% 12 March 1996 \documentstyle[12pt]{article} \setlength\textwidth{6in} \setlength\oddsidemargin{0.25in} \topmargin -20mm \footskip 12mm %\textwidth 38pc \textheight 56pc \def\Bbb#1{{\mathchoice{\mbox{\bf #1}}{\mbox{\bf #1}}% {\mbox{$\scriptstyle \bf #1$}}{\mbox{$\scriptscriptstyle \bf #1$}}}} \def\N{\Bbb N} \def\R{\Bbb R} \def\C{\Bbb C} \def\D{\Bbb D} \def\Z{\Bbb Z} \def\T{\Bbb T} \def\Q{\Bbb Q} \def\E{\Bbb E} \def\O{\Omega} \def\sgn{{\rm sgn}} \def\supp{{\rm supp}} \def\cH{{\cal H}} \def\e{\epsilon} \def\cA{{\cal A}} \def\cE{{\cal E}} \def\cL{{\cal L}} \def\cI{{\cal I}} \def\cB{{\cal B}} \def\cM{{\cal M}} \def\cT{{\cal T}} \def\Linfg{L^\infty (G)} \def\Hinfg{H^\infty (G)} \def\spec{{\rm spec}} \begin{document} \title{A transference theorem for ergodic $H^1$} \author{Nakhl\'e Asmar and Stephen Montgomery--Smith \\ Department of Mathematics\\ University of Missouri--Columbia\\ Columbia, Missouri 65211 U.\ S.\ A. } \date{} \maketitle % % % % % % % % %\begin{abstract} %\baselineskip=10 pt %\end{abstract} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Introduction} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% In this paper, we extend the basic transference theorem for convolution operators on $L^p$ spaces of Coifman and Weiss \cite{cw1} to $H^1$ spaces. For clarity's sake, we start by recalling the Coifman-Weiss transference theorem for a single convolution operator. Suppose that $k\in L^1(G)$, where $G$ is a locally compact abelian group, and let $N_p(k)$ denote the norm of the convolution operator $f\mapsto k*f$, where $f\in L^p(G,\lambda)$, and where $\lambda$ is a fixed Haar measure on $G$. Suppose that $R=\{R_u\}_{u\in G}$ is a strongly continuous, uniformly bounded representation of $G$ acting on a general Lebesgue space $L^p(\cM,\mu)=X_p$ where $1\leq p<\infty$. Let $c_p$ be a positive constant such that $\|R_u\|\leq c_p$ for all $u\in G$. We use the Bochner integral to define the transferred convolution operator for all $f\in X_p$ by $$T_k(f)=k*_Rf=\int_G R_{-u}(f) k(u) du,$$ where here $du=d\lambda (u)$. It is straightforward to obtain the inequality $\|T_k(f)\|_{L^p(\mu)}\leq c_p\|k\|_{L^1(G)} \|f\|_{L^p(\mu)}$. Using the transference methods, one can show that the operator norm of $T_k$ does not exceed $c_p^2 N_p(k)$. In most cases of interest, when $1
0$, and suppose that $g\in L^1(\R)$ has the following
properties:\\
(i)\ $g\geq 0$,\\
(ii)\ $\widehat{g}(0)=\int_\R g d x=1$,\\
(iii) \ $\widehat{g}$ has compact support contained in
$\left(\frac{-\epsilon}{2},\frac{\epsilon}{2}\right)$.\\
Then for any function $f\in H^1(\mu)$ with
$\spec_R (f)\subset [\epsilon,\infty)$, we have that the
function $u\mapsto g(u) R_u f$ is in $H^1(\R, L^1(\mu))$.
\label{lemma-sec3}
\end{lemma-sec3}
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{\bf Proof.} We need to check that for any $s<0$
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$$
\int_\R e^{-isu} g(u) R_u f d u=0\ \mu - a.\ e.
$$
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This will follow if we can show that for any
$A\in \cM$ we have
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$$
\int_A\int_\R e^{-isu} g(u) R_u f d ud\mu=
\int_\R e^{-isu} g(u) \int_A R_u f d\mu du
=0.
$$
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Equivalently, by taking complex conjugates,
it suffices to show that
\begin{equation}
\int_\R e^{isu} g(u)
\overline{ \int_A R_u f d\mu} d u=0.
\label{to-check}
\end{equation}
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By Proposition \ref{proposition1-sec2},
the spectrum of the
the function $u\mapsto \int_A R_u f d\mu$
is contained in $[\epsilon, \infty)$.
