Transference in Spaces of Measures

Nakhlé H. Asmar^{∗}, Stephen J. Montgomery–Smith^{†}, and Sadahiro Saeki^{‡}

1 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 Γ. The symbols ℤ, ℝ and 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 ≤ p < ∞, the space of Haar measurable functions f on G with ∫_{G} |f|^{p}dx < ∞ will be denoted by L^{p}(G). The space of essentially bounded functions on G will be denoted by L^{∞}(G). The expressions “locally null” and “locally almost everywhere” will have the same meanings as in [12, Definition (11.26)].

∗Department of Mathematics, University of Missouri, Columbia, Missouri 65211 U. S. A.

†Department of Mathematics, University of Missouri, Columbia, Missouri 65211 U. S. A.

‡Department of Mathematics, Kansas State University, Manhattan, Kansas 66506 U. S. A.

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 C_{0}(G), the Banach space of continuous functions on G vanishing at infinity.

Let (Ω, Σ) denote a measurable space, where Ω is a set and Σ is a sigma algebra of subsets of Ω. Let M(Σ) denote the Banach space of complex measures on Σ with the total variation norm, and let L^{∞}(Σ) denote the space of measurable bounded functions on Ω.

Let T : t ↦ T_{t} denote a representation of G by isomorphisms of M(Σ). We suppose that T is uniformly bounded, i.e., there is a positive constant c such that for all t ∈ G, we have

∥T_{t}∥ ≤ c.  (1)

Definition 1.1 A measure μ ∈ M(Σ) is called weakly measurable (in symbols, μ ∈ M_{T} (Σ)) if for every A ∈ Σ the mapping t ↦ T_{t}μ(A) is Borel measurable on G.

Given a measure μ ∈ M_{T} (Σ) and a Borel measure ν ∈ M(G), we define the ‘convolution’ ν ∗_{T} μ on Σ by

ν ∗_{T} μ(A) = ∫_{G} T_{−t}μ(A)dν(t)  (2)

for all A ∈ Σ.

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

T_{t}(ν ∗_{T} μ) = ν ∗_{T} (T_{t}μ).  (3)

Condition (3) holds if, for example, for all t ∈ G, the adjoint of T_{t} maps L^{∞}(Σ) into itself. In symbols,

T_{t}^{∗} : L^{∞}(Σ) → L^{∞}(Σ).  (4)

For proofs we refer the reader to [1]. Using (1) and (3), it can be shown that ν ∗_{T} μ is a measure in M_{T} (Σ),

∥ν ∗_{T} μ∥ ≤ c∥ν∥∥μ∥,  (5)

where c is as in (1), and

σ ∗_{T} (ν ∗_{T} μ) = (σ ∗ ν) ∗_{T} μ,  (6)

for all σ, ν ∈ M(G) and μ ∈ M_{T} (Σ) (see [1]).

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

Definition 1.2 A representation T = (T_{t})_{t∈G} of a locally compact abelian group G in M(Σ) 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 μ ∈ M_{T} (Σ) we have

∥μ∥ ≤ C sup {ess sup_{t∈G} |∫_{Ω} hd(T_{t}μ)| : h ∈ L^{∞}(Σ), ∥h∥_{∞} ≤ 1} .  (7)

The fact that the mapping t ↦ ∫_{Ω} hd(T_{t}μ) is measurable is a simple consequence of the measurability of the mapping t ↦ T_{t}μ(A) for every A ∈ Σ.

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) = ⋂_{f∈I} {χ ∈ Γ : f̂(χ) = 0} .

The set Z(I) is called the zero set of I. For a weakly measurable μ ∈ M(Σ), let

I(μ) = {f ∈ L^{1}(G) : f ∗_{T} μ = 0}.

Using properties of the convolution ∗_{T}, it is straightforward to show that I(μ) is a closed ideal in L^{1}(G).

Definition 1.3 The T -spectrum of a weakly measurable μ ∈ M_{T} (Σ) is defined by

spec_{T} (μ) = ⋂_{f∈I(μ)} {χ ∈ Γ : f̂(χ) = 0} = Z(I(μ)).  (8)

If S ⊂ Γ, let

L_{S}^{1} = L_{S}^{1}(G) = {f ∈ L^{1}(G) : f̂ = 0 outside of S} .

Our transference result concerns convolution operators on L_{S}^{1}(G) where S satisfies a special property, described as follows.

Definition 1.4 A subset S ⊂ Γ is a T -set if, given any compact K ⊂ S, each neighborhood of 0 ∈ Γ contains a nonempty open set W such that W + K ⊂ S.

Example 1.5 (a) If Γ is a locally compact abelian group, then any open subset of Γ is a T -set. In particular, if Γ is discrete then every subset of Γ is a T -set.

(b) The set [a, ∞) is a T -subset of ℝ, for all a ∈ ℝ.

(c) Let a ∈ ℝ and ψ : Γ → ℝ be a continuous homomorphism. Then S = ψ^{−1}([a, ∞)) is a T -set.

(d) Let Γ = ℝ^{2} and S = {(x, y) : y^{2} ≤ x}. Then S is a T -subset of ℝ^{2} such that there is no nonempty open set W ⊂ ℝ^{2} such that W + S ⊂ 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(Σ) and let S be a T -subset of Γ. Suppose that ν is a measure in M(G) such that

∥ν ∗ f∥_{1} ≤ ∥f∥_{1}  (9)

for all f in L_{S}^{1}(G). Then for every weakly measurable μ ∈ M(Σ) with spec_{T} (μ) ⊂ S we have

∥ν ∗_{T} μ∥ ≤ c^{3}C∥μ∥,  (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 Ω is a topological space. Usually, the collection of Baire sets, B_{0} = B_{0}(Ω), 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 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∈ℝ} is a group of homeomorphisms of Ω onto itself such that the mapping (t, ω) ↦ T_{t}ω is jointly continuous. The maps T_{t} induce isomorphisms T_{t} on the space of Baire measures via the identity T_{t}μ(A) = μ(T_{t}(A)).

