Decomposition of analytic measures on groups and measure spaces

Nakhlé Asmar and Stephen Montgomery-Smith Department of Mathematics University of Missouri Columbia, MO 65211

          nakhle@math.missouri.edu
         stephen@math.missouri.edu

Dedicated to the memory of Edwin Hewitt

Abstract

In this paper, we consider an arbitrary locally compact abelian group G, with an ordered dual group Γ, acting on a space of measures. Under suitable conditions, we define the notion of analytic measures using the representation of G and the order on Γ. Our goal is to study analytic measures by applying a new transference principle for subspaces of measures, along with results from probability and Littlewood-Paley theory. As a consequence, we will derive new properties of analytic measures as well as extensions of previous work of Helson and Lowdenslager, de Leeuw and Glicksberg, and Forelli.

A.M.S. Subject Classification: 43A17, 43A32,

Keywords: orders, transference, measure space, sup path attaining, F.&M. Riesz Theorem

1 Introduction

This paper is essentially providing a new approach to generalizations of the F.&M. Riesz Theorems, for example, such results as that of Helson and Lowdenslager [16, 17]. They showed that if G is a compact abelian group with ordered dual, and if μ is an analytic measure (that is, its Fourier transform is supported on the positive elements of the dual), then it follows that the singular and absolutely continuous parts (with respect to the Haar measure) are also analytic.

Another direction is that provided by Forelli [12] (itself a generalization of the result of de Leeuw and Glicksberg [9]), where one has an action of the real numbers ℝ acting on a locally compact topological space Ω, and a Baire measure μ on Ω that is analytic (in a sense that we make precise below) with respect to the action. Then again, the singular and absolutely continuous parts of μ (with respect to any so called quasi-invariant measure) are also analytic.

Indeed common generalizations of both these ideas have been provided, for example, by Yamaguchi [23], considering the action of any locally compact abelian group with ordered dual, on a locally compact topological space. For more generalizations we refer the reader to Hewitt, Koshi, and Takahashi [19].

In the paper [4], a new approach to proving these kinds of results was given, providing a transference principle for spaces of measures. In that paper, the action was from a locally compact abelian group into a space of isomorphisms on the space of measures of a sigma algebra. A primary requirement that the action had to satisfy was what was called sup path attaining, a property that was satisfied, for example, by the setting of Forelli (Baire measures on a locally compact topological space). Using this transference principle, the authors were able to give an extension and a new proof of Forelli’s result. This was obtained by using a Littlewood-Paley decomposition of an analytic measure.

In this paper we wish to continue this process, applying this same transference principle to provide the common generalizations of the results of Forelli and Helson and Lowdenslager. What we provide in this paper is essentially a decomposition of an analytic measure as a sum of martingale differences with respect to a filtration defined by the order. For each martingale difference, the action of the group can be described precisely by a certain action of the group of real numbers, and so we can appeal to the results of [4].

In this way, we can reach the following generalization (see Theorem 6.4 below): if P is any bounded operator on the space of measures that commutes with the action (as does, for example, taking the singular part), and if μ is an analytic measure, then Pμ is also an analytic measure.

In the remainder of the introduction, we will establish our notation, including the notion of sup path attaining, and recall the transference principle from [4]. In Section 2, we will describe orders on locally compact abelian groups, including the extension of Hahn’s Embedding Theorem provided in [1]. In Section 3, we define the notions of analyticity. This somewhat technical section continues into Section 4, which examines the role of homomorphism with respect to analyticity. The technical results basically provide proofs of what is believable, and so may be skipped on first reading. It will be seen that the concept of sup path attaining comes up again and again, and may be seen to be an integral part of all our proofs.

In Section 5, we are ready to present the decomposition of analytic measures. This depends heavily on transference of martingale inequalities of Burkholder and Garling, and then using the fact that weakly unconditionally summing series are unconditionally summing in norm for any series in a space of measures [5]. In Section 6, we then give applications of this decomposition, giving the generalizations that we alluded to above.

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 [20, Definition (11.26)].

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

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

(1)  ∥T_{t}∥ ≤ c.

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

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

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,

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

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

(4)  T_{t}^{∗} : L^{∞}(Σ) → L^{∞}(Σ).

For proofs we refer the reader to [2]. Using (1) and (3), it can be shown that ν ∗_{T} μ is a measure in M_{T} (Σ),

(5)  ∥ν ∗_{T} μ∥ ≤ c∥ν∥∥μ∥,

where c is as in (1), and

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

for all σ, ν ∈ M(G) and μ ∈ M_{T} (Σ) (see [2]).

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

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

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.

In [4] were provided many examples of sup path attaining representations. Rather than give this same list again, we give a couple of examples of particular interest.

Example 1.3 (a) (This is the setting of Forelli’s Theorem.) Let G be a locally compact abelian group, and Ω be a locally compact topological space. Suppose that (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. Then the space of Baire measures on Ω, that is, the minimal sigma algebra such that compactly supported continuous functions are measurable, is sup path attaining under the action T_{t}μ(A) = μ(T_{t}(A)), where T_{t}(A) = {T_{t}ω : ω ∈ A}. (Note that all Baire measures are weakly measurable.)

(b) 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, and the representation is sup path attaining with constants c = 1 and C = 1.

Proposition 1.4 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 (see [2]).

We now 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}.

When we need to be specific about the representation, we will use the symbol I_{T} (μ) instead of I(μ).

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

Definition 1.5 The T -spectrum of a weakly measurable μ ∈ M_{T} (Σ) is defined by

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

If S ⊂ Γ, let

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

In order to state the main transference result, we introduce one more definition.

Definition 1.6 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 1.7 (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 [4] is the following transference theorem.

Theorem 1.8 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

(9)  ∥ν ∗ f∥_{1} ≤ ∥f∥_{1}

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

(10)  ∥ν ∗_{T} μ∥ ≤ c^{3}C∥μ∥,

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

2 Orders on locally compact abelian groups

An order P on Γ is a subset that satisfies the three axioms: P + P ⊂ P; P ∪(−P) = Γ; and P ∩(−P)−{0}. We recall from [1] the following property of orders.

Theorem 2.1 Let P be a measurable order on Γ. There are a totally ordered set Π with largest element α_{0}; a chain of subgroups {C_{α}}_{α∈Π} of Γ; and a collection of continuous real-valued homomorphisms {ψ_{α}}_{α∈Π} on Γ such that:

(i) for each α ∈ Π, C_{α} is an open subgroup of Γ; (ii) C_{α} ⊂ C_{β} if α > β. Let D_{α} = {χ ∈ C_{α} : ψ_{α}(χ) = 0}. Then, for every α ∈ Π, (iii) ψ_{α}(χ) > 0 for every χ ∈ P ∩ (C_{α} \ D_{α}), (iv) ψ_{α}(χ) < 0 for every χ ∈ (−P) ∩ (C_{α} \ D_{α}).

(v) When Γ is discrete, C_{α_{0}} = {0}; and when Γ is not discrete, D_{α_{0}} has empty interior and is locally null.

When Γ is discrete, Theorem 2.1 can be deduced from the proof of Hahn’s Embedding Theorem for orders (see [13, Theorem 16, p.59]). The general case treated in Theorem 2.1 accounts for the measure theoretic aspect of orders. The proof is based on the study of orders of Hewitt and Koshi [18].

