0$ \begin{equation*} C^{-1} \|(-\Delta +\Id)^{k/2}\varphi\|_{L^p} \le \|\varphi\|_{W^{k,p}} \le C\|(-\Delta +\Id)^{k/2} \varphi\|_{L^p} \end{equation*} \end{lem} An extension of this characterization will play an important role in our work (see Lemma~\ref{regularity}). We call attention to the fact that this characterization cannot be extended to the cases $p=1$, $p=\infty$ and that indeed there are well known counterexamples (See e.g. the discussion in \cite{Stein70} p. 304 ) When $k$ is an even integer the meaning of $(-\Delta+\Id)^{k/2}$ is clear as a differential operator. When $k$ is an odd integer, it can be interpreted using the calculus of self-adjoint operators and, more importantly, there are integral kernel representations (called the Bessel potentials in \cite{Stein70}). These integral representations are quite explicit in Euclidean spaces. Using the operators $(-\Delta+\Id)^{k/2}$ for non-integer values of $k$, one can define Sobolev spaces of fractional order. $W^{k,p} = \{ u | (-\Delta+\Id)^{k/2}u \in L^p\}$ with the obvious norm. (These spaces are called the ``potential spaces'' in \cite{Stein70}) Indeed, $k$ can be taken to be complex. (Another useful scale of spaces that interpolates the integer valued Sobolev spaces is the so-called Besov spaces.) One useful fact about Sobolev spaces defined in this way is that they are interpolation spaces in the $k$ variable (See \cite{Taylor96} p. 275 ff.) This has the consequence: \begin{lem}\label{lem:interpolation} For every $p$, $1 \le p \le \infty$, every $k < l \in \real$, every $0 \le \theta \le 1$, every $u \in W^{l,p}$ we have. \begin{equation}\label{eq:interpolation} || u||_{W^{\theta k + (1-\theta)l,p}} \le K || u||_{W^{k}}^\theta || u||_{W^{l}}^{1-\theta} \end{equation} Moreover, if a linear operator $T$ maps $W^{k,p}$ into itself and $W^{l,p}$ into itself, we have that it also maps $W^{\theta k + (1-\theta)l,p}$ into itself and we have: \begin{equation}\label{interpolationop} ||T||_{W^{\theta k + (1-\theta)l,p}} \le K ||T||_{W^{k}}^\theta ||T||_{W^{l}}^{1-\theta} \end{equation} \end{lem} Since we are going to be working with dynamical systems and consider compositions with highly iterated diffeomorphisms it will be significantly easier for us to use only differential operators and not integral kernels since the change of variable formulas are much easier to prove and to handle (they reduce to the chain rule). Hence our general regularity results will be restricted to Sobolev spaces of integer (and even) order. On the other hand, our counterexamples will be constructed in easy to understand manifolds such as tori and, hence, for the counterexamples, we will use fractional Sobolev spaces, which make the results sharper. Restricting to integers $k$ in the definition of Sobolev spaces and to even $k$ in Lemma~\ref{characterization} and its analogues will lead to results which are sometimes less than what one can conjecture is optimal by one or two derivatives. (This lack of optimality in our results can be exhibited --- and mitigated --- by applying interpolation as we will detail later in Remark~\ref{interpolation} ). This loss of one derivative in our results on Sobolev regularity in Livsic theorem can presumably be remedied by developing more geometric version of the integral characterizations of Sobolev spaces and studying their behavior under iterates of the dynamical systems. This seems to us quite doable, but we decided not to pursue this research in this note. We will also use the notation from \cite{Stein70} of $\Lambda_r$ for the Lipschitz spaces. These spaces agree with the usual $C^r$ spaces when $r$ is not an integer and $C^r$ means the space of $[r]$-times differentiable functions whose $[r]$ derivatives are H\"older continuous of exponent $r-[r]$. For integer $r$, they are different both from the spaces of $r$ times continuously differentiable functions and from the $r-1$ times differentiable functions with Lipschitz derivatives. A very important fact about Sobolev spaces is the Sobolev imbedding theorem (see \cite{Stein70}, V~2.2 for the first part and \cite{Taylor97} p. 29 Proposition 8 for the second part). We only state it under the hypothesis that we will need it. \begin{lem}\label{lem:imbedding} For compact manifolds of dimension~$n$, \begin{equation}\label{imbedding} W^{k,p} \subset L^q \end{equation} when \begin{equation} \label{imbeddingexponent} \frac1q \le \left[\frac1p -\frac{k}n\right]_+\ . \end{equation} (where we denote by $t_+ = \max( t, 0) $) and the inclusion operator is continuous. (The case that the R.H.S. of \eqref{imbeddingexponent} is zero, can be taken to mean that $q=\infty$ is allowed.) When $ r = k -\frac{k}p > 0$ we have $$ W^{k,p} \subset \Lambda_r$$ \end{lem} This allows us to show: \begin{lem}\label{Banachalgebra} When $\frac1p - \frac{k}n =0$, and the manifold is compact, the space $W^{k,p}$ is a Banach algebra under multiplication. \end{lem} \begin{proof} Recall that, by Leibnitz rule \begin{equation*} D^k (\varphi\cdot \psi) = \sum D^i \varphi D^{k-i}\psi \binom{k}{i}\ . \end{equation*} By Sobolev's theorem \begin{align*} &D^i\varphi \in L^{q_\varphi}\ \text{ with }\ \frac1{q_\varphi} = \frac1p -\frac{k-i}n\ ,\\ &D^{k-i}\psi \in L^{q_\psi}\ \text{ with }\ \frac1{q_\psi} = \frac1p - \frac{k-(k-i)}n \end{align*} By H\"older's inequality $$D^i\varphi D^{k-i} \psi \in L^q$$ with \begin{equation*} \frac1q = \frac1{q_\varphi} + \frac1{q_\psi} = \frac1p - \frac{k}n +\frac{i}n + \frac1p - \frac{k}n + \frac{k-i}n = 2\left[\frac1p - \frac{k}n\right] + \frac{k}n = \frac1p \end{equation*} by our assumption. The case when the number of derivatives is zero is special. We need that we are in a compact manifold to show that $\varphi\in L^\infty$, $\psi \in L^\infty$ and hence, the product is in $L^\infty \subset L^p$. \end{proof} \section{The invariant section theorem} A natural place to start the investigation of Sobolev regularities in dynamical systems is the celebrated invariant section theorem from \cite{HirschP68}. This theorem has become a basic tool in the study of invariant objects in dynamical systems. We reproduce the statement of \cite{HirschP68} supplementing it with one observation c). \begin{thm}\label{invsection} Let $p:Y \rightarrow X$ be a vector bundle over a metric space $X$. Let $D \subset Y$ be the unit ball bundle and $F:D \rightarrow D$ a map covering a homeomorphism $f:X \rightarrow X$. Suppose $0 \le \kappa < 1$ and that, for each $x\in X$, the restriction $F_x: D_x \rightarrow D_{f(x)}$ has Lipschitz constant $ \le \kappa$. Then: \begin{itemize} \item[a)] There is a unique section $g_0: X \rightarrow D$ whose image is invariant under $F$. \item[b)] Let $\Lip(f^{-1}) = \lambda < \infty$. Let $0 < \alpha \le 1$ be such that $\kappa \lambda^\alpha < 1$. \end{itemize} Then, $g_0$ satisfies a H\"older condition of order $\alpha$. Moreover \begin{itemize} \item[c)] Suppose that $X$ is a $C^1$ manifold, $Y$ is an $C^1$ vector bundle, and $F$ and $f$ are $C^1$. If $\kappa \lambda < 1$, then $g_0$ is $C^1$. \item[d)] Suppose that $X$ is a $C^{r+1}$ manifold, $Y$ is an $C^{r+1}$ vector bundle, and $F$ and $f$ are $C^{r+1}$. If $\kappa \lambda^{r+ \alpha} < 1$, then $g_0$ is $C^r$ and the $r$ derivative satisfies a H\"older condition of order $\alpha$. \end{itemize} \end{thm} \begin{proof} The proofs of a), b), c) is in \cite{HirschP68}. The proof of point $d)$ follows by applying the {\sl tangent functor trick} \cite{AbrahamR67}. \end{proof} In the following we would like to explore the possibility that there are some improvements of these results using Sobolev spaces. Note that if the conclusions are that a section is $C^{r}$ then, we conclude automatically that it is in $W^{r,\infty}$ (this remains true even for fractional $r$). Hence, the only improvements that we could hope for is that we obtain that it is in a certain $W^{r',p}$ with an $r' > r$. \begin{remark} It is important to notice that the regularity of the invariant section is limited by $r_0 = \log(\lambda)/\log(\kappa^{-1})$. It does not improve assuming more regularity in $F,f$. We will see later in examples that, as it is well known, this limitation does belong. \end{remark} \begin{remark} \label{spectralradius} Notice that the Lipschitz constants in the hypothesis depend on the distance that we are considering, but nevertheless, the regularity in the conclusions does not. \end{remark} \begin{remark}\label{remark3} Notice that, since we have uniqueness, an invariant section of $F^2$ is an invariant section for $F$. Hence, it suffices to take $\kappa$ in such a way that $\kappa^2 \ge \Lip( F_{f(x)} \circ F) $ and $\lambda$ such that $\lambda^2 \ge \Lip(f^{-2})$. More generally, \begin{equation}\label{iteratedLip} \begin{split} \kappa > \inf_{i} \left( \Lip( F_{f^{i-1}(x)}\circ \cdots F_x ) \right)^{1/i} \qquad \lambda > \inf_i \left( \Lip(f^i) \right)^{1/i} \end{split} \end{equation} Note also that \begin{align*} &\sup_x (\Lip F_{f^{i+k}(x)}\circ \cdots\circ F_x) \\ &\qquad \le \sup_x (\Lip F_{f^{i+k}(x)}\circ\cdots\circ F_{f^i(x)}) \cdot \sup_x (\Lip F_{f^{i-1}(x)}\circ\cdots\circ F_x)\\ &\qquad = \sup_x (\Lip F_{f^k(x)}\circ\cdots\circ F_x) \cdot \sup_x (\Lip F_{f^{i-1}(x)}\circ\cdots\circ F_x) \end{align*} Hence by a subadditivity argument for the logarithms, the infimum in \eqref{iteratedLip} is actually the limit. A similar argument works for $\lambda$. Hence, if we denote \begin{align*} \tilde\lambda (f^{-1}) &= \lim_{i\to\infty} (\Lip