If ν is a Baire measure, we will say that it is quasi-invariant if T_{t}ν and ν are mutually absolutely continuous for all t ∈ ℝ, that is, for all A ∈ B_{0} we have that |ν|(A) = 0 if and only if |T_{t}ν|(A) = 0. If μ is a Baire measure, we will say that μ is T -analytic if

∫_{ℝ} T_{t}μ(A)h(t)dt = 0,

for all A ∈ B_{0} and all h ∈ H^{1}(ℝ), where

H^{1}(ℝ) = {f ∈ L^{1}(ℝ) : f̂(s) = 0 for all s ≤ 0} .

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

Theorem 1.7 Let Ω be a locally compact Hausdorff topological space, and let (T_{t})_{t∈ℝ} be a group of homeomorphisms of Ω onto itself such that the maps (t, ω) ↦ T_{t}ω is jointly continuous. Suppose that μ is a T -analytic Baire measure, and that ν is a quasi-invariant Baire measure. Then both μ_{s} and μ_{a} are T -analytic, where μ_{s} and μ_{a} are the singular and absolutely continuous parts of μ with respect to ν.

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 Ω to be locally compact Hausdorff. Or we might suppose that (T_{t})_{t∈ℝ} is any group of uniformly bounded isomorphisms satisfying (3) on any Lebesgue space (that is, a countably generated sigma algebra).

2 Sup Path Attaining Representations

We first note that for a sup path attaining representation T of G we have

∥μ∥ ≤ C sup {ess sup_{t∈G} |∫_{Ω} hd(T_{t}μ)| : h is a simple function with ∥h∥_{∞} ≤ 1} . (11)

To see this, note that for any bounded measurable function h, there exist a sequence of simple functions s_{n} that converge uniformly to h. Then it is easy to see that

ess sup_{t∈G} |∫_{Ω} s_{n}d(T_{t}μ)| → ess sup_{t∈G} |∫_{Ω} hd(T_{t}μ)| ,

and from this (11) follows easily.

Our first example is related to the setting of Forelli [9].

Example 2.1 Let G be a locally compact abelian group. Suppose that Ω is a topological space and (T_{t})_{t∈G} is a group of homeomorphisms of Ω onto itself such that the mapping

(t, ω) ↦ T_{t}ω

is jointly continuous. Suppose that A is an algebra of bounded continuous complex valued functions on Ω such that if h ∈ A, and if φ : ℂ → ℂ is any bounded continuous function, and if t ∈ G, then φ ∘ h ∘ T_{t} ∈ A. Let σ(A) denote the minimal sigma-algebra so that functions from A are measurable. For any measure μ ∈ M(σ(A)), and A ∈ σ(A), define T_{t}μ(A) = μ(T_{t}(A)), where T_{t}(A) = {T_{t}ω : ω ∈ A}. Note that T satisfies (1) and (3). To discuss the weak measurability of μ, and the sup path attaining property of T, we need that for each h ∈ A that the map t ↦ ∫_{Ω} h d(T_{t}μ) is continuous. This crucial property follows for any measure μ ∈ M(σ(A)), by the dominated convergence theorem, if, for example, G is metrizable. For arbitrary locally compact abelian groups, it is enough to require that μ has sigma-compact support.

Henceforth, we assume that G is metrizable, or that μ ∈ M(σ(A)) has sigma-compact support, and proceed to show that μ is weakly measurable in the sense of Definition 1.1, and that the representation T is sup path attaining. Let

R = {A ∈ σ(A) : 1_{A} = lim_{n} f_{n}, f_{n} ∈ A, ∥f_{n}∥_{∞} ≤ 1},

and let

C = {A ∈ σ(A) : t ↦ T_{t}μ(A) is Borel measurable}.

Clearly, C is a monotone class, closed under nested unions and intersections. Also, R is an algebra of sets, closed under finite unions and set complementation. Furthermore, it is clear that R ⊂ C. Hence, by the monotone class theorem, it follows that C contains the sigma algebra generated by R. Now, if f ∈ A and V ⊂ ℂ is open, then f^{−1}(V) ∈ R, because

1_{f^{−1}(V)}(ω) = lim_{n→∞} min{n dist(f(ω), ℂ \ V), 1}.

Consequently, we have that C = σ(A), that is, μ is weakly measurable.

Next, let us show that A is dense in L^{1}(|μ|). Let g : Ω → ℂ be a bounded σ(A)- measurable function such that ∫_{Ω} hg dμ = 0 for all h ∈ A. Define R as above, and let

C = {A ∈ σ(A) : ∫_{A} g dμ = 0} .

Then C is a monotone class containing R, and so arguing as before, C = σ(A), that is, g = 0 almost everywhere with respect to |μ|. Thus it follows by the Hahn-Banach Theorem that A is dense in L^{1}(|μ|). Hence

∥μ∥ = sup {|∫_{Ω} h dμ| : h ∈ A, ∥h∥_{∞} ≤ 1} ≤ sup {ess sup_{t∈G} |∫_{Ω} h d(T_{t}μ)| : h ∈ A, ∥h∥_{∞} ≤ 1} ,

where the last inequality follows from the fact that the map t ↦ ∫_{Ω} h d(T_{t}μ) is continuous. Hence, T is sup path attaining with C = 1.

