Introduction

Transference theory for $L^p$ spaces is a powerful tool with many fruitful applications to singular integrals, ergodic theory, and spectral theory of operators [4,5]. These methods afford a unified approach to many problems in diverse areas, which before were proved by a variety of methods.

The purpose of this paper is to bring about a similar approach to spaces of measures. Our main transference result is motivated by the extensions of the classical F.&M. Riesz Theorem due to Bochner [3], Helson-Lowdenslager [10,11], de Leeuw-Glicksberg [6], Forelli [9], and others. It might seem that these extensions should all be obtainable via transference methods, and indeed, as we will show, these are exemplary illustrations of the scope of our main result.

It is not straightforward to extend the classical transference methods of Calderón, Coifman and Weiss to spaces of measures. First, their methods make use of averaging techniques and the amenability of the group of representations. The averaging techniques simply do not work with measures, and do not preserve analyticity. Secondly, and most importantly, their techniques require that the representation is strongly continuous. For spaces of measures, this last requirement would be prohibitive, even for the simplest representations such as translations. Instead, we will introduce a much weaker requirement, which we will call `sup path attaining'. By working with sup path attaining representations, we are able to prove a new transference principle with interesting applications. For example, we will show how to derive with ease generalizations of Bochner's theorem and Forelli's main result. The Helson-Lowdenslager theory, concerning representations of groups with ordered dual groups, is also within reach, but it will be treated in a separate paper.

Throughout $G$ will denote a locally compact abelian group with dual group $\Gamma$. The symbols $\mathbb{Z}$, $\mathbb{R}$ and $\mathbb{C}$ denote the integers, the real and complex numbers, respectively. If $A$ is a set, we denote the indicator function of $A$ by $1_A$. For $1\leq p<\infty$, the space of Haar measurable functions $f$ on $G$ with $\int_G\vert f\vert^p dx<\infty$ will be denoted by $L^p(G)$. The space of essentially bounded functions on $G$ will be denoted by $L^\infty(G)$. The expressions ``locally null'' and ``locally almost everywhere'' will have the same meanings as in [12, Definition (11.26)].

Our measure theory is borrowed from [12]. In particular, the space of all complex regular Borel measures on $G$, denoted by $M(G)$, consists of all complex measures arising from bounded linear functionals on ${\cal C}_0(G)$, the Banach space of continuous functions on $G$ vanishing at infinity.

Let $(\O , \Sigma)$ denote a measurable space, where $\O$ is a set and $\Sigma$ is a sigma algebra of subsets of $\O$. Let $M(\Sigma)$ denote the Banach space of complex measures on $\Sigma$ with the total variation norm, and let ${\cal L}^\infty(\Sigma)$ denote the space of measurable bounded functions on $\Omega$.

Let $T:\ t\mapsto T_t$ denote a representation of $G$ by isomorphisms of $M(\Sigma)$. We suppose that $T$ is uniformly bounded, i.e., there is a positive constant $c$ such that for all $t\in G$, we have

$$\Vert T_t\Vert\leq c .$$ (1)

Definition 1.1   A measure $\mu\in M(\Sigma)$ is called weakly measurable (in symbols, $\mu\in{\cal M}_T(\Sigma)$) if for every $A\in \Sigma$ the mapping $t\mapsto T_t\mu(A)$ is Borel measurable on $G$.

Given a measure $\mu\in{\cal M}_T(\Sigma)$ and a Borel measure $\nu \in M(G)$, we define the `convolution' $\nu*_T\mu$ on $\Sigma$ by

$$\nu*_T\mu (A)=\int_G T_{-t}\mu(A) d\nu(t)$$ (2)

for all $A\in \Sigma$.

