1 Introduction

Martingale inequalities occupy an important position in martingale space theory. The topic we shall touch here is the \(\Phi\)-moment martingale inequalities in the framework of Lorentz spaces with variable exponents. In 1970, Burkholder and Gundy [6] first discussed the \(\Phi\)-moment inequalities for martingales. Then the \(\Phi\)-moment version of the Burkholder–Davis–Gundy inequality was proved in [5]. Kikuchi [29] proved the \(\Phi\)-moment martingale inequalities in the framework of rearrangement invariant Banach function spaces. It should be noticed that the functions \(\Phi\) in the articles mentioned above are convex functions. In 2012, Jiao and Yu [24] proved some \(\Phi\)-moment martingale inequalities associated with concave functions. Later, Peng and Li [40] extended the results in [24] to the framework of Lorentz spaces. Wu et al. [47] deduced some modular martingale inequalities in the framework of Orlicz–Karamata spaces. Jiao et al. [18] proved the \(\Phi\)-moment version of Burkholder’s inequalities in rearrangement invariant spaces. Some \(\Phi\)-moment inequalities for noncommutative martingales (see [4]) and independent and freely independent random variables (see [19]) were also deduced. We point out that the results on \(\Phi\)-moment martingale inequalities mentioned above are all in the framework of rearrangement invariant spaces. Hence one natural question arises, that is,

Do \(\Phi\)-moment martingale inequalities also hold in the non-rearrangement invariant setting?

In this article, we give an affirmative answer. More precisely, we shall extend the \(\Phi\)-moment martingale inequalities to the framework of function spaces of variable exponents. As is well known, Lebesgue space with variable exponent is an important kind of non-rearrangement invariant spaces and has been widely used in elasticity, fluid dynamics, calculus of variations, differential equations and so on, see, for example, [2, 12, 42, 51, 52]. Such spaces were first introduced by Orlicz [39] in 1931. Kováčik and Rákosník [30], Fan and Zhao [13] investigated various properties of variable Lebesgue spaces and variable Sobolev spaces. A fundamental breakthrough of the study of Lebesgue spaces with variable exponents is due to Diening [9, 10], who proposed the so-called log-Hölder continuity condition to obtain the boundedness of the Hardy–Littlewood maximal operator. Since then, much progress has been made in variable exponent function space theory. Many important results such as atomic decomposition, boundedness of singular integral operator, Littlewood–Paley characterization, dual theory, and so on, have been extended to the variable exponent setting, see, for example, [7, 8, 11, 38, 43, 49]. As generalizations of variable Lebesgue spaces and variable Hardy spaces, respectively, variable Lorentz spaces and variable Lorentz–Hardy spaces have also been studied by many authors. Kempka and Vybíral [28] showed many important properties of Lorentz spaces with variable exponents. Jiao et al. [27] proved the maximal function characterizations, atomic decompositions, interpolation results, duality results, Littlewood–Paley function characterizations, boundedness of singular integral operators for variable Lorentz–Hardy spaces. We refer the reader to [1, 31,32,33, 48, 53] for more information about variable Lorentz spaces and variable Lorentz–Hardy spaces.

Inspired by the considerable progress of function space theory in the variable exponent setting, the martingale space theory in the variable exponent setting has gained a lot of interests in recent years. Aoyama [3] proved the Doob maximal inequality under some strict restrictions on the variable exponent \(p(\cdot ).\) This is the first attempt to study martingale space theory in the variable exponent setting. However, the condition in [3] is too strong. Indeed, Nakai and Sadasue [37] gave a counterexample to show that the condition in [3] is not necessary for the boundedness of the Doob maximal operator on variable Lebesgue spaces. The major difficulties to study the martingale space theory in the variable exponent setting are, on one hand, abstract probability spaces do not enjoy natural metric structure, and thus the log-Hölder continuity condition cannot be well defined any more; on the other hand, the arguments in the classical setting are no longer efficient and the essential reason is that the variable Lebesgue spaces and variable Lorentz spaces are not rearrangement invariant spaces. To overcome these difficulties, Jiao et al. [25] introduced a condition without metric characterization to replace the log-Hölder continuity condition in some sense. Under this new condition, they proved the weak-type and strong-type estimates of the Doob maximal operator, established the atomic decompositions and obtained duality theorems as well as John–Nirenberg inequalities for the martingale Hardy spaces with variable exponents. Subsequently, the variable martingale Hardy spaces and variable martingale Lorentz–Hardy spaces were systematically studied in [21]. We refer the reader to [15, 20, 22, 26, 45, 46] for more information about martingale Hardy spaces and martingale Lorentz–Hardy spaces in the variable exponent setting.

We consider the \(\Phi\)-moment martingale inequalities associated with concave functions in the framework of variable Lorentz spaces. Our main method is to establish some new atomic decomposition theorems by simple atoms. As far as we know, this is the first paper which deals with the \(\Phi\)-moment martingale inequalities in the variable exponent setting. It should be pointed out that the atomic decomposition theorems of this paper improve the atomic decomposition theorems in [21, 22]. Furthermore, let \(p(\cdot )\equiv p,\) our results greatly broaden the scope of the atomic decomposition theorems in [40]. In [40], the authors just considered the atomic decomposition theorems under the conditions of \(0<q\le 1\) and \(q\le p<\infty .\) They guessed the restricted conditions may be removed via the method in [23]. But the method in [23] is only applicable for “\(\infty\)-atom decompositions”. In this paper, we present “r-atom decompositions” \((1<r\le \infty )\) and remove the restricted conditions of \(0<q\le 1\) and \(q\le p<\infty\) in [40] via some refinement techniques.

The paper is organized as follows. In the next section, some preliminaries are introduced. Atomic decompositions for martingale Lorentz–Hardy spaces with variable exponents are presented in Sect. 3. In the final section, with the help of atomic decompositions, the \(\Phi\)-moment martingale inequalities in the framework of Lorentz spaces with variable exponents are deduced.

At the end of this section, we make some conventions. Throughout this paper, the set of nonnegative integers, the set of integers, the set of real numbers and the set of complex numbers are denoted by \({\mathbb{N}},\) \({\mathbb{Z}},\) \({\mathbb{R}}\) and \({\mathbb{C}},\) respectively. We use C to denote the positive constant that is independent of the essential variables involved but whose value may vary from line to line. The symbol \(f \lesssim g\) stands for the inequality \(f \le C g.\) If \(f \lesssim g \lesssim f,\) then we write \(f \approx g.\)

2 Preliminaries

In this section, we give preliminaries necessary to the whole paper.

2.1 Variable Lebesgue spaces and variable Lorentz spaces

Let \((\Omega , {{\mathcal{F}}}, {\mathbb{P}})\) be a complete probability space and \(\{{{\mathcal{F}}}_n\}_{n\ge 0}\) be a nondecreasing sequence of sub-\(\sigma\)-algebras of \({{\mathcal{F}}}\) such that \({{\mathcal{F}}}=\sigma \big (\bigcup _{n\ge 0}{{\mathcal{F}}}_n\big ).\) A measurable function \(p(\cdot ): \Omega \rightarrow (0, \infty )\) is called a variable exponent. For \(A\in {{\mathcal{F}}},\) we denote

$$\begin{aligned} p_-(A)=\inf _{x\in A}p(x),\quad p_+(A)=\sup _{x\in A}p(x) \end{aligned}$$

and for convenience

$$\begin{aligned} p_-=p_-(\Omega ),\quad p_+=p_+(\Omega ). \end{aligned}$$

Let \({\mathcal{P}}={\mathcal{P}}(\Omega )\) denote the collection of all variable exponents \(p(\cdot )\) such that \(0<p_-\le p_+<\infty .\) The variable Lebesgue space \(L_{p(\cdot )}=L_{p(\cdot )}(\Omega , {{\mathcal{F}}}, {\mathbb{P}})\) is defined as the space of all measurable functions f such that for some \(\lambda >0,\)

$$\begin{aligned} \rho \left( \frac{f}{\lambda }\right) =\int _{\Omega } \left( \frac{|f(x)|}{\lambda }\right) ^{p(x)}{\mathrm{d}}{\mathbb{P}}<\infty . \end{aligned}$$

Then, the space \(L_{p(\cdot )}\) becomes a (quasi-)Banach function space when it is equipped with the (quasi-)norm

$$\begin{aligned} \Vert f\Vert _{p(\cdot )}=\inf \left\{ \lambda >0: \rho \left( \frac{f}{\lambda }\right) \le 1\right\} . \end{aligned}$$

Let \(\underline{p}=\min \{p_-, 1\}.\) For \(p_-\ge 1,\) the conjugate variable exponent \(p'(\cdot )\) is defined pointwise by

$$\begin{aligned} \frac{1}{p(\cdot )}+\frac{1}{p'(\cdot )}=1. \end{aligned}$$

The following facts are well known (see [38]).

