on weighted inequalities for martingale transform operators

Math. Nachr. 251, 52 – 63 (2003) / DOI 10.1002/mana.200310030

On weighted inequalities for martingale transform operators

Teresa Martınez�1

1 Departamento de Matematicas, Universidad Autonoma de Madrid, 28049 Madrid, Spain

Received 3 July 2001, revised 30 January 2002, accepted 1 February 2002Published online 24 February 2003

Key words Weighted inequalities, martingales, martingale transform operators

MSC (2000) Primary: 60G42; Secondary: 60G46, 46B09

For � � � � �, the almost surely finiteness of ��

�

��

�is a necessary and sufficient condition in order

to have almost surely convergence of the sequences �� with � � �� . This condition is alsoequivalent to have weighted inequalities from �� into �� for some weight � for Doob’s maximalfunction, square function and generalized Burkholder martingale transforms. Similarly, �� turnsout to be necessary and sufficient for the above weighted inequalities to hold for some �.

1 Introduction

Let �� be a probability space and �� be an stochastic basis, that is, a nondecreasing sequenceof sub–�–fields of � . Given a positive measurable function � , the conditional expectation of � is denotedby �� and defined for each � � � as the (possibly infinite) Radon–Nikodym density of the measure��

�� , � � ��, see [10]. It is well–known that if � � �� , � � � �, for every � � � the

linear operators �� , defined as �� , are finite almost surely and verify�� almost surely. This result is a consequence of the following inequality, due to Doob, see[12]: �

��

��

��

��

��

where �� is the so–called Doob’s maximal operator defined by �� .We shall say that an �–measurable function is a weight if � � �� almost surely. The initial question

we pose is the following: given , � � � �, and a weight , find necessary and sufficient conditions for such that for all � � �� , the operators �� are finite almost surely and�� almost surely. We shall derive the necessary and sufficient conditions for from solvingone of the two following questions, which, as we will prove, turn out to be equivalent:

Question A. Given , � � � � and a weight , find necessary and sufficient conditions for in order toguarantee the existence of a weight � such that for every � � ��

��

� ��

��

��

��

Question B. Given , � � � � and a weight , find necessary and sufficient conditions for in order toguarantee the existence of a weight � such that �� is bounded from �� into �� .

� e–mail: [email protected], Phone: +34 91 396 76 42, Fax: +34 91 397 48 89

c� 2003 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 0025-584X/03/25103-0052 $ 17.50+.50/0

Math. Nachr. 251 (2003) 53

Question B is a particular case of the following

Problem I. Given , � � � �, a pair of Banach spaces �� and �� and a fixed operator � boundedfrom

��

�� into ��

�� , find necessary and sufficient conditions for a weight in order to guarantee theexistence of a weight � such that � becomes bounded from

��

� �� into ��

�� .

This problem is nowadays classical in the context of Euclidean Harmonic Analysis. It was first formulated inthe scalar case by B. Muckenhoupt and solved for particular operators by several authors, see [22], [8] and [16].The problem has been considered also in Ergodic Theory, see [14]. This problem can be approached in two ways:constructively and non constructively. With the first method, the specific weight � which solves the problem isconstructed, as in [8] and [17]. The second, nonconstructive method, proves the existence of a weight � by usingfactorization results. This last method, that can be applied in the vector valued context of Bochner–Lebesguespaces �

�� , comes from Nikishin, see [26], and was extensively developed by Rubio de Francia, see [28].

In this work, it is showed that, under certain conditions, weighted inequalities and vector valued inequalities areequivalent. Thus, the non constructive method to find a weight and vector valued inequalities are strongly related.

The purpose of this paper is to solve Problem I for several operators, namely Doob’s maximal operator (inparticular, our aim will be to find the answer to Questions A and B) and martingale transform operators as inDefinition 1.1.

