Atomic Decompositions and John-Nirenberg Theorem of Grand Martingale Hardy Spaces with Variable Exponents

Let θ ≥ 0 and p ð · Þ be a variable exponent, and we introduce a new class of function spaces L p ð · Þ , θ in a probabilistic setting which uni ﬁ es and generalizes the variable Lebesgue spaces with θ = 0 and grand Lebesgue spaces with p ð · Þ ≡ p and θ = 1 . 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.

Let 1 < p < ∞, and the grand Lebesgue space L pÞ ðEÞ introduced by Iwaniec and Sbordone [15] is defined as the Banach function space of the measurable functions f on finite E such that f k k L pÞ = sup 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][19][20][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 pð·Þ be a variable exponent and θ ≥ 0. A measurable function a is called a ð1, pð·Þ, θÞ -atom (resp. ð2, pð·Þ, θÞ -atom, ð3, pð·Þ, θÞ -atom) if there exists a stopping time τ such that See Section 2 for the notation L pð·Þ,θ . Denote by A s ðpð·Þ, ∞Þ (resp. A S ðpð·Þ, ∞Þ, A M ðpð·Þ, ∞Þ) the collection of all sequences of triplet ða k , τ k , μ k Þ, where a k are ð1, pð·Þ, θÞ -atoms (resp. ð2, pð·Þ, θÞ-atoms, ð3, pð·Þ, θÞ-atoms), τ k are stopping times satisfying ða1Þ and ða2Þ in Definition 1, and μ k 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 f = ðf n Þ n≥0 , f ∈ H s pð·Þ,θ (resp. Q pð·Þ,θ , D pð·Þ,θ ) iff there exists a sequence of triplet ða k , τ k , μ k Þ ∈ A s ðpð·Þ,∞Þ (resp. A S ðpð·Þ, ∞Þ, A M ðpð·Þ, ∞Þ) so that for each n ≥ 0, 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, ℤ, ℕ, and ℂ denote the integer set, nonnegative integer set, and complex numbers set, respectively. We denote by C the absolute positive constant, which can vary from line to line. The symbol A ≲ B stands for the inequality A ≤ CB. If we write A ≈ B, then it stands for A ≲ B ≲ A.

Preliminaries
2.1. Grand Lebesgue Spaces with Variable Exponents. Let ðΩ, F, ℙÞ be a probability space. An F-measurable function pð·Þ: Ω ⟶ ð0,∞Þ which is called a variable exponent. For convenience, we denote Denote by P = P ðΩÞ the collection of all variable exponents pð·Þ satisfying with 1 < p − ≤ p + < ∞. The variable Lebesgue space L pð·Þ = L pð·Þ ðΩÞ consists of all F-measurable functions f such that for some λ > 0, 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.

Martingale Grand Hardy Spaces via Variable Exponents.
Let fF n g n≥0 be a nondecreasing sequence of sub-σ-algebras of F sets with F = σð S n≥0 F n Þ. The expectation operator and the conditional expectation operator relative to F n are denoted by E and E n , respectively. A sequence f = ð f n Þ n≥0 of random variables is said to be a martingale, if f n is F n -measurable, Eð|f n | Þ < ∞, and E n ð f n+1 Þ = f n for every n ≥ 0: Denote M to be the set of all martingales f = ð f n Þ n≥0 with respect to fF n g n≥0 such that f 0 = 0. For f ∈ M, write its martingale difference by d n f = f n − f n−1 ðn ≥ 0, f −1 = 0Þ. Define the maximal function, the square function, and the conditional square function of f , respectively, as follows: Let Γ be the set of all sequences ðλ n Þ n≥0 of nondecreasing, nonnegative, and adapted functions, and λ ∞ ≔ lim n⟶∞ λ n . For f ∈ M, pð·Þ ∈ P , and θ ≥ 0, denote Now we introduce the grand martingale Hardy spaces associated with variable exponents as follows: Journal of Function Spaces The bounded L pð·Þ,θ -martingale spaces where Remark 3. If θ = 0, then we obtain the definitions of H * pð·Þ , H S pð·Þ , H s pð·Þ , Q pð·Þ , and D pð·Þ , respectively (see [10,12,27]). If we consider the special case θ = 1 and pð·Þ ≡ p with the notations above, we obtain the definitions of H * pÞ , H S pÞ , H s pÞ , Q pÞ , and D pÞ , respectively (see [26]). In addition, if pð·Þ ≡ p and θ = 0, we obtain the martingale Hardy spaces H * q , H S q , H s q , Q q , and D q , respectively (see [28]).
Refer to [29,30] for more information on martingale theory.

