#### Abstract

We discuss martingale transforms between martingale Hardy-amalgam spaces and Let and and let be a martingale in ; then, we show that its martingale transforms are the martingales in for some and similarly for and

#### 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 and , 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 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 spaces , , and introduced in [18]. These spaces, as indicated in [18], are generalizations of the classical martingale Hardy spaces. More precisely if and and is a martingale in , then its martingale transforms are the martingales in and similarly for and

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 and -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 Section 4. 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

Let be an arbitrary nonempty set and let be a sequence of nonempty subsets of such that for , and

For , the classical amalgam of and , denoted on consists of functions which are locally in and have behaviour globally [19]. More precisely, if for and , if is the usual indicator function of the set . 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 be a probability space and let be a nondecreasing sequence of -algebra with respect to the complete ordering on such that

For , and are the expectation operator and the conditional expectation operator, respectively, relative to and , respectively. We denote by the set of all martingales relative to the filtration such that . We recall that for , its martingale difference is denoted , with the convention that .

A martingale is said to be bounded if for all and

We recall that

For a martingale , the quadratic variation, , and the conditional quadratic variation, , of are defined by respectively. In this regard, we have

The maximal function or of the martingale is defined by

The martingale Hardy-amalgam spaces , , and were introduced in [18] but we list them here for completeness purposes. Let and . The spaces are defined as follows:

Let be the set of all sequences of adapted, nondecreasing, nonnegative functions and define also (i)The space called the space of predictive quadratic variations, is the set endowed with the (quasi)-norm (ii)The space , called the space of predictive martingales, is the set endowed with the (quasi)-norm

The sequence is called the predictable sequence. In the subsequent sections, we shall sometimes write , and simply as and , 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 in and norms is attained [23]. Henceforth, such an “infimum” sequence will be referred to as optimal.

#### 3. Martingale Transforms and Their Convergence

Let be a probability space with the filtration Let be an adapted process such that for all is -measurable, normally referred to as multiplier sequence. If is a martingale, then the following process: is called a martingale transform where is the usual martingale difference sequence. The martingale transform need not be a martingale; however, it is a martingale if and only if (i.e., ) [9, 10]. As a trivial example, let be a stopping time. Then, the following process , referred to as stopped process, is an example of a martingale transform as is -measurable;

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. *Let be an bounded martingale and be ’s martingale transform with multiplier sequence defined below:
*

*Then, converges almost everywhere on the set*

#### 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 on followed by and finally

Theorem 2 (Relation between and ). *Let , , , and Let be a martingale define on and suppose that Let be the optimal bounded positive increasing adapted process such that and Then, the process defined by
is a martingale transform of and converges almost everywhere.**Moreover, and
**The next theorem states that if is the martingale transform of , then which in turn is the martingale transform of *

Theorem 3 (Relation between and ). *Let , , , and Let be a martingale define on and let be a bounded positive increasing adapted process such that . Let be a martingale transform of defined by
such that Then,
*(a)* and converges almost everywhere and*(b)* and moreover
**Let us now look at the relations between and .*

Theorem 4 (Relation between and ). *Let*, , , and . Let be a martingale define on . Let be the conditional quadratic variation operator which we assume it is bounded and nonzero. Then, the process defined by
is a martingale transform of and converges almost everywhere. Moreover, if , then and
*The converse of the above theorem is the following theorem that is given that is the martingale transform of , then is the martingale transform of and *

Theorem 5 (Relation between and ). *Let , , , and . Let be a martingale define on and be the conditional quadratic variation operator which we assume it is bounded and nonzero. Let be a martingale transform of defined by
such that Then
*(a)* and converges almost everywhere and*(b)* and moreover
**We now come to the predictive quadratic variation spaces.*

Theorem 6 (Relation between and ). *Let , , , and . Let be a martingale define on and suppose that . Let be the optimal bounded positive increasing adapted process such that and . Then, the process defined by
is a martingale transform of and converges almost everywhere.**Moreover, and
**Similarly, if is the martingale transform of , then , and moreover, is the martingale transform of This is the statement below.*

Theorem 7 (Relation between and ). *Let , , , and . Let be a martingale define on and let be a bounded positive increasing adapted process such that Let be a martingale transform of defined by
such that Then,
*(a)* and converges almost everywhere and*(b)* and moreover
*

#### 5. Proof of Results

*Proof of Theorem 2. *Let By the hypothesis, and ( optimal). From equation (18)
☐

We note that when then which implies that

Therefore, from equation (30), we get that

Indeed since , we get

Hence, since is a bounded sequence. We note that is -measurable as it is adapted and so is . Now, the sequence is a positive decreasing sequence which is bounded above by since Therefore, since for all . Thus, equation (18) is a martingale transform and hence by Theorem 1, converges almost everywhere.

We observe that as , we also have that Therefore,

Let Then, the sequence is also an increasing positive adapted process. Hence, by definition,

Equation (34) then gives us establishing the result.

*Proof of Theorem 3. *Let Then, there exists an optimal positive increasing adapted sequence such that and

Part (a) follows since Since is a martingale and is a bounded positive increasing and adapted, and for all , is -measurable, and the convergence of follows from Theorem 1.

For part (b), we observe that since is increasing,
Let Then, the sequence is also positive increasing and bounded adapted process. Hence, by definition,
Hence,
By definition,
With the choice of , and also observing that , we apply Hölder’s inequality to obtain
Inequality (6) then becomes
and the theorem is proved. ☐

*Proof of Theorem 4. *Let be the conditional quadratic variation operator. Then, by definition, is -measurable. Thus, is -measurable and positive and bounded adapted decreasing process. Hence, is a martingale transform. Since is positive and bounded, is also positive and bounded. That is, Thus, by Theorem 1, converges almost everywhere. Suppose that Then, From equation (2), we get that
Since , we sum both sides of the last equality to obtain
Since is increasing, we observe that for , , and thus,
It follows that
That is,
We deduce from this and (7) that
and hence, we obtain that
which gives us
Now, since
we have that
Thus, by definition,
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 and by measurability and the increasing property of , we get that
Summing both sides, we shall obtain
and thus,
But
Therefore,
The proof is complete. ☐

*Proof of Theorem 6. *Let By hypothesis, and ( optimal). Considering equation (26), we have that
Since is increasing and , we observe from equation (59) that (as ),
Hence, we get that
This implies that Also, is adapted. Thus, is a martingale transform and by Theorem 1, converges almost everywhere since (as ) and is a martingale.

Let Then, the sequence is also an increasing positive and bounded adapted process. Hence, by definition,
In the same manner as we obtained equation (34), we also have that
Therefore,

In other words, (66) and the theorem is proved. ☐

*Proof of Theorem 7. *Similar to the proof of Theorem 3, part (a) is established.

For part (b), let Then, there exist an optimal increasing positive adapted process such that and From equation (59), we have that
We also observe that
and therefore,
Let Then, the sequence is also positive increasing and bounded adapted process. Hence, by definition,
Hence,
By definition,
With the choice of and noting that , we apply Hölder’s inequality to get
Inequality (9) then becomes
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 spaces , , and These are extended results corresponding to Garsia [6] where he considered the space while this paper considers the , the amalgam space of , and spaces, defined in previous pages.

#### Data Availability

No data were used in support of this study.

#### Conflicts of Interest

The author declares that there is no conflict of interests regarding the publication of this paper.