Weighted inequalities for Doob’s maximal function have been considered in the case � � for several authors.In [19] it is proved that the so called �� condition, namely

��

��

�

��

��

�

�

�� (��)

is necessary in order to have boundedness from �� into �� , � � � � of Doob’s maximalfunction. Condition �� is also sufficient in the case of �–adic filtration (see [2]). But in general, an extracondition is needed, see [3], [19], [29]. As well, in [27], it is proved that given two weights � and such that

�� , then ��

��

�

��

�must be finite. For further results

in �� weights, see [9], [18], [20], [21] and [30].Our response to Question A is given in Theorem 3.1. Concretely, we find in a constructive way that the

necessary and sufficient condition for the weight is

��

��

�

�� (1.1)

By using the nonconstructive method developed by Rubio de Francia we solve Question B in Theorem 3.5.Remark 3.6 shows that the condition that we find for Question A is again the same condition (1.1) than forQuestion B and then both questions are equivalent.

Moreover in Theorem 3.3 we prove that given , � � � � the condition (1.1) above for the weight isequivalent to the fact that for each � � �, the operators �� are well defined on �� and continuous inmeasure.

Coming back to classical results in probability, it is well–known that the speed of convergence of the sequence�� for functions � in �� depends on . In fact for each in the range � � � �� and each

� � �� the square function ��

��

is finite almost surely, see [1].Besides, see [6], there exists a constant �� such that �� , and it makes sense to poseProblem I for the square function operator �. We observe that the square function operator can be described as themaximal of an ��–valued martingale transform operator, see Section 3.2 in [25]. Then, our next aim is to considerProblem I in the abstract setting of vector valued martingale transform operators. We recall that given a Banachspace �, a sequence of �–valued random variables �� is called a �–valued martingale relative to ��if each function �� is strongly ��–measurable, Bochner integrable and verifies �� . We observethat the conditional expectation of an integrable vector valued function is well defined by density. For a completestudy of scalar–valued martingales see [24] and [31], and for the vector–valued context, we refer the reader to thebook by Diestel and Uhl [11]. The martingale �� is said to be �–bounded if ��

�� ,

where ��

��

��

. Vector valued martingale transform operators are defined as follows:

54 Martınez: Weighted inequalities for martingale transforms

Definition 1.1 Let �� and �� be two Banach spaces, �� a stochastic basis in a probability space�� , and �� a ��–valued martingale relative to ��. Define �� a sequence of ��–valuedrandom variables such that each �� is ��–measurable, � � �, �� is ��–measurable, and �� .Such a sequence � � �� will be called a multiplying sequence. The martingale �� given by ��

��

�� where �� , is called the martingale transform of �� by the multiplying sequence �.

� will denote the martingale transform operator.

Given , � � � �, we shall say that a vector valued martingale transform operator � as above is �–bounded if there exists a constant �� such that ��

�� , where ��

�� . In [7], Burkholder considered the case where �� and each operator in

the multiplying sequence �� is a scalar valued function. He characterized the spaces in which these martingaletransform operators are �–bounded and called them ��. See [4], [5], [6], [7] and [15] for the main resultsconcerning these spaces.

Let us note that if a function � belongs to ��

�� , with � � � �, then the sequence ��, with�� is an �–bounded martingale. If � is a vector valued martingale transform operator, we have��

�� . Hence, in these cases, it makes sense to pose Problem I for ��

with the target Banach space for � in Problem I being the scalar field.

Observe that the first problem is to define the conditional expectation for functions � in ��

� �� for a givenweight . As the definition of conditional expectation in

��

� �� has to be an extension of the correspondingscalar conditional expectation, it is clear that the condition (1.1) is necessary. In Theorem 4.1 we find that it isalso sufficient.

Problem I in the case �� is solved in Theorem 4.3 by using the nonconstructive method and againwe find that the right condition for is (1.1). In particular this solves Problem I for the square function, seeSection 5.2.

The weight � whose existence is guaranteed in all the former results, can be seen to satisfy the condition�� almost surely. This condition is in fact the solution, found by the nonconstructive method togetherwith some duality arguments, to the following Problem II, in the case �� where � is an �–boundedvector valued martingale transform as above.