  1. (i)

    (Positivity) \(\Vert f\Vert _{p(\cdot )} \ge 0\) and \(\Vert f\Vert _{p(\cdot )}=0 \Leftrightarrow f=0\);

  2. (ii)

    (Homogeneity) \(\Vert cf\Vert _{p(\cdot )}=|c| \cdot \Vert f\Vert _{p(\cdot )},\) \((c\in {\mathbb{C}})\);

  3. (iii)

    (The \(\theta\)-triangle inequality) \(\Vert f+g\Vert _{p(\cdot )}^{\theta } \le \Vert f\Vert _{p(\cdot )}^{\theta } +\Vert g\Vert _{p(\cdot )}^{\theta },\) \((0<\theta \le \underline{p}).\)

We collect some useful lemmas, which will be used in the sequel.

Lemma 2.1

(See [8]) Let \(p(\cdot )\in {\mathcal{P}}\) and \(s>0.\) Then for \(f\in L_{sp(\cdot )},\) we have

$$\begin{aligned} \Vert \,|f|^s\,\Vert _{p(\cdot )}=\Vert f\Vert _{sp(\cdot )}^s. \end{aligned}$$

Lemma 2.2

(See [7]) Let \(p(\cdot ) \in {\mathcal{P}}.\) Then for all \(f\in L_{p(\cdot )}\) and \(\Vert f\Vert _{p(\cdot )}\ne 0,\) we have

$$\begin{aligned} \int _{\Omega } \left| \frac{f(w)}{\Vert f\Vert _{p(\cdot )}} \right| ^{p(w)} {\mathrm{d}}{\mathbb{P}}=1. \end{aligned}$$

Lemma 2.3

(See [7, 13]) Let \(p(\cdot ) \in {\mathcal{P}}\) and \(f\in L_{p(\cdot )}.\) Then we have

  1. (1)

    \(\Vert f\Vert _{p(\cdot )}\le 1(=1)\) if and only if \(\int _{\Omega }|f|^{p(\cdot )}{\mathrm{d}}{\mathbb{P}}\le 1(=1);\)

  2. (2)

    If \(\Vert f\Vert _{p(\cdot )}>1,\) then \(\rho (f)^{1/p_+} \le \Vert f\Vert _{p(\cdot )} \le \rho (f)^{1/p_-};\)

  3. (3)

    If \(0<\Vert f\Vert _{p(\cdot )}\le 1,\) then \(\rho (f)^{1/p_-} \le \Vert f\Vert _{p(\cdot )} \le \rho (f)^{1/p_+}.\)

Lemma 2.4

(Hölder’s inequality, see [7]) Let \(p(\cdot ), q(\cdot ), r(\cdot ) \in {\mathcal{P}}\) such that

$$\begin{aligned} \frac{1}{r(x)}=\frac{1}{p(x)}+\frac{1}{q(x)}. \end{aligned}$$

Then for all \(f \in L_{p(\cdot )}\) and \(g\in L_{q(\cdot )},\) we have \(fg \in L_{r(\cdot )}\) and

$$\begin{aligned} \Vert fg\Vert _{r(\cdot )} \le C\Vert f\Vert _{p(\cdot )}\Vert g\Vert _{q(\cdot )}. \end{aligned}$$

We now present the definition of Lorentz spaces with variable exponents. For more information about such spaces, see [28].

Definition 2.5

Let \(p(\cdot ) \in {\mathcal{P}}\) and \(0 < q \le \infty .\) The variable Lorentz space \(L_{p(\cdot ),q}=L_{p(\cdot ),q}(\Omega , {{\mathcal{F}}}, {\mathbb{P}})\) is defined as the space of all measurable functions f such that

$$\begin{aligned} \big \Vert f\big \Vert _{p(\cdot ),q}= \left\{ \begin{array}{ll} \left( \int _0^\infty t^q \Vert \chi _{\{|f|>t\}}\Vert _{p(\cdot )}^q \frac{{\mathrm{d}}t}{t} \right) ^{\frac{1}{q}} &{}\quad \text{if}\ 0<q<\infty ,\\ \sup _{t>0}t\Vert \chi _{\{|f|>t\}}\Vert _{p(\cdot )} &{}\quad \text{if}\ q=\infty , \end{array} \right. \end{aligned}$$

is finite.

Note that \(L_{p(\cdot ),q}\) is a quasi-Banach space for \(p(\cdot ) \in {\mathcal{P}}\) and \(0 < q \le \infty\) (see [28]). Obviously, \(L_{p(\cdot ),q}\) is the generalization of classical Lorentz space \(L_{p,q}\) and it coincides with \(L_{p,q}\) when \(p(\cdot )\equiv p.\) Moreover, the functional \(\Vert \cdot \Vert _{p(\cdot ), q}\) can be discretized as follows.

Lemma 2.6

(See [28]) Let \(p(\cdot )\in {\mathcal{P}}\) and \(0<q\le \infty .\) If \(f\in L_{p(\cdot ), q},\) then

$$\begin{aligned} \big \Vert f\big \Vert _{p(\cdot ),q}\approx \left\{ \begin{array}{ll} \left( \sum _{k\in {\mathbb{Z}}}2^{kq}\Vert \chi _{\{|f|>2^k\}}\Vert _{p(\cdot )}^q \right) ^{\frac{1}{q}}&{}\quad \text{if}\ 0<q<\infty ,\\ \sup _{k\in {\mathbb{Z}}}2^k\Vert \chi _{\{|f|>2^k\}}\Vert _{p(\cdot )}&{} \quad \text{if}\ q=\infty . \end{array} \right. \end{aligned}$$

2.2 Orlicz functions

Let \({\mathcal{G}}\) be the set of all functions \(\Phi : [0, \infty )\rightarrow [0, \infty )\) such that \(\Phi\) is nondecreasing, \(\Phi (0)=0,\) \(\Phi (t)>0\) for all \(t>0\) and \(\Phi (t)\rightarrow \infty\) as \(t\rightarrow \infty .\) We refer the reader to [16, 36, 41, 50] for more information about properties of functions in \({\mathcal{G}}\) and related functions. We have the following simple but useful lemma.

Lemma 2.7

(See [35, 40]) Let \(\Phi \in {\mathcal{G}}\) be concave.

  1. (1)

    If \(0<s\le 1\) and \(t\ge 0,\) then \(\Phi (st)\ge s\Phi (t);\)

  2. (2)

    If \(s\ge 1\) and \(t\ge 0,\) then \(\Phi (st)\le s\Phi (t).\)

Moreover, \(\Phi\) is subadditive, continuous and bijective from \([0, \infty )\) to \([0, \infty ).\)

We now give the following lemma which will be used in the proof of the atomic decomposition theorems in the next section.