For α ∈ Π with α ≠ α_{0}, let

(11)  S_{α} ≡ P ∩ (C_{α} \ D_{α}) = {χ ∈ C_{α} \ D_{α} : ψ_{α}(χ) ≥ 0}
(12)  = {χ ∈ C_{α} : ψ_{α}(χ) > 0} .

For α = α_{0}, set

(13)  S_{α_{0}} = {χ ∈ C_{α_{0}} : ψ_{α_{0}}(χ) ≥ 0} .

Note that when Γ is discrete, C_{α_{0}} = {0}, and so S_{α_{0}} = {0} in this case.

If A is a subset of a topological space, we will use \overline{A} and A^{∘} to denote the closure, respectively, the interior of A.

Remarks 2.2 (a) It is a classical fact that a group Γ can be ordered if and only if it is torsion-free. Also, an order on Γ is any maximal positively linearly independent set. Thus, orders abound in torsion-free abelian groups, as they can be constructed using Zorn’s Lemma to obtain a maximal positively linearly independent set. (See [18, Section 2].) However, if we ask for measurable orders, then we are restricted in many ways in the choices of P and also the topology on Γ. As shown in [18], any measurable order on Γ has nonempty interior. Thus, for example, while there are infinitely many orders on ℝ, only two are Lebesgue measurable: P = [0, ∞[, and P =] − ∞, 0]. It is also shown in [18, Theorem (3.2)] that any order on an infinite compact torsion-free abelian group is non-Haar measurable. This effectively shows that if Γ contains a Haar-measurable order P, and we use the structure theorem for locally compact abelian groups to write Γ as ℝ^{a} × Δ, where Δ contains a compact open subgroup [20, Theorem (24.30)], then either a is a positive integer, or Γ is discrete. (See [1].)

(b) The subgroups (C_{α}) are characterized as being the principal convex subgroups in Γ and for each α ∈ Π, we have

D_{α} = ⋃_{β>α} C_{β}.

Consequently, we have C_{α} ⊂ D_{β} if β < α. By construction, the sets C_{α} are open. For α < α_{0}, the subgroup D_{α} has nonempty interior, since it contains C_{β}, with α < β. Hence for α ≠ α_{0}, D_{α} is open and closed. Consequently, for α ≠ α_{0}, C_{α} \ D_{α} is open and closed.

(c) Let ψ : Γ_{1} → Γ_{2} be a continuous homomorphism between two ordered groups. We say that ψ is order-preserving if ψ(P_{1}) ⊂ P_{2}. Consequently, if ψ is continuous and order preserving, then ψ(\overline{P_{1}}) ⊂ \overline{P_{2}}.

For each α ∈ Π, let π_{α} denote the quotient homomorphism Γ → Γ/C_{α}. Because C_{α} is a principal subgroup, we can define an order on Γ/C_{α} by setting ψ_{α}(χ) ≥ 0 ⟺ χ ≥ 0. Moreover, the principal convex subgroups in Γ/C_{α} are precisely the images by π_{α} of the principal convex subgroups of Γ containing C_{α}. (See [1, Section 2].)

We end this section with a useful property of orders.

Proposition 2.3 Let P be a measurable order on Γ. Then \overline{P} is a T -set.

Proof. If Γ is discrete, there is nothing to prove. If Γ is not discrete, the subgroup C_{α_{0}} is open and nonempty. Hence the set C_{α_{0}} ∩ {χ ∈ Γ : ψ_{α_{0}}(χ) > 0} is nonempty, with 0 as a limit point. Given an open nonempty neighborhood U of 0, let

W = U ∩ C_{α_{0}} ∩ {χ ∈ Γ : ψ_{α_{0}}(χ) > 0}.

Then W is a nonempty subset of U ∩ P. Moreover, it is easy to see that W + \overline{P} ⊂ P ⊂ \overline{P}, and hence \overline{P} is a T -set.

3 Analyticity

We continue with the notation of the previous section. Using the order structure on Γ we define some classes of analytic functions on G:

(14)  H^{1}(G) = {f ∈ L^{1}(G) : f̂ = 0 on (−P) \ {0}};
(15)  H_{0}^{1}(G) = {f ∈ L^{1}(G) : f̂ = 0 on − P };

and

(16) H^{∞}(G) = {f ∈ L^{∞}(G) : ∫_{G} f(x)g(x)dx = 0 for all g ∈ H_{0}^{1}(G)} .

We clearly have

H^{1}(G) = \{f ∈ L^{1}(G) : f̂ = 0 on \overline{(−P) \ \{0\}}\} .

We can now give the definition of analytic measures in M_{T} (Σ).

Definition 3.1 Let T be a sup path attaining representation of G 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^{∞}(G) for every A ∈ Σ.

Definition 3.2 Recall the T -spectrum of a weakly measurable μ ∈ M_{T} (Σ),

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

A measure μ in M_{T} (Σ) is called T -analytic if spec_{T} (μ) ⊂ \overline{P}.

That the two definitions of analyticity are equivalent will be shown later in this section.

Since I(μ) is translation-invariant, it follows readily that for all t ∈ G,

I(T_{t}μ) = I(μ),

and hence

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

We now recall several basic results from spectral theory of bounded functions that will be needed in the sequel. Our reference is [21, Section 40]. 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

(19)  σ [ϕ] = Z(I(ϕ)).

(See [21, Theorem (40.5)].)

Recall that a closed subset E of Γ is a set of spectral synthesis for L^{1}(G), or an S-set, if and only if I([E]) is the only ideal in L^{1}(G) whose zero set is E.

There are various equivalent definitions of S-sets. Here is one that we will use at several occasions. A set E ⊂ Γ is an S-set if and only if every essentially bounded function g in L^{∞}(G) with σ[g] ⊂ E is the weak-* limit of linear combinations of characters from E. (See [21, (40.23) (a)].) This has the following immediate consequence.

Proposition 3.3 Suppose that B is an S-set, g ∈ L^{∞}(G), and spec(g) ⊂ B. (i) If f is in L^{1}(G) and f̂ = 0 on B, then f ∗ g(x) = 0 for all x in G. In particular,

∫_{G} f(x)g(−x) d x = 0.

(ii) If μ is a measure in M(G) with μ̂ = 0 on B, then μ ∗ g(x) = 0 for almost all x in G.

Proof. Part (i) is a simple consequence of [21, Theorems (40.8) and (40.10)]. We give a proof for the sake of completeness. Write g as the weak- * limit of trigonometric polynomials, ∑_{χ∈E} a_{χ}χ(x), with characters in E. Then

∫_{G} f(x)g(y − x) d x = lim ∫_{G} ∑_{χ∈E} a_{χ}χ(y)f(x)χ(−x) d x
= lim ∑_{χ∈E} a_{χ}χ(y)f̂(χ) = 0

since f̂ vanishes on E.

To prove (ii), assume that μ ∗ g is not 0 a.e.. Then, there is f in L^{1}(G) such that f ∗ (μ ∗ g) is not 0 a.e.. But this contradicts (i), since f ∗ (μ ∗ g) = (f ∗ μ) ∗ g, f ∗ μ is in L^{1}(G), and { \atop f ∗ μ} = 0 on B.

The following is a converse of sorts of Proposition 3.3 and follows easily from definitions.

