The Boundedness of Doob’s Maximal and Fractional Integral Operators for Generalized Grand Morrey-Martingale Spaces

Here 1 ≤ p <∞, 0 ≤ λ ≤N , R+ = ð0,∞Þ, and Bðx, rÞ are a ball in RN centered at x of radius r. This class of functions was first introduced by Morrey [1] in order to study regularity problem arising in Calculus of Variations, describe local regularity more precisely than Lebesgue spaces. In the past, Morrey spaces have been studied heavily, such as the maximal operators, fractional integral operators, and singular operators. The results are extensively applied not only in partial differential equations but also in harmonic analysis. We refer the readers to [2, 3] and the references therein. The Morrey spaces on Euclidean spaces have been developed to the generalization versions, for example, the generalized Morrey spaces [4, 5], the Orlicz-Morrey spaces [6, 7], the Triebel-Lizorkin-Morrey spaces [8], and the variable exponent Morrey spaces [9]. Especially, Meskhi [10] introduced the grand Morrey spaces and established the boundedness of the Hardy-Littlewood maximal, CalderónZygmund, and potential operators in these spaces. The generalized grand Morrey spaces in a general setting of the quasi-metric measure spaces are studied by Kokilashvili et al. [11, 12]. Moreover, in probability theory, Nakai and Sadasue [13] introduced Morrey spaces of martingales as the following: Let ðΩ,F ,PÞ be a probability space and fFngn≥0 be a nondecreasing sequence of sub-σ-algebras of F such that F = σð ∪ n≥0 FnÞ. We assume that every σ-algebra Fn is generated by countable atoms, where B ∈Fn is called an atom, if any A ⊂ B with A ∈Fn satisfies PðAÞ = 0 or PðAÞ =PðBÞ. Denote by AðFnÞ the set of all atoms in Fn. For p ∈ 1⁄21,∞Þ and μ ∈ ð−∞,∞Þ, martingale Morrey space Lp,μðΩÞ consists of all f ∈ L1ðΩÞ having the finite norm


Introduction
A real-valued function f is said to belong to the Morrey space L p,λ ðℝ N Þ on the N-dimensional Euclidean space ℝ N provided the following norm is finite: Here 1 ≤ p < ∞, 0 ≤ λ ≤ N, ℝ + = ð0,∞Þ, and Bðx, rÞ are a ball in ℝ N centered at x of radius r. This class of functions was first introduced by Morrey [1] in order to study regularity problem arising in Calculus of Variations, describe local regularity more precisely than Lebesgue spaces. In the past, Morrey spaces have been studied heavily, such as the maximal operators, fractional integral operators, and singular operators. The results are extensively applied not only in partial differential equations but also in harmonic analysis. We refer the readers to [2,3] and the references therein.
The Morrey spaces on Euclidean spaces have been developed to the generalization versions, for example, the generalized Morrey spaces [4,5], the Orlicz-Morrey spaces [6,7], the Triebel-Lizorkin-Morrey spaces [8], and the variable exponent Morrey spaces [9]. Especially, Meskhi [10] introduced the grand Morrey spaces and established the boundedness of the Hardy-Littlewood maximal, Calderón-Zygmund, and potential operators in these spaces. The generalized grand Morrey spaces in a general setting of the quasi-metric measure spaces are studied by Kokilashvili et al. [11,12].
Moreover, in probability theory, Nakai and Sadasue [13] introduced Morrey spaces of martingales as the following: Let ðΩ, F, ℙÞ be a probability space and fF n g n≥0 be a nondecreasing sequence of sub-σ-algebras of F such that We assume that every σ-algebra F n is generated by countable atoms, where B ∈ F n is called an atom, if any A ⊂ B with A ∈ F n satisfies ℙðAÞ = 0 or ℙðAÞ = ℙðBÞ. Denote by AðF n Þ the set of all atoms in F n . For p ∈ ½1,∞Þ and μ ∈ ð−∞, ∞Þ, martingale Morrey space L p,μ ðΩÞ consists of all f ∈ L 1 ðΩÞ having the finite norm They introduced some basic properties of the martingale Morrey spaces. Furthermore, the Doob maximal inequality was established, and the mapping properties for the fractional integral operators were investigated on these spaces. Two generalized versions of them introduced in [14,15]. Ho [16] presented atomic decompositions of martingale Hardy-Morrey spaces. Later on, he [17] introduced a version of martingale Morrey spaces equipping with Banach function spaces. Jiao et al. [18] studied the maximal operator, atom decompositions, and fractional integral operators on martingale Morrey spaces with variable exponents.
Recently, Deng and Li [19] studied the Doob maximal operator and fractional integral operator in the framework of grand Morrey-martingale spaces associated with an almost decreasing function. Moreover, compared with classical martingale spaces, the grand martingale spaces have not of absolutely continuous norm based on [20]. Consequently, we need a further research about grand martingale spaces. Motivated by the works of this and [11], the paper is to investigate the generalized grand Morrey space theory for the martingale setting. More precisely, we first introduce the generalized grand Morrey-martingale spaces and then establish the Doob maximal inequality in this new framework. As an application, we discuss the boundedness of fractional integral operators for regular martingales in the generalized grand Morrey-martingale spaces.

