operator-valued martingale transforms in rearrangement invariant spaces and applications

SCIENCE CHINAMathematics

. ARTICLES . April 2013 Vol. 56 No. 4: 831–844

doi: 10.1007/s11425-013-4570-8

c© Science China Press and Springer-Verlag Berlin Heidelberg 2013 math.scichina.com www.springerlink.com

Operator-valued martingale transformsin rearrangement invariant spaces and applications

JIAO Yong1,∗, WU Lian1 & POPA Mihai2

1Institute of Probability and Statistics, Central South University, Changsha 410075, China;2Department of Mathematics and Statistics, Queen’s University, Kingston K7L 3N6, Canada

Email: [email protected], [email protected], [email protected]

Received January 4, 2012; accepted May 4, 2012; published online March 4, 2013

Abstract In this paper the operator-valued martingale transform inequalities in rearrangement invariant func-

tion spaces are proved. Some well-known results are generalized and unified. Applications are given to classical

operators such as the maximal operator and the p-variation operator of vector-valued martingales, then we can

very easily obtain some new vector-valued martingale inequalities in rearrangement invariant function spaces.

These inequalities are closely related to both the geometrical properties of the underlying Banach spaces and

the Boyd indices of the rearrangement invariant function spaces. Finally we give an equivalent characterization

of UMD Banach lattices, and also prove the Fefferman-Stein theorem in the rearrangement invariant function

spaces setting.

Keywords operator-valued martingale transforms, i.r. spaces, Boyd indices, uniformly convex (smooth)

MSC(2010) 60G42, 60G46

Citation: Jiao Y, Wu L, Popa M. Operator-valued martingale transforms in rearrangement invariant spaces and

applications. Sci China Math, 2013, 56: 831–844, doi: 10.1007/s11425-013-4570-8

1 Introduction

Martingale transforms were first introduced by Burkholder in [3], as one of the methods to deal with

martingale theory and harmonic analysis, which have been extensively studied; see, for example, [5, 8,

16,22,24]. In 2000, Martinez and Torrea [17] introduced the operator-valued martingale transforms, and

replaced the scalar-valued multiplying sequences by operator-valued multiplying sequences, which are a

generalization of Burkholder’s martingale transforms. Let us first recall the definition of the operator-

valued martingale transforms.

Definition 1.1 (See [17]). Let B1 and B2 be two Banach spaces, f = (fn)n�1 a B1-valued martingale.

Define v = (vn)n�1 a sequence such that

1) (vn)n�1 is Σn−1-measurable for n � 2, v1 is Σ1-measurable;

2) each vn is L(B1, B2)-valued, where L(B1, B2) denotes the set of all bounded operators from B1

to B2;

3) supn�1 ‖vn‖L∞(L(B1,B2)) � 1.

The martingale given by

(Tf)n =

n∑k=1

vkdfk

is called the martingale transform of f by the multiplying sequence v, where dfk = fk − fk−1, f0 = 0. T

will denote the martingale transform operator.

∗Corresponding author

832 Jiao Y et al. Sci China Math April 2013 Vol. 56 No. 4

In [17], it was proved that several bounded inequalities of Lp-norm are equivalent for the operator-

valued martingale transforms, which can be stated as follows (for the notation see Section 2).

Theorem 1.2. Let B1 and B2 be two Banach spaces, and T a martingale transform operator as above.

Then the following statements are equivalent:

(1) There exists a constant C > 0 such that

λP(M(Tf) > λ) � C‖f‖L1(B1), ∀λ > 0.

(2) There exists a constant C > 0 such that

λP(M(Tf) > λ) � C‖M(f)‖L1, ∀λ > 0.

(3) Given any p, 1 � p < ∞, there exists Cp > 0 such that

‖M(Tf)‖Lp � Cp‖M(f)‖Lp .

(4) Given any p, 1 < p < ∞, there exists Cp > 0 such that

‖M(Tf)‖Lp � Cp‖f‖Lp(B1).

(5) There exist p0, 1 < p0 < ∞, and a constant C0 such that

‖M(Tf)‖Lp0� C0‖f‖Lp0(B1).

Some further results and applications on operator-valued martingale transforms were developed in [18,

19, 24]. Very recently, the first author in [9] and [10] successfully applied the operator-valued martingale

transforms to construct the endpoint case of Pisier’s theorem (see [21]) and investigate the embeddings

between weak Orlicz martingale spaces, respectively.

The main goal of the present paper is to give the more general theory of operator-valued martingale

transforms. More precisely, we prove some results similar to Theorem 1.2 in the frame of rearrangement

invariant function spaces which generalize and unify some well-known theorems; see Theorem 3.1. As

applications, in Section 4, we obtain the vector-valued martingale inequalities in rearrangement invariant

function spaces. Classical (scalar-valued) results in rearrangement invariant spaces can be found in

[11–13, 20]. It is well known that the validity of a classical (scalar-valued) result in the vector-valued

setting, i.e., for functions or martingales with values in a Banach space, depends on the geometrical or

topological properties of the underlying Banach space. The relevant geometrical properties are often the

uniform convexity and uniform smoothness. We refer to [14] for the definition and more information about

uniform convexity and uniform smoothness. Our results show that vector-valued martingale inequalities

in rearrangement function spaces heavily rely on both the Boyd indices of the rearrangement function

space and the geometrical properties of the underlying Banach space. In Section 5, we give an equivalent

characterization of UMD Banach lattice by using martingale inequalities in rearrangement invariant

function spaces. In Section 6, we extend the well-known Fefferman and Stein theorem on the Hardy-

Littlewood maximal operator to the martingale version in the frame of rearrangement invariant function

spaces.