Problem II. Given , � � � �, a pair of Banach spaces �� and �� and a fixed operator � boundedfrom

��

�� into ��

�� , find necessary and sufficient conditions for a weight � in order to guarantee theexistence of a weight such that � becomes bounded from

��

� �� into ��

�� .

We expose some preliminary material in Section 2. Doob’s maximal weighted inequalities are studied inSection 3. Section 4 is devoted to vector–valued general results, whose applications to maximal function, squarefunction and Burkholder martingale transforms are collected in Section 5.

2 Preliminaries

For a function � � �� its conditional expectation �� cannot be defined as the Radon–Nikodymdensity of the measure ��

�� , since that quantity need not to even exist for every set � � ��. It is

necessary then to set a proper definition for conditional expectation defined over �� . We shall speak ofa conditional expectation operator �� with respect to a sub–�–algebra �� (which we will shortly denoteby �� ) defined over �� if for every � � �� , �� is an almost surely finite, ��–measurablefunction, such that for � � �� with �� , �� coincides almost surely with the Radon–Nikodymderivative of the measure ��

�� defined on ��. For general � � �� , we define

��

This definition immediately assures that �� is a linear operator and that it verifies ��

��

�� ,

with � � ��, and �� almost surely, for any � � �� .

Math. Nachr. 251 (2003) 55

Classical results are the weak �� and strong �� , � � � �, types for Doob’s maximal function of amartingale, both concepts defined in the Introduction (see [12]). It can also be proved, see [25],��

�

��

��

� � ��

��

�

��

��

� (2.1)

where the norm in �� is defined as ��

��

�� denotes the space of all scalar valued measurable functions on �� , provided with the topologyof the convergence in measure (see [16]). �

�� is the same space, but consisting of vector–valued functions.

Observe that any � � �� is well defined almost surely. A result that will be useful in the sequel is the

following one.

Theorem 2.1 (Nikishin) Let �� be an arbitrary measure space, �� a �–finite measure space, and� � �� a sublinear continuous operator, � � � �. Then, there exists a function � � � a.s.such that

��

� ��

��

��

3 Weighted inequalities for Doob’s maximal function

We start by constructing the weight � that solves Question A.

Theorem 3.1 Given , � � ��, a weight and an stochastic basis ��, there exist all the conditionalexpectation operators �� over � � �� and a weight � such that for some constant ��

��

� ��

��

��

�� (3.1)

for every � � �, if and only if

��

��

�

�� a.s. (3.2)

P r o o f. (3.2) is a consequence of Radon–Nikodym theorem if the measure ��

� �� is �–finite on ��.

Since ��

� � �� and � � � � a.s., we have � � ��

� �

�

�

�� a.s. and therefore � �

��,

��

� �

�

�

��

�. Applying (3.1) we have that

��

� �� is finite for every , and then

Radon–Nikodym theorem and the fact � � � a.s. give � � �� almost surely.

Remark 3.2 Observe that condition �� almost surely can be obtained from a weak inequality likethe following one�

��

� ��

��

��

��

Once we have that �� almost surely, the last step is showing that ��

� has finite integral over allthe sets of the disjoint partition of � given by !� �

��

�� . Indeed, for each we have

��

��

� ��

��

��

��

� ��

� �

��

� " ��


where the supremum is taken over all positive functions " � �� with "�� . By using that!� � ��, we have�

�

��

� " ��

��

� �

�

� "��

� �

�

� ��

� �

�

� "��

� ��

Equation (3.1) and the fact that "�� give

� ��

� �

�

� "��

� ��

��!��

��

� �

�

� "��

��

��!��

�

��

��

Observe that ��!��

��

�� !�� !�� if � � ��!��

�� and it is �� otherwise. Splitting

the last integral according to this, we have� �

��

� " �� !�� , which is finite.

