OPERATOR-VALUED FOURIER HAAR MULTIPLIERS

MARIA GIRARDI

Abstract. Criteria are given to ensure the boundedness of Fourier Haar multiplier operators from L_{p} ([0, 1] , X) to L_{q} ([0, 1] , Y) where the Fourier Haar multiplier sequences come not from ℝ, as in the classical setting, but rather from the space of bounded linear operators from a Banach space X into a Banach space Y.

1. INTRODUCTION

It is well known that the Haar system {h_{j}}_{j∈ℕ} forms an unconditional basis in L_{p} ([0, 1], ℝ) for 1 < p < ∞. Thus the Fourier Haar multiplier operator T, generated by the Fourier Haar multiplier sequence {λ_{j}}_{j∈ℕ} from ℝ, defined on the span of the Haar system {h_{j}}_{j∈ℕ} by

T (∑_{j=1}^{m} c_{j}h_{j}) = ∑_{j=1}^{m} λ_{j}c_{j}h_{j}  where c_{j} ∈ ℝ and m ∈ ℕ ,  (1.1)

extends (uniquely) to a bounded linear operator on the whole of L_{p} ([0, 1], ℝ) provided the multiplier sequence is bounded, in which case,

∥T ∥_{L_{p}([0,1],ℝ)→L_{p}([0,1],ℝ)} ≤ C_{p} sup_{j∈ℕ}|λ_{j}|

for some constant C_{p} for 1 < p < ∞. Much is known (cf. e.g. [19] and the references therein) about the boundedness of such Fourier Haar multiplier operator from L_{p} ([0, 1], ℝ) to L_{q} ([0, 1], ℝ). If 1 < p ≤ q < ∞, then

∥T ∥_{L_{p}([0,1],ℝ)→L_{q}([0,1],ℝ)} ≈ sup_{(n,k)∈Δ} 2^{n( \frac{1}{p} − \frac{1}{q} )} |λ_{2^{n}+k}|  (1.2)

where {h_{k}^{n}}_{(n,k)∈Δ} is the dyadic enumeration of the Haar system. While if 1 < q ≤ p < ∞, then

∥T ∥_{L_{p}([0,1],ℝ)→L_{q}([0,1],ℝ)} ≈ ∥ sup_{j∈ℕ} | λ_{j} h_{j} |∥_{L_{r}([0,1],ℝ)}  (1.3)

where \frac{1}{r} = \frac{1}{q} − \frac{1}{p}. In both cases, the equivalence constants depend only on p and q.

In (1.1), one can replace c_{j} ∈ ℝ by x_{j} in some Banach space X and then consider the boundedness of T on L_{p} ([0, 1], X). Here UMD (unconditionality property for martingale differences) spaces play

Date: 11 July 2005.

2000 Mathematics Subject Classification. 46E40, 46B20, 60G42.

Key words and phrases. Fourier Haar multipliers, UMD, R-boundedness.

Girardi is supported in part by the National Science Foundation Grant DMS-0306750.

2 MARIA GIRARDI

a key role. Indeed, each T generated by a Fourier Haar multiplier sequence {ε_{j}}_{j∈ℕ} from {±1} is bounded (by some constant depending only on X and p) on L_{p} ([0, 1], X) for some (or equivalently, for each) p ∈ (1, ∞) if and only if X is a UMD space.

This paper considers Fourier Haar multiplier operators from L_{p} ([0, 1], X) to L_{q} ([0, 1], Y) where the Fourier Haar multiplier sequence comes not from ℝ but rather from the space B (X, Y) of bounded linear operators from a Banach space X into a Banach space Y. Not surprisely, UMD plays a role. However, an R-boundedness assumption on the multiplier sequence is also used. R-boundedness was introduced by Berkson and Gillespie in [2]. This notion grew out of work of J. Bourgain on vector-valued Fourier transform [3] and has been central to recent results on operator-valued Fourier multipliers and singular integrals with operator-valued kernels on Bochner spaces (e.g. [1, 11, 13, 23]). Through these tools, R-boundedness became important for maximal regularity of parabolic differential equations (e.g. [8, 16, 23]) and the holomorphic functional calculus of sectorial operators (e.g. [14, 15, 16]). It is a key notion in the study [12] of martingales transforms by operator-valued multiplier, which is especially useful for the theory of stochastic integration on Banach spaces which recently was developed in [21] and [22]. For more information on Rboundedness and its properties, see [7, 10, 16].

Theorem 3.3, which covers the case that 1 < q ≤ p < ∞, generalizes (1.3). Its simple short proof, which uses the notions of UMD and R-boundedness, is very different from the usual proof for scalar-valued multiplier sequences, which uses interpolation and is much longer. Theorem 3.4, which covers the case that 1 ≤ p < q < ∞, generalizes (1.2). In this case, the usual proof of the scalar-valued case can be generalized and so no UMD nor R-boundedness assumptions are necessary. It is interesting that in one case UMD and R-boundedness need to be used but in the other case they do not. This work was motivated by a recent paper [12] on martingale transforms where the multiplier sequence is B (X, Y)-valued.

This paper is organized as follows. Section 2 collects the needed definitions and notation. Section 3 contains the main results. Closing examples and remarks are in Section 4.