f^{-i} (x))^{1/i} \\ \tilde\kappa (F) & = \lim_{i\to\infty} (\sup_x \Lip F_{f^i(x)} \circ\cdots\circ F_x)^{1/i} \end{align*} we have that $\tilde\kappa,\tilde\lambda$ can be used in place of $\kappa,\lambda$ in Theorem~\ref{invsection}. Notice that, in contrast with $\kappa,\lambda$ the numbers $\tilde\kappa,\tilde\lambda$ do not change if the distances in $X,Y$ are changed by an equivalent metric. One should also notice that the $\tilde \kappa$, $\tilde \lambda$ are the values of the spectral radius of push-forward operators in continuous vector fields. (See \cite{Mather68}.) Hence, if we are going to produce examples showing that the results of Theorem \ref{invsection} cannot be improved even after the refinements contained in this remark, we will need to make sure that we choose norms in such a way that the above limit is exactly the value after a few steps. \end{remark} Our first example shows that one cannot get any improvements using Sobolev spaces. That is, we cannot get Sobolev spaces that contain a larger number of derivatives than those predicted by the classical result. \begin{example}\label{invsectionexample} Consider the map of $\torus^2$ to itself $A(x,y) = (2 x + y, x + y)$. ( As it is well known, $A$ has eigenvalues $\lambda, \lambda^{-1}$ with $\lambda > 1$, an irrational number. The eigenvectors of $A$ are irrational and orthogonal, since $A$ is symmetric.) Consider now the trivial bundle $\torus^2 \times \complex $ And the map defined on sections by \begin{equation}\label{firstmap} [\L v](x) = \mu v(A x) + P(x,y) \end{equation} for some $0< \mu < 1$ and were $P(x,y)$ is $P(x,y) = \sigma \exp(2 \pi i x)$, with $\sigma$ any number different from zero. Let $r_0 = -\log \mu/\log\lambda$. As predicted by Theorem~\ref{invsection}, the invariant section for this map is in $\Lambda_{r_0 -\delta}\subset W^{r_0 -\delta,\infty}$ for any $\delta > 0$. Nevertheless it is not in $W^{r_0, p}$ for any $1 \le p \le \infty$. (Also, we note it is not in $\Lambda_{r_0+\delta}$.) \end{example} Another sense in with the Sobolev regularity does not lead to improvements will be developed in Section~\ref{sec:bootstrap}. We will show that if an invariant section happens to be in $W^{{r_1},p}$, where $r_1$ is a number similar to $r_0$ depending on expansion rates, etc. then, it is roughly smooth as the map. We postpone a detailed discussion since even a precise formulation will need the development of more notation. \begin{remark} It will be clear after the proof that the example can be considerably extended. One can use other trig polynomials, other Anosov systems. Indeed, the behavior we report should be quite typical. At least, it is $C^\omega$ generic in the spaces of skew products. Of course, one can obtain similar results for real bundles rather than complex just by taking real parts of our example. The complex example has been chosen to simplify the notation. A purely real example can be obtained taking real parts since our example is linear. \end{remark} \begin{remark} Note that if we consider the map $F_0 : \torus^2\times \complex$ \begin{equation*} F_0 (x,y,z) = \left(A\binom{x}y,\mu z\right) \end{equation*} then, the set $\{(x,y,0)\}$ is an invariant normally hyperbolic manifold in the sense of \cite{Fenichel74}, \cite{HirschPS77}. The image of the section invariant by $\L$ in \eqref{firstmap} is an invariant manifold for \begin{equation*} F_\sigma (x,y,z) = A\binom{x}y,\ \mu z+\sigma P(x,y) \end{equation*} Hence, Example~\ref{invsectionexample} also shows that the regularity of the normally hyperbolic manifolds is, in general, limited by the ratios of expansion rates irrespectively of the regularity of the map. We also note that this lack of regularity happens even if $\mu$ is not resonant with the spectrum of $A$. (see the Remark~\ref{nonresonance} later). \end{remark} \begin{proof} In this very simple case, sections are just complex valued functions on the torus. Note that a section $\Psi$ has an invariant image if and only if it satisfies: \begin{equation}\label{invariantsection} \Psi\circ A = \mu \Psi + P \end{equation} We first claim that the invariant section of \eqref{firstmap} is the graph of \begin{equation}\label{invariant} \Psi(x,y) = \sum_{i = 0}^\infty \mu^i P \circ A^i \end{equation} Where the sum is understood in a weak enough sense. For example, in $C^0$. Indeed, to establish the claim, note that the sum in \eqref{invariant} converges uniformly (by Weierstrass M-test) and that the uniform convergence allows one to rearrange terms and verify that indeed it satisfies \eqref{invariant} when substituted in. Note that one of the consequences of the invariant section theorem is that it is the only bounded invariant section. We can compute the Fourier coefficients of $\Psi$ by: $$ \hat \Psi_k = \sum_{i=0}^\infty \mu^i \hat P_{{A^t}^{-i} k} $$ (In the case that we have considered in the example, of course, $A = A^t$.) Since the eigenvectors of $A$ are irrational, any vector $k \in \integer^2 $ has non-zero components both on the stable and unstable components, hence $|(A^{t})^i k | \ge C_k \lambda^i$. Note that in our example for $k^{(i)} = (A^t)^i \left( 1 \atop 0\right) $ we have $\hat \Psi_{k^{(i)}} = \mu^i$. If we consider $ \Phi = (-\Delta +1)^{r/2} \Psi$ -- of course in the sense of distributions -- we see that $\hat \Phi_{k^{(i)}} = \mu^i ( 4 \pi^2 |k^{(i)}|^2 +1)^{r/2}$. We see that if $\lambda^r \mu \ge 1$, that is if $ r \ge r_0$, then $\hat \Phi_{k^{(i)}}$ does not tend to zero as $i$ tends to infinity. This shows, by Riemann--Lebesgue theorem that $\Phi$ cannot be in any $L^p$ space and establishes the conclusions. \end{proof} \begin{remark}\label{nonresonance} Lead by the results in Sternberg theorem one could conjecture that, when the Mather spectrum of the map restricted to the manifold is non resonant with the Mather spectrum of the map in the normal direction, one could improve the persistence theorem of normally hyperbolic manifolds to produce more derivatives. Indeed, if one assumes that $k$ derivatives of $v$ exist, one obtains that they should satisfy: \eqref{invariantsection}, we obtain: \begin{equation}\label{invariantsectionder} D^k v = \mu (D^k v) \circ A\, A^{\otimes k} + D^k P \end{equation} which is equivalent to \begin{equation}\label{invariantsectionder2} \mu^{-1}(D^k v)\circ A^{-1}\, (A^{\otimes k})^{-1} - D^kP\circ A^{-1} = D^k v \end{equation} We note that if we considers \eqref{invariantsectionder} as an equation for the $D^k$ it is not difficult to arrange that it has solutions. (It is easy to arrange that for each partial derivative, either \eqref{invariantsectionder} or \eqref{invariantsectionder2}) define a contraction operator and, therefore, have a continuous solution. Nevertheless, these candidates for a derivative fail to be the true derivatives as it was established in the Example~\ref{invsectionexample}. This is in sharp contrast to what happens in the Sternberg linearization theorem where the solutions of the equations satisfied by the derivatives -- assuming that they exist -- are indeed derivatives. In our case, the solutions of \eqref{invariantsectionder} and of \eqref{invariantsectionder2} exist but are not derivatives. This phenomenon can be understood by noting that the solution of one equation is obtained iterating the system in one direction while the other is obtained by iterating in the opposite. \end{remark} \begin{remark} It is instructive to note that the invariant section that we have produced is rather {\em lacunary}. The coefficients tend to group in different scales. This makes it clear that the spaces to which it belongs are going to be determined by rather simple scaling properties. We believe that this is quite typical from the objects invariant under hyperbolic systems or otherwise related to them. Indeed, it seems that it is this lacunarity what makes it possible certain properties such as decays of correlations, central limit theorem. If this intuition could be made more substantial, it suggests that many of the methods of harmonic analysis developed for lacunary functions could be useful for hyperbolic systems. Note that for these functions, the spaces in a certain scale that they belong to can be ascertained (except for the borderline cases) by simple scaling arguments. \end{remark} \section{The Anosov splitting} One of the most common proofs of the regularity of the invariant Anosov splitting is by an appeal to the invariant section theorem \cite{HirschP68}. Of course, one could have hopes that in this most restrictive situation, the examples for the invariant section theorem did not apply. Indeed, there are other proofs of the H\"older regularity of the Anosov splitting that follow another route \cite{Fenichel74}, \cite{Fenichel77}, which produce better regularity results than the invariant section theorem. Nevertheless, it turns out that one can produce essentially the same phenomena as in Example \ref{invsectionexample}. We will denote by $E^{s,(f)}$, $E^{u,(f)}$ the stable and unstable bundles of the Anosov map $f$. When there is no danger of confusion with the system we are considering, we will suppress the $(f)$ from the index, as it is customary. \begin{example}\label{anosovexample} Consider $\torus^{k+l} = \torus^k \times \torus^{l}$ with $l,k \ge 2$ and write a point in $\torus^{k+l}$ as $(x,y)$ with $x \in \torus^k$, $y \in \torus^{l}$. Take $f$ of the form \begin{equation}\label{anosovformula} f(x,y) = \bigl( Ax,B(y)+e_u\varphi (x)\bigr) \end{equation} where: \begin{itemize} \item[1)] $A$ and $B$ are Anosov linear automorphisms of $\torus^k$ and $\torus^{l}$ respectively. \item[2)] $e_u$ is an unstable eigenvector of $B$. \item[3)] $\varphi : \torus^k \to \real$ is an appropriate function. (We will show that choosing $\varphi$ dense set of analytic functions works.) \end{itemize} We can also arrange -- to make the example more interesting in view of Remark~\ref{spectralradius} -- that \begin{itemize} \item[4)] The matrices $A,A_s,B$ are such that their norm agrees with their spectral radius. (e.g. $A$,$B$ are symmetric) \end{itemize} Denote by $\gamma$ the eigenvalue of $B$ corresponding to $e_u$, and by $A_s, A_u, B_s, B_u$ the restriction of $A$, $B$ to their stable and unstable subspaces. Then, for every $\delta > 0$, the stable splitting of $f$ is in $W^{r_0 -\delta,\infty}$, where $r_0 = (- \log || A_s|| + \log(\gamma))/\log||A^{-1}|| $ Nevertheless, it is not in $W^{r_0, p}$ for any $1 \le p \le \infty$. \end{example} \begin{remark} Note that, as remarked in \cite{delaLlave92}, the eigenvalues of the derivatives at periodic points are just powers of those of $A$ and $B$. If those are chosen to be nonresonant, the return map at every fixed point has a germ which is linearizable. (By Sternberg theorem.) In particular, the obstructions for regularity of foliations found in \cite{Anosov69} p. 193 ff. vanish. Indeed, any local obstruction to regularity of invariant foliations vanishes. (Note that the only local obstructions to regularity need to come from periodic orbits, since aperiodic orbits can be reduced to constants in a neighborhood.) We call attention that, in lower dimensional systems, the vanishing of the Anosov obstructions indeed implies that the foliations are regular (See \cite{HurderK90}.) \end{remark} \begin{remark} This example originated in \cite{delaLlave92} where it was used as a counterexample to some problems in rigidity theory. In \cite{JiangLP95} it was also used as a counterexample to integrability with smooth leaves of distributions. In \cite{NiticaT98}, following a suggestion of A. Katok, it is shown that the derivatives of this example also provide counterexamples to improvements of their results in regularity of transfer operators. We will consider similar remarks in Section \ref{sec:transfer}. \end{remark} \begin{remark} One could hope that by restricting the class of Anosov systems one considers one could find a class of Anosov systems for which the Sobolev regularity indeed leads to better results. We also note that this example starts to work only in dimension 4 or higher. We do not know of similar examples in dimensions 2 and 3. We have not investigated these possibilities in detail, for symplectic and other geometric structures. Note however that Example \ref{anosovexample} is volume preserving. We conjecture, however, that the conclusions of optimality of Sobolev regularity will remain to valid if we consider more restrictive classes of Anosov systems such as symplectic or even geodesic flows or if we consider two or three dimensional maps. One particularly interesting place to study these questions is in the examples constructed in \cite{HasselblattW97}, \cite{HasselblattW99}. We conjecture that these examples also have the same lacunary effect and that the regularity cannot be improved in the Sobolev classes. \end{remark} \begin{remark} We note that, even if in Example \ref{anosovexample} we showed that the results of applying Theorem \ref{invsection} are sharp, there are many interesting situations where the results of \cite{Fenichel74},\cite{Fenichel77} provide better regularity results than the invariant section theorem. The main reason is because the regularity produced in Theorem \ref{invsection} is the ratio of the supremum of logarithm of contraction and the infimum of the expansion in some bundles. The method of \cite{Fenichel74}, \cite{Fenichel77} considers these ratios over individual orbits and then takes the supremum. Of course, taking the supremum after taking the ratios is going to lead to sharper results. Moreover, it happens sometimes in geometrical problems that the presence of a geometric structure forces cancellations along the orbits. These cancellations do not show up when we take suprema over the expansions on the bundle. We should also note that one could also improve further the results in \cite{Fenichel74},\cite{Fenichel77} by noting that one could take the asymptotic ratios \cite{HurderK90}. Of course, for examples as simple as the ones we have been using, these refinements lead to no improvement. \end{remark} \begin{proof} Denote by $f_0$ the linear Anosov map: $$ f_0(x,y) = (A x, B y) $$ Following standard techniques in the perturbation theory for Anosov splittings, we will find the invariant splitting for $f$ as the graph of a linear bundle map from $E^{s, (f_0)}$ to $E^{u, (f_0)}$. Note that, since $f_0$ is a product, we have $E^{s,(f_0)}_{x,y} = E^{s,(A)}_x \oplus E^{s,(B)}_y$ (with the obvious identification of considering $E^{s,(A)}_x$ as a subspace of $T_{x,y} \torus^{k+l}$.) Note moreover, that, since the unperturbed bundles are analytic, the regularity of the splitting is exactly the same as the regularity of the function whose graph gives the splitting. We will search for the invariant graph only among linear bundle maps $$ W_{(x,y)}: E^{s,(f_0)}_{x,y} \mapsto E^{u,(f_0)}_{x,y} $$ of the form \begin{equation}\label{ansatz} W_{(x,y)}(a,b) = (0, e_u <\Psi(x),a> ) \end{equation} where $(a,b)$ correspond to the components of $E^{s,(f_0)}_{(x,y)}$ along the direction of the first factor and the second factor respectively, $\Psi: \torus^k \mapsto \real^k$ and $<\cdot,\cdot>$ denotes the scalar product. Since the fixed point of this graph transform is unique, finding a fixed point of the form \eqref{ansatz} gives us the Anosov splitting. Note that a point in the graph of a function of the form \eqref{ansatz} is transformed to: \begin{equation}\label{transform} DF(x,y)(a,b,0, e_u <\Psi(x), b> ) =(A_s a, B_s b , 0, \gamma <\Psi(x), a> + e_u \nabla \phi(x) a ) \end{equation} In order that the point in the right hand side of \eqref{transform} is in the graph of the the function given by \eqref{ansatz}, it is necessary and sufficient that: \begin{equation}\label{transform1} <\Psi( A x), A_s a> = \gamma <\Psi(x), a> + \nabla \varphi(x) a \end{equation} Equivalently: \begin{equation}\label{transform2} \Psi( A x) A_s^t = \gamma \Psi(x) + \nabla \varphi(x) \end{equation}\label{sumsolution} or: \begin{equation}\label{transform3} \gamma^{-1} \Psi( A x) A_s^t - \nabla \varphi(x) = \Psi(x) \end{equation} We can see by a reasoning very similar to the one we used in Example \ref{invsectionexample}, that \begin{equation}\label{anosovsolution} \Psi(x) = \sum_{i = 0}^\infty \gamma^{-i} (\nabla \varphi)(A^i x) (A_s^t)^i \end{equation} is the only continuous solution of \eqref{transform2}. Namely, we observe that the sum in \eqref{anosovsolution} converges uniformly on $C^0$ by Weierstrass M-test and that substituting the sum in \eqref{transform3}, and rearranging terms, which is allowed by the uniform convergence, we obtain that indeed the $\Psi$ given in \eqref{anosovsolution} is a solution -- the only continuous one -- of \eqref{transform3}. Note that the methods used in the study of the invariant section theorem give us that the $\Psi$ given by \eqref{anosovsolution} is in $\Lambda_{r}$ when $||A||^r ||A_s^t|| \gamma^{-1} < 1$. That is, when $r \le r_0$. Now, to prove the negative results, we note that the Fourier coefficients of $\Psi$ can be computed. \begin{equation}\label{fourieranosov} \hat \Psi_k = \sum_{i = 0 }^\infty \gamma^{-i} \widehat{(\nabla \varphi)}_{(A^t)^{-i} k} (A_s^t)^i \end{equation} If we choose to consider a $k$ such that it has a component over the fastest growing eigendirection of $(A^t)^{-i} k$, we have that $$ | (A^t)^{-i} k| \ge ||A^{-1}||^{i} C $$ If we pick a $\varphi$ in such a way that $ \widehat {(\nabla \varphi)}_k $ has a component on the left eigendirection of $A_s^t$, we will have: $$ |\widehat{(\nabla \varphi)}_{(A^t)^{-i} k} (A_s^t)^i| \ge C ||A^{-1}||^i ||A_s^t||^i $$ If we pick $\varphi$ in such that $ \widehat{(\nabla \varphi)}_{(A^t)^{-i} k} = 0$ when $i \ne 0$, and we consider $\Psi$ as in \eqref{anosovsolution}, we see that $(-\Delta + 1)^{r_0/2} \Psi$ has coefficients that do not go to zero. By Riemann-Lebesgue theorem, this contradicts that it is in any $L^p$. Given the formula \eqref{fourieranosov}, we can see that the fact that \begin{equation} \label{lowerbounds} |\hat \Psi_k| \ge C |k|^{-r_0} \end{equation} holds in an open and dense set of polynomials. Indeed, given \eqref{fourieranosov} then, we see that we have \eqref{lowerbounds} unless some relations -- a linear manifold of positive codimension -- holds among the coefficients. Since, we have several choices for $k$ -- infinitely many in case that the leading eigenvalue is irrational -- we see that we have \eqref{lowerbounds} except in a set of large codimension. \end{proof} \section{Absolute continuity of the Anosov foliations} One of the most important properties of Anosov foliations from the point of view of ergodic theory is that they are absolutely continuous. Moreover, from the point of view of regularity theory, we need to prove some extra properties of the Jacobian and discuss how are they related to the regularity of the leaves. In this section, we will study the properties that we will use