Lemma 2.8

Let \(\Phi \in {\mathcal{G}}\) be concave and \(0<p\le r\le \infty .\) For \(f\in L_r,\) there exists \(A\in {{\mathcal{F}}}\) with \({\mathbb{P}}(A)\ne 0\) such that \(\{f\ne 0\}\subset A,\) then

$$\begin{aligned} \Vert \Phi (|f|)\Vert _p\lesssim {\mathbb{P}}(A)^{\frac{1}{p}}\Phi \left( \frac{\Vert f\Vert _r}{{\mathbb{P}}(A)^{\frac{1}{r}}}\right) . \end{aligned}$$

Proof

It follows from Lemma 2.7 and Hölder’s inequality that

$$\begin{aligned} \Vert \Phi (|f|)\Vert _p&\le \bigg \Vert \Phi \left( |f|+\frac{\Vert f\Vert _r}{{\mathbb{P}} (A)^{\frac{1}{r}}}\right) \chi _A\bigg \Vert _p \\ &\le \left\| \left( \frac{|f|{\mathbb{P}}(A)^{\frac{1}{r}}}{\Vert f\Vert _r}+1\right) \Phi \left( \frac{\Vert f\Vert _r}{{\mathbb{P}}(A)^{\frac{1}{r}}}\right) \chi _A\right\| _p \\ &= \Phi \left( \frac{\Vert f\Vert _r}{{\mathbb{P}}(A)^{\frac{1}{r}}}\right) \left\| \left( \frac{|f|{\mathbb{P}}(A)^{\frac{1}{r}}}{\Vert f\Vert _r}+1\right) \chi _A\right\| _p \\ &\lesssim \Phi \left( \frac{\Vert f\Vert _r}{{\mathbb{P}}(A)^{\frac{1}{r}}}\right) \left( \left\| \frac{|f|{\mathbb{P}}(A)^{\frac{1}{r}}}{\Vert f\Vert _r}\chi _A\right\| _p+ {\mathbb{P}}(A)^\frac{1}{p}\right) \\ &\le \Phi \left( \frac{\Vert f\Vert _r}{{\mathbb{P}}(A)^{\frac{1}{r}}}\right) \left( \frac{{\mathbb{P}}(A)^{\frac{1}{r}}}{\Vert f\Vert _r}\Vert f \Vert _r{\mathbb{P}}(A)^{\frac{1}{p}-\frac{1}{r}}+{\mathbb{P}}(A)^{\frac{1}{p}}\right) \\ &\lesssim {\mathbb{P}}(A)^\frac{1}{p}\Phi \left( \frac{\Vert f\Vert _r}{{\mathbb{P}}(A)^{\frac{1}{r}}}\right) . \end{aligned}$$
(2.1)

\(\square\)

2.3 Martingales

Let us recall some standard notations from martingale theory. We refer the reader to [14, 34, 44] for the theory of classical martingale space theory. Let \((\Omega , {{\mathcal{F}}}, {\mathbb{P}})\) and \(\{{{\mathcal{F}}}_n\}_{n\ge 0}\) be stated as in Sect. 2.1. The expectation operator and the conditional expectation operator related to \({{\mathcal{F}}}_n\) are denoted by \({\mathbb{E}}\) and \({\mathbb{E}}_n,\) respectively. A sequence of measurable functions \(f=(f_n)_{n\ge 0}\subset L_1(\Omega , {{\mathcal{F}}}, {\mathbb{P}})\) is called a martingale with respect to \(\{\mathcal{F}_n\}_{n\ge 0}\) if \({\mathbb{E}}_{n}(f_{n+1})=f_n\) for every \(n\ge 0.\) Let \({\mathcal{M}}\) be the set of all martingale \(f=(f_n)_{n\ge 0}\) relative to \(\{{\mathcal{F}}_n\}_{n\ge 0}\) such that \(f_0=0.\) For \(f= (f_n)_{n\ge 0}\in {\mathcal{M}},\) denote its martingale difference by \({\mathrm{d}}f_n=f_n-f_{n-1}\) (\(n\ge 0,\) with convention \(f_{-1}=0).\) Define the maximal function, the square function and the conditional square function of f, respectively, as follows

$$\begin{aligned}&M_m(f)=\sup _{n\le m}{|f_n|},\quad M(f)=\sup _{n\ge 0} |f_n|;\\ &S_m(f)=\left( \sum _{n=0}^m|{\mathrm{d}}f_n|^2\right) ^{1/2},\quad S(f)=\left( \sum _{n=0}^{\infty }|{\mathrm{d}}f_n|^2\right) ^{1/2};\\ &s_m(f)=\left( \sum _{n=0}^{m}{\mathbb{E}}_{n-1}|{\mathrm{d}}f_n|^2\right) ^{1/2}, \quad s(f)=\left( \sum _{n=0}^{\infty }{\mathbb{E}}_{n-1}|{\mathrm{d}}f_n|^2\right) ^{1/2}. \end{aligned}$$

For \(f=(f_n)_{n\ge 0}\in {\mathcal{M}},\) if \(f_n\in L_{p(\cdot )}\) for every \(n\ge 0,\) then f is called an \(L_{p(\cdot )}\)-martingale. Furthermore, if \(\Vert f\Vert _{p(\cdot )}=\sup _{n\ge 0}\Vert f_n\Vert _{p(\cdot )}<\infty ,\) then f is called a bounded \(L_{p(\cdot )}\)-martingale and it is denoted by \(f\in L_{p(\cdot )}.\)

Let \(\Lambda\) be the collection of all sequences \((\lambda _n)_{n\ge 0}\) of nondecreasing, nonnegative and adapted functions, set \(\lambda _{\infty }=\lim _{n\rightarrow \infty }\lambda _n.\) For \(f\in {\mathcal{M}},\) \(\Phi \in {\mathcal{G}},\) \(p(\cdot ) \in {\mathcal{P}}\) and \(0<q\le \infty ,\) let

$$\begin{aligned} \Lambda [{\mathcal{Q}}_{p(\cdot ),q,\Phi }](f)=\big \{(\lambda _n)_{n\ge 0}\in \Lambda : S_n(f)\le \lambda _{n-1}\quad (n\ge 1),\; \Phi (\lambda _{\infty })\in L_{p(\cdot ),q}\big \} \end{aligned}$$

and

$$\begin{aligned} \Lambda [{\mathcal{D}}_{p(\cdot ),q,\Phi }](f)=\big \{(\lambda _n)_{n\ge 0}\in \Lambda : |f_n|\le \lambda _{n-1} \quad (n\ge 1),\; \Phi (\lambda _{\infty })\in L_{p(\cdot ),q}\big \}. \end{aligned}$$

Set

$$\begin{aligned}&\Vert \Phi (f)\Vert _{{\mathcal{Q}}_{p(\cdot ),q}} =\inf _{(\lambda _n)_{n\ge 0}\in \Lambda [{\mathcal{Q}}_{p(\cdot ),q,\Phi }](f)} \Vert \Phi (\lambda _{\infty })\Vert _{{p(\cdot ),q}},\\ &\Vert \Phi (f)\Vert _{{\mathcal{D}}_{p(\cdot ),q}}= \inf _{(\lambda _n)_{n\ge 0}\in \Lambda [{\mathcal{D}}_{p(\cdot ),q,\Phi }] (f)}\Vert \Phi (\lambda _{\infty })\Vert _{{p(\cdot ),q}}. \end{aligned}$$