Proposition 3.4 Let B be a nonvoid closed subset of Γ. Suppose that f is in L^{∞}(G) and

(20)  ∫_{G} f(x)g(x)dx = 0

for all g in L^{1}(G) such that ĝ = 0 on −B. Then σ[f] ⊂ B.

Proof. Let χ_{0} be any element in Γ \ B. We will show that χ_{0} is not in the spectrum of f by constructing a function h in L^{1}(G) with ĥ(χ_{0}) ≠ 0 and h ∗ f = 0. Let U be an open neighborhood of χ_{0} not intersecting B, and let h be in L^{1}(G) such that ĥ is equal to 1 at χ_{0} and to 0 outside U. Direct computations show that the Fourier transform of the function g : t ↦ h(x − t), when evaluated at χ ∈ Γ, gives \overline{χ(x)}ĥ(−χ), and hence it vanishes on −B. It follows from (20) that h ∗ f = 0, which completes the proof.

A certain class of S-sets, known as the Calderón sets, or C-sets, is particularly useful to us. These are defined as follows. A subset E of Γ is called a C-set if every f in L^{1}(G) with Fourier transform vanishing on E can be approximated in the L^{1}-norm by functions of the form h ∗ f where h ∈ L^{1}(G) and ĥ vanishes on an open set containing E. C-sets enjoy the following properties (see [21, (39.39)] or [22, Section 7.5]).

• Every C-set is an S-set.

• Every closed subgroup of Γ is a C-set.

• The empty set is a C-set.

• If the boundary of a set A is a C-set, then A is a C-set.

• Finite unions of C-sets are C-sets.

Since closed subgroups are C-sets, we conclude that \overline{P} ∩ \overline{(−P)}, and C_{α}, for all α, are C-sets. ¿From the definition of S_{α_{0}}, (13), and the fact that C_{α_{0}} is open and closed, it follows that the boundary of S_{α_{0}} is the closed subgroup ψ_{α_{0}}^{−1}(0) ∩ C_{α_{0}}. Hence S_{α_{0}} is a C-set. For α ≠ α_{0}, the set S_{α} is open and closed, and so it has empty boundary, and thus it is a C-set. Likewise C_{α} \ D_{α} is a C-set for all α ≠ α_{0}. ¿From this we conclude that arbitrary unions of S_{α} and C_{α} \ D_{α} are C-sets, because an arbitrary union of such sets, not including the index α_{0}, is open and closed, and so it is a C-set. We summarize our findings as follows.

Proposition 3.5 Suppose that P is a measurable order on Γ. We have:

(i) \overline{P} and \overline{(−P)} are C-sets; (ii) S_{α} is a C-set for all α; (iii) arbitrary unions of S_{α} and C_{α} \ D_{α} are C-sets.

As an immediate application, we have the following characterizations.

Corollary 3.6 Suppose that f is in L^{∞}(G), then

(i) σ[f] ⊂ S_{α} if and only if ∫_{G} f(x)g(x)dx = 0 for all g ∈ L^{1}(G) such that ĝ = 0 on −S_{α}; (ii) σ[f] ⊂ Γ \ C_{α} if and only if μ_{α} ∗ f = 0; (iii) σ[f] ⊂ \overline{P} if and only if f ∈ H^{∞}(G).

Proof. Assertions (i) and (iii) are clear from Propositions 3.5 and 3.4. To prove (ii), use Fubini’s Theorem to first establish that for any g ∈ L^{1}(G), and any μ ∈ M(G), we have

∫_{G}(μ ∗ f)(t)g(t)dt = ∫_{G} f(t)(μ ∗ g)(t)dt.

Now suppose that σ[f] ⊂ Γ \ C_{α}, and let g be any function in L^{1}(G). From Propositions 3.5 and 3.4, we have that ∫_{G} fgdt = 0 for all g with Fourier transform vanishing on Γ \ C_{α}, equivalently, for all g = μ_{α} ∗ g. Hence, ∫_{G} f(μ_{α} ∗ g)dt = ∫_{G}(μ_{α} ∗ f)gdt = 0 for all g in L^{1}(G), from which it follows that μ_{α} ∗ f = 0. The converse is proved similarly, and we omit the details.

Aiming for a characterization of weakly analytic measures in terms of their spectra, we present one more result.

Proposition 3.7 Let μ be weakly measurable in M(Σ).

(i) Suppose that B is a nonvoid closed subset of Γ and spec_{T} μ ⊂ B. Then σ[t ↦ T_{t}μ(A)] ⊂ B for all A ∈ Σ. (ii) Conversely, suppose that B is an S-set in Γ and that σ[t ↦ T_{t}μ(A)] ⊂ B for all A ∈ Σ, then spec_{T} μ ⊂ B.

Proof. We clearly have I(μ) ⊂ I([t ↦ T_{t}μ(A)]). Hence, spec_{T} μ = Z(I(μ)) ⊃ Z(I([t ↦ T_{t}μ(A)])) = σ[t ↦ T_{t}μ(A)], and (i) follows. Now suppose that B is an S-set and let g ∈ L^{1}(G) be such that ĝ = 0 on −B. Then, for all A ∈ Σ, we have from Proposition 3.4:

∫_{G} g(t)T_{t}μ(A)dt = 0.

Equivalently, we have that

∫_{G} g(−t)T_{−t}μ(A)dt = 0.

Since the Fourier transform of the function t ↦ g(−t) vanishes on B, we see that I(μ) ⊃ {f : f̂ = 0 on B}. Thus Z(I(μ)) ⊂ Z({f : f̂ = 0 on B}) = B, which completes the proof.

Straightforward applications of Propositions 3.5 and 3.7 yield the desired characterization of weakly analytic measures.

Corollary 3.8 Suppose that μ ∈ M_{T} (Σ). Then,

(i) μ is weakly T −analytic if and only if spec_{T} μ ⊂ \overline{P} if and only if σ[t ↦ T_{t}μ(A)] ⊂ \overline{P}, for every A ∈ Σ; (ii) spec_{T} μ ⊂ S_{α} if and only if σ[t ↦ T_{t}μ(A)] ⊂ S_{α} for every A ∈ Σ. (iii) spec_{T} μ ⊂ C_{α} if and only if σ[t ↦ T_{t}μ(A)] ⊂ C_{α} for every A ∈ Σ. (iv) spec_{T} μ ⊂ Γ \ C_{α} if and only if σ[t ↦ T_{t}μ(A)] ⊂ Γ \ C_{α} for every A ∈ Σ.

The remaining results of this section are simple properties of measures in M_{T} (Σ) that will be needed later. Although the statements are direct analogues of classical facts about measures on groups, these generalization require in some places the sup path attaining property of T.

Proposition 3.9 Suppose that μ ∈ M_{T} (Σ) and ν ∈ M(G). Then spec_{T} ν∗_{T} μ is contained in the support of ν̂, and spec_{T} ν ∗_{T} μ ⊂ spec_{T} μ.

Proof. Given χ_{0} not in the support of ν̂, to conclude that it is also not in the spectrum of ν ∗_{T} μ it is enough to find a function f in L^{1}(G) with f̂(χ_{0}) = 1 and f ∗_{T} (ν ∗_{T} μ) = 0. Simply choose f with Fourier transform vanishing on the support of ν̂ and taking value 1 at χ_{0}. By Fourier inversion, we have f ∗ ν = 0, and since f ∗_{T} (ν ∗_{T} μ) = (f ∗ ν) ∗_{T} μ, the first part of the proposition follows. For the second part, we have I(μ) ⊂ I(ν ∗_{T} μ), which implies the desired inclusion.

