Hahn’s Embedding Theorem for orders and harmonic analysis on groups with ordered duals

Nakhlé H. Asmar and Stephen J. Montgomery–Smith Department of Mathematics University of Missouri–Columbia Columbia, Missouri 65211 U. S. A.

Abstract Let G be a locally compact abelian group whose dual group Γ contains a Haar measurable order P. Using the order P we define the conjugate function operator on L^{p}(G), 1 ≤ p < ∞, as was done by Helson [7]. We will show how to use Hahn’s Embedding Theorem for orders and the ergodic Hilbert transform to study the conjugate function. Our approach enables us to define a filtration of the Borel σ-algebra on G, which in turn will allow us to introduce tools from martingale theory into the analysis on groups with ordered duals. We illustrate our methods by describing a concrete way to construct the conjugate function in L^{p}(G). This construction is in terms of an unconditionally convergent conjugate series whose individual terms are constructed from specific ergodic Hilbert transforms. We also present a study of the square function associated with the conjugate series.

1 Introduction

Throughout this paper, G will denote a locally compact abelian group with dual group Γ. A Haar measure on G will be symbolized by μ. For 1 ≤ p < ∞, we denote by L^{p}(G) the Banach space of Haar measurable functions f on G such that |f|^{p} is integrable. The space of essentially bounded Haar measurable functions on G will be denoted by L^{∞}(G).

An order on Γ is a subset P such that P +P ⊆ P; P ∩(−P) = {0}; P ∪(−P) = Γ. Given such a set P we will write α > β to mean that α − β ∈ P \{0}. For f ∈ L^{2}(G) we use the Fourier transform to define the conjugate function f̃ of f with respect to the order P by the Fourier multiplier relation

ˆf̃(χ) = −i sgn_{P} (χ)f̂(χ)  (1)

for almost all χ ∈ Γ, where sgn_{P} (χ) = −1, 0, or 1, according as χ ∈ (−P)\{0}, χ = 0, or χ ∈ P \{0}. When G is compact, these definitions are due to Helson [7] and [8]. In [6], Garling observed that the conjugate function on T^{N}, defined using a lexicographic order on Z^{N} is connected in a natural way to martingale theory. Using this connection, Garling gave simple proofs of basic properties of the conjugate function in this setting. Our goal in this paper is to show how certain notions related to Hahn’s Embedding Theorem for orders ([5, Chapter IV]) can be used to introduce similar tools from probability theory in the study of conjugate functions on groups with an arbitrary measurable order on the dual group. We present the material related to Hahn’s Theorem in Section 2. In particular, we will state a structure theorem (Theorem (2.4) below), which describes an order in terms of a unique chain of convex subgroups of Γ. When Γ is not necessarily discrete, the chain of subgroups may contain elements that are not Haar measurable. In Section 3, we will show how to obtain a structure theorem, similar to the one in Section 2, while avoiding nonmeasurabele subgroups of Γ. This study is based on the work of Hewitt and Koshi [9] concerning measurable orders. In Sections 4, we use the structure theorems for orders to define our construction of the conjugate function as a martingale difference series whose individual terms are constructed by using different ergodic Hilbert transforms. In Section 5, we will show that the conjugate series of f ∈ L^{p}(G) is unconditionally convergent in L^{p}(G) when 1 < p < ∞, and is unconditionally convergent in L^{1,∞}(G) when p = 1. These results yield a concrete way for constructing the conjugate function on abstract groups. We end the paper with a study of the square function associated with the conjugate series.

2 Orders on discrete groups

We start by collecting facts leading to a structure theorem for orders on discrete groups (Theorem 2.4 below). This requisite material is taken from [5, Chapter IV] where it is presented as a background for the proof of Hahn’s Embedding Theorem for orders [5, Theorem 16, p. 59]. Our presentation is simplified by the fact that the groups are all abelian.

Definitions and basic Properties Let Γ denote an infinite torsion-free abelian group. The topology on Γ will play no role in this section. An order on Γ will be denoted by P (Γ) or simply P. Because sometimes we will be dealing with more than one order on a given group, it will be convenient to write (Γ, P) or (Γ, P (Γ)) to denote the group and the given order on it. A subset J ⊂ Γ is called convex if whenever a, b ∈ J, c ∈ Γ and a ≤ c ≤ b, then c ∈ J. As we will see, this notion plays a prominent role in the theory of orders. For the reader’s convenience, we list a few properties of convexity that will be used in the sequel. (See [5, pp. 18-19, and Chapter IV].)

(a) A subgroup C ⊂ Γ is convex if and only if P (Γ) ∩ C is convex in P (Γ).

(b) If B ⊂ C ⊂ Γ, and if B is convex in C and C is convex in Γ, then B is convex in Γ.

(c) The intersection of convex subgroups is again a convex subgroup. Thus, if A is a subset of Γ, there is a smallest convex subgroup containing A. We will denote this subgroup by {A}_{□}. If A is a subgroup, then {A}_{□} = (A + P) ∩ (A − P).

(d) A subgroup C ⊂ Γ is called principal if C = {c}_{□} for some c ∈ Γ.

(e) Let Γ and Γ′ be two ordered groups. A homomorphism ϕ : (Γ, P (Γ)) ⟶ (Γ′, P (Γ′)), is called an order homomorphism if ϕ(P (Γ)) ⊂ P (Γ′). It is clear that if ϕ is an order homomorphism, then ker ϕ is a convex subgroup of Γ. Conversely, if H is a convex subgroup of (Γ, P (Γ)), then we can define an order on the quotient group Γ/H by: a + H ∈ P (Γ/H) ⇔ a ∈ P. To verify this claim, suppose that a + H = b + H, and, say, a ∈ P and b ∈ −P. Then 0 ≤ a ≤ a − b ∈ H. Since H is convex, it follows that a ∈ H, and so a + H = b + H = 0 + H, which shows that P (Γ/H) is indeed an order on Γ/H. It is also clear that the natural homomorphism π : (Γ, P (Γ)) ⟶ (Γ/H, P (Γ/H)) is an order homomorphism. We have thus the following useful theorem. (See [5, Theorem 7, p.21].)