2 Preliminaries

In this section, we give some preliminaries necessary to the whole paper. Let X and Y be normed linear

spaces, we write X ↪→ Y if X is continuously embedded in Y , that is, if X ⊂ Y and the inclusion map is

continuous. Let (Ω,Σ,P) be a complete probability space and f be a real random variable on (Ω,Σ,P).

The non-increasing rearrangement of f , which is denoted by f∗, is a right continuous function on [0, 1]

defined by

f∗(t) = inf{s > 0 : P(|f | > s) � t}, t ∈ [0, 1],

Jiao Y et al. Sci China Math April 2013 Vol. 56 No. 4 833

with the convention that inf Ø = ∞. It is well known that f∗(t) and f have the identical distribution,

that is, for any λ > 0,

P{w ∈ Ω : |f(w)| > λ} = m{t ∈ [0, 1] : f∗(t) > λ},where m denotes the Lebesgue measure on [0, 1].

Definition 2.1 (See [1, Chapter 2]). A Banach function space (X, ‖ · ‖X) over Ω is a real Banach

space of measurable functions (in other words, random variables) on (Ω,Σ,P) satisfying the following

conditions:

(i) L∞ ↪→ X ↪→ L1;

(ii) If |f | � |g| a.s., and g ∈ X , then f ∈ X and ‖f‖X � ‖g‖X ;

(iii) If 0 � fn ↑ f a.s., fn ∈ X and supn ‖fn‖X < ∞, then f ∈ X and ‖f‖X = supn ‖fn‖X .

For the sake of convenience, we adopt the convention that if x ∈ X , then ‖x‖X = ∞. Thus the norm

‖x‖X is defined for any random variable x, and x ∈ X if and only if ‖x‖X < ∞.

Given two random variables f and g, we write fd g to mean that they have the same distribution.

Definition 2.2. A Banach function space (X, ‖ · ‖X) is said to be a rearrangement invariant

function space if fd g and g ∈ X, then f ∈ X and ‖f‖X = ‖g‖X .

A rearrangement invariant function space is simply called an r.i. space in this paper. For example,

Lebesgue spaces, Orlicz spaces and Lorentz spaces are r.i. spaces, while weighted Lebesgue spaces with

suitable weights are Banach function spaces that are not r.i. spaces. It is obvious that if fd g, then

f∗ = g∗. Hence a Banach space (X, ‖ · ‖X) is an r.i. space if f∗ = g∗ and g ∈ X, then f ∈ X and

‖f‖X = ‖g‖X .

We recall the Luxemberg representation theorem; see [1, pp. 62–64] for details. If (X, ‖ · ‖X) is an r.i.

space, then there exists an r.i. space X over [0, 1] with Lebesgue measure m, equipped with the norm

‖ · ‖X such that

‖f‖X = ‖f∗‖X .

We call (X, ‖ · ‖X) the Luxemberg representation of (X, ‖ · ‖X). For instance, for 1 � p � ∞, the

Luxemberg representation of Lp(Ω,Σ,P) is Lp[0, 1], and ‖f‖Lp(Ω,Σ,P) = ‖f∗‖Lp[0,1].

Now we turn to the Banach-valued r.i. spaces X(B). Let B be a Banach space, and f be a

B-valued random variable on (Ω,Σ,P). In this case, we define the non-increasing rearrangement of f as

follows,

f∗(t) = inf{s > 0 : P(‖f‖B > s) � t}, t ∈ [0, 1],

where we follow the convention that inf Ø = ∞. Without danger of confusion, we simply write ‖f‖ to

substitute ‖f‖B for a convenience.

Definition 2.3. Let B be a Banach space and X be an r.i. space over Ω. We define

X(B) = {f with values in B : f∗ ∈ X}

equipped with the norm

‖f‖X(B) = ‖f∗‖X ,

where (X, ‖ · ‖X) is the Luxemberg representation of (X, ‖ · ‖X).

In order to describe our results, we need the Boyd indices of X introduced by Boyd in [2].

We define, for every 0 < c < ∞, the dilation operator Dc acting on the space of measurable functions

on [0, 1] by Dc(f)(t) = f(t/c), if 0 < t < min(1, c); Dc(f)(t) = 0, if c < t < 1. Let Z be an r.i. space on

[0, 1]. The upper Boyd index and the low Boyd index of Z are respectively defined by

qZ = limc→0+

log c

log ‖Dc‖ = inf0<c<1

log c

log ‖Dc‖ ,

pZ = limc→∞

log c

log ‖Dc‖ = supc>1

log c

log ‖Dc‖ ,


where ‖Dc‖ stands for the operator norm of Dc : Z → Z. The upper Boyd index is also the smallest q

such that for all 0 < c < 1,

‖Dc‖ � c1/q;

we refer to [14] for this. If X is an r.i. space on Ω, we set qX = qX and pX = p

X . Noting that for any r.i.

space,

1 � pX � qX � ∞.