later. Since these properties are of a geometric nature, and related to the properties of the map -- which we always measure in $C^r$ spaces --, the regularity properties we discuss are better expressed in $C^r$ spaces. Of course, if we started to consider the results for Anosov systems whose regularity is measured in Sobolev spaces, we would need to express these results in Sobolev classes. We recall that the Anosov foliation theorem (see e.g \cite{Anosov69} ) states that we can find invariant foliations tangent to the stable and unstable spaces. If the map is $C^r$, the foliations consists of $C^r$ leaves, the $r$-jet of the leaves is continuous, the $r-1$ jet is H\"older, and the foliations are absolutely continuous. Moreover, the Jacobian of the foliation is differentiable along the leaves of the foliation. The following result is obtained combining results of \cite{Anosov69} and extensions of them found in \cite{delaLlaveMM86}. \begin{thm}\label{abscont} Let $f$ be a $C^r$ $r = 1,2, \ldots, \infty, \omega$ Anosov diffeomorphism on a compact manifold $M$ of dimension $n$. Denote by $s$ the dimension of the stable bundle. Then, given any point $p \in M$, we can find a sufficiently small open neighborhood $\U$, $p \in \U$ and a mapping (called a local parameterization of the stable foliation) from $$ \Gamma: U \times V \mapsto \U $$ (where $U\subset \real^s$, $V\subset \real^{n-s}$ are open sets) The map $\Gamma$ satisfies that: \begin{itemize} \item[1)] For every $x$ in $U$, $\{ \Gamma(x,y) | y \in V \}$ is the connected component of a leaf of the stable foliation restricted to $\U$. \item[2)] $\partial^\alpha_y \Gamma $ is H\"older in $U\times V$ for $|\alpha | < r$. It is continuous for $\alpha = r$. \end{itemize} More geometrically: \begin{itemize} \item[2')] The $r-1$ jets of the stable foliation are H\"older, the $r$-jet is continuous. \end{itemize} If, moreover, $r \ge 2$ the local parameterization $\Gamma$ can also be chosen to satisfy \begin{itemize} \item[3)] The mapping $\Gamma$ is absolutely continuous. That is $$ | \Gamma( E \times E')| = \int_E dx \int_{E'} dy\, J(x, y) $$ where $| \ \ |$ denotes Lebesgue measure. \item[4)] $\partial^\alpha_y J(x,y)$ is H\"older for $|\alpha| \le r-2$ and continuous for $|\alpha| = r-1$. \end{itemize} Analogous results hold for unstable manifolds and for flows. In the later case, we can also consider the center-stable and center-unstable foliations. \end{thm} \begin{remark} The results 1), 2), 2') are quite standard in stable manifold theory. They can be found in \cite{Anosov69}, \cite{HirschP68}, \cite{HirschPS77}. Result 3) is the fundamental theorem 3) in \cite{Anosov69}. Alternative proofs of related results in terms of holonomy maps appeared in \cite{PughS72},\cite{Mane87}. For us, the formulation in terms of parameterizations is more useful since we need to use also 4), which is hard to even formulate in the holonomy maps language. Result 4) is stated for the $C^\infty$ case in \cite{delaLlaveMM86}. The proof presented there for the $C^\infty$ case is to establish 4) for every $r$. (Hence the proof presented there establishes the result we are stating here.) The idea of the proof is to examine in more detail the proof of 1) in \cite{Anosov69}. Observe that the forms considered in \cite{Anosov69} converge not only in the $C^0$ norm, but also with the derivatives along the foliation. A very transparent reason why 4) is true is that if we had just one parameterization (as we can do in perturbations of linear flows in the torus), we would have \begin{equation}\label{cohomologyjacobian} \log J\circ \Gamma^{-1} - \log J\circ \Gamma^{-1} \circ f = \log \det Df|_{E^s} \end{equation} Note that the right hand side of \eqref{cohomologyjacobian} admits $r-1$ derivatives along the stable directions. The fact that all continuous solutions of an equation of the form \eqref{cohomologyjacobian} when $f$ is an Anosov system and the right hand side admits derivatives along the stable directions is one of the parts of the smooth Livsic theorem in \cite{delaLlaveMM86}. See also \cite{NiticaT98}, where an argument similar to the one in this paragraph is used to study the regularity of the Jacobian starting with the method of \cite{PughS72}. \end{remark} \section{Regularity classes associated with Anosov foliations} Since the regularity of the foliations is quite asymmetric (quite differentiable along leaves, but only H\"older when we move across leaves) it is clear that it will be useful to define regularity classes that incorporate these regularities. Along with these regularity classes we can define differential operators adapted to the geometry of the Anosov foliations. The fact that the leaves of the stable a and unstable foliation are differentiable allows us to define differential operators which are tangent to the foliation and which have smooth coefficients along the manifold and continuous over the whole manifold. The derivatives are to be understood in the usual sense of calculus of differential geometry. Following standard practice, we will refer to this sense of derivatives as the ``strong' sense, even if, as we will see, the use of this word is somewhat misleading. Since we are dealing with operators with continuous coefficients, it could well happen that strong derivatives are not weak derivatives. We will show, that, for our case, indeed, these strong derivatives are weak derivatives, except for those of order exactly $r$. These operators can be composed (provided that the total order is smaller than $r$) even if the coefficients are only continuous. This gives the class of operators tangent to the foliation some algebraic structure. Later we will see that one can also define adjoints when the order is up to $r-1$. \begin{de}\label{tangent} Given a $C^r$ Anosov diffeomorphism $f$, we say that a function $\Phi: M \mapsto \real $ is in $C^{l,(f)}_s$, $(r \le l)$ when the restrictions of $\Phi$ to all the leaves of the stable foliation are uniformly $C^l$ and moreover the $l$-jets are continuous on the manifold. We denote by $\D^{k,l,(f)}_s$ the space of differential operators of order $k$ tangent to the stable foliation with coefficients in $C^{l,(f)}_s$. This makes classical sense when $k,l \le r$. As we will see later (see Remark \ref{adjoint}), we can make sense of these operators in the sense of distributions when $k \le l \le r-1$. When the Anosov system we are referring to is clear from the context --- as it will be most of the time in this paper --- we will suppress the $(f)$ from the notation. Analogous definitions hold for the unstable foliation or for flows rather than diffeomorphisms. In the later case, we also consider the center-stable direction, the center-unstable and the derivatives along the flow. \end{de} \begin{remark}\label{adjoint} One of the consequences of the regularity of the Jacobian is that it is possible to define the adjoint of an operator in $\D^{k,l}$. Indeed, using a partition of unity, we can reduce to the case when the test functions have support in one of the patches covered by the parameterization \begin{equation} \begin{split} \int d \lambda & \Psi (L\varphi) = \int \int \,dx\,dy J(x,y) \, \, \Psi\circ\Gamma(x,y) \sum_{|\alpha|\le k} a_\alpha (x,y) \partial_y^\alpha \varphi\circ \Gamma(x,y) \\ & = \int \int \, dx\,dy \, J(x,y) \, \, \varphi\circ\Gamma(x,y) \\ & \quad \quad \sum_{|\alpha|\le k} (-1)^{|\alpha|}\partial_y^\alpha \left[ a_\alpha (x,y) J^{-1}(x,y) \Psi \circ\Gamma(x,y) J(x,y)\right] \, dx\,dy\ . \end{split} \end{equation} Expanding the derivative we obtain a differential operator in the $y$ variables acting on $\Psi$. Making back the change of variables, we can see that the adjoint is a differential operator. The coefficients of the highest order terms are as differentiable as those of the original operator. The lower order terms are less differentiable since they involve derivatives with respect to the $y$ of the $a_\alpha (x,y)$. This allows us to define the operators directly in the weak sense. This is more satisfactory than to define them as combinations of the partial derivatives with respect to the coordinates (understood, of course in the weak sense) multiplied by continuous coefficients. Note however that since the Jacobian is only $r-1$ times continuously differentiable, this weak definition only makes sense for operators of order $r-1$. We call attention to the fact that, in our case, even if it is true that it is possible to define derivatives in the strong sense of order up to $r$, we can define weak derivatives only up to order $r-1$. (Hence, for this case, it is easier to have strong derivatives than weak derivatives!) Fortunately, as the above argument shows, in the common case, both definitions agree and, more importantly for us, they satisfy the same formulas under changes of variables. \end{remark} \begin{de}\label{partialderivatives} Given a real valued function, we denote by $D_s\varphi$ the derivative restricted to the tangent bundle. That is: $D_s\varphi_{(x)}E_x^s\to\real$ is defined by $(Z\varphi)(x) = D_s\varphi (x)Z(x)$ for every vector field tangent to the stable foliation. Similar definitions hold for functions taking values in manifolds. In particular, it is possible to define $D_s^i$ for $i\le r$. Using Remark~\ref{adjoint}, it is possible to define the derivatives in the weak sense and $D_s^i$ when $i \le r-1$. Since we will not consider the case $i = r$, we will not introduce a different notation for the two different concepts. We will try to explain which one we are using. (Except in some very specific places, it will be the weak derivative.) Of course, analogous definitions hold for unstable foliations or indeed for any foliation with $C^r$ leaves and continuous jets. \end{de} Of fundamental importance from what follows is the next result. \begin{thm}\label{regularity} Let $\F^1, \cdots ,\F^N$ be a finite number of foliations in a compact manifold $M$. Assume: \begin{itemize} \item[i)] The leaves of the manifold are $C^r$, $r$ even. The $r$-jet of the leaves are continuous across $M$. \item[ii)] The foliations are transverse, that is, denoting by $\F_x^i$ the leaf of $\F^i$ passing through $x$, we have \begin{equation*} T_xM = \bigoplus_{i=1}^N T_x \F_x^i \end{equation*} \item[iii)] The foliations are absolutely continuous and it is possible to find local parameterizations as in 3) of Theorem~\ref{abscont} so that $$\partial_y^\alpha J(x,y) \text{ is continuous for } |\alpha|\le k$$ \end{itemize} Let $\varphi$ be any Schwartz distribution such that \begin{itemize} \item[iv)] $D_{\F^i}^j \varphi \in L^p$\quad $i=1,\ldots,N$, $j=0,\ldots,k$, $1

