\documentstyle[12pt]{article}
\newtheorem{defi}{Definition}[section]
\newtheorem{prop}{Proposition}[section]
\newtheorem{theo}{Theorem}[section]
\newtheorem{exam}{Example}[section]
\newtheorem{lemm}{Lemma}[section]
\def\sphere{{\bf S}}
\def\torus{{\bf T}}
\def\bfx{{\bf x}}
\def\natural{{\rm I\kern-.18em N}}
\def\integer{{\rm Z\kern-.32em Z}}
\def\real{{\rm I\kern-.2em R}}
\def\complex{\kern.1em{\raise.47ex\hbox{
$\scriptscriptstyle |$}}\kern-.40em{\rm C}}
\def\Diff{\mathop{\rm Diff}\nolimits}
\def\Sym{\mathop{\rm Sym}\nolimits}
\def\OO{{\cal O}}
\begin{document}
\title{Decomposition theorems for some groups of diffeomorphisms and
regularity of the composition operator.
\thanks{
This preprint is avaliable from
the Math Physics preprints archive.
Send e-mail to {\tt mp\_arc\@math.utexas.edu} for details.
}
}
\author{R. de la Llave \thanks{Supported by N.S.F. grants}
\thanks{e-mail address: {\tt llave\@math.utexas.edu}}\\
Dept. Math. \\
U. Texas at Austin \\
Austin TX 78712\\
\and
{R. Obaya}\\
Dept. Matem\'atica aplicada a la Ingenier\'{\i}a\\
Escuela Superior de Ingenieros Industriales\\
Univ. de Valladolid\\
47011 Valladolid \\
SPAIN
}
\date{}
\maketitle
\begin{abstract}
We study the algebraic structure of several diffeomorphism
groups in $\sphere^n$.
We show that all diffeomorphisms
can be written as a finite product of
diffeomorphisms
which have certain symmetries.
Since the natural group operation among diffeomorphisms
is the composition and H\"older regularities appear naturally in
intermediate steps, we present
an appendix in which optimal results about
regularity of the composition operator are obtained when the
spaces of diffeomorphisms are given the
topology of the classical H\"older norms.
\end{abstract}
\vfill
\eject
\section{Introduction}\label{intro}
The goal of this paper is to prove several decomposition theorems for
diffeomorphism groups.
These theorems, roughly, state that any
diffeomorphism in the group
can be written as a finite product
of diffeomorphisms lying in a
certain subgroup enjoying certain properties such as symmetry.
There are several motivations for the study of this theorems.
For example, in \cite{LS} it is shown that one theorem of this type can be used
to solve the inverse problem for scattering of geodesic
fields in surfaces of genus one. It can also be argued that
these theorems are analogues of the usual
factorization theorems in Lie algebras and
that stronger versions could be useful
in the
problem of computing representations of diffeomorphism groups [Bi].
If we consider these theorems as infinite dimensional versions of
factorization
theorems for Lie groups, one first difficulty
is that for diffeomorphism groups, the corresponding Lie algebra,
the Lie algebra of vector fields, is considerably
worse behaved than those of finite dimensional Lie groups.
For example,
the exponential is not surjective
in any neighborhood of the identity. Hence, it
is not possible to
apply the usual implicit function theorem
in Banach spaces to obtain results in a
small enough neighborhood of the origin using the
{\it ``infinitesimal results'' } using the Lie algebra.
Nevertheles, we will show that it is possible to
obtain similar results to those in the finite dimensional case
by using appropriate {\it ``hard''} implicit function theorems of
the Nash--Moser type.
This seems to be a natural generalization
of the systematic methods of finite dimensional
Lie group theory to groups of diffeomorphisms
and we hope that they have wider applicabilty. e.g.
when the groups of diffeomorphisms are required to preserve a certain geometric structure.
In a first chapter, we will consider decomposition theorems
of the Langer -- Singer type
for diffeomorphisms on $\sphere^n$.
Roughly, we show that $2^{n+1}$ subgroups of diffeomorphisms
consisting of diffeomorphisms which commute with reflection
generate the whole group of diffeomorphisms. That is, any diffeomorphism
can be written as the composition of a finite number of
factors, each of which commutes with reflactions across planes
of codimension one. Moreover, the factors are essentially one dimensional.
If we assume that the factor is differentiable sufficiently often,
we can conclude that the factors are also severl times differentiable.
the factors are
The method we use can obtain the previously known theorems
for diffeomorphisms in $\torus^n$ and obtain results for
diffeomorphisms of $\sphere^n$. From the point of view of
the theory of representations, it would be useful to have global
analogues of the local decomposition theorems [Bi].
We produce some expamples that
show that such global results would be difficult to obtain.
\section{Acknowledgements}\label{acknow}
The work of R.L. has been supported by NSF. grants.
Also R.L wants to ackwoledge the hospitality enjoyed at Valladolid
during several visits. Letters from J. Langer and F. Bien discussing
the implications of these theorems were very useful for us.
\section{Decomposition theorem for spheres.}\label{dos}
\subsection{Notation and statement of results}
We will consider the standard embedding of the sphere
in Euclidean space
$
\sphere^{n-1 } =
\left\{ \bfx \equiv (x_1,\cdots,x_n) \in \real^{n-1} \big|
\sum_{i=1}^{n-1} x_i^2 = 1\right\}
$
If $\bfx$ is a point in $\sphere^{n-1}$, we will denote by $\Pi_x$ the
hyperplane passing through the origin
and consisting of vectors orthogonal to $\bfx$, by ${\cal M}_{x,\theta}$
we will denote the one-dimensional maximum circle going from $\bfx$ to
$-\bfx$ and passing through the point $\theta$ and we will refer to those
sets as {\it meridians\/}. We will also denote by ${\cal P}_{x,\gamma}$
the subsets of $\sphere^{n-1}$ whose points have a constant projection
$\gamma$ along the line joining the origin and the point $\bfx$. The sets
of these form, will be referred to as parallels.
Finally, we denote by $\delta_x$, the reflection across the plane $\Pi_x$.
If $k$ is an integer,
and $\Omega$ is a compact set in $\real^d$
we will say that $f: \Omega \to \real $ is
$C^k$ if it has continuous derivatives
of order $k$. $C^k(\Omega)$, the space of $C^k$ functions in
$\Omega$ endowed with the norm ${\|\ \|_{C^k(\Omega)}}$ defined as
supremum of the derivatives of order up to $k$ is a Banach space.
If $k$ is not an integer, we define as usual
\begin{equation}\label{Holder}
\| f\|_{C^k(\Omega)} = \max( \sup_{x\ne y} \| D^k f(x) - D^k f(y) \| /
|x-y|^{k-[k]}, \| f\|_{C^{[k]}(\Omega)} )
\end{equation}
To cover Lipschitz conditions as well, we introduce the notation
$k^- = (k-1)+1^-$.In this notation,
$1^-$ will be bigger than any number in $[0,1)$
but smaller than $1$.
These notions of regularity can be lifted to geometric objects that
can be expressed in coordinates by saying that an object is $C^k$ if
we can find coordinate patches that cover the manifold on which the
expression is $C^k$.
Furthermore, if we fix a set of coordinate patches that cover the
manifold we can talk about the $C^k$ distance between two objects by
declaring it to be the maximum of the $C^k$ distance between the
coordinate expressions. A moment's reflection will show that
the statement that a geometric object is $C^k$ is independent of the
coordinate patches chosen (that is, if the coordinate expressions
in one patch are $C^k$ so are in all others).
Unfortunately, the $C^k$ distances do depend on the patch. Even if it is
easy to show that the distances obtained using two patches are equivalent,
they do differ and some statements such as that certain operators acting
on $C^k$ functions are contractions require specific choices of coordinate
patches.
For our purposes, this is not a shortcoming since many analytical operations
that we will need to perform
require taking coordinates anyway. We should however point out that,
when $k$ is an integer, there is a geometrically natural notion of $C^k$
distance in Riemannian geometry based on the notion of jets.
(See e.g. \cite{Hi} Ch. 2)
For non-integer $k$, since the definition of H\"older distance
\ref{Holder} requires comparing the values at different points,
there are no geometrically natural notions of $C^k$ distance.
Nevertheless, if we select coordinate charts
covering the manifold, we can define a distance.
In a compact manifold, even if the choice of different charts
leads to different metrics, the resulting metrics are equivalent.
We will denote by $\Diff^k (M,N)$, the set of diffeomorphisms
of $M$ into $N$ and, when $M=N$ we will simply write $\Diff^k (N)$.
When $k$ is finite $k \ge 1$, these sets can be given the structure of Banach
manifolds as follows.
By using charts, we can define a norm
on the space of $C^k$ vector fields on $N$
that makes it
a Banach space.
Given a diffeomorphism $f \in \Diff^k(M,N)$
and a $C^k$ vector field $v$, the map ${\rm Exp}(f,v):M \to N$
defined by ${\rm Exp}(f,v)(x) = \exp_{f(x)}v(f(x))$ --
where $\exp$ denotes the Riemannian geometry exponential associated to a
$C^\infty$ metric -- is a $C^k$ mapping.
The implicit function theorem shows that for sufficiently small
$v$, this mapping is also a diffeomorphism
and, moreover, any diffeomorphism in
a $C^k$ neighborhood of $f$ can be written in this way.
If $k<1$, we cannot apply the implicit function theorem
to conclude that ${\rm Exp}(f,v)$ is a diffeomorphism,
nevertheless, it is possible to show that ${\rm Exp}$
covers a neighborhood of $f$ in the space of
$C^k$ mappings.
(Similar argument show that $\Diff^\infty (M)$ is a Frechet manifold).
In $\Diff^k (M)$ it is natural to define the group operation of composition.
Unfortunately, when $k$ is not an integer, this operation will be
discontinuous
and when $k$ is an integer, even when composition is
continuous, it will not be differentiable. We refer to appendix~A for a
sharp results about the differentiability
of composition in H\"older spaces.
We will denote by $\Diff_0^k (\sphere^{n-1})$ the connected component
of the identity of $\Diff^k (\sphere^{n-1})$ and the subset of
$\Diff^k (\sphere^{n-1})$ commuting with $\delta_x$. $\Sym_{x,0}^k$ will
be the connected component containing the identity.
\par We also have to consider families of diffeomorphisms depending of a
parameter. We will denote the parameters by subindices. We will say that
$f_\lambda$ is a $C^r$ family of diffeomorphisms when for every fixed
$\lambda \in \Lambda$, where $\Lambda \subset \real^m$ is an open set or a
compact manifold, the map $x \rightarrow f_\lambda(x)$ is a $C^r$
diffeomorphism and, moreover $(\lambda,x) \rightarrow f_\lambda(x)$ is also
$C^r$.
\par The composition of diffeomorphisms can be extended in a natural way
to families. We denote by $g_\lambda \circ f_\lambda$ the family
$(\lambda,x) \rightarrow g(\lambda,f_\lambda(x))$.
{\bf Remark.}It is also possible to think of families as mappings from
$\Lambda \times M$ to $\Lambda \times N$ such that the first variable
remains unaltered. This point of view makes it clear that many results
about composition proved for $C^r$ mappings are true for $C^r$ families.
\par Notice also that the notation $f_\lambda$ is somewhat ambiguous since
it denotes at the same time the family and the mapping for a concrete value
of $\lambda$. The notation is, however, standard use and it does not lead to
confusion.\par
{\bf Remark.} Notice that the map $\Lambda \rightarrow C^k(M,N)$ defined by
$\lambda \rightarrow f_\lambda$ is only $C^0$. There are many $C^0$
mappings $\Lambda \rightarrow C^k(M,N)$ which are not $C^k$ families.
