Real interpolation of variable martingale Hardy spaces and BMO spaces

In this paper, we mainly consider the real interpolation spaces for variable Lebesgue spaces defined by the decreasing rearrangement function and for the corresponding martingale Hardy spaces. Let 0<q≤∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0<q\le \infty $$\end{document} and 0<θ<1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0<\theta <1$$\end{document}. Our three main results are the following: (Lp(·)(Rn),L∞(Rn))θ,q=Lp(·)/(1-θ),q(Rn),(Hp(·)s(Ω),H∞s(Ω))θ,q=Hp(·)/(1-θ),qs(Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}{} & {} ({\mathcal {L}}_{p(\cdot )}({\mathbb {R}}^n),L_{\infty }({\mathbb {R}}^n))_{\theta ,q}={\mathcal {L}}_{{p(\cdot )}/(1-\theta ),q}({\mathbb {R}}^n),\\{} & {} ({\mathcal {H}}_{p(\cdot )}^s(\Omega ),H_{\infty }^s(\Omega ))_{\theta ,q}={\mathcal {H}}_{{p(\cdot )}/(1-\theta ),q}^s(\Omega ) \end{aligned}$$\end{document}and (Hp(·)s(Ω),BMO2(Ω))θ,q=Hp(·)/(1-θ),qs(Ω),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} ({\mathcal {H}}_{p(\cdot )}^s(\Omega ),BMO_2(\Omega ))_{\theta ,q}={\mathcal {H}}_{{p(\cdot )}/(1- \theta ),q}^s(\Omega ), \end{aligned}$$\end{document}where the variable exponent p(·)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p(\cdot )$$\end{document} is a measurable function.


Introduction
In the past three decades, as a generalization of the classical Lebesgue spaces L p (R n ), the variable exponent Lebesgue spaces L p(·) (R n ) have attracted much attention. Here, we refer the interested readers to the monographs [4,5] for more information. Kempka and Vybíral investigated the Lorentz spaces L p(·),q (R n ) and L p(·),q(·) (R n ) with variable exponents in [19]. Recently, a new kind of variable Lebesgue spaces and variable Lorentz spaces, which are defined by rearrangement functions, came into sight: Kokilashvili et al. [20] introduced the variable Lebesgue spaces L p(·) (R n ); Ephremidze et al. [7] studied the variable Lorentz spaces L p(·),q(·) (R n ).
The classical martingale theory was systematically studied by Garsia [9], Long [21], Weisz [27] and many others. With the development of variable Lebesgue spaces in harmonic analysis, variable martingale spaces have gained a steadily increasing interest. Let ( , F, P) be a complete probability space and let p(·) be a measurable function on . Similar to L p(·) (R n ) and L p(·),q (R n ), we can define L p(·) ( ) and L p(·),q ( ) (see the definitions of these spaces in Sect. 2.1). Aoyama [1] proved the boundedness of the Doob maximal operator on L p(·) ( ) as p(·) meets certain condition, which was pointed out to be quite strong by Nakai and Sadasue [22]. Jiao et al. [17] introduced variable martingale Hardy spaces associated with L p(·) ( ). Very recently, Jiao et al. provided a relatively complete research on variable Hardy-Lorentz spaces relative to L p(·),q ( ) in [14]. Now let p(·) be a variable exponent on [0, 1]. With the emergence of variable Lebesgue spaces defined by the rearrangement function, Jiao et al. [16] introduced new variable martingale Hardy spaces associated with L p(·) ( ) and Zeng [28] investigated variable martingale Hardy-Lorentz spaces relative to L p(·),q ( ) (for the definitions of these spaces, see Sects. 2.2 and 2.3). The readers may consult the articles [11,15] for more results about variable martingale spaces.
With the help of a new sharp maximal function and a new BMO space, Weisz identified the real interpolation spaces between martingale Hardy and BMO spaces in [26]. For more results about the real interpolation of martingale Hardy spaces, we refer to [12,[23][24][25].
Based on the works of Jiao et al. [16] and Zeng [28], our main purpose of this article is to establish the real interpolation spaces between variable martingale Hardy spaces H s p(·) ( ) defined in [16] and martingale BMO spaces B M O 2 ( ). To this end, we firstly identify the interpolation spaces between variable Lebesgue spaces L p(·) (R n ) and L ∞ (R n ) spaces as variable Lorentz L p(·),q (R n ) spaces, and then via formulating the real interpolation spaces between variable martingale Hardy spaces and applying the sharp maximal functions of martingales, we further prove that the real interpolation spaces between variable martingale Hardy and BMO spaces are just the variable martingale Hardy-Lorentz spaces. More precisely, we obtain the following results: see the definitions in Sect. 2. This paper is organized as follows. In Sect. 2, we present the necessary background and some basic facts that will be used later. Section 3 is devoted to establishing the real interpolation between L p(·) (R n ) spaces and L ∞ (R n ) spaces. In Sect. 4, we aim at identifying the real interpolation spaces between variable martingale Hardy spaces. Finally, we formulate the real interpolation between variable martingale Hardy spaces and martingale BMO spaces in Sect. 5. Now, let us make some conventions to end this section. In the whole article, we use Z and N to denote the integer set and nonnegative integer set, respectively. We denote by C an absolute positive constant that is independent of the main parameters but whose value may differ from line to line, and denote by C p(·) the constant depending only on p(·). The symbol a b stands for the inequality a ≤ Cb or a ≤ C p(·) b. If we write a ≈ b, it means that a b a. The characteristic function of a measurable set A is written as χ A .