Theorem 2.1 Suppose that Γ is an ordered group and H is a subgroup of Γ. If H is convex then the natural homomorphism π : (Γ, P) → (Γ/H, π(P)) is an order homomorphism. Conversely, suppose that ϕ : (Γ, P) → (Γ′, P′) is an order homomorphism with ker ϕ = H. Then H is a convex subgroup of Γ.

Definition An order P on Γ is called Archimedean if given a, b ∈ P \ {0}, then there is a positive integer n such that na > b.

Archimedean orders have a simple characterization in terms of real-valued homomorphisms due to O. Hölder ([5, Theorem 1, p. 45]).

Theorem 2.2 An order P on Γ is Archimedean if and only if Γ is isomorphic to a subgroup of R.

There is another useful characterization of Archimedean orders in terms of convex subgroups ([5, Corollary 5, p. 47]).

Theorem 2.3 Suppose that Γ is an ordered group, then Γ is Archimedean ordered if and only if the only convex subgroups of Γ are {0} and Γ.

Following [5, Chapter IV, Section 3], we let Σ denote the system of all convex subgroups in Γ. This system is in fact a chain containing {0} and Γ. Hence if C and D are in Σ, then either C ⊂ D or D ⊂ C. Also, whenever {C_{λ}}_{λ∈Λ} is a collection from Σ, then ⋂_{λ∈Λ} C_{λ} and ⋃_{λ∈Λ} C_{λ} are again in Σ. By a jump in Σ we mean a pair of subgroups C and D such that D ⊂ C, D ≠ C, and Σ contains no subgroups between C and D. A jump will be denoted by D ≺ C. It is a fact that a subgroup C ∈ Σ is the greater member of a jump (i. e. D ≺ C) if and only if C is a principal convex subgroup. That is,

C = {a}_{□}

for some a ∈ Γ (see p. 54 of [5]). Let Σ_{0} denote the system of principal convex subgroups, and let Π be an indexing set for Σ_{0}. Order Π as follows: for ρ, π ∈ Π, set π ≤ ρ if and only if C_{ρ} ⊂ C_{π}. With this order, Π has a maximal element α_{0} corresponding to {0} ∈ Σ_{0}. Thus C_{α_{0}} = {0}. For notational convenience, let D_{α_{0}} = ∅. Let D_{π} ≺ C_{π} denote a jump in Σ, with π < α_{0}. The quotient group C_{π}/D_{π} has no nontrivial convex subgroups. By Theorem (2.3), C_{π}/D_{π} is Archimedean ordered. Let

ψ_{π} : C_{π}/D_{π} → R  (2)

denote the order isomorphism mapping C_{π}/D_{π} into a subgroup of R, and let

L_{π} : C_{π} → R  (3)

denote the composition of ψ_{π} with the natural homomorphism of C_{π} onto C_{π}/D_{π}. Then L_{π} is an order homomorphism of C_{π} into R with ker L_{π} = D_{π}. Since R is a divisible group, the homomorphism L_{π} can be extended to a homomorphism of the entire group Γ into R. (See [10, Theorem A.7, p. 441].) We keep the same notation for this extension. The next theorem summarizes this discussion. It is a basic result of this section and will be used in defining our construction of the conjugate function.

Theorem 2.4 Let Γ be an infinite discrete (torsion-free) ordered group with order P. Let Σ denote the chain of convex subgroups of Γ and Σ_{0} the subcollection of principal convex subgroups indexed by the ordered set Π. There is a collection of real-valued homomorphisms {L_{π}, π ∈ Π} of Γ into R such that, for every jump D_{π} ≺ C_{π}, we have

(i) L_{π} (D_{π}) = {0}; and (ii) sgn(L_{π}(χ)) = sgn_{P} (χ) for all χ ∈ C_{π} \ D_{π}.

Observe that, in the notation of the previous theorem, we have

Γ = ⋃_{α∈Π} C_{α}\D_{α}.  (4)

In fact, given x ∈ Γ, we have {x}_{□} = C_{α} for some α ∈ Π. Since D_{α} is strictly contained in C_{α}, it follows that x belongs to C_{α}\D_{α} which proves (4). The following example will illustrate many of the results of this section.

Example Suppose that (T, <) is an ordered set and that γ is a limit ordinal. Let Z^{(T,γ)} be the group of sequences (z_{t})_{t∈T} such that the set {t : z_{t} ≠ 0} is reverse well ordered (i. e., every subset has a largest element) with order type less than γ. Define an order as follows: If z = (z_{t})_{t∈T} ∈ Z^{(T,γ)} and t_{0} = max{t : z_{t} ≠ 0}, then z ∈ P if and only if z_{t_{0}} ≥ 0. In this example we see that Σ_{0} is order isomorphic −T, that C_{α} = {(z_{t}) : z_{t} = 0, for t ≥ α}, that D_{α} = {(z_{t}) : z_{t} = 0, for t > α}, and that L_{α}((z_{t})) = z_{α}. We can also construct R^{(T,γ)} similarly.

3 Orders on locally compact abelian groups

In this section we prove a general version of Theorem (2.4) for measurable orders. The difficulty here is due to the fact that, in general, the chain of convex subgroups in Theorem (2.4) may contain nonmeasurable subgroups, or jumps of the form D_{α} ≺ C_{α} with C_{α} \ D_{α} having measure zero. To overcome these measure theoretic problems, we will find a smallest open principal convex subgroup of Γ which will determine when to stop the chain while still being able to separate with continuous real-valued homomorphisms as in Theorem (2.4). This section is based on the study of orders of Hewitt and Koshi [9]. Indeed, Theorem 3.10 below, is a combination of results from [9] and the material from the previous section. Throughout this section Γ will denote an infinite locally compact torsion-free abelian group. The following basic properties of measurable orders will be needed.

Theorem 3.1 (a) If P is a measurable order, then P has a nonvoid interior ([9, Theorem (3.1)]. Consequently, if P is a measurable order, then (−P) has nonvoid interior.