For the converse, let � � ��

��

�

��

��

. This function is finite and positive almost surely, since for any

� � �, the function ��

� has finite integral on the sets of the partition of � given by

��

� �

��

�

�� , for integer. Moreover, the �–martingale convergence

applied locally in each of the ��’s, gives us � �� almost surely. Then, the function � � �� is analmost surely finite and strictly positive function, that is, a weight. Given � � �� , note that by conditionalH�older inequality

��

��

�

� ��

�

��

��

�

��

��

� � ��

Take � � � and # � �� #� , where #� � ��

are disjoint sets. For each $, we have��

� ��

��

��

��

�

��

��

��

��

�

��

��

��

By summing up from $ � � to infinity, the desired Doob’s inequality is obtained. �

In fact, when the weight satisfies (3.2) it is possible to give an alternative formulation of �� , which iscontinuous in measure, as the proof of the next theorem shows.

Theorem 3.3 Given , � � � �, and a weight , there exists a unique continuous in measure conditionalexpectation operator with respect to the sub–�–algebra � � , �� , if and only if

��

��

�

�� almost surely.

P r o o f. Assuming the condition on the weight, let ��

� �

��

�

�� and define for

� � �� , ��

��

��almost surely. Let ��

��

��. Then, by

H�older’s inequality, ��

�

� ��

� �� Thus finite linear combinations of

�� are continuous from �� into �� . In particular, they are continuous in measure. Since ��

��

�� is finite almost surely, by Banach continuity principle (see Proposition VI.1.4 in [16]),

�� is also a continuous in measure operator.Conversely, by Nikishin theorem, we have

��

� ��

��

��

This inequality, used in the same way that inequality (3.1) was in the proof of Theorem 3.1, gives the desiredcondition on the weight . �

As we pointed out in the Introduction, we shall solve Question B by the non–constructive method, for whatwe need the following result that can be found in [13].

Math. Nachr. 251 (2003) 57

Theorem 3.4 Let �� be a measure space, # and % be Banach spaces, and ��be a disjoint parti-tion of � and & and be such that � � & � ��. Assume that � is a sublinear operator satisfying��

�

��

��

� ��

�

��

��

�

for any integer , where �� &� �%� # �. Then, there exists a constant � � ��&� �%� # � and a weight �such that for any � � # ,�

��

holds, where � is a constant depending on # and %.

The answer to Question B is the following theorem, and the remark after it makes clear the equivalencebetween Questions A and B.

Theorem 3.5 Given , � � � �, a weight and a stochastic basis ��, condition

��

��

� ��

�� almost surely is necessary and sufficient in order to have conditional expectation opera-

tors �� continuous in measure for any � � �. Moreover, these operators verify

(i) There exists a constant � � � and a weight � such that, for any � � �� ,��

�� (3.3)

In particular, the weight � satisfies �� almost everywhere.(ii) For any � � �� , � � �� almost everywhere.

Remark 3.6 By a stopping time argument, we can obtain Equation (3.1) from Equation (3.3): given � �� and � � �, define the stopping time ' � �� , with ��Ø� � �. Then, byusing that �� ' ��, that �' � � �

��

��

��

and Equation (3.3), we have

��

� ��

��

�

��

� ��

��

��

��

��

P r o o f of Theorem 3.5. The necessity and sufficiency of condition ��

��

�

�� a.s. in order to have

continuous in measure conditional expectation operators follows from Theorem 3.3. Let ��

� �

��

� ��

��

�� define a disjoint partition of �. Then, for each � � �� ,�

�� and ��

��

��. By using Kolmogorov’s inequality, see Lemma V.2.8 in [16], and

the vector–valued weak type inequality for the maximal operator (2.1), we get, for � � & � ��

�

��

��

��

�

��

�

�� &��

��

�

��

��

��

� ��

��

�

��

��

��

�

Remark 3.7 Conditional expectation is defined for vector–valued random variables by using Bochner integralproperties, see [11] . From these properties, one can verify that for any vector–valued integrable function �

�� . In the particular case of functions � � �� a.s., Bochner integral

properties imply that �� a.s.


