#### Abstract

Let and be a variable exponent, and we introduce a new class of function spaces in a probabilistic setting which unifies and generalizes the variable Lebesgue spaces with and grand Lebesgue spaces with and . Based on the new spaces, we introduce a kind of Hardy-type spaces, grand martingale Hardy spaces with variable exponents, via the martingale operators. The atomic decompositions and John-Nirenberg theorem shall be discussed in these new Hardy spaces.

#### 1. Introduction

The martingale theory is widely studied in the field of mathematical physics, stochastic analysis, and probability. Weisz [1] presented the atomic decomposition theorem for martingale Hardy spaces. Herz [2] established the John-Nirenberg theorem for martingales. Since then, the study of martingale Hardy spaces associated with various functional spaces has attracted a steadily increasing interest. For instance, martingale Orlicz-type Hardy spaces were investigated in [3–6], martingale Lorentz Hardy spaces were studied in [7–9], and variable martingale Hardy spaces were developed in [10–14].

Let , and the grand Lebesgue space introduced by Iwaniec and Sbordone [15] is defined as the Banach function space of the measurable functions on finite such that

Such spaces can be used to integrate the Jacobian under minimal hypotheses [15]. The grand Lebesgue spaces as a kind of Banach function space were investigated in the papers of Capone et al. [16, 17], Fiorenza et al. [18–21], Kokilashvili et al. [22, 23], and so forth. In particular, grand Lebesgue spaces with variable exponents were studied in [24, 25].

We find that the framework of grand Lebesgue spaces with variable exponents has not yet been studied in martingale theory. This paper is aimed at discussing the variable grand Hardy spaces defined on the probabilistic setting and showing the atomic decompositions and John-Nirenberg theorem in these new Hardy spaces. More precisely, we first present the atomic characterization of grand Hardy martingale spaces with variable exponents. To do so, we introduce the following new notations of atom.

*Definition 1. *Let be a variable exponent and . A measurable function is called a *-*atom (resp. *-*atom, *-*atom) if there exists a stopping time such that

,

See Section 2 for the notation . Denote by (resp. , ) the collection of all sequences of triplet , where are -atoms (resp. -atoms, -atoms), are stopping times satisfying and in Definition 1, and are nonnegative numbers and also

Under these definitions, we show the atomic decompositions of the grand Hardy martingale spaces with variable exponents (see Section 3). To be precise, we prove that for any , (resp. , ) iff there exists a sequence of triplet (resp. , ) so that for each ,

Moreover, we extend the classical John-Nirenberg theorem to the grand variable Hardy martingale spaces. To be precise, under suitable conditions, we present the following one:

See Theorem 11 for the details. This conclusion improves the recent results [12, 26], respectively.

Throughout this paper, , , and denote the integer set, nonnegative integer set, and complex numbers set, respectively. We denote by the absolute positive constant, which can vary from line to line. The symbol stands for the inequality . If we write , then it stands for .

#### 2. Preliminaries

##### 2.1. Grand Lebesgue Spaces with Variable Exponents

Let be a probability space. An -measurable function which is called a variable exponent. For convenience, we denote

Denote by the collection of all variable exponents satisfying with . The variable Lebesgue space consists of all -measurable functions such that for some ,

This leads to a Banach function space under the Luxemburg-Nakano norm

Based on this, we begin with the definition of the grand Lebesgue space with variable exponent.

*Definition 2. *Suppose that and . We define the grand Lebesgue space with variable exponent as the set of all *-*measurable functions satisfying

The Grand Lebesgue space with variable exponent can unify and generalize the various function spaces. To be precise, if , degenerates to the variable Lebesgue space . If and , becomes the grand Lebesgue space .

##### 2.2. Martingale Grand Hardy Spaces via Variable Exponents

Let be a nondecreasing sequence of sub--algebras of sets with . The expectation operator and the conditional expectation operator relative to are denoted by and , respectively. A sequence of random variables is said to be a martingale, if is -measurable, , and for every Denote to be the set of all martingales with respect to such that . For , write its martingale difference by . Define the maximal function, the square function, and the conditional square function of , respectively, as follows:

Let be the set of all sequences of nondecreasing, nonnegative, and adapted functions, and . For , , and , denote

Now we introduce the grand martingale Hardy spaces associated with variable exponents as follows:

The bounded -martingale spaces where

*Remark 3. *If , then we obtain the definitions of , , , , and , respectively (see [10, 12, 27]). If we consider the special case and with the notations above, we obtain the definitions of , , , , and , respectively (see [26]). In addition, if and , we obtain the martingale Hardy spaces , , , , and , respectively (see [28]).

Refer to [29, 30] for more information on martingale theory.

#### 3. Atomic Characterization

The method of atomic characterization plays an useful tool in martingale theory (see for instance [1, 4, 6, 31–33]). We shall construct the atomic characterizations for grand Hardy martingale spaces with variable exponents in this section.