1$, $k$ as in the hypothesis is significantly more subtle. First we show that using the properties of the Jacobian we can construct elliptic operators $\square_i$ of order $k$, which are tangent to the foliation $\F_i$ (here is where we use the fact that $k$ is even). Then $\square = \sum_i \square_i$ will be elliptic (by the transversality of the foliations). Then, we need to apply the following well known result from elliptic regularity theory. \begin{thm} \label{ellipticreg} Let $\square$ be an elliptic operator of degree $k$ with continuous coefficients. Assume that $u\in \D'(M)$ --- the space of distributions --- and that $$\square u = \eta \in L^p\ .$$ Then, $$u\in W^{k,p}\ .$$ In case that $\eta \in C^0$, then $$u\in \Lambda_k$$ \end{thm} \begin{proof} The proof of this theorem can be found in \cite{AgmonDN59}, Chapter V. but the proof for our case is significantly easier since we do not have to discuss the boundary. A pedagogical exposition of the argument can be found in \cite{Taylor97}, p.379 ff. In this book, however, one can only find the result for $H^k\equiv W^{2,p}$ spaces. The only difference in the proof of the result claimed here with the proof in \cite{Taylor97} is that the estimates (11.9), (11.25) have to be replaced by the $W^{k,p}$ estimates (see \cite{Stein70}, p.77) or the $\Lambda_k$ estimates, which follow from the theory of Riesz potentials (see \cite{Stein70}, p.143). \end{proof} Applying Theorem~\ref{ellipticreg} to the operators $\square$, we obtain the desired result in Theorem~\ref{regularity} in the $W^{k,p}$ scale. For the $\Lambda_r$ scales, this is weaker that the result claimed in Theorem~\ref{regularity} since the elliptic regularity approach needs assumption iii). For the $\Lambda_r$ regularity, there is another approach which does not use elliptic theory and which, therefore, does not require $iii)$. It is based on approximation and interpolation \cite{Journe88}. (An earlier version of this approach, which included extra hypothesis is \cite{Journe86}.) \end{proof} \begin{remark} In several respects, the approach of \cite{Journe88} is superior to the elliptic approach. It can work with any number of derivatives while the elliptic regularity approach --- as it is written here --- needs an even integer number. Also, it can deal with derivatives up to order $r$, while the elliptic regularity approach can deal only with derivatives up to order $r-1$. (We need to be able to integrate by parts to make sure that we can define weak derivatives.) More importantly, since the approach of \cite{Journe88} does not use the Jacobian, it does not need that the foliations are absolutely continuous. This allows to consider several situations in dynamics where the foliations are not absolutely continuous. In \cite{delaLlave92}, there are extensions of the approach to situations where the functions are not defined on manifolds, but rather in Cantor sets enjoying certain geometric properties. On the other hand, the elliptic regularity approach can deal with any number of foliations, while the approach of \cite{Journe88} is restricted to two foliations. Also, the approach in \cite{Journe88} cannot deal with Sobolev spaces. A third approach to regularity lemmas is the approach of \cite{HurderK90}. This approach is based on estimating very simply the Fourier coefficients just by expressing the integrals defining the Fourier coefficients in the coordinates giving the trivialization of the foliations. In order to be able to carry out this step, one also needs the smoothness of the Jacobian. Since the characterization of regularity in $C^r$ by the size of the Fourier coefficients is not very sharp, this method leads to a loss of the number of derivatives that is roughly the same as the dimension of the manifold. The situation for Sobolev spaces is significantly worse since there are no characterizations of $L^p$ spaces (except $p=2$) by sizes of Fourier coefficients. On the other hand, being simple, the approach in \cite{HurderK90}, can work in a variety of circumstances where elliptic theory is somewhat cumbersome. In \cite{delaLlave97} it was shown how to use it to establish analytic regularity. In that case, losing a finite number of derivatives is of no consequence, and elliptic theory is somewhat difficult for analytic regularity since some basic tools such as cut-offs, stop working. Nevertheless, there are textbook studies of elliptic equations and analytic regularity, which we think can be extended to cover the problems considered here (See \cite{Friedman69} p. 199 ff.) Some extensions of the above methods that I believe would be welcome are: \begin{itemize} \item We think that it would be quite interesting if the approach in \cite{HurderK90} could be supplemented by some more sophisticated harmonic analysis (e.g., instead of estimating just the sizes of Fourier coefficients estimating the sizes of Littlewood-Paley decompositions or, perhaps wavelets). Of course, the Littlewood-Paley decomposition is closely related to the elliptic estimates. \item Develop the analytic regularity results using the analytic elliptic regularity. \item Extend the method of \cite{Journe88} to a larger number of foliations. \item Extend the method of \cite{Journe88} to the Cantor sets that appear as basic sets in Axiom A systems. \item Extend the method of \cite{Journe88} to analytic regularity. \end{itemize} Of course, introducing another method would be even more welcome. \end{remark} \section{The smooth Livsic theorem} In this section we discuss the regularity of solutions of cohomology equations These are equations where the unknown is the function $\varphi: M \to \real$ and take the form: \begin{eqnarray} \varphi\circ f - \varphi &= \eta \label{cohomologydiff} \\ X \varphi &= \eta \label{cohomologyvec} \end{eqnarray} where $f$, (resp . $X$) is a given Anosov diffeomorphism (resp. vector field) in a compact manifold $M$, $\eta:M \mapsto \real $ is a given function. Similarly, the non-commutative analogues (again the unknown is $\varphi: M \to G$, $G$ a Lie group with whose operation we will denote by just juxtaposing the elements and we will denote the inverse Lie operation by a superindex of $-1$. We will denote by $\varphi^{-1}$ the function that to a point $x$ associates $(\varphi(x))^{-1}$. \begin{eqnarray} \varphi\circ f \varphi^{-1} &= \eta \label{nccohomologydiff} \\ X \varphi &= \eta \label{nccohomologyvec} \end{eqnarray} where in the case (\ref{nccohomologydiff}), $\eta, \varphi$ are functions on $M$ taking values on a Lie group $G$. In the case of (\ref{nccohomologydiff}), $\varphi$ takes values in a Lie group $G$ and and $\eta$ takes values in its Lie algebra $\G$. We want to establish that if $f$ -- or $X$ -- and $\eta$ are regular, then any continuous $\varphi$ solving \eqref{cohomologydiff}, \eqref{cohomologyvec}, \eqref{nccohomologydiff} or \eqref{nccohomologyvec}, has to be smooth. The main result of this section will be: \begin{thm}\label{smoothLivsic} Let $f$ be a $C^r$ Anosov diffeomorphism of the compact manifold $M$, $r \in 2,\cdots$. (resp. $X$ a $C^r$ Anosov vector field on $M$). Let $\eta$ be a function in $W^{l,p}$, $l \in \natural$ $(0 < l \le r)$. $1 \le p \le \infty$. Assume that $f$ preserves Lebesgue measure or, more generally that there is a $0 < \lambda < 1$ such that: \begin{eqnarray*} \|J_f \|_{L^\infty}^{1/p} || D_s f||_{L^\infty} \le \lambda \\ \|J_{f^{-1}} \|_{L^\infty}^{1/p} || (D_u f)^{-1}||_{L^\infty} \le \lambda \end{eqnarray*} where $J_f$ denotes the Jacobian of $f$ with respect to the Lebesgue measure. (In the case of flows, we need the same hypothesis when $f$ the time one map of the flow.) Assume that $\varphi$ is a continuous function that satisfies \eqref{cohomologydiff} (resp. \eqref{cohomologyvec}, \eqref{nccohomologydiff} \eqref{cohomologyvec}). Then, $\varphi \in W^{l',p}$, where $l' = l $ when $l$ is an even integer or $l=1$, $l' = l -1 $ when $l$ is odd. \end{thm} \begin{remark} The result is not optimal in the case of odd $l$. One can certainly take this result and improve it using interpolation. (see Remark~\ref{interpolation}.) Nevertheless, it seems quite possible that one can construct a direct proof of these improved results. This seems, however to require the development of certain techniques such as fractional