We next prove a property of L^{∞}(G) functions similar to the characterization of L^{1} functions which are constant on cosets of a subgroup [21, Theorem (28.55)].

Proposition 3.10 Suppose that f is in L^{∞}(G) and that Λ is an open subgroup of Γ. Let λ_{0} denote the normalized Haar measure on the compact group A(G, Λ), the annihilator in G of Λ (see [20, (23.23)]. Then, σ[f] ⊂ Λ if and only if f = f ∗λ_{0} a. e. This is also the case if and only if f is constant on cosets of A(G, Λ).

Proof. Suppose that the spectrum of f is contained in Λ. Since Λ is an S-set, it follows that f is the weak-* limit of trigonometric polynomials with spectra contained in Λ. Let {f_{α}} be a net of such trigonometric polynomials converging to f weak-*. Note that, for any α, we have λ_{0} ∗ f_{α} = f_{α}. For g in L^{1}(G), we have

lim_{α} ∫_{G} f_{α}\overline{g}dx = ∫_{G} f\overline{g}dx.

In particular, we have

lim_{α} ∫_{G} f_{α}(λ_{0} ∗ \overline{g})dx = ∫_{G} f(λ_{0} ∗ \overline{g})dx,

and so

lim_{α} ∫_{G}(f_{α} ∗ λ_{0})\overline{g}dx = ∫_{G}(f ∗ λ_{0})\overline{g}dx.

Since this holds for any g in L^{1}(G), we conclude that λ_{0} ∗ f_{α} converges weak-* to λ_{0} ∗ f. But λ_{0} ∗ f_{α} = f_{α}, and f_{α} converges weak-* to f, hence f ∗ λ_{0} = f. The remaining assertions of the lemma are easy to prove. We omit the details.

In what follows, we use the symbol μ_{α} to denote the normalized Haar measure on the compact subgroup A(G, C_{α}), the annihilator in G of C_{α}. This measure is also characterized by its Fourier transform:

μ̂_{α} = 1_{C_{α}}

(see [20, (23.19)]).

Corollary 3.11 Suppose that μ ∈ M_{T} (Σ). Then,

(i) spec_{T} μ ⊂ C_{α} if and only if μ = μ_{α} ∗_{T} μ; (ii) spec_{T} μ ⊂ Γ \ C_{α} if and only if μ_{α} ∗_{T} μ = 0.

Proof. (i) If μ = μ_{α} ∗_{T} μ, then, by Proposition 3.10, σ[t ↦ μ_{α} ∗_{T} T_{t}μ(A)] ⊂ C_{α}. Hence by Corollary 3.8, spec_{T} μ ⊂ C_{α}. For the other direction, suppose that spec_{T} μ ⊂ C_{α}. Then by Corollary 3.8 we have that the spectrum of the function t ↦ T_{t}μ(A) is contained in C_{α} for every A ∈ Σ. By Proposition 3.10, we have that

T_{t}μ(A) = ∫_{G_{α}} T_{t−y}μ(A)dμ_{α} = T_{t}(μ_{α} ∗ μ)(A)

for almost all t ∈ G. Since this holds for all A ∈ Σ, the desired conclusion follows from Proposition 1.4. Part (ii) follows from Corollary 3.6 (ii), Proposition 3.7(ii), and the fact that Γ \ C_{α} is an S-set.

Corollary 3.12 Suppose that μ ∈ M_{T} (Σ) and spec_{T} μ ⊂ C_{α}, and let y ∈ G_{α} = A(G, C_{α}). Then T_{y}μ = μ.

Proof. For any A ∈ Σ, we have from Corollary 3.11

T_{y}μ(A) = T_{y}(μ_{α} ∗ μ)(A) = μ_{α} ∗ T_{y}μ(A)
= ∫_{G_{α}} T_{y−x}μ(A)dμ_{α}(y)
= ∫_{G_{α}} T_{−x}μ(A)dμ_{α}(y) = μ_{α} ∗ μ(A) = μ(A).

4 Homomorphism theorems

We continue with the notation of the previous section: G is a locally compact abelian group, Γ the dual group of G, P is a measurable order on Γ, T is a sup path attaining representation of G acting on M(Σ). Associated with P is a collection of homomorphisms ψ_{α}, as described by Theorem 2.1. Let ϕ_{α} denote the adjoint of ψ_{α}. Thus, ϕ_{α} is a continuous homomorphism of ℝ into G. By composing the representation T with the ϕ_{α}, we define a new representation T_{ϕ_{α}} of ℝ acting on M(Σ) by: t ∈ ℝ ↦ T_{ϕ_{α}(t)}. If μ in M(Σ) is weakly measurable with respect to T then μ is also weakly measurable with respect to T_{ϕ_{α}}. We will further suppose that T_{ϕ_{α}} is sup path attaining for each α. This is the case with the representations of Example 1.7.

Our goal in this section is to relate the notion of analyticity with respect to T to the notion of analyticity with respect to T_{ϕ_{α}}. More generally, suppose that G_{1} and G_{2} are two locally compact abelian groups with dual groups Γ_{1} and Γ_{2}, respectively. Let

ψ : Γ_{1} → Γ_{2}

be a continuous homomorphism, and let ϕ : G_{2} → G_{1} denote its adjoint homomorphism. Suppose ν is in M(G_{2}). We define a Borel measure Φ(ν) in M(G_{1}) on the Borel subsets A of G_{1} by:

(21)  Φ(ν)(A) = ∫_{G_{2}} 1_{A} ∘ ϕ(t) dν(t) = ∫_{G_{1}} 1_{A}dΦ(ν),

where 1_{A} is the indicator function of A. We have ∥Φ(g)∥_{M(G_{1})} = ∥ν∥_{M(G_{2})} and, for every Borel measurable bounded function f on G_{1}, we have

(22)  ∫_{G_{1}} fdΦ(ν) = ∫_{G_{2}} f ∘ ϕ(t) dν(t).

In particular, if f = χ, a character in Γ, then (23)

{ \atop Φ(ν)}(χ) = ∫_{G_{1}} \overline{χ}dΦ(ν) = ∫_{G_{2}} \overline{χ} ∘ ϕ(t) dν(t) = ∫_{G_{2}} \overline{ψ(χ)}(t) dν(t) = ν̂(ψ(χ)),

where ψ is the adjoint homomorphism of ϕ. So,

(24)  { \atop Φ(ν)} = ν̂ ∘ ψ.

Our first result is a very useful fact from spectral synthesis of bounded functions. The proof uses in a crucial way the fact that the representation is sup path attaining, or, more precisely, satisfies the property in Proposition 1.4.

Lemma 4.1 Suppose that T is a sup path attaining representation of G_{1} acting on M(Σ), ϕ is a continuous homomorphism of G_{2} into G_{1} such that T_{ϕ} is a sup path attaining representation of G_{2}. Suppose that B is a nonempty closed S-subset of Γ_{1} and that μ is in M(Σ) with spec_{T} μ ⊂ B. Suppose that C is an S-subset of Γ_{2} and ψ(B) ⊂ C. Then spec_{T_{ϕ}}μ ⊂ C.