\par The main theorem in the paper will be
\begin{theo}\label{main}
For any $n\in \natural$ and any $k\ge 7$,
$k \in \natural$, we can find points
$x_1,\cdots,x_{2^n}$ in such a way that any
$f\in Diff_0^k({\bf S}^{n-1})$
can be written $f=f_1\circ f_2\circ \cdots \circ f_N$ with each $f_i \in
Sym^{k-2}_{x_j}$ for some $j$.
\end{theo}
\par For the cases of $n=1$, $k=\infty$ this theorem was proved in \cite{LS}.
Also for $n=1$ and finite differentiability a different proof was given
in \cite{Ll}. The latter proof also provided with an analogue for ${\bf T}^n$.
\par We remark that as a corollary of the proof we will obtain that
the diffeomorphisms appearing as factors can be
identified with families of diffeomorphisms of the interval.
\subsection{Proof of the theorem}
We start by observing that it suffices to prove the result in an
arbitrary small neighborhood of the identity. In effect, given any
open neighborhood $U$ of the identity,
it is possible to write any diffeomorphism in the
connected component of the identity as the composition of a finite number
of diffeomorphisms in $U$.\par\medskip
We will establish the local theorem by induction on the dimension.
The result for $n=1$ is stated explicitly in \cite{Ll}\par
\medskip
Our first goal will be to decompose diffeomorphisms of {\bf S}$^n$ into
simpler diffeomorphisms that preserve lower dimensional spheres, so that
we can apply induction.\par\medskip
It will turn out that the induction will proceed in such a way that we
prove the theorem for each parallel and we consider the results as a
diffeomorphism parametrized by the height.\par\medskip
We will start by proving a parametrized version of the well known
{\it fragmentation lemma}.
\begin{lemm}\label{frag}
Let $M$ be a compact manifold, $n\in \natural$ and
$\{K_i\}_{i=1}^n$ be a collection of
compact subsets of $M$ such that
\[\bigcap_{i=1}^{n} K_i=\emptyset\;.\]
Then, we can find a neighborhood ${\cal U}$ of the identity in the space of
$C^l$ and a map ${\cal F}_0$ that, to each family $f_\lambda$ in ${\cal U}$,
associates $C^l$ families $f_\lambda^1,\cdots,f_\lambda^n$, such that
\begin{itemize}
\item[i)] $f_\lambda= f_\lambda^1\circ \cdots\circ f_\lambda^n\,$.
\item[ii)] $f_\lambda^i|_{K_i}=Id\,$.
\item[iii)] If for some $x$, $f_\lambda(x)=x$ for all $\lambda$, then
$f_\lambda^i(x)=x$ for $i=1,\dots,n$ and all values of $\lambda$.
\end{itemize}
The mapping ${\cal F}_0$ is continuous when we give its domain and its range
the topology of $C^l$ families.
\end{lemm}
{\bf Remark.} Properties i)-iii) do not determine ${\cal F}_0$ uniquely. So that
the claim of continuity refers only to the one we construct.\par\medskip
\noindent
{\bf Proof.} We will assume that ${\cal U}$ is small enough so that
$f_\lambda(x)$ is whitin the injectivity radius of the exponential. We
will also need that some intermediate steps in our construction lie in this
domain. Since those are finite number of conditions, it will be easy to check
that they are satisfied for a non-trivial neighborhood.\par\medskip
We will discuss first the case $n=2$. We can find a $C^\infty$-function
$\varphi$ such that
\[\varphi|_{K_1}\equiv 1\,;\quad \varphi|_{K_2} \equiv 0\;.\]
\par\medskip
If $f_\lambda$ can be represented as the exponential of the vector field
$F_\lambda=E^{-1}(f_\lambda)$, we see that $f_\lambda^2=E(F_\lambda \varphi)$
will be the identity restricted to $K_2$. Moreover, since $f_\lambda(x)=x\,
\Longleftrightarrow E^{-1}(f_\lambda)(x)=0$, the set of fixed points of
$f_\lambda^2$ is bigger than those of $f_\lambda$. Provided that $f_\lambda$
is sufficiently close to the identity -- depending only on the $K$'s and
$\varphi$ -- then $f_\lambda^2|_{K_1}=f_\lambda$ and
$f_\lambda^2|_{K_2}=Id\,$. \par\medskip
Hence, setting $f_\lambda^1=f_\lambda\circ(f_\lambda^2)^{-1}$, the claims of the
lemma are satisfied. Notice that since the choice of $\varphi$ depends only on
the compacts $K_1$ and $K_2$ the continuity follows.
\par\medskip
\noindent
{\bf Remark.} Notice that the previous argument works even in the case that
one of the compacts is the empty set! \par\medskip
To prove the general case, we will proceed by induction in the number of
compacts set $n$. We will assume that it is true when $n\leq n_0$,
$n_0 \geq 2$, and will prove it for $n=n_0+1\,$.\par\medskip
If $K_1,\dots, K_{n_0+1}$ are compact sets with empty intersection, we can
find open sets $V_1,\dots, V_{n_0+1}$ such that $K_i\supset V_i$, and
$\cap_{i=1}^{n_0} \overline{V}_i=\emptyset\,$.\par\medskip
Set $L_{n_0+1}=K_{n_0+1}$, $L_0=\cap_{i=1}^{n_0} \overline{V}_i\,.$ Since
$L_{n_0+1}\cap L_0=\emptyset$ we apply the previous result with $n_0=2$ to
obtain families $f_\lambda^0$, $f_\lambda^{n_0+1}$ which are the
identity on $L_0$, $L_{n_0+1}$ respectively and with not less fixed points than
$f_\lambda$ such that $f_\lambda=f_\lambda^0\circ f_\lambda^{n_0+1}$.
\par\medskip
We can now consider the compact sets $L_i=K_i-\cap_{i=1}^{n_0} V_i\,$. They
have empty intersection because $\cap_{i=1}^{n_0} K_i \subset
\cap_{i=1}^{n_0} V_i\,$, so we can apply the induction hypothesis to
$f_\lambda^0$ to write $f_\lambda^0=f_\lambda^1\circ \cdots\circ
f_\lambda^{n_0}\,$. Each one of the $f_\lambda^i$ will be the identity on a
neighborhood of $K_{n_0+1}$. This gives us the representation
$f_\lambda=f_\lambda^1\circ \cdots \circ f_\lambda^{n_0}$ that verify all
the conditions of the statement. \par\medskip
The continuity of ${\cal F}_0$ is also a consequence of the induction process.
\begin{lemm}\label{vert}
Let $x$ be a point in {\rm \bf S}$^n$, $-x$ the antipodal point and
$U$, $U'$ two
open neighborhoods around $x$, $-x$ respectively. Then, for every
$r\in\,\natural\; \cup \{\infty\}$, we can find a neighborhood ${\cal U}$ of the
identity in the space of $C^r$ families of diffeomorphisms in such a way
that for any family $f_\lambda$ in ${\cal U}$, such that $f_\lambda |_U= Id$,
$f_\lambda|_{U'}= Id$ we can find $C^r$ families $f_\lambda^h$, $f_\lambda^v$
verifying:
\begin{itemize}
\item[i)] $f_\lambda=f_\lambda^h \circ f_\lambda^v\,$.
\item[ii)] $f_\lambda^v\,{\cal M}_{x,\theta}= {\cal M}_{x,\theta}\,$.
\item[iii)] $f_\lambda^h {\cal P}_{x,\gamma}={\cal P}_{x,\gamma}\,$.
\item[iv)] If $x\in$ { \rm \bf S}$^n$ and $f_\lambda(x)=x$ then
$f_\lambda^h(x)=f_\lambda^v(x)=x$. In particular
$f_\lambda^h|_{U\cup U'}=f_\lambda^v|_{U\cup U'}=Id$.
\end{itemize}
Moreover, if we restrict $f_\lambda^h$, $f_\lambda^v$ to
be in a neighborhood of the identity, they are unique, and the mapping
${\cal F}_1$ that assigns to each $f_\lambda$ the $f_\lambda^h$,
$f_\lambda^v$ is continuous.
\end{lemm}
\indent We will henceforth refer to diffeomorphisms and families that preserve
the parallels as {\it horizontal} and those that preserve the
meridians as {\it vertical}.\par\medskip
\noindent {\bf Proof.} This lemma says that any family can be factored
into a family that preserves the parallels and another one that preserves
the meridians. Obviously, we should set $f_\lambda^v(p)$ to be the point
in the meridian of $p$ and on the parallel of $f_\lambda(p)$. Then,
$f_\lambda^h$ would move along the
parallels so as to get to $ f_{\lambda} (p) $.
\par\medskip
More formally, using the notation for parallels and meridians
introduced in \ref{intro}, we have:
\[ \begin{array}{l}
f_{\lambda} ^v (p) = {\cal M} _{x,p} \cap {\cal P} _{x,f_\lambda (p)} \\[.2cm]
f^h _{\lambda} (p) = {\cal M} _{x,f_\lambda ((f^v _\lambda)^{-1} (p))} \cap
{\cal P} _{x,p}
\end{array} \]
Using the uniform transversality of the meridian and parallel foliations
outside a neighborhood of the poles, the usual implicit function theorem,
establishes that $f^v _\lambda (p)$ $f^h _ \lambda (p) $ depend jointly
$C^r$ on $p$ and on the parameters $ \lambda $.
\vspace{.3cm} \par\medskip
{\bf Remark.} If $f_\lambda(\varphi,\theta)=(\varphi
+v_\lambda(\varphi,\theta), \theta)$ denotes a vertical diffeomorphism
in a small neighborhood of the identity with $v_\lambda \in C^r({\bf T}^m)$,
$r \geq 1$, then $f_\lambda^{-1}$ is also a vertical diffeomorphism
$f_\lambda^{-1}(\varphi,\theta)=(\varphi
+{\bf v}_\lambda(\varphi,\theta), \theta)$ and
${\bf v}_\lambda \in C^r({\bf T}^m)$
The induction in the dimension
to prove the main theorem \ref{main} will proceed as follows.
For $ \varepsilon $
sufficiently small,
we will consider the sets $ K^{\varepsilon} _i = \{ x \in
\mbox{\bf S}^n \,/\, |x_i| \leq \varepsilon \} $.
\par\medskip
Using Lemma~\ref{frag} we will be able to write any
diffeomorphism $f $ as composition
of diffeomorphisms $f_i$, each of which is the identity restricted to $ K^
\varepsilon _i $.
\par\medskip
Furthermore, for $f_i$ we will pick a polar axis going through the points
$ x_i = \pm 1 $, $x_j =0$, $ j \neq i $ and using Lemma~\ref{vert} will
write it $ f_i = f^h_i \circ f_i^v $.
\par\medskip
If $f_i$ was sufficiently close to the identity then $ f^h_i$, $f^v_i $
would be the identity in $ K^\varepsilon _i $.
\par\medskip
We will show how to factor $ f^v_i $ and will apply the induction hypothesis to
$f^h_i$.\par\medskip
To apply the induction hypothesis, we will think of $f_i^h$ as a family
of maps sending {\bf S}$^{n-1}$ into {\bf S}$^{n-1}$, the parameter being the
height of the parallel.\par\medskip
Notice how we are lead to consider families of diffeomorphisms even
if we start with a single one. With hindsight, this is the reason why
we formulated Lemma~\ref{frag} and Lemma~\ref{vert} in terms of families.
