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\topmatter
\title\nofrills
Transfer operators acting on Zygmund functions
\endtitle
\author
Viviane Baladi, Yunping Jiang,
and Oscar E. Lanford III
\endauthor
\abstract
We obtain a formula for the essential spectral radius $\rho_{\text{ess}}$
of
transfer-type operators associated
with families of $C^{1+\delta}$ diffeomorphisms
of the line and Zygmund, or H\"older,
weights acting on Banach spaces of Zygmund
(respectively H\"older) functions.
In the uniformly contracting case the essential spectral radius is strictly
smaller than
the spectral radius when the weights are positive.
\endabstract
\date
March 1995
\enddate
\address
V. Baladi: ETH Zurich, CH-8092 Zurich, Switzerland\newline
\phantom{vb}
(on leave from CNRS, UMR 128, ENS Lyon, France)\newline
\phantom{vb}
Current address:
Math\'ematiques, Universit\'e de Gen\`eve, 1211 Geneva 24, Switzerland
\endaddress
\email
baladi\@sc2a.unige.ch
\endemail
\address
Y. Jiang: Department of Mathematics, Queens College\newline
\phantom{vb}
The City University of New York, Flushing, NY 11367-1597, U.S.A.
\endaddress
\email
yunqc\@qcunix.acc.qc.edu
\endemail
\address
O.E. Lanford III: ETH Zurich, CH-8092 Zurich, Switzerland\newline
\endaddress
\email
lanford\@math.ethz.ch
\endemail
\thanks \noindent Y.~Jiang is partially supported by an NSF grant (contract
DMS-9400974), and PSC-CUNY awards.
\endthanks
\endtopmatter
\heading 1. Introduction
\endheading
During the last decade, a generalised theory of Fredholm
determinants has been obtained using tools from statistical
mechanics, often in a dynamical setting. Typically, one considers
\item
{-}
a transformation $f$, with finitely or countably
many inverse branches, of a metric
space $M$ to itself,
\item
{-}
a weight $g: M \to \complex$;
\noindent and one defines the associated {\it transfer operator}
$$
\LL \varphi (z) = \sum_{f(w)=z} g(w) \varphi(w)
$$
acting on a Banach space of functions $\varphi : M \to \complex$.
Transfer operators are useful in the study of ``interesting''
invariant measures for $f$. They sometimes arise in a
surprising fashion: It has been proved that the
period-doubling renormalization spectrum is exactly the spectrum of a
suitably defined transfer operator (see e.g. Jiang-Morita-Sullivan [1992]).
Transfer operators are usually bounded but non-compact; however,
it has been possible in many cases to compute an upper bound, or
even an exact value
for the {\it essential spectral radius} $\rho_{\text{ess}}$
of $\LL$. This is
the first step towards a generalised Fredholm theory.
The second step is to introduce a {\it generalised Fredholm determinant,}
which is often closely connected to weighted {\it dynamical zeta
functions} (see Section 5).
One then shows under suitable
assumptions that the determinant is an analytic function
in a subset of the complex plane, or that the zeta function is
meromorphic in some domain, where its zeroes (respectively poles)
describe exactly the spectrum of $\LL$ outside of a disc of
radius $r \ge \rho_{\text{ess}}$.
This program has been successfully
carried out in an Axiom A framework with various degrees of smoothness
(H\"older, analytic, differentiable: see Parry-Pollicott [1990];
and Rugh [1994] for more recent developments),
for families of contractions on finite
dimensional manifolds and $C^{k+\alpha}$ smoothness,
$0 \le k \le \omega$, $0 \le \alpha \le 1$
(Ruelle [1989, 1990], Fried [1993]).
In dimension one, one may consider test functions of bounded variation
(see Ruelle [1994] and references therein, Baladi-Ruelle [1994]),
and under Markov-type assumptions also $C^{k}$ Banach spaces
(Collet-Isola [1991]).
One Banach space which had not yet been investigated in this
context
is the space $Z(I)$ of Zygmund functions on an interval or circle
$I$ (see Section 2 for definitions).
The space $Z(I)$, which has been much used
in dynamical systems in recent years, notably
in Sullivan's analysis of renormalisation
(Sullivan [1992]), is interesting not only because
$\Lambda^1 \subsetneq Z \subsetneq \Lambda^\alpha$ for all $0< \alpha <1$,
where $\Lambda^\alpha$ denotes the space of $\alpha$-H\"older functions
($\Lambda^1=\text{Lip(I)}$)
but also because
it arises in the
study of quasiconformal mappings and Teichm\"uller theory as we explain now.
Let $I$ denote the circle $\real/\integer$, and choose three points
$p_{1}0$ a generic constant (in particular we admit identities such as
$C=2C$).
\subhead Zygmund functions
\endsubhead
\smallskip
The Zygmund space $Z$ on $I$ (Zygmund [1945])
is the complex vector space of continuous
(or equivalently locally bounded) functions
$\varphi : I \to \complex$ such that
$$
Z(\varphi)=
\sup \Sb x \in I \\ t > 0: x\pm t \in I\endSb
|Z(\varphi,x,t) |< \infty \, .
$$
where
$Z(\varphi,x,t)= (\varphi(x+t)+\varphi(x-t)-2\varphi(x))/t$.
The vector space $Z$ becomes a Banach space when endowed with
the norm $\|\varphi\|_Z =\max(\sup_I |\varphi|,Z(\varphi))$.
For $0 < \alpha \le 1$,
let $\Lambda^\alpha$ denote the
space of $\alpha$-H\"older functions, i.e. functions
$\varphi : I \to \complex$ satisfying
$$
|\varphi|_{\alpha}=
\sup_{ x\ne y \in I}
{|\varphi(x)-\varphi(y)|\over |x-y|^\alpha}
< \infty \, .
$$
In particular, $\Lambda^1$ is the space of Lipschitz functions.
Each $\Lambda^\alpha$ is a Banach space for the norm
$\|\varphi\|_{\alpha}
=\max(\sup_I|\varphi|,|\varphi|_{\alpha})$;
$Z
\raise.3ex\hbox{$ \mathrel{\mathop{\kern0pt\subset}\limits_{\neq}^{}}$ }
\Lambda^\alpha$
for
$0 < \alpha < 1$; and $\Lambda^1\,
\raise.3ex\hbox{$ \mathrel{\mathop{\kern0pt
\subset}\limits_{\neq}^{}}$ } Z$.
(For a proof of the second assertion,
see e.g. de Melo-van Strien [1993, p. 293]; for an example
showing that $\Lambda^1 \neq Z$, see the remark following the
proof of Lemma 1.)
We shall also consider the Banach space $\BB$ of bounded
functions on $I$ endowed with the supremum norm.
Note that the norms
$
\|\varphi\|_{Z,\alpha}=\max(\sup_I |\varphi| ,
Z(\varphi) ,|\varphi|_{\alpha})
$
for $0 < \alpha < 1$
on $Z$ are all equivalent with
the norm $\|\cdot \|_Z$.
(Indeed, for
each $0\le\alpha < 1$ the space $Z$
is a Banach space for the norm $\|\cdot\|_{Z,\alpha}$;
the open mapping theorem may then be applied to the identity maps
$(Z,\|\cdot\|_{Z,\alpha}) \to (Z,\|\cdot\|_Z)$.)
In other words, for each $0 \le \alpha < 1$, there is a constant
$K=K(\alpha)$ such that
$$
|\varphi|_\alpha \le K(\alpha) \, (\sup|\varphi| +
Z(\varphi))\, , \quad \forall \varphi \in Z\, .
$$
\noindent
The following key lemma may
be proved by direct computation:
\proclaim{Zygmund derivation of a product}
For all $\varphi, \psi$ in $Z(I)$, $x \in I$, and $t > 0$,
$$
\eqalign
{
Z(\varphi \psi, x,t)
&=\varphi(x) Z(\psi, x,t)
+ \psi(x) Z(\varphi, x,t)\cr
&\qquad + t \cdot\Delta_+ (\varphi,x,t) \Delta_+(\psi,x,t)
+t \cdot \Delta_-(\varphi,x,t)\Delta_-(\psi,x,t) \, ,
}\tag{2.1}
$$
where
$\Delta_+(\upsilon,x,t)=(\upsilon(x+t)-\upsilon(x))/ t$
and $\Delta_-(\upsilon,x,t)=(\upsilon(x)-\upsilon(x-t))/ t$.