Proof. Since C is an S-subset of Γ_{2}, it is enough to show that for every ‘A ∈ Σ, spec_{T_{ϕ}}(x ↦ T_{ϕ(x)}μ(A)) ⊂ C, by Proposition 3.7. For this purpose, it is enough by [21, Theorem (40.8)], to show that

g ∗ T_{ϕ(·)}μ(A) = 0

for every g in L^{1}(G_{2}) with ĝ = 0 on C. For r ∈ G_{2} and x ∈ G_{1}, consider the measure

T_{x}(g ∗_{T_{ϕ}} T_{ϕ(r)}μ) = g ∗_{T_{ϕ}} T_{x+ϕ(r)}μ.

For A ∈ Σ, we have

g ∗_{T_{ϕ}} T_{x+ϕ(r)}μ(A) = ∫_{G} T_{−t+x}(T_{ϕ(r)}μ)(A) dΦ(g)(t)
= Φ(g) ∗ [t ↦ T_{t}(T_{ϕ(r)}μ)(A)](x)
= 0

for almost all x ∈ G_{1}. To justify the last equality, we appeal to Proposition 3.3 and note that { \atop Φ(g)} = ĝ ∘ ψ and so { \atop Φ(g)} = 0 on B ⊂ ψ^{−1}(C). Moreover, σ[t ↦ T_{t}(T_{ϕ(r)}μ)(A)] ⊂ spec_{T} (μ) ⊂ B. Now, using Proposition 1.4 and the fact that, for every A ∈ Σ,

T_{x}[g ∗_{T_{ϕ}} T_{ϕ(r)}]μ(A) = g ∗_{T_{ϕ}} T_{x+ϕ(r)}μ(A) = 0,

for almost all x ∈ G_{1}, we conclude that the measure g ∗_{T_{ϕ}} T_{ϕ(r)}μ is the zero measure, which completes the proof.

Given C, a collection of elements in L^{1}(G_{1}) or M(G_{1}), let

Z(C) = ⋂_{δ∈C} {χ : δ̂(χ) = 0} .

This is the same notation for the zero set of an ideal in L^{1}(G) that we introduced in Section 1. Given a set of measures S in M(G_{2}), let

Φ(S) = {Φ(ν) : ν ∈ S} ⊂ M(G_{1}).

Lemma 4.2 In the above notation, if μ ∈ M(Σ) is weakly measurable, then

Z (Φ(I_{T_{ϕ}}μ)) = ψ^{−1} (Z(I_{T_{ϕ}}μ)) = ψ^{−1} (spec_{T_{ϕ}}μ) .

Proof. It is enough to establish the first equality; the second one follows from definitions. We have

Z (Φ(I_{T_{ϕ}}μ)) = ⋂_{δ∈Φ(I_{T_{ϕ}}(μ))} {χ ∈ Γ : δ̂(χ) = 0}

= ⋂_{g∈I_{T_{ϕ}}(μ)} \{χ ∈ Γ : { \atop Φ(g)}(χ) = 0\}

= ⋂_{g∈I_{T_{ϕ}}(μ)} {χ ∈ Γ : ĝ(ψ(χ)) = 0}

= ⋂_{g∈I_{T_{ϕ}}(μ)} ψ^{−1} (Z(g))

= ψ^{−1} ( ⋂_{g∈I_{T_{ϕ}}(μ)} (Z(g)))

= ψ^{−1} (Z (I_{T_{ϕ}}(μ))) = ψ^{−1} (spec_{T_{ϕ}}μ) .

Lemma 4.3 Suppose that C is a nonempty closed S-subset of Γ_{2} and that ψ^{−1}(C) is an S-subset of Γ_{1}. Suppose that μ is in M(Σ) and spec_{T_{ϕ}}(μ) ⊂ C. Then spec_{T} μ ⊂ ψ^{−1}(C).

Proof. We will use the notation of Lemma 4.2. If f ∈ I_{T_{ϕ}}(μ) and t ∈ G_{1}, then f ∈ I_{T_{ϕ}}(T_{t}μ). So, for A ∈ Σ, we have f ∗_{T_{ϕ}} (T_{t}μ)(A) = 0. But

f ∗_{T_{ϕ}} (T_{t}μ)(A) = ∫_{ℝ} T_{t−ϕ(x)}μ(A)f(x) dx
= ∫_{G} T_{t−x}μ(A) dΦ(f),

where Φ(f) is the homomorphic image of the measure f(x) dx. Hence, Φ(f) ∈ I_{T} ([t ↦ T_{t}μ(A)]), and so Φ(I_{T_{ϕ}}(μ)) ⊂ I_{T} ([t ↦ T_{t}μ(A)]), which implies that

Z (Φ(I_{T_{ϕ}}(μ))) ⊃ Z (I_{T} ([t ↦ T_{t}μ(A)])) = spec_{T} (t ↦ T_{t}μ(A)).

By Lemma 4.2,

Z (Φ(I_{T_{ϕ}}(μ))) = ψ^{−1} (spec_{T_{ϕ}}μ) ⊂ ψ^{−1}(C).

Hence, spec_{T} (t ↦ T_{t}μ(A)) ⊂ ψ^{−1}(C) for all A ∈ Σ, which by Proposition 3.7 implies that spec_{T} (μ) ⊂ ψ^{−1}(C).

Taking G_{1} = G, G_{2} = ℝ and ψ = ψ_{α} to be one of the homomorphisms in Theorem 2.1, and using the fact that [0, ∞[, S_{α}, C_{α} \ D_{α} are all S-sets, we obtain useful relationships between different types of analyticity.

Theorem 4.4 Let G be a locally compact abelian group with ordered dual group Γ, and let P denote a measurable order on Γ. Suppose that T is a sup path attaining representation of G by isomorphisms of M(Σ), such that T_{ϕ_{α}} is sup path attaining, where ϕ_{α} is the adjoint of ψ_{α} in Theorem 2.1.

(i) If μ ∈ M(Σ) and spec_{T} (μ) ⊂ C_{α} \ D_{α}. Then

spec_{T} (μ) ⊂ S_{α} ⇔ spec_{T_{ϕα}}(μ) ⊂ [0, ∞[.

(ii) If μ ∈ M(Σ) and spec_{T} (μ) ⊂ C_{α_{0}}. Then

spec_{T} (μ) ⊂ S_{α_{0}} ⇔ spec_{T_{ϕα_{0}}} (μ) ⊂ [0, ∞[.

We can use the representation T_{ϕ} to convolve a measure ν ∈ M(G_{2}) with μ ∈ M(G_{1}):

ν ∗_{T_{ϕ}} μ(A) = ∫_{G_{2}} T_{−ϕ(x)}μ(A)dν(x) = ∫_{G_{2}} μ(A − ϕ(x)) dν(x),

for all Borel A ⊂ G_{1}.

Alternatively, we can convolve Φ(ν) in the usual sense of [20, Definition 19.8] with μ to yield another measure in M(G_{1}), defined on the Borel subsets of G_{1} by

Φ(ν) ∗ μ(A) = ∫_{G_{1}} ∫_{G_{1}} 1_{A}(x + y)dΦ(ν)(x)dμ(y).

Using (22), we find that

Φ(ν) ∗ μ(A) = ∫_{G_{1}} ∫_{G_{2}} 1_{A}(ϕ(t) + y)dν(t)dμ(y)
= ∫_{G_{2}} μ(A − ϕ(t))dν(t) = ν ∗_{T_{ϕ}} μ(A).