We will assume throughout this paper that the representation $T$ commutes with the convolution (2) in the following sense: for each $t\in G$,

$$T_t(\nu*_T\mu)=\nu*_T(T_t\mu).$$ (3)

Condition (3) holds if, for example, for all $t\in G$, the adjoint of $T_t$ maps ${\cal L}^\infty(\Sigma)$ into itself. In symbols,

$$T_t^*: {\cal L}^\infty(\Sigma) \rightarrow {\cal L}^\infty(\Sigma).$$ (4)

For proofs we refer the reader to [1]. Using (1) and (3), it can be shown that $\nu*_T\mu$ is a measure in ${\cal M}_T(\Sigma)$,

$$\Vert\nu*_T\mu\Vert\leq c\Vert\nu\Vert\Vert\mu\Vert,$$ (5)

where $c$ is as in (1), and

$$\sigma*_T(\nu*_T\mu)=(\sigma*\nu)*_T\mu,$$ (6)

for all $\sigma , \nu \in M(G)$ and $\mu\in{\cal M}_T(\Sigma)$ (see [1]).

We come now to a definition which is fundamental to our work.

Definition 1.2   A representation $T=(T_t)_{t\in G}$ of a locally compact abelian group $G$ in $M(\Sigma)$ is said to be sup path attaining if it is uniformly bounded, satisfies property (3), and if there is a constant $C$ such that for every weakly measurable $\mu\in{\cal M}_T(\Sigma)$ we have

$$\Vert \mu\Vert \leq C\sup \left\{ {\rm ess\ sup}_{t\in G} \left\v... ...t\vert :\ \ h\in {\cal L}^\infty(\Sigma),\ \Vert h\Vert _\infty\leq 1 \right\}.$$ (7)

The fact that the mapping $t\mapsto \int_\O h d (T_t\mu)$ is measurable is a simple consequence of the measurability of the mapping $t\mapsto T_t\mu(A)$ for every $A\in \Sigma$.

Examples of sup path attaining representations will be presented in the following section. Proceeding toward the main result of this paper, we recall some basic definitions from spectral theory.

If $I$ is an ideal in $L^1(G)$, let

$$Z(I)=\bigcap_{f\in I} \left\{ \chi\in\Gamma:\ \ \widehat{f}(\chi)=0 \right\}.$$

The set $Z(I)$ is called the zero set of $I$. For a weakly measurable $\mu\in M(\Sigma)$, let

$${\cal I}(\mu)=\{f\in L^1(G):\ \ f*_T\mu =0\}.$$

Using properties of the convolution $*_T$, it is straightforward to show that ${\cal I}(\mu)$ is a closed ideal in $L^1(G)$.

Definition 1.3   The $T$-spectrum of a weakly measurable $\mu\in{\cal M}_T(\Sigma)$ is defined by

$${\rm spec}_T (\mu)= \bigcap_{f\in {\cal I}(\mu)} \left\{ \chi\in\Gamma:\ \ \widehat{f}(\chi)=0 \right\}=Z({\cal I}(\mu)).$$ (8)

If $S\subset \Gamma$, let

$$L_S^1=L_S^1(G)=\left\{f\in L^1(G):\ \widehat{f}=0\ \mbox{outside of}\ S\right\}\,.$$

Our transference result concerns convolution operators on $L^1_S(G)$ where $S$ satisfies a special property, described as follows.

Definition 1.4   A subset $S\subset \Gamma$ is a ${\cal T}$-set if, given any compact $K\subset S$, each neighborhood of $0\in\Gamma$ contains a nonempty open set $W$ such that $W+K\subset S$.

Example 1.5   (a) If $\Gamma$ is a locally compact abelian group, then any open subset of $\Gamma$ is a ${\cal T}$-set. In particular, if $\Gamma$ is discrete then every subset of $\Gamma$ is a ${\cal T}$-set.
(b) The set $\left[ a,\infty\right. )$ is a ${\cal T}$-subset of $\mathbb{R}$, for all $a\in\mathbb{R}$.
(c) Let $a\in\mathbb{R}$ and $\psi:\ \Gamma \rightarrow \mathbb{R}$ be a continuous homomorphism. Then $S=\psi^{-1}([a,\infty))$ is a ${\cal T}$-set.
(d) Let $\Gamma=\mathbb{R}^2$ and $S=\{(x,y):\ y^2\leq x\}$. Then $S$ is a ${\cal T}$-subset of $\mathbb{R}^2$ such that there is no nonempty open set $W\subset \mathbb{R}^2$ such that $W+S\subset S$.