Now let us recall the notion of the log-Hölder continuity condition. Given a function \(p(\cdot )\in {\mathcal{P}}({\mathbb{R}}^n),\) let \(p_{\infty }=\lim _{x\rightarrow \infty }p(x),\) then \(p(\cdot )\) is said to satisfy the log-Hölder continuity condition, if for all \(x, y\in {\mathbb{R}}^n,\)

$$\begin{aligned} |p(x)-p(y)|\le \frac{C}{\log (e+\frac{1}{|x-y|})} \end{aligned}$$

and

$$\begin{aligned} |p(x)-p_{\infty }|\le \frac{C}{\log (e+|x|)}. \end{aligned}$$

We mention that if \(p_->1\) and \(p(\cdot )\) satisfies the log-Hölder continuity condition in the Euclidean spaces \({\mathbb{R}}^n,\) then the Hardy–Littlewood maximal operator is bounded on \(L_{p(\cdot )}({\mathbb{R}}^n)\) and the inverse Hölder’s inequality holds for the characteristic functions defined on cubes in \(L_{p(\cdot )}({\mathbb{R}}^n)\) (see [8, 38]). Compared with the Euclidean space \({\mathbb{R}}^n,\) the probability space \((\Omega , {{\mathcal{F}}}, {\mathbb{P}})\) has no natural metric structure. Fortunately, Jiao et al. [15, 25, 26] found the following condition without metric characterization to replace the log-Hölder continuity condition in some sense. That is, there exists an absolute constant \(K_{p(\cdot )}\ge 1\) depending only on \(p(\cdot )\) such that

$$\begin{aligned} {\mathbb{P}}(B)^{p_-(B)-p_+(B)}\le K_{p(\cdot )},\quad \forall \;B\in \bigcup _{n\ge 0} A({{\mathcal{F}}}_n), \end{aligned}$$
(2.2)

where \({{\mathcal{F}}}_n\) is generated by countable atoms (\(B\in {{\mathcal{F}}}_n\) is called an atom if any \(A\subset B\) with \(A\in {{\mathcal{F}}}_n\) satisfies \({\mathbb{P}}(A)={\mathbb{P}}(B)\) or \({\mathbb{P}}(A)=0\)) and \(A({{\mathcal{F}}}_n)\) is the set of all atoms in \({{\mathcal{F}}}_n.\) We always assume that every \({{\mathcal{F}}}_n\) is generated by at most countable many atoms in the sequel.

Lemma 2.9

(See [15, 21, 26]) Let \(p(\cdot ) \in {\mathcal{P}}\) satisfy (2.2). Then

  1. (1)

    For \(p_->1\) and \(B\in \bigcup _{n\ge 0} A({{\mathcal{F}}}_n),\) we have

    $$\begin{aligned} {\mathbb{P}}(B)\approx \Vert \chi _B\Vert _{p(\cdot )}\Vert \chi _B\Vert _{p'(\cdot )}. \end{aligned}$$
  2. (2)

    For \(q(\cdot ) \in {\mathcal{P}}\) satisfying (2.2) and \(r(\cdot )\in {\mathcal{P}}.\) If \(r(\cdot )\) satisfies

    $$\begin{aligned} \frac{1}{r(x)}=\frac{1}{p(x)}+\frac{1}{q(x)}, \end{aligned}$$

    then \(r(\cdot )\) also satisfies (2.2). Moreover, for \(B\in \bigcup _{n\ge 0} A({{\mathcal{F}}}_n),\) we have

    $$\begin{aligned} \Vert \chi _B\Vert _{r(\cdot )}\approx \Vert \chi _B\Vert _{p(\cdot )}\Vert \chi _B\Vert _{q(\cdot )}. \end{aligned}$$
  3. (3)

    For \(p_->1\) and \(f\in L_{p(\cdot )},\) we have

    $$\begin{aligned} \Vert M(f)\Vert _{p(\cdot )}\lesssim \Vert f\Vert _{p(\cdot )}. \end{aligned}$$

3 Atomic decompositions

In this section, we construct some new atomic decomposition theorems. The atoms we used here are simple atoms. Note that there are two notions of atoms here, one is a measurable set described in Sect. 2, the other is a measurable function defined as follows.

Definition 3.1

Let \(p(\cdot )\in {\mathcal{P}},\) \(1<r\le \infty\) and \(\Phi \in {\mathcal{G}} .\) A measurable function a is called a simple \((\Phi ,p(\cdot ), r)^s\)-atom (resp. \((\Phi ,p(\cdot ), r)^S\)-atom, \((\Phi ,p(\cdot ), r)^M\)-atom) if there exists \(I\in A({{\mathcal{F}}}_m)\) for some \(m\in {\mathbb{N}}\) such that

  1. 1.

    the support of a is contained in I; 

  2. 2.

    \(\Vert s(a)\Vert _r\) (resp. \(\Vert S(a)\Vert _r,\) \(\Vert M(a)\Vert _r)\) \(\le {\mathbb{P}}(I)^{1/r}\Phi ^{-1}\Big ( \frac{1}{\Vert \chi _I\Vert _{p(\cdot )}}\Big );\)

  3. 3.

    \({\mathbb{E}}_m(a)=0.\)

Given \(p(\cdot )\in {\mathcal{P}}\) and \(0<q \le \infty .\) Denote by \({\mathcal{A}}^s(\Phi ,p(\cdot ),q ,r)\) (resp. \({\mathcal{A}}^S(\Phi ,p(\cdot ),q ,r),\) \({\mathcal{A}}^M(\Phi ,p(\cdot ),q ,r)\)) the collection of all sequences of triples \((a^{k,m,i},I_{k,m,i},\mu _{k,m,i})_{k\in {\mathbb{Z}},m,i\in {\mathbb{N}}},\) where \((a^{k,m,i})_{k\in {\mathbb{Z}}, m, i\in {\mathbb{N}}}\) are simple \((\Phi ,p(\cdot ), r)^s\)-atoms (resp. \((\Phi ,p(\cdot ), r)^S\)-atoms, \((\Phi ,p(\cdot ), r)^M\)-atoms), \((I_{k,m,i})_{k\in {\mathbb{Z}}, m, i\in {\mathbb{N}}}\subset A({{\mathcal{F}}}_m)\) are disjoint for fixed k and satisfying (1),  (2) and (3) in Definition 3.1. Moreover,

$$\begin{aligned} \mu _{k,m,i}=\frac{2\Phi ^{-1}(2^{k+1})}{\Phi ^{-1}\left( \frac{1}{\Vert \chi _{I_{k,m,i}}\Vert _{p(\cdot )}}\right) } \end{aligned}$$

and

$$\begin{aligned} \sum _{k\in {\mathbb{Z}}}\left\| \sum _{m=0}^\infty \sum _i\Phi \left[ \mu _{k,m,i}\Phi ^{-1}\left( \frac{1}{\Vert \chi _{I_{k,m,i}}\Vert _{p(\cdot )}}\right) \right] \chi _{I_{k,m,i}}\right\| ^q_{p(\cdot )}<\infty , \quad (0<q<\infty ), \end{aligned}$$

(with the usual modification for \(q = \infty\)).

Remark 3.2

  1. (i)

    Note that if we consider the special case \(\Phi (t)=t,\) then the simple \((\Phi ,p(\cdot ),r)^{\star }\)-atom \((\star =s,S,M)\) is the same as Definition 3.1 in Jiao et al. [22]. Moreover, if \(\Phi (t)=t,\) \(p(\cdot )\equiv p\) and \(r=\infty ,\) then the simple \((\Phi ,p(\cdot ),r)^{\star }\)-atom \((\star =s,S,M)\) is the same as Definition 2.4 in Weisz [44].

  2. (ii)

    It follows from [22] that if a is a simple \((\Phi ,p(\cdot ),r)^{\star }\)-atom (\(\star =s,S,M,\) resp.) associated with \(I\in A({{\mathcal{F}}}_m)\) for some \(m\in {\mathbb{N}},\) then

    $$\begin{aligned} s(a)=s(a)\chi _I,\quad S(a)=S(a)\chi _I\quad \text{and}\quad M(a)=M(a)\chi _I. \end{aligned}$$
  3. (iii)

    Let \(\Phi \in {\mathcal{G}}\) be concave. If \((a^{k,m,i},I_{k,m,i},\mu _{k,m,i}) \in {\mathcal{A}}^s(\Phi ,p(\cdot ),q ,r),\) then

    $$\begin{aligned} \Phi \left( \frac{\mu _{k,m,i}\Vert s(a^{k,m,i})\Vert _{r}}{{\mathbb{P}}(I_{k,m,i})^{1/r}}\right) \le 2^{k+2}. \end{aligned}$$