Thus,

(25)  Φ(ν) ∗ μ = ν ∗_{T_{ϕ}} μ.

We end the section with homomorphism theorems, which complement the well-known homomorphism theorems for L^{p}-multipliers (see Edwards and Gaudry [11, Appendix B]). In these theorems, we let G_{1} act on M(G_{1}) by translation. That is, if μ ∈ M(G_{1}), x ∈ G_{1}, and A is a Borel subset of G_{1}, then

T_{x}μ(A) = μ(A + x).

Let ϕ : G_{2} → G_{1} be a continuous homomorphism. By Example 1.3, T and T_{ϕ} are sup path attaining. (Recall that if t ∈ G_{2}, μ ∈ M(G_{1}), then T_{ϕ(t)}μ(A) = μ(A + ϕ(t)).) A simple exercise with definitions shows that for μ ∈ M(G_{1})

spec_{T} μ = suppμ̂.

Theorem 4.5 Suppose that Γ_{1} and Γ_{2} contain measurable orders P_{1} and P_{2}, respectively, and ψ : Γ_{1} → Γ_{2} is a continuous, order-preserving homomorphism (that is, ψ(\overline{P_{1}}) ⊂ \overline{P_{2}}). Suppose that there is a positive constant N(ν) such that for all f ∈ H^{1}(G_{2})

(26)  ∥ν ∗ f∥_{1} ≤ N(ν)∥f∥_{1}.

Then

(27)  ∥Φ(ν) ∗ μ∥ ≤ N(ν)∥μ∥

for all Borel measures in M(G_{1}) such that μ̂ is supported in \overline{P_{1}}.

Proof. We have Φ(ν) ∗ μ = ν ∗_{T_{ϕ}} μ. Also \overline{P_{2}} is a T -set. So (27) will follow from Theorem 1.8 once we show that spec_{T_{ϕ}}μ ⊂ \overline{P_{2}}. For that purpose, we use Lemma 4.1. We have

spec_{T} μ = suppμ̂ ⊂ \overline{P_{1}},

and ψ(\overline{P_{1}}) ⊂ \overline{P_{2}} is an S-set. Hence spec_{T_{ϕ}}μ ⊂ \overline{P_{2}} by Lemma 4.1. The following special case of Theorem 4.5 deserves a separate statement.

Theorem 4.6 With the above notation, suppose that there is a positive constant N(ν) such that for all f ∈ H^{1}(G_{2})

(28)  ∥ν ∗ f∥_{1} ≤ N(ν)∥f∥_{1}.

Then for all f ∈ H^{1}(G_{1}) we have

(29)  ∥Φ(ν) ∗ f∥_{1} ≤ N(ν)∥f∥_{1}.

Theorem 4.7 Suppose that there is a positive constant N(ν) such that for all f ∈ H^{1}(ℝ)

(30)  ∥ν ∗ f∥_{1} ≤ N(ν)∥f∥_{1}.

Then for all μ ∈ M(G) with support of μ̂ contained in C_{α} \ D_{α}, where α < α_{0}, we have

(31)  ∥Φ_{α}(ν) ∗ μ∥_{1} ≤ N(ν)∥μ∥.

Proof. The proof is very much like the proof of Theorem 4.5. We have Φ_{α}(ν) ∗ μ = ν ∗_{T_{ϕ}} μ. Apply Theorem 1.8, taking into consideration that

spec_{T} μ = suppμ̂ ⊂ C_{α} \ D_{α}

is an S-set and so

spec_{T_{ϕα}}μ ⊂ ψ_{α}(C_{α} \ D_{α}) ⊂ [0, ∞[.

5 Decomposition of Analytic Measures

Define measures μ_{α_{0}} and d_{α} by their Fourier transforms: μ̂_{α_{0}} = 1_{C_{α_{0}}}, and d̂_{α} = 1_{C_{α}\D_{α}}. Then we have the following decomposition theorem.

Theorem 5.1 Let G be a locally compact abelian group with an ordered dual group Γ. Suppose that T is a sup path attaining representation of G in M(Σ). Then for any weakly analytic measure μ ∈ M(Σ) we have that the set of α for which d_{α} ∗_{T} μ ≠ 0 is countable, and that

(32)  μ = μ_{α_{0}} ∗_{T} μ + ∑_{α} d_{α} ∗_{T} μ,

where the right side converges unconditionally in norm in M(Σ). Furthermore, there is a positive constant c, depending only upon T, such that for any signs ϵ_{α} = ±1 we have

(33)  ∥∑_{α} ϵ_{α}d_{α} ∗_{T} μ∥ ≤ c∥μ∥.

One should compare this theorem to the well-known results from Littlewood-Paley theory on L^{p}(G), where 1 < p < ∞ (see Edwards and Gaudry [11]). For L^{p}(G) with 1 < p < ∞, it is well-known that the subgroups (C_{α}) form a Littlewood-Paley decomposition of the group Γ, which means that the martingale difference series

f = μ_{α_{0}} ∗ f + ∑_{α} d_{α} ∗ f

converges unconditionally in L^{p}(G) to f. Thus, Theorem 5.1 above may be considered as an extension of Littlewood-Paley Theory to spaces of analytic measures.

The next result, crucial to our proof of Theorem 5.1, is already known in the case that G = 𝕋^{n} with the lexicographic order on the dual. This is due to Garling [15], and is a modification of the celebrated inequalities of Burkholder. Our result can be obtained directly from the result in [15] by combining the techniques of [3] with the homomorphism theorem 4.5. However, we shall take a different approach, in effect reproducing Garling’s proof in this more general setting.

Theorem 5.2 Suppose that G is a locally compact group with ordered dual Γ. Then for f ∈ H^{1}(G), for any set {α_{j}}_{j=1}^{n} of indices less than α_{0}, and for any numbers ϵ_{j} ∈ {0, ±1} (1 ≤ j ≤ n), there is an absolute constant a > 0 such that

(34)  ∥∑_{j=1}^{n} ϵ_{j}d_{α_{j}} ∗ f∥_{1} ≤ a∥f∥_{1}.

Furthermore,

(35)  f = μ_{α_{0}} ∗ f + ∑_{α} d_{α} ∗ f,

where the right hand side converges unconditionally in the norm topology on H^{1}(G).

Proof. The second part of Theorem 5.2 follows easily from the first part and Fourier inversion.

Now let us show that if we have the result for compact G, then we have it for locally compact G. Let π_{α_{0}} : Γ → Γ/C_{α_{0}} denote the quotient homomorphism of Γ onto the discrete group Γ/C_{α_{0}} (recall that C_{α_{0}} is open), and define a measurable order on Γ/C_{α_{0}} to be π_{α_{0}}(P). By Remarks 2.2 (c), the decomposition of the group Γ/C_{α_{0}} that we get by applying Theorem 2.1 to that group, is precisely the image by π_{α_{0}} of the decomposition of the group Γ. Let G_{0} denote the compact dual group of Γ/C_{α_{0}}. Thus if Theorem 5.2 holds for H^{1}(G_{0}), then applying Theorem 4.5, we see that Theorem 5.2 holds for G.

Henceforth, let us suppose that G is compact. We will suppose that the Haar measure on G is normalized, so that G with Haar measure is a probability space.

