OPERATOR-VALUED MARTINGALE TRANSFORMS AND R-BOUNDEDNESS

MARIA GIRARDI AND LUTZ WEIS

Abstract. Banach space X-valued martingale transforms by a B(X)-valued multiplier sequence are bounded on L_{p}(X), where 1 < p < ∞ and X is a UMD space, if and only if the multiplier sequence is pointwise R-bounded. This is also true for unconditionally convergent martingales in arbitrary Banach spaces.

1. INTRODUCTION

Let X be a Banach space. The martingale transform of an X-valued martingale {f_{n}}_{n∈ℕ} by a ℝ-valued, predictable, uniformly bounded sequence {v_{n}}_{n∈ℕ} is the martingale {g_{n}}_{n∈ℕ} where

g_{n} := ∑_{k=1}^{n} v_{k}d_{k}  and  f_{n} := ∑_{k=1}^{n} d_{k} ;  (1.1)

so {d_{n}}_{n∈ℕ} is the martingale difference sequence of {f_{n}}_{n∈ℕ}.

Burkholder [6] introduced UMD (unconditionality property for martingale differences) Banach spaces: for 1 < p < ∞, 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)}  (1.2)

for each X-valued martingale difference sequence {d_{n}}_{n∈ℕ} with respect to some filtration {F_{n}}_{n∈ℕ}, choice {ε_{n}}_{n∈ℕ} 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 [29], for each) p ∈ (1, ∞).

In this setting, the underlying probability space (unless it is nonatomic) and filtration must vary. Burkholder [6] showed that (1.2) holds, with the same constant β_{p}(X), if one replaces the choices {ε_{n}}_{n∈ℕ} of signs by {F_{n}}-predictable sequences {v_{n}}_{n∈ℕ} of functions valued in [−1, 1].

Over the years, the interplay between probability and harmonic analysis has been very fruitful (see, e.g., [10, 11]) Indeed, the study of the martingale transform uses, for example, Doob’s maximal function (f^{∗}(ω) = sup_{n} |f_{n} (ω)|) and the square function (Sf = (∑_{n∈ℕ} |d_{n}|^{2})^{1/2}). Also [4, 8], X

Date: 16 October 2004.

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

Key words and phrases. Operator-valued martingale transforms, R-boundedness, UMD.

Girardi is supported in part by the National Science Foundation Grant DMS-0306750 and the Deutscher Akademischer Austausch Dienst Gastdozenten Programm. She thanks Universität Karlsruhe, which is where this paper was written. Weis is supported in part by Deutsche Forschungs Gesellschaft Az We 2847/1-1.

2 GIRARDI AND WEIS

has UMD if and only if the Hilbert transform is bounded on L_{p}(ℝ, X) for some (or equivalently for each) p ∈ (1, ∞).

Martínez and Torrea [27] studied operator-valued martingale transforms where the multiplier sequences {v_{n}}_{n∈ℕ} are valued in B (X, Y) instead of ℝ. They derived a theory that parallels the ℝ-valued case. For example, they obtained a martingale version of the well-known theorem of Fefferman and Stein [17] for Hardy-Littlewood maximal operator.

However, they did not give a criteria on a fixed B (X, Y)-valued multiplier sequence {v_{n}}_{n∈ℕ} to ensure that, for some C_{p} ∈ ℝ,

∥v_{1}d_{1} + … + v_{m}d_{m}∥_{L_{p}(Ω,Y)} ≤ C_{p} ∥d_{1} + … + d_{m}∥_{L_{p}(Ω,X)}  (1.3)

for each admissible X-valued martingale difference sequence {d_{n}}_{n∈ℕ} and m ∈ ℕ and for some (or for each) p ∈ (1, ∞). This paper gives such a criteria, in which R-bounded plays a key role. Indeed, Theorems 3.2, 3.3, and 4.1 led to the following crystallizing corollary.

Corollary 1.1. Let (Ω, F, μ) be a probability space with filtration {F_{n}}_{n∈ℕ_{0}} and p ∈ (1, ∞). Let X and Y be UMD spaces. Let {v_{n}}_{n∈ℕ} be a B (X, Y)-valued {F_{n}}-multiplier sequence.

(A) For arbitrary filtrations, the following are equivalent. (1) There exists R_{p} ∈ ℝ so that R_{p} ({v_{n} (ω) : n ∈ ℕ}) ≤ R_{p} for a.e. ω ∈ Ω.

(2) There exists C_{p} ∈ ℝ so that for each (uniformly bounded) X-valued martingale difference sequence \{d_{n}\}_{n=1}^{m} with respect to some subfiltration \left\{{ \atop F_{n}}\right\}_{n=1}^{m} of \left\{{ \atop F_{n}}\right\}_{n∈ℕ}, where (Ω̂, F̂, μ̂) is an extension of (Ω, F, μ),

∥∑_{n=1}^{m} v̂_{n}d_{n}∥_{L_{p}(Ω̂,Y)} ≤ C_{p} ∥∑_{n=1}^{m} d_{n}∥_{L_{p}(Ω̂,X)} .

(B) For atomic filtrations satisfying (4.1), the following are equivalent.

(1) There exists R_{p} ∈ ℝ so that R_{p} ({v_{n} (ω) : n ∈ ℕ}) ≤ R_{p} for each (or equivalently, for a.e.) ω ∈ Ω.

(2) There exists C_{p} ∈ ℝ so that for each (uniformly bounded) X-valued martingale difference sequence {d_{n}}_{n=1}^{m} with respect to some subfiltration {F_{n}}_{n=1}^{m} of {F_{n}}_{n∈ℕ},

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

If (1 ) holds, then C_{p} in (2 ) can be taken to be β_{p}(X) β_{p}(Y) R_{p}. If (2 ) holds, then R_{p} in (1 ) can be taken to be C_{p}. (Needed definitions and notation to come.)

OPERATOR-VALUED MARTINGALE TRANSFORMS AND R-BOUNDEDNESS 3

R-boundedness was introduced by Berkson and Gillespie in [2]. The notion grew out of work of J. Bourgain on vector-valued Fourier transform [5] 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, 20, 19, 33]). Through these tools, R-boundedness became important for maximal regularity of parabolic differential equations (e.g. [13, 14, 25, 33]) and the holomorphic functional calculus of sectorial operators (e.g. [21, 22, 25]). Results of the present paper are especially useful for the theory of stochastic integration on Banach spaces, which recently was developed in [31] and [32]. For more information on R-boundedness and its properties, see [12, 18, 25].

This paper is organized as follows. Section 2 collects the needed definitions and notation. The main results are in Sections 3 and 4. Section 5 gives further corollaries to these main theorems. Section 6 gives a technical proof of a lemma needed 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 while S (X) is the unit sphere of X. The space B (X, Y) of bounded linear operators from X into Y is endowed with the usual operator norm topology. ([0, 1] , M, m) is the usual Lebesgue measure space. (Ω, F, μ) is an arbitrary (complete) probability measure space; corresponding to it is the usual Bochner-Lebesgue space L_{p} (Ω, X) of measurable functions from Ω into X with finite L_{p} (Ω, X)-norm where 1 ≤ p ≤ ∞. A sequence {d_{n}}_{n∈ℕ} of functions from Ω into X is uniformly bounded (by M ∈ ℝ) provided

sup_{n∈ℕ} sup_{ω∈Ω} ∥d_{n} (ω)∥_{X} ≤ M .

Following Burkholder [7], a sequence {d_{n}}_{n=1}^{m} of functions in L_{p} (Ω, X) is called τ-unconditional in L_{p} (Ω, X) provided

∥∑_{n=1}^{m} ε_{n}λ_{n}d_{n}∥_{L_{p}(Ω,X)} ≤ τ ∥∑_{n=1}^{m} λ_{n}d_{n}∥_{L_{p}(Ω,X)}

for each choice {ε_{n}}_{n=1}^{m} of signs from {±1} and choice {λ_{n}}_{n=1}^{m} of scalars.

ℕ is the set of natural numbers while ℕ_{0} = ℕ ∪ {0}.

Let (Ω, F, μ) be a probability space with a filtration {F_{n}}_{n∈ℕ_{0}} (i.e., {F_{n}}_{n∈ℕ_{0}} is a nondecreasing sequence of sub-σ-fields of F). Let m ∈ ℕ ∪ {∞}. A sequence {f_{n}}_{n=1}^{m} of functions from Ω into X is a martingale with respect to {F_{n}}_{n=1}^{m} provided f_{n} ∈ L_{1} ((Ω, F_{n}, μ) , X) and 𝔼 (f_{n+1} | F_{n}) = f_{n} for each admissible n. A sequence {d_{n}}_{n=1}^{m} of functions from Ω into X is a martingale difference

4 GIRARDI AND WEIS

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}  and  d_{n} = f_{n} − f_{n−1}

where f_{0} ≡ 0. Note that for a finite (i.e. m ∈ ℕ) X-valued martingale difference sequence {d_{n}}_{n=1}^{m} and p ∈ [1, ∞), each d_{n} is in L_{p} (Ω, X) if and only if ∑_{n=1}^{m} d_{n} is in L_{p} (Ω, X). A sequence {v_{n}}_{n∈ℕ} of functions from Ω into Z is predictable with respect to {F_{n}}_{n∈ℕ_{0}} (in short, {F_{n}}-predictable) provided v_{n} is F_{n−1}-measurable for each n ∈ ℕ. Note that if {v_{n}}_{n∈ℕ} is predictable with respect to {F_{n}}_{n∈ℕ_{0}}, then it is predictable with respect to each subfiltration (i.e. subsequence) {F_{j_{n}}}_{n∈ℕ_{0}} of {F_{n}}_{n∈ℕ_{0}}.

Definition 2.1. To ease the statements of theorems to come, for a probability space (Ω, F, μ) with filtration {F_{n}}_{n∈ℕ_{0}}, let

M ({F_{n}} , X) := { {f_{n}}_{n∈ℕ} : {f_{n}}_{n∈ℕ} is an X-valued martingale with respect to {F_{n}}_{n∈ℕ} }

and

D ({F_{n}} , X) := { {d_{n}}_{n=1}^{m} : {d_{n}}_{n=1}^{m} is an X-valued martingale difference sequence with respect to some subfiltration {F_{j_{n}}}_{n=1}^{m} of {F_{n}}_{n∈ℕ} and m ∈ ℕ} .

Definition 2.2. Let (Ω, F, μ) be a probability space with filtration {F_{n}}_{n∈ℕ_{0}}.

(1) A B (X, Y)-valued {F_{n}}-multiplier sequence is a sequence {v_{n}}_{n∈ℕ} of functions from Ω into B (X, Y) that is predictable with respect to {F_{n}}_{n∈ℕ_{0}} and is uniformly bounded by one.

(2) For such a multiplier sequence v := {v_{n}}_{n∈ℕ}, the martingale transform of a martingale f := {f_{n}}_{n∈ℕ} ∈ M ({F_{n}} , X) is the martingale {(T_{v}f)_{n}}_{n∈ℕ} ∈ M ({F_{n}} , Y) where

f_{n} := ∑_{k=1}^{n} d_{k}  and  (T_{v}f)_{n} := ∑_{k=1}^{n} v_{k}d_{k}

for each n ∈ ℕ.