\endproclaim
\smallskip
\noindent The following result is also useful
(the constant $1/2$ is not optimal):
\proclaim{Skewed Zygmund bound}
For all $\varphi \in Z$, $x, y \in I$, $0 < t < 1$,
$$
\left\vert \bigl((1-t) \varphi(x) + t \varphi(y)\bigr) -
\varphi((1-t) x + t y) \right\vert \leq \frac{1}{2} Z(\varphi) \vert x-y
\vert.
$$
\endproclaim
\demo{Proof of the skewed Zygmund bound}
Fix $x$ and $y$.
There is nothing to prove
if $Z(\varphi)=0$. Otherwise, by subtracting off an affine function, making
an
affine change of variables, and multiplying by a constant, we can
reduce to the case $x=0$, $y=1$, $\varphi(0) = \varphi(1) = 0$, $Z(\varphi)=4$.
We then have
$$
\left\vert \frac{1}{2} \varphi(u) + \frac{1}{2} \varphi(v) -
\varphi(\frac{1}{2} u + \frac{1}{2} v ) \right \vert \leq \vert u - v
\vert,
$$
and we want to prove that
$
\vert \varphi(t) \vert \leq 2$ for $0 \leq t \leq 1$.
By continuity, it is enough to prove the desired bound for $t$ a dyadic
rational. We will construct recursively an increasing sequence of
bounds $\gamma_n$ such that
$
\vert \varphi(t) \vert \leq \gamma_n$
for $t$ of the
form $\frac{j}{2^n}$.
We start with $\gamma_1 = 1$. For the induction step, it is evidently
enough to consider
$$
t = \frac{2j+1}{2^{n+1}} = \frac{1}{2} \frac{j}{2^n} +
\frac{1}{2} \frac{j+1}{2^n}.
$$
By the induction hypothesis,
$
\varphi(\frac{j}{2^n}) \leq \gamma_n$ and
$\varphi(\frac{j+1}{2^n}) \leq \gamma_n$;
by the Zygmund condition
$$
\left\vert \varphi(t) - \bigl( \frac{1}{2} \varphi(\frac{j}{2^n}) +
\frac{1}{2}\varphi(\frac{j+1}{2^n})\bigr) \right\vert \leq \frac{1}{2^n}.
$$
Hence, the bound holds inductively if we set
$
\gamma_{n+1} = \gamma_n + \frac{1}{2^n}
$,
and, since $\lim_{n \to \infty} \gamma_n = 2$, the assertion
follows.\qed
\enddemo
\smallskip
\subhead The transfer operator
\endsubhead
\smallskip
The basic data entering into the definition of the transfer
operator are a dynamical system and a weight. Let $\II$ be a finite
or countable set and $0 \le \delta < 1$.
The {\it dynamical system} here is a family
of $C^{1+\delta}$
diffeomorphisms,
$f_i : I \to J_i$, for $i \in \II$, where the intervals
$J_i \subset I$
have disjoint interiors.
We assume further that
$\sup_i \|f_i'\|_{\delta} < \infty$,
in particular $\lambda := 1/\sup_{i,x} |f'_i(x)| > 0$.
The {\it weight} is a
family of functions $g_i :I \to \complex$, $i \in \II$. Such a family
$g_i$ is called {\it summably
bounded} if
$
\supp^\Sigma |g|=\sum_i \sup |g_i| < \infty
$.
A summably bounded family is called {\it summably $\Lambda^\alpha$}
if
$
|g |^\Sigma_\alpha=\sum_i |g_i|_\alpha < \infty
$
for some
$0 < \alpha \le 1$; it is called {\it summably Zygmund} if
$
Z(g )^\Sigma=\sum_i Z(g_i) < \infty
$.
\smallskip
Define formally the transfer operator
$\LL$ associated with the families $f_i$ and $g_i$, and acting
on functions $\varphi : I \to \complex$, by
$$
\LL \varphi (x)
=\sum_{i\in \II} g_i(x) \varphi(f_i(x)) \, .\tag{2.2}
$$
A typical example is when the $f_i$ are the finitely many {\it inverse
branches}
of a piecewise expanding,
piecewise surjective interval map $f$, or
the finitely many inverse branches
of a one-dimensional hyperbolic repeller, and $g_i=|f_i'|$.
\noindent The following lemma is a ``warm-up":
\proclaim{Lemma 1}
The linear operator $\LL$ is bounded
when acting on $\BB$ (respectively $\Lambda^\alpha$,
for any $0< \alpha \le 1$) if the family $g_i$ is
summably bounded (respectively summably $\Lambda^\alpha$)
and $\delta \ge 0$;
the operator $\LL$ is bounded when acting on $Z$ if the family is summably
Zygmund and $\delta > 0$.
\endproclaim
\demo{Proof of Lemma 1}
It follows immediately from the definitions that
$$
\sup_I |\LL \varphi| \le \sup_I|\varphi| \sum_{i\in \II} \sup_I|g_i|
\le \supp^\Sigma |g| \sup_I|\varphi| \, .
$$
To bound the $\alpha$-H\"older seminorm,
we use $|x-y|\ge \lambda |f_i x-f_i y|$ for all $i$ and get
$$
\eqalign
{
| \LL \varphi |_\alpha&=
\sup_{x,y\in I}
{|\sum_i g_i(x) \varphi(f_i x)-g_i(y)\varphi(f_i y)| \over |x-y|^\alpha }\cr
&\le
\sup_{x,y\in I}
{\sum_i |g_i(x) (\varphi(f_i x)-\varphi(f_i y))|+
|\varphi(f_i y)(g_i(x)-g_i(y))|
\over |x-y|^\alpha }\cr
&\le
\supp^\Sigma |g| {|\varphi|_\alpha
\over \lambda^\alpha} + |g|_\alpha^\Sigma \sup_I |\varphi|\, .
}\tag{2.3}
$$
For the Zygmund bound,
we first note that
for each $x\in I$ and $t> 0$ with $x\pm t\in I$, the Zygmund derivation
formula
yields for any $0 < \alpha < 1$:
$$
\eqalign
{
\bigl |Z(\LL \varphi,x,t) \bigr |
&=\biggl |\sum_{i\in \II} Z(g_i \cdot (\varphi \circ f_i) ,x,t) \biggr |\cr
&\le \sup{}^\Sigma|g| \sup_i \bigl |Z(\varphi \circ f_i,x,t) \bigr | +
Z^\Sigma(g) \sup |\varphi|
+ {2 \over \lambda^\alpha} |g|_{1-\alpha}^\Sigma |\varphi|_\alpha\, .
}\tag{2.4}
$$
Defining $0 < | t_i |\le t/\lambda$ for each $i \in \II$ by
$f_i(x+t)=f_i (x)+t_i$,
we observe that,
since $\delta > 0$,
there is a constant $C > 0$ such that for all $i$, and
all $x \in I$, $t > 0$ with $x\pm t\in I$
$$
\eqalign
{
|f_i(x-t) - (f_i(x) -t_i)|
&=|(f_i(x+t)-f_i(x)) -(f_i(x)-f_i(x-t))|\cr
&= |f_i'(x+u) t - f_i'(x-v) t| \le |f_i'|_\delta 2^\delta t^{1+\delta}
\le C t^{1+\delta}\, ,
}\tag{2.5}
$$
where we used $0\le u+v \le 2 t$ and $\sup_i |f'_i|_\delta< \infty$.