2. DEFINITIONS and NOTATION

Throughout this paper, the Banach spaces that appear are over the fixed scalar field of either the real or complex numbers. X, Y, and Z are Banach spaces. B (X) is the closed unit ball of X. The space B (X, Y) of bounded linear operators from X into Y is endowed with the usual operator norm topology. For a measure space (Ω, F, μ), the Bochner-Lebesgue space L_{p} (Ω, X) consists of the measurable functions from Ω into X with finite L_{p} (Ω, X)-norm where 1 ≤ p ≤ ∞. The weak-L_{p}

OPERATOR-VALUED FOURIER HAAR MULTIPLIERS 3

space L_{p}^{wk} (Ω, X), for 1 ≤ p < ∞, consists of the measurable functions from Ω into X that satisfy

∥f∥_{L_{p}^{wk}(Ω,X)} := sup_{λ>0} λ [μ (\{ω ∈ Ω: ∥f (ω)∥_{X} > λ\})]^{\frac{1}{p}} < ∞ .

It is well-known that the above expression ∥·∥_{L_{p}^{wk}(Ω,X)} is a quasi-norm on L_{p}^{wk} (Ω, X) with

∥f + g∥_{L_{p}^{wk}(Ω,X)} ≤ 2 [∥f∥_{L_{p}^{wk}(Ω,X)} + ∥g∥_{L_{p}^{wk}(Ω,X)}] .

The balls with respect to ∥·∥_{L_{p}^{wk}(Ω,X)} define a linear topology on L_{p}^{wk} (Ω, X) and L_{p}^{wk} (Ω, X), endowed with this topology, is a quasi-Banach space.

ℕ is the set of natural numbers while ℕ_{0} = ℕ ∪ {0}. Nonnumerical subscripts on constants indicate dependency.

Let (Ω, F, μ) be a probability space with a filtration {F_{n}}_{n=1}^{m} (i.e., {F_{n}}_{n=1}^{m} is a nondecreasing sequence of sub-σ-fields of F) where m ∈ ℕ. A sequence {d_{n}}_{n=1}^{m} of functions from Ω into X is a (stationary) martingale difference sequence with respect to {F_{n}}_{n=1}^{m} provided d_{n} ∈ L_{1} ((Ω, F_{n}, μ) , X) and 𝔼 (d_{n+1} | F_{n}) = 0 for each admissible n. There is a one-to-one correspondence between martingales {f_{n}}_{n=1}^{m} and martingale difference sequence {d_{n}}_{n=1}^{m} given by f_{n} = ∑_{k=1}^{n} d_{k}. A sequence {v_{n}}_{n=1}^{m} of functions from Ω into Z is {F_{n}}_{n=1}^{m}-predictable provided v_{n} is F_{n−1}-measurable for each n ∈ {1, 2, … m} (where F_{0} := F_{1}). The martingale transform of an X-valued martingale {∑_{k=1}^{n} d_{k}}_{n=1}^{m} with respect to {F_{n}}_{n=1}^{m} by a B (X, Y)-valued {F_{n}}_{n=1}^{m}-predictable sequence {v_{n}}_{n=1}^{m} is the Y -valued martingale {∑_{k=1}^{n} v_{k}d_{k}}_{n=1}^{m} with respect to {F_{n}}_{n=1}^{m}. Burkholder [4] introduced UMD Banach spaces.

Definition 2.1. The UMD constant of X is the smallest β_{p} (X) ∈ [1, ∞] so that

∥ε_{1}d_{1} + … + ε_{m}d_{m}∥_{L_{p}(Ω,X)} ≤ β_{p} (X) ∥d_{1} + … + d_{m}∥_{L_{p}(Ω,X)}

for each X-valued martingale difference sequence {d_{n}}_{n=1}^{m} with respect to some filtration {F_{n}}_{n=1}^{m}, choice {ε_{n}}_{n=1}^{m} of signs from {±1}, and m ∈ ℕ. A Banach space X is UMD provided that its UMD constant is finite for some (or equivalently, by Pisier [18], for each) p ∈ (1, ∞).

One obtains an equivalent formulation of UMD spaces (with the same UMD constant) if, in Definition 2.1, one replaces choice {ε_{n}}_{n=1}^{m} of signs from {±1} with [−1, 1]-valued {F_{n}}_{n=1}^{m}-predictable sequence {v_{n}}_{n=1}^{m}.

Notation 2.2. Henceforth, (Ω, F, μ) denotes the usual Lebesgue measure space on [0, 1].

Let

Δ_{1} = {(n, k) ∈ ℕ_{0} × ℕ: 1 ≤ k ≤ 2^{n}}

4 MARIA GIRARDI

Δ_{0} = {(0, 0)} and Δ = Δ_{0} ∪ Δ_{1} .

There is a bijection from Δ onto ℕ given by (n, k) → 2^{n} +k, which provides a linear ordering on Δ. Thus any sequence {Θ_{j}}_{j∈ℕ} of objects can also be denoted by {Θ_{(n,k)}}_{(n,k)∈Δ} where Θ_{(n,k)} = Θ_{2^{n}+k}. This identification will be used freely throughout this paper.

