A NOTE ON UMD SPACES AND TRANSFERENCE IN VECTOR-VALUED FUNCTION SPACES

Nakhlé H. Asmar, Brian P. Kelly, and Stephen Montgomery-Smith

Abstract. A Banach space X is called an HT space if the Hilbert transform is bounded from L^{p}(X) into L^{p}(X), where 1 < p < ∞. We introduce the notion of an ACF Banach space, that is, a Banach space X for which we have an abstract M. Riesz Theorem for conjugate functions in L^{p}(X), 1 < p < ∞. Berkson, Gillespie, and Muhly [5] showed that X ∈ HT ⟹ X ∈ ACF. In this note, we will show that X ∈ ACF ⟹ X ∈ UMD, thus providing a new proof of Bourgain’s result X ∈ HT ⟹ X ∈ UMD.

1. Introduction

Recent structure theorems of orders and results in abstract harmonic analysis on groups with ordered dual groups have shown that abstract harmonic analysis on such groups captures martingale theory and classical harmonic analysis (see [4]). Our goal in this note is to illustrate this further by obtaining Bourgain’s result [6] as a consequence of a generalized version of M. Riesz’s theorem due to Berkson, Gillespie, and Muhly [5].

We use the usual notation L^{p}(Ω, μ, X) = L^{p}(X) to denote the set of all strongly measurable functions on the measure space (Ω, F, μ) with values in the Banach space X such that ∫_{Ω} ∥f(ω)∥^{p} dμ(ω) < ∞. Recall that X is a UMD space if, for some (equivalently, all) p ∈ (1, ∞), there exists a constant C_{p}(X) such that for all n ∈ ℕ,

(1.1)  ∥ ∑_{j=1}^{n} ϵ_{j}d_{j}∥_{L^{p}(X)} ≤ C_{p}(X)∥ ∑_{j=1}^{n} d_{j}∥_{L^{p}(X)}

Typeset by AMS-TEX 2 NAKHLÉ H. ASMAR, BRIAN P. KELLY, AND STEPHEN MONTGOMERY-SMITH

1991 Mathematics Subject Classification. Primary: 43A17, Secondary: 42A50, 60G46.

The work of the first and third authors was partially funded by NSF grants. The second and third authors’ work was partially funded by the University of Missouri Research Board

for every X-valued martingale difference sequence (d_{j}) and for every (ϵ_{j}) ∈ {−1, 1}^{ℕ}. (For further background regarding this property, see [7], [8] and [9].)

We say that X is an HT space if for some p ∈ (1, ∞), there exists a constant N_{p}(X) such that for all n ∈ ℕ, we have ∥H_{n}f∥_{L^{p}(X)} ≤ N_{p}(X)∥f∥_{L^{p}(X)} for all f ∈ L^{p}(ℝ, X), where

H_{n}f(t) = \frac{1}{π} ∫_{1/n≤|s|≤n} \frac{f(t − s)}{s} ds .

Let G be a compact abelian group with dual group Γ. Let P ⊂ Γ be an order on Γ, that is P + P ⊂ P, P ∩ (−P) = {0}, and P ∪ (−P) = Γ. Define a signum function sgn_{P} on Γ by sgn_{P} (χ) = 1, 0,or −1 according as χ ∈ P \ {0}, χ = 0, or χ ∈ (−P) \ {0}. Define a conjugate function operator T_{P} on the X-valued trigonometric polynomials by

(1.2)  T_{P} (∑_{χ∈Γ} a_{χ} χ) = −i ∑_{χ∈Γ} sgn_{P} (χ)a_{χ} χ .

Then we will say that a Banach space X has the ACF (abstract conjugate function) property if for some p ∈ (1, ∞), there is a constant A_{p}(X) such that for all compact abelian groups G with ordered dual groups, the operator T_{P} extends to L^{p}(G, X), and for all f ∈ L^{p}(G, X), ∥T_{P} f∥_{L^{p}(X)} ≤ A_{p}(X)∥f∥_{L^{p}(X)}.

In [8], Burkholder showed that if X is a UMD space, it is an HT space. He also conjectured that the converse was true, and this was soon answered affirmatively by Bourgain [6]. We will show the same result by first showing that every HT space is an ACF space, and then showing that every ACF space is a UMD space.

This new concept of ACF provides a natural bridge between the concepts of UMD and HT, thus demonstrating how the study of functions on abstract abelian groups enhances and solidifies the connections between harmonic analysis and martingale theory.

2. Transference and the ACF Property The following generalized version of M. Riesz’s theorem follows from [5], Theo-

rem 4.1 (see also [2], Theorem 6.3). We will sketch its proof to show the role of

UMD SPACES AND VECTOR-VALUED TRANSFERENCE 3

transference and the HT property. (2.1) Theorem. If X has the HT property, then it has the ACF property. Furthermore, for each p ∈ (1, ∞), we have that A_{p}(X) ≤ N_{p}(X).