(b) If Γ is an infinite compact torsion-free group, then every order on Γ is dense and has void interior ([9, Theorem (3.2)]. Consequently, every order on a compact infinite group is nonmeasurable.

Remark 3.2 Suppose that Γ is a locally compact abelian group, and that P is a measurable order on Γ. Use the structure theorem for locally compact abelian groups to write Γ as Γ = R^{a} × Ω where a is a nonnegative integer and Ω contains a compact open subgroup Ω_{0} ([10, Theorem (24.30)]). The fact that P is measurable automatically implies that either Γ is discrete or a > 0. In fact, if a = 0, then Γ = Ω and so Ω_{0} is a compact open subgroup of Γ. The restriction of P to Ω_{0} is a measurable order in Ω_{0}. But since Ω_{0} is compact, it follows from Theorem (3.1)(b) that Ω_{0} = {0}, and so Γ is discrete.

Henceforth to avoid the cases treated in the previous section, we will assume that Γ = R^{a} × Ω where a > 0. For use in the sequel, we need the following result due to Hewitt and Koshi [9, Theorem (3.12)].

Theorem 3.3 Let P be an order on R^{a} × B where a > 0 and B is an infinite torsion-free locally compact abelian group that is the union of its compact open subgroups. Suppose that P has nonempty interior. Then there is a continuous real-valued homomorphism L : R^{a} ⟶ R such that

L^{−1}(]0, ∞[) × B ⊂ int(P) ⊂ P ⊂ L^{−1}([0, ∞[) × B.  (5)

The order P ∩ (L^{−1}({0}) × B) is arbitrary.

Remarks 3.4 (a) It follows from Theorems (3.1) and (3.3) that an order P on R^{a} is measurable if and only if it is not dense in R^{a}.

(b) Let P be a measurable order on R^{a} × Ω, and let B denotes the union of all the compact open subgroups of Ω. The restriction of P to R^{a}×B is a nondense order, and so by Theorem (3.3) there is a continuous real-valued homomorphism L : R^{a} ⟶ R such that

L^{−1}(]0, ∞]) × B ⊂ P.

We can now describe our candidate for a smallest open convex principal subgroup of Γ. Let

H = {y ∈ Ω : R^{a} × {y} has nonvoid intersections with P and (−P)} . (6)

Proposition 3.5 Let P be a measurable order on R^{a} × Ω, and let H be as in (6). Then H is a subgroup of Ω that contains all the compact open subgroups of Ω; and R^{a} × H is an open convex subgroup of Γ.

Proof. The fact that H contains all the compact open subgroups of Ω follows from Theorem (3.3). Also, the fact that H is a subgroup is easily verified. Since R^{a} × H is a subgroup with nonvoid interior it follows immediately that the subgroup is open. To establish the convexity of R^{a} × H, suppose that

0 < (t, y) < (t′, y′)

with y′ ∈ H, t, t′ ∈ R^{a}. To show that (t, y) ∈ R^{a} × H, it is enough to find x ∈ R^{a} with (x, y) ∈ −P. Since y′ ∈ H, we can find x ∈ R^{a} such that (x, y′) ∈ −P. Hence (t − t′, y − y′) + (x, y′) ∈ (−P), or, (t − t′ + x, y) ∈ (−P).

Lemma 3.6 Let P be a measurable order in Γ and let H be as in (6). For y ∈ H, let A_{y} = (R^{a} × {y}) ∩ P and B_{y} = (R^{a} × {y}) ∩ (−P). Then A_{y} and B_{y} are nondense in R^{a} × {y}.

Proof. It is enough to deal with the set A_{y} with y ≠ 0. Assume that (R^{a}×{y})∩P is dense in R^{a}×{y}. Let (s, y) be any element of R^{a}×{y}, and let (s_{0}, y) ∈ (R^{a}×{y})∩P be such that L(s_{0}) > L(s). We have L(s_{0} − s) > 0, and so from Remark (3.4) (b) we see that (s_{0} − s, 0) ∈ P. Hence (s_{0}, y) + (s − s_{0}, 0) = (s, y) ∈ P implying that R^{a}×{y} ⊂ P which contradicts the fact that y ∈ H. Thus P ∩(R^{a}×{y}) is nondense in R^{a} × {y}.

We can now prove a special separation theorem for orders. (Compare with [9, Theorem 3.7].)

Theorem 3.7 Let P be a measurable order on Γ, let H be as in (6), and let L be as in Remarks (3.4) (b). For every y ∈ H, there is a real number α(y) such that

(i) L^{−1} (] − ∞, α(y)[) × {y} ⊂ −P, (ii) L^{−1} (]α(y), ∞[) × {y} ⊂ P. Moreover, the mapping y ↦ α(y) is a continuous real-valued homomorphism from H.

Proof. We will write the elements of Γ = R^{a} × Ω as (x, y) where x ∈ R^{a}, and y ∈ Ω. Suppose that (x_{1}, y) ∈ P and L(x_{2}) > L(x_{1}). Then, L(x_{2} − x_{1}) > 0, and so from Remarks (3.4) (b) we have that (x_{2} − x_{1}, 0) ∈ P \{0}, and consequently, (x_{2}, y) ∈ P \{0}. Similarly, if (x_{1}, y) ∈ (−P), and L(x_{2}) < L(x_{1}), then (x_{2}, y) ∈ (−P)\{0}. ¿From these observations and the definition of H, we see that the following holds for every y ∈ H:

−∞ < sup {L(x) : x ∈ R^{a}, (x, y) ∈ (−P)}
= inf {L(x) : x ∈ R^{a}, (x, y) ∈ P)} < ∞.  (7)

For y ∈ H, let α(y) be defined by either the inf or the sup in (7). That α is a continuous homomorphism follows exactly as in the proof of [9, Theorem 3.7]. We omit the details.

We restate Theorem (3.7) using separating homomorphisms that reflect the order on R^{a} × H.