\par\medskip
Clearly, if the induction result obtains in its conclusions that the factors
are a smooth family, this will imply that the factors are really diffeomorphisms
in {\bf S}$^n$.\par\medskip
We now proceed to establish the factorization of $f_i^v$ into symmetric
diffeomorphisms.\par\medskip
To simplify the notation and without any loss of generality, we will assume
that $i=1$ and that the axis is going through the points $x_\pm=(0,\ldots,\pm 1)$.
We also can think about the height as a periodic variable and consider the
parameters in a set $\Lambda=${\bf T}$^{m-2}$.
\begin{lemm}\label{div}
We can find two points $y_1$, $y_2$ in the sphere such that there is a
neighborhood ${\cal U}$ of the identity in the space of $C^r$ $r\geq 7$
families of diffeomorphisms such that for every family $f_\lambda^v\,\in
{\cal U}$ and satisfying
\begin{itemize}
\item[i)] $f_\lambda^v\,{\cal M}_{x_+,\theta}={\cal M}_{x_+,\theta} \qquad
\forall \theta$.
\item[ii)] $f_\lambda^v |_{K_1^\varepsilon}=Id$,
\end{itemize}
then, we can find two $C^{r-2}$ families $f_\lambda^{v,1}$, $f_\lambda^{v,2}$
satisfying
\begin{itemize}
\item[a)] $f_\lambda^{v,1}\,{\cal M}_{x_+,\theta}=
f_\lambda^{v,2}\,{\cal M}_{x_+,\theta}={\cal M}_{x_+,\theta} \qquad
\forall \theta$.
\item[b)] $f_\lambda^{v,1}\circ \Pi_{y_1}= \Pi_{y_1} \circ f_\lambda^{v,1}$.
\item[] $f_\lambda^{v,2}\circ \Pi_{y_2}= \Pi_{y_2} \circ f_\lambda^{v,2}$.
\item[c)] $f_\lambda^v=f_\lambda^{v,1}\circ f_\lambda^{v,2}$.
\end{itemize}
\end{lemm}
{\bf Proof.} We will take polar coordinates in {\bf S}$^n-K_1^\varepsilon$
defined by:
\[ \begin{array}{rcl}
x_{n+1} &=&\cos 2\pi\varphi \\
x_n &=& \sin 2\pi\varphi \; \cos 2\pi w_1 \\
x_{n-1} &=& \sin 2\pi\varphi \; \sin 2\pi w_1 \; \cos 2\pi w_2 \\
\vdots & & ~~~~~~\vdots \\
x_3 &=& \sin 2\pi\varphi \; \sin 2\pi w_1 \cdots \sin 2\pi w_{n-2}\\
x_2 &=& \sin 2\pi\varphi \; \sin 2\pi w_1 \cdots \sin 2\pi w_{n-2}
\; \cos 2\pi \theta \\
x_1 &=& \sin 2\pi\varphi \; \sin 2\pi w_1 \cdots \sin 2\pi w_{n-2}
\; \sin 2\pi \theta \end{array}\]
for $\varphi$, $w_1\,,\dots,$ $w_{n-2} \in (0,1/2)$, $\theta \in (0,1)$.\par
\medskip
Notice that if $|x_1| \geq \varepsilon$, then $|x_{n+1}| \leq 1-\varepsilon '$
so that $\varphi$ is defined univocally, and, since
$|x_n|^2+|x_{n+1}|^2 \leq 1-\varepsilon^2$, $w_1$ is likewise outside of the
critical values of $\sin$ and the same argument is applicable to
$w_2\,,\dots,$ $w_{n-2}$. In others words, this polar coordinate system
is a good coordinate system admitting a differentiable inverse defined
in a open set $V \subset$ {\bf T}$^{n+1}$.\par\medskip
We observe that when we express a vertical diffeomorphism in these coordinates,
the value of $\varphi$ changes but the others coordinates remain unaltered.
Similarly, a reflection across the plane $\Pi_{y_1}$ with $y_1=(1,0,\dots,0)$
is expressed as a change in sign for the $\theta$ coordinate while all the
others remain fixed. More generally, a reflection across $\Pi_{y_2}$
with $y_2=(\cos \rho,\sin \rho, 0,\dots,0)$ can be expressed by sending
$\theta$ into $2\rho -\theta$ and leaving all the other variables
unchanged. \par\medskip
If we take coordinates $(\varphi,\theta,w)$ the family $f_\lambda^v$
will be written as
\[ (\varphi,\theta,w) \rightarrow
(\varphi+v_\lambda(\varphi,\theta,w),\theta,w)\]
and we will look for $f_\lambda^{v,1}$ $f_\lambda^{v,2}$ that can be written
as
\[ (\varphi,\theta,w) \rightarrow
(\varphi+v_\lambda^i(\varphi,\theta,w),\theta,w)\]
with
\begin{equation}\label{2.2}
\begin{array}{l}
v_\lambda^1 (\varphi,\theta,w)=v_\lambda^1(\varphi,-\theta,w)\\[.2cm]
v_\lambda^2 (\varphi,\theta,w)=v_\lambda^2(\varphi,-\theta+2\rho,w)
\end{array}
\end{equation}
The equation that $v_\lambda^1$ $v_\lambda^2$ have to solve is, as
an straightforward calculation shows:
\begin{equation}\label{2.3}
v_\lambda(\varphi,\theta,w)
=v_\lambda^1(\varphi+v_\lambda^2(\varphi,\theta,w),\theta,w)+
v_\lambda^2(\varphi,\theta,w)
\end{equation}
Notice that there is an open set of values of $w$ for which
the point correspond $(\varphi,\theta,w)$ is in $K_1^{\varepsilon}$
hence $v_\lambda(\varphi,\theta,w)=0$. For these values of $w$, (\ref{2.3})
admits the trivial solution $v_\lambda^1(\varphi,\theta,w)=
v_\lambda^2(\varphi,\theta,w)=0$. This implies that $f_\lambda^{v,1}=Id$,
$f_\lambda^{v,2}=Id$ for the points in the sphere satisfying this condition.
\par\medskip
Clearly, its solutions can be extended by the identity to all points of the
sphere in $K_1^{\varepsilon}$ even for those for which our coordinate system
does not work. Under conditions of proximity to the identity of
$f_\lambda^v$, $f_\lambda^{v,1}$, $f_\lambda^{v,2}$ these will be solutions
of our original problem.\par\medskip
Since, except for these considerations $w$ does not enter either~(\ref{2.3})
or in~(\ref{2.2}) we can consider these variables as parameters in the problem
in the same way as $\lambda$.
We will, hence assume that $\lambda$ refers both to the parameters in the
family and to $w$ and omit $w$ from the notation.\par\medskip
In the enunciate of the theorem $y_1$, $y_2$ were left at our choice. The
choices of $\rho$ -- the angle between the planes
passing through the origin and $y_1$, $y_2$ respectively--
that are acceptable for the theorem are those satisfying
the so-called {\it Diophantine} conditions.\par\medskip
Those are conditions of the form
\begin{equation}\label{2.4}
\left|\rho- \frac{p}{q} \right|^{-1} \leq C q^{2 + \delta}
\end{equation}
It is well known that for $\delta >0$, the numbers satisfying
inequalities of this form have full measure. For $\delta=0$, the set of numbers
satisfying these inequalities is called the of constant type numbers.
(\ref{2.4}) is equivalent to $\rho$ having a bounded continued fraction
expansion. All irrational numbers that satisfy a quadratic equation with
integer coefficients are constant type. In particular constant type
numbers are dense.\par\medskip
Even if all Diophantine $\rho$'s would lead to a theorem of the type we want
and for $\delta$ small with the conditions of differentiability required in
the statement, constant type numbers will lead to the sharpest
differentiability conclusions and we will, from now on assume we have chosen
a constant type number. The actual choice will affect the sizes of
the neighborhoods but not the differentiability properties.\par\medskip
Proceeding heuristically for the moment, the importance of the Diophantine
properties becomes apparent if we consider the {\it linearized equations}
obtained by expanding formally (\ref{2.3}) in the unknowns and keeping
only the linear terms. We obtain:
\begin{equation}\label{2.5}
v_\lambda(\varphi,\theta)=v_\lambda^1(\varphi,\theta)+
v_\lambda^2(\varphi,\theta)
\end{equation}
We expand in Fourier coefficients in $\theta$ $v_\lambda(\varphi,\theta)=
\sum_k \hat{v}_{\lambda,k}(\varphi) e^{ 2\pi i k \theta}$ and
analogously for $v_\lambda^1$, $v_\lambda^2$ and obtain that (\ref{2.5})
is equivalent to:
\begin{equation}\label{2.6}
\hat{v}_{\lambda,k}(\varphi)=\hat{v}_{\lambda,k}^1(\varphi)
+\hat{v}_{\lambda,k}^2(\varphi)
\end{equation}
Similarly, the symmetry conditions
for $v^1_\lambda$, $v^2_\lambda$ can be expressed as:
\begin{equation}\label{2.7}
\begin{array}{c}
\hat{v}^1_{\lambda,-k}(\varphi)=\hat{v}_{\lambda,k}^1(\varphi)\\[.2cm]
\hat{v}_{\lambda,-k}^2(\varphi)e^{4 \pi i\rho}=\hat{v}_{\lambda,-k}^2(\varphi)
\end{array}
\end{equation}
When $k\neq 0$, then we can group the equations for $k$, $-k$ in~(\ref{2.6})
and, using~(\ref{2.7}) we obtain:
\[ \begin{array}{c}
\hat{v}_{\lambda,k}(\varphi)=\hat{v}_{\lambda,k}^1(\varphi)+
\hat{v}_{\lambda,k}^2(\varphi) \\[.2cm]
\hat{v}_{\lambda,-k}(\varphi)=\hat{v}_{\lambda,-k}^1(\varphi)+
\hat{v}_{\lambda,k}^2(\varphi) e^{-4 \pi i\rho}
\end{array} \]
When $k \ne 0$, these equations admit the solutions:
\begin{equation}\label{2.8}
\begin{array}{c}
\hat{v}_{\lambda,k}^1(\varphi)=(\hat{v}_{\lambda,k}(\varphi) e^{-4\pi i \rho k}
-\hat{v}_{\lambda,-k}(\varphi))/(e^{-4\pi i\rho k}-1) \\[.2cm]
\hat{v}_{\lambda,k}^2(\varphi)=(\hat{v}_{\lambda,k}(\varphi)
-\hat{v}_{\lambda,-k}(\varphi))/(e^{-4\pi i\rho k}-1) \\[.2cm]
\end{array}
\end{equation}
When $k=0$, there are many solutions. We just choose:
\begin{equation}\label{2.9}
\hat{v}_{\lambda,0}^1 (\varphi)=\hat{v}_{\lambda,0}^2 (\varphi)= \frac{1}{2}
\hat{v}_{\lambda,0} (\varphi)
\end{equation}
Notice that, if $\rho$ were rational, the denominators would become zero,
that is, there would be no solution for the equation. Even if $\rho$ is
irrational the denominators would become arbitrarily close to zero.