We follow the proof of [5], Theorem 4.1. We will need the following separation theorem for discrete groups (see [3], Theorem 5.14). As shown recently, this result follows easily from the basic background required for the proof of Hahn’s Embedding Theorem for orders ([12], Theorem 16, p. 16). For details, see [4]. (2.2) Theorem. Let P be an order on a discrete abelian group Γ. Given a finite subset F ⊂ Γ, there is a homomorphism ψ : Γ → ℝ such that for all χ ∈ F,

(2.2.1)  sgn_{P} (χ) = sgn(ψ(χ)) .

(2.3) Proof of Theorem 2.1. For f an X-valued trigonometric polynomial, let F be a finite subset of Γ such that f = ∑_{χ∈F} a_{χ} χ where a_{χ} ∈ X. Apply Theorem 2.2 and obtain a real-valued homomorphism ψ such that sgn(ψ(χ)) = sgn_{P} (χ) for all χ ∈ F. With this choice of ψ, T_{P} (f) = −i ∑_{χ∈F} sgn(ψ(χ))a_{χ} χ.

Let ϕ : ℝ → G be the adjoint homomorphism of ψ which is defined by the relation ψ(χ)(t) = χ(ϕ(t)) for all t ∈ ℝ and all χ ∈ Γ. It is easy to see that for any χ ∈ Γ, we have

(2.3.1) lim_{n→∞} \frac{1}{π} ∫_{1/n≤|t|≤n} \frac{χ(x − ϕ(t))}{t} dt = −i sgn(ψ(χ))χ(x)

for all x ∈ G. Consequently, for an X-valued trigonometric polynomial, we have

(2.3.2) lim_{n→∞} \frac{1}{π} ∫_{1/n≤|t|≤n} \frac{f(x − ϕ(t))}{t} dt = T_{P} (f)(x)

for all x ∈ G. Thus, it is enough to show that, for all n ∈ ℕ,

(2.3.3) ∥\frac{1}{π} ∫_{1/n≤|t|≤n} \frac{f(x − ϕ(t))}{t} dt∥_{L^{p}(G,X)} ≤ N_{p}(X)∥f∥_{L^{p}(G,X)} .

This last inequality follows by adapting the transference argument of Calderón [10], Coifman and Weiss [11], to the setting of vector-valued functions, and requires the HT property of X. We omit its proof and refer the reader to the proof of Theorem 2.8 in [5] for details. ■

4 NAKHLÉ H. ASMAR, BRIAN P. KELLY, AND STEPHEN MONTGOMERY-SMITH

3. Proof of Bourgain’s Result

As noted in [9], to show that X ∈ UMD, it is enough to consider dyadic martingale difference sequences defined on [0, 1] using the Rademacher functions (r_{n})_{n=1}^{∞}. (A proof of this reduction can be obtained by adapting Remarque 3 in Maurey [14] to the vector-valued setting.) Also, as noted by Burkholder in [9], it suffices to consider martingale difference sequences such that d_{1} = 0. In [6], Bourgain implicitly uses the fact that to prove X ∈ UMD, it is enough to consider dyadic martingale difference sequences on the infinite dimensional torus, 𝕋^{ℕ}. We will give a precise version of this, setting the notation in the process.

Let ℤ^{∞ ∗} = ∏_{n∈ℕ}^{∗} ℤ denote the weak direct product of ℕ copies of ℤ. When endowed with the discrete topology, ℤ^{∞ ∗} is topologically isomorphic to the dual group of 𝕋^{ℕ}. For each J = (j_{n}) ∈ ℤ^{∞ ∗}, we denote the corresponding character by χ_{J}, that is, χ_{J}(θ_{1}, θ_{2}, …) = ∏_{n=1}^{∞} e^{ij_{n}θ_{n}} where, except for finitely many factors, e^{ij_{n}θ_{n}} ≡ 1. For each J = (j_{n}) ∈ ℤ^{∞ ∗} \{0}, define n(J) to be the largest n ∈ ℕ such that j_{n} ≠ 0, and let n(0) = 0. With a slight abuse of notation, we let ℤ^{n} denote {J ∈ ℤ^{∞ ∗} : n(J) ≤ n} for each n ≥ 0. Note that ℤ^{∞ ∗} = ∪_{n=1}^{∞}ℤ^{n}.