Therefore, by using this remark and H�older’s inequality, we have

��

��

��

��

��

��

� ��

�

��

��

�

Pasting together the last inequalities and Theorem 3.4 we get the desired weight �, that satisfies �� a.s. by applying Remark 3.2. In order to prove (ii), first step is to observe that ��, which is the supremum ofcontinuous in measure operators and verifies (i), is also continuous in measure by Banach continuity principle.Then the result comes easily by noting that (ii) is true for functions in �� , which is a densesubspace of �� . �

4 Vector–valued results

The discussion in Section 2 in order to define the scalar–valued conditional expectation operator cannot be ex-tended to the vector–valued case. In the case of �–valued integrable functions, its conditional expectation isdefined by density, as an operator continuous in �

�. In particular, it verifies that if � is �–valued integrable

function and � is a �–measurable function whose values are linear continuous operators acting on �, then�� . In the case of a function in

�� , its conditional expectation will also be defined as

a continuous operator: we shall say that the operator �� is a vector–valued conditional expectation operatorwith respect to the sub–�–algebra � � over �

�� if for any � �

��

�� , �� coincides

almost surely with its conditional expectation as a function of �� , and �� is continuous in measure. In

particular, Egorov’s theorem allows us to consider also almost surely convergence, and this gives that �� isa linear operator, �� is �–measurable and for any � � �, �

��

�� Also, for any � � �

such that ��

�� then

��

� ��

��

��

since for this set � � �, �� . In the Banach valued case, (3.2) is also a necessary and sufficient

condition in order to have conditional expectation operators.

Theorem 4.1 Given , � � ��, a weight and� a Banach space, there exists a vector valued conditionalexpectation operator with respect to the sub–�–algebra � � , �� , defined on

�� if and only if the

weight satisfies ��

��

�

�� almost surely.

P r o o f. Suppose ��

��

�

�� almost surely and define the disjoint partition of � given by the sets

��

� �

��

�

�� . For every � �

�� , ��

� �� and the operator

��

��

��

is well defined. By Banach continuity principle, used as in the proof ofTheorem 3.3, this operator is the continuous in measure conditional expectation operator that we were lookingfor.

Conversely, let ( � � with (�

� � be fixed. Then, for � � �� , (� � �� and ��(� ��

( �� in measure, where �� is a sequence converging to � in�� . Thus, �� in measure defines a continuous in measure conditional expec-tation operator in �� . By applying Theorem 3.3, the condition for the weight is obtained. �

Observe that the former theorem gives us an explicit formula for vector valued conditional expectation opera-

tors. For a stochastic basis ��, we will denote by �� the operator�� . Condition ��

��

�

��

� a.s. is again the right condition in order to have conditional expectation operators �� for every � � �.

Corollary 4.2 Given , � � � �, a Banach space �, a weight and a stochastic basis ��,

condition ��

��

�

�� a.s. is necessary and sufficient in order to have conditional expectation operators

Math. Nachr. 251 (2003) 59

��

�� for any � � �. Moreover, if we define ��

��

� �

��

�

��

��, these operators have the form

��

��

��

�

As we announced in the Introduction, next step is solving Problem I for for �� , when � is an�–bounded martingale transform operator, � � � �, as in Definition 1.1. In this case, � has an ��–valuedextension, � � � � �, in such a way that given a �

��

–valued martingale ��

��

��

almost surely, we have (see Corollary 1 in [25])��

�

��

��

� � ��

��

�

��

��

� (4.1)

The vector valued version of inequality (2.1) also holds, since it is a particular case of Equation (4.1) (see theproof of Theorem 4.5).

Theorem 4.3 Consider , � � � �, a weight satisfying ��

��

�

�� almost surely and an

�–bounded martingale transform operator � , with multiplying sequence ��. Then, there exist a constant� � � and a weight � � � almost surely such that�

��

��

��

��

� ��

In particular, if �� is a constant operator, �� almost surely.