Theorem 3.3

Let \(p(\cdot )\in {\mathcal{P}}\) satisfy (2.2), \(\Phi \in {\mathcal{G}}\) be a concave function, \(0 < q \le \infty\) and \(1 < r \le \infty .\) If the martingale \(f=(f_n)_{ n \ge 0 }\) satisfies \(\big \Vert \Phi \big (s(f)\big )\big \Vert _{p(\cdot ),q}<\infty ,\) then there exists a sequence of triples \(( a^{k,m,i}, I_{k,m,i}, \mu _{k,m,i})_{k\in {\mathbb{Z}},m,i\in {\mathbb{N}}}\subset {\mathcal{A}}^s(\Phi ,p(\cdot ),q ,r)\) such that for each \(n \ge 0 ,\)

$$\begin{aligned} f_n= \sum _{ k \in {\mathbb{Z}} }\sum _{m=0}^{n-1}\sum _i \mu _{k,m,i} {\mathbb{E}}_n (a^{k,m,i}) \quad \text{a.e.}, \end{aligned}$$
(3.1)

and

$$\begin{aligned}&\left\{ \sum _{k\in {\mathbb{Z}}}\left\| \sum _{m=0}^\infty \sum _i\Phi \left[ \mu _{k,m,i}\Phi ^{-1}\left( \frac{1}{\Vert \chi _{I_{k,m,i}}\Vert _{p(\cdot )}}\right) \right] \chi _{I_{k,m,i}}\right\| ^q_{p(\cdot )}\right\} ^{1/q}\\ &\quad \lesssim \Vert \Phi (s(f))\Vert _{{p(\cdot ),q}}, \quad (0<q<\infty ), \end{aligned}$$
(3.2)

(with the usual modification for \(q = \infty ).\)

Conversely, if \(\max \{p_{+},1\} <r\le \infty\) and the martingale \(f=(f_n)_{ n \ge 0 }\) has a decomposition of type (3.1), then for \(0<q<\infty ,\)

$$\begin{aligned}&\Vert \Phi (s(f))\Vert _{{p(\cdot ),q}}\\ &\quad \lesssim \inf \left\{ \sum _{k\in {\mathbb{Z}}}\left\| \sum _{m=0}^\infty \sum _i\Phi \left[ \mu _{k,m,i}\Phi ^{-1}\left( \frac{1}{\Vert \chi _{I_{k,m,i}} \Vert _{p(\cdot )}}\right) \right] \chi _{I_{k,m,i}}\right\| ^q_{p(\cdot )}\right\} ^{1/q}, \end{aligned}$$

(with the usual modification for \(q=\infty ),\) where the infimum is taken over all the preceding decompositions of f of the form (3.1).

Proof

Let \(f=(f_n)_{ n \ge 0 }\) be a martingale with \(\big \Vert \Phi \big (s(f)\big )\big \Vert _{p(\cdot ),q}<\infty .\) For every \(k\in {\mathbb{Z}},\) define

$$\begin{aligned} \tau _k=\inf \big \{n\in {\mathbb{N}}: s_{n+1}(f)>\Phi ^{-1}(2^k)\big \}\quad (\inf \emptyset =\infty ). \end{aligned}$$

Apparently, \(\tau _k\) is a stopping time and \(\tau _k\le \tau _{k+1}\) for each \(k\in {\mathbb{Z}}.\) It is easy to see that for each \(n\in {\mathbb{N}},\)

$$\begin{aligned} f_n = \sum _{k\in {\mathbb{Z}}} \big (f_n^{\tau _{k+1}}-f_n^{\tau _{k}}\big ) \quad \text{a.e.}, \end{aligned}$$

where \(f^{\tau _{k}}:=(f_{n\wedge \tau _{k}})_{n\ge 0}.\) Note that for fixed \(k\in {\mathbb{Z}}\) and \(m\in {\mathbb{N}},\) \(\{\tau _k=m\}\in {{\mathcal{F}}}_m.\) Then there exist disjoint atoms \((I_{k,m,i})_i\subset A({{\mathcal{F}}}_m)\) such that

$$\begin{aligned} \{\tau _k=m\}=\bigcup _i I_{k,m,i}. \end{aligned}$$
(3.3)

For each \(n\ge 0,\) set

$$\begin{aligned} \mu _{k,m,i}=\frac{2\Phi ^{-1}(2^{k+1})}{\Phi ^{-1}\left( \frac{1}{\Vert \chi _{I_{k,m,i}}\Vert _{p(\cdot )}}\right) }\quad \text{ and }\quad a_n^{k,m,i}= \frac{ f_n^{\tau _{k+1}}-f_n^{\tau _{k}} }{\mu _{k,m,i}}\chi _{I_{k,m,i}}. \end{aligned}$$

Hence, for each \(n\ge 0,\) \(f_n\) can be represented as follows

$$\begin{aligned} f_n&=\sum _{k\in {\mathbb{Z}}} \big (f_n^{\tau _{k+1}}-f_n^{\tau _{k}}\big )\chi _{\{\tau _k<n\}}\\ &=\sum _{k\in {\mathbb{Z}}} \sum _{m=0}^{n-1}\big (f_n^{\tau _{k+1}}-f_n^{\tau _{k}}\big )\chi _{\{\tau _k=m\}}\\ &=\sum _{k\in {\mathbb{Z}}} \sum _{m=0}^{n-1}\sum _i\big (f_n^{\tau _{k+1}}-f_n^{\tau _{k}}\big )\chi _{I_{k,m,i}}\\ &=\sum _{ k \in {\mathbb{Z}} } \sum _{m=0}^{n-1}\sum _i\mu _{k,m,i} a_n^{k,m,i}. \end{aligned}$$

For fixed k,  m and i,  \((a_n^{k,m,i})_{n \ge 0}\) is a martingale. Moreover, in view of the definition of \(\tau _{k},\) we have \(s(f^{\tau _k}) = s_{\tau _k}(f)\le \Phi ^{-1}(2^k).\) Hence, using the definition of \(\mu _{k,m,i},\) we obtain

$$\begin{aligned} s\big ((a_n^{k,m,i})_{n\ge 0}\big )\le \frac{s(f_n^{\tau _{k+1}})+s(f_n^{\tau _{k}})}{\mu _{k,m,i}} \le \Phi ^{-1}\left( \frac{1}{\Vert \chi _{I_{k,m,i}}\Vert _{p(\cdot )}}\right) . \end{aligned}$$

That is, \((a_n^{k,m,i})_{n\ge 0}\) is an \(L_2\)-bounded martingale. Thus there exists an \(a^{k,m,i}\in L_2\) such that \({\mathbb{E}}_n(a^{k,m,i})=a_n^{k,m,i}\) and

$$\begin{aligned} \Vert s(a^{k,m,i})\Vert _r\le {\mathbb{P}}(I_{k,m,i})^{1/r}\Phi ^{-1}\left( \frac{1}{\Vert \chi _{I_{k,m,i}}\Vert _{p(\cdot )}}\right) . \end{aligned}$$

Furthermore,

$$\begin{aligned} (f_n^{\tau _{k+1}}-f_n^{\tau _{k}})\chi _{I_{k,m,i}}&= \left( \sum ^n_{l=0}\chi _{\{l\le \tau _{k+1}\}}{\mathrm{d}}f_l- \sum ^n_{l=0}\chi _{\{l\le \tau _{k}\}}{\mathrm{d}}f_l\right) \chi _{I_{k,m,i}}\\ &=\chi _{I_{k,m,i}}\sum ^n_{l=0}{\mathrm{d}}f_l\chi _{\{\tau _k<l\le \tau _{k+1}\}}\\ &=\chi _{I_{k,m,i}}\sum ^n_{l=m+1}{\mathrm{d}}f_l\chi _{\{\tau _k<l\le \tau _{k+1}\}}. \end{aligned}$$

Hence, \({\mathbb{E}}_m(a^{k,m,i})=0.\) Thus, we conclude that \(a^{k,m,i}\) is really a simple \((\Phi ,p(\cdot ), r)^s\)-atom.