For each $i \in \II$, we decompose
$$
Z( \varphi \circ f_i,x,t)
={t_i \over t} Z( \varphi,f_i(x), t_i) -
{ \varphi (f_i(x)-t_i) - \varphi (f_i(x-t)) \over t} = \text{I}_i+
\text{II}_i\, .
\tag{2.6}
$$
Clearly,
$$
\sup_{i\in \II} |\text{I}_i| \le
{1\over \lambda} Z(\varphi) \, .\tag{2.7}
$$
Now, using \thetag{2.5}, we get for all $i$ with $\text{II}_i \ne 0$:
$$
\eqalign
{
|\text{II}_i|&\le
C \,
{ | \varphi (f_i(x)-t_i)- \varphi (f_i(x-t)) |
\over | f_i(x) -t_i- f_i(x-t) |^{1/(1+\delta)}}
\le C \, |\varphi|_{1/(1+\delta)} \, .\cr
}
\tag{2.8}
$$
To finish, put
\thetag{2.4}, and \thetag{2.6}-\thetag{2.8}
together, observing that for any $(1+\delta)^{-1} \le \alpha < 1$
there is a constant $K(\alpha)$ with
$|\varphi|_{1/(1+\delta)}\le |\varphi|_{\alpha}\le K(\alpha) \|\varphi\|_Z$,
and $|g|_\alpha^\Sigma \le K(\alpha) Z^\Sigma(g)$.
\qed
\enddemo
\remark{Remark}
We would like to point out that the transfer operator
$\LL$ acting on $Z$ may be unbounded
if $\delta =0$ (even for constant
weights). Indeed, it is well known that there exist Zygmund
functions $\varphi$ and $C^1$ diffeomorphisms $f$ such that
$\varphi \circ f$ is not Zygmund. For example,
let $I=[-\epsilon, \epsilon]$ be a small neighbourhood of $0$, let
$\varphi(x) = x \log|x|$ on $I$, and
let $f:I\to f(I)\subset I$ be a $C^1$ diffeomorphism
with $f(0)=0$, $f'(x)=1$ for $x \le 0$ and $f'(x)=1-1/\sqrt{|\log(x)|}$
for $x > 0$ (in particular, there is a constant $C> 0$ with
$C < f'(x) \le 1$ on $I$). To check that $\varphi\circ f$
is not Zygmund, we first show -- by straightforward computation --
that
$$
Z(\varphi \circ f,0,t) =
\biggl ({f(t)\over t} -1 \biggr ) \log(t) +
{f(t) \over t} \log {f(t)\over t} \, , \quad
\text{for $t > 0$} \, .
$$
The second term on the right goes to zero as $t \to 0^+$;
the first on the other hand is unbounded since
$$
\biggl ( {f(t) \over t} -1\biggr )
\sqrt {|\log t|} =
-{\sqrt{|\log t|}\over t} \int_0^t {ds \over \sqrt{|\log s|}}
\to -1 \, \text{ when $t\to 0^+$.}
$$
\endremark
\smallskip
\subhead The essential spectral radius of the transfer operator
\endsubhead
\smallskip
For each $n \ge 1$ and $i_\ell \in \II$, $1\le \ell\le n$,
introduce the maps
$f_{\vec \imath}^{(n)}=f_{i_n} \circ \cdots \circ f_{i_1}$,
and the weights $g_{\vec \imath}^{(n)} (x)
= g_{i_n} (f_{i_{n-1}} \cdots f_{i_1} (x))
\cdots g_{i_2} (f_{i_1} (x))\cdot g_{i_1}(x)
$. Note that for all $n \ge 1$
$$
\LL^n \varphi (x) = \sum_{\vec \imath \in \II^n}
g_{\vec \imath}^{(n)} ( x)
\varphi (f_{\vec \imath}^{(n)} x)\, .
$$
Our main result is:
\proclaim{Theorem 1}
\roster
\item
Assume that the family $g_i$ is summably Zygmund and that $\delta > 0$.
The essential spectral radius $\rho_{\text{ess}}(\LL)$
of the operator $\LL$ acting on $Z$
is equal to
$$
\rho_{\text{ess}}(\LL)=
\lim_{n \to \infty}
\left [
\sup_{x \in I}
\sum_{\vec \imath \in \II^n} |g^{(n)}_{\vec \imath} (x) |\,
|{f^{(n)}_{\vec \imath}}'(x)|
\right ]^{1/n}
$$
(in particular, the limit on the right exists).
\item
If the family $g_i$ is summably $\Lambda^\alpha$
for some $0 < \alpha \le 1$,
the essential spectral radius $\rho_{\text{ess}}(\LL)$
of the operator $\LL$ acting on $\Lambda^\alpha$
is equal to
$$
\rho_{\text{ess}}(\LL)=
\lim_{n \to \infty}
\left [
\sup_{x \in I}
\sum_{\vec \imath \in \II^n} |g^{(n)}_{\vec \imath} (x) |\,
|{f^{(n)}_{\vec \imath}}'(x)|^\alpha
\right ]^{1/n}\, .
$$
\endroster
\endproclaim
The proof of Theorem 1 is based on the following result of Nussbaum [1970],
which
holds for any bounded linear operator $\LL$ on a Banach space:
$$
\rho_{\text{ess}}(\LL) =
\lim_{n \to \infty}
\left ( \inf \{ \|\LL^n -\KK\| \mid \KK \text{ compact } \}
\right )^{1/n} \, .
$$
Indeed, using the above equality and the expression of
$\LL^n_g$ as a sum over $\II^n$, the theorem will be an immmediate
consequence of the two following lemmas:
\proclaim{Lemma 2} (Upper bound.)
There is a universal constant $C > 0$ so that, for any
family $f_i$ with $\delta > 0$ and summably Zygmund $g_i$,
$$
\inf \{ \|\LL -\KK\|_{Z} \mid \KK :Z \to Z\text{ compact } \}
\le C \cdot
\sup_{x \in I}\sum_{i\in \II} |g_i (x) | \, |f_i'(x)| \, ;
$$
and for any family $f_i$ with $\delta \ge 0$
and summably $\Lambda^\alpha$ weights
$g_i$
$$
\inf \{ \|\LL -\KK\|_\alpha \mid \KK :\Lambda^\alpha
\to \Lambda^\alpha \text{ compact } \}
\le C \cdot
\sup_{x \in I}\sum_{i \in \II} |g_i (x)| \, |f_i'(x)|^\alpha \, .
$$
\endproclaim
\proclaim{Lemma 3} (Lower bound.)
For any
family $f_i$ with $\delta > 0$ and summably Zygmund $g_i$,
$$
\inf \{ \|\LL -\KK\|_Z \mid \KK:Z \to Z\ \text{ compact } \}
\ge
\sup_{x \in I}\sum_{i \in \II} |g_i (x)| \, |f_i'(x)| \, .
$$
For any family $f_i$ with $\delta \ge 0$
and summably $\Lambda^\alpha$ weights $g_i$,
$$
\inf \{ \|\LL -\KK\|_\alpha \mid \KK:\Lambda^\alpha
\to \Lambda^\alpha \text{ compact } \}
\ge
\sup_{x \in I}\sum_{i \in \II} |g_i (x)|\, |f_i'(x)|^\alpha \, .
$$
\endproclaim
\smallskip
\subhead The essential spectral radius of restrictions of linear operators
\endsubhead
\smallskip
If the family $g_i$ is summably Zygmund and $\delta >0$,
it follows from Theorem 1 that the essential spectral radius
of $\LL$ acting on $Z(I)$ is the limit of its essential spectral
radii on $\Lambda^\alpha$ as $\alpha \to 1$.