The main result of this paper is the following transference theorem.

Theorem 1.6   Let $T$ be a sup path attaining representation of a locally compact abelian group $G$ by isomorphisms of $M(\Sigma)$ and let $S$ be a ${\cal T}$-subset of $\Gamma$. Suppose that $\nu$ is a measure in $M(G)$ such that

$$\Vert\nu*f\Vert _1\leq \Vert f\Vert _1$$ (9)

for all $f$ in $L_S^1(G)$. Then for every weakly measurable $\mu\in M(\Sigma)$ with ${\rm spec}_T ( \mu )\subset S$ we have

$$\Vert\nu*_T\mu\Vert\leq c^3 C \Vert\mu\Vert,$$ (10)

where $c$ is as in (1) and $C$ is as in (7).

To state Forelli's main result in [9], let us recall two definitions of Baire sets. Suppose that $\Omega$ is a topological space. Usually, the collection of Baire sets, ${\cal B}_0={\cal B}_0(\O )$, is defined as the sigma algebra generated by sets that are compact and also countable intersections of open sets. A second definition is to define ${\cal B}_0$ as the minimal sigma algebra so that compactly supported continuous functions are measurable. For locally compact Hausdorff topological spaces, these two definitions are equivalent.

Suppose that $(T_t)_{t\in \mathbb{R}}$ is a group of homeomorphisms of $\Omega$ onto itself such that the mapping $(t,\omega) \mapsto T_t \omega$ is jointly continuous. The maps $T_t$ induce isomorphisms $T_t$ on the space of Baire measures via the identity $T_t \mu(A) = \mu(T_t(A))$.

If $\nu$ is a Baire measure, we will say that it is quasi-invariant if $T_t\nu$ and $\nu$ are mutually absolutely continuous for all $t \in \mathbb{R}$, that is, for all $A \in {\cal B}_0$ we have that $\vert\nu\vert(A) = 0$ if and only if $\vert T_t\nu\vert(A) = 0$. If $\mu$ is a Baire measure, we will say that $\mu$ is $T$-analytic if

$$\int_{\mathbb{R}} T_t\mu(A) h(t) dt = 0,$$

for all $A \in {\cal B}_0$ and all $h \in H^1(\mathbb{R})$, where

$$H^1(\mathbb{R})=\left\{f\in L^1(\mathbb{R}):\ \widehat{f}(s)=0\ \mbox{for all}\ s\leq 0\right\}\,.$$

The main result of Forelli [9] says the following.

Theorem 1.7   Let $\Omega$ be a locally compact Hausdorff topological space, and let $(T_t)_{t\in \mathbb{R}}$ be a group of homeomorphisms of $\Omega$ onto itself such that the maps $(t,\omega) \mapsto T_t \omega$ is jointly continuous. Suppose that $\mu$ is a $T$-analytic Baire measure, and that $\nu$ is a quasi-invariant Baire measure. Then both $\mu_s$ and $\mu_a$ are $T$-analytic, where $\mu_s$ and $\mu_a$ are the singular and absolutely continuous parts of $\mu$ with respect to $\nu$.

The proofs of both this result in [9], and also its predecessor [6], are long and difficult. Furthermore, it is hard to understand why one must use Baire measures instead of Borel measures. As it turns out, the mystery of why we need to use Baire measures in Forelli's work is reduced to the fact that such representations as described above on the Baire measures are sup path attaining. By working with sup path attaining representations, we are able to prove a more general version of Theorem 1.7. We do not need $\Omega$ to be locally compact Hausdorff. Or we might suppose that $(T_t)_{t\in \mathbb{R}}$ is any group of uniformly bounded isomorphisms satisfying (3) on any Lebesgue space (that is, a countably generated sigma algebra).