Since each one of the subgroups C_{α}, and D_{α} (α < α_{0}) is open, it follows that their annihilators in G, G_{α} = A(G, C_{α}), and A(G, D_{α}), are compact. Let μ_{α} and ν_{α} denote the normalized Haar measures on A(G, C_{α}) and A(G, D_{α}), respectively. We have μ̂_{α} = 1_{C_{α}} (for all α), and ν̂_{α} = 1_{D_{α}} (for all α ≠ α_{0}), so that d_{α} = μ_{α} − ν_{α}.

For each α, let B_{α} denote the σ-algebra of subsets of G of the form A + G_{α}, where A is a Borel subset of G. We have B_{α_{1}} ⊂ B_{α_{2}}, whenever α_{1} > α_{2}. It is a simple matter to see that for f ∈ L^{1}(G), the conditional expectation of f with respect to B_{α} is equal to μ_{α} ∗ f (see [11, Chapter 5, Section 2]).

We may suppose without loss of generality that α_{1} > α_{2} > … > α_{n}. Thus the σ-algebras B_{α_{k}} form a filtration, and the sequence (d_{α_{1}} ∗ f, d_{α_{2}} ∗ f, …, d_{α_{n}} ∗ f) is a martingale difference sequence with respect to this filtration.

In that case, we have the following result of Burkholder [7, Inequality (1.7)], and [8]. If 0 < p < ∞, then there is a positive constant c, depending only upon p, such that

(36)  ∥ sup_{1≤k≤n} (∑_{j=1}^{k} ϵ_{j}d_{α_{j}} ∗ f)∥_{p} ≤ c ∥ sup_{1≤k≤n} (∑_{j=1}^{k} d_{α_{j}} ∗ f)∥_{p} .

Lemma 5.3 For any index α, 0 < p < ∞, and f ∈ H^{1}(G) ∩ L^{p}(G), we have almost everywhere on G

(37)  |μ_{α} ∗ f|^{p} ≤ μ_{α} ∗ |f|^{p} ,

where μ_{α} is the normalized Haar measure on the compact subgroup G_{α} = A(G, C_{α}).

Proof. The dual group of G_{α} is Γ/C_{α} and can be ordered by the set π_{α}(P), where π_{α} is the natural homomorphism of Γ onto Γ/C_{α}.

Next, by convolving with an approximate identity for L^{1}(G) consisting of trigonometric polynomials, we may assume that f is a trigonometric polynomial. Then we see that for each x ∈ G that the function y ↦ f(x+y), y ∈ G_{α}, is in H^{1}(G_{α}). To verify this, it is sufficient to consider the case when f is a character in P. Then

f(x + y) = f(x)π_{α}(f)(y),

and by definition π_{α}(f) is in H^{1}(G_{α}).

Now we have the following generalization of Jensen’s Inequality, due to Helson and Lowdenslager [16, Theorem 2]. An independent proof based on the ideas of this section is given in [3]. For all g ∈ H^{1}(G)

(38)  |∫_{G} g(x)dx| ≤ exp ∫_{G} log |g(x)|dx.

Apply (38) to y ↦ f(x + y), y ∈ G_{α} to obtain

|∫_{G_{α}} f(x + y)dμ_{α}(y)| ≤ exp ∫_{G_{α}} log |f(x + y)|dμ_{α}(y).

Extending the integrals to the whole of G (since μ_{α} is supported on G_{α}), raising both sides to the pth power, and then applying the usual Jensen’s inequality for the logarithmic function on finite measure spaces, we obtain

|∫_{G} f(x + y)dμ_{α}(y)|^{p} ≤ exp ∫_{G} log |f(x + y)|^{p}dμ_{α}(y)
≤ ∫_{G} |f(x + y)|^{p}dμ_{α}(y).

Changing y to −y, we obtain the desired inequality.

Let us continue with the proof of Theorem 5.2. We may suppose that f is a mean zero trigonometric polynomial, and that the spectrum of f is contained in ⋃_{j=1}^{n} C_{α_{j}} \ D_{α_{j}}, that is to say

f = ∑_{j=1}^{n} d_{α_{j}} ∗ f.

By Lemma 5.3, we have that

sup_{1≤k≤n} |μ_{α_{k}} ∗ f| = ( sup_{1≤k≤n} |μ_{α_{k}} ∗ f|^{1/2})^{2}
(39)  ≤ ( sup_{1≤k≤n} μ_{α_{k}} ∗ |f|^{1/2})^{2} .

Also, we have that (μ_{α_{j}} ∗ |f|^{1/2})_{j=1}^{n} is a martingale with respect to the filtration (B_{j})_{j=1}^{n}. Hence, by Doob’s Maximal Inequality [10, Theorem (3.1), p. 317] we have that

∥ sup_{1≤k≤n′} μ_{β_{k}} ∗ |f|^{1/2}∥_{2}^{2} ≤ 4 ∥μ_{β_{n′}} ∗ |f|^{1/2}∥_{2}^{2}
(40)  ≤ 4 ∥|f|^{1/2}∥_{2}^{2} = 4∥f∥_{1}.

The desired inequality follows now upon combining Burkholder’s Inequality (36) with (39), and (40).

Proof of Theorem 5.1. Transferring inequality (34) by using Theorem 1.8, we obtain that for any set {α_{j}}_{j=1}^{n} of indices less than α_{0}, and for any numbers ϵ_{j} ∈ {0, ±1} (1 ≤ j ≤ n), there is a positive constant c, depending only upon the representation T, such that

(41)  ∥∑_{j=1}^{n} ϵ_{j}d_{α_{j}} ∗_{T} μ∥ ≤ c∥μ∥.

Now suppose that {α_{j}}_{j=1}^{∞} is a countable collection of indices less than α_{0}. Then by Bessaga and Pełczyński [5], the series ∑_{j=1}^{∞} d_{α_{j}} ∗_{T} μ is unconditionally convergent. In particular, for any δ > 0, for only finitely many k do we have that ∥d_{α_{k}} ∗_{T} μ∥ > δ. Since this is true for all such countable sets, we deduce that the set of α for which d_{α} ∗_{T} μ ≠ 0 is countable.

Hence we have that ∑_{α} d_{α} ∗_{T} μ is unconditionally convergent to some measure, say ν. Clearly ν is weakly measurable. To prove that μ = ν, it is enough by Proposition 1.4 to show that for every A ∈ Σ, we have T_{t}μ(A) = T_{t}ν(A) for almost all t ∈ G.

We first note that since for every f ∈ L^{1}(G) the series μ_{α_{0}} ∗f +∑_{α} d_{α} ∗f converges to f in L^{1}(G), it follows that, for every g ∈ L^{∞}(G), the series μ_{α_{0}} ∗ g + ∑_{α} d_{α} ∗ g converges to g in the weak-* topology of L^{∞}(G). Now on the one hand, for t ∈ G and A ∈ Σ, we have μ_{α_{0}} ∗_{T} T_{t}μ(A) + ∑_{α} d_{α} ∗_{T} T_{t}μ(A) = T_{t}ν(A), because of the (unconditional) convergence of the series μ_{α_{0}} ∗_{T} μ + ∑_{α} d_{α} ∗_{T} μ to ν. On the other hand, by considering the L^{∞}(G) function t ↦ T_{t}(A), we have that μ_{α_{0}} ∗_{T} T_{t}μ(A) + ∑_{α} d_{α} ∗_{T} T_{t}μ(A) = μ_{α_{0}} ∗ T_{t}μ(A) + ∑_{α} d_{α} ∗ T_{t}μ(A) = T_{t}μ(A), weak *. Thus T_{t}μ(A) = T_{t}ν(A) for almost all t ∈ G, and the proof is complete.