The dyadic intervals {I_{k}^{n}: (n, k) ∈ Δ_{1}} are given by

I_{1}^{n} = [0, \frac{1}{2^{n}} ] and I_{k}^{n} = (\frac{k − 1}{2^{n}}, \frac{k}{2^{n}} ] for k > 1 .

The Haar system {h_{k}^{n}}_{(n,k)∈Δ} is given by h_{0}^{0} = 1_{I_{1}^{0}} and, for (n, k) ∈ Δ_{1},

h_{k}^{n} = 1_{I_{2k−1}^{n+1}} − 1_{I_{2k}^{n+1}} .

The Haar filtration {H_{j}}_{j∈ℕ} is defined by

H_{j} = σ {h_{1}, …, h_{j}} .

The Rademacher functions {r_{n}}_{n∈ℕ_{0}} take the form r_{0} = h_{0}^{0} and, for n ∈ ℕ,

r_{n} = ∑_{k=1}^{2^{n−1}} h_{k}^{n−1} .

Let

E (Ω, X) := {f: Ω → X | f = ∑_{j=1}^{n} x_{j}h_{j} for some n ∈ ℕ , x_{j} ∈ X} .

E (Ω, X) is norm dense in L_{p} (Ω, X) for 1 ≤ p < ∞; indeed, X-valued simple functions are dense in L_{p} (Ω, X) and the Haar system is a basis for L_{p} (Ω, ℝ). Also, the representation of functions in E (Ω, X) is unique: if f = ∑_{j=1}^{n} x_{j}h_{j} then x_{j} = ∥h_{j}∥_{L_{1}}^{−1} ∫_{Ω} f (ω) h_{j} (ω) dω.

Definition 2.3. The Fourier Haar multiplier operator T, generated by a Fourier Haar multiplier sequence {T_{j}}_{j∈ℕ} from B (X, Y), is the linear mapping from E (Ω, X) to E (Ω, Y) given by

T (∑_{j∈ℕ} x_{j}h_{j}) = ∑_{j∈ℕ} T_{j}x_{j}h_{j} for ∑_{j∈ℕ}x_{j}h_{j} ∈ E (Ω, X) .

For 1 ≤ p, q < ∞ define

∥T ∥_{L_{p}(Ω,X)→L_{q}(Ω,Y)} := sup_{\begin{matrix}f∈E(Ω,X) \\ f≠0 \end{matrix}} \begin{matrix}∥T f∥ & L_{q}(Ω,Y) \\ ∥f∥ & L_{p}(Ω,X) \end{matrix} .  (2.1)

If the supremum in (2.1) is finite, the T is called a bounded Fourier Haar multiplier operator (from L_{p} (Ω, X) to L_{q} (Ω, Y)).

OPERATOR-VALUED FOURIER HAAR MULTIPLIERS 5

In Definition 2.3, if T is a bounded Fourier Haar multiplier operator, then T: E (Ω, X) → E (Ω, Y) extends uniquely to a bounded linear operator from L_{p} (Ω, X) to L_{q} (Ω, Y), with norm the supremum in (2.1). In Definition 2.3, one can replace L_{q} (Ω, Y) with L_{q}^{wk} (Ω, Y) for 1 ≤ q < ∞. All remains valid except, in the bounded case, the norm of the extension is at most twice the supremum in (2.1).

Loosely speaking, a set τ of operators is R-bounded provided Kahane’s Contraction Principle holds for operator coefficients from τ. The precise definition is as follows.

Definition 2.4. Let τ be a subset of B (X, Y) and p ∈ [1, ∞). Let R_{p}(τ) be the smallest constant R ∈ [0, ∞] with the property that for each n ∈ ℕ and subset {T_{j}}_{j=1}^{n} of τ and subset {x_{j}}_{j=1}^{n} of X,

∥∑_{j=1}^{n} r_{j}(·)T_{j}(x_{j})∥_{L_{p}([0,1],Y)} ≤ R ∥∑_{j=1}^{n} r_{j}(·)x_{j}∥_{L_{p}([0,1],X)} .

The set τ is R-bounded provided R_{p}(τ) is finite for some (and thus then, by Kahane’s inequality, for each) p ∈ [1, ∞).

Pisier [1] showed that each (norm) bounded subset of B (X, Y) is R-bounded if and only if X has cotype 2 and Y has type 2 (cf. e.g. [17] for needed definitions). Note that if X and Y are q-concave Banach lattices for some finite q (e.g. X = Y = L_{q} (Ω, ℂ) where 1 ≤ q < ∞) then R-boundedness is equivalent to the square function estimate

∥(∑_{j=1}^{m} |T_{j}x_{j}|^{2})^{1/2}∥_{Y} ≤ R ∥(∑_{j=1}^{n} |x_{j}|^{2})^{1/2}∥_{X}

known from harmonic analysis (cf. [17, Thm. II.1.d.6]). For basic properties of R-bounded sets and further references, see [7, 10, 16, 23].

All notation and terminology, not otherwise explained, are as in [6, 9, 17].

3. MAIN RESULTS

Consider a Fourier Haar multiplier operator T generated by {T_{j}}_{j∈ℕ} from B (X, Y). This section gives conditions on {T_{j}}_{j∈ℕ} that guarantee that T is bounded from L_{p} (Ω, X) to L_{q} (Ω, Y). Remark 3.1 relates the boundedness of T to the boundedness of certain martingale transforms.

