# Martingale Transforms between Martingale Hardy-amalgam Spaces

Authors: Justice Sam Bansah
Journal: Abstract and Applied Analysis
 (2021)

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## Abstract

We discuss martingale transforms between martingale Hardy-amalgam spacesHp,qs,Qp,q and Pp,q. Let 0<p<q<∞,p1<p and q1<q and let f=fn,n∈ℕ be a martingale in Pp1,q1; then, we show that its martingale transforms are the martingales in Pp,q for some p,q and similarly for Hp,qs and Qp,q.


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## Body

## 1. Introduction

Martingale came to existence through Doob as part of his seminal work [1]. The concept of classical martingale Hardy spaces came to light when Burkholder and Gundy extended an inequality due to R. E. A. C. Paley in 1970 [2]. This was made possible due to the introduction of the operators sf and Sf, by Burkholder, which will be made clear in the sequel. Afterwards, playmakers including Burkholder himself, Garcia, Cairoli, and Davis contributed to the growth of the classical martingale Hardy space theory [2–8].However, in 1966, Burkholder had already introduced the notion of martingale transforms [9] which became an indispensable tool in the study of some relations between classical martingale Hardy spaces, mostly the predictive spaces Pp in the classical settings [6, 10]. In the past years, various authors have generalized the classical martingale Hardy spaces of the classical Lebesgue spaces to Lorentz spaces, Orlicz spaces, and Orlicz-Musielak spaces [11–17].The interest in this paper is to discuss the martingale transforms between the martingale Hardy-amalgam spacesHp,qs, Qp,q, and Pp,q introduced in [18]. These spaces, as indicated in [18], are generalizations of the classical martingale Hardy spaces. More precisely if p1<p and q1<q and f is a martingale in Pp1,q1, then its martingale transforms are the martingales in Pp,q and similarly for Hp,qs and Qp,q.The motivation to look for the various martingale transforms in these spaces comes from the various applications of martingale transforms in general. Especially, with the use of martingale transforms, the upcrossing theorem of martingales was established; the convergence of martingales has also been proved using martingale transforms andL1-characterization of martingales [9, 10].The next section introduces the basic definitions and the notations needed in subsequent sections. This is followed by a presentation on martingale transforms and their convergence. Presentation of the main results in this paper is given in Section4. In Section 5, we prove the results and finally we conclude.


## 2. Basic Definitions and Notations

We introduce in this section some function spaces that will be relevant in the subsequent sections of this paper.


### 2.1. Wiener Amalgam Spaces

LetΩ be an arbitrary nonempty set and let Ωjj∈ℤ be a sequence of nonempty subsets of Ω such that Ωj∩Ωi=∅ for j≠i, and
(1)⋃j∈ℤΩj=Ω.For0<p,q≤∞, the classical amalgam of Lp and lq, denoted Lp,q, on Ω consists of functions which are locally in Lp and have lq behaviour globally [19]. More precisely, f∈Lp,q if for 0<p<∞ and 0<q<∞,
(2)fLp,qΩ≔∑j∈ℤ∫Ωfp1Ωjdℙq/p1/q<∞,fLp,∞Ω≔supj∈ℤ∫Ωfp1Ωjdℙ1/p<∞,if q=∞.1A is the usual indicator function of the set A. For more on amalgam spaces, we refer the reader to [19–22].