Atomic Characterization
The method of atomic characterization plays an useful tool in martingale theory (see for instance [1,4,6,[31][32][33]). We shall construct the atomic characterizations for grand Hardy martingale spaces with variable exponents in this section. Theorem 4. Let pð·Þ ∈ P and θ ≥ 0. If the martingale f ∈ H s pð·Þ,θ , then there exists a sequence of triplet ða k , τ k , μ k Þ ∈ A s ðpð·Þ,∞Þ so that for each n ≥ 0, Conversely, if the martingale f has a decomposition of (14), then where the infimum is taken over all the admissible representations of (14).
Proof. Let f ∈ H s pð·Þ,θ . Now consider the stopping time for each k ∈ ℤ: It is easy to see that the sequence of these stopping times is nondecreasing. For each stopping time τ, denote f τ n = f n∧τ . It is easy to write that For each k ∈ ℤ, let μ k = 3 · 2 k kχ fτ k <∞g k If μ k = 0, we set a k n = 0 for each n ∈ ℕ. For each fixed k We can easily check that ða k n Þ n≥0 is a bounded L 2 -martingale. Hence, there exists an element a k ∈ L 2 such that E n a k = a k n . If n ≤ τ k , then a k n = 0, and sða k Þ ≤ kχ fτ k <∞g k −1 L pð·Þ,θ .

Journal of Function Spaces
This means For the converse part, according to the definition of ð1, pð·Þ, θÞ-atom, we easily conclude where a k is the ð1, pð·Þ, θÞ-atom and τ k is the stopping time associated with a k which, when combined with the subadditivity of the operator s, yields This implies Taking over all the admissible representations of (14) for f , we obtain the desired result.
Next, we will characterize Q pð·Þ,θ and D pð·Þ,θ by atoms, respectively. The proof is similar to the proof of Theorem 4. For the completeness of this paper, we provide some details.
For the converse part, write Clearly, ðλ n Þ n≥0 is a nonnegative, nondecreasing, and adapted sequence with S n+1 ð f Þ ≤ λ n (resp.|f n | ≤λ n ). Thus, we get Taking over all the admissible representations of (27) for f , we obtain the desired result. Remark 6. Suppose pð·Þ ∈ P and θ ≥ 0. We conclude that the sum ∑ N k=M μ k a k in Theorem 4 converges to f in H s pð·Þ,θ as M ⟶ −∞, N ⟶ ∞. Indeed, it follows by the subadditive of the operator s, we get, for any M, N ∈ ℤ with M < N, 4 Journal of Function Spaces Moreover, sð f − f τ N+1 Þ is decreasing and convergent to 0 (a.e.) as N ⟶ ∞, and sðf τ M Þ is decreasing and convergent to 0 (a.e.) as M ⟶ −∞. From this and the dominated convergence theorem in L pð·Þ−ε for 0 < ε < p − − 1 (see [34], Theorem 2.62), it follows that Furthermore, we can also show the norm convergence of the summation ∑ N k=M μ k a k in Theorems 5 as M ⟶ −∞, N ⟶ ∞.

The Generalized John-Nirenberg Theorem
In the sequel of this section, we will often suppose that every F n is generated by countably many atoms. Recall that B ∈ F n is called an atom, and if for any A ⊆ B with A ∈ F n satisfying ℙðAÞ < ℙðBÞ, we have ℙðAÞ = 0. We denote by AðF n Þ the set of all atoms in F n . We shall present the generalized John-Nirenberg theorem on grand Lebesgue spaces with variable exponents. For each 1 ≤ p < ∞, the Banach space BMO p (bounded mean oscillation [35]) is defined as It can be easily shown that the norm of BMO p is equivalent to where T consists of all stopping times relative to fF n g n≥0 . It follows immediately from the John-Nirenberg theorem [2,30] that What is more, in [2], there has Definition 7. For pð·Þ ∈ P and θ ≥ 0, the generalized BMO martingale space is defined by where Remark 8. If θ = 0, BMO pð·Þ,θ degenerates to the variable exponent BMO martingale space BMO pð·Þ introduced and studied in [12]. If θ = 1 and pð·Þ ≡ p, BMO pð·Þ,θ becomes the grand BMO martingale space BMO pÞ studied in [26].
In order to establish the generalized John-Nirenberg theorem in the framework of BMO pð·Þ,θ , we need the following lemmas and notations.
Then, for every f ∈ BMO 1 , there has Proof. If pð·Þ ∈ P satisfies (43), then we clearly get that pð·Þ − η also satisfies (43) for 0 < η < p − − 1. It follows from Lemmas 9 and 10 that for any 0 < η < p − − 1. Here, the variable exponent ðpð·Þ − ηÞ ′ 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 L pð·Þ,θ , we get It follows from Lemma 9 that where qð·Þ 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 θ = 0, then the following result holds: Corollary 12. If pð·Þ satisfies (43) with 1 < p − ≤ p + < ∞, then for f ∈ BMO 1 , And especially for θ = 1 and pð·Þ ≡ p, we get the conclusion as follows.

Data Availability
No data is used in the manuscript.

Conflicts of Interest
The authors declare that they have no conflicts of interest.