Theorem 3.8 Let P be a measurable order on Γ, and let H, L, and α be as Theorem (3.7). Define the homomorphism τ on R^{a} × H by

τ(x, y) = L(x) − α(y).

Then

(i) τ is continuous on R^{a} × H; (ii) τ^{−1} (] − ∞, 0[) ⊂ (−P); (iii) τ^{−1} (]0, ∞[) ⊂ P. (iv) The kernel of τ is locally null (i. e. , if K is any compact subset of τ^{−1} ({0}), then μ_{Γ}(K) = 0).

Proof. Assertions (i)-(iii) follow from the definitions of the homomorphisms τ and α. Now suppose that K is a compact subset of τ^{−1} ({0}) and μ_{Γ}(K) > 0, where μ_{Γ} is a Haar measure on Γ.. Then K −K contains an open neighborhood of the identity in Γ, and hence τ^{−1} ({0}) is an open subgroup of Γ which implies that R^{a}×{0} ⊂ τ^{−1} ({0}). This is plainly a contradiction, and so (iv) holds.

Theorem 3.9 Let H be as in (6). Then R^{a} × H is a principal convex open subgroup of Γ.

Proof. Because of Proposition (3.5), all we need to show is that R^{a} ×H is a principal subgroup. For this purpose, let z be any element in R^{a} × H such that τ(z) > 0. We claim that R^{a} × H = {z}_{□}. Since by definition {z}_{□} is the smallest convex subgroup containing z, and since R^{a} × H is convex, it is enough to show that {z}_{□} ⊃ R^{a} × H. Let x be any element of P ∩ (R^{a} × H). ¿From Theorem (3.8), τ(x) ≥ 0. Choose a positive integer n such that τ(nz) > τ(x) ≥ 0. Again, from Theorem (3.8) we see that 0 ≤ x ≤ nz, and since {z}_{□} is convex, it follows that x ∈ {z}_{□}. Hence {z}_{□} ⊃ P ∩ (R^{a} × H), and consequently {z}_{□} ⊃ R^{a} × H.

We are now ready to state the main result of this section. We write the group Γ as R^{a} ×Ω, where a > 0. Let Σ(H) denote the chain of convex subgroups of Γ containing R^{a} × H, and let Σ_{0}(H) denote the chain of principal convex subgroups containing R^{a}×H. Let Π(H) be an indexing set for Σ_{0}(H). Order Π(H) as we did in the discrete case: for ρ, π ∈ Π(H), set π ≤ ρ if and only if C_{ρ} ⊂ C_{π}. With this order, Π(H) has a maximal element α_{0} corresponding to R^{a} × H ∈ Σ_{0}(H). Hence C_{α_{0}} = R^{a} × H. As in the discrete case, we denote a jump in Σ(H) by D_{π} ≺ C_{π}, where C_{π} is a principal convex subgroup of Γ containing R^{a} × H. Note that for C_{α_{0}} = R^{a} × H the jump occurs with an element outside of Σ_{0}(H). We set, by definition, D_{α_{0}} = ker τ where τ is the homomorphism of Theorem (3.8). Hence D_{α_{0}} is locally null by Theorem (3.8). Note that

Γ = D_{α_{0}} ∪ ⋃_{α∈Π(H)} C_{α}\D_{α}.  (8)

With the exception of D_{α_{0}}, each set on the right side of (8) is open.

Theorem 3.10 With the above notation, for every α ∈ Π(H), α ≠ α_{0}, there is a continuous real-valued homomorphism L_{α} on Γ such that

(i) L_{α} (D_{α}) = {0}; (ii) sgn_{P} (χ) = sgn(L_{α}(χ)) for all χ ∈ C_{α} \ D_{α}. When α = α_{0}, there is a real-valued homomorphism L_{α_{0}} on Γ, such that (iii) L_{α_{0}} (D_{α_{0}}) = {0}; (iv) sgn_{P} (χ) = sgn(L_{α_{0}}(χ)) for all χ ∈ C_{α_{0}} \ D_{α_{0}}. (Since D_{α_{0}} is locally null, (iv) holds for locally almost all χ ∈ C_{α_{0}}.)

Proof. We treat first the case α = α_{0}. Consider the homomorphism τ provided by Theorem (3.8). Since τ maps into R and R is a divisible group, then τ can be extended to a homomorphism on all of Γ ([10, Theorem (A.7)]. We denote the extended homomorphism by L_{α_{0}}. By the properties of τ from Theorem (3.8), it is clear that (iii) and (iv) hold. Now since L_{α_{0}} is continuous on an open subgroup of Γ, it follows by linearity that L_{α_{0}} is continuous on all Γ. This proves the theorem in this case. We now treat the remaining cases. Since R^{a} × H is convex and open, the group Γ/(R^{a} × H) is discrete and can be ordered as in Theorem (2.1). Let Φ denote the natural homomorphism of Γ onto Γ/(R^{a} × H). It is easy to see that C is a convex subgroup of Γ if and only if Φ(C) is a convex subgroup of Γ/(R^{a} × H). Consequently, if D_{α} ≺ C_{α} is a jump in Γ, with α ≠ α_{0}, then Φ(D_{α}) ≺ Φ(C_{α}) is a jump in Γ/(R^{a} × H). The theorem follows now by composing Φ with the homomorphisms provided by Theorem (2.4) for the discrete ordered group Γ/(R^{a} × H). With Theorem (3.10) in hand we can give a simple proof of a separation theorem for measurable orders [1, Theorem (5.14)]. The statement here slightly improves on [1].

Theorem 3.11 Let P be a measurable order on Γ and let K be an arbitrary compact subset of Γ. Let N = ∅ if Γ is discrete and N = D_{α_{0}} if Γ is not discrete, where D_{α_{0}} is as in Theorem (3.10). Then there is a continuous real-valued homomorphism ψ of Γ such that

sgn_{P} (χ) = sgn(ψ(χ))

for all χ ∈ K \ D_{α_{0}}.