### 2.2. Martingale Hardy Spaces via Amalgams

The same notations used in [18] is adopted in this paper. Let Ω,F,ℙ be a probability space and let Fnn≥0≔Fnn∈ℤ+ be a nondecreasing sequence of σ-algebra with respect to the complete ordering on ℤ+=0,1,2⋯ such that
(3)σ⋃n∈ℕFn=F.Forn∈ℤ+, E and En are the expectation operator and the conditional expectation operator, respectively, relative to F and Fn, respectively. We denote by M the set of all martingales f=fnn≥0 relative to the filtration Fnn≥0 such that f0=0. We recall that for f∈M, its martingale difference is denoted dnf=fn−fn−1, n≥0 with the convention that d0f=0.A martingalef=fnn≥0 is said to be Lp bounded 0<p≤∞ if fn∈Lp for all n∈ℤ+ and
(4)fp≔supn∈ℕfnp<∞.We recall that(5)fp=Efp1/p.For a martingalef=fnn≥0, the quadratic variation, Sf, and the conditional quadratic variation, sf, of f are defined by
(6)Sf=∑n∈ℕdnf21/2,sf=∑n∈ℕEn−1dnf21/2,respectively. In this regard, we have
(7)Snf=∑i=1ndif21/2,snf=∑i=1nEi−1dif21/2.The maximal functionf∗ or Mf of the martingale f is defined by
(8)Mf=f∗=supn∈ℕfn.The martingale Hardy-amalgam spacesHp,qs, Qp,q, and Pp,q were introduced in [18] but we list them here for completeness purposes. Let 0<p<∞ and 0<q≤∞. The spaces are defined as follows:
(9)Hp,qsΩ=f∈M:fHp,qsΩ≔sfp,q<∞LetΠ be the set of all sequences β=βnn≥0 of adapted, nondecreasing, nonnegative functions and define also
(10)β∞≔limn→∞βn.(i)
The spaceQp,qΩ, called the space of predictive quadratic variations, is the set
(11)f∈M:forβ∈Π,Snf≤βn−1andβ∞∈Lp,qΩ,endowed with the (quasi)-norm
(12)fQp,qΩ≔infβ∈Πβ∞p,q(ii)
The spacePp,qΩ, called the space of predictive martingales, is the set
(13)f∈M:forβ∈Π,fn≤βn−1andβ∞∈Lp,qΩ,endowed with the (quasi)-norm
(14)fPp,qΩ≔infβ∈Πβ∞p,qThe sequenceβ=βnn≥0 is called the predictable sequence. In the subsequent sections, we shall sometimes write Hp,qsΩ, Qp,qΩ,Pp,qΩ and Lp,qΩ simply as Hp,qs,Qp,q,Pp,q and Lp,q, respectively, and we shall do the same for their respective (quasi)-norms. The atomic decompositions and duality characterizations of these spaces are in [18].We note that the “infimum” taken inPp,q and Qp,q norms is attained [23]. Henceforth, such an “infimum” sequence will be referred to as optimal.


## 2.1. Wiener Amalgam Spaces

LetΩ be an arbitrary nonempty set and let Ωjj∈ℤ be a sequence of nonempty subsets of Ω such that Ωj∩Ωi=∅ for j≠i, and
(1)⋃j∈ℤΩj=Ω.For0<p,q≤∞, the classical amalgam of Lp and lq, denoted Lp,q, on Ω consists of functions which are locally in Lp and have lq behaviour globally [19]. More precisely, f∈Lp,q if for 0<p<∞ and 0<q<∞,
(2)fLp,qΩ≔∑j∈ℤ∫Ωfp1Ωjdℙq/p1/q<∞,fLp,∞Ω≔supj∈ℤ∫Ωfp1Ωjdℙ1/p<∞,if q=∞.1A is the usual indicator function of the set A. For more on amalgam spaces, we refer the reader to [19–22].