Nevertheless, if $\rho$ is a number of constant type, we have the bounds
\begin{equation}\label{2.10}
|(e^{-2 \pi i k \rho}-1)^{-1}| \leq C'|k|
\end{equation}
that allow us to control the new Fourier expansions.\par
If we think of (\ref{2.3}) as an equation in spaces of functions, (\ref{2.5})
would be the derivative -- in some sense that will have to be specified --
at zero. The calculation shows that the derivative, even if
invertible is not bounded,
hence the usual implicit function theorem in Banach spaces does not apply.
\par\medskip
We also emphasize that the calculation above shows only existence of the
inverse of the derivative zero. If we had started making perturbations
around a non-zero, $v_\lambda^1$, $v_\lambda^2$ we would have been lead to an
equation that cannot be readily analyzed in terms of Fourier components.
\par\medskip
The above heuristic discussion suggests that, among the several {\it hard}
implicit function theorems can cope with unbounded inverses, the one that
will be useful for this problem is that of Zehnder \cite{Ze1} which
requires existence
of inverse only at one point, but which also requires a {\it group}
structure for the equation.\par\medskip
We continue the proof by recalling Zehnder's theorem, which we have taken
from \cite{Ze1} in the version we are going to apply.
\begin{defi}\label{regu}
We say $(X_\alpha)_{\alpha\geq0}$ is a family of regular spaces when
$X_\alpha$ are real Banach spaces with norm $\| \quad \|_\alpha$ and, whenever
for $\alpha'\geq \alpha$ we have:
\[ \begin{array}{l}
X_0\supset X_\alpha \supset X_{\alpha '}\supseteq X_\infty =
{\displaystyle \bigcap _{\alpha \geq 0}} X_\alpha \\[.3cm]
\|u\|_\alpha \leq \|u\|_{\alpha '} \quad \forall u\in X_{\alpha '}
\end{array} \]
Furthermore, it is possible to define on them a $C^\infty$-regularization, that
is, a family $(S_t)_{t>0}$ of linear mappings $ S_t\,:\,X_0 \rightarrow
X_\infty$ and constants $C(\alpha,\alpha ')$ satisfying:
\[ \begin{array}{ll}
i) & {\displaystyle \lim_{t\rightarrow \infty}} \|S_t u-u\|_0=0 \quad
\forall u \in X_0 \\[.3cm]
ii) & \| S_t u\|_\alpha \leq t^{\alpha-\alpha '} C(\alpha,\alpha ')
\|u\|_{\alpha '} \quad \forall u \in X_{\alpha '} \quad
0\leq \alpha ' \leq \alpha \\[.3cm]
iii) & \|S_t u-u\|_{\alpha '}\leq t^{\alpha '-\alpha} C(\alpha,\alpha ')
\|u\|_\alpha \quad \forall u\in X_\alpha \quad
0\leq \alpha \leq \alpha ' \end{array} \]
\end{defi}
We now consider several families of regular spaces and denote by the same
symbol their norms which are probably different. However the confusion
is not possible because the kind of vector that we are evaluating
indicate the norm we are using.
\begin{theo}\label{zehn}
Let $X_\alpha$, $Y_\alpha$, $Z_\alpha$ be families of regular spaces.\par
\medskip
Let $F\,:\, X_0 \times Y_0 \rightarrow Z_0$ be such that $F(x_0,y_0)=0$
and continuous on a set $B_0$. (For any $\alpha$, we will write $B_\alpha
= \{ x,y \in X_0 \times Y_0 \; /\; \|x-x_0\|_\alpha \leq 1\,, \;
\|y-y_0\|_\alpha \leq 1\}$.)\par\medskip
Assume
\begin{description}
\item[{\rm (H.1.)}] $F(x,\cdot)\,:\; Y_0\rightarrow Z_0$ is twice differentiable and
\[ \|d_2 F(x,y)\|\leq M_0\,, \quad \|d_2^2F(x,y)\|\leq M_0 \quad
\forall x,y \in B_0 \;.\]
\item[{\rm (H.2.)}] $F$ is uniformly Lipschitz in $X_0$. That is, for
all $(x,y) \in B_0$, $(x',y)\in B_0$
\[ \|F(x,y)-F(x',y)\|_{0} \leq M_0 \|x-x'\|_{0}\;.\]
\item[{\rm (H.3.)}] The triple $(F,x_0,y_0)$ is of order $\infty$. That is
\begin{itemize}
\item[{\rm (H.3.1.)}] $(x_0,y_0) \in X_\infty \times Y_\infty$.
\item[{\rm (H.3.2.)}] $F(B_0 \cap X_\alpha\times Y_\alpha) \subset Z_\alpha
\quad 0\leq \alpha$.
\item[{\rm (H.3.3.)}] There exist constants $M_\alpha$, $1\leq \alpha$
such that if $(x,y)\in X_\alpha\times Y_\alpha\cap B_1$ satisfies
$\|x-x_0\|_\alpha \leq K,$ $\|y-y_0\|\leq K$, then $\|F(x,y)\|_\alpha
\leq K\,M_\alpha$.
\end{itemize}
\item[{\rm (H.4.)}] There exist an approximate right-inverse of loss $\gamma$,
$1\leq \gamma$. That is, for every $(x,y) \in B_\alpha$ there exist
a linear mapping $\eta (x,y) \in {\cal L}(Z_\alpha,Y_{\alpha-\gamma})$
such that:
\begin{itemize}
\item[{\rm (H.4.1.)}] $\|\eta(x,y)z\|_0\leq M_0 \|z\|_\gamma$.
\item[{\rm (H.4.2.)}] $\|d_2F(x,y)\, \eta(x,y)-1)z\|_0 \leq M_0
\|F(x,y)\|_\gamma\,\|z\|_\gamma$.
\end{itemize}
\end{description}
Then if $\varepsilon$ is a small positive number and
$\lambda=2\gamma+\varepsilon$, we can find an open neighborhood
$D_\lambda=\{ x\in X_\lambda \;/\; \|x-x_0\|_\lambda < C_\lambda \}$
and a mapping $\psi\,:\, D_\lambda \rightarrow Y_\gamma$ such that
\begin{itemize}
\item[a)] $F(x,\psi(x))=0$.
\item[b)] $\|\psi(x)-y_0\|_\gamma \leq C_\lambda ^{-1} \|x -x_0\|_\lambda$.
\end{itemize}
\noindent In particular $\psi(D_\lambda \cap X_\infty) \subset
Y_\infty$.\par\medskip
Moreover, if $\eta(x,y)$ depends continuously on $(x,y)$, then the mapping
$\psi\,:\, D_\lambda \rightarrow Y_\gamma$ is continuous.
\end{theo}
To apply the theorem to solve (\ref{2.3}) we have to choose an appropriate
functional and appropriate spaces of families functions.\par\medskip
We will consider Banach spaces of $C^r$-families of mappings. Let
$\delta \in \real$, we stand by $C^r_\delta(${\bf T}$^m)$ the space of
functions $v\in C^r(${\bf T}$^m)$ which verify:
\[v_\lambda(\varphi,\theta)=v_\lambda(\varphi,-\theta+2\rho), \quad
(\lambda,\varphi,\theta)\in {\mbox{\rm \bf T}}^m\,.\]
Let us fix $s\geq0$ and a small parameter $\varepsilon > 0$. We take
$X_\alpha=Z_\alpha= C^{s+\alpha}(${\bf T}$^m)$,
$Y_\alpha=C^{s+\alpha+2+\varepsilon}_0(${\bf T}$^m)
\times C_\rho^{s+\alpha+2+\varepsilon} (${\bf T}$^m)$ with
the corresponding H\"{o}lder norms and define the operators
\[ \begin{array}{lcl}
F\;: & X_\alpha\times Y_\alpha &\longrightarrow Z_\alpha\\
& (v_\lambda,v_\lambda^1, v_\lambda^2) & \longrightarrow
v_\lambda^1(\varphi+v_\lambda^2(\varphi,\theta),\theta)+
v_\lambda^2(\varphi,\theta)-v_\lambda(\varphi,\theta)
\end{array} \]
We will show that $F$ verify all the conditions of the
previous statement around $(0,0,0)$. In our case Zehnder's theorem is
not anymore than a hidden inverse function theorem.\par\medskip
Most of its conditions are deduced in the Appendix \ref{Appen}.
The hypothesis
(H.1.) (H.2.) (H.3.)
are immediate consequences of
XXXXXX .\par\medskip
The most crucial hypothesis is that about existence of an approximate
right-inverse of the linearized operator,
which we now discuss.
We will start the construction of such approximate inverse
$(0,0,0)$. To study
its differentiability properties,
we will use the following result proved in R\"ussmann \cite{Ru}.
\begin{lemm}\label{rus}
Let $\rho$ be a constant type number and
\[ h(z)=\sum_{k=-\infty}^{\infty} {\hat h}_k e^{i k z}\]
an analytic complex function in $B_r=\{ z \in {\mbox {\rm \bf C}}\,/\;
|{\mbox {\rm Im}} z| < r\}$ with mean value zero and
$|h(z)|\leq M$ $\forall z\in B_r$. If we take
\[u(z)=\sum_{k=-\infty}^{\infty} \frac{{\hat h}_k}{e^{2 \pi i k \rho -1}} e^{i k z}\]
there is a constant $K$ such that if $r' 1$,
with mean value zero.
The function:
\[u(z)=\sum_{k=-\infty}^{\infty} \frac{{\hat h}_k}{e^{2 \pi i k \rho -1}} e^{i k z}\]
satisfies:
\[ || u||_{C^{r-1} } \le K ||h||_{C^r} \]
\end{lemm}
where $K$ is a constant depending only on $\rho$.
Using \ref{mos} to estimate the
solution of the approximate inverse that
we are considering, we obtain:
\begin{lemm}\label{pde}
Let $\gamma>3$ and $\alpha \geq \gamma$. For every $v\in Z_{\alpha}$
there exist
functions $(v^1,v^2) \in Y_{\alpha-\gamma}$ such that $v=v^1+v^2$. Moreover the map
\[ \begin{array}{rlc}
\eta\;:&Z_{\alpha}\longrightarrow & Y_{\alpha-\gamma}\\
&v \longrightarrow &(v^1,v^2)
\end{array} \]
is continuous.
\end{lemm}
\par Now a approximate right-inverse in a complete neighborhood of $(0,0,0)$ is
required. An important observation is that one exact right-inverse
can be obtained thanks to the
group action. See Hamilton \cite{Ham}, pg 198], Zehnder \cite{Ze1}, pg. 133].
\par We proceed heuristically in the following.
Consider for instance a smooth manifold ${\cal M}$, which
locally can be identified with a Banach space, and a differentiable action
$\Phi :{\cal G} \times {\cal M} \rightarrow {\cal M}$ where ${\cal G}$ is an
infinite dimensional group. Assume that
\par i) $F:{\cal G} \rightarrow {\cal M}$ is defined as $F(g)=\Phi(g,m_0) $
for $m_0 \in {\cal M}$.
\par ii) $dF(id) :G=T_{id}{\cal G} \rightarrow T_{m_0}{\cal M}$
is surjective with right inverse $\eta$.
\par
If $L_g:{\cal G} \rightarrow {\cal G} $
denotes the left multiplication by $g$ on ${\cal G}$ and $\Phi_g:{\cal M}
\rightarrow {\cal M}$ denotes the action of $g$ on ${\cal M}$ then
\begin{equation}
\Phi_g \circ F(h)= F \circ L_g(h)
\end{equation}
for every $g,h \in {\cal G}$, which yields
\begin{equation} \label{2.12}
d\Phi_g(m_0) \circ dF(id)= dF(g) \circ dL_g(id)
\end{equation}
Hence the map $\eta(g)= dL_g(id) \circ \eta \circ d\Phi_{g^{-1}}(F(g))$
defines a right-inverse of $dF(g)$.