Now we show (3.2). For \(0<q<\infty ,\) since \(\chi _{\{\tau _k<\infty \}}=\sum _{m=0}^\infty \sum _i\chi _{I_{k,m,i}},\) it follows from \(\{\tau _k<\infty \}=\big \{s(f)>\Phi ^{-1}(2^k)\big \}\) and Lemma 2.7 that

$$\begin{aligned}&\sum _{k\in {\mathbb{Z}}}\left\| \sum _{m=0}^\infty \sum _i\Phi \left[ \mu _{k,m,i}\Phi ^{-1}\left( \frac{1}{\Vert \chi _{I_{k,m,i}}\Vert _{p(\cdot )}}\right) \right] \chi _{I_{k,m,i}}\right\| ^q_{p(\cdot )}\\ &\quad =\sum _{k\in {\mathbb{Z}}}\left\| \sum _{m=0}^\infty \sum _i\Phi \big (2\Phi ^{-1}(2^{k+1})\big ) \chi _{I_{k,m,i}}\right\| ^q_{p(\cdot )}\le \sum _{ k \in {\mathbb{Z}} }2^{(k+2)q}\Vert \chi _{\{\tau _k<\infty \}}\Vert ^q_{p(\cdot )}\\ &\quad =\sum _{ k \in {\mathbb{Z}} }2^{(k+2)q}\Vert \chi _{\{\Phi (s(f))>2^k\}}\Vert ^q_{p(\cdot )} \lesssim \sum _{ k \in {\mathbb{Z}} }\Vert \chi _{\{\Phi (s(f))>2^k\}} \Vert ^q_{p(\cdot )}\int _{2^{k-1}}^{2^k}y^{q}\frac{{\mathrm{d}}y}{y}\\ &\quad \le \sum _{ k \in {\mathbb{Z}} }\int _{2^{k-1}}^{2^k} \Big (y\Vert \chi _{\{\Phi (s(f))>y\}}\Vert _{p(\cdot )}\Big )^{q}\frac{{\mathrm{d}}y}{y} =\int _{0}^{\infty }\Big (y\Vert \chi _{\{\Phi (s(f))>y\}}\Vert _{p(\cdot )}\Big )^{q}\frac{{\mathrm{d}}y}{y}\\ &\quad =\Vert \Phi (s(f))\Vert _{{p(\cdot ),q}}^q. \end{aligned}$$

For \(q=\infty ,\) similarly to the proof of the case \(0<q<\infty ,\) we have

$$\begin{aligned}&\left\| \sum _{m=0}^\infty \sum _i\Phi \left[ \mu _{k,m,i}\Phi ^{-1} \left( \frac{1}{\Vert \chi _{I_{k,m,i}}\Vert _{p(\cdot )}}\right) \right] \chi _{I_{k,m,i}}\right\| _{p(\cdot )}\\ &\quad \le 2^{k+2}\Vert \chi _{\{\tau _k<\infty \}}\Vert _{p(\cdot )} =2^{k+2}\Vert \chi _{\{\Phi (s(f))>2^k\}}\Vert _{p(\cdot )}\\ &\quad \lesssim \Vert \Phi (s(f))\Vert _{{p(\cdot ),\infty }}. \end{aligned}$$

Thus, we conclude that \((a^{k,m,i}, I_{k,m,i}, \mu _{k,m,i})_{k\in {\mathbb{Z}},m,i\in {\mathbb{N}}} \subset {\mathcal{A}}^s(\Phi ,p(\cdot ),q ,r)\) and (3.2) holds.

To prove the converse part, let \(f=(f_n)_{n \ge 0}\) have a decomposition of type (3.1). It follows from the sublinearity of the conditional square operator s and subadditivity of \(\Phi\) that

$$\begin{aligned} \Phi \big (s(f)\big )&\le \Phi \left( \sum _{ k \in {\mathbb{Z}} } \sum _{m=0}^\infty \sum _i\mu _{k,m,i} s(a^{k,m,i})\right) \\ &\le \sum _{ k \in {\mathbb{Z}} }\sum _{m=0}^\infty \sum _i \Phi \big (\mu _{k,m,i} s(a^{k,m,i})\big )\quad \text{a.e.} \end{aligned}$$
(3.4)

For an arbitrary integer \(k_0,\) let

$$\begin{aligned} \sum _{ k \in {\mathbb{Z}} }\sum _{m=0}^\infty \sum _i \Phi \big (\mu _{k,m,i} s(a^{k,m,i})\big )=T_{ 1}+T_{2}, \end{aligned}$$

where

$$\begin{aligned} T_{1}=\sum _{ k =-\infty }^{k_0-1}\sum _{m=0}^\infty \sum _i \Phi \big (\mu _{k,m,i} s(a^{k,m,i})\big ) \end{aligned}$$

and

$$\begin{aligned} T_{2}=\sum _{ k =k_0 }^{\infty }\sum _{m=0}^\infty \sum _i \Phi \big (\mu _{k,m,i} s(a^{k,m,i})\big ). \end{aligned}$$

For \(0<q<\infty .\) Let \(0<\theta < \min \{p_-,q,1\}.\) We can choose \(\lambda\) such that \(1<\lambda < \min \{1/\theta , r/p_+\}.\) Using Hölder’s inequality, we obtain

$$\begin{aligned} T_1&\le \left( \sum _{ k =-\infty }^{k_0-1}2^{k\sigma \lambda '} \right) ^{\frac{1}{\lambda '}} \left\{ \sum _{ k =-\infty }^{k_0-1}2^{-k\sigma \lambda } \left[ \sum _{m=0}^\infty \sum _i\Phi \big (\mu _{k,m,i} s(a^{k,m,i})\big ) \right] ^\lambda \right\} ^{\frac{1}{\lambda }}\\ &=\left( \frac{2^{k_0\sigma \lambda '}}{2^{\sigma \lambda '}-1} \right) ^{\frac{1}{\lambda '}}\left\{ \sum _{ k =-\infty }^{k_0-1}2^{-k\sigma \lambda } \left[ \sum _{m=0}^\infty \sum _i\Phi \big (\mu _{k,m,i} s(a^{k,m,i})\big ) \right] ^\lambda \right\} ^{\frac{1}{\lambda }}, \end{aligned}$$

where \(0<\sigma <1-1/\lambda\) and \(1/\lambda +1/\lambda '=1.\) By the \(\theta\)-triangle inequality and Lemma 2.1, we obtain

$$\begin{aligned}&\Vert \chi _{\{T_1>2^{k_0}\}}\Vert _{p(\cdot )}\le \frac{1}{2^{k_0\lambda }}\Vert T_1^\lambda \Vert _{p(\cdot )}\\ &\quad \lesssim 2^{k_0\lambda (\sigma -1)} \left\| \sum _{ k =-\infty }^{k_0-1}2^{-k\sigma \lambda } \left[ \sum _{m=0}^\infty \sum _i\Phi \big (\mu _{k,m,i} s(a^{k,m,i})\big ) \right] ^\lambda \right\| _{p(\cdot )} \\ &\quad \lesssim 2^{k_0\lambda (\sigma -1)} \left\{ \sum _{ k =-\infty }^{k_0-1}2^{-k\sigma \lambda \theta } \left\| \left[ \sum _{m=0}^\infty \sum _i\Phi \big (\mu _{k,m,i} s(a^{k,m,i})\big ) \right] ^{\lambda \theta }\right\| _{\frac{p(\cdot )}{\theta }} \right\} ^{1/\theta } \\ &\quad \le 2^{k_0\lambda (\sigma -1)} \left\{ \sum _{ k =-\infty }^{k_0-1}2^{-k\sigma \lambda \theta } \left\| \sum _{m=0}^\infty \sum _i\Phi \big (\mu _{k,m,i} s(a^{k,m,i}) \big )^{\lambda \theta }\right\| _{\frac{p(\cdot )}{\theta }}\right\} ^{1/\theta }. \end{aligned}$$
(3.5)