Identifying 𝕋 with the interval [−π, π), define a sequence, (s_{n})_{n=1}^{∞}, of functions on 𝕋^{ℕ} by s_{1} ≡ 1, and for n ≥ 2, s_{n}(θ_{1}, θ_{2}, …) = sgn(θ_{n−1}). Suppose (d_{n}) is a dyadic martingale difference sequence on [0, 1] with d_{1} = 0 and d_{n} = v_{n}(r_{1}, …, r_{n−1})r_{n} where v_{n} : {−1, 1}^{n−1} → X for all n ≥ 2. Letting d′_{1} = 0 and d′_{n} = v_{n}(s_{1}, …, s_{n−1})s_{n} for n ≥ 2 we obtain a martingale difference sequence on 𝕋^{ℕ} such that the sequences (d_{n}) and (d′_{n}) are identically distributed. Call such a sequence a dyadic martingale difference sequence on 𝕋^{ℕ}. Therefore, to prove that X ∈ UMD, it suffices to show that there exists a constant satisfying (1.1) for dyadic martingale difference sequences on 𝕋^{ℕ}. Since we may approximate each d′_{n} by a function with finite spectrum in ℤ^{n} \ ℤ^{n−1}, to show that X has the UMD property, we see that it is sufficient to show the following. For p ∈ (1, ∞), there exists a constant C_{p}(X) such that if K_{j} is a finite subset of ℤ^{j} \ ℤ^{j−1}, and a_{J} ∈ X for all

UMD SPACES AND VECTOR-VALUED TRANSFERENCE 5

J ∈ K_{j}, and (ϵ_{n}) ∈ {−1, 1}^{ℕ}, then

(3.1)  ∥ ∑_{k=1}^{n} ϵ_{k}( ∑_{J∈K_{j}} a_{J} χ_{J})∥_{L^{p}(𝕋^{ℕ},X)} ≤ C_{p}(X)∥ ∑_{j=1}^{n} ∑_{J∈K_{j}} a_{J} χ_{J} ∥_{L^{p}(𝕋^{ℕ},X)} .

This reduction appears in Bourgain [6]. We will show that if X is an HT space, (3.1) follows from Theorem 2.1 with specific choices of the order P on ℤ^{∞ ∗}. For this purpose, define a reversed lexicographic order on ℤ^{∞ ∗} as follows: P = {J = (j_{n}) ∈ ℤ^{∞ ∗} : j_{n(J)} > 0} ∪ {0} where as previously, if J = (j_{n}), then j_{n(J)} is the last non-zero coordinate of J. Thus, sgn_{P} (χ_{J}) = sgn(j_{n(J)}). Observe that if ϵ = (ϵ_{n}) ∈ {−1, 1}^{ℕ}, then the set P (ϵ) = {J = (j_{n}) ∈ ℤ^{∞ ∗} : ϵ_{n(J)}j_{n(J)} > 0} ∪ {0} is also an order on ℤ^{∞ ∗}. In this case, sgn_{P (ϵ)}(χ_{J}) = ϵ_{n(J)}sgn(j_{n(J)}). We now state a simple identity that links the unconditionality of martingale difference sequences to harmonic conjugation with respect to orders: for every n ≥ 1, and all J = (j_{n}) ∈ ℤ^{n} \ ℤ^{n−1}, we have

(3.2)  ϵ_{n} = sgn_{P (ϵ)}(χ_{J}) sgn_{P} (χ_{J}).

To verify (3.2), simply note that for each n ∈ ℕ, if J ∈ ℤ^{n} \ ℤ^{n−1}, then n(J) = n. From (3.2), one immediately obtains that

(3.3)  T_{P} ∘ T_{P (ϵ)}( ∑_{k=1}^{n} ∑_{J∈K_{k}} a_{J} χ_{J}) = ∑_{k=1}^{n} ϵ_{k}( ∑_{J∈K_{k}} a_{J} χ_{J}),

which expresses the martingale transform on the right side as a composition of two conjugate function operators. Applying Theorem 2.1 twice yields (3.1) and implies our next and last result. (3.1) Theorem. Suppose X is a Banach space, and let 1 < p < ∞. If X has ACF, then X is a UMD space and (1.1) holds with C_{p}(X) ≤ (A_{p}(X))^{2}. (3.2) Remarks. (a) Combining the implications above, we see that for a Banach space X, the properties UMD, ACF, and HT are equivalent.

(b) Burkholder also proved in [8] that when X is a UMD space, the Hilbert transform is weak-type bounded on L^{1}(ℝ, X). This weak-type estimate also transfers to

6 NAKHLÉ H. ASMAR, BRIAN P. KELLY, AND STEPHEN MONTGOMERY-SMITH

the conjugate function operator on L^{1}(G, X) defined with respect to an arbitrary

order on Γ. For a proof see [13]. Thus, when X ∈ UMD, we have vector-valued

analogs of the classical results of M. Riesz and Kolmogorov.

One may ask for an analog of Privalov’s classical theorem for UMD-valued func-

tions in L^{1}(G, X). For the case of X = ℂ, this is done in [4].

Transference of operators on vector-valued L^{p}-spaces can be carried out in much

greater generality than that shown here. A representation satisfying the vector-

valued version of the distributional control condition introduced in [1] will transfer

strong-type and weak-type bounds for maximal operators. This work will appear

in [13]. The contents of this article are part of the second author’s dissertation.

References

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Department of Mathematics, University of Missouri-Columbia, Columbia, Mis-

souri

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65211 U.S.A.