\vspace{.3cm} \par We now make precise the spaces and the domains in which the
above calculations are valid.
\begin{lemm}
Let $\alpha \geq \gamma$ and take $(v_{\lambda}',v_\lambda^1,v_\lambda^2) \in B_
\alpha$. There exists $\eta(v_\lambda^1,v_\lambda^2) \in {\cal L}(Z_\alpha,
Y_{\alpha-\gamma})$ such that
\[d_2F(v_{\lambda}',v_\lambda^1,v_\lambda^2) \circ \eta(v_\lambda^1,v_\lambda^2)
(v_\lambda)=v_\lambda\]
for every $v_\lambda \in Z_{\alpha}$.
\end{lemm}
{\bf Proof.} Let us consider $F_2:Y_\alpha \rightarrow Z_\alpha$,
$(v_\lambda^1,v_\lambda^2) \rightarrow F(0,v_\lambda^1,v_\lambda^2)$. It is
obvious that $d_2F(v_{\lambda}',v_\lambda^1,v_\lambda^2)=dF_2(v_\lambda^1,
v_\lambda^2)$.
\par We can realize the sets $X_\alpha, Y_\alpha, Z_\alpha$ as the
corresponding tangent spaces at the identity of groups of diffeomorphisms in
${\bf T}^m$, where the functional equation defined in lemma (\ref{div},c) is
the composition.
\par Let $(v_\lambda^1,v_\lambda^2)\in Y_\alpha$ and $v_\lambda=F_2(v_\lambda^1,
v_\lambda^2)$. In our problem the group operation and differentiable
action which appear in (\ref{2.12}) are defined by the relations
\[ \begin{array}{clcl}
\Phi:\;&Z_\alpha &\longrightarrow &Z_\alpha\\
&v_\lambda &\longrightarrow &\begin{array}{l}
{\bf v}_\lambda^1(\varphi+v_\lambda(\varphi+{\bf v}_\lambda^2(\varphi,
\theta),\theta)+{\bf v}_\lambda^2(\varphi,\theta),\theta)+\\
v_\lambda(\varphi+{\bf v}_\lambda^2(\varphi,\theta),\theta)
+{\bf v}_\lambda^2(\varphi,\theta).
\end{array}
\end{array} \]
and
\[ \begin{array}{clcl}
L:\;&\:Y_{\alpha-\gamma} & \longrightarrow &Y_{\alpha-\gamma}\\
&(w_\lambda^1,w_\lambda^2)& \longrightarrow &\begin{array}{l}
( v_\lambda^1(\varphi+w_\lambda^1(\varphi,\theta),\theta)
+ w_\lambda^1(\varphi,\theta),\\
w_\lambda^2(\varphi+v_\lambda^2(\varphi,\theta),\theta)
+ v_\lambda^2(\varphi,\theta)).\end{array}
\end{array} \]
\par It follows from proposition (\ref{dip}) that $\Phi$ is continuously
differentiable and $d\Phi(v_\lambda) \in {\cal L}(Z_\alpha,Z_\alpha)$ and
\[d\Phi(v_\lambda)(w_\lambda)=
w_\lambda(\varphi+{\bf v}_\lambda^2(\varphi,\theta),\theta)+\]
\[\frac{\partial{\bf v}_\lambda^1}{\partial \varphi}
(\varphi+v_\lambda(\varphi+{\bf v}_\lambda^2(\varphi,\theta),\theta)
w_\lambda(\varphi+{\bf v}_\lambda^2(\varphi,\theta),\theta).\]
Similarly, $L$ is continuously differentiable, $dL(0) \in {\cal L}(Y_{\alpha-
\gamma},Y_{\alpha-\gamma})$ and
\[dL(0)(w_\lambda^1,w_\lambda^2)=(\frac{\partial v_\lambda^1}{\partial \varphi}
(\varphi,\theta)w_\lambda^1(\varphi,\theta)+w_\lambda^1(\varphi,\theta),
w_\lambda^2(\varphi+v_\lambda^2(\varphi,\theta),\theta))\]
\par The map $\eta(v_\lambda^1,v_\lambda^2)=dL(0) \circ \eta \circ d\Phi
(v_\lambda) \in {\cal L}(Z_\alpha,Y_{\alpha-\gamma})$ and defines a right-inverse
of $dF_2$.
\vspace{.3cm} \par We have verified all the hypotheses of theorem
(\ref{zehn}).
Therefore, given $v_\lambda \in C^s({\bf T}^m)$,$ s>6$ close to $0$ there
are $(v_\lambda^1,v_\lambda^2) \in C^{s-1-\varepsilon}_0({\bf T}^m) \times
C^{s-1-\varepsilon}_\rho({\bf T}^m)$ such that $F(v_\lambda,v_\lambda^1,
v_\lambda^2)=0$, and hence the conclusions of the lemma (\ref{div}) hold.
Therefore, we have shown that in a $C^r$ neighborhood of
the identity $r > 6$ all diffeomorphisms can be factored.
Unfortunately, the set of $C^r$ diffeomorphisms
is not a group unless $r \in \natural$.
This is the reason why the statement
of \ref{main}
is made
using only $C^7$ regularity.
\appendix\section{Differentiability of composition in H\"older spaces}\label{apen}\label{Appen}
\subsection{{\bf Introduction}}\label{introd}
In this appendix we collect some theorems and counterexamples that clarify the
differentiability properties of composition in the classical spaces of
H\"{o}lder functions.
\vspace{.1cm} We have emphasized the methods based on elementary estimates, triangle
inequality , mean value theorem and the like. Therefore, the results presented
here will be valid in a great generality - e.g., open sets of Banach spaces.
In finite dimensions, it is sometimes possible to obtain similar results using
characterizations of H\"{o}lder spaces by approximation properties by analytic
functions \cite{ST}, \cite{Kr}, \cite{Ni}. As intermediate steps
we will also prove elementary
versions of interpolation inequalities that work
in infinite dimensional
domains with good geometric properties.
\par Since the composition operator is the basic operator of
dynamical systems, it
is clear that the study of this operator acting on different functional spaces
should be very useful. The systematic study of differentiability properties of
the composition operator in the spaces of continuously differentiable functions
can be found in \cite{Ir}. In order to deduce good properties from this operator
uniform continuity of the factors is required. Irwin formulates his version
on compact sets where continuity automatically implies uniform continuity. \\
In this appendix, we have studied only classical H\"{o}lder spaces. Since
H\"{o}lder character assures
uniform continuity we consider maps defined on open subsets. Notice that
some of the results for these spaces, even on compact sets,
are different from the straightforward
extensions of those for continuously differentiable functions. Unfortunately,
the literature abounds with confusions caused by this fact.
Notice also that, as stated in \ref{intro}, many of the results
discussed here carry over to manifolds.
\subsection{{\bf Basic definitions and notation}}\label{dfn}
Let $E, F$ be Banach spaces. We will denote the space of bounded
linear functions from $E$ to $F$ by ${\cal L}(E,F)$ and in general
${\cal L}_n(E,F)={\cal L}(E,{\cal L}_{n-1}(E,F))$.
\par \vspace{.1cm} Let $ U \subset E $ be an open set. A function $ f: U
\rightarrow F $
is Frechet differentiable at $x_0$ if we can find a bounded linear function
$ df(x_0) $ such that
\[ \lim_{h \rightarrow 0} \frac{ \parallel f(x_0+h) - f(x_0) -df(x_0) h
\parallel}{ \parallel h \parallel } = 0 \]
We say that $f$ is differentiable in $U$ if $f$ is differentiable at every
point $ x_0 \in U $. We say that $f$ is of class $C^1$ if it is differentiable
and the mapping $ df: U \rightarrow{\cal L}(E,F) $ that to each $x_0 \in U $
associates $df(x_0) $ is continuous.
\vspace{.1cm} Proceeding by induction, we define $d^n f$ to be the
differential of
$ d^{n-1} f$ $(d^1f=df) $ and we say that a function $f$ is $n$ times
continuously differentiable, $ df: U \rightarrow {\cal L}(E,F) $ is $(n-1) $
times continuously differentiable.
\vspace{.1cm} We will denote by ${\cal L}_n (E_1, \ldots , E_n, F) $ the space of
$n$-multilinear functions having arguments in $ E_1, \ldots , E_n $ and values
in $F$.
It is standard that we can identify ${\cal L}_n (E,F) $ with the
space of
bounded multilinear functions with $ n$ arguments in $E$ taking values in $F$.
We will think of derivatives as multilinear functions.
\vspace{.1cm} We will say that $ f \in C^n(U,F) $ if $f$ is $n$ times
continuously
differentiable and $ \parallel d^nf(x) \parallel $ is uniformly bounded by
$ x \in U$. In that case, we denote
\[ \parallel f \parallel_{C^n} = \sup _{0 \leq k \leq n} \;\;\;\;
\sup_{x} \parallel d^k f(x) \parallel . \]
Notice that, according to our definition there could be functions which are $n$
times continuously differentiable but not in $ C^n(U,F).$
\par \vspace{.3cm} If $f \in C^n(U,F)$, $V$ is an open subset of $F$ with $f(U) \subset V$
and $g \in C^n(V,G)$ then $ g\circ f \in C^n(U,G)$. Moreover by a repeated
application of the chain rule we can find constants $c_{n,j_1,\ldots ,j_n}$
such that
\[d^n(g \circ f)(x)=\sum_{k=1}^{n}c_{k,j_1,\ldots ,j_n}d^kg(f(x))\{d^{j_1}f(x)
,\ldots,d^{j_k}f(x)\}\]
\vspace{.1cm} {\bf Remark.} Weaker notions of differentiability spaces have
been proposed
and the one we consider here is sometimes called "strongly differentiable
functions." Moreover, if the manifold under consideration is not compact,
- e.g., $\real ^d $ or a bounded subset of an infinite dimensional
Banach space - there may be functions which are continuously differentiable
but whose derivative is unbounded. In those cases, besides the topology
induced by the supremum norm we consider here, there are other natural
topologies, e.g., the Whitney topology associated to uniform convergence on
compact sets. This topology is an inductive limit of the ones we consider
here and, hence, the results of continuity can be immediately lifted.
Since this topology does not come from a norm only weak concepts of
differentiability can be introduced for composition and
similar operators, when we consider such topologies.
\begin{defi}
If $ r = n + \alpha $ with $ n \in \natural $ and $ 0 < \alpha
\leq 1^- $ we say that $ f \in C^r(U,F) $ if $ f \in C^n(U,F) $
and moreover we have
\[ H_{\alpha}(d^nf)=\sup_{x\neq y}\frac{
\parallel d^nf(x) - d^n f(y) \parallel }{ \parallel x-y
\parallel ^{\alpha} } < \infty \]
\end{defi}
With the introduction of
\[ \parallel f \parallel _{C^r} =
\sup( \parallel f \parallel _{C^n} , H_{\alpha}(d^nf))\]
$C^r(U,F)$ is a Banach space. \par
Notice that with this definition, we have
\[ \parallel f \parallel _{C^r} \leq \sup( \parallel f \parallel _{C^0},
\parallel df \parallel _{C^{r-1}})\]
this makes it easy to carry out several inductions and hence, this definition
is preferable to other equivalent norms.
