
from the fact that $\vp \in \ccL^p_\nu$ with $\nu \ge 0$, $1 \le p \le \infty$
we conclude that $\vp$ is continuous.
\subsection{Results for flows}
There are very similar results for flows. We note that
Theorem \ref{main} applied to the time one map, gives
a result for the strong stable direction of the flow.
We also note that, for the cohomology equations for
flows, regularity along the flow is automatic.
We also note that Proposition \ref{decomposition} is
elementary and, hence, holds whenever we have
a number of transversal foliations satisfying an
analogue of Definition \ref{uniformmeasure}.
Another rather straightforward case is cohomology
over Cartan actions (we refer to \cite{Hurder94}
for the definitions) provided that we have an
analogous of Definition \ref{uniformmeasure}.
\subsection{Some possible future extensions and open problems}
The main idea of this paper that we sometimes we can
obtain information on Sobolev regularity by examining modulus of
continuity in some integral norm.
This is useful in the study of cohomology
equations because information on
modulus of continuity can be obtained by iterating the equations
and deriving asymptotic formulas for increments
when we displace along stable or unstable manifolds.
We point out that our study is very preliminary and we have
only started with the easiest cases where uniform moduli
of continuity are used. There are many more sophisticated
spaces which could be useful in these studies.
(In particular the spaces that \cite{Stein70} calls
$\Lambda^{p,q}_\alpha$.)
It seems to us that the main obstacle in using these spaces is
the somewhat technical problem that many of the proofs use the
Euclidean structure.
One important result that is still missing
is an analogue of \eqref{decomposition} for higher differentiability properties.
Since the methods of proof the regularity results Lemma 2.5
in \cite{LlaveMM86}
are based in Schauder estimates, which generalize well to
Sobolev spaces, it seems that this program of study has good
chances of success.
Of course, the regularity results for systems with
Definition \ref{uniformlyaccessible} remain wide open.
\section{Acknowledgments}
The work of the author has been supported by NSF grants.
%\bibliographystyle{alpha}
%\bibliography{llave99}
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