6 Generalized F. and M. Riesz Theorems

Throughout this section, we adopt the notation of Section 5, specifically, the notation and assumptions of Theorem 5.1.

Suppose that T is a sup path attaining representation of ℝ by isomorphisms of M(Σ). In [4], we proved the following result concerning bounded operators P from M(Σ) into M(Σ) that commute with the representation T in the following sense:

P ∘ T_{t} = T_{t} ∘ P

for all t ∈ ℝ.

Theorem 6.1 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.

To describe an interesting application of this theorem from [4], let us recall the following.

Definition 6.2 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.

Using Theorem 6.1 we obtained in [4] the following extension of results of de Leeuw-Glicksberg [9] and Forelli [12], concerning quasi-invariant measures.

Theorem 6.3 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, ∞).

Our goal in this section is to extend Theorems 6.1 above to representations of a locally compact abelian group G with ordered dual group Γ. More specifically, we want to prove the following theorems.

Theorem 6.4 Suppose that T is a sup path attaining representation of G by isomorphisms of M(Σ) such that T_{ϕ_{α}} is sup path attaining for each α. Suppose that P commutes with T in the sense that

P ∘ T_{t} = T_{t} ∘ P

for all t ∈ G. Let μ ∈ M(Σ) be weakly analytic. Then Pμ is also weakly analytic.

As shown in [4, Theorem (4.10)] for the case G = ℝ, an immediate corollary of Theorem 6.4 is the following result.

Theorem 6.5 Suppose that T is a sup path attaining representation of G by isometries of M(Σ), such that T_{ϕ_{α}} is sup path attaining for each α. Suppose that μ ∈ M(Σ) is weakly analytic with respect to T, and σ is quasi-invariant with respect to T. 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 with respect to T. In particular, the T -spectra of μ_{a} and μ_{s} are contained in \overline{P}.

Proof of Theorem 6.4. Write

μ = μ_{α_{0}} ∗_{T} μ + ∑_{α} d_{α} ∗_{T} μ,

as in (5.1), where the series converges unconditionally in M(Σ). Then

(42)  Pμ = P(μ_{α_{0}} ∗_{T} μ) + ∑_{α} P(d_{α} ∗_{T} μ).

It is enough to show that the T -spectrum of each term is contained in \overline{P}. Consider the measure μ_{α_{0}} ∗_{T} μ. We have spec_{T} (μ_{α_{0}} ∗_{T} μ) ⊂ S_{α_{0}}. Hence by Theorem 4.4, μ_{α_{0}} ∗_{T} μ is T_{ϕ_{α_{0}}}-analytic. Applying Theorem 6.1, we see that

(43)  spec_{T_{ϕα_{0}}} (P(μ_{α_{0}} ∗_{T} μ)) ⊂ [0, ∞[.

Since P commutes with T, it is easy to see from Proposition 3.10 and Corollary 3.11 that

spec_{T} (P(μ_{α_{0}} ∗_{T} μ)) ⊂ C_{α_{0}}.

Hence by (43) and Theorem 4.4,

spec_{T} (P(μ_{α_{0}} ∗_{T} μ)) ⊂ S_{α_{0}},

which shows the desired result for the first term of the series in (42). The other terms of the series (42) are handled similarly.

Acknowledgments The second author is grateful for financial support from the National Science Foundation (U.S.A.) and the Research Board of the University of Missouri.

References

[1] N. Asmar and S. Montgomery-Smith, Hahn’s embedding theorem for orders and harmonic analysis on groups with ordered duals, Colloq. Math. 70 (1996), 235–252.

[2] N. Asmar and S. Montgomery-Smith, Analytic measures and Bochner measurability, Bull. Sci. Math. 122 (1998), 39–66.

[3] N. Asmar and S. Montgomery-Smith, Hardy martingales and Jensen’s Inequality, Bull. Austral. Math. Soc. 55 (1997), 185–195.

[4] N. Asmar, S. Montgomery-Smith, and S. Saeki, Transference in spaces of measures, J. Functional Analysis, 165 (1999), 1–23.

[5] C. Bessaga and A. Pełczyński, On bases and unconditional convergence of series in Banach spaces, Studia Math. 17 (1958), 151– 164.

[6] S. Bochner, Boundary values of analytic functions in several variables and almost periodic functions, Ann. of Math., 45 (1944), 708–722.

[7] D. L. Burkholder, A geometrical characterization of Banach spaces in which martingale difference sequences are unconditional, Ann. Math. Statist., 37 (1966), 1494–1504.

[8] D. L. Burkholder, Martingale transforms, The Annals of Probability, 9 (1981), 997–1011.

[9] K. De Leeuw and I. Glicksberg, Quasi-invariance and measures on compact groups, Acta Math., 109 (1963), 179–205.

[10] J. L. Doob, “Stochastic Processes”, Wiley Publications in Statistics, New York 1953.

[11] R. E. Edwards, and G. I. Gaudry, “Littlewood-Paley and Multiplier Theory”, Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer-Verlag, No. 90, Berlin, 1977.

[12] F. Forelli, Analytic and quasi-invariant measures, Acta Math., 118 (1967), 33–59.

[13] L. Fuchs, “ Partially ordered algebraic systems”, Pergamon Press, Oxford, New York, 1960.

[14] D. J. H. Garling, On martingales with values in a complex Banach space, Math. Proc. Camb. Phil. Soc. 104 (1988), 399–406.

[15] D. J. H. Garling, Hardy martingales and the unconditional convergence of martingales, Bull. London Math. Soc. 23 (1991), 190– 192.

[16] H. Helson and D. Lowdenslager, Prediction theory and Fourier series in several variables, Acta Math. 99 (1958), 165–202.

[17] H. Helson and D. Lowdenslager, Prediction theory and Fourier series in several variables II, Acta Math. 106 (1961), 175–212.

[18] E. Hewitt and S. Koshi, Orderings in locally compact Abelian groups and the theorem of F. and M. Riesz, Math. Proc. Camb. Phil. Soc. 93 (1983), 441–457.

[19] E. Hewitt, S. Koshi, and Y. Takahashi, The F. and M. Riesz Theorem revisited, Math. Scandinav., 60 (1987), 63–76.

[20] E. Hewitt and K. A. Ross, “Abstract Harmonic Analysis I,” 2^{nd} Edition, Grundlehren der Math. Wissenschaften, Band 115, Springer–Verlag, Berlin 1979.

[21] E. Hewitt and K. A. Ross, “Abstract Harmonic Analysis II,” Grundlehren der Math. Wissenschaften in Einzeldarstellungen, Band 152, Springer–Verlag, New York, 1970.

[22] W. Rudin, “Fourier Analysis on Groups,” Interscience Tracts in Pure and Applied Mathematics, No. 12, John Wiely, New York, 1962.

[23] H. Yamaguchi, A property of some Fourier-Stieltjes transforms, Pac. J. Math. 108 (1983), 243–256.