Moreover, if the family $g_i$ is summably Lipschitz,
$\LL$ has the same essential spectral radius
when acting on $\Lambda^1$ or $Z$. Although this is hardly
surprising, we believe that
Theorem 1 \therosteritem{1} cannot be easily deduced from
Theorem 1 \therosteritem{2}, i.e., that the Zygmund result cannot be deduced
immediately
from the $\Lambda^\alpha$, $0 < \alpha \le 1$, results: The essential
spectrum of a bounded operator contains its residual spectrum, which
can be very badly behaved under restriction
(see e.g. Dowson [1978]).
In this respect, we recall the very well known example of the shift
operator acting
on the Hilbert space
$\ell^2=\{ (x_k)_{k \in \integer} \mid x_k \in \complex
\, , \sum_k |x_k|^2 < \infty \}$ by
$( T (\vec x))_j = x_{j-1}$,
whose spectrum is the unit circle, but which has the property that
the spectrum of its restriction to the closed invariant
space of sequences
$\{(x_k)_{k \in \integer} \in \ell^2 \mid x_k =0 \, , k \le 0\}$
fills the whole unit {\it disc.}
It can happen that
the essential spectral radius {\it decreases} when one lets $\LL$
act on the bigger spaces
$\Lambda^\alpha$ for $\alpha < 1$ instead of $Z$.
A simple example can be constructed as follows:
We take $I=[0,1]$ and the index set $\II$
to have one member $1$. We then take for $f_1$
an analytic diffeomorphism $I \to I$ satisfying $f_1''< 0$, and having
exactly two fixed points $0$ and $1$,
with $f_1'(0) > 1$ and $f_1'(1) < 1$.
If $g_1$ is analytic and satisfies $g_1(0)=1$
and $0 \le g_1(x)\le 1$ for all $x\in I$, then Theorem 1 yields that
the essential spectral radius of $\LL$ acting
on $Z$ or $\Lambda^1$ is $f_1'(0)> 1$, but shrinks to
$f_1'(0)^\alpha$
when $\LL$ acts on
$\Lambda^\alpha$ for $0 <\alpha < 1$.
If $\sup|f'_i|\le 1$ for all $i$ this shrinking phenomenon
is of course not possible.
\smallskip
\head 3. The upper bound
\endhead
\smallskip
To prove the upper bound we consider an
explicit sequence of compact projections. Assuming that
$I=[0,1]$ to fix ideas, define for
integers $n\ge 1$
$$
\tau_j^{(n)}= {j \over n}\, , j=0, \ldots, n\tag{3.1}
$$
and let $P^{(n)}$ be the compact operator of piecewise
affine interpolation at the $\tau_j^{(n)}$.
(I.e., $P^{(n)} \varphi$ is the unique function which is
affine on each interval $[\tau^{(n)}_{j-1}, \tau^{(n)}_j]$
and which agrees with $\varphi$ at the points $\tau^{(n)}_j$.)
We write $Q^{(n)} = 1-P^{(n)}$, where $1$ denotes the identity
operator.
For simplicity, we often drop the subscript $(n)$.
We will use the compact operators $\KK=\KK^{(n)}=
\LL -Q^{(n)} \LL Q^{(n)}$.
For each fixed $n\ge 1$,
it will be convenient to use the auxiliary seminorms
$$
\eqalign
{
|\varphi|^{(n)}_\alpha&=
\sup_{0\le j\le n-1} \, \sup_{ \tau_{j} \le x < y \le \tau_{j+1}}
{|\varphi(x)-\varphi(y)|\over |x-y|^\alpha}
\, , \, \, \text{for }\varphi \in \Lambda^\alpha(I) \, , \, 0 < \alpha \le 1 \, , \cr
Z^{(n)}(\varphi)&=
\sup_{0 \le j\le n-2} \, \sup \Sb x \in I \\ t > 0: x\pm t \in [\tau_{j},
\tau_{j+2}]\endSb
|Z(\varphi,x,t)| \, , \, \text{ for }\varphi \in Z(I) \, .
}\tag{3.2}
$$
Obviously, $|\varphi|^{(n)}_\alpha\le |\varphi|_\alpha$ and
$Z^{(n)}(\varphi)\le Z(\varphi)$.
We summarize properties of the operators $Q^{(n)}$ and the
seminorms $|\cdot|^{(n)}_\alpha$ and $Z^{(n)}(\cdot)$:
\proclaim{Sublemma 4}
For any $n \ge 1$ and $0 < \alpha \le 1$:
\roster
\item
$\sup |Q ^{(n)}\varphi| \le 2 \sup |\varphi|$ and
$\sup |Q ^{(n)}\varphi| \le (2n)^{-\alpha}|Q^{(n)} \varphi|^{(n)}_\alpha$,
for each $\varphi \in \Lambda^\alpha$.
\item
$|Q^{(n)}\varphi|^{(n)}_\alpha \le 2 |\varphi|^{(n)}_\alpha$
and
$|Q^{(n)}\varphi|_\alpha \le 2 |Q^{(n)}\varphi|^{(n)}_\alpha$,
for each $\varphi \in \Lambda^\alpha$.
\item
$
Z(Q^{(n)} \varphi) \le 4 Z^{(n)}(\varphi)
$, for each $\varphi \in Z$.
\endroster
\endproclaim
\demo{Proof of Sublemma 4}
Clearly, $\sup|P \varphi|
\le \sup |\varphi|$ which yields the first bound
by the definition of $Q$. The other
claim is immediate too, since $Q\varphi$ vanishes at the
$\tau_j$ and any point $x$ is within distance at most $1/(2n)$
of some $\tau_j$.
To prove the first bound for the $\alpha$-H\"older seminorm
it suffices to control $P$.
Consider a pair of points $x < y$ belonging to
the same interval $[\tau_{j-1}, \tau_j]$. Then
$(P \varphi(y) - P\varphi(x))/(y-x)
=(\varphi(\tau_j)-\varphi(\tau_{j-1}))/(\tau_j -\tau_{j-1})$.
Therefore
$$
{|P \varphi(y) - P\varphi(x)|\over |y-x|^\alpha}
= {|\varphi(\tau_j)-\varphi(\tau_{j-1})|\over |\tau_j -\tau_{j-1}|^\alpha}
\cdot \left (
|y-x| \over |\tau_j -\tau_{j-1}| \right ) ^{1-\alpha}
\le |\varphi|^{(n)}_\alpha \, . \tag3.3
$$
To prove the second bound, write $\psi = Q\varphi$ and consider
$x 0$,
$$
0 \leq \frac{|x+t| - |x|}{t} - \frac{\vert x \vert - \vert x-t
\vert}{t} \leq 1 - (-1) = 2,
$$
so that
$
Z(\vert \,.\, \vert) = 2
$, proving \thetag{3.4}.
We now show that
$Z(Q^{(n)}\varphi) \le 4 Z^{(n)} (\varphi)$.
Recall that $Z(Q^{(n)}\varphi)$ is defined as the supremum of
$\vert Z(Q^{(n)} \varphi, x, t)\vert$ over an appropriate set of pairs $x$,
$t$; $Z^{(n)}(Q^{(n)}\varphi)$ as the supremum of the same quantity over
the set of pairs such that $x \pm t$ lie in the union of some pair of
successive subintervals. By \thetag{3.4},
this latter supremum can be majorized by
$
Z^{(n)}(P^{(n)}\varphi) + Z^{(n)}(\varphi) \leq 2 Z^{(n)}(\varphi)
$,
so the asserted bound holds when $x \pm t$ lie in the union
of a pair of successive intervals.
If, on the other hand, $x \pm t$ do not lie in the union of two
successive subintervals, then $\vert t \vert$ must be $> 1/2n$. By the skewed
Zygmund bound -- and the fact that $Q^{(n)}(\varphi)$ vanishes at the division
points --
$$
|Q^{(n)}(\varphi)(s)| \leq \frac{1}{2} Z^{(n)}(\varphi)
\frac{1}{n}\quad\text{for all relevant $s$,}
$$
so we can estimate
$$
\vert Z(Q^{(n)}\varphi, x, t) \vert \leq 4 \cdot \frac{1}{2} Z^{(n)}(\varphi)
\frac{1}{n} \cdot \frac{1}{\vert t \vert} \leq 4 Z^{(n)}(\varphi),
$$
using
$
\vert t \vert \geq 1/(2n)
$.