## 2.2. Martingale Hardy Spaces via Amalgams

The same notations used in [18] is adopted in this paper. Let Ω,F,ℙ be a probability space and let Fnn≥0≔Fnn∈ℤ+ be a nondecreasing sequence of σ-algebra with respect to the complete ordering on ℤ+=0,1,2⋯ such that
(3)σ⋃n∈ℕFn=F.Forn∈ℤ+, E and En are the expectation operator and the conditional expectation operator, respectively, relative to F and Fn, respectively. We denote by M the set of all martingales f=fnn≥0 relative to the filtration Fnn≥0 such that f0=0. We recall that for f∈M, its martingale difference is denoted dnf=fn−fn−1, n≥0 with the convention that d0f=0.A martingalef=fnn≥0 is said to be Lp bounded 0<p≤∞ if fn∈Lp for all n∈ℤ+ and
(4)fp≔supn∈ℕfnp<∞.We recall that(5)fp=Efp1/p.For a martingalef=fnn≥0, the quadratic variation, Sf, and the conditional quadratic variation, sf, of f are defined by
(6)Sf=∑n∈ℕdnf21/2,sf=∑n∈ℕEn−1dnf21/2,respectively. In this regard, we have
(7)Snf=∑i=1ndif21/2,snf=∑i=1nEi−1dif21/2.The maximal functionf∗ or Mf of the martingale f is defined by
(8)Mf=f∗=supn∈ℕfn.The martingale Hardy-amalgam spacesHp,qs, Qp,q, and Pp,q were introduced in [18] but we list them here for completeness purposes. Let 0<p<∞ and 0<q≤∞. The spaces are defined as follows:
(9)Hp,qsΩ=f∈M:fHp,qsΩ≔sfp,q<∞LetΠ be the set of all sequences β=βnn≥0 of adapted, nondecreasing, nonnegative functions and define also
(10)β∞≔limn→∞βn.(i)
The spaceQp,qΩ, called the space of predictive quadratic variations, is the set
(11)f∈M:forβ∈Π,Snf≤βn−1andβ∞∈Lp,qΩ,endowed with the (quasi)-norm
(12)fQp,qΩ≔infβ∈Πβ∞p,q(ii)
The spacePp,qΩ, called the space of predictive martingales, is the set
(13)f∈M:forβ∈Π,fn≤βn−1andβ∞∈Lp,qΩ,endowed with the (quasi)-norm
(14)fPp,qΩ≔infβ∈Πβ∞p,qThe sequenceβ=βnn≥0 is called the predictable sequence. In the subsequent sections, we shall sometimes write Hp,qsΩ, Qp,qΩ,Pp,qΩ and Lp,qΩ simply as Hp,qs,Qp,q,Pp,q and Lp,q, respectively, and we shall do the same for their respective (quasi)-norms. The atomic decompositions and duality characterizations of these spaces are in [18].We note that the “infimum” taken inPp,q and Qp,q norms is attained [23]. Henceforth, such an “infimum” sequence will be referred to as optimal.


## 3. Martingale Transforms and Their Convergence

LetPs≔Ω,F,ℙ,Fnn≥0 be a probability space with the filtration Fnn≥0. Let ν=νnn≥0 be an adapted process such that for all n=0,1,2,⋯,νn is Fn−1-measurable, normally referred to as multiplier sequence. If f=fn,n∈ℕ is a martingale, then the following process:
(15)gn=∑k=1nνkdkf,g0=0,is called a martingale transform where dkf is the usual martingale difference sequence. The martingale transform need not be a martingale; however, it is a martingale if and only if gn∈L1 (i.e., Egn<∞) [9, 10]. As a trivial example, let τ be a stopping time. Then, the following process fnτ, referred to as stopped process, is an example of a martingale transform as 1k≤τ is Fk−1-measurable;
(16)fnτ=fn∧τ=∑k=1n1k≤τdkf.The usefulness of martingale transforms has aided various authors to study the relations of the predictive spaces in their various generalizations such as the martingale transforms between Hardy-Orlicz spaces among others [24–26]. Martingale transforms also share some properties with fractional integrals [27–29]. Burkholder has established the almost everywhere convergence of the martingale transform based on the condition that the maximal function of the multiplier sequence is finite. In fact, we can find the proof of the following convergence result in [2], [Theorem 1].Theorem 1.
Letf=fn,n∈ℕ,f0=0 be an L1 bounded martingale and g be f’s martingale transform with multiplier sequence ν=νkk≥0 defined below:
(17)gn=∑k=1nνkdkf,n≥1,g0=0.
Then,g=gnn≥0 converges almost everywhere on the set supkνk<∞.


