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} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% {\bf Proof.} We need to check that for any $s<0$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% $$ \int_\R e^{-isu} g(u) R_u f d u=0\ \mu - a.\ e. $$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% This will follow if we can show that for any $A\in \cM$ we have %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% $$ \int_A\int_\R e^{-isu} g(u) R_u f d ud\mu= \int_\R e^{-isu} g(u) \int_A R_u f d\mu du =0. $$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 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} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% By Proposition \ref{proposition1-sec2}, the spectrum of the the function $u\mapsto \int_A R_u f d\mu$ is contained in $[\epsilon, \infty)$. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 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}. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% The proof of (\ref{main-inequality}) will be facilitated by the following two reductions. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% {\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 $f\in H^1(\mu)$. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% {\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, %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \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} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 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. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% {\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 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \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} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% We now proceed to show that (\ref{main-inequality'}) is a consequence of (\ref{*}). We have %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \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} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Using the fact that $R$ is a uniformly bounded and strongly continuous representation by separation-preserving operators, we obtain %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \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*} %%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 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 %%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% $$ |R_{-x}| \left( \max_{1\leq n\leq N} \left| \int_\R F(x-t)k_n(t) dt \right| \right) \geq \max_{1\leq n\leq N} \left| \int_\R (R_{-t} f) g(x-t) k_n(t) dt \right|. $$ Hence, using the last inequality and the uniform boundedness of $R$, we obtain %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \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} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Integrating both sides of (\ref{*22-2}) over $\R$ in the $x$ variable, and using (\ref{*}) and (\ref{**}), we obtain %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \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} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 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 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \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*} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% which proves (\ref{main-inequality'}). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \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|