Remark 3.1. Note that {d_{n}}_{n=1}^{m} is an X-valued martingale difference sequence with respect to the Haar filtration {H_{n}}_{n=1}^{m} if and only if it takes the form d_{n} = x_{n}h_{n} for some x_{n} ∈ X. Let

v_{j} (·) := T_{j} |h_{j} (·)| .

Then {v_{n}}_{n=1}^{m} is a B (X, Y)-valued {H_{n}}_{n=1}^{m}-predictable sequence. Furthermore, the martingale transform of {∑_{k=1}^{n} d_{k}}_{n=1}^{m} by {v_{n}}_{n=1}^{m} has the form

∑_{n=1}^{m} v_{n} (·) d_{n} (·) = ∑_{n=1}^{m} T_{n} |h_{n} (·)| x_{n}h_{n} (·) = ∑_{n=1}^{m} T_{n} x_{n}h_{n} (·) .

Thus T is bounded (by some constanst C_{XY pq}) if and only if

∥∑_{n=1}^{m} v_{n}d_{n}∥_{L_{q}(Ω,Y)} ≤ C_{XY pq} ∥∑_{n=1}^{m} d_{n}∥_{L_{p}(Ω,X)}

for each X-valued Haar martingale difference sequence {d_{n}}_{n=1}^{m}.

Motivated by Remark 3.1, define u_{p}: Ω → [0, ∞] by

u_{p} (·) := R_{p} ({T_{j} |h_{j} (·)| : j ∈ ℕ})  (3.1)

for 1 ≤ p < ∞. Thus

u_{p} (ω) = R_{p} ({T_{0}^{0}} ∪ {T_{k}^{n} : (n, k) ∈ Δ_{1} , ω ∈ I_{k}^{n}}) .

Clearly u_{p} is measurable since it is the pointwise limit of the sequence {s_{n}}_{n∈ℕ} where

s_{n} (·) := ∑_{k=1}^{2^{n}} R_{p} ({T_{0}^{0}} ∪ {T_{j}^{m}: (m, j) ∈ Δ_{1} , I_{k}^{n} ⊂ I_{j}^{m}}) 1_{I_{k}^{n}} (·) .

The case p = q is a direct consequence of results in [12].

Theorem 3.2. Let T be the Fourier Haar multiplier operator generated by {T_{j}}_{j∈ℕ} from B (X, Y). Let X and Y be UMD spaces. Let u_{p} be as in (3.1).

(a) If 1 < p < ∞ then ∥T ∥_{L_{p}(Ω,X)→L_{p}(Ω,Y)} ≤ β_{p} (X) β_{p} (Y) ∥u_{p}∥_{L_{∞}(Ω,[0,∞])} .

(b) There exists a constant A_{XY} so that ∥T ∥_{L_{1}(Ω,X)→L_{1}^{wk}(Ω,Y)} ≤ A_{XY} ∥u_{1}∥_{L_{∞}(Ω,[0,∞])} .

Proof. Theorem 3.2 follows easily from Remark 3.1 and [12, Theorem 3.2 and Fact 5.1]). □

The next theorems covers the case q ≤ p. Its rather simple proof is quite different from the usual proof for scalar-valued multiplier sequences (see [19, Theorem 12.2]), which uses interpolation.

Theorem 3.3. Let 1 < q ≤ p < ∞. Let X and Y be UMD spaces. Let T be the Fourier Haar multiplier operator generated by {T_{j}}_{j∈ℕ} from B (X, Y). Then

∥T ∥_{L_{p}(Ω,X)→L_{q}(Ω,Y)} ≤ β_{p} (X) β_{q} (Y) ∥u_{q}∥_{L_{r}(Ω,[0,∞])} where r ∈ (1, ∞] is given by \frac{1}{r} = \frac{1}{q} − \frac{1}{p}.

Recall that u_{q} is defined in (3.1).

Proof. Fix ∑_{j∈ℕ} x_{j}h_{j} ∈ E (Ω, X). Note that for each fixed t ∈ [0, 1],

∥∑_{j∈ℕ} r_{j} (t) x_{j}h_{j}∥_{L_{p}(Ω,X)}^{q} ≤ β_{p}^{q} (X) ∥∑_{j∈ℕ} x_{j}h_{j}∥_{L_{p}(Ω,X)}^{q}
∥∑_{j∈ℕ} T_{j}x_{j}h_{j}∥_{L_{q}(Ω,Y)}^{q} ≤ β_{q}^{q} (Y) ∥∑_{j∈ℕ} r_{j} (t) T_{j}x_{j}h_{j}∥_{L_{q}(Ω,Y)}^{q} .