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Since the support of the Fourier transform
of the function $u\mapsto e^{isu} g(u)$
is contained in $(-\frac{\epsilon}{4}+s,\frac{\epsilon}{4}+s)$,
and $s\leq 0$, we have that
$e^{isu} g(u)\in \Im([\epsilon,\infty))$,
and (\ref{to-check}) follows from
Remark \ref{useful}.
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The proof of
(\ref{main-inequality}) will be facilitated
by the following
two reductions.
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{\bf First reduction} In proving (\ref{main-inequality}),
it is enough to assume that the sequence $\{k_n\}$ is finite.
This is a simple consequence of Monotone Convergence.
Henceforth, we assume
that $n$ ranges from $1$ to $N$, where $N$ is a fixed positive
integer and, instead of (\ref{main-inequality}),
prove the inequality
\begin{equation}
\|\max_{ 1\leq n\leq N} |k_n*_Rf|\|_{L^1(\mu)}\leq
c^2 N(\{k_n\}) \|f\|_{L^1(\mu)},
\label{main-inequality'}
\end{equation}
for all $f\in H^1(\mu)$.
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{\bf Second reduction} In proving (\ref{main-inequality'}),
it is enough
to consider functions $f\in H^1(\mu)$ with
$\spec_R (f)\subset [\epsilon,\infty)$, where $\epsilon >0$.
To justify this reduction,
suppose that (\ref{main-inequality'})
holds whenever a representation $R$
is separation-preserving,
strongly continuous, uniformly bounded with constant $c$,
and $f$ has its spectrum contained in $[\epsilon,\infty)$ where
$\epsilon >0$.
Let $\alpha>0$, and consider the representation
$e^{i\alpha (\cdot )} R$. It is clear that
this representation enjoys all the stated properties of $R$
(strong continuity, uniform boundedness with the
same constant $c$, and separation-preserving).
Moreover, if $f\in H^1(\mu)$,
then
$\spec_{e^{i\alpha (\cdot )} R} f\subset [\alpha,\infty)$,
by Lemma \ref{lemma2}(i).
Hence, by our assumption,
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\begin{equation}
\|\max_{1\leq n\leq N}
|k_n*_{e^{i\alpha (\cdot )} R} f|\|_{L^1(\mu)}
\leq c^2 N(\{k_n\}) \|f\|_{L^1(\mu)}.
\label{assumption-second}
\end{equation}
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Letting $\alpha\downarrow 0$, and using Lemma \ref{lemma2},
we have that,
for each $n\in \{1,2,\ldots, N\}$,
$k_n*_{e^{i\alpha (\cdot )} R} f\rightarrow k_n*_R f$
in $L^1(\mu)$. From this and (\ref{assumption-second}),
the inequality (\ref{main-inequality'})
follows easily, establishing the second reduction.
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{\bf Proof of Theorem \ref{main-theorem}}
Suppose that $f\in L^1(\mu)$ with
$\spec_R (f)\subset [\epsilon, \infty)$ where $\epsilon $
is a fixed positive number.
Let $g$ be as in Lemma \ref{lemma-sec3} and let
$F(t)=g(t)R_tf$. By Lemma \ref{lemma-sec3},
$F\in H^1(\R, L^1(\mu))$, and so, by
Lemma \ref{vector-version}, we have
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\begin{equation}
\int_\R \left\|
\max_{1\leq n\leq N}
\left| \int_\R
F(x-t)k_n(t) dt \right|
\right\|_{L^1(\mu)}dx
\leq N(\{k_n\}) \|F\|_{L^1(\R,L^1(\mu))}.
\label{*}
\end{equation}
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We now proceed to show that (\ref{main-inequality'})
is a consequence of (\ref{*}). We have
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\begin{eqnarray}
\|F\|_{L^1(\R,L^1(\mu))}
&=&
\int_\R \int_\O |g(t)R_tf|d|\mu | dt \nonumber\\
&=&
\int_\R g(t) \int_\O|R_tf|d | \mu | dt \leq c \|f\|_{L^1(\mu)}.
\label{**}
\end{eqnarray}
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Using the fact that $R$ is a uniformly bounded and strongly
continuous representation by separation-preserving
operators, we obtain
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\begin{eqnarray*}
\max_{1\leq n\leq N}
\left|
\int_\R F(x-t)k_n(t) dt
\right|
&=&
\max_{1\leq n\leq N}
\left|
\int_\R
(R_{x-t} f) g(x-t) k_n(t) dt
\right| \\
&=&
\max_{1\leq n\leq N}
|R_x|\left(\left|
\int_\R
(R_{-t} f) g(x-t) k_n(t) dt
\right|\right).