Thus, the asserted bound also holds when $x \pm t$ do not lie in the
union of two successive subintervals.\qed
\enddemo
For each fixed $n \ge 1$ and each $0 < \alpha \le 1$,
we define
$$
\beta^{(n)}_\alpha =
\sup_{0\le j\le n-1}
\sup_{x,y \in [\tau^{(n)}_{j}, \tau^{(n)}_{j+1}]}\sum_{i \in \II} |g_i (x) |
\, |f_i'(y)|^\alpha \, .
$$
For $0 < \alpha \le 1$,
and large enough $n$, $\beta^{(n)}_\alpha$ is
arbitrarily close to
$$
\sup_{x \in I}\sum_{i \in \II} |g_i (x) |\,
|f_i'(x)|^\alpha\, .
$$
The next sublemma shows the usefulness of the seminorms
$|\cdot|^{(n)}_\alpha$, $Z^{(n)}(\cdot)$:
\proclaim{Sublemma 5}
If $g_i$ is summably $\Lambda^\alpha$ for $0 < \alpha \le 1$,
then for each $n \ge 1$ and $\varphi \in \Lambda^\alpha$
$$
| \LL \varphi|^{(n)}_\alpha \le |g|^\Sigma_\alpha \sup|\varphi|
+\beta^{(n)}_\alpha |\varphi|_\alpha \, .
$$
If $g_i$ is summably Zygmund
and $\delta > 0$, there are constants $K > 0$ and
$\epsilon > 0$,
depending only on the families $f_i$ and $g_i$, such that for
any $n \ge 1$, and $\varphi \in Z$
$$
Z^{(2n)}(\LL \varphi) \le K \sup |\varphi|
+ \bigl ( \beta^{(n)}_1 + {K \over n^{\epsilon}} \bigr ) Z(\varphi)
\, .
$$
\endproclaim
\demo{Proof of Sublemma 5}
We first prove the bound on the $\Lambda^\alpha$ seminorm
by refining \thetag{2.3}. Let
$\varphi \in \Lambda^\alpha$ and
$\tau_{j-1} \le x < y \le \tau_j$. Then there are points $z_i \in [x,y]$ with
$$
\eqalign
{
| \LL \varphi (y) - \LL \varphi(x)|
&\le \sum_{i\in \II} \biggl ( |g_i(y)-g_i(x)| \, |\varphi(f_i(y))|
+ |g_i(x)| \, |\varphi(f_i(y))-\varphi(f_i(x))| \biggr ) \cr
&\le
|g|^\Sigma_\alpha \sup |\varphi| \, |x-y|^\alpha
+ \sum_{i\in \II} |g_i(x)| \, |\varphi|_\alpha |f_i(y)-f_i(x)|^\alpha \cr
&=
\biggl (
|g|^\Sigma_\alpha \sup |\varphi|+ \sum_{i \in \II} |g_i(x)|
\, |f_i'(z_i)|^\alpha |\varphi|_\alpha
\biggr ) |x-y|^\alpha\cr
&\le
\biggl (
|g|^\Sigma_\alpha \sup |\varphi|+ \beta^{(n)}_\alpha |\varphi|_\alpha
\biggr ) |x-y|^\alpha \, , \cr
}
$$
as claimed.
To prove the Zygmund bound, we fix
$0<\alpha < 1$ and
consider $x$, $x\pm t$ in some $[\tau^{(2n)}_j, \tau^{(2n)}_{j+2}]$.
We first rewrite \thetag{2.4} more carefully
$$
\bigl |Z(\LL \varphi,x,t) \bigr |
\le \sum_{i\in \II} |g_i(x)| \, |Z(\varphi \circ f_i,x,t) |
+ Z^\Sigma(g) \sup |\varphi|
+ {2 \over \lambda^\alpha} \left ( {1\over n} \right )^\epsilon
|g|_{1-\alpha+\epsilon}^\Sigma |\varphi|_\alpha\, ,\tag{3.5}
$$
where $0<\epsilon<\delta$ is such that $1-\alpha+\epsilon < 1$, and we used
$t < 1/n$.
To bound the first term in the right-hand-side
of \thetag{3.5}, we may use
the decomposition \thetag{2.6}
of $Z(\varphi \circ f_i,x,t)$ into $\text{I}_i+\text{II}_i$.
Then, by definition of
the $t_i$, there are points
$z_i \in [x,x+t]$ so that
$$
\text{I}_i = f_i'(z_i) Z(\varphi, f_i(x),t_i) \, .\tag{3.6}
$$
Using again $t<1/n$ we may rewrite \thetag{2.8} as
$$
|\text{II}_i| \le C\left ({1\over n}\right )^{\epsilon}
|\varphi|_{{1+\epsilon\over 1+\delta}} \, .\tag{3.7}
$$
Setting $\alpha=(1+\epsilon)/(1+\delta)<1$,
the bounds \thetag{3.5}--\thetag{3.7}
yield a constant $C > 0$, depending only
on the $f_i$, with
$$
\eqalign
{
Z^{(n)}(\LL \varphi) &\le
Z^\Sigma(g) \sup |\varphi| + \beta^{(n)}_1 Z(\varphi) \cr
&\qquad+ \left ({1\over n} \right ) ^{\epsilon}
\bigl ( {2\over \lambda^{1+\epsilon\over 1+\delta} }
|g|^\Sigma_{\delta\cdot {1+\epsilon\over 1+\delta}}
+ C\sup{}^\Sigma|g| \bigr ) |\varphi|_{1+\epsilon\over 1+\delta}
\, ,
}
$$
To finish the proof, we
proceed as in Lemma 1 to bound the $\Lambda^\alpha$ seminorms. \qed
\enddemo
\demo{Proof of Lemma 2}
It suffices to show that there is a universal constant $C>0$
so that for each
$n\ge1$
$$
\limsup_{n \to \infty}
\| Q^{(n)} \LL Q^{(n)} \|_{\alpha}
\le C \sup_{x \in I}\sum_{i \in \II} |g_i (x) |\, |f_i'(x)|^\alpha \, ,
$$
when the $g_i$ are summably $\Lambda^\alpha$ and $\delta \ge 0$, and
$$
\limsup_{n \to \infty}
\| Q^{(2n)} \LL Q^{(2n)} \|_{Z}
\le C \sup_{x \in I}\sum_{i \in \II} |g_i (x) | \, |f_i'(x)| \,,
$$
when the $g_i$ are summably Zygmund and $\delta > 0$.
Applying Sublemma 4, we get
for each $\varphi \in \Lambda^\alpha$, $n \ge 1$:
$$
\eqalign
{
\sup | Q^{(n)} \LL Q^{(n)} \varphi|
&\le C\sup |\LL Q^{(n)} \varphi |
\le C \sum_{i\in \II} \sup|g_i|\, \sup |Q^{(n)} \varphi |\cr
&\le C \sup{}^\Sigma |g| \, {|\varphi |_\alpha
\over( 2n)^\alpha} \, .