\vspace{.1cm} {\bf Remark.} If $U=U_1 \cup U_2$, $U_1\cap U_2 \neq
\emptyset $, we can write
$C^r(U)=C^r(U_1)\oplus C^r(U_2)$ and it is possible to study $C^r(U_i)$
separately. Similarly, the study of composition operators can be broken up
into the study of the components.
So, we will henceforth assume that the domains in which we study the
functions are connected by arcs.
\subsection{{\bf Geometry of domains and
Interpolation properties}}\label{geom}
\vspace{.1cm} The following are immediate.
\begin{prop} \label{inc}
Let $0\leq \alpha < \beta \leq 1^-, $ and $n\in \natural$.Then
\begin{itemize}
\item[i)] $C^{n+ \beta}(U) \subset C^{n+ \alpha}(U) \subset C^n(U) $.
\item[ii)] $C^{n+ \beta}(U) \mbox{is a closed subset of} \:C^{n+ \alpha}(U)$.
\end{itemize}
\end{prop}
{\bf Remark.} For most spaces useful in analysis the inclusion in
Proposition \ref{inc} is strict. It suffices to consider $ |x|^{\alpha + n} $ on
[0,1] which is in $ C^{n+ \alpha} $ but not on $ C^{n+ \beta} $.
\vspace{.1cm}
In the following lemmas we develop
some interpolation
properties of the norms that we will
use later in the discussions of the composition operator.
\begin{lemm} \label{int}
Let $r=n+\alpha,\: s=n+\beta, \:t=n+\gamma$ with
$0\leq \alpha <\beta <\gamma \leq 1$ and $\mu =(t-s)/(t-r)$
. There is a constant $M_{r,t}$ such that if $ f \in C^t (U,F), $ then
\begin{equation}\label{exp}
\parallel f \parallel _{C^s} \leq M_{r,t}\parallel f \parallel _{C^r}
^{\mu } \parallel f \parallel _{C^t}^{1-\mu }
\end{equation}
\end{lemm}
{\bf Proof:} Notice that $1-\mu=(s-r)/(t-r)$ and $s=\mu r+(1-\mu) t$.
We will establish the claim using induction on $n$.
We will first describe the case $n=0$
If we define for $\beta\in (0,1^-]$:
\[H_\beta(f)=\sup \frac{\parallel f(x)-f(y)\parallel}
{\parallel x-y \parallel^\beta} \]
we have
\[ H_\beta = \sup \frac{\parallel f(x)-f(y)\parallel^\mu}{\parallel x-y
\parallel^{\alpha \mu}}\sup \frac{\parallel f(x)-f(y)\parallel^{1-\mu}}
{\parallel x-y \parallel^{\gamma(1-\mu)} }
=H_r(f)^{\mu }H_t(f)^{1-\mu }.\]
From here it is obvious that
\[ \parallel f \parallel _{C^s} \leq M_{r,t}\parallel f \parallel _{C^r}
^{\mu } \parallel f \parallel _{C^t}^{1-\mu } \]
where
\[M_{r,t}=\left\{ \begin{array}{ll}
1 & \mbox{if $\:\:0< \alpha <\gamma \leq 1^-$}\\
2 & \mbox{if $\:\:0= \alpha <\gamma \leq 1^-$}\\
\end{array} \right.\]
\indent For $n \geq 1$ using the inductive hypothesis we find
\[\parallel df \parallel _{C^{s-1}} \leq M_{r,t} \parallel df \parallel
_{C ^{r-1}} ^{\mu} \parallel df \parallel _{C^{t-1}}^{1- \mu} \leq
M_{r,t} \parallel f \parallel _{C ^r}
^{\mu} \parallel f \parallel _{C^t}^{1-\mu} \]
and thus
\[\parallel f\parallel_{C^s}\leq M_{r,t} \parallel f \parallel _{C ^r}
^{\mu} \parallel f \parallel _{C^t}^{1-\mu} \]
Notice that $\ln\parallel f\parallel_{C^r}$ is a convex
, therefore continuous, function
in each open interval $(n,n+1)$ with $n \in \natural$.\par
\vspace{.3cm}
\begin{prop}
If $ 0 \leq \alpha < \beta \leq 1^- $ and $ U $ is separable then bounded sets
of $ C^{n+ \beta}(U) $ are compact in $ C^{n+\alpha}(U) $.
\end{prop}
{\bf Proof.} For $\alpha=0$ this is a consequence of Ascoli-Arzela theorem.\par
For $\alpha>0$ we use the result for $\alpha=0$ and the
interpolation formula (\ref{exp}).
\vspace{.1cm} Notice that in Proposition (\ref{inc}) we have not claimed that
if $ r = n + \alpha < s = m + \beta $ then $ C^s \subset C^r $.
\vspace{.1cm} The fact that this is not unfortunately true in general can
be seen in the following.
\begin{exam}\label{cas}
{\rm Let $ D= \{ z \in C / |z| < 1 \}$, $S = \{ z \in C / \Im z = 0,
\;0\leq \Re z \leq 1 \} $ and $ U = D - S $.
\par Denote by $dist_U (z,z')$ the infimum of the length of smooth paths
contained in $ U $ joining $ z $ to $ z' $ and define
\[ f(x,y) = dist_U (x + iy, \frac{1}{2} + \frac{i}{2} ) \]
Then, $f$ is continuously differentiable in $ U $ and $ \parallel df \parallel_{C^0}
\leq 1 $. Nevertheless, there are no constants $ k > 0, \alpha > 0 $ such that
\[ |f(x,y) - f(x',y') | \leq k |(x,y) - (x',y') | ^{ \alpha} \]
Hence, in the space $ U \;\; C^1(U) \not \subseteq C^{ \alpha} (U) $ for any
$ 0 < \alpha \leq 1^- $. }
\end{exam}
\vspace{.2cm} In many respects this example is a "pathology." Its origins are clear
if one tries to use the mean value theorem.
\begin{equation}\label{vme}
f({\bf x}) - f({\bf y}) = \int ^1_0 df( \gamma(s)) \dot{ \gamma} (s) ds
\end{equation}
where $ \gamma $ is a path joining $ {\bf x},{\bf y} $.
\vspace{.2cm} This immediately gives $ | f({\bf x}) - f({\bf y}) |
\leq \parallel f
\parallel _{C^1} dist_U({\bf x},{\bf y}) $ where the symbol
$ dist_U({\bf x},{\bf y}) $ denotes the infimum of
the lengths of paths joining ${\bf x} $ to ${\bf y} $. In the space $ U $ of the example
this could be much larger than $ |{\bf x}-{\bf y}| $. Clearly, if we want to use arguments
based on the mean value formula (\ref{vme}) we have to restrict ourselves to
spaces in which $ dist_U (x,y) $ is commensurate to $| (x,y)| $.
\vspace{.1cm} When we consider problems involving composition, we
have to ensure that the intermediate ranges also have this property.
\begin{defi}{balanced}
We say that an open set $ U $ is balanced if there exists a constant $ k_U $
such that $ d_U(x,y) \leq k_U \parallel x-y \parallel \forall x, y \in U $.
\end{defi}
This property depends on the shape of $U$.
For example, a convex set of a Banach space is balanced or more generally, a
connected set which is a finite union of convex sets, is balanced. Notice that,
as stated, the definition only makes sense for connected sets. This is not a
shortcoming in view of the
fact that , as remarked in the introduction, it suffices to
consider spaces of
function on connected sets.
Notice also that a subset of a balanced subset needs not be balanced.
Unfortunately, $ C^{ \infty} $ functions could transform balanced
domains into unbalanced ones.
\begin{exam}
{\rm Let $ D_+ = \{ z \in C / |z| < 1, \Im z > 0 \} $ the map $ z \rightarrow z^2 $
transforms the balanced open $ D_+ $ on the unbalanced $ U $ defined in the example
(\ref{cas})}
\end{exam}
\par \vspace{.3cm} The following results are useful in the representation
of the sucesive
derivatives of the composition. We omit their proofs which are routine
\begin{prop} \label{lin}
Let $ l \in L(F,G) $. Then $ l_{\star} : C^r(U,F) \rightarrow C^r(U,G) $ defined by
$ l_{\star} f = l \circ f $ is linear and bounded. Moreover, $ \parallel l_{\star}
\parallel= \parallel l \parallel $.
\end{prop}
\begin{prop} \label{bil}
Let $ r, s, t \in \real $, $ r \geq t, s \geq t $. Let $ b \in {\cal L}_2
(F,G;H) $ . Then the map $ b_{\star} : C^r(U,F) \times C^s (U,G) \rightarrow C^t (U,G) $
defined by $ b_{\star} (f,g)(x) = b(f(x), g(x)) $ is continuous and bilinear.
Moreover $ \parallel b_{\star} \parallel \leq 2 \parallel b \parallel $.
\end{prop}
\par We specially employ these relations when
\[\begin{array}{ccc}
b : {\cal L}(F,G)\times {\cal L}(G,H)& \longrightarrow& {\cal L}(F,H) \\
(l,m)& \longrightarrow& l \circ m
\end{array} \]
represents the composition of linear operators. If $F=K$ then
$ {\cal L}(F,G)=G$, $ {\cal L}(F,H)=H$. We will stand by $ g\cdot f$ the map
$b_{\star}(f,g)$ defined as
\[\begin{array}{ccc}
b : C^r(U,G)\times C^s(U,{\cal L}(G,H))& \longrightarrow& C^t(U,H) \\
(f,g)& \longrightarrow& [x \rightarrow g(x)(f(x))]
\end{array} \]
\begin{lemm} \label{com}
Let $ F,G $ be Banach spaces $ U \subset F, V \subset G $ be balanced
subsets. \\
Let $ r = n+ \alpha , s =m + \beta$ with $m, n \in \natural, \alpha, \beta \in
[0,1^-]. $ If $ f \in C^r (U,F)$, $f(U) \subset V, g \in C^s (V,G) $ then $ g
\circ f $ belongs to $ C^t (U,G) $ where we can take $t$ as follows:\\
\noindent i) If $ n=m = 0,t = rs $
\vspace{.1cm} Moreover
\begin{itemize}
\item[i.1)] $ \parallel g \circ f \parallel _{C^{rs}} \leq \parallel g
\parallel _{C^s}
\parallel f \parallel ^s _{C^r} + \parallel g \parallel _{C^0} $
\end{itemize}
ii) If $ \max (r,s) \geq 1,t = \min (r,s) $
\vspace{.1cm} Moreover
\begin{itemize}
\item[ii.1)] if $ r \geq 1,0 < s \leq 1^- $
\[ \parallel g\circ f \parallel _{C^s} \leq \parallel g \parallel _{C^s}
\parallel f \parallel ^s _{C^1} k_U ^s + \parallel g \parallel _{C^0} \]
\item[ii.2)] If $ 0< r \leq 1^-, s \geq 1 $
\[ \parallel g\circ f \parallel _{C^r} \leq \parallel g \parallel _{C^{1}}
\parallel f \parallel _{ \dot{C} ^r} k_V + \parallel g \parallel _{C^0} \]
\item[ii.3)] If $ r \geq 1, s\geq 1, \exists M_t $ such that
\[ \parallel g\circ f \parallel _{C^t} \leq M_{t} \parallel g \parallel _{t}
(1+ \parallel f \parallel ^t _t) \]
\end{itemize}
\end{lemm}
\noindent {\bf Proof.} If $n = m = 0, $ we have
\[ \parallel g \circ f(x) - g \circ f(y)\parallel \leq \parallel g
\parallel _{C^{ \alpha}} \parallel f(x)
- f(y) \parallel ^{ \alpha} \leq
\parallel g \parallel _{C^{ \alpha}} \parallel f \parallel _{C^{ \beta}}
\parallel x-y \parallel ^{ \alpha \beta}\]
This establishes i), i.1). The cases ii.1) and ii.2) can be obtained in a
similar way.