Now, we first estimate

$$\begin{aligned} \Xi _{k,\alpha }:=\left\| \sum _{m=0}^\infty \sum _i\Phi \big (\mu _{k,m,i} s(a^{k,m,i})\big )^{\alpha \theta }\right\| _{\frac{p(\cdot )}{\theta }}, \end{aligned}$$

where \(k\in {\mathbb{Z}}\) and \(0<\alpha <\min \{1/\theta , r/p_+\}.\) Since \(0<\theta < \min \{p_-,q,1\},\) we know that \(\Vert \cdot \Vert _{\frac{p(\cdot )}{\theta }}\) is a norm. By the duality, there exists a nonnegative measurable function \(h\in L_{\zeta (\cdot )}\) with \(\Vert h\Vert _{\zeta (\cdot )}\le 1\) such that

$$\begin{aligned} \Xi _{k,\alpha }\le 2\int _{\Omega }\sum _{m=0}^\infty \sum _i\Phi \big (\mu _{k,m,i} s(a^{k,m,i})\big )^{\alpha \theta }h {\mathrm{d}}{\mathbb{P}}, \end{aligned}$$

where

$$\begin{aligned} \frac{1}{\zeta (\omega )}+\frac{\theta }{p(\omega )}=1 \quad \text{a.e.}\ \omega \in \Omega . \end{aligned}$$

By Lemma 2.9(2), we know that \(\zeta (\cdot )\) satisfies (2.2). Due to \(0<\alpha < 1/\theta ,\) \(r>1,\) Hölder’s inequality and Lemma 2.8, we get

$$\begin{aligned} \Xi _{k,\alpha }&\lesssim \sum _{m=0}^\infty \sum _i \left\| \Phi \big (\mu _{k,m,i} s(a^{k,m,i})\big )^{\alpha \theta }\right\| _{\frac{r}{\alpha \theta }} \big \Vert h\chi _{I_{k,m,i}}\big \Vert _{\frac{r}{r-\alpha \theta }}\\ &=\sum _{m=0}^\infty \sum _i\Big \Vert \Phi \big (\mu _{k,m,i} s(a^{k,m,i})\big ) \Big \Vert ^{\alpha \theta }_{r}\big \Vert h\chi _{I_{k,m,i}}\big \Vert _{\frac{r}{r-\alpha \theta }}\\ &\lesssim \sum _{m=0}^\infty \sum _i\Phi \left( \frac{\mu _{k,m,i}\Vert s(a^{k,m,i})\Vert _{r}}{{\mathbb{P}}(I_{k,m,i})^{1/r}}\right) ^{\alpha \theta } {\mathbb{P}}(I_{k,m,i})^{\frac{\alpha \theta }{r}} \big \Vert h\chi _{I_{k,m,i}}\big \Vert _{\frac{r}{r-\alpha \theta }}\\ &=\sum _{m=0}^\infty \sum _i\Phi \left( \frac{\mu _{k,m,i}\Vert s(a^{k,m,i})\Vert _{r}}{{\mathbb{P}}(I_{k,m,i})^{1/r}}\right) ^{\alpha \theta } {\mathbb{P}}(I_{k,m,i})\\ &\quad \times \left( \frac{1}{{\mathbb{P}}(I_{k,m,i})} \int _{I_{k,m,i}} h^{\frac{r}{r-\alpha \theta }}{\mathrm{d}}{\mathbb{P}} \right) ^{{\frac{ r-\alpha \theta }{r}}}\\ &\le \sum _{m=0}^\infty \sum _i\Phi \left( \frac{\mu _{k,m,i}\Vert s(a^{k,m,i})\Vert _{r}}{{\mathbb{P}}(I_{k,m,i})^{1/r}}\right) ^{\alpha \theta } \int _{\Omega } \chi _{I_{k,m,i}}\left[ M(h^{\frac{r}{r-\alpha \theta }}) \right] ^{{\frac{ r-\alpha \theta }{r}}}{\mathrm{d}}{\mathbb{P}}\\ &=\int _{\Omega } \sum _{m=0}^\infty \sum _i\Phi \left( \frac{\mu _{k,m,i}\Vert s(a^{k,m,i})\Vert _{r}}{{\mathbb{P}}(I_{k,m,i})^{1/r}}\right) ^{\alpha \theta }\chi _{I_{k,m,i}} \left[ M(h^{\frac{r}{r-\alpha \theta }})\right] ^{{\frac{ r-\alpha \theta }{r}}}{\mathrm{d}}{\mathbb{P}}\\ &\lesssim \left\| \sum _{m=0}^\infty \sum _i\Phi \left( \frac{\mu _{k,m,i}\Vert s(a^{k,m,i})\Vert _{r}}{{\mathbb{P}}(I_{k,m,i})^{1/r}} \right) ^{\alpha \theta }\chi _{I_{k,m,i}}\right\| _{\frac{p(\cdot )}{\theta }}\\ &\quad \times \left\| \left[ M(h^{\frac{r}{r-\alpha \theta }})\right] ^{{\frac{ r-\alpha \theta }{r}}} \right\| _{\zeta (\cdot )}. \end{aligned}$$

Since \(\alpha p_+<r,\) we have

$$\begin{aligned} \text{ess}\inf _{\omega \in \Omega }\zeta (\omega )\frac{r-\alpha \theta }{r} =\frac{p_+}{p_+-\theta }\frac{r-\alpha \theta }{r}>1. \end{aligned}$$

Using Lemma 2.1 and Lemma 2.9(3), we have

$$\begin{aligned} \left\| \left[ M(h^{\frac{r}{r-\alpha \theta }})\right] ^{{\frac{ r-\alpha \theta }{r}}} \right\| _{\zeta (\cdot )}&=\left\| M(h^{\frac{r}{r-\alpha \theta }}) \right\| _{\zeta (\cdot )\frac{r-\alpha \theta }{r}}^{\frac{r-\alpha \theta }{r}} \lesssim \left\| h^{\frac{r}{r-\alpha \theta }}\right\| _{\zeta (\cdot ) \frac{r-\alpha \theta }{r}}^{\frac{r-\alpha \theta }{r}}\\ &=\Vert h\Vert _{\zeta (\cdot )}\le 1. \end{aligned}$$

According to Remark 3.2(iii) and the disjointness of \((I_{k,m,i})_{m,i\in {\mathbb{N}}}\) for fixed \(k\in {\mathbb{Z}},\) we have

$$\begin{aligned} \Xi _{k,\alpha }&\lesssim \left\| \sum _{m=0}^\infty \sum _i\Phi \left( \frac{\mu _{k,m,i}\Vert s(a^{k,m,i})\Vert _{r}}{{\mathbb{P}}(I_{k,m,i})^{1/r}} \right) ^{\alpha \theta }\chi _{I_{k,m,i}}\right\| _{\frac{p(\cdot )}{\theta }}\\ &\le 2^{(k+2)\alpha \theta }\left\| \sum _{m=0}^\infty \sum _i\chi _{I_{k,m,i}} \right\| _{\frac{p(\cdot )}{\theta }} \\ &=2^{(k+2)\alpha \theta }\left\| \sum _{m=0}^\infty \sum _i\chi _{I_{k,m,i}}\right\| ^\theta _{p(\cdot )}. \end{aligned}$$
(3.6)

For \(1/\lambda<\eta <1-\sigma ,\) taking \(\alpha =\lambda\) in (3.6), it follows from (3.5), (3.6) and Hölder’s inequality that

$$\begin{aligned}&\Vert \chi _{\{T_1>2^{k_0}\}}\Vert _{p(\cdot )}\\ &\quad \lesssim 2^{k_0\lambda (\sigma -1)} \left\{ \sum _{ k =-\infty }^{k_0-1}2^{-k\sigma \lambda \theta } \Xi _{k,\lambda } \right\} ^{1/\theta } \\ &\quad \lesssim 2^{k_0\lambda (\sigma -1)} \left\{ \sum _{ k =-\infty }^{k_0-1}2^{k\lambda \theta (1-\sigma )} \left\| \sum _{m=0}^\infty \sum _i\chi _{I_{k,m,i}}\right\| ^\theta _{p(\cdot )} \right\} ^{1/\theta } \\ &\quad =2^{k_0\lambda (\sigma -1)} \left\{ \sum _{ k =-\infty }^{k_0-1}2^{k\lambda \theta (1-\sigma - \eta )}2^{k\lambda \theta \eta } \left\| \sum _{m=0}^\infty \sum _i\chi _{I_{k,m,i}} \right\| ^\theta _{p(\cdot )} \right\} ^{1/\theta } \\ &\quad \le 2^{k_0\lambda (\sigma -1)}\left\{ \sum _{ k =-\infty }^{k_0-1} 2^{k\lambda \theta (1-\sigma -\eta )\frac{q}{q-\theta }}\right\} ^{\frac{q-\theta }{q\theta }} \\ &\qquad \times \; \left\{ \sum _{ k =-\infty }^{k_0-1}2^{k\lambda q\eta } \left\| \sum _{m=0}^\infty \sum _i\chi _{I_{k,m,i}}\right\| ^q_{p(\cdot )} \right\} ^{1/q} \\ &\quad \lesssim 2^{-k_0\lambda \eta }\left\{ \sum _{ k =-\infty }^{k_0-1} 2^{k\lambda q\eta } \left\| \sum _{m=0}^\infty \sum _i\chi _{I_{k,m,i}} \right\| ^q_{p(\cdot )} \right\} ^{1/q}. \end{aligned}$$
(3.7)