Thus

∥∑_{j∈ℕ} T_{j}x_{j}h_{j}∥_{L_{q}(Ω,Y)}^{q} ≤ β_{q}^{q} (Y) ∫_{[0,1]} ∥∑_{j∈ℕ} r_{j} (t) T_{j}x_{j}h_{j} (·)∥_{L_{q}(Ω,Y)}^{q} dt
= β_{q}^{q} (Y) ∫_{Ω} ∥∑_{j∈ℕ} r_{j} (·) (T_{j} |h_{j} (ω)|) x_{j}h_{j} (ω)∥_{L_{q}([0,1],Y)}^{q} dω
≤ β_{q}^{q} (Y) ∫_{Ω} R_{q}^{q} ({T_{j} |h_{j} (ω)| : j ∈ ℕ}) ∥∑_{j∈ℕ} r_{j} (·) x_{j}h_{j} (ω)∥_{L_{q}([0,1],X)}^{q} dω
= β_{q}^{q} (Y) ∫_{[0,1]} ∥u_{q} (·) (∑_{j∈ℕ} r_{j} (t) x_{j}h_{j} (·))∥_{L_{q}(Ω,X)}^{q} dt
≤ β_{q}^{q} (Y) ∫_{[0,1]} ∥u_{q}∥_{L_{r}(Ω,[0,∞])}^{q} ∥∑_{j∈ℕ} r_{j} (t) x_{j}h_{j} (·)∥_{L_{p}(Ω,X)}^{q} dt
≤ β_{q}^{q} (Y) ∥u_{q}∥_{L_{r}(Ω,[0,∞])}^{q} β_{p}^{q} (X) ∥∑_{j∈ℕ} x_{j}h_{j}∥_{L_{p}(Ω,X)}^{q} .

This completes the proof. □

The next theorem covers the case p < q. It gives a vector-valued analogue of (1.2). In this case, the usual proof of the scalar-valued case can be generalized and so no UMD nor R-boundedness assumptions are necessary.

Theorem 3.4. Let 1 ≤ p < q < ∞. Let T be the Fourier Haar multiplier operator generated by {T_{j}}_{j∈ℕ} from B (X, Y). Define

A_{pq} := sup_{(n,k)∈Δ} 2^{n( \frac{1}{p} − \frac{1}{q} )} ∥T_{k}^{n}∥_{B(X,Y)} .

(a) If 1 < p then

A_{pq} ≤ ∥T ∥_{L_{p}(Ω,X)→L_{q}(Ω,Y)} ≤ C_{pq} A_{pq} .

(b) If 1 = p then

∥T ∥_{L_{1}(Ω,X)→L_{q}^{wk}(Ω,Y)} ≤ C_{q} A_{1q} .

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Proof. The lower bound in part (a) follows from Remark 4.1.

Set

α = \frac{1}{p} − \frac{1}{q} ;

thus, 0 < α < 1. Define J: ℝ → ℝ via

J (t) := \begin{cases}|t|^{α−1} & if t ≠ 0 \\ 0 & if t = 0 . \end{cases}

By the Hardy-Littlewood-Sobolev theorem (cf. [20, page 119]), for each g ∈ L_{p} (ℝ, ℝ), the integral

(Sg) (t) := ∫_{ℝ} \frac{g (s)}{|t − s|^{1−α}} ds = (J ∗ g) (t)

converges absolutely for a.e. t ∈ ℝ and the operator S satisfies

∥Sg∥_{L_{q}(ℝ,ℝ)} ≤ C_{pq} ∥g∥_{L_{p}(ℝ,ℝ)} if p > 1
∥Sg∥_{L_{q}^{wk}(ℝ,ℝ)} ≤ C_{pq} ∥g∥_{L_{1}(ℝ,ℝ)} if p = 1

for some constants C_{pq}.

Define K: Ω × Ω → B (X, Y) via

K (t, s) = \begin{cases}∑_{(n,k)∈Δ} 2^{n}T_{k}^{n}h_{k}^{n} (t) h_{k}^{n} (s) & if t ≠ s \\ 0 & if t = s . \end{cases}  (3.2)

Note that, for each fixed t and s with t ≠ s, only a finite number of terms in the summand in (3.2) are nonzero. Fix f = ∑_{(n,k)∈Δ} x_{k}^{n}h_{k}^{n} ∈ E (Ω, X). Thus, for each t ∈ Ω,

∫_{Ω} K (t, s) f (s) ds = ∫_{Ω\{t}} [ ∑_{(m,j)∈Δ} 2^{m}T_{j}^{m}h_{j}^{m} (t) h_{j}^{m} (s)] [ ∑_{(n,k)∈Δ} x_{k}^{n}h_{k}^{n} (s)] ds

= ∑_{(n,k)∈Δ} ∫_{Ω} ∑_{(m,j)∈Δ} [T_{j}^{m}x_{k}^{n}h_{j}^{m} (t)] [2^{m}h_{j}^{m} (s) h_{k}^{n} (s)] ds

= ∑_{(n,k)∈Δ} T_{k}^{n}x_{k}^{n}h_{k}^{n} (t) = (T f) (t) .

Fix t, s ∈ Ω with t ≠ s. Find the unique m ∈ ℕ so that

2^{−m} < |t − s| ≤ 2^{−m+1} .