}
$$
Applying again Sublemma 4, and also Sublemma 5, we get for
any $\varphi \in \Lambda^\alpha$, $n \ge 1$:
$$
\eqalign
{
| Q^{(n)} \LL Q^{(n)} \varphi|_\alpha
&\le C |\LL Q^{(n)} \varphi|^{(n)}_\alpha
\le C \cdot \bigl ( |g|^\Sigma_\alpha \sup|Q^{(n)} \varphi|+
\beta_\alpha^{(n)} |Q^{(n)} \varphi|_\alpha
\bigr )\cr
&\le C \cdot
\bigl ( |g|^\Sigma_\alpha {|\varphi|_\alpha \over( 2n)^\alpha}+
\beta_\alpha^{(n)} |\varphi|_\alpha
\bigr ) \, .\cr
}
$$
Finally, with Sublemmas 4 and 5, we obtain for each $\varphi \in Z(I)$,
$0 < \alpha < 1$, and $n \ge 1$:
$$
\eqalign
{
Z( Q^{(2n)} \LL Q^{(2n)} \varphi)
&\le C Z^{(2n)}(\LL Q^{(2n)} \varphi)
\le
C \bigl ( K \sup |Q^{(2n)}\varphi|
+ ( \beta^{(n)}_1 + {K \over n^{\epsilon}} ) Z(Q^{(2n)}\varphi)\bigr ) \cr
&\le C \cdot
\bigl ( { K
\over n^{\alpha } } |\varphi|_\alpha
+(\beta^{(n)}_1+{K\over n^{\epsilon} }) Z(\varphi)
\bigr ) \, ,
}
$$
where $C>0$ is universal and $ K>0$, $\epsilon > 0$
depend on the $f_i$ and $g_i$ (but not on $n$). \qed
\enddemo
\smallskip
\head 4. The lower bound
\endhead
The idea for the argument yielding the lower bound on the Banach
spaces $\Lambda^\alpha$ ($0 < \alpha \le 1$)
is originally due to A.~Davies (Lanford [1992]). The Zygmund
case can be treated similarly as will be shown now.
\demo{Proof of Lemma 3}
To prove the Zygmund
claim, we introduce the continuous function
$$
\beta_1 (x)=
\sum_{i \in \II} |g_i (x) | \, |f_i'(x)| \, .
$$
Writing $\bar \beta_1=
\sup_{x \in I} \beta_1(x)$,
the first assertion of Lemma 3 is that the infimum
of $\| \LL -\KK\|_Z$ for $\KK$ compact is not less than
$\bar \beta_1$. Fix $\epsilon >0$
small. We may assume that $\II$ is finite,
since otherwise replacement of $\II$ by a large finite subset of $\II$
in the definition of $\beta_1(x)$
yields a supremum arbitrarily close to $\bar\beta_1$.
The strategy is now to construct
an infinite-dimensional subspace $\chi_\epsilon
\subset Z(I)$ (with, in fact, $\chi_\epsilon \subset \Lambda^1$)
such that
$
\|\LL \varphi \|_Z \ge (\bar \beta_1 -\epsilon) \|\varphi\|_Z
$
for each $\varphi \in \chi_\epsilon$.
Then, if $\KK$ is a compact operator on $Z(I)$,
there is a function $\varphi \in \chi_\epsilon$ with $\|\varphi\|_Z=1$
and such that $\|\KK \varphi\|_Z \le \epsilon$,
and hence such that
$\| (\LL -\KK) \varphi \|_Z \ge (\bar \beta_1 -2\epsilon)$.
Therefore
the norm of $\LL -\KK$ cannot be less than $\bar \beta_1 -
3\epsilon$.
The construction of these subspaces goes as follows: We take
a point $x_\infty=x_\infty$
where $\beta_1(x_\infty)=\bar \beta_1$
and choose -- with some care -- a sequence $x_1$, $x_2, \ldots$
of distinct points in $I$ converging to $x_\infty$. We then construct
a sequence of functions $\psi_1$, $\psi_2, \ldots$ in
$\Lambda^1(I)$ such that
\roster
\item "(P1)"
$\| a_1 \psi_1 + \cdots + a_N \psi_N \|_Z =
\max_j \{ |a_j|\}$ for any $N\ge 1$, and complex numbers $a_1, \ldots, a_N$.
In particular $\| \psi_j\|_Z=1$ for every $j$;
\item "(P2)"
$\limsup_{t \to 0}
|Z(\LL \psi_j, x_j, t)| = \beta_1(x_j)
\rightarrow \bar \beta_1$
as $j \to \infty$;
\item "(P3)"
$\LL \psi_j$ vanishes on a neighbourhood of $x_\ell$
for all $j\ne \ell$.
\endroster
>From (P2) and (P3) we get
$
\| \LL (a_1 \psi_1 + \cdots + a_N \psi_N) \|_Z \ge
\max_{1 \le j\le N} \{ \beta_1(x_j) |a_j|\}
$,
and hence, using (P1), we get for any $\varphi$ in the
linear span of $\psi_k$, $\psi_{k+1}, \cdots$
$$
\| \LL \varphi \|_Z \ge \inf_{j \ge k}
\{ \beta_1 (x_j) \} \| \varphi \|_Z \, .
$$
Thus, we can take $\chi_\epsilon$ to be the closed linear
span of the $\psi_j$'s with $j \ge k$ for any
sufficiently large $k=k(\epsilon)$.
The problem is therefore reduced to constructing $(x_j)$, $(\psi_j)$ so that
(P1), (P2) and (P3) hold.
We first specify how to choose the $x_j$'s.
For $x_\infty$ as defined above, we choose inductively a sequence of $x_j$'s
converging to, but distinct from, $x_\infty$, assuming furthermore
that the $f_i(x_j)$, for $i \in \II$ and $j \ge 1$, are distinct from
each other and from the $f_i(x_\infty)$. Suppose $x_1$, $\ldots$,
$x_k$ have been chosen so that the $f_i(x_j)$ for $1 \le j \le k$
are distinct from each other and from the $f_i(x_\infty)$.
We then choose a point $x_{k+1}'$ near enough to $x_\infty$ so that
each $f_i(x_{k+1}')$ is nearer to its
one or two neighbours in the set of $\{ f_\ell (x_\infty) \}$
than {\it any} previous $f_\ell(x_j)$ but still not in this set.
(We use here that no $f_i$ can be locally
constant.) Then, by moving $x_{k+1}'$ a little --
and using the fact that no two $f_i$s coincide on any non-trivial
interval -- we find $x_{k+1}$ so that the $f_i(x_{k+1})$ are distinct
from each other, but so that the preceding ``inequalities''
still hold. Constructed in this way, the $f_i(x_j)$
for $i \in \II$ and $j \ge 1$ are all different
and no $f_i(x_j)$ is an accumulation point of the others.
Now, let $\phi \in \Lambda^1(]-1,1[)$ be of Zygmund norm one, with compact
support, and such that
$$
\phi(t)=|t|/2 \quad \text{for small $t$.}\tag{4.1}
$$
For any
$\gamma$ between $0$ and $1$, the rescaled function
$\gamma \phi(t/\gamma)$ has the same properties, and by taking
$\gamma$ small we can make its support and supremum norm as small
as we like. We simply construct $\psi_j$ as a sum of functions
$\psi_{i,j}$, for
$i \in \II$, each of which is a rescaled $\phi$ translated to
$f_i(x_j)$ (up to a complex phase), i.e., has the form
$$
\psi_{i,j} (x) = \omega_{i,j} \cdot \gamma_{i,j}\cdot
\phi( (x-f_i(x_j))/\gamma_{i,j})
$$
where $|\omega_{i,j}|=1$ will be chosen later,
and $\gamma_{i,j}>0$ is such that the support
of $\psi_{i,j}$ is a subset of the interior of $f_i(I)$,
and may be reduced further in Sublemma 6 below.
Now $\LL \psi_j(x)$ is non-zero only if some $f_i(x)$ is in
the support of some $\psi_{k,j}$. Since we can make the
supports of the $\psi_{k,j}$ disjoint by making the $\gamma_{k,j}$
small enough, for any $\ell\ne j$
there is a neighbourhood of $x_\ell$ on which no
$f_i(x)$ is in the support of any $\psi_{k,j}$. Thus, assertion (P3)
holds.