The dyadic sigma-fields {D_{n}}_{n∈ℕ_{0}} are given by

D_{n} = σ (I_{k}^{n}: 1 ≤ k ≤ 2^{n})

and the Rademacher functions {r_{n}}_{n∈ℕ} are given by

r_{n} = ∑_{k=1}^{2^{n}}(−1)^{k+1}1_{I_{k}^{n}}

where, for n ∈ ℕ_{0}, I_{1}^{n} = [\frac{0}{2^{n}} , \frac{1}{2^{n}} ] and

I_{k}^{n} = (\frac{k − 1}{2^{n}}, \frac{k}{2^{n}} ]

if k ∈ ℕ and 1 < k ≤ 2^{n}.

A proof of the next fact can be found at [16, Contraction Principle 12.2]. In the special case of when the independent symmetric sequence is the Rademacher functions {r_{n}}_{n∈ℕ}, it is known as Kahane’s Contraction Principle.

Fact 2.3 (Contraction Principle). Let {d̃_{n}^{o}}_{n∈ℕ} be a sequence of independent, symmetric, ℝ-valued random variables on a probability space (Ω, F, μ). If {z_{n}}_{n=1}^{m} is a sequence in any Banach space Z and {λ_{n}}_{n=1}^{m} is a sequence from ℝ, then

∥∑_{n=1}^{m} λ_{n} z_{n} d̃_{n}^{o}∥_{L_{p}(Ω,Z)} ≤ [ max_{1≤n≤m} |λ_{n}|] ∥∑_{n=1}^{m} z_{n} d̃_{n}^{o}∥_{L_{p}(Ω,Z)}

for each p ∈ [1, ∞).

R-boundedness is the central notion of this paper.

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 [16], for each) p ∈ [1, ∞).

Thus a set τ is R-bounded provided Kahane’s Contraction Principle holds for operator coefficients from τ. Pisier [1] showed that X is isomorphic to a Hilbert space if and only if each (norm) bounded subset of B (X, X) is R-bounded. 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}

6 GIRARDI AND WEIS

known from harmonic analysis (cf. [26, Thm. II.1.d.6]). For basic properties of R-bounded sets and further references, see [12, 33].

All notation and terminology, not otherwise explained, are as in [9, 15, 26].

3. MAIN RESULTS for ARBITRARY FILTRATIONS

Part (A) of Corollary 1.1 follows easily from Theorems 3.2 and 3.3.

For arbitrary filtrations, the notion of an extension (cf. eg. [23]) of a probability space is used.

Definition 3.1. Let (Ω, F, μ) and (Ω′, F′, μ′) be probability spaces with filtrations {F_{n}}_{n∈ℕ_{0}} and {F′_{n}}_{n∈ℕ_{0}} respectively. The extension of (Ω, F, μ) by (Ω′, F′, μ′) is their product probability space (Ω̂, F̂, μ̂), along with the filtration {F̂_{n}}_{n∈ℕ_{0}} where F̂_{n} = σ(F_{n} × F′_{n}). For h ∈ L_{0} (Ω, Z), define ĥ ∈ L_{0} (Ω̂, Z) by

ĥ(ω, ω′) := h(ω) .

In the special case that (Ω′, F′, μ′) = ([0, 1], M, m) and {F′_{n}}_{n∈ℕ_{0}} = {D_{n}}_{n∈ℕ_{0}}, one calls (Ω̂, F̂, μ̂) the dyadic extension of (Ω, F, μ).

Note that if h ∈ L_{p} (Ω, Z) then ∥h∥_{L_{p}(Ω,Z)} = ∥ĥ∥_{L_{p}(Ω̂,Z)} for 1 ≤ p ≤ ∞. Also, if {v_{n}}_{n∈ℕ} is a \{F_{n}\}-multiplier sequence then \{v̂_{n}\}_{n∈ℕ} is a \left\{{ \atop F_{n}}\right\}-multiplier sequence.

Theorem 3.2. Let (Ω, F, μ) be a probability space with filtration {F_{n}}_{n∈ℕ_{0}} and p ∈ [1, ∞). Let {v_{n}}_{n∈ℕ} be a B (X, Y)-valued {F_{n}}-multiplier sequence that satisfies, for some C_{p} ∈ ℝ,

R_{p} ({v_{n} (ω) : n ∈ ℕ}) ≤ C_{p}

for a.e. ω ∈ Ω.

(a) If {d_{n}}_{n=1}^{m} ∈ D ({F_{n}} , X) is so that {d_{n}}_{n=1}^{m} is α_{p}-unconditional in L_{p} (Ω, X) and {v_{n}d_{n}}_{n=1}^{m} is β_{p}-unconditional in L_{p} (Ω, Y), then

∥∑_{n=1}^{m} v_{n}d_{n}∥_{L_{p}(Ω,Y)} ≤ α_{p} β_{p} C_{p} ∥∑_{n=1}^{m} d_{n}∥_{L_{p}(Ω,X)} .

(b) If \{d_{n}\}_{n=1}^{m} ∈ D (\left\{{ \atop F_{n}}\right\} , X), for some extension (Ω̂, F̂, μ̂) of (Ω, F, μ), is so that \{d_{n}\}_{n=1}^{m} is α_{p}-unconditional in L_{p} (Ω̂, X) and {v̂_{n}d_{n}}_{n=1}^{m} is β_{p}-unconditional in L_{p} (Ω̂, Y ), then

∥∑_{n=1}^{m} v̂_{n}d_{n}∥_{L_{p}(Ω̂,Y)} ≤ α_{p} β_{p} C_{p} ∥∑_{n=1}^{m} d_{n}∥_{L_{p}(Ω̂,X)} .

OPERATOR-VALUED MARTINGALE TRANSFORMS AND R-BOUNDEDNESS 7

Proof. Part (a) follows easily from (b). Towards (b), note that for each fixed t ∈ [0, 1]

∥∑_{n=1}^{m} r_{n} (t) d_{n}∥_{L_{p}(Ω̂,X)}^{p} ≤ α_{p}^{p} ∥∑_{n=1}^{m} d_{n}∥_{L_{p}(Ω̂,X)}^{p}
∥∑_{n=1}^{m} v̂_{n}d_{n}∥_{L_{p}(Ω̂,Y)}^{p} ≤ β_{p}^{p} ∥∑_{n=1}^{m} r_{n} (t) v̂_{n}d_{n}∥_{L_{p}(Ω̂,Y)}^{p} .

Thus

∥∑_{n=1}^{m} v̂_{n}d_{n}∥_{L_{p}(Ω̂,Y)}^{p} ≤ β_{p}^{p} ∫_{[0,1]} ∥∑_{n=1}^{m} r_{n} (t) v̂_{n}d_{n}∥_{L_{p}(Ω̂,Y)}^{p} dt

= β_{p}^{p} ∫_{Ω} ∫_{Ω′} ∫_{[0,1]} ∥∑_{n=1}^{m} r_{n} (t) v_{n} (ω) d_{n} (ω, ω′)∥_{Y}^{p} dt dμ′(ω′) dμ(ω)

≤ β_{p}^{p} C_{p}^{p} ∫_{Ω} ∫_{Ω′} ∫_{[0,1]} ∥∑_{n=1}^{m} r_{n} (t) d_{n} (ω, ω′)∥_{X}^{p} dt dμ′(ω′) dμ(ω)

= β_{p}^{p} C_{p}^{p} ∫_{[0,1]} ∥∑_{n=1}^{m} r_{n} (t) d_{n}∥_{L_{p}(Ω̂,X)}^{p} dt

≤ β_{p}^{p} C_{p}^{p} α_{p}^{p} ∥∑_{n=1}^{m} d_{n}∥_{L_{p}(Ω̂,X)}^{p} .

This finishes the proof of (b). □ Theorem 3.3. Let (Ω, F, μ) be a probability space with filtration {F_{n}}_{n∈ℕ_{0}} and p ∈ [1, ∞).

Let {v_{n}}_{n∈ℕ} be a B (X, Y)-valued {F_{n}}-multiplier sequence that satisfies, for some C_{p} ∈ ℝ,

for each uniformly bounded \{d_{n}\}_{n=1}^{m} ∈ D (\left\{{ \atop F_{n}}\right\} , X) , where (Ω̂, F̂, μ̂) is the dyadic extension of (Ω, F, μ) ,

∥∑_{n=1}^{m} v̂_{n}d_{n}∥_{L_{p}(Ω̂,Y)} ≤ C_{p} ∥∑_{n=1}^{m} d_{n}∥_{L_{p}(Ω̂,X)} .

(3.1)

Then

R_{p} ({v_{n} (ω) : n ∈ ℕ}) ≤ C_{p}

for a.e. ω ∈ Ω.

Remark 3.4. Condition (3.1) can be replaced by the (apparently) weaker Condition (3.1′). Condition (3.1′):

for each ℝ-valued finite martingale difference sequence {d_{n}}_{n=1}^{m} with respect to {F̂_{m−1+n}}_{n=1}^{m}, where (Ω̂, F̂, μ̂) is the dyadic extension of (Ω, F, μ), of the form d_{n}(ω, t) = r_{m−1+n} (t) 1_{A} (ω) where A ∈ F_{m−1}, one has that

∥∑_{n=1}^{m} v̂_{n}x_{n}d_{n}∥_{L_{p}(Ω̂,Y)} ≤ C_{p} ∥∑_{n=1}^{m} x_{n}d_{n}∥_{L_{p}(Ω̂,X)}

for each choice {x_{n}}_{n=1}^{m} from B(X).

Thus Condition (3.1′) reduces, from Condition (3.1), the class of martingale difference sequences that one must test. Note that for such a martingale difference sequence {d_{n}}_{n=1}^{m} in Condition (3.1′), if {z_{n}}_{n=1}^{m} is from any Banach space Z, then {z_{n}d_{n}}_{n=1}^{m} is 1-unconditional in L_{p} (Ω̂, Z) by Fact 2.3.

Proof of Theorem 3.3. Assume condition (3.1′) of Remark 3.4 holds (but not that condition (3.1) necessarily holds). Let {ε_{j}}_{j∈ℕ} be a sequence of real numbers tending to zero.

For each n ∈ ℕ, since v_{n} ∈ L_{∞} ((Ω, F_{n−1}, μ), B (X, Y)), there is a sequence {v_{n}^{j}}_{j∈ℕ} of countablyvalued functions in L_{∞} ((Ω, F_{n−1}, μ), B (X, Y)) so that lim_{j→∞} ∥v_{n} − v_{n}^{j}∥_{L_{∞}} = 0. Note that for any sub-σ-field G_{n}^{j} containing σ(v_{n}^{j})

∥v_{n} − 𝔼 (v_{n} | G_{n}^{j})∥_{L_{∞}} ≤ ∥v_{n} − 𝔼 (v_{n}^{j} | G_{n}^{j})∥_{L_{∞}} + ∥𝔼 (v_{n}^{j} − v_{n} | G_{n}^{j})∥_{L_{∞}} ≤ 2 ∥v_{n} − v_{n}^{j}∥_{L_{∞}} .