Proof. We treat the discrete case first. Without loss of generality, we may assume that K is a finite subset of Γ not containing 0. We appeal to Theorem (2.4) and use its notation. Let

D_{α_{1}} ⊂ C_{α_{1}} ⊂ D_{α_{2}} ⊂ C_{α_{2}} ⊂ … D_{α_{n}} ⊂ C_{α_{n}}

be a finite collection in Σ such that K ∩ (C_{α_{j}}\D_{α_{j}}) ≠ ∅ for all j = 1, 2, …, n, and

K ⊂ ⋃_{j=1}^{n} (C_{α_{j}}\D_{α_{j}} ) ,

and let L_{α_{j}} be the real-valued homomorphism of Γ corresponding to α_{j}. We have from Theorem (2.4)

L_{α_{j}}(D_{α_{j}}) = {0};  (9)

L_{α_{j}} (K ∩ P ∩ (C_{α_{j}}\D_{α_{j}})) ⊂]0, ∞[;  (10)

and

L_{α_{j}} (K ∩ (−P) ∩ (C_{α_{j}}\D_{α_{j}})) ⊂] − ∞, 0[.  (11)

We construct the homomorphism ψ as a linear combination

ψ = ∑_{j=1}^{n} a_{j}L_{α_{j}}

where the coefficients a_{j} are defined inductively as follows. Set a_{1} = 1. If a_{j} is defined for j = 1, …, k − 1, let

A_{k} = max_{x∈K} ∑_{j=1}^{k−1} |a_{j}L_{α_{j}}(x)|,

B_{k} = min_{x∈K∩(C_{α_{k}} \D_{α_{k}})} |L_{α_{k}}(x)|.

Note that B_{k} is positive. Choose a_{k} so that a_{k}B_{k} > A_{k}. Using (9)–(11), it is straightforward to check that the homomorphism ψ has the desired property. To treat the general case, we appeal to Theorem (3.10), and borrow its notation. Let Φ denote the quotient homomorphism of Γ onto the discrete group Γ/(R^{a} ×H) (recall that R^{a} × H is open). Since K is compact, Φ(K) is a finite subset of Γ/(R^{a} × H). Order Γ/(R^{a} × H) as in Theorem (2.1). By the case we just treated, we can find a homomorphism ψ^{∗} of Γ/(R^{a} × H) separating the set Φ(K). It is clear that the homomorphism ψ^{∗} ∘ Φ separates the set K\(R^{a} × H). If K ∩ ((R^{a} × H)\D_{α_{0}}) = ∅ then we are done. If not, consider the homomorphism

ψ = L_{α_{0}} + \frac{a}{b}ψ^{∗} ∘ Φ

where a = max_{x∈K} |L_{α_{0}}(x)| and b = min_{x∈K\R^{a}×H} |ψ^{∗} ∘ Φ(x)|. A simple argument that we omit shows that ψ has the desired property.

4 The conjugate function and its basic properties

In this section, we use the structure of orders that we derived to define the conjugate series of a function in L^{p}(G), 1 ≤ p < ∞. For use in the following section, we also recall some basic properties of the conjugate function operator, such as generalized versions of M. Riesz’s and Kolmogorov’s Theorems.

Let Γ denote a locally compact abelian group containing a measurable order P, and let G denote the dual group of Γ. Recall that the conjugate function operator f ↦ f̃ is defined on L^{2}(G) by the multiplier relation (1). It is convenient to write H_{P} (f) as an alternative notation for the conjugate function. We appeal to Theorems (2.4) and (3.10) and write the group Γ as a disjoint union of open sets

Γ = C_{α_{0}} ∪ ⋃_{α≠α_{0}} C_{α} \ D_{α}.  (12)

For each α ≠ α_{0}, let L_{α} denote the continuous homomorphism from Γ into R such that

sgn(L_{α}(χ)) = sgn_{P} (χ)  (13)

for all χ ∈ C_{α} \ D_{α}, and let L_{α_{0}} denote the continuous homomorphism from Γ into R such that

sgn(L_{α_{0}}(χ)) = sgn_{P} (χ)  (14)

for locally almost all χ ∈ C_{α_{0}}. To simplify notation, let us write Π for the indexing set in both cases of Theorems (2.4) and (3.10). For each α ∈ Π, the subgroup C_{α} is open, and similarly, D_{α} is open for α ≠ α_{0}. Hence the annihilators in G, A(G, C_{α}) and A(G, D_{α}), are compact. Let μ_{α}, respectively, ν_{α}, denote the normalized Haar measure on A(G, C_{α}), respectively, A(G, D_{α}). We have

μ̂_{α} = 1_{C_{α}} and ν̂_{α} = 1_{D_{α}}  (15)

where, if A is a set, 1_{A} is the indicator of A. For f ∈ L^{p}(G), 1 ≤ p < ∞, we have

∥f ∗ μ_{α}∥_{p} ≤ ∥f∥_{p}, and ∥f ∗ ν_{α}∥_{p} ≤ ∥f∥_{p},

([10, Theorem (20.12)]). For α ≠ α_{0}, we let

d_{α}f = f ∗ μ_{α} − f ∗ ν_{α},

and

d_{α_{0}}f = f ∗ μ_{α_{0}}.

It is clear that if f ∈ L^{2}(G), then the support of f̂ is σ−compact. Hence it has a nonvoid intersection with only countably many of the sets appearing on the right side of (12). It follows from (15) that, except for countably many α’s, d_{α}f is zero almost everywhere. By approximating with functions in L^{2}(G), we see that the same is true for any f ∈ L^{p}(G), 1 ≤ p < ∞. As a convention, when d_{α}f = 0 a.e., we take it to be identically 0. With this convention, the formal difference series

∑_{α∈Π} d_{α}f  (16)

has only countably many nonzero terms. The conjugate function will be defined by a series conjugate to (16). Central to our construction is the ergodic Hilbert transform which we introduce next. This transform has been systematically studied by Cotlar [4], Calderón [2], and Coifman and Weiss [3].