Taking A to be the uniformly continuous functions on a group, we then derive the following useful example.

Example 2.2 Suppose that G_{1} and G_{2} are locally compact abelian groups and that ϕ : G_{2} → G_{1} is a continuous homomorphism. Define an action of G_{2} on M(G_{1}) (the regular Borel measures on G_{1}) by translation by ϕ. Hence, for x ∈ G_{2}, μ ∈ M(G_{1}), and any Borel subset A ⊂ G_{1}, let T_{x}μ(A) = μ(A + ϕ(x)). Then every μ ∈ M(G_{1}) is weakly measurable, the representation is sup path attaining with constants c = 1 and C = 1.

In the following example no topology is required on the measure space.

Example 2.3 Let (X, Σ) be an abstract Lebesgue space, that is, Σ is countably generated. Then any uniformly bounded representation T of G by isomorphisms of M(Σ) is sup path attaining. To see this, note that since Σ is countably generated, there is a countable subset A of the unit ball of L^{∞}(Σ) such that for any μ ∈ M(Σ) we have

∥μ∥ = sup {|∫_{X} hdμ| : h ∈ A} .

If μ is weakly measurable, then for g ∈ A and for locally almost all u ∈ G, we have

|∫_{X} gd(T_{u}μ)| ≤ ess sup_{t∈G} |∫_{X} gd(T_{t}μ)| .  (12)

Since A is countable we can find a subset B of G such that the complement of B is locally null, and such that (12) holds for every u ∈ B and all g ∈ A. For u ∈ B, take the sup in

(12) over all g ∈ A, and get

∥T_{u}μ∥ ≤ sup_{g∈A} ess sup_{t∈G} |∫_{X} gd(T_{t}μ)| .

But since ∥μ∥ ≤ c∥T_{u}μ∥, it follows that T is sup path attaining with C = c.

Sup path attaining representations satisfy the following property, which was introduced in [1] and was called hypothesis (A).

Proposition 2.4 Suppose that T is sup path attaining and μ is weakly measurable such that for every A ∈ Σ we have

T_{t}μ(A) = 0

for locally almost all t ∈ G. Then μ = 0.

The proof is immediate and follows from (11).

The key in Proposition 2.4 is that the set of t ∈ G for which T_{t}μ(A) = 0 depends on A. If this set were the same for all A ∈ Σ, then the conclusion of the proposition would trivially hold for any representation by isomorphisms of M(Σ).

For further motivation, we recall the following example from [1].

Example 2.5 (a) Let Σ denote the sigma algebra of countable and co-countable subsets of ℝ. Define ν ∈ M(Σ) by

ν(A) = \{ \begin{matrix}1 & if A is co-countable, \\ 0 & if A is countable. \end{matrix}

Let δ_{t} denote the point mass at t ∈ ℝ, and take μ = ν − δ_{0}. Consider the representation T of ℝ given by translation by t. Then: μ is weakly measurable; ∥μ∥ > 0; T_{t}(μ) = T_{t}(ν − δ_{0}) = ν − δ_{t}; for every A ∈ Σ, T_{t}(μ)(A) = 0 for almost all t ∈ ℝ. It now follows from Proposition 2.4 that the representation T is not sup path attaining.

(b) Let α be a real number and let Σ, μ, ν, δ_{t}, and T_{t} have the same meanings as in (a). Define a representation T^{α} by

T_{t}^{α} = e^{iαt}T_{t}.

Arguing as in (a), it is easy to see that T^{α} is not sup path attaining.

3 Proof of the Main Theorem

Throughout this section, G denotes a locally compact abelian group with dual group Γ; M(Σ) is a space of measures on a set Ω; and T = (T_{t})_{t∈G} is a sup path attaining representation of G by isomorphisms of M(Σ).

If ϕ is in L^{∞}(G), write [ϕ] for the smallest weak-* closed translation-invariant subspace of L^{∞}(G) containing ϕ, and let I([ϕ]) = I(ϕ) denote the closed translation-invariant ideal in L^{1}(G):

I(ϕ) = {f ∈ L^{1}(G) : f ∗ ϕ = 0}.

It is clear that I(ϕ) = {f ∈ L^{1}(G) : f ∗ g = 0, ∀g ∈ [ϕ]}. The spectrum of ϕ, denoted by σ [ϕ], is the set of all continuous characters of G that belong to [ϕ]. This closed subset of Γ is also given by

σ [ϕ] = Z(I(ϕ)).  (13)

(See [14, Chapter 7, Theorem 7.8.2, (b), (c), and (d)].)

Remarks 3.1 (a) For a weakly measurable μ ∈ M(Σ), since I(μ) is a closed ideal in L^{1}(G), it is translation-invariant, by [14, Theorem 7.1.2]. It follows readily that for all t ∈ G,

I(T_{t}μ) = I(μ),

and hence

spec_{T} (T_{t}(μ)) = spec_{T} (μ).  (14)

(b) Let μ ∈ M(Σ) be weakly measurable and let E ∈ Σ. It is clear that I(μ) ⊃ I(t ↦ (T_{t}μ)(E)). Thus σ [t ↦ (T_{t}μ)(E)] ⊂ spec_{T} (μ).

Lemma 3.2 Let μ ∈ M_{T} (Σ).

(a) If g ∈ L^{1}(G) and E ∈ Σ, then

∫_{G} g(t)T_{t}μ(E)\overline{χ(t)} dt = 0,

for all χ not in supp(ĝ) + spec_{T} (μ).