Theorem 4. *Let and . If the martingale , then there exists a sequence of triplet so that for each ,
**Conversely, if the martingale has a decomposition of (14), then
where the infimum is taken over all the admissible representations of (14).*

*Proof. *Let . Now consider the stopping time for each :
It is easy to see that the sequence of these stopping times is nondecreasing. For each stopping time , denote . It is easy to write that
For each , let . If , we set
If , we set for each . For each fixed , is a martingale. Since , we get
We can easily check that is a bounded -martingale. Hence, there exists an element such that . If , then , and . Consequently, it implies that is really a -atom.

Denote . Knowing that and is nondecreasing for each , we obtain . Now, we point out that
Indeed, for a fixed , there is so that and , then we have
This means
For the converse part, according to the definition of -atom, we easily conclude
where is the -atom and is the stopping time associated with which, when combined with the subadditivity of the operator , yields
This implies
Taking over all the admissible representations of (14) for , we obtain the desired result.

Next, we will characterize and by atoms, respectively. The proof is similar to the proof of Theorem 4. For the completeness of this paper, we provide some details.

Theorem 5. *Suppose and . If the martingale (resp. ), then there exists a sequence of triplet (resp. ) so that for each ,
**Conversely, if the martingale has admissible representation (27), then (resp. ) and
where the infimum is taken over all the admissible representations of (27).*

*Proof. *The proof follows the ideas in Theorem 4, so we omit some details. Suppose (resp. ). We define stopping times as follows:
where is an adapted, nondecreasing sequence such that almost everywhere (resp.) and

Let and be defined as in the proof of Theorem 4. And replace by the . Then, we obtain that and (28) still hold.

For the converse part, write
Clearly, is a nonnegative, nondecreasing, and adapted sequence with (resp.). Thus, we get

Taking over all the admissible representations of (27) for , we obtain the desired result.

*Remark 6. *Suppose and . We conclude that the sum in Theorem 4 converges to in as , . Indeed, it follows by the subadditive of the operator , we get, for any with ,

Moreover, is decreasing and convergent to (a.e.) as , and is decreasing and convergent to (a.e.) as . From this and the dominated convergence theorem in for (see [34], Theorem 2.62), it follows that

Furthermore, we can also show the norm convergence of the summation in Theorems 5 as , .

#### 4. The Generalized John-Nirenberg Theorem

In the sequel of this section, we will often suppose that every is generated by countably many atoms. Recall that is called an atom, and if for any with satisfying , we have . We denote by the set of all atoms in . We shall present the generalized John-Nirenberg theorem on grand Lebesgue spaces with variable exponents. For each , the Banach space (bounded mean oscillation [35]) is defined as

It can be easily shown that the norm of is equivalent to where consists of all stopping times relative to . It follows immediately from the John-Nirenberg theorem [2, 30] that

What is more, in [2], there has

*Definition 7. *For and , the generalized BMO martingale space is defined by
where

*Remark 8. *If , degenerates to the variable exponent BMO martingale space introduced and studied in [12]. If and , becomes the grand BMO martingale space studied in [26].

In order to establish the generalized John-Nirenberg theorem in the framework of , we need the following lemmas and notations.

Lemma 9 (Hölder’s inequality, see [34]). *Let satisfy
**Then, there exists a constant such that for all and , we have and
*

We mention that if the variable exponent meets the log-Hölder continuity condition in Euclidean spaces, the inverse Hölder’s inequality holds for characteristic functions in (see [36]). Compared with Euclidean space , the probability space has no natural metric structure. Fortunately, Jiao et al. [11, 27] put forward the following condition: there exists an absolute constant depending only on such that

Lemma 10 (see [27]). *Suppose satisfying (43).
*(1)*For each , we get**(2)**Let satisfy (43). If satisfies**then also satisfies condition (43). Moreover, for each , we deduce
*

Theorem 11. *Suppose that satisfies (43) and . Then, for every , there has
*

*Proof. *If satisfies (43), then we clearly get that also satisfies (43) for . It follows from Lemmas 9 and 10 that
for any . Here, the variable exponent is defined by
This is equivalent to the following inequality:
Hence, we have
Taking the supremum over all stopping times, we deduce
Conversely, from the definition of , we get
It follows from Lemma 9 that
where satisfies
Hence, by (38), we deduce that
From what has been discussed above, we draw the conclusion that
Theorem 11 improves the recent results [12, 26], respectively. More precisely, if we consider the case , then the following result holds:

Corollary 12. *If satisfies (43) with , then for ,
*

And especially for and , we get the conclusion as follows.

Corollary 13 (see [26]). *Suppose , then for ,
*

#### Data Availability

No data is used in the manuscript.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.

#### Acknowledgments

This project is partially supported by the National Natural Science Foundation of China (Grant Nos. 11801001 and 12101223), Scientific Research Fund of Hunan Provincial Education Department (Grant No. 20C0780), and Scientific Research Fund of Hunan University of Science and Technology (Grant Nos. E51997 and E51998).