Let L_{α} be as in (14) or (13), and let ϕ_{α} denote its adjoint homomorphism. Thus ϕ_{α} is a continuous homomorphism mapping R into G and satisfying

χ ∘ ϕ_{α}(r) = L_{α}(χ)(r)  (17)

for all r ∈ R and all χ ∈ Γ ([10, Section 24]). The truncated Hilbert transform in the direction of L_{α} is the operator defined on L^{p}(G), 1 ≤ p < ∞, by

H_{L_{α},n}f(x) = \frac{1}{π} ∫_{\frac{1}{n} ≤|t|≤n} f(x − ϕ_{α}(t))\frac{1}{t}dt.  (18)

The (ergodic) Hilbert transform in the direction of L_{α} is the operator defined on L^{p}(G), 1 ≤ p < ∞, by

H_{L_{α}}f(x) = lim_{n→∞} H_{L_{α},n}f(x).  (19)

The fact that this limit exits μ−a.e. on G follows from [4] (see also [2] or [3]). In fact, several other properties of this transform follow from those of the Hilbert transform on R and the transference methods of [2] and [3]. For ease of reference, we state some properties that are needed in the sequel. We let L denote an arbitrary continuous nonzero homomorphism from Γ into R, and let ϕ denote its adjoint homomorphism. The operator H_{L,n} is defined as in (18).

Theorem 4.1 Let f ∈ L^{p}(G), where 1 ≤ p < ∞.

(i) The limit

H_{L}f(x) = lim_{n→∞} H_{L,n}f(x)

exists μ−a.e. (ii) If 1 < p < ∞, then the limit converges in L^{p}(G), and

∥H_{L}f∥_{p} ≤ A_{p}∥f∥_{p}

where A_{p} is the bound of the Hilbert transform operator on L^{p}(R). (iii) For f ∈ L^{1}(G), we have

μ (\{x ∈ G : |H_{L}f(x)| > y\}) ≤ \frac{A}{y}∥f∥_{1}

for all y > 0, where A is the weak type (1, 1) norm of the Hilbert transform on L^{1}(R). (iv) For f ∈ L^{2}(G), we have

Ĥ_{L}f(χ) = −isgn(L(χ))f̂(χ)

for almost all χ ∈ Γ.

The usefulness of this theorem is due in great part to the fact that all the estimates are independent of L or G. Property (iv) justifies using the terminology ”the Hilbert transform in the direction of L” and shows a clear connection between the ergodic Hilbert transform and the conjugate function on groups. The proof of (iv) is straightforward, using (ii) and (17). (See [1, Theorem (6.7)].) For use in the sequel, we recall the generalizations of M. Riesz’s Theorem and Kolmogorov’s Theorem from [1]. (These results are due to Helson [7] and [8], when G is compact.) Also, having all the necessary ingredients to prove these results, we will sketch short proofs to make the paper more self contained and to illustrate the use of the separation theorems.

Theorem 4.2 Let G be a locally compact abelian group with dual group Γ, and let P denote an arbitrary measurable order on Γ. For all f ∈ L^{p}(G), 1 < p < ∞, we have

∥H_{P} f∥_{p} ≤ A_{p}∥f∥_{p}

where A_{p} is the norm of the Hilbert transform on L^{p}(R).

Theorem 4.3 Let G be a locally compact abelian group with dual group Γ, and let P denote an arbitrary measurable order on Γ. For all f ∈ L^{2} ∩ L^{1}(G) and all y > 0, we have

μ (\{x ∈ G : |H_{P} f(x)| > y\}) ≤ \frac{A}{y}∥f∥_{1}

where A is the weak type (1, 1) norm of the Hilbert transform on L^{1}(R).

Both theorems are proved in a similar way. It is enough to consider f ∈ L^{2}(G) with compactly supported Fourier transform. Let K ⊂ Γ denote the compact support of f̂. Apply Theorem (3.11) and obtain a real-valued homomorphism L of Γ such that

sgn_{P} (χ) = sgn(L(χ))

for almost all χ ∈ K. Thus, from Theorem (4.1)(iv) and the fact that f̂ is supported in K, it follows from the uniqueness of the Fourier transform that

H_{P} f = H_{L}f

a.e. on G. The inequalities in Theorems (4.2) and (4.3) follow now from the corresponding ones for H_{L} in Theorem (4.1). Because of Theorem (4.3), the operator H_{P} extends from L^{2} ∩ L^{1}(G) to an operator on L^{1}(G) satisfying the same weak type (1, 1) estimate. We keep the same notation for the extended operator.

The next theorem is our first step toward building the conjugate function. We continue with the notation leading to (16).

Theorem 4.4 Let f ∈ L^{p}(G) where 1 ≤ p < ∞, and let α ∈ Π. Then

(i) H_{P} (d_{α}f) = H_{L_{α}}(d_{α}f) μ−a.e. If f ∈ L^{2} ∩ L^{p}(G), then we also have (ii) H_{P} (d_{α}f) = d_{α}(H_{P} f) and H_{L_{α}}(d_{α}f) = d_{α}(H_{L_{α}}f) μ−a.e.

Proof. The equalities in (ii) are clear since all operators in question are multiplier operators and so they commute. To prove (i) we note that since d_{α} is a bounded operator from L^{1}(G) into L^{1}(G), and since H_{P} and H_{L_{α}} are bounded from L^{1}(G) into L^{1,∞}(G), it is enough to consider f ∈ L^{2}(G). Since sgn_{P} and sgn(L_{α}(·)) agree a.e. on C_{α} \ D_{α}, and since d_{α} projects the Fourier transform on C_{α} \ D_{α}, it is easy to see that the Fourier transforms of H_{P} (d_{α}f) and H_{L_{α}}(d_{α}f) agree almost everywhere on Γ, and so (i) follows. As we argued for (16), we will agree that, for f ∈ L^{p}(G) (1 ≤ p < ∞), the formal series

∑_{α∈Π} H_{L_{α}}(d_{α}f)  (20)

has only countably many terms. We will refer to (20) as the conjugate (difference) series of f.