derivatives and Bessel potentials associated to the Anosov foliations. Even if it looks to us that this is reasonable straightforward to do, it would be somewhat lengthy. \end{remark} \begin{remark} Notice that, in the same way as in the more familiar $C^r$ cases, these regularity results do not require that the system is transitive or, in the non-commutative case that $\eta$ is close to the identity. Such assumptions are needed for the well known Livsic theorem that states that conditions on periodic orbits are necessary and sufficient for the existence of a solution. \end{remark} The bootstrap of regularity will be done in different stages. Each stage will require a different argument. Furthermore, we will need different arguments for the commutative and non-commutative case. We develop these arguments and the corresponding theorems in the rest of this section. We will study first the (somewhat simpler) commutative case and start by proving some theorems that accomplish different stages of the bootstrap of regularity. \begin{lem}\label{bootstrap1} Let $f$ be a transitive $C^r$ Anosov diffeomorphism in a compact manifold $M$, $r \ge 2$. Denote by $\mu_+,\mu_-$ the forward and backwards SRB measure for $f$. Let $\varphi: M\to \real$, $\varphi\in L^1(\mu^+)\cap L^1(\mu^-)$ be such that \begin{equation}\label{boundary} \varphi -\varphi\circ f=\eta \end{equation} where $\eta(x)$ is a continuous function with a modulus of continuity $\omega$ (i.e., $|\eta (x)-\eta(y)| \le\omega (d(x,y))$, $\omega:\real^+\to\real^+$ non-decreasing $\omega(0)=0$) satisfying for some $\lambda<1$ (and hence for all $\lambda <1$) \begin{equation}\label{omega} \begin{split} \sum_{i=0}^\infty \omega (\lambda^i t) &= \Gamma (t)<\infty \\ \lim_{t\to0} \Gamma (t) & = 0 \end{split} \end{equation} Then $\varphi$ agrees Lebesgue a.e. with a continuous function of modulus of continuity $\Gamma$. If $\omega(t)=Ct^\alpha$ for some $0<\alpha\le1$, (i.e., $\eta$ is H\"older) of then, $\varphi$ agrees a.e. with a H\"older function of the same exponent as $\eta$. \end{lem} \begin{remark} Notice that a consequence of $\varphi$ agreeing a.e. with a continuous function is that then we can conclude that $\eta$ satisfies the usual condition on periodic orbits $0=\sum_{i=0}^{T-1} \eta \circ f^i(x)$ when $f^T(x)=x$. \end{remark} \begin{proof} If $\varphi$ satisfies \eqref{boundary}, then $$\varphi (x) = \eta (x) + \cdots + \eta \circ f^i(x) + \varphi\circ f^{i+1}(x)$$ Summing the above equations for $i=0$ to $N-1$ and dividing by $N$, we obtain $$\varphi(x) = \frac1N \sum_{i=0}^{N-1} (N-i)\eta\circ f^i (x) + \frac1N \sum_{i=1}^N \varphi \circ f^i(x)$$ Substituting the above expression evaluated for $x$ and $y$, taking limits as $N\to\infty$ and recalling that for Lebesgue almost all points $x$, $\lim_{N\to\infty}\frac1N \sum_{i=0}^{N-1} \varphi\circ f^i(x) = \int\varphi\,d\mu^+ $ we have: \begin{equation}\label{cancellation} \varphi (x) - \varphi(y) = \lim_{N\to\infty} \biggl[ \sum_{i=0}^{N-1}\eta\circ f^i(x) - \eta \circ f^i (y) - \sum_{i=0}^{N-1} \frac{i}N [\eta \circ f^i(x) - \eta \circ f^i(y)]\biggr] \end{equation} for Lebesgue almost all $x,y$. We recall that, by the absolute continuity of the stable foliation there is a dense set of stable leaves in which the formula \eqref{cancellation} for $x,y$ a.e. with respect to the Riemannian measure on that leaf. If $x,y$ are on the same stable leaf and \eqref{cancellation} is valid for them we note that \begin{equation} \label{cesaroestimate} \begin{split} \Big|\sum_{i=0}^{N-1} \frac{i}N (\eta\circ f^i(x)-\eta\circ f^i(y))\Big| &\le \sum_{i=0}^{N-1} \frac{i}N \omega (\lambda^i d_s(x,y))\\ &\le \sum_{i=0}^{\sqrt{N}} \frac{i}N \omega (\lambda^i d_s(x,y)) + \sum_{\sqrt{N}+1}^{N-1} \frac{i}N \omega (\lambda^i d_s(x,y))\\ &\le \frac1{\sqrt{N}} \sum_{i=0}^\infty \omega (\lambda^i d_s(x,y)) + \sum_{\sqrt{N}+1}^N \omega (\lambda^i d_s(x,y)) \end{split} \end{equation} where $d_s(x,y)$ denotes the distance of $x,y$ on the stable leaf on which they lie. Both terms of the R.H.S. of \eqref{cesaroestimate} tend to zero as $N\to\infty$ provided $\sum^\infty_i \omega (\lambda^i d_s(x,y)) <\infty$. Similarly, we can bound $$\Big| \sum_{i=0}^{N-1} \eta\circ f^i(x) - \eta\circ f^i(y)\Big| \le \sum^N_i \omega (\lambda^i d_s (x,y)) = \Gamma (d(x,y))$$ Hence, we have established that for a dense set of stable leaves \begin{equation}\label{continuitys} |\varphi (x)-\varphi (y)| \le\Gamma (d_s(x,y)) \end{equation} almost everywhere on the leaf. Hence, we conclude that the function $\varphi$ restricted to this leaf, agrees almost everywhere on the leaf with a continuous function of modulus of continuity $\Gamma$. Since we have that $\Gamma$ is independent of the leaf, then, the result can be improved from a dense set of leaves to all the leaves. An identical argument with $f^{-1}$ in place of $f$ and $\mu^-$ in place of $\mu^+$, we get that for a dense set of unstable leaves, for almost all points (and, hence, for all leaves and all points) we have: \begin{equation}\label{continuityu} |\varphi (x)-\varphi (y) | \le\Gamma (d_u(x,y)) \end{equation} when $x,y$ are in the same unstable leaf and $d_u$ is the distance along this leaf. Using the uniform transversality of the Anosov foliations, we have $$C^{-1} \max (d_s (x,y),d_u (x,y)) \le d(x,y) \le C\max (d_s (x,y),d_u(x,y))$$ when $d(x,y) \le\delta$ for some conveniently chosen $\delta>0$. {From} this and \eqref{continuitys}, \eqref{continuityu}, the desired result follows. The result about H\"older continuity follows because when $\omega (t) = ct^\alpha$ $$\sum \omega (\lambda^i (t) = \left( \frac{C}{1-\lambda^\alpha}\right) t^\alpha$$ \end{proof} Now we state the main results of this section. They allow us to bootstrap the regularity in Sobolev classes for solutions of the cohomology equations. We call attention to the fact that there is a difference between the commutative and the non commutative cases. \begin{thm}\label{bootstrap2} Let $f$ be a $C^r$ Anosov diffeomorphism (resp. $X$ a $C^r$ Anosov vector field) $r\ge2$. Assume that $\varphi,\eta$ satisfy \eqref{cohomologydiff} (resp. \eqref{cohomologyvec}) and that: \begin{itemize} \item[i)] $\varphi\in C^0$ and $\eta\in W^{k,p}$, $1\le k\le r$, $1

p\ge \frac{n}k$
\item[ii)] or ii) of Theorem~\ref{bootstrap2} hold.
\end{itemize}
Then $\varphi\in W^{k',p}$ where $k'=k$ if $k$ even or $1$,
$k'=k-1$ when $k$ is odd and not 1.
In case that $k=1$, we can allow $1\le p\le\infty$, $r\ge1$ in the
other hypotheses.
\end{thm}
\begin{remark}\label{interpolation}
We note that the fact that we can only deal with
even $k$ presumably does not belong. This is
due to the fact that we want to use positive
operators elliptic operators in Theorem\ref{regularity} and,
at the same time, we want to use
differential operators in our work so as to have
easy change of variable formulas.
As we conjectured before, it seems doable to
develop a theory of pseudo-differential operators
and their changes of variables. (Perhaps it is
even done in the literature but we are not aware
of it.)
Nevertheless, it is easy to use interpolation
to deduce from Theorem
\ref{bootstrap2} results that
have less restrictions in the $k$.
In the commutative case, when $f$ is transitive, the fundamental
Livsic theorem \cite{Livsic71} tells us that a necessary and
sufficient condition for a H\"older
function $\eta$ to be a coboundary
(i.e. to be such that \eqref{cohomologydiff} has a
solution) is that the sum over all periodic orbits is
zero.
Then, when $k,p$ are such that the Sobolev
imbedding theorem implies that the $W^{k,p}$
norm dominates a H\"older norm, the
set of coboundaries is
closed in $W^{k,p}$. We denote this space
by $B^{k,p}$
If $k$ is even, $1 < p < \infty$ we can use
Theorem \ref{bootstrap2} to
produce a bounded linear operator $T$ that
maps $B^{k,p}$ into $W^{k,p}$.
The operator $T$ is defined as the operator
that to a function $\eta \in B^{k,p}$
associates the function
$\varphi$ solving \eqref{cohomologydiff}
and $\int \varphi d \mu = 0$.
Note also that, since the solutions of
\eqref{cohomologydiff} and the normalization
condition are unique, we can consider that the
operator $T$ is the same in all the $B^{k,p}$
spaces.
If we can define this operator and apply theorem
\ref{bootstrap2} for two different values of
$k$ (say $k_1, k_2$ such that the Sobolev imbedding
gives that the spaces are H\"older), then,
applying Lemma~\ref{lem:interpolation},
the operator $T$ will also be bounded in
all the Sobolev spaces with intermediate $k$.
Similar considerations apply for flows.
\end{remark}
\subsection{Proof of Theorem \protect{\ref{bootstrap2} }}
Now, we start the proof of Theorem~\ref{bootstrap2}. This will
require to study different bootstrap arguments that we will
state as separate lemmas since they will appear again in the
proof of other theorems.
We first study the case of \eqref{cohomologydiff} in the non-transitive
case and $r=1$.
The first stage of bootstrap is to obtain the existence of
one derivative.
\begin{lem}\label{stable1}
Assume that $f$ is a $C^1$ mapping preserving an stable bundle --- and
hence, having an stable foliation.
Assume that $\varphi$ and $\eta$ are as in \eqref{cohomologydiff}
and that $\eta\in W^{1,p}$.
Then $\varphi$ is differentiable in the sense of distributions along the
stable direction and $D_s\varphi \in L^p$.
Moreover, the formula
\begin{equation}\label{derstableformula}
D_s\varphi = \sum_{i=0}^\infty D_s \eta \circ f^i D_s f^i
\end{equation}
holds.