(b) If ν ∈ M(G), then

spec_{T} (ν ∗_{T} μ) ⊂ supp(ν̂) ∩ spec_{T} (μ) .

(c) If (k_{α}) is an approximate identity for L^{1}(G), then

∥μ∥_{M(Σ)}/C ≤ lim inf_{α} ∥t ↦ ∥k_{α} ∗_{T} T_{t}μ∥_{M(Σ)}∥_{L^{∞}(G)} ,

where C is as in (7).

Proof. (a) Fix E ∈ Σ and χ ∈ Γ, and let f(t) = (T_{−t}μ)(E)χ(t). Then σ [f] ⊂ χ − spec_{T} (μ). Hence, for g ∈ L^{1}(G) with supp(ĝ) ∩ (χ − spec_{T} μ) = ∅, we have

∫_{G} g(t)(T_{t}μ)\overline{χ}(t)dt = (g ∗ f)(0) = 0 .

(b) Immediate from I(ν ∗_{T} μ) ⊃ I(ν) ∪ I(μ) .

(c) Fix h on Ω with |h| ≤ 1 and let f(t) = ∫_{Ω} h(x)d(T_{t}μ(x)). Then f ∈ L^{∞}(G) and so, if (k_{α}) is an approximate identity for L^{1}(G), then k_{α} ∗f → f in the weak ∗-topology of L^{∞}(G). Hence

∥f∥_{∞} ≤ lim inf ∥k_{α} ∗ f∥_{∞} = lim inf ∥t → ∫_{Ω} h(x)d(k_{α} ∗_{T} T_{t}μ)∥_{∞}
≤ lim inf ∥t → ∥k_{α} ∗_{T} T_{t}μ∥∥_{∞} .

The proof is completed by taking the sup of ∥f∥_{∞} over all h and using (7).

Lemma 3.3 Let f : G → M(Σ) be bounded and continuous, and let ν ∈ M(G). Suppose that

(i) for all E ∈ Σ, t ↦ f(t)E is in L_{S}^{1}(G); (ii) for all g ∈ L_{S}^{1}(G), ∥ν ∗ g∥_{1} ≤ ∥g∥_{1} . Then

∫_{G} ∥(ν ∗ f)(t)∥_{M(Σ)}dt ≤ ∫_{G} ∥f(t)∥_{M(Σ)}dt .  (15)

Proof. We suppose throughout the proof that the right side of (15) is finite; otherwise there is nothing to prove. Write f_{E}(t) = f(t)E. Thus

(ν ∗ f)(t)E = (ν ∗ f_{E})(t)

is continuous on G. By (i) and (ii),

∫_{G} |(ν ∗ f)(t)E|dt ≤ ∫_{G} |f(t)E|dt,  (16)

for all E ∈ Σ. Now let P = {E_{j}} be a finite measurable partition of Ω, and define

h(P, t) = ∑_{j} |(ν ∗ f)(t)E_{j}| .

Then h(P, ·) is continuous on G and, by (16),

∫_{G} h(P, t)dt = ∑_{j} ∫_{G} |(ν ∗ f)(t)E_{j}|dt ≤ ∫_{G} ∥f(t)∥_{M(Σ)}dt .  (17)

If R is a common refinement of P and Q, then, for all t,

max{h(P, t), h(Q, t)} ≤ h(R, t) .

It follows from [12, (11.13)] that

∫_{G} ∥ν ∗ f∥_{M(Σ)}dt = ∫_{G} sup_{P} h(P, t)dt
= sup_{P} ∫_{G} h(P, t)dt ≤ ∫_{G} ∥f(t)∥_{M(Σ)}dt

by (17).

Proof of Theorem 1.6. Let f(t) = T_{t}μ and suppose first that f is continuous and that there is a neighborhood V of 0 ∈ Γ such that V + spec_{T} (μ) ⊂ S. Choose a continuous g ∈ L^{1}(G) such that

g ≥ 0;
ĝ(0) = ∫_{G} g(t)dt = 1;

and

supp(ĝ) ⊂ V .

Then, for all E ∈ Σ, the mapping t ↦ g(t)f(t)E is bounded and continuous on G, and, by Lemma 3.2, belongs to

L_{supp(ĝ)+spec_{T} (μ)}^{1} ⊂ L_{V +spec_{T} (μ)}^{1} ⊂ L_{S}^{1} .

Hence Lemma 3.3 implies that

∥ν ∗ (gf)∥_{L^{1}(G,M(Σ))} ≤ ∥gf∥_{L^{1}(G,M(Σ))} .  (18)

We now proceed to show that (10) is a consequence of (18). We start with the right side of (18):

∥gf∥_{1} = ∫_{G} g(t)∥f(t)∥dt
≤ c∥μ∥ ∫_{G} g(t)dt = c∥μ∥.  (19)

Consider now the left side of (18). We have

∥ν ∗ (gf)∥_{1} = ∫_{G} ∥ν ∗ (gf)(t)∥dt
= ∫_{G} ∥ ∫_{G}(gf)(t − s)dν(s)∥dt
= ∫_{G} ∥ ∫_{G} g(t − s)T_{t−s}μdν(s)∥dt
≥ \frac{1}{c} ∫_{G} ∥ ∫_{G} T_{−t} [g(t − s)T_{t−s}μ] dν(s)∥dt
= \frac{1}{c} ∫_{G} ∥ ∫_{G} g(t − s)T_{−s}μdν(s)∥dt
≥ \frac{1}{c}∥ ∫_{G} ∫_{G} g(t − s)dtT_{−s}μdν(s)∥
= \frac{1}{c}ĝ(0)∥ ∫_{G} T_{−s}μdν(s)∥
= \frac{1}{c}∥ν ∗_{T} μ∥.  (20)

Inequalities (18), (19), and (20) imply that

∥ν ∗_{T} μ∥ ≤ c^{2}∥μ∥ .  (21)

Now fix ϵ > 0 and let k ∈ L^{1}(G) be such that k̂ has compact support in Γ. We have by Lemma 3.2

spec_{T} (k ∗_{T} μ) ⊂ supp(k̂) ∩ spec_{T} (μ) .