Let {Σn}n�1 be a nondecreasing sequence of sub-σ-fields of Σ such that Σ = ∨Σn. Given a Banach space

B, by a B-valued martingale relative to {Σn}n�1 we mean a sequence f = (fn)n�1 of B-valued random

variables such that fn is Σn-measurable, E(‖fn‖B) < ∞ and E(fn+1|Σn) = fn for every n � 1. For a

B-valued martingale f = (fn)n�1, we denote by df = (dfn)n�1 the martingale difference sequence of f ,

that is, dfn = fn − fn−1, f0 = 0. For a background on vector-valued martingales, see [6,15,21]. We adopt

the standard notions of its maximal function, q-variation function (1 � q < ∞) as follows, respectively:

Mn(f) = sup1�i�n

‖fi‖, M(f) = supn�1

‖fn‖,

S(q)n (f) =

( n∑i=1

‖dfi‖q)1/q

, S(q)(f) =

( ∞∑n=1

‖dfn‖q)1/q

.

Remark 2.4. For convenience, we also use Mn(df) to denote sup1�k�n ‖dfk‖.For any Banach space B and any B-valued random variable f defined on (Ω,Σ,P), for 1 � p < ∞, let

‖f‖Lp(B) = (E‖f‖pB)1/p

be its Lp(B)-norm, and in the case p = ∞, set

‖f‖L∞(B) = esssup ‖f‖B.

The space Lp(B) is the space of functions with finite Lp(B)-norm. When the Banach space B is the

scalar field, Lp(B) will be simply written as Lp.

Definition 2.5. Given a B-valued martingale f = (fn)n�1 and an r.i. space X , we define

‖f‖X(B) = supn�1

‖fn‖X(B).

Analogously to the scalar case, see [8] or [16], the so-called Doob inequalities also hold in the vector-valued

setting, and for any 1 < p � ∞, we have λP(M(f) > λ) � C‖f‖L1(B) and ‖M(f)‖Lp � Cp‖f‖Lp(B) or

‖Mn(f)‖Lp � Cp‖fn‖Lp(B).

We close the section by mentioning that throughout the paper the notation like Cp, CX , CX,q, . . .

will respectively denote a positive constant, which only depend on the subindex but never on the

martingales in consideration, and which may change from line to line.

3 Martingale transforms

Theorem 3.1. Let B1 and B2 be two Banach spaces, and T a martingale transform operator defined

as in Section 1. Then the following statements are equivalent:

(1) There exists an absolute constant C > 0 such that

λP(M(Tf) > λ) � C‖M(f)‖L1, ∀λ > 0.

(2) For any r.i. space X with qX < ∞, there exists CX such that

‖M(Tf)‖X � CX‖M(f)‖X .


(3) For any r.i. space X with 1 < pX � qX < ∞, there exists CX such that

‖M(Tf)‖X � CX‖f‖X(B1).

(4) For any (or equivalently, for some) 1 � p < ∞, there exists Cp such that

‖M(Tf)‖Lp � C‖M(f)‖Lp .

Lemma 3.2. Let {An}n�1 be a nonnegative, increasing and adapted sequence, and Y be a nonnegative

random variable. If for any stopping time τ ,

E(A∞ −Aτ−1|Στ ) � E(Y |Στ ),

then for any r.i. space X with qX < ∞, there exists CX such that

‖A∞‖X � CX‖Y ‖X .

Proof. For any λ > 0, set stopping time τ = inf{n : An > λ}, then Aτ−1 � λ, and {A∞ > λ} = {τ <

∞} ∈ Στ . Since E(A∞ −Aτ−1|Στ ) � E(Y |Στ ), it follows that∫{A∞>λ}

(A∞ − λ)dP �∫{A∞>λ}

Y dP,

or ∫{A∞>λ}

A∞dP �∫{A∞>λ}

Y dP + λP{A∞ > λ} . (3.1)

As λ → 0, from (3.1) we obtain

‖A∞‖L1 � ‖Y ‖L1. (3.2)

On the other hand, for t ∈ [0, 1], setting λ = A∗∞(t), and noting that A∗

∞(t) and A∞ have the identical

distribution, the inequality (3.1) also implies∫ t

0

A∗∞(s)ds �

∫ t

0

Y ∗(s)ds+ tA∗∞(t). (3.3)

Note that qX < ∞,

limt→0

‖Dt‖ � limt→0

t1/qX = 0.

We can choose t0 ∈ (0, 1) such that ‖Dt0‖ < 1. Observe that, for every t ∈ [0, 1],∫ t

0

A∗∞(s)ds �

∫ t0t

0

A∗∞(s)ds+ (1− t0)tA

∗∞(t) = t0

∫ t

0

D1/t0A∗∞(s)ds + (1− t0)tA

∗∞(t). (3.4)

It follows from the inequalities (3.3) and (3.4) that∫ t

0

D1/t0A∗∞(s)ds � 1

t0

∫ t

0

Y ∗(s)ds+ tA∗∞(t) �

∫ t

0

(A∗

∞(s) +1

t0Y ∗(s)

)ds,

which results from Proposition 2.a.8 in [14] that

‖D1/t0A∗∞‖

X � ‖A∗∞‖

X +1

t0‖Y ∗‖

X = ‖A∞‖X +1

t0‖Y ‖X . (3.5)

Noting that Dt0D1/t0A∗∞ = A∗

∞ · �[0,t0) and A∗∞(t) is non-increasing, we have

‖A∞‖X � ‖A∗∞ · �[0,t0)‖ X +A∗

∞(t0)

� ‖Dt0‖ · ‖D1/t0A∗∞‖

X +A∗∞(t0)

� t1/qX0 ‖D1/t0A

∗∞‖

X +A∗∞(t0). (3.6)


Since L∞ ↪→ X ↪→ L1, it follows from (3.2) that

A∗∞(t0) �

1

t0

∫ t0

0

A∗∞(t)dt � 1

t0‖A∞‖L1 � 1

t0‖Y ‖L1 � 1

t0‖Y ‖X . (3.7)

According to (3.5)–(3.7), we finally have

‖A∞‖X � CX‖Y ‖X ,

where CX = (1 + t1/qX0 )/(t0(1− t

1/qX0 )).