## 4. Presentation of Main Results

In this section, we present the results that will be discussed in this paper. Their proofs are discussed in the next section. We start with the results onPp,q followed by Hp,qs and finally Qp,q.Theorem 2 (Relation betweenPp,q and Pp1,q1).
Let0<p<q<∞, 0<p1<p, 0<q1=p1/pq<q, and α=1−p1/p. Let f=fn,n∈ℕ be a martingale define on Ps and suppose that f∈Pp1,q1. Let ν=νkk≥0 be the optimal bounded positive increasing adapted process such that fn≤νn−1 and ν∞∈Lp1,q1. Then, the process defined by
(18)gn=∑k=1n1νk−1αdkf,g0=0,is a martingale transform of f and converges almost everywhere.
Moreover,g=gn,n∈ℕ∈Pp,q and
(19)gPp,qq≤pp1+1qfPp1,q1q1.
The next theorem states that ifg∈Pp,q is the martingale transform of f, then f∈Pp1,q1 which in turn is the martingale transform of g.Theorem 3 (Relation betweenPp,q and Pp1,q1).
Let0<p<q<∞, 0<p1<p, 0<q1=p1/pq<q, and α=1−p1/p. Let f=fn,n∈ℕ be a martingale define on Ps and let ν=νkk≥0 be a bounded positive increasing adapted process such that ν∞∈Lp1,q1. Let g=gn,n∈ℕ∈Pp,q be a martingale transform of f defined by
(20)gn=∑k=1n1νk−1αdkf,g0=0,such that Egn<∞. Then,
(a)
fn=∑k=1nνk−1αdkg and converges almost everywhere and(b)
f∈Pp1,q1 and moreover
(21)fPp1,q1≲xfPp,qν∞p1,q11−p1/p.
Let us now look at the relations betweenHp,qs and Hp1,q1s.Theorem 4 (Relation betweenHp,qs and Hp1,q1s).
Let0<p<q<∞, 0<p1<p, 0<q1=p1/pq<q, and α=1−p1/p. Let f=fn,n∈ℕ be a martingale define on Ps. Let sf be the conditional quadratic variation operator which we assume it is bounded and nonzero. Then, the process defined by
(22)gn=∑k=1n1skαfdkf,g0=0,is a martingale transform of f and converges almost everywhere. Moreover, if f∈Hp1,q1s, then g∈Hp,qs and
(23)gHp,qsq≤pp1q/2fHp1,q1sq1.
The converse of the above theorem is the following theorem that is given thatg∈Hp,qs is the martingale transform of f, then f is the martingale transform of g and f∈Hp1,q1s.Theorem 5 (Relation betweenHp,qs and Hp1,q1s).
Let0<p<q<∞, 0<p1<p, 0<q1=p1/pq<q, and α=1−p1/p. Let f=fn,n∈ℕ be a martingale define on Ps and sf be the conditional quadratic variation operator which we assume it is bounded and nonzero. Let g=gn,n∈ℕ∈Hp,qs be a martingale transform of f defined by
(24)gn=∑k=1n1skαfdkf,g0=0,such that Egn<∞. Then
(a)
fn=∑k=1nskαfdkg and converges almost everywhere and(b)
f∈Hp1,q1s and moreover
(25)fHp1,q1sq1/q≤gHp,qs.
We now come to the predictive quadratic variation spaces.Theorem 6 (Relation betweenPp,q and Pp1,q1).
Let0<p<q<∞, 0<p1<p, 0<q1=p1/pq<q, and α=1−p1/p. Let f=fn,n∈ℕ be a martingale define on Ps and suppose that f∈Qp1,q1. Let ν=νkk≥0 be the optimal bounded positive increasing adapted process such that Snf≤νn−1 and ν∞∈Lp1,q1. Then, the process defined by
(26)gn=∑k=1n1νk−1αdkf,g0=0,is a martingale transform of f and converges almost everywhere.
Moreover,g=gn,n∈ℕ∈Qp,q and
(27)gQp,qq≤pp1+1q/2fQp1,q1q1.
Similarly, ifg∈Qp,q is the martingale transform of f, then f∈Qp1,q1, and moreover, f is the martingale transform of g. This is the statement below.Theorem 7 (Relation betweenPp,q and Pp1,q1).
Let0<p<q<∞, 0<p1<p, 0<q1=p1/pq<q, and α=1−p1/p. Let f=fn,n∈ℕ be a martingale define on Ps and let ν=νkk≥0 be a bounded positive increasing adapted process such that ν∞∈Lp1,q1. Let g=gn,n∈ℕ∈Qp,q be a martingale transform of f defined by
(28)gn=∑k=1n1νk−1αdkf,g0=0,such that Egn<∞. Then,
(a)
fn=∑k=1nνk−1αdkg and converges almost everywhere and(b)
f∈Qp1,q1 and moreover
(29)fQp1,q1≲gQp,qν∞p1,q11−p1/p