Preliminaries
Now we recall some standard notations from martingale theory. Refer to [21,22] for more information on martingale theory. The expectation is denoted by E with respect to ðΩ , F, ℙÞ. Recall that the conditional expectation operator relative to F n is denoted by E n , i.e., Eð f jF n Þ = E n ð f Þ. A sequence of measurable functions f = ð f n Þ n≥0 ⊂ L 1 ðΩÞ is called a martingale with respect to ðF n Þ n≥0 if E n ð f n+1 Þ = f n for every n ≥ 0: Let M be the set of all martingale f = ð f n Þ n≥0 relative to ðF n Þ n≥0 such that f 0 = 0. For f ∈ M, denote its martingale difference by d n f = f n − f n−1 (n ≥ 0, with convention d 0 f = 0).
The maximal function of f ∈ M is defined by For p > 1 and f ∈ L p ðMÞ, we have which is well known in the literature as the Doob maximal inequality (see [22]). Hence, it follows from the above inequality that if p ∈ ð 1,∞Þ, then L p -bounded martingale converges in L p . Moreover, if p ∈ ½1,∞Þ, then, for any f ∈ L p , its corresponding martingale ðf n Þ n≥0 with f n = E n f is an L p -bounded martingale and converges to f in L p (see [21]). For this reason, a function f ∈ L 1 and the corresponding martingale ðf n Þ n≥0 will be denoted by the same symbol f .
It is convenient for us to state the generalized grand Morrey-martingale spaces, we first need to recall the definition of martingale Morrey spaces L p,λ = L p,λ ðΩÞ as follows.

2
Journal of Function Spaces The stochastic basis fF n g n≥0 is said to be regular, if there exists a constant R ≥ 2 such that holds for all nonnegative martingale ðf n Þ n≥0 adapted to fF n g n≥0 . For regular stochastic basis, there has the following property, proved in [13].
Lemma 5. Let fF n g n≥0 be regular. Then, for every sequence we have where R is the positive constant in (9).

The Doob Maximal Operator
In this section, we present the boundedness of Doob's maximal operator on generalized grand Morrey-martingale spaces.
In order to prove Theorem 6, we need the following useful lemma: Proof. For any B ∈ AðF m Þ and m ≥ 0, suppose that f = g + h and g = f χ B . Then, according to the well-known Doob's maximal inequality, that is, we have Hence, Next, take B n ∈ AðF n Þ, If n < m, according to Jensen's inequality, then where the last inequality dues to 0 ≤ λ < 1 and ℙðBÞ ≤ ℙðB n Þ. This means Then, we obtain Combining inequalities (17) and (21) and Mf ≤ Mg + Mh, we can get The proof is complete.
Note that Nakai and Sadasue [13] proved that, for 1 < p < ∞ and −1/p ≤ μ < 0, The proof of Lemma 7 is devoted to determination of the constant C p . Now we prove Theorem 6:

Journal of Function Spaces
Proof. Let 0 < θ < s, and we have where α = sup fx > 0 : φðxÞ ≤ λg. Let Note that the function hðεÞ ≔ ε δ/ðp−εÞ is increasing in 0 < ε < p, which means where Note that for θ ≤ ε, Taking the infimum over all θ, we obtain that where

The Fractional Integral Operator
In this section, we present the boundedness of the fractional integral operator in the new type grand Morrey-martingale spaces. In martingale theory, Chao and Ombe [26] introduced the fractional integrals for dyadic martingales. The fractional integrals in this section are defined for more general martingale setting as in [13,14] (see also [15,[27][28][29][30][31][32]).
Definition 8. Let f = ð f n Þ n≥0 ∈ M and ι > 0, and the fractional integral I ι f = ððI ι f Þ n Þ n≥0 of martingale f is defined by where b k is an F k -measurable function such that Remark 9. Obviously, b k is bounded in above definition; there The following lemma was shown in [13]. Here we focus on more accurate upper boundedness.
Proof. Since k f k L q 1 ,λ ≤ k f k L q 2 ,λ for q 1 ≤ q 2 by Hölder's inequality, it is enough to prove it in the case where q = ðv/ uÞp. Without loss of generality, we let kf k L p,v ≠ 0.
First, we prove the following inequality holds for any n ≥ 1, and any B n ∈ AðF n Þ, Choose B k ∈ AðF k Þ and 0 ≤ k < n, such that B n ⊂ B n−1 ⊂ ⋯ ⊂ B 0 , and let where 0 = k 0 < k 1 < k 2 < ⋯ < k h . Since fF n g n≥0 is regular, according to Lemma 5, we have So, for k ∉ K, we have b k = b k−1 and d k f = 0. Hence, we obtain For ω ∈ B n , Then, for ω ∈ B n and when 0 < k ≤ m where m ≤ n, For ω ∈ B n and when m + 1 ≤ k < n, let jðkÞ = min fj : k < k j g, and we have Now let Next we estimate ðI ι f Þ n from the following cases. For the first case, if ω ∈ Λ 1 ∩ B n and then, by formula (40) and u = v + ι < 0, we have

Journal of Function Spaces
For the second case, if ω ∈ Λ 1 ∩ B n and then there exists m such that Combining (46) with formulas (40) and (41), we have For the third case, if ω ∈ Λ 2 ∩ B n , then by (41), we have Formulas (44), (47), and (48) give that which implies that Moreover, by Lemma 7, we have It follows from the above inequality and (50) that that is to say, The proof is complete.