Moreover, the set {γ ∈ Γ : ∫_{G} |1 − γ|d|ν| < ϵ} is a neighborhood of 0 ∈ Γ. Hence, by the hypothesis on S, there is a neighborhood V of 0 in Γ and γ ∈ Γ such that

∫_{G} |1 − γ|d|ν| < ϵ and V + spec_{T} (k ∗_{T} μ) ⊂ S − γ .

If h ∈ L_{S−γ}^{1}, then γh ∈ L_{S}^{1}; hence

∥(\overline{γ}ν) ∗ h∥_{1} = ∥ν ∗ (γh)∥_{1} ≤ ∥γh∥_{1} = ∥h∥_{1},

for all h ∈ L_{S−γ}^{1}. Since translation in L^{1}(G) is continuous, it follows that the map t ↦ T_{t}(k ∗_{T} μ) is continuous, and hence we obtain from (21),

∥(\overline{γ}ν) ∗_{T} (k ∗_{T} μ)∥ ≤ c^{2}∥k ∗_{T} μ∥ .

As ∥\overline{γ}ν − ν∥ < ϵ and ϵ is arbitrary, we get

∥k ∗_{T} (ν ∗_{T} μ)∥ = ∥ν ∗_{T} (k ∗_{T} μ)∥ ≤ c^{2}∥k ∗_{T} μ∥ ≤ c^{3}∥μ∥ .

Letting k run through an approximate identity of L^{1}(G) and using Lemma 3.2, we obtain ∥ν ∗_{T} μ∥ ≤ Cc^{3}∥μ∥.

4 Transference in Spaces of Analytic measures

In the remainder of this paper, we transfer a result from Littlewood-Paley theory in H^{1}(ℝ) to spaces of measures on which ℝ is acting. We then show how this transferred result implies with ease several of the main results of Bochner [3], de Leeuw and Glicksberg [6], and Forelli [9].

We start by setting our notation. Let

H^{∞}(ℝ) = {f ∈ L^{∞}(ℝ) : ∫_{ℝ} f(t)g(t) dt = 0 for all g ∈ H^{1}(ℝ)} .

Let M(Σ) = M(Ω, Σ) denote a space of measures and let T = (T_{t})_{t∈ℝ} denote a sup path attaining representation of ℝ by isomorphisms of M(Σ). According to Forelli [9], a measure μ ∈ M(Σ) is called T -analytic if spec_{T} (μ) ⊂ [0, ∞). We now introduce another equivalent definition.

Definition 4.1 Suppose that T is a sup path attaining representation of ℝ by isomorphisms of M(Σ). A measure μ ∈ M_{T} (Σ) is called weakly analytic if the mapping t ↦ T_{t}μ(A) is in H^{∞}(ℝ) for every A ∈ Σ.

It is easy to see that if μ is T -analytic then it is weakly analytic. The converse is also true. The proof is based on the fact that [0, ∞) is a set of spectral synthesis (see [1, Proposition 1.7]).

For n ∈ ℤ, let m_{n} ∈ L^{1}(ℝ) be the function whose Fourier transform is piecewise linear and satisfies

m̂_{n}(s) = \{ \begin{matrix}0 & if s ∉ [2^{n−1}, 2^{n+1}]; \\ 1 & if s = 2^{n}. \end{matrix}  (22)

Let h ∈ L^{1}(ℝ) be the function whose Fourier transform is piecewise linear and satisfies

ĥ(s) = \{ \begin{matrix}0 & if s ∉ [−1, 1], \\ 1 & if |s| ≤ \frac{1}{2}. \end{matrix}  (23)

It is easy to check that

ĥ(s) + ∑_{n=0}^{∞} m̂_{n}(s) = \{ \begin{matrix}1 & if s ≥ −\frac{1}{2}, \\ 0 & if s ≤ −1, \end{matrix}  (24)

and that the left side of (24) is continuous and piecewise linear. The following theorem is a consequence of standard facts from Littlewood-Paley theory. We postpone its proof to the end of this section.

Theorem 4.2 (i) Let h and m_{n} be as above, f be any function in H^{1}(ℝ), and N be any nonnegative integer. Then there is a positive constant a, independent of f and N, such that

∥h ∗ f + ∑_{n=0}^{N} ϵ_{n}m_{n} ∗ f∥_{1} ≤ a∥f∥_{1}  (25)

for any choice of ϵ_{n} = −1 or 1. (ii) For f ∈ H^{1}(ℝ), we have

h ∗ f + lim_{N→∞} ∑_{n=0}^{N} m_{n} ∗ f = f

unconditionally in L^{1}(ℝ). (iii) For f ∈ H^{∞}(ℝ), we have

h ∗ f + lim_{N→∞} ∑_{n=0}^{N} m_{n} ∗ f = f  (26)

almost everywhere on ℝ.

Our main theorem is the following.