Preliminaries
In this section, we mainly provide some preparations for the follow-up work. We divide this section into four subsections. Throughout the paper, we always assume that ( , F, P) is a complete probability space. Let (R, μ) be a complete measure space. In this section, (R, μ) could be ( , F, P) or the Euclidean space (R n , m) (n ≥ 1), where m denotes the Lebesgue measure. In Sect. 2.1, we give the definitions of variable Lebesgue spaces L p(·) (R) and variable Lorentz spaces L p(·),q (R). In Sect. 2.2, we introduce the variable Lebesgue spaces L p(·) (R), the variable Lorentz spaces L p(·),q (R) and L p(·),q (R), which are defined by rearrangement function, and give some useful properties about these spaces. To our surprise, the variable Lorentz space L p(·),q ( ) is equivalent to the classical Lorentz space L p(0),q ( ) if p(·) satisfies the locally log-Hölder condition. In Sect. 2.3, we introduce the variable martingale Hardy spaces and variable martingale Hardy-Lorentz spaces associated with variable Lebesgue spaces L p(·) ( ) and variable Lorentz spaces L p(·),q ( ), respectively. Finally, we recall some basic notations and results about real interpolation in Sect. 2.4.

Variable Lebesgue spaces L p(·) (R) and variable Lorentz spaces L p(·),q (R)
The so-called variable exponent function (or simply variable exponent) on R is a measurable function p(·) : R → (0, ∞). Let P(R) be the collection of all variable exponents on R. For a variable exponent p(·) ∈ P(R) and a set E ⊂ R, we denote For simplicity, we adopt the following notation in the sequel: Moreover, we recall the definition of locally log-Hölder continuous condition for the variable exponent p(·) defined on R n .

Remark 2.2
Let p(·) ∈ B(R n ). It was proved in [4, Proposition 2.3] that p(·) satisfies the locally log-Hölder continuous condition is equivalent to 1 p(·) satisfies the locally log-Hölder continuous condition. Now the definitions of variable Lebesgue spaces L p(·) (R) and variable Lorentz spaces L p(·),q (R) are given as follows.

Definition 2.3
Given p(·) ∈ B(R), the variable Lebesgue space L p(·) (R) is defined to be the collection of all measurable functions f on R such that f L p(·) (R) < ∞, where

Definition 2.4
Let p(·) ∈ B(R) and 0 < q ≤ ∞. Then the variable Lorentz space L p(·),q (R) is defined as the set of all measurable functions f on R such that If p(·) ≡ p (0 < p < ∞), then the variable Lebesgue space L p(·) (R) and variable Lorentz space L p(·),q (R) reduce to the classical Lebesgue space L p (R) and Lorentz space L p,q (R), respectively.

Variable Lebesgue spaces L p(·) (R) and variable Lorentz spaces L p(·),q (R), L p(·),q (R)
Let f be a measurable function defined on (R, μ). The distribution function of f is given by While the decreasing rearrangement function of f is defined as In addition, as the maximal function of f * , the function f * * is defined by It is well known that f * * is non-negative, non-increasing and continuous on (0, ∞). Moreover, We refer to [2] for these properties. Next, we use the rearrangement function to define respectively the variable Lebesgue spaces L p(·) (R) and variable Lorentz spaces L p(·),q (R).