So, for each j ∈ ℕ, there is a sequence {G_{n}^{j}}_{n∈ℕ_{0}} of sub-σ-fields of F so that (1) ∥v_{n} − w_{n}^{j}∥_{L_{∞}(Ω,B(X,Y))} < \frac{ε_{j}}{2^{n}} where w_{n}^{j} := 𝔼 (v_{n} | G_{n−1}^{j})

(2) G_{n−1}^{j} ⊂ F_{n−1} and G_{n−1}^{j} ⊂ G_{n}^{j}

(3) G_{n−1}^{j} is generated by a partition of Ω into (finitely or countably many) sets of (strictly)

positive measure for each n ∈ ℕ. So there exists G ∈ F so that μ(G) = 1 and

∥v_{n} (u) − w_{n}^{j} (u)∥_{B(X,Y)} < \frac{ε_{j}}{2^{n}}

for each u ∈ G and j, n ∈ ℕ.

OPERATOR-VALUED MARTINGALE TRANSFORMS AND R-BOUNDEDNESS 9

Fix u ∈ G. Fix {x_{n}}_{n=1}^{m} from B(X). Fix j ∈ ℕ. It suffices to show

∥∑_{n=1}^{m} r_{n}v_{n} (u) x_{n}∥_{L_{p}([0,1],Y)} ≤ C_{p} ∥∑_{n=1}^{m} r_{n}x_{n}∥_{L_{p}([0,1],X)} + 2ε_{j} .  (3.2)

Find the atom A of G_{m−1}^{j} so that u ∈ A. Note that

w_{n}^{j} (u) = w_{n}^{j} (ω) for each ω ∈ A , n ∈ {1, …, m} .

So

∥ ∑_{n=1}^{m} r_{n}v_{n} (u)x_{n}∥_{L_{p}([0,1],Y)} ≤ ∥∑_{n=1}^{m} w_{n}^{j} (u) x_{n}r_{m−1+n}∥_{L_{p}([0,1],Y)} + ε_{j}

= [∫_{A} ∫_{[0,1]} ∥∑_{n=1}^{m} w_{n}^{j} (ω) x_{n}r_{m−1+n} (t)∥_{Y}^{p} dt \frac{dμ (ω)}{μ(A)}]^{1/p} + ε_{j}

≤ \frac{1}{μ^{1/p}(A)} [∫_{Ω} ∫_{[0,1]} ∥∑_{n=1}^{m} v_{n} (ω) x_{n}r_{m−1+n} (t) 1_{A} (ω)∥_{Y}^{p} dt dμ (ω)]^{1/p} + 2ε_{j}

≤ \frac{C_{p}}{μ^{1/p}(A)} [∫_{Ω} ∫_{[0,1]} ∥∑_{n=1}^{m} x_{n}r_{m−1+n} (t) 1_{A} (ω)∥_{X}^{p} dt dμ (ω)]^{1/p} + 2ε_{j}

= \frac{C_{p}}{μ^{1/p}(A)} [μ(A) ∫_{[0,1]} ∥∑_{n=1}^{m} x_{n}r_{m−1+n} (t)∥_{X}^{p} dt]^{1/p} + 2ε_{j}

= C_{p} ∥∑_{n=1}^{m} r_{n}x_{n}∥_{L_{p}([0,1],X)} + 2ε_{j} .

Thus (3.2) holds. □

4. MAIN RESULTS for ATOMIC FILTRATIONS

Now consider a probability space (Ω, F, μ) with a filtration {F_{n}}_{n∈ℕ_{0}} satisfying:

each F_{n} is generated by (finitely or countably many) atoms of (strictly) positive measure
and lim_{n→∞} sup {μ(B): B is an atom of F_{n}} = 0 .

(4.1)

Part (B) of Corollary 1.1 follows easily from Theorems 4.1 and 3.2.

10 GIRARDI AND WEIS

Theorem 4.1 is the atomic version of the general filtration Theorem 3.3. Note that Theorem 4.1 reduces the test class of martingale difference sequences from the test class needed in Theorem 3.3 in that, for the atomic case, one need not have to pass to extensions.

Theorem 4.1. Let (Ω, F, μ) be a probability space with a filtration {F_{n}}_{n∈ℕ_{0}} satisfying (4.1) and p ∈ [1, ∞).

Let {v_{n}}_{n∈ℕ} be a B (X, Y)-valued {F_{n}}-multiplier sequence that satisfies, for some C_{p} ∈ ℝ, for each uniformly bounded {d_{n}}_{n=1}^{m} ∈ D ({F_{n}} , X)

∥∑_{n=1}^{m} v_{n}d_{n}∥_{L_{p}(Ω,Y)} ≤ C_{p} ∥∑_{n=1}^{m} d_{n}∥_{L_{p}(Ω,X)} . (4.2)

Then

R_{p} ({v_{n} (ω) : n ∈ ℕ}) ≤ C_{p}

for each ω ∈ Ω.

Remark 4.2. Condition (4.2) can be replaced by the (apparently) weaker Condition (4.2′). Condition (4.2′):

there exists τ > 1 so that for each uniformly bounded {d_{n}}_{n=1}^{m} ∈ D ({F_{n}} , ℝ) satisfying that

if {z_{n}}_{n=1}^{m} is from any Banach space Z

then {z_{n}d_{n}}_{n=1}^{m} is τ-unconditional in L_{p} (Ω, Z)

(4.3)

one has that

∥∑_{n=1}^{m} v_{n}x_{n}d_{n}∥_{L_{p}(Ω,Y)} ≤ C_{p} ∥∑_{n=1}^{m} x_{n}d_{n}∥_{L_{p}(Ω,X)}  (4.4)
for each choice {x_{n}}_{n=1}^{m} from X.

Thus Condition (4.2′) reduces, from Condition (4.2), the class of martingale difference sequences that one must test.

The proof of Theorem 4.1 uses the following lemma, whose long technical proof is in Section 6.

Lemma 4.3. Let (Ω, F, μ) be a probability space with a filtration {F_{n}}_{n∈ℕ_{0}} satisfying (4.1).

Let A be an atom of some F_{j}. Let p ∈ [1, ∞) and τ, τ_{1} > 1 and ε > 0 and m ∈ ℕ. Then there exists a uniformly bounded ℝ-valued martingale difference sequence {d_{n}}_{n=1}^{m} with respect to some subfiltration {F_{j_{n}}}_{n=1}^{m} such that

OPERATOR-VALUED MARTINGALE TRANSFORMS AND R-BOUNDEDNESS 11

(1) j_{1} > j

(2) supp d_{n} ⊂ A for each n ∈ {1, …, m}

(3) 1 ≤ |d_{n} (ω)| ≤ τ_{1} for each n ∈ {1, …, m} and ω ∈ G_{m} := supp d_{m} and furthermore, for any choice {z_{n}}_{n=1}^{m} from any Banach space Z,

(4) {z_{n}d_{n}}_{n=1}^{m} is τ-unconditional in L_{p} (Ω, Z)

(5) if z_{n_{0}} ≠ 0, then

∫_{A} ∥∑_{n=1}^{m} d_{n} (ω) z_{n}r_{n}∥_{L_{p}([0,1],Z)}^{p} dμ (ω) ≤ [1 + εM^{p}] ∫_{G_{m}} ∥∑_{n=1}^{m} d_{n} (ω) z_{n}r_{n}∥_{L_{p}([0,1],Z)}^{p} dμ (ω)

where M ∥z_{n_{0}}∥_{Z} = τ_{1} ∑_{n=1}^{m} ∥z_{n}∥_{Z} .

Proof of Theorem 4.1. Assume condition (4.2′) of Remark 4.2 holds (thus giving τ > 1) (but not that condition (4.2) necessarily holds).

Fix u ∈ Ω. Fix m ∈ ℕ and {x_{n}}_{n=1}^{m} from X. Let τ_{1}, τ_{2} > 1. It suffices to show

∥∑_{n=1}^{m} r_{n}v_{n} (u) x_{n}∥_{L_{p}([0,1],Y)} ≤ τ_{1} τ_{2} C_{p} ∥∑_{n=1}^{m} r_{n}x_{n}∥_{L_{p}([0,1],X)} .  (4.5)

Without loss of generality, there exists n_{0} ∈ {1, … m} so that x_{n_{0}} ≠ 0.

Find the atom A of F_{m−1} so that u ∈ A. Note that

v_{n} (u) = v_{n} (ω) for each ω ∈ A , n ∈ {1, … m} .  (4.6)

Find ε > 0 so that

1 + ε [\frac{τ_{1} ∑_{n=1}^{m} ∥x_{n}∥_{X}}{∥x_{n_{0}}∥_{X}} ]^{p} < τ_{2}^{p} .  (4.7)

Apply Lemma 4.3 (with F_{j} := F_{m−1} and other notation consistent) to find the corresponding uniformly bounded {d_{n}}_{n=1}^{m} ∈ D ({F_{n}} , ℝ). Let G_{m} := supp d_{m}. Note that {d_{n}}_{n=1}^{m} satisfies condition (4.3) and so (4.4) holds for the choice {r_{n} (t) x_{n}}_{n=1}^{m} from X for each fixed t ∈ [0, 1]. By Kahane’s Contraction Principle (Fact 2.3), for each fixed w ∈ G_{m},

∥∑_{n=1}^{m} d_{n} (ω) x_{n}r_{n}∥_{L_{p}([0,1],X)} ≤ τ_{1} ∥∑_{n=1}^{m} x_{n}r_{n}∥_{L_{p}([0,1],X)}  (4.8)
∥∑_{n=1}^{m} v_{n} (u) x_{n}r_{n}∥_{L_{p}([0,1],Y)} ≤ ∥∑_{n=1}^{m} d_{n} (ω) v_{n} (u) x_{n}r_{n}∥_{L_{p}([0,1],Y)}  (4.9)

by (3) of Lemma 4.3. Thus

∥∑_{n=1}^{m} r_{n}v_{n} (u) x_{n}∥_{L_{p}([0,1],Y)}^{p} ≤ ∫_{G_{m}} ∥∑_{n=1}^{m} d_{n} (ω) r_{n}v_{n} (u) x_{n}∥_{L_{p}([0,1],Y)}^{p} \frac{dμ (ω)}{μ(G_{m})}

≤ ∫_{[0,1]} ∫_{Ω} ∥∑_{n=1}^{m} v_{n} (ω) [r_{n} (t) x_{n}] d_{n} (ω)∥_{Y}^{p} \frac{dμ (ω)}{μ(G_{m})} dt

≤ C_{p}^{p} ∫_{[0,1]} ∫_{Ω} ∥∑_{n=1}^{m} r_{n} (t) x_{n}d_{n} (ω)∥_{X}^{p} \frac{dμ (ω)}{μ(G_{m})} dt

= C_{p}^{p} ∫_{A} ∥∑_{n=1}^{m} d_{n} (ω) [x_{n}r_{n}]∥_{L_{p}([0,1],X)}^{p} \frac{dμ (ω)}{μ(G_{m})}

≤ C_{p}^{p} τ_{2}^{p} ∫_{G_{m}} ∥∑_{n=1}^{m} d_{n} (ω) [x_{n}r_{n}]∥_{L_{p}([0,1],X)}^{p} \frac{dμ (ω)}{μ(G_{m})}

≤ C_{p}^{p} τ_{2}^{p} τ_{1}^{p} ∥∑_{n=1}^{m} x_{n}r_{n}∥_{L_{p}([0,1],X)}^{p}

where the inequalities (in order) follow from: (4.9), the monotonicity of the integral for nonnegative functions and (4.6), (4.4), Lemma 4.3 and (4.7), and (4.8). So (4.5) holds. □

The next example shows that the condition

lim_{n→∞} sup {μ(B): B is an atom of F_{n}} = 0

of (4.1) in Theorem 4.1 is necessary.