Theorem 4.3 Let T be a representation of ℝ in M(Σ) that is sup path attaining, and let h and m_{n}, n = 0, 1, 2, … be as in Theorem 4.2. Suppose that μ ∈ M(Σ) is weakly analytic. Then

∥h ∗_{T} μ + ∑_{n=0}^{N} ϵ_{n}m_{n} ∗_{T} μ∥ ≤ ac^{3}C∥μ∥  (27)

for any choice of ϵ_{n} ∈ {−1, 1}, where a is as in (25), c as in (1) and C as in (7). Moreover,

μ = h ∗_{T} μ + ∑_{n=0}^{∞} m_{n} ∗_{T} μ,  (28)

where the series converges unconditionally in M(Σ).

Proof. To prove (27), combine Theorems 1.6 and 4.2. Inequality (27) states that the partial sums of the series h ∗_{T} μ + ∑_{n=0}^{∞} m_{n} ∗_{T} μ are unconditionally bounded. To prove that they converge unconditionally, we recall the Bessaga-Pełczyński Theorem from [2]. This theorem tells us that for any Banach space, every unconditionally bounded series is unconditionally convergent if and only if the Banach space does not contain an isomorphic copy of c_{0}. Now since M(Σ) is weakly complete and c_{0} is not (see [7, Chap IV.9, Theorem 3, and IV.13.9]), we conclude that M(Σ) does not contain c_{0}. Applying the Bessaga-Pełczyński Theorem, we infer that there is a measure η ∈ M(Σ) such that

η = h ∗_{T} μ + ∑_{n=0}^{∞} m_{n} ∗_{T} μ  (29)

unconditionally in M(Σ). Moreover, η is weakly measurable, because of (29). It remains to show that η = μ. By Proposition 2.4, it is enough to show that for every A ∈ Σ, we have

T_{t}μ(A) = T_{t}η(A)  (30)

for almost every t ∈ ℝ. Since μ is weakly analytic, the function t ↦ T_{t}μ(A) is in H^{∞}(ℝ). By Theorem 4.2 (iii), we have

T_{t}μ(A) = h ∗ T_{t}μ(A) + ∑_{n=0}^{∞} m_{n} ∗ T_{t}μ(A)  (31)

for almost every t ∈ ℝ. On the other hand, from the unconditional convergence of the series in (29), (1), and (3), it follows that

T_{t}η(A) = h ∗ T_{t}μ(A) + ∑_{n=0}^{∞} m_{n} ∗ T_{t}μ(A)  (32)

for all t ∈ ℝ. Comparing (31) and (32), we see that (30) holds, completing the proof of the theorem.

Remarks 4.4 It is interesting to note that Theorem 4.3 implies the classical F. and M. Riesz theorem for measures defined on the real line. To see this, consider the representation of ℝ acting by translation on the Banach space of complex regular Borel measures on ℝ. It is easy to see that a regular Borel measure is analytic if and only if its Fourier-Stieltjes transform is supported in [0, ∞). In this case, each term in (28) belongs to L^{1}(ℝ), being the convolution of an L^{1}(ℝ) function with a regular Borel measure. Thus, the unconditional convergence of the series in (28) implies that the measure μ is absolutely continuous.

This argument provides a new proof of the F. and M. Riesz Theorem, based on Littlewood-Paley theory and the result of Bessaga-Pełczyński [2]. Also, it can be used to prove the following version of Bochner’s generalization of the F. and M. Riesz Theorem.

Theorem 4.5 Suppose that G is a locally compact abelian group with dual group Γ, and ψ : Γ → ℝ is a continuous homomorphism. Suppose that ν ∈ M(G) is such that, for every real number s, ψ^{−1}((−∞, s]) ∩ supp(ν̂) is compact. Then ν is absolutely continuous with respect to Haar measure on G. That is, ν ∈ L^{1}(G).

Proof. Let ϕ : ℝ → G denote the continuous adjoint homomorphism of ψ. Define a representation T = (T_{t})_{t∈ℝ} of ℝ on the regular Borel measures M(G) by

T_{t}(μ)(A) = μ(A + ϕ(t)),

for all μ ∈ M(G) and all Borel subsets A ⊂ G. By Example 2.2, T is sup path attaining, and every measure μ ∈ M(G) is weakly measurable. Moreover, μ ∈ M(G) is weakly analytic (equivalently, T -analytic) if and only if supp(μ̂) ⊂ ψ^{−1}([0, ∞)) (see [6]).

To prove the theorem, we can, without loss of generality, suppose that supp(ν̂) ⊂ ψ^{−1}([0, ∞)). Otherwise, we consider the measure \overline{χ}ν, where ψ(χ) + ψ(supp(ν̂)) ⊂ [0, ∞).

Let S = ψ^{−1}([0, ∞)). Then S is a T -set. Applying Theorem 4.3, we see that

ν = h ∗_{T} ν + ∑_{n=1}^{∞} m_{n} ∗_{T} ν,  (33)

unconditionally in M(G). For f ∈ L^{1}(ℝ), a straightforward calculation shows that { \atop f ∗_{T} ν}(χ) = f̂(ψ(χ))ν̂(χ) for all χ ∈ Γ. Since ψ^{−1}((−∞, s]) ∩ supp(ν̂) is compact for every s ∈ ℝ, it follows that supp({ \atop h ∗_{T} ν}) and supp({ \atop m_{n} ∗_{T} ν}) are compact. Thus h ∗_{T} ν and m_{n} ∗_{T} ν are in M(G) ∩ L^{2}(G), and hence they belong to L^{1}(G). As a consequence, (33) implies that ν ∈ L^{1}(G).