Remark 3.3. (1) In Lemma 3.2, it is sufficient to verify E(A∞ − Aτ−1|Fτ ) � E(Y |Fτ ) for stopping

time taking constant values n, see for example [16].

(2) If we take the i.r. space X = Lp, 1 � p < ∞, then Lemma 3.2 comes back to the Garsia-Neveu

lemma.

Lemma 3.4. Let X be an r.i. space with qX < ∞, then for any B-valued martingale f = (fn)n�1

with ‖M(f)‖X < ∞, we have f = g + h, where g = (gn)n�1 and h = (hn)n�1 are martingales satisfying

the following conditions:

(1) ‖dgn‖ � 4Mn−1(df);

(2) ‖∑∞n=1 ‖dhn‖‖X � CX‖M(f)‖X.

Proof. Set Fi = dfi · �{‖dfi‖�2Mi−1(df)}, Gi = dfi · �{‖dfi‖>2Mi−1(df)}. Now we let

dgi = Fi − E(Fi|Σi−1), gn =

n∑i=1

dgi; dhi = Gi − E(Gi|Σi−1), hn =

n∑i=1

dhi.

It is obvious that g = (gn)n�1 and h = (hn)n�1 are martingales, and ‖dgn‖ � 4Mn−1(df). Noting that

‖Gn‖ = 2‖Gn‖ − ‖Gn‖ � 2Mn(df)− 2Mn−1(df),

we deduce that

E

( ∞∑i=n

‖dhi‖|Σn

)� 4E(M(df)|Σn) � 8E(M(f)|Σn).

It follows from Lemma 3.2 that ∥∥∥∥∞∑n=1

‖dhn‖∥∥∥∥X

� CX‖M(f)‖X. �

Based on the lemmas above we are now on a position to prove Theorem 3.1. We will adopt the methods

used in [17] with some modifications.

Proof of Theorem 3.1. (1) ⇒ (2). Considering a martingale f satisfying Mf ∈ X, by Lemma 3.4, we

can decompose f = g + h. Since the maximal operator M is sublinear, it follows that

‖M(Tf)‖X � ‖M(Tg)‖X + ‖M(Th)‖X. (3.8)

According to the boundedness of the sequence vk, we get

‖M(Th)‖X =

∥∥∥∥ supn

∥∥∥∥n∑

k=1

vkdhk

∥∥∥∥B2

∥∥∥∥X

�∥∥∥∥ sup

n

n∑k=1

(‖vk‖L(B1,B2)‖dhk‖B1)

∥∥∥∥X

�∥∥∥∥ sup

n

n∑k=1

‖dhk‖B1

∥∥∥∥X

=

∥∥∥∥∞∑k=1

‖dhk‖B1

∥∥∥∥X


� CX‖M(f)‖X .

Set Wn = 4Mn−1(df), then ‖dgn‖ � Wn, and Wn is nondecreasing and Σn−1-measurable. Fix λ > 0. For

β > 0, δ > 0 satisfying β > δ + 1, define the following stopping times,

μ = inf{n : ‖(Tg)n‖B2 > λ},v = inf{n : ‖(Tg)n‖B2 > βλ},σ = inf{n : ‖gn‖B1 ∨Wn+1 > δλ}.

These are clearly stopping times and μ � υ. Denote un = χ(μ<n�v∧σ), then {μ < n � v ∧ σ} is

Σn−1-measurable. Now we consider the martingale an =∑n

k=1 ukdgk and its martingale transform

(Ta)n =∑n

k=1 vkukdgk. Obviously, M(a) � 2δλ in the set {μ < ∞} and M(a) = 0 in {μ = ∞}. Thus‖M(a)‖L1 � 2δλP(μ < ∞) = 2δλP(M(Tg) > λ).

By the condition (1), we get

P(M(Ta) > (β − δ − 1)λ) � C‖M(a)‖L1

(β − δ − 1)λ� 2Cδ

β − δ − 1P(M(Tg) > λ).

It is also easy to check that

P(M(Tg) > βλ,M(W ) � δλ) � P(M(Ta) > (β − δ − 1)λ),

and

P(M(Tg) > βλ) � P(M(Tg) > βλ,M(W ) � δλ) + P(M(W ) > δλ)

� 2Cδ

β − δ − 1P(M(Tg) > λ) + P(M(W ) > δλ).

Denote ρ = 2Cδβ−δ−1 . Since ρP(M(Tg) > λ) = P(DρM(Tg) > λ), it follows from Lemma 4 in [11] that

‖β−1M(Tg)‖X � ‖DρM(Tg)‖X + ‖δ−1(M(W ))‖X� ρ1/qX‖M(Tg)‖X + δ−1‖M(W )‖X� ρ1/qX‖M(Tg)‖X + δ−1‖4M(df)‖X� ρ1/qX‖M(Tg)‖X +

8

δ‖M(f)‖X.