So h_{k}^{n} (t) h_{k}^{n} (s) = 0 if n ≥ m and (n, k) ∈ Δ_{1}. Thus

\begin{aligned}∥K (t, s)∥_{B(X,Y)} & ≤ ∑_{(n,k)∈Δ} 2^{n} ∥T_{k}^{n}∥_{B(X,Y)} |h_{k}^{n} (t) h_{k}^{n} (s)| \\ & ≤ ∑_{(n,k)∈Δ} 2^{n}A_{pq}2^{−nα} |h_{k}^{n} (t) h_{k}^{n} (s)| \\ & ≤ A_{pq} [1 + ∑_{n=0}^{m−1} (2^{1−α})^{n}] \\ & = A_{pq} [1 + \frac{2^{(1−α)m} − 1}{2^{1−α} − 1}] \frac{2^{α−1}}{2^{α−1}} \\ & ≤ \frac{A_{pq}}{1 − 2^{α−1}} \frac{1}{(2^{−m+1})^{1−α}} \\ & ≤ \frac{A_{pq}}{1 − 2^{α−1}} \frac{1}{|t − s|^{1−α}} \\ & = \frac{A_{pq}}{1 − 2^{α−1}} J (t − s) . \end{aligned}  (3.3)

Fix f ∈ E (Ω, X). Define g ∈ L_{∞} (ℝ, ℝ) via

g (t) := \begin{cases}∥f (t)∥_{X} & if t ∈ Ω \\ 0 & if t ∉ Ω . \end{cases}

Towards part (b), now let 1 = p < q < ∞. For each t ∈ Ω

∥(T f) (t)∥_{Y} = ∥∫_{Ω} K (t, s) f (s) ds∥_{Y} ≤ \frac{A_{1q}}{1 − 2^{α−1}} ∫_{Ω} J (t − s) g (s) ds

by (3.3). Thus, for each λ > 0,

λμ^{1/q} ({t ∈ Ω: ∥(T f) (t)∥_{Y} > λ})

≤ \frac{A_{1q}}{1 − 2^{α−1}} \frac{λ (1 − 2^{α−1})}{A_{1q}} μ^{1/q} (\{t ∈ ℝ: |(J ∗ g) (t)| > \frac{λ (1 − 2^{α−1})}{A_{1q}}\})

≤ \frac{A_{1q}}{1 − 2^{α−1}} ∥J ∗ g∥_{L_{q}^{wk}(ℝ,ℝ)}

≤ \frac{C_{1q}}{1 − 2^{α−1}} A_{1q} ∥g∥_{L_{1}(ℝ,ℝ)} = \frac{C_{1q}}{1 − 2^{α−1}} A_{1q} ∥f∥_{L_{1}(Ω,X)} .

Thus part (b) holds.

Towards part (a), now let 1 < p < q < ∞. By (3.3)

∥T f∥_{L_{q}(Ω,Y)}^{q} = ∫_{Ω} ∥∫_{Ω} K (t, s) f (s) ds∥_{Y}^{q} dt

≤ ∫_{Ω} [∫_{Ω} ∥K (t, s)∥_{B(X,Y)} ∥f (s)∥_{X} ds]^{q} dt

10 MARIA GIRARDI

≤ [\frac{A_{pq}}{1 − 2^{α−1}} ]^{q} ∫_{Ω} [∫_{ℝ} J (t − s) g (s) ds]^{q} dt

= [\frac{A_{pq}}{1 − 2^{α−1}} ]^{q} ∫_{Ω} |(Sg) (t)|^{q} dt

≤ [\frac{A_{pq}}{1 − 2^{α−1}} ]^{q} ∥Sg∥_{L_{q}(ℝ,ℝ)}^{q} ≤ [\frac{A_{pq}}{1 − 2^{α−1}} ]^{q} C_{pq}^{q} ∥g∥_{L_{p}(ℝ,ℝ)}^{q}

= [\frac{C_{pq}}{1 − 2^{α−1}} A_{pq} ∥f∥_{L_{p}(Ω,X)}]^{q} .

Thus part (a) holds. □

4. EXAMPLES AND REMARKS

A lower bound on the norm of a Fourier Haar multiplier operator is easy. Remark 4.1. Let T be the Fourier Haar multiplier operator generated by {T_{j}}_{j∈ℕ} from B (X, Y). Then

∥T ∥_{L_{p}(Ω,X)→L_{q}(Ω,Y)} ≥ sup_{(n,k)∈Δ} 2^{n( \frac{1}{p} − \frac{1}{q} )} ∥T_{k}^{n}∥_{B(X,Y)}

for each 1 ≤ p, q < ∞. Proof. Fix (n, k) ∈ Δ. Then

∥T ∥_{L_{p}(Ω,X)→L_{q}(Ω,Y)} ≥ sup_{x∈B(X)} \begin{matrix}∥T_{k}^{n}xh_{k}^{n}∥ & L_{q}(Ω,Y) \\ ∥xh_{k}^{n}∥ & L_{p}(Ω,X) \end{matrix}

= sup_{x∈B(X)} \begin{matrix}∥T_{k}^{n}x∥ & Y \\ ∥x∥ & X \end{matrix} \begin{matrix}∥h_{k}^{n}∥ & L_{q}(Ω,Y) \\ ∥h_{k}^{n}∥ & L_{p}(Ω,X) \end{matrix} = ∥T_{k}^{n}∥_{B(X,Y)} \begin{matrix}(2^{−n}) & 1/q \\ (2^{−n}) & 1/p \end{matrix} .