With Theorem 4.3 in hand, we can derive with ease several fundamental properties of analytic measures that were obtained previously by de Leeuw-Glicksberg [6], and Forelli [9]. We note however, that the techniques in [6] and [9] do not apply in our more general settings.

Theorem 4.6 Let T be a representation of ℝ in M(Σ) that is sup path attaining, and let μ be a weakly analytic measure in M(Σ). Then the mapping t ↦ T_{t}μ is continuous from ℝ into M(Σ).

Proof. Using the uniform continuity of translation in L^{1}(ℝ), it is a simple matter to show that for any function f ∈ L^{1}(ℝ), and any weakly measurable μ ∈ M(Σ), the mapping t ↦ f ∗_{T} T_{t}μ is continuous. Now use Theorem 4.3 to complete the proof.

Theorem 4.6 is very specific to representations of ℝ or 𝕋, in the sense that no similar result holds on more general groups. To see this, consider the group G = 𝕋 × 𝕋 with a lexicographic order on the dual group ℤ × ℤ. Let μ_{0} denote the normalized Haar measure on the subgroup {(x, y) : y = 0}, and consider the measure e^{−ix}μ_{0}. Its spectrum is supported on the coset {(m, 1) : m ∈ ℤ} and thus it is analytic with respect to the regular action of G by translation in M(G). Clearly, the measure e^{−ix}μ_{0} does not translate continuously, and so a straightforward analog of Theorem 4.6 fails in this setting.

The following application concerns bounded operators P from M(Σ) into M(Σ) that commute with T in the following sense:

P ∘ T_{t} = T_{t} ∘ P

for all t ∈ ℝ.

Theorem 4.7 Suppose that T is a representation of ℝ that is sup path attaining, and that P commutes with T. Let μ ∈ M(Σ) be weakly analytic. Then Pμ is also weakly analytic.

Proof. First note that by Theorem 4.6, the mapping t ↦ T_{t}μ(A) is continuous, and hence measurable.

Now suppose that g ∈ H^{1}(ℝ). Again, by Theorem 4.6, the map t ↦ g(t)T_{t}μ is Bochner integrable. Let

ν = ∫_{ℝ} g(t)T_{t}μdt.

Then by properties of the Bochner integral, and since μ is weakly analytic, we have that for all A ∈ Σ

ν(A) = ∫_{ℝ} g(t)T_{t}μ(A)dt = 0.

Hence ν = 0. Therefore, for all A ∈ Σ we have

∫_{ℝ} g(t)T_{t}(Pμ)(A)dt = ∫_{ℝ} g(t)P(T_{t}μ)(A)dt = Pν(A) = 0.

Since this is true for all g ∈ H^{1}(ℝ), it follows that Pμ is weakly analytic.

Definition 4.8 Let T be a sup path attaining representation of G in M(Σ). A weakly measurable σ in M(Σ) is called quasi-invariant if T_{t}σ and σ are mutually absolutely continuous for all t ∈ G. Hence if σ is quasi-invariant and A ∈ Σ, then |σ|(A) = 0 if and only if |T_{t}(σ)|(A) = 0 for all t ∈ G.

We can use Theorem 4.7 to generalize a result of de Leeuw-Glicksberg [6] and Forelli [9], concerning quasi-invariant measures. In this application, it is necessary to restrict to sup path attaining representations given by isometries of M(Σ). We need a lemma.

Lemma 4.9 Suppose that T is a linear isometry of M(Σ) onto itself. Let μ, ν ∈ M(Σ). Then,

(a) μ and σ are mutually singular (in symbols, μ ⊥ σ) if and only if T μ ⊥ T σ;

(b) μ << σ if and only if T μ << T σ.

Proof. For (a), simply recall that two measures μ and σ are mutually singular if and only if ∥μ + σ∥ = ∥μ∥ + ∥σ∥, and ∥μ − σ∥ = ∥μ∥ + ∥σ∥. For (b), it is clearly enough to prove the implication in one direction. So suppose that μ << σ and write μ = μ_{1} + μ_{2} where T μ_{1} << T σ and T μ_{2} ⊥ T σ. Then T μ_{1} ⊥ T μ_{2}. Hence μ_{1} ⊥ μ_{2}, and hence μ_{2} << μ << σ. But T μ_{2} ⊥ T σ implies that μ_{2} ⊥ σ. So μ_{2} = 0. Thus T μ = T μ_{1} << T σ.

Theorem 4.10 Suppose that T is a sup path attaining representation of ℝ by isometries of M(Σ). Suppose that μ ∈ M(Σ) is weakly analytic, and σ is quasi-invariant. Write μ = μ_{a} + μ_{s} for the Lebesgue decomposition of μ with respect to σ. Then both μ_{a} and μ_{s} are weakly analytic. In particular, the spectra of μ_{a} and μ_{s} are contained in [0, ∞).

Proof. Let P(μ) = μ_{s}. Since σ is quasi-invariant, the operator P commutes with T by Lemma 4.9. Now apply Theorem 4.7.

Let us finish with an example to show that the hypothesis of sup path attaining is required in these results. The next example is a variant of Example 2.5.

Example 4.11 Let Σ_{1} denote the sigma algebra of countable and co-countable subsets of ℝ, let Σ_{2} denote the Borel subsets of ℝ, and let Σ = Σ_{1} ⊗ Σ_{2} denote the product sigma algebra on ℝ × ℝ. Let ν_{1} : Σ_{1} → {0, 1} be the measure that takes countable sets to 0 and co-countable sets to 1, let δ_{t} : Σ_{1} → {0, 1} be the measure that takes sets to 1 if they contain t, and to 0 otherwise, and let ν_{2} denote the measure on Σ_{2} given by

ν_{2}(A) = ∫_{A} exp(−x^{2}) dx.