Definition 2.5
Let p(·) ∈ B([0, ∞)). We define the variable Lebesgue space L p(·) (R) as the collection of all measurable functions f on R such that This is a quasi-Banach space with respect to the quasi-norm

Definition 2.6
Let p(·) ∈ B([0, ∞)) and 0 < q ≤ ∞. Then the variable Lorentz space L p(·),q (R) is defined as the set of all measurable functions f on R such that Obviously, when p(·) ≡ p (0 < p < ∞), the variable Lebesgue spaces L p(·) (R) and variable Lorentz spaces L p(·),q (R) respectively go back to the classical Lebesgue spaces L p (R) and Lorentz spaces L p,q (R).
The following two useful lemmas can be founded in [16] and [28], respectively.
The following result tells us that is locally log-Hölder continuous.

Definition 2.10
Let p(·) ∈ B([0, 1]) and 0 < q ≤ ∞. Then we define the variable Lorentz space L p(·),q ( ) as the space of all measurable functions f on such that Based on this, we further prove that L p(·),q ( ) = L p(0),q ( ) with equivalent quasi-norms under the same condition as in the lemma above.
By (2.1), it is easy to see that Now let us consider the functions Then is bounded on [0, 1/2). Hence, there exist two positive constants C and C such that Now the proof is complete.

Variable martingale Hardy spaces
In this subsection, we give some basic notions and notations for martingales. We refer the interested readers to monographs [9,21,27] for further study. Let ( , F, P) be a complete probability space and (F n ) n≥0 be an increasing sequence of sub-σ -algebras of F such that F = σ n≥0 F n . The expectation operator and the conditional expectation operator relative to F n are written as E and E n , respectively. A sequence f = ( f n ) n≥0 of adapted and integrable functions is said to be a martingale with respect to (F n ) n≥0 if E n ( f n+1 ) = f n for all n ≥ 0. For a martingale f = ( f n ) n≥0 , the martingale differences are given by d n f = f n − f n−1 (with the convention d 0 f = 0 and f −1 = 0). Let T be the set of all stopping times with respect to (F n ) n≥0 . For a martingale f = ( f n ) n≥0 and a stopping time τ ∈ T , we denote the stopped martingale by The maximal function, the square function and the conditional square function of a martingale f = ( f n ) n≥0 are defined respectively as follows: Denote by the set of all sequences (λ n ) n≥0 of non-negative, non-decreasing and adapted functions with λ ∞ = lim n→∞ λ n . Let p(·) ∈ B [0, 1] and let 0 < q ≤ ∞. Denote variable exponent p(·) or ( p(·), q) by θ . Define the variable martingale Hardy spaces associated with the variable Lebesgue spaces L θ ( ) as follows: In [16,Theorem 2.22], the authors obtained the Doob's inequality for variable martingale spaces L p(·) ( ).

Lemma 2.13
Let p(·) ∈ B([0, 1]) satisfy the locally log-Hölder continuous condition. Then there exists a positive constant C p(·) such that, for every martingale f ∈ L p(·) ( ), To establish the interpolation theorems, we need the atomic characterizations of variable martingale Hardy-Lorentz spaces. First, let us recall the definition of an atom. where (a k ) k∈Z is a sequence of (i, p(·), ∞)-atoms with respect to the stopping time sequence (τ k ) k∈Z and μ k = 3 · 2 k χ {τ k <∞} L p(·) ( ) (k ∈ Z). Endow this space with the following quasi-norm where the infimum is taken over all the decompositions of f by the form (2.7).
with equivalent quasi-norms.

Real interpolation
In this subsection, we collect some basic concepts and results about real interpolation theory. For the details, we refer to the monographs [2,3]. Let (Y 0 , Y 1 ) be a compatible couple of quasi-normed spaces, namely, Y 0 and Y 1 can be embedded continuously into a topological vector space Y . Define the sum of Y 0 and Y 1 as For any t ∈ (0, ∞) and f ∈ Y 0 + Y 1 , the Peetre K -functional is defined by is finite. We adopt the conventions (Y 0 , Y 1 ) 0,q = Y 0 and (Y 0 , Y 1 ) 1,q = Y 1 for each 0 < q ≤ ∞. Now we give two basic properties of K and (Y 0 , Y 1 ) θ,q , which may be used in the sequel.
where → denotes the continuous embedding relationship.
We take the following reiteration lemma and Wolff's lemma from [2].