This finishes the proof. □

Example 4.2 shows that R-bounded is a natural assumption in Section 3. Example 4.2. Consider a sequence {S_{n}}_{n∈ℕ_{0}} from B (X, Y). Define {T_{k}^{n}}_{(n,k)∈Δ} by T_{0}^{0} = S_{0} and T_{k}^{n} = S_{n+1} for (n, k) ∈ Δ_{1}. Then

C_{pq} R_{q} ({S_{n}: n ∈ ℕ_{0}}) ≤ ∥T ∥_{L_{p}(Ω,X)→L_{q}(Ω,Y)}

for 1 ≤ p, q < ∞. Indeed

∥T ∥_{L_{p}(Ω,X)→L_{q}(Ω,Y)} ≥ sup_{\begin{matrix}N∈ℕ \\ x_{n}∈X \\ x_{n}≠0 \end{matrix}} \begin{matrix}∥T_{0}^{0}x_{0}h_{0}^{0} + ∑_{n=0}^{N} ∑_{k=1}^{2^{n}} T_{k}^{n}x_{n+1}h_{k}^{n}∥ & L_{q}(Ω,Y) \\ ∥x_{0}h_{0}^{0} + ∑_{n=0}^{N} ∑_{k=1}^{2^{n}} x_{n+1}h_{k}^{n}∥ & L_{p}(Ω,X) \end{matrix}

= sup_{\begin{matrix}N∈ℕ \\ x_{n}∈X \\ x_{n}≠0 \end{matrix}} \begin{matrix}∥S_{0}x_{0}r_{0} + ∑_{n=0}^{N} S_{n+1}x_{n+1}r_{n+1}∥ & L_{q}(Ω,Y) \\ ∥x_{0}r_{0} + ∑_{n=0}^{N} x_{n+1}r_{n+1}∥ & L_{p}(Ω,X) \end{matrix}

= sup_{\begin{matrix}N∈ℕ \\ x_{n}∈X \\ x_{n}≠0 \end{matrix}} \begin{matrix}∥∑_{n=0}^{N} S_{n}x_{n}r_{n}∥ & L_{q}(Ω,Y) \\ ∥∑_{n=0}^{N} x_{n}r_{n}∥ & L_{q}(Ω,X) \end{matrix} \begin{matrix}∥∑_{n=0}^{N} x_{n}r_{n}∥ & L_{q}(Ω,X) \\ ∥∑_{n=0}^{N} x_{n}r_{n}∥ & L_{p}(Ω,X) \end{matrix}

≥ R_{q} (\{S_{n}: n ∈ ℕ_{0}\}) inf_{\begin{matrix}N∈ℕ \\ x_{n}∈X \\ x_{n}≠0 \end{matrix}} \begin{matrix}∥∑_{n=0}^{N} x_{n}r_{n}∥ & L_{q}(Ω,X) \\ ∥∑_{n=0}^{N} x_{n}r_{n}∥ & L_{p}(Ω,X) \end{matrix}

≥ C_{pq} R_{q} ({S_{n}: n ∈ ℕ_{0}})

for some constant C_{pq} ∈ (0, ∞).

Example 4.2 also sheds light on the proper generalization of (1.3).

Example 4.3. Now let X and Y be UMD spaces and 1 < q ≤ p < ∞.

Theorem 3.3 generalizes (1.3) via the function

u_{q} (·) = R_{q} ({T_{j} |h_{j} (·)| : j ∈ ℕ}) .

Also consider the function

ũ (·) = sup_{j∈ℕ}∥T_{j} |h_{j} (·)| ∥_{B(X,Y)} .

Clearly, ũ ≤ u_{q}. If X has cotype 2 and Y has type 2, then u_{q} ≤ C_{XY q}ũ for some constant C_{XY q} ∈ (0, ∞).

Note that, in Example 4.2, the functions u_{q} and ũ are constant:

u_{q} (ω) = R_{q} ({S_{n}: n ∈ ℕ_{0}}) and ũ (ω) = sup_{n∈ℕ_{0}} ∥S_{n}∥_{B(X,Y)}

for each ω ∈ Ω. Thus, for this example, the bounds in Theorems 3.2 and 3.3 are of the proper order; that is,

∥T ∥_{L_{p}(Ω,X)→L_{q}(Ω,Y)} ≈ ∥u_{q}∥_{L_{r}(Ω,[0,∞])}

where \frac{1}{r} = \frac{1}{q} − \frac{1}{p}.

However, if X does not have cotype 2 or Y does not have type 2, then there exists a sequence {S_{n}}_{n∈ℕ_{0}} from the unit sphere of B (X, Y) that is not R-bounded. Consider the corresponding Fourier Haar multiplier operator T as defined in Example 4.2. Then T is not bounded, indeed, u_{q} (ω) = ∞ for each ω ∈ Ω. However, ũ (ω) = 1 for each ω ∈ Ω.

Thus, in generalizing (1.3), R-boundedness is needed.