Example 4.4. Consider any filtration \{F_{n}\}_{n∈ℕ_{0}} on ([0, 1] , M, m) satisfying that (\frac{1}{2}, 1] is an atom of F_{n} for each n ∈ ℕ_{0}. Let v_{n}: Ω → B (X, X) have the form

v_{n} (ω) = \begin{cases}T_{n} & if ω ∈ (\frac{1}{2}, 1] \\ 0 & if ω ∈ [0, \frac{1}{2}] . \end{cases}

Any \{d_{n}\}_{n=1}^{m} ∈ D (\{F_{n}\} , X) satisfies d_{n} (ω) = 0 if ω ∈ (\frac{1}{2}, 1] for n > 1. So (4.2) holds. But if X is not Hilbertian, then there is a non-R-bounded set {T_{n}}_{n∈ℕ} in B (X, X).

5. COROLLARIES to the MAIN RESULTS

As in the scalar case, boundedness of operator-valued martingale transforms in one sense is equivalent to other notions of boundedness. To be precise, for a Z-valued martingale f := {f_{n}}_{n∈ℕ}, define

∥f∥_{L_{p}(Ω,Z)} := sup_{n∈ℕ}∥f_{n}∥_{L_{p}(Ω,Z)}  for 1 ≤ p < ∞
f_{n}^{∗} (ω) := sup_{1≤k≤n} ∥f_{k} (ω)∥_{Z} for n ∈ ℕ

f^{∗} (ω) := sup_{n∈ℕ}∥f_{n} (ω)∥_{Z}  Doob’s maximal function .

Let’s keep with the notation in Definitions 2.1 and 2.2.

Fact 5.1. Let (Ω, F, μ) be a probability space with filtration {F_{n}}_{n∈ℕ_{0}}. Let {v_{n}}_{n∈ℕ} be a B (X, Y)- valued {F_{n}}-multiplier sequence. Then conditions (1) through (5) are equivalent.

(1) For each (or equivalently, for some) p ∈ (1, ∞) there exists C_{p} ∈ ℝ so that

∥(T_{v}f)_{m}∥_{L_{p}(Ω,Y)} ≤ C_{p} ∥f_{m}∥_{L_{p}(Ω,X)}

for each f := {f_{n}} ∈ M ({F_{n}} , X) and m ∈ ℕ.

(2) For each (or equivalently, for some) p ∈ (1, ∞) there exists C_{p} ∈ ℝ so that

∥(T_{v}f)^{∗}∥_{L_{p}(Ω,ℝ)} ≤ C_{p} ∥f∥_{L_{p}(Ω,X)}

for each f := {f_{n}} ∈ M ({F_{n}} , X).

(3) For each p ∈ [1, ∞) there exists C_{p} ∈ ℝ so that

∥(T_{v}f)^{∗}∥_{L_{p}(Ω,ℝ)} ≤ C_{p} ∥f^{∗}∥_{L_{p}(Ω,ℝ)}

for each f := {f_{n}} ∈ M ({F_{n}} , X).

(4) There exists C ∈ ℝ so that

λ μ [(T_{v}f)^{∗} > λ] ≤ C ∥f^{∗}∥_{L_{1}(Ω,ℝ)}

for each f := {f_{n}} ∈ M ({F_{n}} , X) and λ > 0.

(5) There exists C ∈ ℝ so that

λ μ [(T_{v}f)^{∗} > λ] ≤ C ∥f∥_{L_{1}(Ω,X)}

for each f := {f_{n}} ∈ M ({F_{n}} , X) and λ > 0. If, furthermore, Y has the Radon-Nikodym property, then (3) implies (6).

(6) For each f := {f_{n}} ∈ M ({F_{n}} , X), if ∥f∥_{L_{1}(Ω,X)} is finite then (T_{v}f) converges a.e..

Martínez and Torrea [27] showed the equivalence of (2) through (5) and the implication to (6) indicated above. Of course, that (1) implies (2) follows from standard techniques (such as those found in [27, Remark 1]) while that (2) implies (1) follows easily from the definitions.

Corollary 5.2. Let (Ω, F, μ) be a probability space with filtration {F_{n}}_{n∈ℕ_{0}}. Let X and Y be UMD spaces. Let {v_{n}}_{n∈ℕ} be a B (X, Y)-valued {F_{n}}-multiplier sequence that satisfies, for some C ∈ ℝ,

R_{2} ({v_{n} (ω) : n ∈ ℕ}) ≤ C for a.e. ω ∈ Ω .

14 GIRARDI AND WEIS

Then (1) through (6) of Fact 5.1 hold (with the constants appearing depending also on the UMD constants of X and Y).

Proof. Let f := {f_{n}} ∈ M ({F_{n}} , X) and m ∈ ℕ. It follows from Theorem 3.2 that

∥(T_{v}f)_{m}∥_{L_{2}(Ω,Y)} ≤ β_{2}(X) β_{2}(Y) C ∥f_{m}∥_{L_{2}(Ω,X)} .

Now apply Fact 5.1. □

Remark 5.3. Martínez and Torrea [28] showed the equivalence of (1) in Fact 5.1 to the boundedness of the martingale transform on various Banach space valued BMO and Hardy spaces. Thus, similar to Corollary 5.2, a pointwise R-bounded B (X, Y)-valued multiplier sequence {v_{n}}_{n∈ℕ}, where X and Y are UMD spaces, yields bounded martingale transform operators between BMO and Hardy spaces.

Burkholder [6] showed that if X is a UMD space then (1.2) holds, with the same constant β_{p}(X), if one replaces the choices {ε_{n}}_{n∈ℕ} of signs by [−1, 1]-valued {F_{n}}-multiplier sequences {v_{n}}_{n∈ℕ}. A similar result is true for operator-valued multiplier sequences.

Corollary 5.4. Let (Ω, F, μ) be a probability space with filtration {F_{n}}_{n∈ℕ_{0}} and p ∈ [1, ∞). Assume that there is τ_{p} ({F_{n}} , X) ∈ ℝ so that for each {d_{n}}_{n=1}^{m} ∈ D ({F_{n}} , X)

∥∑_{n=1}^{m} ε_{n}d_{n}∥_{L_{p}(Ω,X)} ≤ τ_{p} ({F_{n}} , X) ∥∑_{n=1}^{m} d_{n}∥_{L_{p}(Ω,X)}

for each choice {ε_{n}}_{n=1}^{m} of signs from {±1}.

If a B (X, X)-valued {F_{n}}-multiplier sequence {v_{n}}_{n∈ℕ} satisfies, for some C_{p} ∈ ℝ,

R_{p} ({v_{n} (ω) : n ∈ ℕ}) ≤ C_{p} for a.e. ω ∈ Ω

then

∥∑_{n=1}^{m} v_{n}d_{n}∥_{L_{p}(Ω,X)} ≤ [τ_{p} ({F_{n}} , X)]^{2} C_{p} ∥∑_{n=1}^{m} d_{n}∥_{L_{p}(Ω,X)}

for each {d_{n}}_{n=1}^{m} ∈ D ({F_{n}} , X).

Note that if {v_{n}}_{n∈ℕ} is a [−1, 1]-valued {F_{n}}-multiplier sequence then {v_{n}1_{X}}_{n∈ℕ} is a B (X, X)- valued {F_{n}}-multiplier sequence and R_{p} ({v_{n} (ω) 1_{X}: n ∈ ℕ}) = sup {|v_{n} (ω)| : n ∈ ℕ}.

Proof. The result follows directly from Theorem 3.2. □

This section closes with a special case of Theorem 4.1: note that here one must only test condition (4.2) for translated filtration rather than for all subfiltration.

OPERATOR-VALUED MARTINGALE TRANSFORMS AND R-BOUNDEDNESS 15

Proposition 5.5. Consider the Lebesgue measure space ([0, 1], M, m) along with its dyadic filtration {D_{n}}_{n∈ℕ_{0}}. Let p ∈ [1, ∞).

Let {v_{n}}_{n∈ℕ} be a B (X, Y)-valued {D_{n}}-multiplier sequence that satisfies, for some C_{p} ∈ ℝ,

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

with respect to {D_{m−1+n}}_{n=1}^{m}
of the form d_{n} = x_{n}r_{m−1+n}1_{I_{k}^{m−1}}  (5.1)

for some {x_{n}}_{n=1}^{m} ⊂ B(X) and k ∈ {1, …, 2^{m−1}}

∥∑_{n=1}^{m} v_{n}d_{n}∥_{L_{p}([0,1],Y)} ≤ C_{p} ∥∑_{n=1}^{m} d_{n}∥_{L_{p}([0,1],X)} .

Then

R_{p} ({v_{n} (u) : n ∈ ℕ}) ≤ C_{p}

for each u ∈ [0, 1].

Note that any martingale difference sequence of the above form is 1-unconditional in L_{p} ([0, 1], X).

Proof. Fix u ∈ [0, 1].

Fix m ∈ ℕ and {x_{n}}_{n=1}^{m} from B(X). Find k ∈ {1, …, 2^{m−1}} so that u ∈ I_{k}^{m−1}. Note that, for n ∈ {1, …, m}, each v_{n} is constant on I_{k}^{m−1}. Thus, by changes of variables and (5.1),

∫_{[0,1]} ∥∑_{n=1}^{m} r_{n} (t) v_{n} (u) x_{n}∥_{Y}^{p} dt = 2^{m−1} ∫_{I_{k}^{m−1}} ∥∑_{n=1}^{m} v_{n} (u) [x_{n}r_{n} (2^{m−1}t − k + 1)]∥_{Y}^{p} dt

= 2^{m−1} ∫_{[0,1]} ∥∑_{n=1}^{m} v_{n} (t) [x_{n}r_{m−1+n} (t) 1_{I_{k}^{m−1}} (t)]∥_{Y}^{p} dt

≤ 2^{m−1} C_{p}^{p} ∫_{[0,1]} ∥∑_{n=1}^{m} x_{n}r_{m−1+n} (t) 1_{I_{k}^{m−1}} (t)∥_{X}^{p} dt

= 2^{m−1} C_{p}^{p} ∫_{I_{k}^{m−1}} ∥∑_{n=1}^{m} x_{n}r_{n} (2^{m−1}t − k + 1)∥_{X}^{p} dt

= C_{p}^{p} ∫_{[0,1]} ∥∑_{n=1}^{m} x_{n}r_{n} (t)∥_{X}^{p} dt .