\vspace{.1cm} To verify the last case, we assume that
$s = r = t = k + \gamma $ with $ k ,\in \natural \; k \geq 1 $ and
$ \gamma \in [0,1^-] $. We proceed by induction in $ k $.
\vspace{.1cm} For $ k=1$, $d(g \circ f) = d g \circ f\cdot df $. Applying Proposition [\ref{bil}] and
the i) of Proposition [\ref{com}] we obtain
\[\parallel d g \circ f \parallel _{C^{ \gamma}} \leq k_U^\gamma \parallel dg
\parallel _{C^{ \gamma}} \parallel f \parallel ^{\gamma} _{C^1} + \parallel
dg \parallel _{C^0} \leq \]
\[ M \parallel g \parallel _{C^{1+ \gamma}} (1 + \parallel f \parallel
^{1 + \gamma} _{C^{1 + \gamma}})\]
Let us assume that the above inequality holds until the integer $ k-1 $ and
consider $ t= k + \gamma $.
Then by induction we have that $ d(g \circ f) = dg \circ f\cdot df \in C^{t-1} (U) $
and
\[ \parallel d(g \circ f) \parallel _{C^{t-1}} \leq 2 \parallel d g \circ f
\parallel _{C^{t-1}} \parallel df \parallel _{C^{t-1}} \leq \]
\[ 2M \parallel dg \parallel _{C^{t-1}} ( 1 + \parallel df \parallel ^{t-1}
_{C^{t-1}} ) \parallel df \parallel _{C^{t-1}} \]
\noindent since $M_t (1 + |x|^t) \geq M(1 +|x|^{t-1} ) |x|\: $ for some
$ M_t > 1 $, we obtain
\[ \parallel g \circ f \parallel _{C^t} \leq \sup ( \parallel g \circ f
\parallel _{C^0} , \parallel d (g \circ f) \parallel _{C^{t-1}}) \leq
M_t \parallel g \parallel _{C^t} ( 1 + \parallel f \parallel _{C^t} ^t ) \]
This completes the proof of Lemma \ref{com}
\vspace{.1cm} The following examples show that the hypotheses in proposition
(\ref{com}) are sharp.
\begin{exam}
{\rm Let $ \alpha > 0 $ and $ f_{ \alpha } : [-1,1] \rightarrow \real, \;
x \rightarrow |x|^{ \alpha} $ \\
i) If $ 0 < \alpha, \beta \leq 1^-$, then $ f_{ \alpha} \circ f_{ \beta} \in
C ^{ \gamma} ([-1,1],\real) $ if and only if $ \gamma \leq \alpha \cdot \beta $. \\
ii) If $ \alpha/2 \not \in \natural $ then $ f_ {\alpha} \circ Id \in C^{ \gamma}
([-1,1],\real) $ if and only if $ \gamma \leq (\alpha -1)+1^-$.}
\end{exam}
\vspace{.3cm} As a corollary of proposition (\ref{com}) we obtain that
if $U$ is a balanced open set then the formula (\ref{exp})
also holds for $t=n+1$ taking
\[M_{r,t}=\left\{ \begin{array}{ll}
k_U & \mbox{if $\:\:0< \alpha <\gamma =1$}\\
\max(2,k_U) & \mbox{if $\:\:0= \alpha <\gamma =1$}
\end{array} \right.\]
However, unless we impose some condition
on the domain, these inequalities are not possible for exponents
with different number of derivatives as shown in the following
example.
\begin{exam}\label{noint}
{\rm Let $U=\{(x,y) \in \real^2/x>0, y>1, xy<1\}$. We take $\varphi$ as a $C^{\infty}$
cut-off function with $\parallel \varphi'\parallel_{C^0} \leq 4$ which takes
the value $1$ on $[-\frac{1}{3},\frac{1}{3}]$ and
zero outside of $[-1,1]$. We define on $U$ the following functions
\[f_n(x,y)=\log(1+x)\varphi(y-n)\]
For $ n\in \natural$ big enough we find
\[ \begin{array}{l}
\frac{ \partial F_n}{\partial x} (x,y) = \frac{1}{1+x} \varphi (y-n) \\
\frac{ \partial F_n}{\partial y} (x,y) = \log(1+x) \varphi ' (y-n) \\
\frac{ \partial^2 F_n}{\partial x^2} = - \frac{1}{(1+x)^2} \varphi (y-n) \\
\frac{ \partial^2 F_n}{\partial y^2} = \log(1+x) \varphi '' (y-n) \\
\frac{ \partial^2 F_n}{\partial x \partial y} = \frac{1}{1+x} \varphi '
(y-n)
\end{array} \]
Observe that since the support of $ \varphi (y-n) $ is centereed around
$ y = n $, which in the domain we are considering implies $ x \approx 1/n $ we have:
\[ \begin{array}{l}
| \frac{ \partial F_n}{\partial x} | = \OO(1) ; | \frac{\partial F_n}
{\partial y} | = \OO (1/n) ; \\
| \frac{ \partial ^2 F_n}{\partial x^2} | = \OO (1) ; | \frac{\partial
^2 F}{\partial y^2} | = \OO (1/n) ; \;\;\;\; | \frac{\partial ^2 F}
{\partial x \partial y} | = \OO (1) \\
|| F ||_{C^0} = \OO (1/n)
\end{array} \]
This implies that $ \parallel F \parallel_{C^2} $ remains bounded and also $ || F ||_{C^0} $
converges to zero. \par
Since $ \frac{ \partial F_n}{ \partial x} $ evaluated at $ x = \frac{1}{zn},
y=n $ does not converge to zero, we conclude that there cannot be an inequality
of the form
\[ || F_n || _{C^1} \leq M || F_n ||^{1/2}_{C^0} || F_n || ^{1/2} _{C^2} \]
More generally, we observe that for any $ s \in [ 1^-, \,\,\, 1+1^-) \,\,\,\,\,
|| F_n || _{C^s} $ does not tend to zero. For any $ r \in [0,1^-) \,\,\,\,\,\,
|| F_n || _{C^r} $ tends to zero and for any $ t \leq 2 , \,\,\, || F_n ||
_{C^t} $ remains bounded.
Nevertheless, as it is well known, interpolation inequalities for spaces of
functions defined on $ \real ^n $ can be proved those ranges of
regularities (see e.g. \cite{ST}). \par
\indent This shows that it should be neccessary to include some conditions in
the domain $ U $ if interpolation inequalities are to hold.
\vspace{2.cm} {\bf Remark} We observe that there is an abstract
argument (see \cite{Ze1} )
that shows that if there are smoothing operators, then interpolation
inequalities hold. Hence, we conclude that it is impossible to define
smooting operators for H\"{o}lder spaces of functions in $ U $.
Notice that $ U $ in Example \ref{noint} is a subset of $ \real ^2 $ and that
there are interpolation formulas for $ C^{ \alpha} ( \real ^2 ) $ when
$ \alpha $ is not an integer (see e.g. \- $ [K_r] $ ). Hence, the fact that
there are interpolation formulas is for $ C^{ \alpha} $ spaces is not
inherited by subsets.
This is somewhat surprising since a naive argument would suggest that we can
apply a Whitney extension theorem to extend our function $F$ in $U$ to a
function in $ \real ^2 $, apply interpolation to the extended functions
and then restrict.
The reason why this naive argument is wrong is that the extension operators used
in Whitney theorem depend on the regularity considered.
In \cite{ST} VI, 2 one can
find a construction of extension operators $ \xi _k $ which extend
$ C^{ \alpha}$ for $ k \alpha < k+1 $. Example \ref{noint} shows that it is
impossible to find a unique extension operator working simultaneously in
$ C^{ \alpha _1} \,\,\,\, C^{ \alpha _2} $ when the integer parts of $ \alpha _1
$ and $ \alpha _2 $ are not the same.
This situation is in contrast with regularities measured in other scales of
spaces - e.g. Sobolev spaces $ L^p _k $ where it is possible to find extension
operators that work for all the scale of spaces simultaneously ( see. \cite{ST}
VI 3).
In the following, we will develop son conditions that
guarantee that one can obtain interpolation inequalities.
We do not know whether condition \ref{regular} on $U$ is neccessary for the validity
of interpolation inequalities in $ U $ . We do not know conditions which
ensure that there is a unique extension operator for all $ C^{ \alpha} $
regularities.
\begin{defi}\label{regular}
Let $U$ a balanced open set and $0<\beta \leq 1^-$. We say that $U$ is
$\beta$ -regular if there is a constant $M_0$ such that for
every $x_0 \in U$ and $h \in E$ with $\parallel h\parallel =1$ we can find
$\gamma \in C^{1+ \beta}([0,1],U)$ with $\gamma(0)=x_0$,
$\gamma'(0)=\tau h$ with $\tau=1$ or $-1$
and $\parallel \gamma \parallel_{C^{1+\beta}} \leq M_0$.
\par We say that $U$ is regular if it is $\beta$-regular for some $\beta$.
\end{defi}
\par Notice that the union of $\beta$-regular subsets is also $\beta$-regular.
\par \vspace{.2cm}The relevance of regularity lies in the following lemma.
\begin{lemm} \label{itt}
Assume that $U$ is regular.
Let $r,s,t$ be positive numbers
with $0\leq r~~1$ we can deduce the inequality (\ref{ep2}) by induction.\par
To obtain the complete result we only have to apply the previous lemma.
\par \vspace{.2cm} Using the inequality (\ref{itp}) and arguing
by induction
as in the finite dimensional case, considered in \cite{Ham}, we obtain:
\begin{lemm}
Assume that $U \subset F$, $V \subset G$ are regular subsets. Let $t\geq 1$,
$f \in C^t(U,F)$, $g \in C^t(V,G)$ with $f(U) \subset V$. If $\parallel f
\parallel_{C^1} \leq K$ there exist a constant $M_t$, depending on $K$ such
that
\[\parallel g \circ f \parallel_{C^t} \leq M_t(\parallel g \parallel_{C^t} +
\parallel g \parallel_{C^1} \parallel f \parallel_{C^t})\]
\end{lemm}
\subsection{{\bf Differentiability properties of composition}}\label{prco}
We will systematically study differentiability properties of
\[\mbox{comp}:\:C^r(U,F) \times C^s(V,G) \rightarrow C^t(U,G)\]
First we will study continuity and differentiability with respect to each of
the components and then, join continuity and differentiability.
In order to obtain $g\circ f$ we need $ f(U)\subset V $. We assume this
condition in proposition (\ref{pa1}). To consider this composition as a function of
the first factor it is necessary that $ d(f(U) , V^c) \geq \lambda > 0 $
to make it possible in a complete neighborhood ${\cal U} $ of $ f $. We assume this
condition from lemma \ref{pa2} on.
The following is a direct consequence of lemma (\ref{com})
\begin{prop}\label{pa1}
Let $ r,s\geq 0 $ and
\[ t \leq \left \{ \begin{array}{lcl}
rs \;\;\;& \mbox{if} & 0\leq s<1, 0 \leq r<1 \\
\min(r,s) & \mbox{if} & 1 \leq \max (r,s)
\end{array}
\right. \]
Let $ f \in C^r (U,F) $. Then the map
\[ \begin{array}{ccc}
f^*: C^s(V,G)& \longrightarrow &C^t(U,G) \\
g &\longrightarrow &g \circ f
\end{array} \]
is linear and continuous .