Noting that qX < ∞, we can take δ small enough to satisfy 1− βρ1/qX > 0, then

‖M(Tg)‖X � 8β

δ(1− βρ1/qX )‖M(f)‖X.

Finally it follows from (3.8) that

‖M(Tf)‖X � CX‖M(f)‖X .

(2) ⇒ (3). For any t > 1, ‖Dt‖ � t1/qX . It follows from qX > 1 that

limt→∞

‖Dt‖t

� limt→∞

t1/qX

t= 0,

by Theorem 1 in [20], which is equivalent to

‖M(f)‖X � CX‖f‖X(B1).

Then the condition (2) gives ‖M(Tf)‖X � CX‖f‖X(B1).

(3) ⇒ (1). Taking X = Lp, 1 < p < ∞, then 1 < pX = qX = p < ∞. Thus

‖M(Tf)‖Lp � Cp‖M(f)‖Lp � Cp‖f‖Lp(B1),

which is an equivalent condition of (1) by Theorem 1 in [17].

(4) ⇔ (1). It is due to Theorem 1 in [17].


4 Vector-valued martingale inequalities

Regarding the maximal operator and p-variation operator as two martingale transform operators respec-

tively, applying Theorem 2.1, by handling the two concrete martingale transform operators, we can very

easily obtain some new vector-valued inequalities in r.i. spaces. The results extend those in [11, 12, 20].

It should be mentioned that in the scalar case, the upper Body index pX < ∞ is equivalent to

C−1X ‖S(2)(f)‖X � ‖M(f)‖X � CX‖S(2)(f)‖X .

However as we can see, the geometrical properties of the underlying Banach space are important in the

vector-valued case.

We refer to [15] and [21] for the following two lemmas.

Lemma 4.1. Let B be a Banach space. For 2 � q < ∞, the following statements are equivalent:

(1) B is isomorphic to a q-uniformly convex space;

(2) there exists Cp,q such that for any (or equivalently, for some) 1 � p < ∞,

‖S(q)(f)‖Lp � Cp,q‖M(f)‖Lp,

where f = (fn)n�1 is any B-valued martingale.

Lemma 4.2. Let B be a Banach space. For 1 < q � 2, the following statements are equivalent:

(1) B is isomorphic to a q-uniformly smooth space;

(2) there exists Cp,q such that for any (or equivalently, for some) 1 � p < ∞,

‖M(f)‖Lp � Cp,q‖S(q)(f)‖Lp ,


Theorem 4.3. Let B be a Banach space. For 2 � q < ∞, the following statements are equivalent:

(1) B is isomorphic to a q-uniformly convex space;

(2) For any r.i. space X with qX < ∞, there exists CX,q such that

‖S(q)(f)‖X � CX,q‖M(f)‖X ,

where f = (fn)n�1 is any B-valued martingale;

(3) For any r.i. space X with 1 < pX � qX < ∞, there exists a constant CX,q such that

‖S(q)(f)‖X � CX,q‖f‖X(B),


Proof. We consider a martingale transform operator Q from the family of B-valued martingales to that

of �q(B)-valued martingales. Let vk ∈ L(B, �q(B)) be the operator defined by vkx = (xj)j�1 for x ∈ B,

where xj = x if j = k and xj = 0 otherwise. Namely,

vkx = (

k−1︷︸︸︷0, . . . , 0, x, 0, 0, . . .), ∀x ∈ B.

Q is the martingale transform associated to the sequence (vk):

(Qf)n =

n∑k=1

vkdfk = (df1, df2, . . . , dfn, 0, . . .),

where f is any B-valued martingale. Then

M(Qf) = supn

‖(Qf)n‖�q(B) = supn

‖(df1, df2, . . . , dfn, 0, 0, . . .)‖�q(B) = S(q)(f).


B is isomorphic to a q-uniformly convex space, by Lemma 4.1, which is equivalent to

‖M(Qf)‖Lp = ‖S(q)(f)‖Lp � Cp,q‖M(f)‖Lp, ∀ 1 � p < ∞.

Thus the martingale transform operator Q satisfies (4) in Theorem 3.1. Then the equivalence is obtained

immediately.

Theorem 4.4. Let B be a Banach space. For 1 < p � 2, the following statements are equivalent:

(1) B is isomorphic to a p-uniformly smooth space;

(2) For any r.i. space X with qX < ∞, there exists CX,p such that

‖M(f)‖X � CX,p‖S(p)(f)‖X ,

where f = (fn)n�1 is any B-valued martingale;

(3) For any r.i. space X with 1 < pX � qX < ∞, there exists CX,p such that

‖f‖X(B) � CX,p‖S(p)(f)‖X ,


Proof. Let �p(B)-valued martingale F = (Fn)n�1, Fn =∑n

k=1 Dk, Dk = (Djk)j�1, where Dj

k∈B, j�1.

We now consider a martingale transform operatorR from the family of �p(B)-valued martingales to that of

B-valued martingales. Let vk ∈ L(�p(B), B) be the operator defined by vkx = xk for x = (xj)j�1 ∈ �p(B).

Namely,

vk(x1, . . . , xk, xk+1, . . .) = xk, ∀x = (xk)k�1 ∈ �p(B).

Define the martingale transform operator R associated to the sequence vk by

(RF )n =n∑

k=1

vkdFk =n∑

k=1

vkDk =n∑

k=1

Dkk .