The sum on the R.H.S. converges in the $L^p$ sense.
\end{lem}
\begin{proof}
Given a vector field $Z$ tangent to the stable foliation and with
coefficients in $C_s^1(f)$ we can define a flow $Z_t$ by integrating
the differential equations restricted to the stable leaf.
If $\eta\in W^{1,p}$, we know that $Z\eta$ can be defined in the sense
of distributions on each leaf and $Z\eta \in L^p(M)$ and (appealing to
the absolute continuity of the Anosov foliations)
for almost all leaves.
As in one of the standard proofs of Sobolev imbedding
(\cite{Stein70}, V~2.2) we have:
$$\eta (Z_t(x)) - \eta (x) = \int_0^t (Z\eta) \circ Z_s(x)\,ds
= \int_0^t \left[ (D_s \eta) Z\right] \circ Z_s(x)\,ds$$
where the equality should be understood as equality
between $L^p$ functions.
(Note that $Z_a$ is absolutely continuous with respect
to Lebesgue since it is Lipschitz on each leaf
and the foliation is absolutely continuous.)
Similarly:
\begin{equation} \label{iteratedidentity}
\begin{split}
\eta \circ f^i(Z_t(x)) - \eta\circ f^i(x)
& = \int_0^t Z(\eta\circ f^i) \circ Z_a (x)\,da\\
& = \int_0^t \left[ (D_s \eta)\circ f^i (D_sf^iZ)\right]\circ Z_a(x) \, da
\end{split}
\end{equation}
We note that if \eqref{cohomologydiff} holds, then
\begin{equation}\label{iterateidentity}
\varphi (x) = \eta (x) +\cdots + \eta \circ f^i(x) + \varphi \circ f^{i+1}(x)
\end{equation}
Subtracting \eqref{iterateidentity} for $x$ and $y$
and applying \eqref{iteratedidentity}, we have:
\begin{equation}\label{fundamentaltheorem}
\begin{split}
\varphi\circ Z_t (x) -\varphi (x)
&= \int_0^t \sum_{i=0}^N \left[ (D_s\eta) \circ f^i
(D_s f^i Z)\circ f^i\right]\circ Z_a (x)\,da \\
&\qquad + \varphi \circ Z_t \circ f^{N+1}(x) - \varphi\circ f^{N+1}(s)
\end{split}
\end{equation}
We note that, since $x$, $Z_t(x)$ are in the same stable manifold and
$\varphi$ is uniformly continuous,
$$\varphi \circ f^{N+1}\circ Z_t(x) - \varphi \circ f^{N+1}(x)$$
converges to zero as $N\to\infty$ uniformly in $x$.
We also note that
$$\|\Gamma\circ f\|_{L^p} \le \|J_f\|_{L^\infty}^{1/p} \|\Gamma\|_{L^p}$$
Hence, the general term in the series \eqref{fundamentaltheorem} can be
bounded by
\begin{equation}\label{generalterm}
\begin{split}
\|(D_s\eta D_s f^i Z)\circ f^i\|_{L^p}
&\le \|J_f\|_{L^\infty}^{i/p} \|D_s \eta \|_{L^p} \|D_sf^iZ\|_{L^\infty}\\
&\le (\|J_f\|_{L^\infty}^{1/p} \|D_sf\|_{L^\infty})^i \|D_s\eta\|_{L^p}
\end{split}
\end{equation}
We have, therefore, established that the series in
the R.H.S. of \eqref{derstableformula} converges in $L^p$.
To show that it is indeed a derivative,
we note that, taking limits $N \to \infty$ in \eqref{fundamentaltheorem}
we have:
$$\varphi (Z_t(x)) - \varphi(x) = \int_0^t \Gamma\circ Z_a (x)\,da$$
where $\Gamma$ is the series in the R.H.S. of \eqref{derstableformula}.
Again, using the absolute continuity
of the foliation, we obtain that
$\Gamma = Z\varphi$ in the sense of distributions.
Since $Z$ is arbitrary, we have established
\eqref{derstableformula}
This finishes the proof of Lemma~\ref{stable1}.
\end{proof}
Higher stable derivatives are much easier (at least conceptually).
We will just show that the series obtained taking derivatives along the
stable direction term by term in \eqref{derstableformula} converges in $L^p$.
This implies that indeed this is the distributional derivative and that the
distributional derivative is in $L^p$.
This is the content of the following lemma.
We call attention that the bootstrap of regularity
can be done independently for stable derivatives and unstable
derivatives. This will be relevant later.
\begin{lem} \label{highderivatives}
Let $f$ be a $C^r$ map with a stable direction.
Assume that $\varphi,\eta$ satisfy \eqref{cohomologydiff},
$D_s^j \eta\in L^p$ for $j \le \ell$.
(This is clearly implied by $\eta \in W^{\ell,p}$.)
If either
\begin{itemize}
\item $\varphi \in C^0$ and $f$ satisfies the first of \eqref{Jacobianbounds1}
\item $\varphi \in W^{k,p}$ and $f$ satisfies the first of
\eqref{Jacobianbounds2}
\end{itemize}
Then,
$$D_s^\ell\varphi \in L^p$$
\end{lem}
\begin{proof}
Note that, by Faa-di-Bruno formula
\begin{equation}\label{FdB}
D_s^{k-1} (D_s\eta \circ f^i D_sf^i)
= \sum_{j=1}^k\enspace
\sum_{i_1+\cdots+i_j=j}\sigma_{j;\, i_i,\ldots,i_j} \, \,
D_s^j \eta \circ f^i D_s^{i_1} f^i\cdots D_s^{i_j} f^i
\end{equation}
where $\sigma_{j;\, i_1,\ldots,i_j}$ are combinatorial coefficients
whose explicit expression
(which we will not need in these notes) can be found in many places, including
\cite{AbrahamR67}.
Note that, by the bounds in \cite{delaLlaveMM86}, p.547, we have
\begin{equation}\label{LMM}
\|D_s^i f^l\|_{L^\infty} \le \|D_sf\|_{L^\infty}^{l\, i} C_i
\end{equation}
Using that
\begin{equation}\label{Jacobian}
\|D^j \eta \circ f^l\|_{L^p}
\le \|D_s^j \eta\|_{L^p} \|J_f\|_{L^\infty}^{1/p}
\end{equation}
we obtain:
\begin{equation}\label{termbound}
\|D_s^\ell (\eta \circ f^j)\|_{L^p}
\le \|D^\ell \eta \| (\|J_f\|^{1/p} \|D_sf\|^\ell)^j
\end{equation}
The reason why we can use ii$'$) instead of ii) is the following:
Again if we start from \eqref{iteratedidentity} and take $\ell$ derivatives
we obtain
$$D_s^\ell \varphi
= \sum_{i=0}^N D_s^\ell (\eta\circ f^i) (x)
+ D_s^\ell \varphi \circ f^{N+1} (x)$$
Using the same argument as before
(i.e., use \eqref{FdB}, \eqref{LMM}, \eqref{Jacobian})
we can bound
$$\|D_s^\ell \varphi\circ f^{N+1} \|_{L^p}
\le \|J_f\|^{N\frac1p} \|D_s^\ell \varphi\|_{L^p}
\|D_S f\|_{L^\infty}^{Ni}$$
which under assumption ii) converges to zero in $L^p$.
We are then left with the formula
$$D_s^\ell \varphi = \sum_{i=0}^\infty D_s^\ell (\eta\circ f^i)$$
where the sum converges in $L^p$ sense by \eqref{termbound}.
We have, therefore, established Lemma~\ref{highderivatives}.
\end{proof}
Note that if $f$ is an Anosov system and it satisfies \eqref{Jacobianbounds1}
and \eqref{Jacobianbounds2} then we can apply
Lemma~\ref{highderivatives} to both $f$ and $f^{-1}$ (the stable direction
for $f^{-1}$ is the unstable for $f$).
Then, we can apply Lemma~\ref{regularity} and establish
Theorem~\ref{bootstrap2}
for the case of commutative cohomology equations over Anosov diffeomorphisms.
The result for commutative equations for flows can be established using
the results we have already established.
Note that \eqref{cohomologyvec} implies
\begin{equation}\label{integratedvector}
\varphi\circ X_t(x) - \varphi (x)
= \int_0^t \eta\circ X_a (x)\,da
\end{equation}
If we fix $t$ (to, for the sake of simplicity, 1), we see that the mapping
$X_1$ has stable directions, which are integrable and absolutely continuous.
If we apply Lemma~\ref{highderivatives} to $X_1$ and to $X_1^{-1} =X_{-1}$
we obtain the existence of $D_s^k\varphi$, $D_u^k\varphi$.
The existence of derivatives along the flow follows from the equation.
Then, again, applying Lemma~\ref{regularity} we establish
Theorem~\ref{bootstrap2} for the case of commutative cohomology equation
over flows.
\begin{remark}
Note that the bootstrap of regularity along the stable directions is valid
for $p=1,\infty$.
Nevertheless, when we need to apply Theorem~\ref{regularity} we need to
exclude these values for $p$.
\end{remark}
\subsection{Proof of Theorem \protect{\ref{bootstrap3}} }
The proof proceeds, roughly, along the same lines as the proof of
Theorem~\ref{bootstrap2}. We will bootstrap the derivatives along
the stable and unstable directions separately and then apply
Theorem~\ref{regularity} to conclude regularity.
Nevertheless, the actual bootstraps will require different
arguments since the non-commutativity will prevent some
of the rearrangements that we did in the commutative case.
Indeed, we do not have any analogue of Lemma~\ref{bootstrap1}.
We will assume, for the sake of simplifying the notation that the groups
we consider are groups of matrices.
This will allow us to use norms in the estimates of products and additive
notation.