## 5. Proof of Results

Proof of Theorem 2.
Letf∈Pp1,q1. By the hypothesis, fn≤νn−1 and ν∞∈Lp1,q1 (ν=νnn≥0 optimal). From equation (18)
(30)gn=fnνn−1α+∑k=1n−1fk1νk−1α−1νkα☐We note that whenνk−1≤t≤νk then 1/νkα≤1/tα which implies that
(31)∫νk−1νk1νkαdt≤∫νk−1νk1tαdt⇒νk−νk−1νkα≤∫νk−1νk1tαdt.Therefore, from equation (30), we get that
(32)gn≤∫0νn−11tαdt=11−α+1νn−11−α.Indeed since1/νk−1α−1/νkα≥0, we get
(33)gn≤fnνn−1α+∑k=1n−1fk1νk−1α−1νkα≤νn−11−α+∑k=1n−1νk−11νk−1α−1νkα=ν01−α+∑k=1n−1νk1−α−νk−11−α+∑k=1n−1νk−11−α−νk−1νkα=ν01−α+∑k=1n−1νk1−α+∑k=1n−1−νk−1νkα=ν01−α+∑k=1n−1νkνkα−νk−1νkα≤ν01−α+∑k=1n−1∫νk−1νk1tαdt≤ν01−α+∫0νn−11tαdt≤11−α+1νn−11−αHence,supngn<∞ since νnn≥0 is a bounded sequence. We note that νk is Fk−1-measurable as it is adapted and so is νk−1−α. Now, the sequence wk=νk−1−α is a positive decreasing sequence which is bounded above by ν0−α since ν0>0. Therefore, supkνk−α<∞ since νkα≠0 for all k. Thus, equation (18) is a martingale transform and hence by Theorem 1, gn converges almost everywhere.We observe that as1−αp=p1, we also have that ν∞1−α1Ωjp=ν∞1Ωjp1p1/p. Therefore,
(34)ν∞1−αp,q=ν∞p1,q1q1/q≔fPp1,q1q1/q.Letβn−1=1/1−α1−α+1vn−11−α. Then, the sequence β=βnn≥0 is also an increasing positive adapted process. Hence, by definition,
(35)gPp,q≤11−α+1v∞1−αp,q=11−α+1v∞1−αp,q.Equation (34) then gives us
(36)gPp,q≤pp1+1fPp1,q1q1/q,establishing the result.Proof of Theorem 3.
Letg∈Pp,q. Then, there exists an optimal positive increasing adapted sequence u=unn≥0, such that gn≤un−1 and u∞∈Lp,q.
Part (a) follows sincednf=νn−1αdng. Since g is a martingale and ν is a bounded positive increasing and adapted, and for all n, νn is Fn−1-measurable, and the convergence of fn follows from Theorem 1.
For part (b), we observe that sinceunn≥0 is increasing,
(37)fn≤gnνn−1α+∑k=1n−1gkdkνα≤un−1νn−1α+∑k=1n−1uk−1dkνα≤un−1νn−1α+un−1∑k=1n−1dkνα≤2un−1νn−1α≤2u∞νn−1α.
Letγn−1=2u∞νn−1α. Then, the sequence γ=γnn≥0 is also positive increasing and bounded adapted process. Hence, by definition,
(38)fPp1,q1≤γ∞p1,q1=2u∞ν∞αp1,q1.
Hence,(39)fPp1,q1≤2u∞ν∞αp1,q1.
By definition,(40)u∞ν∞αp1,q1=∑j∈ℤu∞ν∞α1Ωjp1q11/q1.