Interpolation between L p(·) (R n ) and L ∞ (R n )
Before we formulate the interpolation between variable martingale Hardy spaces, we need to establish the real interpolation between L p(·) (R n ) and L ∞ (R n ) spaces. The main result of this section is stated as follows.

Remark 3.2
With only minor modifications to the following proof of Theorem 3.1, we can easily find that Theorem 3.1 is also true for general measure spaces (R, μ).
To prove the theorem above, we need the following technical lemma.

Lemma 3.3 For any t
Proof We show the above inequality in two steps.
Step 1: In this step, we show that Then we have f * 0 ∈ L p(·) ([0, ∞)) and From the property of the K -functional, we obtain

Taking the infimum over all decompositions
Step 2: By Step 1, it is enough to prove that We can show easily (see also the proof of [19, Theorem 4.1]) that First, we note that, for every g ∈ L p(·) ([0, ∞)),
Now we are ready to prove Theorem 3.1.

Proof of Theorem 3.1
Step 1: In this step, we shall prove For this, it suffices to show that where the last inequality is referred to [19, p. 948]. On the other hand, it follows from Lemma 3.3 that see also [19, p. 948] for the details. Thus (3.7) is valid.
Step 2: In this step, let us verify the reverse embedding, that is, By (2.4), it is enough to check the following inequality: To this end, we begin with a reformulation of K (t, f , L p(·) (R n ), L ∞ (R n )). Similarly to (3.3), By Proposition 2.9, we have For fixed t > 0, we choose μ = μ(t) by Thus, Applying (3.9), one can deduce that and choose δ ∈ 0, θ 1−θ . A similar approach combined with Hölder's inequality gives that On the other hand, using (2.3), we find that Hence, (3.8) holds. The proof is complete.

Interpolation between variable martingale Hardy spaces
This section is devoted to identifying the interpolation spaces between variable martingale Hardy spaces. The following theorem is the main result of this section. B([0, 1]) satisfy the locally log-Hölder continuous condition,

Remark 4.4
Similarly, by Lemma 2.17, one can deduce that and Proof of Theorem 4. 1 We are going to show the theorem for H s p(·) ( ) only, since the proofs for the other two spaces are similar. Firstly, let us prove Consider the operator T : f → s( f ). Note that both T : H s p(·) ( ) → L p(·) ( ) and H s ∞ ( ) → L ∞ ( ) are bounded. It follows from the interpolation theorem and Theorem 3.1 that is bounded as well, which means that Conversely, we show that or, equivalently, Let f ∈ H sp (·),q ( ). Applying Lemma 4.3, we have, for every t > 0, Then Now an argument similar to Step 2 of the proof in Theorem 3.1 allows us to prove (4.2) as well as the theorem.

Interpolation between variable martingale Hardy spaces and BMO spaces
In this section, we aim at formulating the real interpolation between variable martingale Hardy spaces and martingale BMO spaces. Recall that, for any r ∈ [1, ∞), the space B M O r ( ) is defined to be the collection of all martingales f ∈ L r ( ) such that To be precise, we mainly obtain the following theorem. Proof Note that Since r ∈ (0, p − ), it follows immediately from (2.3) and Lemma 2.13 that which completes the proof.
To prove this proposition, we need the following lemma.
Based on the results above, we are ready to prove Theorem 5.1.

Proof of Theorem 5.1
Let us prove this theorem in two steps.
Step 1: In this step, we verify Theorem 5.1 for the case p − > 1. From (5.1) and the boundedness of the Doob maximal operator on L ∞ ( ), it follows that In other words, Since r ≥ 1, from this and Proposition 5.4, one can conclude that Consequently, (H s p(·) ( ), B M O 2 ( )) θ,q = H sp (·),q ( ) for the case p − > 1, which finishes the proof of Step 1.
Recall that the stochastic basis (F n ) n≥0 is said to be regular, if for every n ≥ 0 and A ∈ F n , there exists B ∈ F n−1 such that A ⊂ B and P(B) ≤ K P(A), where K is a positive constant independent of n and the choices A and B; see [21]. From [16,Theorem 5.4], it follows that, if (F n ) n≥0 is regular and p(·) ∈ B([0, 1]) satisfies the locally log-Hölder continuous condition, then, with equivalent quasi-norms; see [28,Theorem 4.4]. A combination of these results and Theorem 5.1 immediately yields the following corollary. Applying Lemma 2.18 together with Theorem 5.1 and Corollary 5.6, one can further obtain the real interpolation between variable martingale Hardy-Lorentz spaces and martingale BMO spaces.
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