Let ν = ν_{1} ⊗ ν_{2}, let θ = δ_{0} ⊗ ν_{2}, and let μ = ν − θ. Finally, let T_{t} be the representation given by T_{t}(x, y) = (x + t, y + t).

Then, we see that ν is quasi-invariant, and that θ and ν are mutually singular. Arguing as in Example 2.5, we see that μ is weakly analytic. However, the singular part of μ with respect to ν is −θ, and it may be readily seen that this is not weakly analytic, for example

T_{t}θ(ℝ × [−1, 1]) = ∫_{−1}^{1} exp(−(x − t)^{2}) dx,

is not in H^{∞}(ℝ).

We end this section by proving Theorem 4.2. We have

∑_{n=−∞}^{∞} m̂_{n}(s) = \{ \begin{matrix}1 & if s > 0; \\ 0 & if s ≤ 0. \end{matrix}  (34)

Recall the Fejér kernels {k_{a}}_{a>0}, where

k̂_{a}(s) = \{ \begin{matrix}1 − \frac{|s|}{a} & if |s| < a; \\ 0 & otherwise. \end{matrix}  (35)

By Fourier inversion, we see that

m_{n}(x) = exp(i2^{n}x)k_{2^{n−1}}(x) + \frac{1}{2} exp(i3 2^{n−1}x)k_{2^{n−1}}(x).  (36)

Theorem 4.12 (i) Let f be any function in H^{1}(ℝ), and let M and N be arbitrary positive integers. Then there is a positive constant a, independent of f, M, and N such that

∥ ∑_{n=−M}^{N} ϵ_{n}m_{n} ∗ f∥_{1} ≤ a∥f∥_{1}  (37)

for any choice of ϵ_{n} = −1 or 1. (ii) Moreover, for f ∈ H^{1}(ℝ),

lim_{M,N→∞} ∑_{n=−M}^{N} m_{n} ∗ f = f

unconditionally in L^{1}(ℝ).

Proof. The proof of (ii) is immediate from (i) and (34), by Fourier inversion. For part (i), use (36), to write

∑_{n=−M}^{N} ϵ_{n}m_{n} = ∑_{n=−M}^{N} ϵ_{n} exp(i2^{n}x)k_{2^{n−1}}(x) + \frac{1}{2} exp(i3 2^{n−1}x)k_{2^{n−1}}(x)

= ∑_{n=−M}^{−1} ϵ_{n} exp(i2^{n}x)k_{2^{n−1}}(x) + ∑_{n=0}^{N} ϵ_{n} exp(i2^{n}x)k_{2^{n−1}}(x)

+ \frac{1}{2} ∑_{n=−M}^{−1} ϵ_{n} exp(i3 2^{n−1}x)k_{2^{n−1}}(x) + \frac{1}{2} ∑_{n=0}^{N} ϵ_{n} exp(i3 2^{n−1}x)k_{2^{n−1}}(x)
= K_{1}(x) + K_{2}(x) + K_{3}(x) + K_{4}(x).

Hence, to prove (37) it is enough to show that there is a positive constant a, independent of f such that ∥K_{j} ∗ f∥_{1} ≤ a∥f∥_{1}, for j = 1, 2, 3, 4. Appealing to [15, Theorem 3, p. 114], we will be done once we establish that:

|K̂_{j}| ≤ A,  (38)

and the Hörmander condition

sup_{y>0} ∫_{|x|>2y} |K_{j}(x − y) − K_{j}(x)|dx ≤ B,  (39)

where A and B are absolute constants. Inequality (38) holds with A = 1, since the Fourier transforms of the summands defining the kernels K_{j} have disjoint supports and are bounded by 1. Condition (39), is well-known. For a proof, see [8, pp. 138-140, and 7.2.2, p. 142].

Proof of Theorem 4.2. Parts (i) and (ii) follow as in Theorem 4.12, so we only prove (iii). For notational convenience, let

κ_{N}(x) = ∑_{n=0}^{N} m_{n}(x), N = 0, 1, 2 …  (40)

Let V_{2^{N}} denote the de la Vallée Poussin kernels on ℝ of order 2^{N}. Its Fourier transform is continuous, piecewise linear, and satisfies

V̂_{2^{N}}(s) = \{ \begin{matrix}0 & if s ∉ [−2^{n+1}, 2^{n+1}]; \\ 1 & if |s| ≤ 2^{n}. \end{matrix}  (41)

It is well-known that V_{n} is a summability kernel for L^{1}(ℝ) and, in particular, that V_{2^{N}} ∗ f converges pointwise almost everywhere to f for all f ∈ L^{p}(ℝ), for 1 ≤ p ≤ ∞. Thus, we will be done, if we can show that for f ∈ H^{∞}(R),

V_{2^{N}} ∗ f = h ∗ f + ∑_{n=0}^{N} m_{n} ∗ f.  (42)

Write V_{2^{N}} = (h + κ_{N}) + (V_{2^{N}} − (h + κ_{N})), where κ_{N} is as in (40). For N ≥ 1, the Fourier transform of (V_{2^{N}} − (h + κ_{N})), vanishes on [−\frac{1}{2}, ∞). Thus, for f ∈ H^{∞}(ℝ), we have (V_{2^{N}} − (h + κ_{N})) ∗ f = 0, and so (42) follows, and the proof is complete.

Acknowledgements The work of the second author was supported by a grant from the National Science Foundation (U.S.A.).

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