With the choice ofα, and also observing that q1=1−αq, we apply Hölder’s inequality to obtain
(41)u∞ν∞αp1,q1≤u∞p,qν∞p1,q11−p1/p.
Inequality (6) then becomes(42)fPp1,q1≤2u∞p,qν∞p1,q11−p1/p=2gPp,qν∞p1,q11−p1/p,and the theorem is proved. ☐Proof of Theorem 4.
Letsf be the conditional quadratic variation operator. Then, by definition, skf is Fk−1-measurable. Thus, sk−αf is Fk−1-measurable and positive and bounded adapted decreasing process. Hence, gn is a martingale transform. Since skf is positive and bounded, sk−αf is also positive and bounded. That is, supksk−αf<∞. Thus, by Theorem 1, gn converges almost everywhere. Suppose that f∈Hp1,q1s. Then, sfp1,q1<∞. From equation (2), we get that
(43)dk2g=dk2fsk2αf⇒Ek−1dk2g=Ek−1dk2fsk2αf.
SinceEk−1dk2f=sk2f−sk−12f, we sum both sides of the last equality to obtain
(44)sn2g=∑k=1nsk2f−sk−12fsk2αf.
Sinceskf is increasing, we observe that for t>0, sk−12f≤t≤sk2f, and thus,
(45)1sk2αf≤1tα.
It follows that(46)∫sk−12fsk2f1sk2αfdt≤∫sk−12fsk2f1tαdt.
That is,(47)sk2f−sk−12fsk2αf≤∫sk−12fsk2f1tαdt.
We deduce from this and (7) that(48)sn2g≤∑k=1n∫sk−12fsk2f1tαdt,and hence, we obtain that
(49)sn2g≤∫0sn2f1tαdt≤11−αsf2−2α,which gives us
(50)sg≤11−α1/2s1−αf.
Now, since(51)s1−αfp,q=sfp1,q1q1/q,we have that
(52)sgp,q≤pp11/2sfp1,q1q1/q.
Thus, by definition,(53)gHp,qs≤pp11/2fHp1,q1sq1/q.and the theorem is proved. ☐Proof of Theorem 5.
The proof of part (a) follows obviously. For part (b), we have from equation (22) that dkf=dkgskαf and by measurability and the increasing property of s·, we get that
(54)Ek−1dk2f≤s2αfEk−1dk2g.
Summing both sides, we shall obtain(55)s2f≤s2αfs2g,and thus,
(56)s1−αf≤sg.
But(57)s1−αfp,q=sfp1,q1q1/q.
Therefore,(58)sfp1,q1q1/q≤sgp,q.
The proof is complete. ☐Proof of Theorem 6.
Letf∈Qp1,q1. By hypothesis, Snf≤νn−1 and ν∞∈Lp1,q1 (ν=νnn≥0 optimal). Considering equation (26), we have that
(59)dng2=νn−1−2αdnf2.
Sinceν=νkk≥0 is increasing and Skf≤νk−1, we observe from equation (59) that (as S0f=0),
(60)Sn2g=∑k=1nSk2f−Sk−12fνk−12α=Sn2fνn−12α−Sn−12fνn−12α+∑k=1n−1Sk2f−Sk−12fνk−12α=Sn2fνn−12α−∑k=1n−1Sk2fνk2α−Sk−12fνk−12α+∑k=1n−1Sk2f−Sk−12fνk−12α=Sn2fνn−12α+∑k=1n−1Sk2fνk−12α−Sk2fνk2α≤νn−12−2α+∑k=1n−1Sk2f1νk−12α−1νk2α≤νn−12−2α+∑k=1n−1νk−121νk−12α−1νk2α=ν02−2α+∑k=1n−1νk2−2α−νk−12−2α+∑k=1n−1νk−12−2α−νk−12νk2α=ν02−2α+∑k=1n−1νk2−2α+∑k=1n−1−νk−12νk2α=ν02−2α+∑k=1n−1νk2νk2α−νk−12νk2α≤ν02−2α+∑k=1n−1∫νk−12νk21tαdt≤ν02−2α+∫0νn−121tαdt≤11−α+1νn−12−2α.