Thus R_{p} ({v_{n} (u) : n ∈ ℕ}) ≤ C_{p}. □

16 GIRARDI AND WEIS

6. PROOF of LEMMA 4.3

A tree-structured sequence {Γ_{n}^{∗}}_{n∈ℕ_{0}} of indexing sets is needed. Let Γ_{0}^{∗} := {∅} and, for n ∈ ℕ,

Γ_{n}^{∗} = ((0, ±1) × ℕ_{0})^{n} .

There is a natural identification of Γ_{n}^{∗} with Γ_{n−1}^{∗} × ((0, ±1) × ℕ_{0}) and so one can express Γ_{n}^{∗} as

Γ_{n}^{∗} = {((δ_{1}, k_{1}) , … (δ_{n}, k_{n})) : δ_{j} ∈ {0 ± 1} and k_{j} ∈ ℕ_{0} for each j ∈ {1, …, n}}
= {(γ, (δ, k)) : γ ∈ Γ_{n−1}^{∗} , δ ∈ {0, ±1} , k ∈ ℕ_{0}}

for n ∈ ℕ. The notation

A ⊎ B = C

indicates that C is the disjoint union of A and B.

Lemma 6.1. Let (Ω, F, μ) be a probability space with a filtration {F_{n}}_{n∈ℕ_{0}} satisfying (4.1). Let A be an atom of some F_{j} and 0 < δ < \frac{1}{2}. Let n ∈ ℕ satisfy

sup {μ (B) : B ⊂ A , B is an atom of F_{n}} < δμ (A) .  (6.1)

(Note n > j). Then there exists A_{1} and A_{−1} in F_{n} so that, for ε = ±1, (1) A_{1} ⊎ A_{−1} ⊂ A

(2) A_{ε} is a finite union of atoms of F_{n}

(3) \frac{1}{2} − δ < \frac{μ(A_{ε})}{μ(A)} < \frac{1}{2} and so

(4) (1 − 2δ) μ (A) < μ (A_{1} ∪ A_{−1})

(5) 1 < \frac{μ(A)}{2μ(A_{ε})} < \frac{1}{1−2δ}

(6) |\frac{εμ(A)}{2μ(A_{ε})} − ε| < \frac{2δ}{1−2δ} .

Proof. One can express A as

A = ⨄_{k=1}^{m} A_{(0,k)}

where the A_{(0,k)}’s are (disjoint) atoms of F_{n} and m ∈ ℕ ∪ {∞}. Note that

\frac{μ (A_{(0,k)})}{μ (A)} < δ < \frac{1}{2}

and so m ≥ 3.

So there exists l_{1} ∈ ℕ (with 1 + l_{1} < m) so that

∑_{k=1}^{l_{1}} \frac{μ (A_{(0,k)})}{μ (A)} < \frac{1}{2} ≤ ∑_{k=1}^{1+l_{1}} \frac{μ (A_{(0,k)})}{μ (A)} .

Let

A_{1} := ⋃_{k=1}^{l_{1}} A_{(0,k)} .

Note

\frac{1}{2} − δ < \frac{1}{2} − [∑_{k=1}^{1+l_{1}} \frac{μ (A_{(0,k)})}{μ (A)} − ∑_{k=1}^{l_{1}} \frac{μ (A_{(0,k)})}{μ (A)}] ≤ ∑_{k=1}^{l_{1}} \frac{μ (A_{(0,k)})}{μ (A)} = \frac{μ (A_{1})}{μ (A)} < \frac{1}{2}

and

\frac{1}{2} < ∑_{k=1+l_{1}}^{m} \frac{μ (A_{(0,k)})}{μ (A)} = 1 − ∑_{k=1}^{l_{1}} \frac{μ (A_{(0,k)})}{μ (A)} < \frac{1}{2} + δ .

So there exists l_{−1} ∈ ℕ (with 1 + l_{−1} + l_{1} ≤ m) so that

∑_{k=1+l_{1}}^{l_{−1}+l_{1}} \frac{μ (A_{(0,k)})}{μ (A)} < \frac{1}{2} ≤ ∑_{k=1+l_{1}}^{1+l_{−1}+l_{1}} \frac{μ (A_{(0,k)})}{μ (A)} .

Let

A_{−1} := ⋃_{k=1}^{l_{−1}} A_{(0,k+l_{1})} .

Note

\frac{1}{2} − δ < \frac{1}{2} − [ ∑_{k=1+l_{1}}^{1+l_{−1}+l_{1}} \frac{μ (A_{(0,k)})}{μ (A)} − ∑_{k=1+l_{1}}^{l_{−1}+l_{1}} \frac{μ (A_{(0,k)})}{μ (A)}] ≤ ∑_{k=1+l_{1}}^{l_{−1}+l_{1}} \frac{μ (A_{(0,k)})}{μ (A)} = \frac{μ (A_{−1})}{μ (A)} < \frac{1}{2} .

Thus (1), (2), and (3) hold, from which (4), (5), and (6) follow easily. □

The ultimate goal of (the long) Lemma 6.2 is to find the functions mentioned in Remark 6.3 along with some sets {G_{n}}_{n∈ℕ}, all of which satisfy conditions (F7) through (F11) of Lemma 6.2.

Lemma 6.2. Let (Ω, F, μ) be a probability space with a filtration {F_{n}}_{n∈ℕ_{0}} satisfying (4.1). Let A be an atom of F_{j} and A^{o} ∈ M be so that μ(A) = m(A^{o}). Let \{δ_{n}\}_{n∈ℕ} be a sequence from (0, \frac{1}{2}).

Then there exists, for n ∈ ℕ_{0},

(E1) good sets Γ_{n}^{G} ⊂ Γ_{n}^{∗}

(E2) bad sets Γ_{n}^{B} ⊂ Γ_{n}^{∗}

(E3) subsets {l_{γ}}_{γ∈Γ_{n}^{G}} of ℕ

(E4) expansions of the good sets Γ⃗_{n} ⊂ Γ_{n+1}^{∗} (E5) j_{n} ∈ ℕ

18 GIRARDI AND WEIS

(E6) subsets {A_{γ}}_{γ∈Γ_{n}^{G}} of F_{j_{n}} (E7) atoms {A_{γ}}_{γ∈Γ⃗_{n}} of F_{j_{n}} (E8) subsets {A_{γ}^{o}}_{γ∈Γ_{n}^{G}∪Γ⃗_{n}} and {B_{γ}^{o}}_{γ∈Γ_{n}^{G}∪Γ_{n}^{B}} of M

where the items in (E1) through (E8) corresponding to n = 0 are (Z1) Γ_{0}^{G} = {∅} (Z2) Γ_{0}^{B} = ∅ (Z3) l_{γ} = 1 for γ ∈ Γ_{0}^{G} (Z4) Γ⃗_{0} = {(∅, (0, 1))} ⊂ Γ_{1}^{∗} (Z5) j_{0} = j (Z6) if γ ∈ Γ_{0}^{G}, then A_{γ} = ∅ and A_{γ}^{o} = ∅ = B_{γ}^{o} (Z7) if γ ∈ Γ⃗_{0}, then A_{γ} = A and A_{γ}^{o} = A^{o}

and the indexing sets take the form, for n ∈ ℕ, (I1) Γ_{n}^{G} = {(γ, (ε, k)) ∈ Γ_{n}^{∗}: γ ∈ Γ_{n−1}^{G} , ε = ±1 , 1 ≤ k ≤ l_{γ}} (I2) Γ_{n}^{B} = {(γ, (ε, 0)) ∈ Γ_{n}^{∗}: γ ∈ Γ_{n−1}^{G} ∪ Γ_{n−1}^{B} , ε = ±1}

(the zero is a notationally convenient way to ensure Γ_{n}^{G} ∩ Γ_{n}^{B} = ∅) (I3) Γ⃗_{n} = {(γ, (0, l)) ∈ Γ_{n+1}^{∗}: γ ∈ Γ_{n}^{G} , 1 ≤ l ≤ l_{γ}} (which also holds for n = 0)

and so one can write (I3′) Γ⃗_{n} = {(γ, (ε, k) , (0, l)) ∈ Γ_{n+1}^{∗}: γ ∈ Γ_{n−1}^{G} , ε = ±1 , 1 ≤ k ≤ l_{γ} , 1 ≤ l ≤ l_{(γ,(ε,k))}}

and it easily follows that (I1′) Γ_{n}^{G} = {(γ, (ε, k)) ∈ Γ_{n}^{∗}: ε = ±1 , (γ , (0, k)) ∈ Γ⃗_{n−1}}

so that, for n ∈ ℕ, (C0) Γ_{n}^{G}, Γ_{n}^{B}, and Γ⃗_{n} each have finitely many elements (C1) j_{n} > j_{n−1} (C2) if γ_{1}, γ_{2} ∈ Γ⃗_{n} and γ_{1} ≠ γ_{2}, then A_{γ_{1}} ∩ A_{γ_{2}} = ∅ and A_{γ_{1}}^{o} ∩ A_{γ_{2}}^{o} = ∅ (C3) if γ ∈ Γ_{n}^{G} then

A_{γ} = ⋃_{(γ,(0,l))∈Γ⃗_{n}} A_{(γ,(0,l))} and A_{γ}^{o} = ⋃_{(γ,(0,l))∈Γ⃗_{n}} A_{(γ,(0,l))}^{o}

(C4) if (γ, (±1, k)) ∈ Γ_{n}^{G} then for ε = ±1

A_{(γ,(1,k))} ⊎ A_{(γ,(−1,k))} ⊂ A_{(γ,(0,k))} and \frac{1}{2} − δ_{n} < \frac{μ (A_{(γ,(ε,k))})}{μ (A_{(γ,(0,k))})} < \frac{1}{2}

(C5) if γ ∈ Γ⃗_{n} then m(A_{γ}^{o}) = μ(A_{γ})

OPERATOR-VALUED MARTINGALE TRANSFORMS AND R-BOUNDEDNESS 19

(C6) if (γ, (±1, k)) ∈ Γ_{n}^{G} then for ε = ±1 [A_{(γ,(1,k))}^{o} ⊎ A_{(γ,(−1,k))}^{o}] ⊎ [B_{(γ,(1,k))}^{o} ⊎ B_{(γ,(−1,k))}^{o}] = A_{(γ,(0,k))}^{o}

m (A_{(γ,(ε,k))}^{o} ∪ B_{(γ,(ε,k))}^{o}) = \frac{1}{2}m (A_{(γ,(0,k))}^{o})

(C7) if (γ, (±1, 0)) ∈ Γ_{n}^{B} then

B_{(γ,(1,0))}^{o} ⊎ B_{(γ,(−1,0))}^{o} = B_{γ}^{o} and m (B_{(γ,(1,0))}^{o}) = m (B_{(γ,(−1,0))}^{o})

(C8) the family M_{n} := {A_{γ}^{o}}_{γ∈Γ_{n}^{G}} ∪ {B_{γ}^{o}}_{γ∈Γ_{n}^{G}∪Γ_{n}^{B}} is pairwise disjoint (C9) [∪_{γ∈Γ_{n}^{G}}A_{γ}^{o}] ∪ [∪_{γ∈Γ_{n}^{G}∪Γ_{n}^{B}}B_{γ}^{o}] = A^{o} .