Now for any B-valued martingale f = (fn)n�1, fn =∑n

k=1 dfk, we can choose �p(B)-valued martingale

Fn =∑n

k=1 Dk, Dk = (Djk)j�1 with Dj

k = dfk if j = k and Djk = 0 otherwise. Namely,

Dk = (

k−1︷︸︸︷0, . . . , 0, dfk, 0, 0, . . .).

Then

(RF )n =

n∑k=1

Dkk =

n∑k=1

dfk = fn, M(RF ) = M(f)

and

‖Fn‖�p(B) = ‖(df1, df2, . . . , dfn, 0, . . .)‖�p(B) = S(p)n (f), M(F ) = S(p)(f).

Based on Lemma 4.2 and Theorem 3.1, we can deduce the desired results.

Corollary 4.5. Let B be a Banach space. Then the following statements are equivalent:

(1) B is isomorphic to a Hilbert space.

(2) For any r.i. space X with qX < ∞ and any martingale f with values in B, there exists CX such

that

C−1X ‖S(2)(f)‖X � ‖M(f)‖X � CX‖S(2)(f)‖X .

(3) For any r.i. space X with 1 < pX � qX < ∞ and any martingale f with values in B, there exists

CX such that

C−1X ‖S(2)(f)‖X � ‖f‖X(B) � CX‖S(2)(f)‖X .

Proof. It is well known that a space which is 2-uniformly smooth and 2-uniformly convex is isomorphic

to a Hilbert space.

Remark 4.1. If taking respectively X = Lp spaces and Orlicz spaces with proper conditions, we get

the results in [15,21,24]; furthermore, if taking B = R or C, we come back to the results in [8,16,23]; so

some well-known results are unified.


5 UMD Banach lattice

Definition 5.1. A Banach space B is said to be UMD if for any p, 1 < p < ∞, there exists Cp such

that

‖ε1df1 + · · ·+ εndfn‖Lp(B) � Cp‖df1 + · · ·+ dfn‖Lp(B), ∀n � 1

for any B-valued martingale f and all numbers ε1, ε2, . . . in {−1, 1}.This definition is due to Burkholder, see [4]. It is known that the existence of one p0 satisfying the

inequality is enough to assure the existence of the rest of p, 1 < p < ∞.

B will denote a Banach lattice in this section. Without loss of generality we assume that B is a

Banach lattice of measurable functions on some measure spaces (Θ, dθ). We refer to [14] for information

on Banach lattices. In the Banach lattice case, it is natural to consider the following variant of square

function S(2)(f).

Definition 5.2. Let B be a Banach lattice and f = {fn}n�1 a B-valued martingale. We define the

operators

Sf(ω) =

( ∞∑k=1

|dfk(ω)|2)1/2

.

On the one hand for every fixed ω ∈ Ω, Sf(ω) can be regarded as a B-valued function defined on Θ;

on the other hand ‖Sf‖ can be seen as the norm of the element (df1, df2, . . .) in the Banach space

B(�2) =

{(x1, x2, . . .) :

∥∥∥∥( ∞∑

k=1

|xk|2)1/2∥∥∥∥

B

< ∞}.

B(�2) is also a Banach lattice when B is a Banach lattice.

The following lemma is well known; see [22].

Lemma 5.3. Given a Banach lattice B, the following statements are equivalent:

1) B satisfies the UMD property.

2) For any B-valued martingale f = (fn)n�1, there exists Cp such that

C−1p ‖M(f)‖Lp(B) � ‖Sf‖Lp(B) � Cp‖M(f)‖Lp(B), 1 � p < ∞.

Now we can prove the following characterization of UMD Banach lattices.

Theorem 5.4. Let B be a Banach lattice. Then the following statements are equivalent:

(1) B satisfies the UMD property.

(2) For any r.i. space X with qX <∞, there exists CX such that for any B-valued martingale f =

(fn)n�1,

C−1X ‖M(f)‖X � ‖Sf‖X(B) � CX‖M(f)‖X.

(3) For any r.i.space X with 1 < pX � qX < ∞, there exists CX such that for any B-valued martingale

f = (fn)n�1,

C−1X ‖f‖X(B) � ‖Sf‖X(B) � CX‖f‖X(B).

Proof. The proof follows the argument on those of the preceding theorems with some technical modi-

fications. In order to see that (1) ⇒ (2), let vk ∈ L(B,B(�2)) be the operator defined by vkx = {xj}∞j=1

for x ∈ B, where xj = x if j = k and xj = 0 otherwise. We consider the martingale transform operator

Qf = {(Qf)n}n�1, defined by

(Qf)n =

n∑k=1

vkdfk = (df1, df2, . . . , dfn, 0, . . .).

Then

M(Qf) = supn

‖(Qf)n‖B(�2) = ‖Sf‖B.


B satisfies the UMD property, by Lemma 5.3,

C−1p ‖M(f)‖Lp(B) � ‖M(Qf)‖Lp(B) � Cp‖M(f)‖Lp(B), 1 < p < ∞.

Hence the martingale transform operator Qf satisfies (4) of Theorem 3.1, it follows the right inequalities

in (2). For the left inequalities, we now consider the martingale transform operator from the family

of B(�2)-valued martingales to that of B-valued martingales. Let vk ∈ L(B(�2), B) be the operator

defined by vkx = xk for x = (xj)j�1 ∈ B(�2). For any B(�2)-valued martingale F = (Fn)n�1, Fn =∑nk=1 Dk, Dk = (Dj

k)j�1, where Djk ∈ B, j � 1, define the martingale transform operator R:

(RF )n =

n∑k=1

vkDk =

n∑k=1

Dkk .