5 Unconditional convergence of conjugate difference series

We will show that the conjugate series (20) converges unconditionally in L^{p}(G) when 1 < p < ∞ and unconditionally in L^{1,∞}(G) when p = 1. This will further justify our notation in (20) since the order of summation will become irrelevant in (20). For use with weak type estimates, we recall a few facts about the Lorentz spaces L^{p,∞}(G). All details can be found in [11, Chapter V, Section 3]. Although the presentation in the cited reference is confined to σ−finite measure spaces, the results that we need on locally compact abelian groups follow easily by restricting a given function to its σ−compact support. Given a measurable function f on G, let λ_{f} denote its distribution function, and let f^{∗} denote the decreasing rearrangement of f. Define

∥f∥_{p,∞}^{∗} = sup_{y>0} y (λ_{f}(y))^{\frac{1}{p}} = sup_{y>0} y^{1/p}f^{∗}(y),  (21)

and

∥f∥_{p,∞} = sup_{y>0} y^{\frac{1}{p}} m_{f}(y)  (22)

where

m_{f}(y) = \frac{1}{y} ∫_{0}^{y} f^{∗}(u)du.

Let L^{p,∞}(G) consist of all measurable functions on G such that ∥f∥_{p,∞}^{∗} < ∞. It is well-known that, when 1 < p < ∞, (21) and (22) are equivalent and define a norm on L^{p,∞}(G). In fact, ∥f∥_{p,∞} is a norm for all 1 ≤ p < ∞, and when 1 < p < ∞, we also have

∥f∥_{p,∞}^{∗} ≤ ∥f∥_{p,∞} ≤ \frac{p}{p − 1}∥f∥_{p,∞}^{∗}.  (23)

(See [11, Chap. V, Theorem 3.21].)

Let ϵ ∈ {−1, 1}^{Π}. We will write ϵ(P) for the subset of Γ obtained from P by changing the sign on C_{α} \ D_{α} according to ϵ(α). That is, if x ∈ C_{α} \ D_{α} and α ≠ α_{0}, or, if x ∈ C_{α_{0}}, then x ∈ ϵ(P) if and only if ϵ(α)x ∈ P. It is easy to see that ϵ(P) is an order on Γ. Suppose that η ∈ {0, 1}^{Π}. Define a projection operator P_{η} on L^{2}(G) by

P̂_{η}(f) = f̂1_{⋃_{α∈Π, η(α)=1} C_{α}\D_{α}}.

Define the conjugate projection operator P̃_{η,P} by

P̃_{η,P} f = H_{P} (P_{η}f).  (24)

Thus,

ˆP̃_{η,P} f(χ) = \{ \begin{matrix}0 & if χ ∉ ⋃_{α,η(α)=1} C_{α} \ D_{α} \\ −isgn_{P} (χ)f̂(χ) & otherwise \end{matrix}  (25)

To establish the unconditional convergence of the conjugate series, the following result is fundamental. It is a simple consequence of Theorems (4.2) and (4.3).

Theorem 5.1 Let η be any element of {0, 1}^{Π}, and let P be an arbitrary order on Γ. (i) The operator P̃_{η,P} is bounded on L^{p}(G) for 1 < p < ∞ with norm ≤ A_{p} where A_{p} is as in Theorem (4.2). (ii) The operator P̃_{η,P} is of weak type (1, 1) on L^{2} ∩ L^{1}(G) with norm ≤ 2A where A is the weak type norm in Theorem (4.3).

Proof. Define ϵ ∈ {−1, 1}^{Π} by ϵ(π) = 1 if η(π) = 1 and ϵ(π) = −1 if η(π) = 0. It is easy to check using the Fourier transform that for all f ∈ L^{2}(G), we have

P̃_{η,P} f = \frac{1}{2}(H_{P} f + H_{ϵ(P)}f).

The theorem follows now from Theorems (4.2) and (4.3) applied to the operators H_{P} and H_{ϵ(P)}.

As a simple corollary, we have the following.

Corollary 5.2 Let {α_{1}, α_{2}, …, α_{n}} be a finite subset of Π. Then, the operator

f ↦ ∑_{j=1}^{n} H_{L_{α_{j}}} (d_{α_{j}}f)

is of weak type (1, 1) on L^{1}(G) with norm ≤ 2A and is bounded from L^{p}(G) into L^{p}(G) with norm ≤ A_{p}, where A and A_{p} are as in Theorems (4.2) and (4.3).

Proof. Define η ∈ {0, 1}^{Π} by η(α_{j}) = 1 for j = 1, 2, …, n and and η(π) = 0 otherwise. Then

∑_{j=1}^{n} H_{L_{α_{j}}} (d_{α_{j}}f) = P̃_{η,P} f.  (26)

Now apply Theorem (5.1). We are now ready to establish the unconditional convergence of the conjugate series (20).

Theorem 5.3 Let f ∈ L^{p}(G), 1 ≤ p < ∞, and let {α_{j}} ⊂ Π be an arbitrary enumeration of the countable set of α ∈ Π such that d_{α}f ≢ 0.

(i) If p = 1, the series ∑_{j} H_{L_{α_{j}}} d_{α_{j}}f converges in L^{1,∞}(G) to H_{P} f. (ii) If 1 < p < ∞, the series ∑_{j} H_{L_{α_{j}}} d_{α_{j}}f converges in L^{p}(G) to H_{P} f.

Proof. We will deal with the case p = 1 only. The other case is done similarly. The assertions of the theorem are clear if f̂ is compactly supported, since in this case only finitely many d_{α}f are nonzero. Suppose that f is an arbitrary function in L^{1}(G), and approximate f in L^{1}(G) by functions with compactly supported Fourier transforms, say {g_{n}}. Then using Corollary (5.2) and Theorem (4.3), we get

∥ ∑_{j=1}^{N} H_{L_{α_{j}}} d_{α_{j}}f − H_{P} f∥_{1,∞}^{∗} ≤ 2∥ ∑_{j=1}^{N} H_{L_{α_{j}}} d_{α_{j}}(f − g_{n}) − H_{P} (f − g_{n})∥_{1,∞}^{∗}

+2∥ ∑_{j=1}^{N} H_{L_{α_{j}}} d_{α_{j}}g_{n} − H_{P} g_{n}∥_{1,∞}^{∗}

≤ 12A∥f − g_{n}∥_{1,∞}^{∗} + 2∥ ∑_{j=1}^{N} H_{L_{α_{j}}} d_{α_{j}}g_{n} − H_{P} g_{n}∥_{1,∞}^{∗}.