\end{eqnarray*}
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Since $|R_{\pm x}|$ is positivity-preserving
and since $|R_x|^{-1}=|R_{-x}|$, we obtain
after applying $|R_{-x}|$ to both sides of
the last equality
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$$
|R_{-x}|
\left(
\max_{1\leq n\leq N}
\left|
\int_\R F(x-t)k_n(t) dt
\right|
\right)
\geq
\max_{1\leq n\leq N}
\left|
\int_\R
(R_{-t} f) g(x-t) k_n(t) dt
\right|.
$$
Hence, using the last inequality
and the uniform boundedness of $R$,
we obtain
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\begin{equation}
c\left\|
\max_{1\leq n\leq N}
\left|
\int_\R F(x-t)k_n(t) dt
\right|
\right\|_{L^1(\mu)}
\geq
\left\|
\max_{1\leq n\leq N}
\left|
\int_\R
(R_{-t} f) g(x-t) k_n(t) dt
\right|
\right\|_{L^1(\mu)}.
\label{*22-2}
\end{equation}
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Integrating both sides of (\ref{*22-2})
over $\R$ in the $x$ variable,
and using (\ref{*}) and (\ref{**}),
we obtain
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\begin{equation}
N(\{k_n\}) c^2 \|f\|_{L^1(\mu)}\geq
\int_\R \int_\O
\max_{1\leq n\leq N}
\left|
\int_\R
(R_{-t} f) g(x-t) k_n(t) dt
\right|
d|\mu|dx .
\label{**22-2}
\end{equation}
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Obvious manipulations with
(\ref{**22-2}),
Fubini's Theorem and
the fact that $g\geq 0$ and $\widehat{g}(0)=\int_\R g =1$,
yield
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\begin{eqnarray*}
N(\{k_n\}) c^2 \|f\|_{L^1(\mu)}
&\geq&
\int_\O \int_\R
\max_{1\leq n\leq N}
\left|
\int_\R
(R_{-t} f) g(x-t) k_n(t) dt
\right|
dx d|\mu| \\
&\geq&
\int_\O
\max_{1\leq n\leq N}
\left| \int_\R g(x-t)dx
\int_\R
(R_{-t} f) k_n(t) dt
\right|
d|\mu | \\
&=&
\int_\O
\max_{1\leq n\leq N}
\left|
\int_\R
(R_{-t} f) k_n(t) dt
\right|
d|\mu| \\
&=&
\|\max_{1\leq n\leq N} |k_n*_Rf|\|_{L^1(\mu)},
\end{eqnarray*}
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which proves
(\ref{main-inequality'}).
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\section{$H^1(\mu)$ and the ergodic Hilbert transform}
\newtheorem{th1}{Theorem}[section]
\newtheorem{sec4prop1}[th1]{Proposition}
\newtheorem{sec4prop2}[th1]{Proposition}
\newtheorem{sec4prop3}[th1]{Proposition}
\newtheorem{sec4prop4}[th1]{Proposition}
\newtheorem{remark1}[th1]{Remark}
\newtheorem{lemma-th1}[th1]{Lemma}
\newtheorem{th2}[th1]{Theorem}
\newtheorem{corollary-th2}[th1]{Corollary}
In this section we will investigate a connection
between $H^1(\mu)$,
the space ergodic $H^1$ of \cite{cw2},
and spaces of functions introduced in
\cite{abg2} (Theorem \ref{th1} below).
Throughout, $u\rightarrow R_u$ will denote a
strongly continuous
representation of $\R$ by measure-preserving transformations
on an finite measure space $(\O, \cM,\mu)$.
In particular, $R$ is
separation-preserving and uniformly bounded with
$c=1$. (The results of this section apply as well
in the more general
setting of distributionally controlled
representations that were introduced in \cite{abg2}.
For clarity's sake, we will only discuss
representations given by measure-preserving transformations.)
Since the measure $\mu$ is finite,
we have the following useful direct sum decomposition
of $L^1(\mu)$:
$$L^1(\mu)=Y \bigoplus Z,$$
where
$$Y=\{f\in L^1(\mu):\ R_uf=f,\ {\rm for\ all}\ u\in\R\},$$
and $Z$ is the $L^1(\mu)$-closure of the linear subspace of
$L^1(\mu)$ spanned by the ranges of the
operators $f\mapsto g*_R f$, for all $g\in L^1(\R)$
such that $\R\setminus \{0\}$ contains the
support of $\widehat{g}$. (See \cite[Proposition (3.19)]{abg2}.)
Let $h(t)=\frac{1}{\pi t}$ for $t\neq 0$ denote the
Hilbert kernel. For each $n$, let $h_n$ denote the $n$th truncate
$h_n(t)=\frac{1}{\pi t}$ if $\frac{1}{n}<|t|