We next check that by making the $\gamma_{i,j}$
sufficiently small we can guarantee that (P1) is satisfied. To carry
out the verification it is convenient to relabel our objects:
We label the pairs $(i,j)$, $i \in \II$,
$j \ge 1$, with a positive integer $m$ and we
write $\xi_m = f_i(x_j)$. It suffices to prove the following Sublemma:
\enddemo
\proclaim{Sublemma 6}
Let $\xi_m$, $m \ge 1$, be distinct points in $I$ such that no
$\xi_m$ is an accumulation point of the others,
and let $\gamma_m$ be a sequence of positive numbers. For $\phi$
as defined in \thetag{4.1}, we set $\phi_m(x) = \omega_m \gamma_m
\phi((x-\xi_m)/\gamma_m)$ for arbitrary $|\omega_m|=1$.
If the $\gamma_m$'s are small enough, then, for any $N\ge 1$, and
any $b_1, \ldots, b_N$,
$$
\| b_1 \phi_1 + \cdots + b_N \phi_N \|_Z = b_{\text{max}} =
\max \{ |b_m|
\mid 1 \le m \le N \} \, .
\tag{4.2}
$$
\endproclaim
\demo{Proof of Sublemma 6}
We define $\eta =b_1 \phi_1 + \cdots + b_N \phi_N$ and set
$
d_m = \inf \{ |\xi_m - \xi_{m'}| \mid m \ne m' \} >0
$.
We claim that it suffices to take $\gamma_m$ small enough
so that
$$
\phi_m(x) \text{ vanishes for } |x-\xi_m| \ge d_m /4
\text { and } \sup|\phi_m| < d_m/8 \tag{4.3}
$$
to get \thetag{4.2} to hold.
Half of \thetag{4.2} is immediate: If we take $x=\xi_m$
and $t>0$ very small, we have (since the supports
of the $\phi_m$ are disjoint)
$$
\eta(x+t)+\eta(x-t) -2\eta(x)=\eta(\xi_m+t)+\eta(\xi_m-t)
= \omega_m b_m \cdot t \, ,
$$
so that $\|\eta\|_Z \ge Z(\eta) \ge |b_m|$ for each $m$.
To prove the opposite inequality, we first observe that the disjointness
of the supports and the fact that $\sup|\phi_m | \le 1$ imply
$\sup|\eta| \le b_{\text{max}}$. To prove the corresponding
estimate for $Z(\eta)$ we consider general $x\in I$ and $t > 0$
with $x \pm t \in I$. We need to show that
$$
|\eta(x+t)+\eta(x-t)-2\eta(x)| \le t b_{\text{max}} \, .
$$
Since $Z(\phi_m)=1$ for all $m$ this is immediate unless $\{x,
x+ t, x-t\}$ intersects the supports of at least
two {\it different} $\phi_m$'s.
Assume thus that $x$ is in the support of $\phi_{m_1}$,
$x-t$ in the support of $\phi_{m_0}$, and $x+t$ in the support
of $\phi_{m_2}$, where the set $\{ m_0, m_1, m_2\}$ is
not a singleton. We leave to the reader the easier
case where this set has only two elements and suppose
that $m_0 \ne m_1 \ne m_2$. By our assumption \thetag{4.3}
$$
\eqalign
{
|x-t - \xi_{m_0}| &\le d_{m_0}/4 \le |\xi_{m_0} - \xi_{m_1}|/4
\quad \text{and} \quad
|x- \xi_{m_1}| \le d_{m_1}/4 \le |\xi_{m_1} - \xi_{m_0}|/4 \, .\cr
}
$$
Therefore
$$
t= |x - t - x| \ge |\xi_{m_0} -\xi_{m_1}|/2 \, . \tag{4.4}
$$
Similarly, we get $t \ge |\xi_{m_2}-\xi_{m_1}|/2$.
On the other hand
$$
\eqalign
{
|\eta(x)| &= |b_{k_1}| |\phi_{m_1} (x)|
\le b_{\text{max}} \sup|\phi_{m_1}|
\le b_{\text{max}} d_{m_1} /8 \le
b_{\text{max}} |\xi_{m_0}-\xi_{m_1}| /8\, .
}
$$
Analogously $|\eta(x+ t)|\le b_{\text{max}}
|\xi_{m_2} - \xi_{m_1}|/8$, and $|\eta(x- t)|\le b_{\text{max}}
|\xi_{m_0} - \xi_{m_1}|/8$. Finally,
recalling \thetag{4.4},
$$
|\eta(x\pm t) -\eta(x)| \le |\eta(x\pm t)| +|\eta(x)|
\le {b_{\text{max}}\over 4}
\max(|\xi_{m_0} - \xi_{m_1}|,|\xi_{m_2} - \xi_{m_1}|)
\le b_{\text{max}} {t\over 2 }\, .
$$
Since
$$
|\eta(x+t)+\eta(x-t)-2\eta(x)| \le |\eta(x+t) - \eta(x)| +
|\eta(x-t) - \eta(x)| \, ,
$$
this ends the proof of Sublemma 6 and therefore of assertion (P1).\qed
\smallskip
Going back to
the notation with pairs $i\in \II$ and $j\ge 1$,
it remains to choose the $\omega_{i,j}$ so that (P2) holds.
To do this, we fix $j$ and
we decompose as before, using \thetag{2.1}, \thetag{2.6},
and the property of the support of $\psi_{i,j}$:
$$
\eqalign
{
t Z(\LL& \psi_j, x_j, t) \cr
&=\sum_{i\in \II} t_i g_i (x_j) Z(\psi_j,f_i (x_j),t_i)
-\sum_{i \in \II} g_i (x_j) (\psi_j(f_i(x_j)-t_i)-\psi_j(f_i(x_j-t))\cr
&\quad +\sum_{i\in \II} (g_i(x_j+t) -g_i(x_j) ) (\psi_j(f_i(x_j+t)))\cr
&\quad+\sum_{i\in \II} (g_i(x_j) -g_i(x_j-t) ) (-\psi_j(f_i(x_j-t))) \cr
&= Z_a - Z_b + Z_c + Z_d \, ,\cr
}
$$
(we used the $t_i=t_{i,j}$
defined by $f_i(x_j+t)=f_i (x_j)+t_i$, and the fact that
$\psi_j(f_i (x_j))=0$ for all $i$).
Now, since each $\psi_j \in \Lambda^1$ (use e.g. $\# \II < \infty$),
we have $|\psi_j(f_i(x_j\pm t))|\le |\psi_j|_1 |f_i(x_j \pm t)-f_i(x_j)|
\le |\psi_j|_1 |f'_i(u)| t$, for some $u$, by construction.
Therefore, using $|g|^\Sigma_\alpha<\infty$, for any $0 < \alpha < 1$,
we get $Z_c+Z_d=o(t)$ when $t\to 0$.
Since $\sup^\Sigma|g|< \infty$, we get,
using again $\psi_j \in \Lambda^1$, that
$Z_b=o(t)$ by applying \thetag{2.5} once more (recall that $\delta>0$).
By definition,
$Z(\psi_j,f_i (x_j),u)=\omega_{i,j} + o(u)$ for $u \to 0$,
uniformly in $i \in \II$ (using $\# \II < \infty$). Finally,
$t_i/t =f_i'(x_j+u)=f'_i(x_j)+O(|u|^\delta)$ for some $|u|\le t$.
Therefore, if we choose the complex phases $\omega_{i,j}$ properly, we find:
$$
t Z(\LL \psi_j, x_j, t)=
t \sum_{i\in \II} \omega_{i,j} g_i(x_j)f_i'(x_j) + o(t)
= t \sum_{i\in \II} |g_i(x_j)f_i'(x_j)| + o(t)\, , \quad t\to 0\, ,
$$
which gives assertion (P2) above, and thus the first claim of Lemma 3.
\smallskip
A simple modification of the construction in the proof
yields the second claim of Lemma 3:
Instead of $\phi(t) =|t|/2$ for small $t$, we take $\phi(t)=
|t|^\alpha$ (assuming that $\|\phi\|_\alpha=1$)
and we rescale by $\gamma^\alpha(\phi(t/\gamma))$,
i.e., we have $\phi_m(x) = \omega_m (\gamma_m)^\alpha
\phi((x-\xi_m)/\gamma_m)$,
where
$|\omega_m|=1$ and the points $\xi_m$ are chosen exactly as above.