\end{prop}
Now we consider the variation of the first factor.
\begin{prop} \label{pa2}
Let $ r,s,t \geq 0$, $U$, $V$ balanced subsets.
and $ g \in C^s (V,G) $
Let ${\cal U} \subset C^r(U,F)$ be such that
for every $f \in {\cal U}$ we have $d(f(U, v^c) > \epsilon > 0$.
The map
\[\begin{array}{ccc}
g_* : {\cal U} \subset C^r(U,F)& \longrightarrow& C^t (U,G) \\
f& \longrightarrow& g \circ f
\end{array} \]
is continuous in the following cases
\begin{itemize}
\item[i)] for $t=0\:$ when $\: r\geq 0 ,\; s>0.$
\item[ii)] for $\: 0t, \; r\geq t,\:rs>t.$
\item[iii)] for $\: t=k+ \gamma$ with $k\geq 1$, $\gamma \in [0,1)$
when $ s>t,\: r \geq t.$
\item[iv)] for $t=k+1^-$ with $k\geq 0$ when $s>k+1$, $r\geq t$
\end{itemize}
\end{prop}
{\bf Proof:}
i) It is similar to (i) of proposition (\ref{com}). Thus if $0\leq s \leq 1^-$
then
\[\parallel g_*(f_1)-g_*(f_2) \parallel_{C^0} \leq \parallel
g\parallel_{C^s}\parallel f_1-f_2\parallel_{C^0}^s\]
ii.1) First we consider that $0~~~~0 $. We take $ x,y \in U $ and make $ z_1 = f_1(x),
w_1 = f_1 (y) $. For $\epsilon>0$ fixed there is
a path $ \gamma _1 $ joining $ z_1 $ to $ w_1 $ such that
\[ g(z_1) - g(w_1) = \int ^1 _0 dg (\gamma _1 (t)) \dot{ \gamma _1}(t) dt \]
and
\[\int_0^1\parallel \dot{\gamma}_1(t)\parallel dt \leq (k_V+\epsilon)\parallel
z_1-w_1\parallel\]
Let $ f_2 \in {\cal U}$ with $ \parallel f_2-f_1 \parallel _r < \lambda $. If
$ z_2 = f_2(x), w_2 = f_2(y) $ then
\[ \gamma _2 (t) = \gamma _1 (t) + (1-t) (z_2-z_1) + t(w_2-w_1) \]
is a path contained in $ U $ joining $z_2$ to $w_2$ and
\[ g(z_2) - g(w_2) = \int ^1_0 dg (\gamma _2 (t)) \dot{\gamma}_2 (t) dt \]
We take
\[S_1= \parallel \int ^1_0 [dg(\gamma _1 (t)) - dg (\gamma _2(t))]
\dot{ \gamma}_1 (t)dt \parallel \leq\]
\[ \parallel g \parallel _{C^{1 + \beta}} (\parallel z_1 - z_2 \parallel +
\parallel w_1 -w_2 \parallel ) ^{\beta}( k_V+\epsilon) \parallel z_1 - w_1 \parallel\]
\[S_2= \parallel \int^1_0 dg(\gamma _2 (t))
( \dot{ \gamma}_1 (t)- \dot{ \gamma} _2 (t)) dt \parallel \leq
\parallel g \parallel _1 \parallel z_2 - z_1 + w_1 - w_2 \parallel \]
thus
\[ \parallel g(f_1(x)) - g(f_2(x)) - g(f_1(y)) + g(f_2(y)) \parallel
\leq S_1+S_2\leq \]
\[ 2 \parallel g \parallel _{C^{1+ \beta}} \parallel f_1 - f_2 \parallel
^{\beta}
_{C^0} k_V \parallel f_1 \parallel_{C^r} \parallel x-y \parallel ^r + \parallel g
\parallel _{C^1} \parallel f_1-f_2 \parallel _{C^r}
\parallel x-y \parallel ^r \]
This shows that
\[\parallel g \circ f_1 - g \circ f_2 \parallel _{C^r} \leq 2 k_V \parallel g
\parallel _{C^{1+ \beta}}(\parallel f_1 \parallel _{C^r} \parallel f_1 - f_2
\parallel ^{\beta} _{C^0} + \parallel f_1 - f_2
\parallel_{C^r}) \]
hence there is a constant $M$ such that
\begin{equation}\label{li2}
\parallel g \circ f_1 - g \circ f_2 \parallel _{C^r} \leq M \parallel g
\parallel_ {C^{1+\beta}}\parallel f_1 - f_2 \parallel ^{\beta} _{C^r}
\end{equation}
iii) Let us assume that $ t=r = k + \alpha , s = k+ \beta$ with
$0 \leq \alpha <\beta < 1 $.
The case $ \alpha = 0 $ is included in \cite{Ir} , so we consider
$ \alpha > 0$.\par
Let $ \lambda > 0 $ and $ f_1, f_2 \in {\cal U} $ with $ \parallel f_i \parallel _r
< \lambda \;\;\; i=1,2 $. For $ k=1 $ we have
\[ \parallel g_* (f_1) - g_* (f_2) \parallel _{C^0} \leq \parallel g
\parallel _{C^1} k_V \parallel f_1 - f_2 \parallel _{C^0} \]
Moreover $ d(g \circ f ) \in C^{r-1} (U,{\cal L}(E,G)) $ and
\[ \parallel d ( g \circ f_1 ) - d ( g \circ f_2 ) \parallel _{C^{\alpha}} =
\parallel d g \circ f_1 \cdot d(f_1-f_2 ) - (dg \circ f_2 - dg \circ f_1 )
\cdot df_2
\parallel _{C^{\alpha}} \leq \]
\begin{equation}\label{li3}
2 \parallel dg \circ f_1 \parallel _{C^{\alpha}} \parallel d (f_1-f_2) \parallel
_{C^{\alpha}} + 2 \parallel dg \circ f_2 - dg \circ f_1 \parallel _{C^{\alpha}}
\parallel df_2 \parallel _{C^{\alpha}}
\end{equation}
On the other hand
\[ \parallel dg \circ f_2 - dg \circ f_1 \parallel _{C^{\alpha}} \leq 2 \parallel
dg \circ f_2 - dg \circ f_1 \parallel ^{\frac{s-r}{ \beta}} _{C^0}(\parallel
dg \circ f_2 \parallel _{C^{\beta}} + \parallel dg \circ f_1 \parallel _
{C^{\beta}})^{\frac{\alpha}{\beta}} \leq \]
\begin{equation}\label{li4}
2\parallel g \parallel _{C^{1+ \beta}} (
\parallel f_1 \parallel ^{ \beta}_{C^1} k_U^\beta+ \parallel
f_2 \parallel ^{ \beta} _{C^1}k_U^{\beta} + 2 )^{\frac{\alpha}{\beta}} \parallel f_1 - f_2
\parallel ^{s-r} _{C^0}
\end{equation}
Substituting (\ref{li4}) in (\ref{li3}) we find a constant $M$ such that
\begin{equation}\label{ldf}
\parallel g \circ f_1 - g \circ f_2 \parallel _{C^t} \leq M \parallel g
\parallel_ {C^{1+\beta}}\parallel f_1 - f_2 \parallel ^{s-r} _{C^r}
\end{equation}
The verification of the inequality (\ref{ldf}) for every $n$ is a
routine induction argument.
iv) Is a consequence of the arguments used in (ii) and (iii).
\begin{defi}\label{ind}
Let $ \Omega \subset \real ^3_+ $ be the set of indices $ (r,s,t) $
which satisfy some of the conditions stated in proposition \ref{pa2}.
We say that $ \Omega $ is the set of the exponents of continuity
\end{defi}
\begin{prop}\label{dis}
Let $ (r,s,t) \in \real^3 - \Omega $ and $\rho>0$. There are Banach spaces
$E$, $F$ and $ g \in C^s (B_{2\rho}\subset F, \real ) $ such that
\[ g_* :\:{\cal U}\subset C^r (B_{\rho}\subset E, F ) \longrightarrow
C^t (B_\rho, \real) \]
is discontinuous.
\end{prop}
\indent The continuity in (i) requires the uniform continuity of $g$. In the
remaining cases if $(r,s,t) \not \in \Omega$ it satisfies some of the conditions
considered below.
If $ 0< s\leq 1^-, 0 m $
be integers, $ m, n $ of different parity. Then $ |g \circ f_n - g \circ f_m
| (x) = 2 |x|^{rs} $ when $ x \in [-2^{ \frac{-n-1}{r}}, 2^{\frac{-n-1}{r}}] $
so that $ \parallel g\circ f_n - g \circ f_m \parallel _{C^{rs}} \geq 2 $.
Hence, it is impossible that $ g \circ f_n $ converges in $ C^{rs} $.
\vspace{.2cm} If we consider in (ii) or (iii) the option $ s=t $, $r\geq 1$
the continuity also fails. This fact is well known for integer exponents and
we will sharpen it below. If $ s= t=
k + \gamma $ with $ k \in \natural , 0 < \gamma \leq 1^- $, it suffices to take
$ g = x^k |x| ^{ \gamma}, f_n = x+\frac{1}{n} $. Clearly $ \parallel f_n - Id
\parallel _{C^r} \rightarrow 0 $ for any $r$.
\[ ( g \circ f_n)^{(k)} (x) = | x+ \frac{1}{n} |^{ \gamma} (k + \gamma)
\cdots ( \gamma + 1) \]
Hence,
\[ |(g \circ f_n - g \circ Id)^{(k)} (\frac{-1}{n})
- (g \circ f_n - g \circ Id)^{(k)} (0) |
= 2 \frac{1}{n^{ \gamma}} (k + \gamma) \cdots ( \gamma + 1) \]
so that $ \parallel g \circ f_n - g \circ Id \parallel _{C^t} \geq K > 0 $.
We deal the other cases in infinite-dimensional Banach spaces. For
$p\geq 1$ we introduce the usual space of summable sequences
\[l^p=\{{\bf x}=(x_n)_{n=1}^{\infty} \in \real^\natural/ \sum_{n=1}^{\infty}
|x_n|^p<\infty \}\]
with the norm
\[\parallel {\bf x} \parallel=(\sum_{n=1}^{\infty} |x_n|^p)^{1/p}\]
which converts it into a Banach space. We denote by $e_n$ the sequence with
all the components $0$ except the $n$-th one which takes the value $1$.
First, we take $00$. In this example there is not continuity for $s=1$ and $t=r$. This
completes (ii).
Finally if we consider $r=1$ and $h(x)=x\varphi(x-1)$ in the above
relations then $\parallel{\bf g}_{k+1}\circ{\bf f}_n-{\bf g}_{k+1}
\circ{\bf f}\parallel_{C^{k+1^-}}\geq K>0$. In this example there is not
continuity for $s=k+1$, $t=k+1^-$.
\vspace{.3cm}{\bf Remark} If $F$ is a finite dimensional space then
$\overline{f(U)}$ is a compact subset of $V$. Every continuous function on $V$
is uniformly continuous on $f(U)$. For this reason under the additional hypothesis
that $F$ is finite dimensional, we can supplement the cases in proposition
(\ref{pa2}) with
\begin{itemize}
\item[v)] $0~~