The use of norms in products can be avoided using the local Lipschitz
nature of the product in Lie groups and the additive notation can be
avoided using the
exponential mapping, but this is only a typographical matter.
As in the commutative case, we start by studying the case $r=1$, establish
that under the assumptions there is a derivative and that this derivative
is given by an explicit formula.
\begin{lem}\label{ncstable1}
Let $f\in C^r$, $r\ge 2$, and preserves an stable direction.
Assume $\varphi,\eta$ satisfy \eqref{nccohomologydiff}
$$\varphi \in C^0\ ,\quad D_s \eta\in L^{p}\qquad \ 1\le p\le
\infty$$
and that ii$'$) of Theorem~\ref{bootstrap1} holds.
Then
\begin{itemize}
\item[a)] $D_s\varphi \in L^p$.
\end{itemize}
and we have
\begin{itemize}
\item[b)] $D_s\varphi (x) = \sum_{i=0}^\infty \varphi\ \varphi^{-1}\circ
f^i (x) D_s\eta \circ f^i (x)\varphi\circ f^{i+1}(x)$
\end{itemize}
Of course, similar results hold for the unstable derivatives.
\end{lem}
\begin{remark}\label{compactrange}
Note that since $\varphi$ takes values in a compact set and $g\to g^{-1}$
is uniformly differentiable in a compact set, the same conclusions of
regularity are reached for $\varphi^{-1}$ in place of $\varphi$.
This remark also applies to higher derivatives.
\end{remark}
\begin{remark}
The only reason we need to assume $r=2$ in this Lemma
and not just $r=1$ is that
we need to ensure that the stable manifold is absolutely continuous
and that we can define the first derivatives in the weak sense.
\end{remark}
\begin{proof}[Proof of Lemma~\ref{ncstable1}]
We start by observing that, if \eqref{nccohomologydiff} holds then,
\begin{equation}\label{iteratedidentity2}
\varphi (x) = \eta (x) \cdots \eta \circ f^N (x)\varphi \circ f^{N+1}(x)
\end{equation}
Hence, we can write
\begin{equation}\label{ladder}
\begin{split}
\varphi (x) - \varphi (y)
& = [\eta (x)-\eta (y)] \eta\circ f(x)\cdots \eta\circ f^N(x)
\varphi\circ f^{N+1}(x) +\\
&\qquad + \eta (y) [\eta\circ f(x) -\eta\circ f(y)] \eta\circ f^2 (x)
\cdots \eta \circ f^N(x)\varphi \circ f^{N+1}(x) +\\
&\qquad +\cdots +\\
&\qquad + \eta (y) \cdots \eta \circ f^{N-1}(y) [\eta\circ f^N(x)
- \eta \circ f^N(y)] \varphi \circ f^{N+1}(x)\\
&\qquad + \eta (y) \cdots\eta\circ f^N(y)[\varphi \circ f^{N+1} (x)
- \varphi \circ f^{N+1}(y)]\\
&= [\eta (x) -\eta (y)] \varphi \circ f(x)\\
&\qquad + \varphi (y) \varphi^{-1}\circ f(y) [\eta\circ f(x)-\eta\circ f(y)]
\varphi \circ f^2 (x)\\
&\qquad + \varphi (y)\varphi^{-1}\circ f^2(y)
[\eta \circ f^2 (x)-\eta \circ f^2(y)] \varphi\circ f^3 (x)\\
&\qquad +\cdots + \\
&\qquad + \varphi (y)\varphi^{-1}\circ f^{N-1}(y)
[\eta\circ f^N(x) - \eta\circ f^N(y)]\varphi \circ f^{N+1}(x)\\
&\qquad + \varphi(y) \varphi^{-1}\circ f^N(y)
[\varphi\circ f^{N+1}(x) - \varphi \circ f^{N+1} y]
\end{split}
\end{equation}
This remarkable cancellation was used in \cite{NiticaT96}.
It will be important to observe that the terms in this sum except for the last
one are independent of $N$.
That is, if we take the limit as $N\to\infty$ of \eqref{ladder} except for the
last term, we obtain a series.
If $Z$ is a vector field tangent to the stable foliation, we can write
\begin{equation}\label{ncinterpolation}
\begin{split}
&\varphi\circ Z_t(x) - \varphi (x) =\\
&\qquad + \biggl[ \int_0^1 [(D_s\eta)Z]\circ Z_a (x)\,da\biggr]
\varphi \circ f(x)\\
&\qquad + \varphi (y) \varphi^{-1}\circ f(y)
\bigg[ \int_0^1 [D_s (\eta\circ f)Z] Z_a (x)\,da\ \bigg]
\varphi \circ f^2(x)\\
&\qquad +\cdots +
\varphi (y)\varphi^{-1}\circ f^{N-1}(y)
\bigg[ \int_0^1 (D_s\eta\circ f)Z] \circ Z_a (x)\,dz \biggr]
\varphi \circ f^N (x)\\
&\qquad + \varphi (y) \varphi^{-1}\circ f^N(y)
[\varphi \circ f^{N+1}(x) - \varphi^{N+1}(y)]
\end{split}
\end{equation}
Note also that, because $\varphi(y) \varphi^{-1}\circ f^i(y)$ are matrices,
using the linearity, we can change the order of the integration with
the first factors in
the term, which do not depend on the integration variable.
Of course, in more general groups similar formulas
exist, but we have to introduce the derivative of
the group operation.
We note that, since $\varphi$,$\varphi^{-1}$ are continuous are bounded and
$x,y$ are in the same stable manifold we have that the last
term in \eqref{ncinterpolation} converges to zero uniformly.
We also can estimate the general term of \eqref{ncinterpolation}.
The term in braces already estimated in \eqref{generalterm} and the factors
involving $\varphi,\varphi^{-1}$ are uniformly bounded.
Hence, we have established that
$$\varphi \circ Z_t(x) - \varphi (x) = \int_0^t \Gamma\circ Z_a(x)Z$$
with $\Gamma$ the $L^p$ function defined by
\begin{equation}
\Gamma(x) = \sum_{i=0}^\infty \varphi (x)\varphi^{-1} \circ f^i (x),\
D_s\eta \circ f^i(x),\ \varphi \circ f^{i+1}(x)
\end{equation}\label{derivativeformula}
Hence, this is the stable derivative and this establishes
Lemma~\ref{ncstable1}.
\end{proof}
{From} Lemma~\ref{ncstable1} the results that we claimed for $k=1$ in
Theorem~\ref{bootstrap3} follow immediately.
When $f$ is a diffeomorphism apply Lemma~\ref{ncstable1} to
$f^{-1}$ to obtain the derivative along the unstable foliation and apply
Theorem~\ref{regularity}.
For flows, applying Lemma~\ref{ncstable1} to $X_1,X_{-1}$ we obtain
existence of the stable and unstable derivatives.
The derivative along the flow follows from the equation and, then, we
can apply Theorem~\ref{regularity}.
To finish the proof of Theorem~\ref{bootstrap3} we just argue by induction
in the number of derivatives
that the series in \eqref{derivativeformula} can be differentiated
repeatedly term
by term and that the result converges in an appropriate $L^q$ space.
The induction stops when the term by term derivative
would require derivatives of $\eta$ not included in the
assumptions.
Eventually, we will show that $D_s^k \varphi \in L^p$.
This shows that the derivative in the sense of distributions exists and
that it belongs to the appropriate space.
The argument that we have to use is significantly more complicated
than in the commutative case, since in this case, we use
just all the intermediate derivatives are in $L^p$.
Note that using the Sobolev imbedding
(Lemma \ref{lem:imbedding})
we expect that if the function is indeed in $W^{k,p}$ derivatives
of lower order are in a $L^q$ space with a $q > p$.
The reason why we have to pay attention to the exponent $L^q$ depending
on the order is that products of derivatives belong to an $L^q$ spaces with a
lower exponent than each of the factors.
We need to make the induction argument without loosing
too much in each step in order
to avoid ending up with higher loses than necessary.
As it turns out, the right hypothesis to carry
out the induction is that the
derivatives are in the $L^q$ space predicted by the
Sobolev imbedding theorem from the fact that $\varphi \in W^{k,p}$.
We will formulate our induction step after we explain some of
the more technical results, which will motivate the definitions.
The following elementary result is the key to the induction procedure
alluded to above.
We express the derivatives of the general term of b) in Lemma~\ref{ncstable1}.
\begin{prop}\label{Leibnitz}
The $k-1$ stable derivative of the general term of \eqref{derivativeformula}
\begin{equation}\label{derivativegeneral}
D_s^{i_0-1} [\varphi\varphi^{-1}\circ f^i D_s\eta\circ f^i \varphi \circ
f^{i+1}]
\end{equation}
can be expressed, applying Leibnitz rule, as a sum of terms satisfying
the following properties.
\begin{itemize}
\item[a)] The term is a product of four factors of the form
$$D_s^{i_1}\varphi\ ,\quad
D_s^{i_2}(\varphi^{-1}f^i)\ ,\quad
D_s^{i_3}(\eta\circ f^i)\ ,\quad
D_s^{i_4}(\varphi \circ f^{i+1})$$
\item[b)] All the derivatives of $\varphi$, $\varphi^{-1}$ appearing in the
terms above are of order smaller or equal than $i_0-1$.
\item[c)] The sum of the orders of the derivatives of each factor is
precisely $i_0$.
\end{itemize}
\end{prop}
The proof of this proposition is quite well known. It is just
the Leibnitz theorem for derivatives of the product. One just
needs to verify that it holds in the weak sense that we are using.
But this is quite well known from the theory of distributions.
The induction lemma is
\begin{lem}\label{ncinduction}
Assume that we are in the conditions of Theorem~\ref{bootstrap3}.
Assume inductively that for $i