Furthermore, if for n ∈ ℕ one defines (D1) d_{n} := ∑_{(γ,(ε,k))∈Γ_{n}^{G}} \frac{ε μ(A_{(γ,(0,k))})}{2 μ(A_{(γ,(ε,k))})} 1_{A_{(γ,(ε,k))}} : Ω → ℝ

(D2) d_{n}^{o} := ∑_{(γ,(ε,k))∈Γ_{n}^{G}} \frac{ε m(A_{(γ,(0,k))}^{o})}{2 m(A_{(γ,(ε,k))}^{o})} 1_{A_{(γ,(ε,k))}^{o}}: [0, 1] → ℝ

(D3) d̃_{n}^{o} := ∑_{(γ,(ε,k))∈Γ_{n}^{G}} ε 1_{A_{(γ,(ε,k))}^{o}∪B_{(γ,(ε,k))}^{o}} + ∑_{(γ,(ε,0))∈Γ_{n}^{B}} ε 1_{B_{(γ,(ε,0))}^{o}}: [0, 1] → ℝ (D4) F_{j_{n}}^{o} := σ ({A_{γ}^{o}: γ ∈ Γ⃗_{n}}) and F_{j_{0}}^{o} := σ ({A_{γ}^{o}: γ ∈ Γ⃗_{0}}) (D5) G_{n} := ∪_{γ∈Γ_{n}^{G}}A_{γ}

then for each n ∈ ℕ (F1) d_{n} is F_{j_{n}}-measurable (F2) if B is an atom of F_{j_{n−1}}, then ∫_{B} d_{n} dμ = 0 (F3) d_{n}^{o} is F_{j_{n}}^{o}-measurable (F4) if B^{o} is an atom of F_{j_{n−1}}^{o}, then ∫_{B^{o}} d_{n}^{o} dm = 0 (F5) m [d̃_{n}^{o} = 1] = \frac{m(A^{o})}{2} = m [d̃_{n}^{o} = −1] (F6) for each choice {ε_{l}}_{l=1}^{n} of signs there exists Γ̃_{n}^{G} ⊂ Γ_{n}^{G} and Γ̃_{n}^{B} ⊂ Γ_{n}^{B} such that

• ∩_{l=1}^{n} [d̃_{l}^{o} = ε_{l}] = [∪_{γ∈Γ̃_{n}^{G}}A_{γ}^{o}] ∪ [∪_{γ∈Γ̃_{n}^{G}∪Γ̃_{n}^{B}}B_{γ}^{o}]

• m (∩_{l=1}^{n} [d̃_{l}^{o} = ε_{l}]) = (\frac{1}{2})^{n} m (A^{o}) (F7) G_{n} ⊂ G_{n−1} where G_{0} := A (F8) 1 < |d_{n} (ω)| < \frac{1}{1−2δ_{n}} if ω ∈ G_{n} and d_{n} (ω) = 0 if ω ∈ Ω \ G_{n} (F9) μ (G_{n}) > [ ∏_{k=1}^{n} (1 − 2δ_{k}) ] μ (A) (F10) if 1 ≤ p < ∞ and {z_{n}}_{k=1}^{n} are from any Banach space Z then

∥∑_{k=1}^{n} z_{k}d_{k}∥_{L_{p}(Ω,Z)} = ∥∑_{k=1}^{n} z_{k}d_{k}^{o}∥_{L_{p}([0,1],Z)} (F11) if 1 ≤ p < ∞ then

∥d_{n}^{o} − d̃_{n}^{o}∥_{L_{p}([0,1],ℝ)}^{p} < [(\frac{2δ_{n}}{1−2δ_{n}} )^{p} + (1 − ∏_{k=1}^{n} (1 − 2δ_{k}))] μ (A) .

20 GIRARDI AND WEIS

Remark 6.3. Note that, in Lemma 6.2, (1) {d_{n}}_{n∈ℕ} is a martingale difference sequence with respect to the filtration {F_{j_{n}}}_{n∈ℕ}

(2) {d_{n}^{o}}_{n∈ℕ} is a martingale difference sequence with respect to the filtration {F_{j_{n}}^{o}}_{n∈ℕ}

(3) {d̃_{n}^{o} 1_{A^{o}}}_{n∈ℕ} is an independent sequence of {±1}-valued symmetric random variables on the probability space (A^{o}, \{B ∈ M: B ⊂ A^{o}\} , \frac{m(·)}{m(A^{o})}).

Proof of Lemma 6.2. Let the desired items in (E1) through (E8) corresponding to n = 0 be as in (Z1) through (Z7). The proof now continues by induction on n.

Let n = 1. Let

Γ_{1}^{G} := {(∅, (ε, 1)) ∈ Γ_{1}^{∗}: ε = ±1} ≡ {(ε, 1) ∈ Γ_{1}^{∗}: ε = ±1}
Γ_{1}^{B} := {(∅, (ε, 0)) ∈ Γ_{1}^{∗}: ε = ±1} ≡ {(ε, 0) ∈ Γ_{1}^{∗}: ε = ±1} .

Recall Γ⃗_{0} = {(∅, (0, 1))} ⊂ Γ_{1}^{∗}.

Since A = A_{(∅,(0,1))} is an atom of F_{j_{0}}, there exists j_{1} > j_{0} so that

sup {μ (B) : B ⊂ A_{(∅,(0,1))} , B is an atom of F_{j_{1}}} < δ_{1} μ (A_{(∅,(0,1))}) .

By Lemma 6.1, there are sets A_{(∅,(±1,1))} ∈ F_{j_{1}} so that

A_{(∅,(1,1))} ⊎ A_{(∅,(−1,1))} ⊂ A_{(∅,(0,1))}  (6.2)

and, for ε = ±1, each A_{(∅,(ε,1))} is a finite union of atoms of F_{j_{1}}, say

A_{(∅,(ε,1))} = ⨄_{l=1}^{l_{(∅,(ε,1))}} A_{(∅,(ε,1),(0,l))} ,  (6.3)

and

\frac{1}{2} − δ_{1} < \frac{μ (A_{(∅,(ε,1))})}{μ (A_{(∅,(0,1))})} < \frac{1}{2} .  (6.4)

Let

Γ⃗_{1} := {(∅, (ε, 1) , (0, l)) ∈ Γ_{2}^{∗}: ε = ±1, 1 ≤ l ≤ l_{(∅,(ε,1))}} .

This completes the construction of the desired items in (E1) through (E7) that satisfy their conditions in (C0) through (C4).

Since A_{(∅,(0,1))} = A and A_{(∅,(0,1))}^{o} = A^{o} and μ (A) = m (A^{o}), by (6.2), for ε = ±1, there exists A_{(∅,(ε,1))}^{o} ∈ M so that

A_{(∅,(1,1))}^{o} ⊎ A_{(∅,(−1,1))}^{o} ⊂ A_{(∅,(0,1))}^{o}
m (A_{(∅,(ε,1))}^{o}) = μ (A_{(∅,(ε,1))}) .  (6.5)

So by (6.4), for ε = ±1,

\frac{1}{2} − δ_{1} < \frac{m (A_{(∅,(ε,1))}^{o})}{m (A_{(∅,(0,1))}^{o})} < \frac{1}{2} .

So, for ε = ±1, there exists B_{(∅,(ε,1))}^{o} ∈ M so that (C6) holds. It follows from (6.3) and (6.5) that, for ε = ±1, there exists A_{(∅,(ε,1),(0,l))}^{o} ∈ M so that

A_{(∅,(ε,1))}^{o} = ⨄_{l=1}^{l_{(∅,(ε,1))}} A_{(∅,(ε,1),(0,l))}^{o}

m (A_{(∅,(ε,1),(0,l))}^{o}) = μ (A_{(∅,(ε,1),(0,l))}) .

If γ ∈ Γ_{1}^{B}, let B_{γ}^{o} = ∅. This completes the construction of the desired items in (E8) that satisfy their conditions in (C2), (C3), and (C5) through (C9).

Note that

d_{1} = \frac{μ (A_{(∅,(0,1))})}{2 μ (A_{(∅,(1,1))})}1_{A_{(∅,(1,1))}} − \frac{μ (A_{(∅,(0,1))})}{2 μ (A_{(∅,(−1,1))})}1_{A_{(∅,(−1,1))}}
d_{1}^{o} = \frac{m (A_{(∅,(0,1))}^{o})}{2 m (A_{(∅,(1,1))}^{o})}1_{A_{(∅,(1,1))}^{o}} − \frac{m (A_{(∅,(0,1))}^{o})}{2 m (A_{(∅,(−1,1))}^{o})}1_{A_{(∅,(−1,1))}^{o}}
d̃_{1}^{o} = 1_{A_{(∅,(1,1))}^{o}∪B_{(∅,(1,1))}^{o}} − 1_{A_{(∅,(−1,1))}^{o}∪B_{(∅,(−1,1))}^{o}} .

So, clearly, (F1) through (F7) along with (F10) hold. A quick look at Lemma 6.1 gives (F8) and (F9) and also that

∥d_{1}^{o} − d̃_{1}^{o}∥_{L_{p}([0,1],ℝ)}^{p} < ∑_{γ∈Γ_{n}^{G}} (\frac{2δ_{1}}{1 − 2δ_{1}} )^{p} m (A_{γ}^{o}) + ∑_{γ∈Γ_{n}^{G}} m (B_{γ}^{o})

= (\frac{2δ_{1}}{1 − 2δ_{1}} )^{p} μ (G_{1}) + μ (A \ G_{1}) .

So (F11) now follows from (F9).

This completes the n = 1 base step.

Fix n ∈ ℕ with n ≥ 2 and assume that the desired items in (E1) through (E8) have been found for k ∈ {0, 1, …, n − 1}. Let

Γ_{n}^{G} := {(γ, (ε, k)) ∈ Γ_{n}^{∗}: γ ∈ Γ_{n−1}^{G} , ε = ±1 , 1 ≤ k ≤ l_{γ}}
Γ_{n}^{B} := {(γ, (ε, 0)) ∈ Γ_{n}^{∗}: γ ∈ Γ_{n−1}^{G} ∪ Γ_{n−1}^{B} , ε = ±1} .