P r o o f. By Theorem 4.1, the operators ��

� ��

�� are continuous in measure. Let

��

� �

��

�

�� . For � � & � �, Kolmogorov’s inequality (Lemma V.2.8 in [16])

gives us ��

�

��

��

��

��

�

��

�

�� &��

��

�

��

��

��

�

Using that �� for any � � � and that the functions involved are almost surely finite, we have that��

��

�� Now, ��

� �� and � is �–bounded, so Equation (4.1) gives us��

�

��

��

��

� ��

��

�

��

��

��

��

� ��

�

��

� � �

��

also by using Remark 3.7. Thus, Theorem 3.4 gives us the weight �. If �� is a constant operator,there exists � � �� such that ��

� �. With fixed �, the function � ��

� � ��

� �� and there-

fore ��

��

�

�

��

�

��

� � almost surely, that is, � � ��

� �

�

�

�� a.s. Since��

��

�

�

��

� ��

� �

�

�

��

� �

�

�

�� the sets #� �

��

��

��

�

�

��

� ��

��, � �, are a disjoint partition of �. Since, by Theorem 3.4,

��

� ��

��

��

�

�

��

� ��

��

��

��

�

��

��

an application of Radon–Nikodym theorem gives � � �� almost surely. �


Remark 4.4 Observe that the following weak inequality, when it is verified for every � � �� , would

have been enough in order to prove that �� almost surely:��

��

�

��

��

��

Finally, we will deal with the dual problem, that is, Problem II in the case �� , with � an�–bounded vector–valued martingale transform operator as in Definition 1.1. For such a martingale trans-form operator, given � � �, the operator ��, defined as �� is bounded from

��

into ��

� Let

��

��

�

��

��be its adjoint operator, that is, ��"�� "� �� for � �

��

and " � ��

��

�

. Theproperties of conditional expectation and measurability conditions on � imply

�"� ��

��

�� "� ��

�

��

��"��"��

�

� ��

��

��"��"��

��

where �� associates to each � the adjoint of ��. It is clear that ��

��

��

�� Moreover �� is ��–measurable: for any ) � �� and ( � ��, the properties of conditional

expectation give,

�� )�� (� � ��)�� (�� )��(�� )�� (��

These computations show that �� is a multiplying sequence, and that ��" is the function

��" �

��

��"��"��

Moreover if we denote by � the martingale transform operator whose multiplying sequence is �� then��"�� "� for each � � � and martingale �"��. It is clear that � is a martingale transform opera-tor as in Definition 1.1 with associated Banach spaces �� and ��, somehow the “dual” martingale transformoperator of � .

The solution to Problem II in the case �� is given in the following theorem.

Theorem 4.5 Given , � � � �, a weight � and a martingale transform operator � as in Definition 1.1with multiplying sequence ��, such that �� is a constant operator, the following sentences areequivalent:

(i) �� a.s.,(ii) There exists a weight and a conditional expectation operator �� defined on

��

� �� for every� � �, such that for some constant � � �,� �

��

��

��

��

(iii) There exists a weight and a conditional expectation operator �� defined on ��

� �� for every� � �, such that for some constant � � �,

��

��

��

�

��

��

��

In particular, the weight in each statement verifies ��

��

�

�� a.s.

Math. Nachr. 251 (2003) 61

P r o o f. Since (iii) � (i) comes from Remark 4.4, we shall prove just (i) � (ii). To this aim, define � to bethe “dual” martingale transform operator considered above. Since �� is a constant operator and

��

��

�

��

�� a.s., we can apply Theorem 4.3 to the martingale transform operator �

and we get a weight * such that��"��

�

* ��

�"

��

��

�

��

� �� (4.2)

Let be such that � � ��

� � * � � almost surely. In particular, by Remark 4.4, we have ��*��

��

��

�

�� a.s., and by Corollary 4.2,�� is a well defined finite a.s. function for any � � ��

� ��

By duality ��

��

� ��

��

��

��

�

� �"� ��

�

�

��

�� (4.3)

since for such a ", by H�older’s inequality and Equation (4.2) we get� �"� ��

�

�

��

� ��

�"�

�

�

� �

�

� � � �

�

��

��

Now, consider� any Banach space and the martingale transform operator � given by the multiplying sequence

�*��, where *��

��

�for � � �. Then, for a �–valued martingale � �

��, bounded in �, � � ��, Doob’s inequality gives us

��

� ��

� ��

� (4.4)

that is, � is an �–bounded martingale transform operator as in Definition 1.1, with �� , �� This

implies that the martingale transform operator �� , given by ��

�� *�� , verifies thehypothesis in Theorem 4.5, and therefore, by using the part already proved (see Equation (4.3)), there exists aweight such that

��

��

��

� ��

��

��

��

By (4.4), the left–hand side of this inequality equals�� and (ii) is proved. Finally, Corollary 4.2

applied to the weight in each of the statements in this theorem gives us the condition ��

��

�

�� a.s. �

5 Applications

5.1 Doob’s maximal function

Theorem 5.1 The necessary and sufficient condition on the weight � in order to have some weight such that

inequalities (3.1) and (3.3) hold, is �� a.s. Moreover, the weight is such that ��

��

�

�� a.s.

P r o o f. The ideas developed in the proof of (i) � (ii) of Theorem 4.5 allow us to see Doob’s maximalfunction as a maximal of a ��–valued martingale transform operator �� see (4.4). The key point in this proofis that �� and �� By applying Theorem 4.5 to the martingale transform operator ��

we have that �� a.s. is equivalent to �3.3�, which in particular implies ��

��

�

�� almost

surely. We could repeat the constructive argument given in the proof of Theorem 3.1 to obtain (3.1). From Doob’sinequality (3.1), Remark 4.4 gives us condition �� a.s. �

Theorem 5.1 remains valid when Banach–valued functions are considered. The proof is straightforward.


5.2 Square function

Given an scalar–valued martingale � � �� the martingale square function of � is defined as

��

��

��

��

�

��

��

��

By orthogonality of the differences �� , it is clear that

��

��

��

��

��

�

��

��

��

��

��

� �� (5.1)

Given the scalar–valued martingale ��, we consider the vector–valued martingale transform operator +

whose multiplying sequence is defined as ��

�for real �. Then �+��

�� for any real valued martingale �� and �+�� .Therefore, �� is the maximal of a martingale transform operator as in Definition 1.1 that, by (5.1), is bounded in�. Thus it is �–bounded for � � � �, see [25]. Since �� is a constant operator, Theorems 4.3 and 4.5 are

applied to square function and then we have that the conditions ��

��

�

�� a.s. and �� a.s.

are necessary and sufficient in order to have boundedness of the square function � from �� into �� in any of the senses contained in those theorems. These results may be extended to the vector–valued setting incase that the space� is isomorphic to a Hilbert space, since the key fact of them is the � boundedness of squarefunction (see [23]).

5.3 Burkholder martingale transform operators

Burkholder defined in [7] the spaces�� as the class of Banach spaces� such that there exists a , � � ��,such that the inequality ,�� ,��

�

� ��

holds for all�–valued martingales, all numbers,�� ,�� in �� , and all � � �. Theorems 4.3 and 4.5 can be applied to these transforms and again we getthat, when � is a UMD space, the necessary and sufficient condition on the weight � (resp. ) in order to find aweight (resp. �) such that the following inequality

,�� ,��

��

holds, is �� a.s.�

resp. ��

��

�

�� a.s.�

.

Acknowledgements The author would like to thank the referees for their helpful comments, that contributed to the im-provement of the manuscript.

Partially supported by DGICYT, Spain, under grant BFM2001–0188 and European Comission via the TMR network“Harmonic Analysis”.

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on weighted inequalities for martingale transform operators

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