B satisfies the UMD property, we know that for any B-valued martingale f, ‖f‖L2(B) � ‖Sf‖L2(B). In

particular,

‖RF‖L2(B) � ‖SRF‖L2(B).

By Lemma 2.9 in [19],

‖RF‖L2(B) � ‖F‖L2(B(�2)).

Hence

‖M(RF )‖L2(B) � C‖M(F )‖L2(B(�2)),

which shows that the martingale transform operator R satisfies (4) of Theorem 3.1. Now for any B-

valued martingale f, fn =∑n

k=1 dfk, we choose B(�2)-valued martingale Fn =∑n

k=1 Dk, Dk = (Djk)j�1

with Djk = dfk if j = k and Dj

k = 0 otherwise. Then

(RF )n = fn, M(RF ) = M(f); ‖Fn‖B(�2) = Snf, M(F ) = Sf.

Again according to Lemma 5.3 and Theorem 3.1, we get the left inequalities in (2).

The proof of (2) ⇒ (3) is almost plain, which is similar to that of (2) ⇒ (3) in Theorem 3.1.

(3) ⇒ (1) is easy, and just take X = Lp, 1 < p < ∞. The proof is complete.

6 Fefferman and Stein theorem

We first recall the well-known Fefferman and Stein theorem on the Hardy-Littlewood maximal operator

(see [7]). Let f = (f1, f2, . . .) be a sequence of random variables on Rn, then for 1 < r, p < ∞,

∥∥∥∥( ∞∑

j=1

|M(fj)(·)|r)1/r∥∥∥∥

Lp(Rn)

� Cr,p

∥∥∥∥( ∞∑

j=1

|fj(·)|r)1/r∥∥∥∥

Lp(Rn)

,

where the Hardy-Littlewood maximal function M(fj) is defined by

M(fj)(x) = sup1

|Q|∫Q

|fj(y)|dy,

where the “ sup ” is taken over all cubes Q centered at x. In this section we will obtain the martingale

version of the Fefferman and Stein theorem in the r.i. spaces setting.

Theorem 6.1. Let X be an r.i. function space with 1 < pX � qX < ∞, f j = (f jn)n�1, j ∈ N, a

sequence of B-valued martingales. Then for any 1 < p < ∞, there exists CX,p such that

∥∥∥∥( ∞∑

j=1

(M(f j))p)1/p∥∥∥∥

X

� CX,p supn

∥∥∥∥( ∞∑

j=1

‖f jn‖pB

)1/p∥∥∥∥X

.


In order to prove this theorem, we need to introduce a new martingale transform operator T . Let

B1 and B2 be two Banach spaces. Suppose that T is a martingale transform operator, f j = (f jn)n�1 a

B1-valued martingale, j = 1, 2, . . . , and F = (Fn)n�1 an �p(B1)-valued martingale, where Fn = (f jn)j�1

and 1 < p < ∞. Define the new operator T : �p(B1) → �p(B2) by

(TF )n = {(Tf j)n}j�1.

Lemma 6.2. The operator T defined as above is a martingale transform. Moreover, if T satisfies any

of the statements in Theorem 3.1, then for any r.i. space X with 1 < pX � qX < ∞, there exists CX,p

such that

‖M(TF )‖X � CX,p‖F‖X(�p(B1)), 1 < p < ∞.

Proof. Let υk ∈ L(B1, B2) be the multiplying sequence of the martingale transform operator T. Given

Fn =∑n

k=1 Dk = (∑n

k=1 dfjk)j�1 = (f1

n, f2n, f

3n, . . .) ∈ �p(B1), we have

(TF )n = {(Tf j)n}∞j=1 =

{ n∑k=1

υkdfjk

}∞

j=1

=n∑

k=1

VkDk,

where Vk = {vk, vk, vk, . . .} ∈ �∞(L(B1, B2)) ⊂ L(�p(B1), �p(B2)), and ‖Vk‖�∞(L(B1,B2)) = ‖υk‖L(B1,B2).

Therefore, T is a martingale transform operator. Furthermore, if T satisfies any of the statements in

Theorems 3.1, then

‖T (f j)‖Lp(B2) � ‖M(T (f j))‖Lp � Cp‖f j‖Lp(B1).

Consequently,

‖TF‖Lp(�p(B2)) = supn

‖(TF )n‖Lp(�p(B2))

= supn

‖{(Tf j)n}∞j=1‖Lp(�p(B2))

= supn

∥∥∥∥( ∞∑

j=1

‖(Tf j)n‖pB2

)1/p∥∥∥∥Lp

� Cp supn

∥∥∥∥( ∞∑

j=1

‖f jn‖pB1

)1/p∥∥∥∥Lp

= Cp‖F‖Lp(�p(B1)).

According to the Doob inequality,

‖M(TF )‖Lp � Cp‖M(F )‖Lp .

It follows that T satisfies (4) of Theorem 3.1 with B1 replaced by �p(B1) and B2 by �p(B2). Then the

lemma follows.

Proof of Theorem 6.1. We consider a martingale transform operator T from the family of B-valued

martingales to that of �∞(B)-valued martingales. Let vk ∈ L(B, �q(B)) be the operator defined by

vkx = (xj)j�1 for x ∈ B, where xj = x if j � k and xj = 0 otherwise; namely,

vkx = (

k−1︷︸︸︷0, . . . , 0, x, x, x, . . .).