The scalars $\gamma_m$ are then chosen similarly
as in Sublemma 6, condition \thetag{4.3} being naturally replaced by
$$
\phi_m(x) \text{ vanishes for } |x-\xi_m| \ge d_m /4
\text { and } \sup|\phi_m| < (d_m)^\alpha/4 \, .
$$
A slight variation on the above arguments (replacing the Zygmund
seminorm by $|\cdot |_\alpha$, and using the decomposition
\thetag{2.3} as a starting point) then yields
\roster
\item "(P1${}_\alpha$)"
$\| a_1 \psi_1 + \cdots + a_N \psi_N \|_\alpha =
\max_j \{ |a_j|\}$ for any $N$, $a_1, \ldots a_N$.
In particular $\| \psi_j\|_\alpha=1$ for every $i$;
\item "(P2${}_\alpha$)"
$\limsup_{x\to x_j}
|\LL\psi_j(x)- \LL\psi_j(x_j)|/|x_j-x|^\alpha=
\beta^{\II}_\alpha(x_j) \rightarrow \bar \beta^{\II}_\alpha$
as $j \to \infty$
(we set
$\beta^{\II}_\alpha (x)=
\sum_{i \in \II} |g_i (x) | \, |f_i'(x)|^\alpha$
and $\bar \beta^{\II}_\alpha=
\sup_{x \in I} \beta^{\II}_\alpha(x)$);
\item "(P3${}_\alpha$)"
$\LL \psi_j$ vanishes on a neighbourhood of $x_\ell$
for all $j\ne \ell$.\qed
\endroster
\enddemo
\smallskip
\head 5. The spectral radius and two conjectures
\endhead
In this section, it is convenient to use the notation $\LL_g$ instead of $\LL$.
We have the following result
(the statements for $\Lambda^\alpha$ and $\BB$ were obtained previously by
Ruelle [1989, 1990]):
\proclaim{Theorem 2}
If the family $g_i$ is summably bounded, the spectral radius of
$\LL_{|g|}$ acting on $\BB$ is equal to
$$
e^P:=\lim_{n \to \infty} \left (
\sup_{x \in I} \sum_{\vec \imath \in \II^n}
|g_{\vec \imath}^{(n)} (x)| \right )^{1/n} \, ,
$$
and the spectral radius of $\LL_g$ on $\BB$ is bounded above
by $e^P$.
If the $g_i$ are summably Zygmund
and $\delta > 0$, respectively $\Lambda^\alpha$ and $\delta \ge 0$,
the spectral radius of
$\LL_{|g|}$ acting
on $Z$ (respectively $\Lambda^\alpha$) is equal to
$\max(e^P, \rho_{\text{ess}} (\LL_{|g|}))$, and
the spectral radius of $\LL_g$ acting on
$Z$, respectively $\Lambda^\alpha$,
is bounded above by $\max(e^P, \rho_{\text{ess}} (\LL_{|g|}))$.
\endproclaim
Under the additional assumption that
$\lambda > 1$, Theorem 2 together with
Theorem 1 yields
that $\rho_{\text{ess}} (\LL_{|g|})$ is strictly smaller than the spectral
radius of $\LL_{|g|}$ (except when both vanish) acting on $Z$
(respectively $\Lambda^\alpha$).
\demo {Proof} Since
$ e^{n(P-\epsilon)}\le \sup \LL_{|g|}^n \psi \le e^{n(P+\epsilon)}$
for $\psi\equiv 1$, $\epsilon > 0$, and $n\ge n(\epsilon)$,
the proof of Theorem 2
for the Banach space $\BB$ is an immediate
consequence of the spectral radius formula together with
the easy inequality
$
\sup |\LL^n_g \varphi| \le \sup |\varphi | \sup \LL^n_{|g|} \psi
$, for all $\varphi \in \BB$.
For the other Banach spaces, use the definition of the
essential spectral radius.\qed
\enddemo
\smallskip
\subhead Maximal eigenfunctions, zeta functions, and two conjectures
\endsubhead
\smallskip
In this subsection, we assume throughout that $\lambda > 1$.
Consider $\LL_{|g|}$ acting on $\Lambda^\alpha$:
When our family $f_i$ consists of the
finitely many inverse branches of a (mixing)
map $f:I\to I$, it is known that $e^P$ is the only point in the spectrum
of modulus $e^P$, that it is a simple
eigenvalue, and that
$\LL_{|g|}$ admits a positive maximal eigenfunction $\varphi$
(i.e., $\LL_{|g|} \varphi =e^P \varphi$).
Finally
$P$ is the topological pressure of $\log|g|$ and $f$.
For all these results, and a theory
of equilibrium states, see Ruelle [1989], where it was
proven that the essential spectral radius of $\LL_g$
acting on $\Lambda^\alpha$ is not bigger than $e^P/\lambda^\alpha$,
a result which follows from our Theorem 1 \therosteritem{2}.
By Theorem 1 \therosteritem{1},
the essential spectral radius of $\LL_{|g|}$ acting on
$Z$ is smaller than $e^P/\lambda< e^P$. Since each
eigenfunction of $\LL_{|g|}$
in $Z$ is also an eigenfunction in $\Lambda^\alpha$, the eigenvalue
$e^P$ is the unique point in the spectrum with modulus
$e^P$, and it is simple with a positive eigenfunction
$\varphi \in Z$.
The case of countably many branches can be treated
similarly.
\smallskip
In Ruelle [1989, 1990] and Fried [1993], zeta functions
associated with $\Lambda^\alpha$ (for $0 < \alpha \le 1$) systems of finitely
or countably many branches $f_i$ and weights
$g_i$ were studied
(in a slightly different setting -- in particular the dimension
was not limited to one). In our case, the zeta function is defined by
$$
\zeta_g(z) = \sum_{n \ge 1} {z^n \over n}
\sum \Sb \vec \imath \in \II^n \\
x \, :\, f^{(n)}_{\vec \imath} (x) = x
\endSb
\prod_{k=0}^{n-1} g_{i_{k+1}} (f_{i_{k}} \cdots f_{i_1}) (x)\, .
$$
The poles $\omega$ of $\zeta_g(z)$ in the disc of radius
$\lambda^{\alpha} e^{-P}$ (where the function was shown
to be meromorphic) were proved to be in bijection with the
eigenvalues $\nu=1/\omega$ of $\LL_g$ acting on $\Lambda^\alpha$ of modulus
$> e^P/\lambda^{\alpha}$.
For summably $\Lambda^\alpha$
weights $g$, we {\it conjecture}
that $\zeta_g(z)$ is meromorphic
in the disc of radius
$\rho_{\text{ess}}^{-1}(\LL_g)$, where $\rho_{\text{ess}}(\LL_g)$
is given by Theorem 1 \therosteritem{2},
and that its poles there are the inverses of
the $\Lambda^\alpha$-eigenvalues
of $\LL_g$ of modulus $> \rho_{\text{ess}}(\LL_g)$.
Also, if $g$ is summably
Zygmund and $\delta > 0$, we {\it conjecture} that
$\zeta_g(z)$ is meromorphic
in the disc of radius $\rho_{\text{ess}}^{-1}(\LL_g)$
for $\LL_g$ acting on $Z(I)$, and that its poles there
are in bijection $\omega=1/\nu$ with the eigenvalues of
$\LL_g: Z(I)\to Z(I)$ of modulus
$> \rho_{\text{ess}}(\LL_g)$.
The $\Lambda^\alpha$ conjectures do not immediately
imply the Zygmund one since $Z$ is a strict
subset of $\cap_{\alpha < 1} \Lambda^\alpha$.
The proof of these two conjectures would complete
the second part of the program described in the introduction.
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\enddocument