If (γ, (0, k)) ∈ Γ⃗_{n−1}, then A_{(γ,(0,k))} is an atom of F_{j_{n−1}}; find j_{n} > j_{n−1} so that sup {μ (B) : B ⊂ A_{(γ,(0,k))} , B is an atom of F_{j_{n}}} < δ_{n} μ (A_{(γ,(0,k))})

22 GIRARDI AND WEIS

for each (γ, (0, k)) ∈ Γ⃗_{n−1}.

Fix (γ, (0, k)) ∈ Γ⃗_{n−1} (and so γ ∈ Γ_{n−1}^{G} and 1 ≤ k ≤ l_{γ}). By Lemma 6.1, there are sets A_{(γ,(±1,k))} ∈ F_{j_{n}} so that

A_{(γ,(1,k))} ⊎ A_{(γ,(−1,k))} ⊂ A_{(γ,(0,k))}  (6.6)

and, for ε = ±1, each A_{(γ,(ε,k))} is a finite union of atoms of F_{j_{n}}, say

A_{(γ,(ε,k))} = ⨄_{l=1}^{l_{(γ,(ε,k))}} A_{(γ,(ε,k),(0,l))} ,  (6.7)

and

\frac{1}{2} − δ_{n} < \frac{μ (A_{(γ,(ε,k))})}{μ (A_{(γ,(0,k))})} < \frac{1}{2} .  (6.8)

Let

Γ⃗_{n} := {(γ, (ε, k) , (0, l)) ∈ Γ_{n+1}^{∗}: γ ∈ Γ_{n−1}^{G} , ε = ±1 , 1 ≤ k ≤ l_{γ} , 1 ≤ l ≤ l_{(γ,(ε,k))}} .

Towards (C2), note that for distinct elements (γ_{1}, (ε_{1}, k_{1}) , (0, l_{1})) and (γ_{2}, (ε_{2}, k_{2}) , (0, l_{2})) from Γ⃗_{n}

A_{(γ_{1},(ε_{1},k_{1}),(0,l_{1}))} ∩ A_{(γ_{2},(ε_{2},k_{2}),(0,l_{2}))} = ∅ ;  (6.9)

indeed, it follows from (6.6) and (6.7) that, for i ∈ {1, 2},

A_{(γ_{i},(ε_{i},k_{i}),(0,l_{i}))} ⊂ A_{(γ_{i},(ε_{i},k_{i}))} ⊂ A_{(γ_{i},(0,k_{i}))}

and so if γ_{1} ≠ γ_{2} or k_{1} ≠ k_{2} then (6.9) follows from the inductive hypothesis (specifically (C2)) while if γ_{1} = γ_{2} and k_{1} = k_{2} then (6.9) follows from (6.6) if ε_{1} ≠ ε_{2} and from (6.7) if ε_{1} = ε_{2}. This completes the construction of the desired items in (E1) through (E7) that satisfy their conditions in (C0) through (C4).

Towards (E8), fix (γ, (0, k)) ∈ Γ⃗_{n−1}. Thus m (A_{(γ,(0,k))}^{o}) = μ (A_{(γ,(0,k))}). By (6.6), for ε = ±1, there exists A_{(γ,(ε,k))}^{o} ∈ M so that

A_{(γ,(1,k))}^{o} ⊎ A_{(γ,(−1,k))}^{o} ⊂ A_{(γ,(0,k))}^{o}  (6.6′)
m (A_{(γ,(ε,k))}^{o}) = μ (A_{(γ,(ε,k))}) .  (6.10)

So by (6.8), for ε = ±1,

\frac{1}{2} − δ_{n} < \frac{m (A_{(γ,(ε,k))}^{o})}{m (A_{(γ,(0,k))}^{o})} < \frac{1}{2} .  (6.8′)

So, for ε = ±1, there exists B_{(γ,(ε,k))}^{o} ∈ M so that (C6) holds. It follows from (6.7) and (6.10) that, for ε = ±1, there exists A_{(γ,(ε,k),(0,l))}^{o} ∈ M so that

A_{(γ,(ε,k))}^{o} = ⨄_{l=1}^{l_{(γ,(ε,k))}} A_{(γ,(ε,k),(0,l))}^{o}  (6.7′)
m (A_{(γ,(ε,k),(0,l))}^{o}) = μ (A_{(γ,(ε,k),(0,l))}) .

Fix (γ, (±1, 0)) ∈ Γ_{n}^{B}. Thus γ ∈ Γ_{n−1}^{G} ∪ Γ_{n−1}^{B}. Find B_{(γ,(±1,0))}^{o} ∈ M so that (C7) holds. This completes the construction of the items in (E8). Clearly, their conditions in (C3), (C5), (C6) and (C7) hold. As (C2) holds for the A_{γ}’s follows from the inductive hypothesis, (6.6), and (6.7), that (C2) holds for the A_{γ}^{o}’s follows from the inductive hypothesis, (6.6′), and (6.7′).

Towards (C9), note that by (C6) and (C3)

⋃_{γ∈Γ_{n}^{G}} (A_{γ}^{o} ∪ B_{γ}^{o}) = ⋃_{(γ,(0,k))∈Γ⃗_{n−1}} ⋃_{ε=±1} [A_{(γ,(ε,k))}^{o} ∪ B_{(γ,(ε,k))}^{o}]
= ⋃_{(γ,(0,k))∈Γ⃗_{n−1}} A_{(γ,(0,k))}^{o} = ⋃_{γ∈Γ_{n−1}^{G}} ⋃_{(γ,(0,k))∈Γ⃗_{n−1}} A_{(γ,(0,k))}^{o} = ⋃_{γ∈Γ_{n−1}^{G}} A_{γ}^{o}

and by (C7)

⋃_{γ∈Γ_{n}^{B}} B_{γ}^{o} = ⋃_{(γ,(1,0))∈Γ_{n}^{B}} [B_{(γ,(1,0))}^{o} ∪ B_{(γ,(−1,0))}^{o}]

= ⋃_{(γ,(1,0))∈Γ_{n}^{B}} B_{γ}^{o} = [ ⋃_{γ∈Γ_{n−1}^{G}} B_{γ}^{o}] ∪ [ ⋃_{γ∈Γ_{n−1}^{B}} B_{γ}^{o}] .

So (C9) holds by the inductive hypothesis.

Now to show (C8). Note that the family

M_{n}^{1} := {A_{γ}^{o}}_{γ∈Γ_{n}^{G}} ∪ {B_{γ}^{o}}_{γ∈Γ_{n}^{G}}  is pairwise disjoint.  (6.11)

Indeed, if γ ∈ Γ_{n}^{G} then A_{γ}^{o} ∩ B_{γ}^{o} = ∅. So fix γ̃_{i} = (γ_{i}, (ε_{i}, k_{i})) ∈ Γ_{n}^{G} with γ̃_{1} ≠ γ̃_{2} and consider C_{γ_{i}} ∈ M_{n}^{1}. If γ_{1} = γ_{2} and k_{1} = k_{2}, then (6.11) follows from (C6). If γ_{1} ≠ γ_{2} or k_{1} ≠ k_{2}, then (6.11) follows from (C2) since

C_{(γ_{i},(ε_{i},k_{i}))} ⊂ A_{(γ_{i},(0,k_{i}))}^{o} .

and (γ_{i}, (0, k_{i})) ∈ Γ⃗_{n−1}. Next note that the family

M_{n}^{2} := {B_{γ}^{o}}_{γ∈Γ_{n}^{B}}  is pairwise disjoint.  (6.12)

24 GIRARDI AND WEIS

Indeed, if γ̃_{i} = (γ_{i}, (ε_{i}, 0)) ∈ Γ_{n}^{B} then

B_{(γ_{i},(ε_{i},0))}^{o} ⊂ B_{γ_{i}}^{o} .

If γ_{1} = γ_{2}, then (6.12) follows from (C7). If γ_{1} ≠ γ_{2}, then (6.12) follows by the inductive hypothesis (specifically, (C8)) since γ_{i} ∈ Γ_{n−1}^{G} ∪ Γ_{n−1}^{B}. Now if

C_{(γ_{1},(ε_{1},k_{1}))} ∈ M_{n}^{1} and B_{(γ_{2},(ε_{2},0))}^{o} ∈ M_{n}^{2}

then C_{(γ_{1},(ε_{1},k_{1}))} ⊂ A_{γ_{1}}^{o} with γ_{1} ∈ Γ_{n−1}^{G} and B_{(γ_{2},(ε_{2},0))}^{o} ⊂ B_{γ_{2}}^{o} with γ_{2} ∈ Γ_{n−1}^{G} ∪ Γ_{n−1}^{B}. So by the inductive hypothesis on (C8), C_{(γ_{1},(ε_{1},k_{1}))} ∩ B_{(γ_{2},(ε_{2},0))}^{o} = ∅. So (C8) holds.

Now to show that (F1) through (F11) hold. (F1) follows from (E6). Towards (F2), rewrite d_{n} as

d_{n} = ∑_{(γ,(0,k))∈Γ⃗_{n−1}} [\frac{μ (A_{(γ,(0,k))})}{2 μ (A_{(γ,(1,k))})}1_{A_{(γ,(1,k))}} − \frac{μ (A_{(γ,(0,k))})}{2 μ (A_{(γ,(−1,k))})}1_{A_{(γ,(−1,k))}}]

and note that, by (C4) and (E7)

A_{(γ,(1,k))} ⊎ A_{(γ,(−1,k))} ⊂ A_{(γ,(0,k))} ∈ {B: B is an atom of F_{j_{n−1}}} .

So (F2) holds. (F3) follows from (C3). Note that

d_{n}^{o} = ∑_{(γ,(0,k))∈Γ⃗_{n−1}} [\frac{m (A_{(γ,(0,k))}^{o})}{2 m (A_{(γ,(1,k))}^{o})}1_{A_{(γ,(1,k))}^{o}} − \frac{m (A_{(γ,(0,k))}^{o})}{2 m (A_{(γ,(−1,k))}^{o})}1_{A_{(γ,(−1,k))}^{o}} ]

and by (C6) and the definition of F_{j_{n−1}}^{o}

A_{(γ,(1,k))}^{o} ⊎ A_{(γ,(−1,k))}^{o} ⊂ A_{(γ,(0,k))}^{o} ∈ {B: B is an atom of F_{j_{n−1}}^{o}} .

So (F4) holds. Towards (F5), note that by (C8), (C9), (C6), and (C7)

m [d̃_{n}^{o} = 1] = ∑_{(γ,(1,k))∈Γ_{n}^{G}} m (A_{(γ,(1,k))}^{o} ∪ B_{(γ,(1,k))}^{o}) + ∑_{(γ,(1,0))∈Γ_{n}^{B}} m (B_{(γ,(1,0))}^{o})
= ∑_{(γ,(−1,k))∈Γ_{n}^{G}} m (A_{(γ,(−1,k))}^{o} ∪ B_{(γ,(−1,k))}^{o}) + ∑_{(γ,(−1,0))∈Γ_{n}^{B}} m (B_{(γ,(−1,0))}^{o})
= m [d̃_{n}^{o} = −1] .