T is the martingale transform associated to the sequence (vk):

(Tf)n =

n∑k=1

vkdfk =

(df1, df1 + df2, . . . ,

n∑k=1

dfk,

n∑k=1

dfk, . . .

)= (f1, f2, . . . , fn, fn, . . .),

where f is any B-valued martingale. Then

M(Tf) = supn�1

‖(Tf)n‖�∞(B) = supn�1

‖fn‖B = M(f).


By the weak (1, 1) type inequality of the Doob maximal operator,

λP(M(f) > λ) � ‖f‖L1(B) � ‖M(f)‖L1, ∀λ > 0,

which implies that T satisfies (1) of Theorem 3.1 with B1 = B,B2 = �∞(B), so T defined as above

satisfies Lemma 6.2. Namely, for 1 < p < ∞,

‖M(TF )‖X � CX,p‖F‖X(�p(B))

= CX,p supn

‖Fn‖X(�p(B))

= CX,p supn

‖(f1n, f

2n, . . . , f

jn, . . .)‖X(�p(B))

= CX,p supn

∥∥∥∥( ∞∑

j=1

‖f jn‖pB

)1/p∥∥∥∥X

.

On the other hand, by the definition of T ,

M(TF ) = supn�1

‖(TF )n‖�p(�∞(B))

= supn�1

( ∞∑j=1

‖(Tf j)n‖p�∞(B)

)1/p

=

( ∞∑j=1

(M(f j))p)1/p

= ‖M(f j)‖�p .

Thus we get the desired result. The proof is complete.

Acknowledgements This work was supported by National Natural Science Foundation of China (Grant

No. 11001273), Research Fund for International Young Scientists (Grant No. 11150110456), Research Fund

for the Doctoral Program of Higher Education of China (Grant No. 20100162120035) and Postdoctoral Science

Foundation of China and Central South University. The authors acknowledge the two anonymous referees for

their useful comments and the first author is grateful to Professor J. L. Torrea for some helpful communications.

References

1 Bennett C, Sharpley R. Interpolation of Operators. New York: Academic Press, 1988

2 Boyd D. Indices of function spaces and their relationship to interpolation. Canad J Math, 1969, 21: 1245–1254

3 Burkholder D L. Martingale transform. Ann Math Stat, 1966, 37: 1494–1504

4 Burkholder D L. A geometrical characterization of Banach spaces in which martingale difference sequence are uncon-

ditional. Ann Probab, 1981, 9: 997–1011

5 Burkholder D L, Gundy R F. Extrapolation and interpolation of quasilinear operators on martingales. Acta Math,

1970, 124: 249–304

6 Diestel J, Uhl J J. Vector Measures. Mathematical Surveys and Monographs, vol. 15. Providence, RI: Amer Math

Soc, 1977

7 Fefferman C, Stein E. Some maximal inequalities. Amer J Math, 1971, 93: 107–115

8 Garsia A M. Martingale Inequalities: Seminar Notes on Recent Progress. Math Lecture Note Series. New York:

Benjamin, 1973

9 Jiao Y. Carleson measures and vector-valued BMO martingales. Probab Theory Related Fields, 2009, 145: 421–434

10 Jiao Y. Embeddings between weak Orlicz martingale spaces. J Math Anal Appl, 2011, 378: 220–229

11 Johnson W B, Schechtman G. Martingale inequalities in rearrangement invariant function spaces. Israel J Math, 1988,

64: 267–275

12 Ktkuchi M. New martingale inequalities in rearrangement function spaces. Proc Edinb Math Soc, 2004, 47: 633–657

13 Ktkuchi M. On the Davis inequalities in Banach function spaces. Math Nachr, 2008, 281: 697–709

14 Lindenstrauss J, Tzafriri L. Classical Banach Spaces II. Berlin: Springer-Verlag, 1979


15 Liu P D. Martingale and the Geometry of Banach Spaces. Beijing: Science Press, 2007

16 Long R L. Martingale Spaces and Inequalities. Beijing: Peking University Press, 1993

17 Martinez T, Torrea J L. Operator-valued martingale transforms. Tohoku Math J, 2000, 52: 449–474

18 Martinez T, Torrea J L. Boundedness of vector-valued martingale transforms on extreme points and applications. J

Aust Math Soc, 2004, 76: 207–221

19 Martinez T, Torrea J L. Uniform convexity in terms of martingale H1 and BMO spaces. Houston J Math, 2001, 27:

461–478

20 Novikov I Ya. Martingale inequalities in rearrangement invariant function spaces. In: Function Spaces. Teubner-Texte

Math, 120. Stuttgart: Teubner, 1991, 120–127

21 Pisier G. Martingale with values in uniformly convex spaces. Israel J Math, 1975, 20: 326–350

22 Rubio de Francia J L. Martingale and integral transform of Banach space valued functions. In: Lecture Notes in

Mathematics, vol. 1221. Berlin-New York: Springer-Verlag, 1986

23 Weisz F. Martingale Hardy Spaces and Their Applications in Fourier Analysis. In: Lecture Notes in Mathematics, vol.

1568. New York: Springer-Verlag, 1994

24 Yu L. Orlicz norm inequalities for operator-valued martingale transforms. Stat Probab Lett, 2008, 78: 2664–2670

operator-valued martingale transforms in rearrangement invariant spaces and applications

Documents