Hence, we get that(61)Sng≤11−α+11/2νn−11−α.
This implies thatSg≤p/p1+11/2ν∞p1/p<∞. Also, νk−1−α is adapted. Thus, gn is a martingale transform and by Theorem 1, gn converges almost everywhere since supkνk−α<∞ (as ν0>0) and f is a martingale.
Letβn−1=1/1−α+1νn−11−α. Then, the sequence β=βnn≥0 is also an increasing positive and bounded adapted process. Hence, by definition,
(62)gQp,q≤β∞p,q=11−α+1νn−11−αp,q=11−α+1ν∞1−αp,q.
In the same manner as we obtained equation (34), we also have that
(63)ν∞1−αp,q=ν∞p1,q1q1/q≔fQp1,q1q1/q.
Therefore,
In other words,gQp,qq≤p/p1+1q/2fQp1,q1q1, (66) and the theorem is proved. ☐Proof of Theorem 7.
Similar to the proof of Theorem3, part (a) is established.
For part (b), letg∈Qp,q. Then, there exist an optimal increasing positive adapted process u=ukk≥0 such that Sng≤un−1 and u∞∈Lp,q. From equation (59), we have that
(64)dnf2=νn−12αdng2⇒Sn2f=∑k=1nνk−12αSk2g−Sk−12g.
We also observe that(65)Sn2f=Sn2gνn−12α−∑k=1n−1Sk2gdkν2α,and therefore,
(66)Snf≤2u∞νn−1α.
Letγn−1=2u∞νn−1α. Then, the sequence γ=γnn≥0 is also positive increasing and bounded adapted process. Hence, by definition,
(67)fQp1,q1≤γ∞p1,q1=2u∞ν∞αp1,q1.
Hence,(68)fQp1,q1≤2u∞ν∞αp1,q1.
By definition,(69)u∞ν∞αp1,q1=∑j∈ℤu∞ν∞α1Ωjp1q11/q1.
With the choice ofα, and noting that q1=1−αq, we apply Hölder’s inequality to get
(70)u∞ν∞αp1,q1≤u∞p,qν∞p1,q11−p1/p.
Inequality (9) then becomes(71)fQp1,q1≤2u∞p,qν∞p1,q11−p1/p=2gQp,qν∞p1,q11−p1/p,and the theorem is proved. ☐


## 6. Conclusion

We are thus able to locate where the martingale transform of a particular martingale is, and also given a martingale, we are able to find its martingale transform in the spacesPp,q, Qp,q, and Hp,qs. These are extended results corresponding to Garsia [6] where he considered the Lp space while this paper considers the Lp,q, the amalgam space of ℓq, and Lp spaces, defined in previous pages.



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