So (F5) holds (again by using (C8) and (C9)).

Towards (F6), fix a choice {ε_{l}}_{l=1}^{n} of signs. Find Γ̃_{n−1}^{G} ⊂ Γ_{n−1}^{G} and Γ̃_{n−1}^{B} ⊂ Γ_{n−1}^{B} such that

⋂_{l=1}^{n−1} [d̃_{l}^{o} = ε_{l}] = [ ⋃_{γ∈Γ̃_{n−1}^{G}} A_{γ}^{o}] ∪ [ ⋃_{γ∈Γ̃_{n−1}^{G}∪Γ̃_{n−1}^{B}} B_{γ}^{o}] .

OPERATOR-VALUED MARTINGALE TRANSFORMS AND R-BOUNDEDNESS 25

Let

Γ̃_{n}^{G} = {(γ, (ε_{n}, k)) ∈ Γ_{n}^{G}: γ ∈ Γ̃_{n−1}^{G}}
Γ̃_{n}^{B} = {(γ, (ε_{n}, 0)) ∈ Γ_{n}^{B}: γ ∈ Γ̃_{n−1}^{G} ∪ Γ̃_{n−1}^{B}} .

It follows from (C3), (C6), and (C7) that

[d̃_{n}^{o} = ε_{n}] ∩ ( ⋃_{γ∈Γ̃_{n−1}^{G}} A_{γ}^{o}) = ⋃_{γ∈Γ̃_{n}^{G}} (A_{γ}^{o} ∪ B_{γ}^{o})

[d̃_{n}^{o} = ε_{n}] ∩ ( ⋃_{γ∈Γ̃_{n−1}^{G}∪Γ̃_{n−1}^{B}} B_{γ}^{o}) = ⋃_{γ∈Γ̃_{n}^{B}} B_{γ}^{o} .

Thus

⋂_{l=1}^{n} [d̃_{l}^{o} = ε_{l}] = ( ⋃_{\begin{matrix}(γ,(ε_{n},k))∈Γ_{n}^{G} \\ γ∈Γ̃_{n−1}^{G} \end{matrix}} (A_{(γ,(ε_{n},k))}^{o} ∪ B_{(γ,(ε_{n},k))}^{o})) ∪ ( ⋃_{\begin{matrix}(γ,(ε_{n},0))∈Γ_{n}^{B} \\ γ∈Γ̃_{n−1}^{G}∪Γ̃_{n−1}^{B} \end{matrix}} B_{(γ,(ε_{n},0))}^{o}) . (6.13)

By (C6) and (C7), for the set on the right-hand side of (6.13), replacing ε_{n} by −ε_{n} does not change its measure. So (F6) holds.

(F7) follows from (C3) and (C4) while (F8) follows from (C4).

Fix γ ∈ Γ_{n−1}^{G} and 1 ≤ k ≤ l_{γ}. So (γ, (0, k)) ∈ Γ⃗_{n−1} and (γ, (±1, k)) ∈ Γ_{n}^{G}. It follows from (C4) that

(1 − 2δ_{n}) μ (A_{(γ,(0,k))}) < μ (A_{(γ,(1,k))} ⊎ A_{(γ,(−1,k))}) .

Taking the double sum ∑_{γ∈Γ_{n−1}^{G}} ∑_{1≤k≤l_{γ}} of both sides gives (via (C3))

(1 − 2δ_{n}) μ (G_{n−1}) < μ (G_{n}) .

So (F9) holds. (F10) is clear. Since

∥d_{n}^{o} − d̃_{n}^{o}∥_{L_{p}([0,1],ℝ)}^{p} < ∑_{γ∈Γ_{n}^{G}} (\frac{2δ_{n}}{1 − 2δ_{n}} )^{p} m (A_{γ}^{o}) + ∑_{γ∈Γ_{n}^{G}∪Γ_{n}^{B}} m (B_{γ}^{o})

= (\frac{2δ_{n}}{1 − 2δ_{n}} )^{p} μ (G_{n}) + μ (A \ G_{n}) .

(F11) now follows from (F9). □

The next lemma follows easily from the Contraction Principle (Fact 2.3) and a standard [3, 24] perbutation argument. A proof is included for completeness sake.

26 GIRARDI AND WEIS

Lemma 6.4. Let p ∈ [1, ∞) and 0 < δ < 1.

Let {d̃_{n}^{o}}_{n=1}^{m} be a sequence of independent, symmetric, {±1}-valued random variables on a probability space (Ω, F, μ) and {d_{n}^{o}}_{n=1}^{m} be a sequence in L_{p} (Ω, ℝ). If

∑_{n=1}^{m} ∥d̃_{n}^{o} − d_{n}^{o}∥_{L_{p}(Ω,ℝ)} ≤ \frac{δ}{2}

then for any choice \{x_{n}\}_{n=1}^{m} from a Banach space X, \{x_{n}d_{n}^{o}\}_{n=1}^{m} is a (\frac{1+δ}{1−δ} )-unconditional sequence in L_{p} (Ω, X).

Proof. Fix {x_{n}}_{n=1}^{m} from some Banach space X, a choice {ε_{n}}_{n=1}^{m} of signs from {±1}, and scalars {λ_{n}}_{n=1}^{m}. It needs to be shown that

∥∑_{n=1}^{m} ε_{n} λ_{n} x_{n} d_{n}^{o}∥_{L_{p}(Ω,X)} ≤ (\frac{1 + δ}{1 − δ} ) ∥∑_{n=1}^{m} λ_{n} x_{n} d_{n}^{o}∥_{L_{p}(Ω,X)} .  (6.14)

Find {x̃_{n}}_{n=1}^{m} from S(X) and {λ̃_{n}}_{n=1}^{m} from ℝ so that λ_{n} x_{n} = λ̃_{n}x̃_{n} for each n ∈ {1, …, m}.

It follows from Fact 2.3 that {x̃_{n}d̃_{n}^{o}}_{n=1}^{m} is a (normalized) 1-unconditional basic sequence in L_{p} (Ω, X). Since ∑_{n=1}^{m} ∥x̃_{n}d̃_{n}^{o} − x̃_{n}d_{n}^{o}∥_{L_{p}(Ω,X)} ≤ \frac{δ}{2}, for any choice \{α_{n}\}_{n=1}^{m} of scalars

(1 − δ) ∥∑_{n=1}^{m} α_{n}x̃_{n}d̃_{n}^{o}∥_{L_{p}(Ω,X)} ≤ ∥∑_{n=1}^{m} α_{n}x̃_{n}d_{n}^{o}∥_{L_{p}(Ω,X)} ≤ (1 + δ) ∥∑_{n=1}^{m} α_{n}x̃_{n}d̃_{n}^{o}∥_{L_{p}(Ω,X)}

(cf., eg., [26, Prop. I.1.a.9]) Thus \{x̃_{n}d_{n}^{o}\}_{n=1}^{m} is a (\frac{1+δ}{1−δ})-unconditional basic sequence. So (6.14) holds. □

Proof of Lemma 4.3. Let’s keep with the notation in Lemma 6.2.

Pick {δ_{n}}_{n=1}^{m} so that

(a) 0 < δ_{n} < \frac{1}{2} for each n ∈ \{1, …, m\}

(b) \frac{1}{1−2δ_{n}} ≤ τ_{1} for each n ∈ \{1, …, m\}

(c) ∑_{n=1}^{m} [(\frac{2δ_{n}}{1−2δ_{n}} )^{p} + (1 − ∏_{k=1}^{m} (1 − 2δ_{k}))]^{\frac{1}{p}} ≤ \frac{1}{2} \frac{τ−1}{τ+1}

(d) ∏_{n=1}^{m} (1 − 2δ_{n})^{−1} ≤ 1 + ε .

Apply Lemma 6.2 to find {d_{n}}_{n=1}^{m} ∈ L_{p} (Ω, ℝ), along with everything else. Thus (1) of Lemma 4.3 holds.

Condition (b) above along with (F7) and (F8) of Lemma 6.2 imply (2) and (3) of Lemma 4.3. By (c) above and (F11) of Lemma 6.2,

∑_{n=1}^{m} [∫_{A^{o}} |d_{n}^{o} − d̃_{n}^{o}|^{p} \frac{dm}{m(A^{o})}]^{\frac{1}{p}} = ∑_{n=1}^{m} ∥d_{n}^{o} − d̃_{n}^{o}∥_{L_{p}([0,1],ℝ)} [μ (A)]^{\frac{−1}{p}} ≤ \frac{1}{2} (\frac{τ − 1}{τ + 1}) .

So (4) of Lemma 4.3 holds by Lemma 6.4, Remark 6.3, and (F10) of Lemma 6.2.

Towards (5) of Lemma 4.3, let z_{n_{0}} ≠ 0. Then for each ω ∈ G_{m}, by Kahane’s Contraction

Principle (Fact 2.3) and (3) of Lemma 4.3

∥∑_{n=1}^{m} d_{n} (ω) z_{n}r_{n}∥_{L_{p}([0,1],Z)} ≥ ∥d_{n_{0}} (ω) z_{n_{0}}r_{n_{0}}∥_{L_{p}([0,1],Z)} ≥ ∥z_{n_{0}}∥_{Z} .

Thus, by (b) and (F8), along with (d) and (F9)

∫_{A\G_{m}} ∥ ∑_{n=1}^{m} d_{n} (ω)z_{n}r_{n}∥_{L_{p}([0,1],Z)}^{p} dμ (ω) ≤ ∫_{A\G_{m}} [∑_{n=1}^{m} ∥d_{n} (ω) z_{n}r_{n}∥_{L_{∞}([0,1],Z)}]^{p} dμ (ω)

≤ ∫_{A\G_{m}} [∑_{n=1}^{m} τ_{1} ∥z_{n}∥_{Z}]^{p} dμ (ω)

= \frac{μ (A \ G_{m})}{μ (G_{m})} [\frac{τ_{1} ∑_{n=1}^{m} ∥z_{n}∥_{Z}}{∥z_{n_{0}}∥_{Z}} ]^{p} μ (G_{m}) ∥z_{n_{0}}∥_{Z}^{p}

≤ [\frac{μ (A)}{μ (G_{m})} − 1] M^{p} ∫_{G_{m}} ∥∑_{n=1}^{m} d_{n} (ω) z_{n}r_{n}∥_{L_{p}([0,1],Z)}^{p} dμ (ω)

≤ ε M^{p} ∫_{G_{m}} ∥∑_{n=1}^{m} d_{n} (ω) z_{n}r_{n}∥_{L_{p}([0,1],Z)}^{p} dμ (ω) .

So (5) of Lemma 4.3 holds. □

Acknowledgement. The authors thank Dr. W. Elch for his careful thorough check of Section 6.

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

Mathematisches Institut I, Universität Karlsruhe, Englerstraße 2, 76128 Karlsruhe, Germany

E-mail address: Lutz.Weis@math.uni-karlsruhe.de