Given ϵ > 0, we can make the left side < ϵ by first choosing n so that ∥f −g_{n}∥_{1,∞}^{∗} < \frac{ϵ}{12A} and then choosing N = N(n) so that ∥ ∑_{j=1}^{N} H_{L_{α_{j}}} d_{α_{j}}g_{n} − H_{P} g_{n}∥_{1,∞}^{∗} = 0. This completes the proof.

The conjugate square function We end this section with a study of the square function associated with the conjugate series (20). We start with a definition. For f ∈ L^{p}(G), 1 ≤ p < ∞, let

S̃f = ( ∑_{α∈Π} |H_{L_{α}}(d_{α}f)|^{2})^{\frac{1}{2}} ,  (27)

where the index of summation runs over those α’s for which d_{α}f ≢ 0.

Theorem 5.4 (i) Let 1 < p < ∞. There is a constant B_{p}, depending only on p, such that for all f ∈ L^{p}(G), we have

∥S̃f∥_{p} ≤ B_{p}∥f∥_{p}.

(ii) There is an absolute constant B such that, for all f ∈ L^{1}(G), and all y > 0, we have

μ (\{x ∈ G : |S̃f(x)| > y\}) ≤ \frac{B}{y} ∥f∥_{1}.

Proof. Part (i) is a well-known consequence of Theorem (5.3) (ii). We will omit the proof. (In fact one can prove it by reproducing the argument that we present for part (ii)). To prove (ii), let p be an arbitrary but fixed number in ]0, 1[. We will need Kintchine’s Inequality ([12, Theorem V. 8.4, p.213], which we will cite here in a notation convenient for our proof. Let a_{1}, a_{2}, …, a_{N} be arbitrary complex numbers, and write E for the expected value over the probability space {−1, 1}^{N}. Then, Kintchine’s Inequality asserts that there are constants α_{p} and β_{p}, depending only on p, such that

α_{p} \{∑_{j=1}^{N} |a_{j}|^{2}\}^{\frac{1}{2}} ≤ \{E |∑_{j=1}^{N} a_{j}ϵ_{j}|^{p}\}^{\frac{1}{p}} ≤ β_{p} \{∑_{j=1}^{N} |a_{j}|^{2}\}^{\frac{1}{2}} .  (28)

Returning to the proof of (ii), we note by monotone convergence that it is enough to consider a finite sum

(∑_{j=1}^{N} |H_{L_{α_{j}}} d_{α_{j}}f|^{2})^{\frac{1}{2}} .

Applying Kintchine’s Inequality, we see that, pointwise on G, we have

(∑_{j=1}^{N} |H_{L_{α_{j}}} d_{α_{j}}f|^{2})^{\frac{1}{2}} ≤ C_{p} (E| ∑_{j=1}^{N} ϵ_{j}H_{L_{α_{j}}} d_{α_{j}}f|^{p})^{\frac{1}{p}} .

We think of each ϵ ∈ {−1, 1}^{N} as an element of {−1, 1}^{Π} by setting ϵ(α_{j}) = ϵ(j) for j = 1, 2, …, N, and ϵ(π) = 1 for π ∉ {α_{1}, α_{2}, …, α_{N}}. Let η = η(ϵ) be defined as in the proof of Corollary (5.2), (see (26)) so that

∑_{j=1}^{N} ϵ_{j}H_{L_{α_{j}}} d_{α_{j}}f = P̃_{η(ϵ),ϵ(P)}f.

Then

∥ (∑_{j=1}^{N} |H_{L_{α_{j}}} d_{α_{j}}f|^{2})^{\frac{1}{2}} ∥_{1,∞}^{∗} ≤ C_{p}∥ (E|P̃_{η(ϵ),ϵ(P)}f|^{p})^{\frac{1}{p}} ∥_{1,∞}^{∗}.  (29)

It is easy to prove from definitions that, for any s > 0 and for any measurable function f on G, ∥|f|^{s}∥_{p,∞}^{∗} = ∥f∥_{sp,∞}^{∗ s}. The fact that ∥ · ∥_{\frac{1}{p} ,∞}^{∗} is equivalent to a norm (see (23)), implies that

∥Ef∥_{\frac{1}{p} ,∞}^{∗} ≤ \frac{1}{1 − p}E∥f∥_{\frac{1}{p} ,∞}^{∗}.

We can now estimate the right side of (29) as follows

C_{p}∥ (E|P̃_{η(ϵ),ϵ(P)}f|^{p})^{\frac{1}{p}} ∥_{1,∞}^{∗} = C_{p} (∥E|P̃_{η(ϵ),ϵ(P)}f|^{p}∥_{\frac{1}{p} ,∞}^{∗})^{\frac{1}{p}}
≤ C_{p} (\frac{1}{1 − p}E∥|P̃_{η(ϵ),ϵ(P)}f|^{p}∥_{\frac{1}{p} ,∞}^{∗})^{\frac{1}{p}}

= C_{p} (\frac{1}{1 − p}E∥P̃_{η(ϵ),ϵ(P)}f∥_{1,∞}^{∗ p})^{\frac{1}{p}}

≤ C_{p} (\frac{2^{p}A^{p}}{1 − p}E∥f∥_{1}^{p})^{\frac{1}{p}}

= 2AC_{p} (\frac{1}{1 − p})^{\frac{1}{p}} ∥f∥_{1}.

The penultimate inequality follows from Corollary (5.2). This completes the proof of the theorem.

Acknowledgements The research of the authors was supported by grants from the National Science Foundation (U. S. A.) and the Research Board of the University of Missouri. Both authors are grateful for conversations with Professors Nigel Kalton, and Saleem Watson.

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