12 MARIA GIRARDI

Remark 4.4. In Theorems 3.2 and 3.3, it is essential that X and Y be UMD spaces. Indeed, one obtains ([18], see [5]) an equivalent formulation of UMD spaces (with the same constant) if, in Definition 2.1, one replaces the arbitrary filtration {F_{n}}_{n=1}^{m} with the Haar filtration {H_{n}}_{n=1}^{m}. Thus (see Remark 3.1), X is a UMD space if and only if each Fourier Haar multiplier operator T generated by a multiplier sequence of the form {ε_{j}1_{X}}_{j∈ℕ} for some choice {ε_{j}}_{j∈ℕ} of signs {±1} is bounded from L_{p} (Ω, X) to L_{p} (Ω, X) by a constant depending only on X and p for some (or equivalently, for each) p ∈ (1, ∞). Note that for such an operator T

u_{p} (ω) = 1

for each ω ∈ Ω.

Remark 4.5. Theorem 3.2 part (a) fails for p = 1 (or equivalently: Theorem 3.2 part (b) fails if L_{1}^{wk} (Ω, Y) is replaces with L_{1} (Ω, Y)). Indeed, let X = Y = ℝ and assume that there is a constant C such that each Fourier Haar multiplier operator T generated by {T_{k}^{n}}_{(n,k)∈Δ} satisfies

∥T ∥_{L_{1}(Ω,ℝ)→L_{1}(Ω,ℝ)} ≤ C ∥u_{1}∥_{L_{∞}(Ω,[0,∞])} .  (4.1)

By considering {T_{k}^{n}}_{(n,k)∈Δ} of the form T_{k}^{n} = ε_{k}^{n} where ε_{k}^{n} ∈ {±1}, equation (4.1) would imply that the {h_{n}^{k}}_{(n,k)∈Δ} is an unconditional basis for L_{1} (Ω, ℝ), which is not true (cf. [17]).

Remark 4.6. Theorem 3.4 part (a) fails for 1 = p < q < ∞ (or equivalently: Theorem 3.4 part (b) fails if L_{q}^{wk} (Ω, Y) is replaced with L_{q} (Ω, Y)). Indeed, let X = Y = ℝ and consider the Fourier Haar multiplier operator T generated by \{λ_{k}^{n}\}_{(n,k)∈Δ} where λ_{k}^{n} := 2^{− \frac{n}{q′}} with \frac{1}{q} + \frac{1}{q′} = 1. Clearly

A_{pq} := sup_{(n,k)∈Δ} 2^{n( \frac{1}{p} − \frac{1}{q} )} |λ_{k}^{n}| = 1 .

If T where bounded from L_{1} (Ω, ℝ) to L_{q} (Ω, ℝ), then its adjoint T^{∗} would be bounded from L_{q′} (Ω, ℝ) to L_{∞} (Ω, ℝ). But

∥∑_{n=1}^{m} 2^{\frac{n}{q′}} \frac{1}{n} h_{1}^{n}∥_{L_{q′}(Ω,ℝ)} ≈ ∥\{\frac{1}{n}\}_{n=1}^{m}∥_{ℓ_{q′}}

∥T^{∗} (∑_{n=1}^{m} 2^{\frac{n}{q′}} \frac{1}{n} h_{1}^{n})∥_{L_{∞}(Ω,ℝ)} = ∥\{\frac{1}{n}\}_{n=1}^{m}∥_{ℓ_{1}} .

for each m ∈ ℕ.

Remark 4.7. Theorem 3.4 part (a), it is essential that p ≠ q. Indeed, let 1 < p = q < ∞. Let X and Y be UMD spaces such that X does not have cotype 2 or Y does not have type 2. Then there exists a sequence {S_{n}}_{n∈ℕ_{0}} from the unit sphere of B (X, Y) that is not R-bounded. Consider the

corresponding Fourier Haar multiplier operator T as defined in Example 4.2. Clearly,

A_{pq} := sup_{(n,k)∈Δ} 2^{n( \frac{1}{p} − \frac{1}{q} )} ∥T_{k}^{n}∥_{B(X,Y)} = 1 .

However, as noted in Example 4.3, T is not bounded from L_{p} (Ω, X) to L_{p} (Ω, Y).

Remark 4.8. Let {v_{j}}_{j∈ℕ} be a B (X, Y)-valued {H_{j}}_{j∈ℕ}-predictable sequence and 1 ≤ p, q < ∞.

Let’s consider the following natural question.

When does there exist a constant C_{XY pq} so that

∥∑_{n=1}^{m} v_{n}d_{n}∥_{L_{q}(Ω,Y)} ≤ C_{XY pq} ∥∑_{n=1}^{m} d_{n}∥_{L_{p}(Ω,X)}

for each X-valued Haar martingale difference sequence {d_{n}}_{n=1}^{m}?

(4.2)

This question reduces to Fourier Haar multipliers.

Indeed, by predictability, each v_{j} is constant on the support of h_{j} and so there exists a unique

sequence {T_{j}}_{j∈ℕ} from B (X, Y) so that

v_{j} (·) |h_{j} (·)| = T_{j} |h_{j} (·)|

for each j ∈ ℕ.

It now follows from Definition 2.3 and Remark 3.1 that question (4.2) is true if and only if

the Fourier Haar multiplier generated by the Fourier Haar multiplier sequence {T_{j}}_{j∈ℕ} is bounded

from L_{p} (Ω, X) to L_{q} (Ω, Y).

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Department of Mathematics, University of South Carolina, Columbia, SC 29208, U.S.A.

E-mail address: girardi@math.sc.edu