Then, it follows from the Abel transformation and the monotonicity of \(\Phi ^{-1}\) that

$$\begin{aligned}&\sum _{ k_0 =-\infty }^\infty 2^{k_0q}\Vert \chi _{\{T_1>2^{k_0}\}}\Vert _{p(\cdot )}^q\\ &\quad \lesssim \sum _{ k_0 =-\infty }^\infty 2^{k_0q(1-\lambda \eta )} \sum _{ k =-\infty }^{k_0-1}2^{k\lambda q\eta } \left\| \sum _{m=0}^\infty \sum _i\chi _{I_{k,m,i}}\right\| ^q_{p(\cdot )} \\ &\quad =\sum _{ k =-\infty }^\infty 2^{k\lambda q\eta }\left\| \sum _{m=0}^\infty \sum _i\chi _{I_{k,m,i}}\right\| ^q_{p(\cdot )} \sum _{ k_0 =k+1 }^\infty 2^{k_0q(1-\lambda \eta )} \\ &\quad \lesssim \sum _{ k =-\infty }^\infty 2^{k q}\left\| \sum _{m=0}^\infty \sum _i\chi _{I_{k,m,i}}\right\| ^q_{p(\cdot )} \\ &\quad \lesssim \sum _{k\in {\mathbb{Z}}}\left\| \sum _{m=0}^\infty \sum _i\Phi \left[ \mu _{k,m,i}\Phi ^{-1}\left( \frac{1}{\Vert \chi _{I_{k,m,i}}\Vert _{p(\cdot )}}\right) \right] \chi _{I_{k,m,i}}\right\| ^q_{p(\cdot )}. \end{aligned}$$
(3.8)

Now, we estimate \(T_2.\) For the above symbol \(\theta ,\) let \(0<\beta<\xi <1.\) Taking \(\alpha =\beta\) in (3.6), it follows from Minkowski’s inequality and (3.6) that

$$\begin{aligned}&\Vert \chi _{\{T_2>2^{k_0}\}}\Vert _{p(\cdot )}\le \frac{1}{2^{k_0\beta }}\Vert T_2^\beta \Vert _{p(\cdot )}\\ &\quad =\frac{1}{2^{k_0\beta }}\Bigg \{\Bigg \Vert \bigg [\sum _{ k =k_0 }^{\infty } \sum _{m=0}^\infty \sum _i\Phi \big (\mu _{k,m,i} s(a^{k,m,i})\big )\bigg ]^{\beta \theta }\Bigg \Vert _{\frac{p(\cdot )}{\theta }}\Bigg \}^{1/\theta } \\ &\quad \le \frac{1}{2^{k_0\beta }} \Bigg \{\sum _{ k =k_0 }^{\infty } \Bigg \Vert \sum _{m=0}^\infty \sum _i\Phi \big (\mu _{k,m,i} s(a^{k,m,i})\big )^{\beta \theta }\Bigg \Vert _{\frac{p(\cdot )}{\theta }} \Bigg \}^{1/\theta } \\ &\quad =\frac{1}{2^{k_0\beta }} \bigg \{\sum _{ k =k_0 }^{\infty } \Xi _{k,\beta } \bigg \}^{1/\theta } \lesssim \frac{1}{2^{k_0\beta }} \Bigg \{\sum _{ k =k_0 }^{\infty }2^{(k+2)\beta \theta }\Big \Vert \sum _{m=0}^\infty \sum _i\chi _{I_{k,m,i}}\Big \Vert ^\theta _{p(\cdot )} \Bigg \}^{1/\theta } \\ &\quad \approx \frac{1}{2^{k_0\beta }} \Bigg \{\sum _{ k =k_0 }^{\infty }2^{k\theta (\beta -\xi )}2^{k\xi \theta } \Big \Vert \sum _{m=0}^\infty \sum _i\chi _{I_{k,m,i}}\Big \Vert ^\theta _{p(\cdot )} \Bigg \}^{1/\theta } \\ &\quad \le \frac{1}{2^{k_0\beta }} \Bigg \{\sum _{ k =k_0 }^{\infty }2^{k\theta (\beta -\xi )\frac{q}{q-\theta }}\Bigg \}^{\frac{q-\theta }{q\theta }} \Bigg \{\sum _{ k =k_0 }^{\infty }2^{kq\xi }\Big \Vert \sum _{m=0}^\infty \sum _i\chi _{I_{k,m,i}}\Big \Vert _{ p(\cdot )}^{q}\Bigg \}^{1/q} \\ &\quad \lesssim \frac{1}{2^{k_0\xi }}\Bigg \{\sum _{ k =k_0 }^{\infty }2^{kq\xi }\Big \Vert \sum _{m=0}^\infty \sum _i\chi _{I_{k,m,i}}\Big \Vert _{ p(\cdot )}^{q}\Bigg \}^{1/q}. \end{aligned}$$
(3.9)

By the Abel transformation and the monotonicity of \(\Phi ^{-1},\) we have

$$\begin{aligned} \quad \quad&\sum _{ k_0 =-\infty }^\infty 2^{k_0q}\Vert \chi _{\{T_2>2^{k_0}\}}\Vert _{p(\cdot )}^q\\ &\quad \lesssim \sum _{ k_0 =-\infty }^\infty 2^{k_0q(1-\xi )}\sum _{ k =k_0 }^{\infty }2^{kq\xi }\Big \Vert \sum _{m=0}^\infty \sum _i\chi _{I_{k,m,i}}\Big \Vert _{ p(\cdot )}^{q} \\ &\quad =\sum _{ k =-\infty }^\infty 2^{kq\xi }\Big \Vert \sum _{m=0}^\infty \sum _i\chi _{I_{k,m,i}}\Big \Vert _{ p(\cdot )}^{q} \sum _{ k_0 =-\infty }^k2^{k_0q(1-\xi )} \\ &\quad \lesssim \sum _{ k =-\infty }^\infty 2^{kq}\Big \Vert \sum _{m=0}^\infty \sum _i\chi _{I_{k,m,i}}\Big \Vert _{ p(\cdot )}^{q} \\ &\quad \lesssim \sum _{k\in {\mathbb{Z}}}\bigg \Vert \sum _{m=0}^\infty \sum _i\Phi \bigg [\mu _{k,m,i}\Phi ^{-1}\Big ( \frac{1}{\Vert \chi _{I_{k,m,i}}\Vert _{p(\cdot )}}\Big )\bigg ] \chi _{I_{k,m,i}}\bigg \Vert ^q_{p(\cdot )}. \end{aligned}$$
(3.10)

Combining (3.4), (3.8) and (3.10), we conclude that

$$\begin{aligned} \big \Vert \Phi (s(f))\big \Vert _{{p(\cdot ),q}}^q&\approx \sum _{k_0\in {\mathbb{Z}}}2^{(k_0+1)q}\big \Vert \chi _{\{\Phi (s(f))>2^{k_0+1}\}}\big \Vert _{p(\cdot )}^q \\ {}&\le \sum _{k_0\in {\mathbb{Z}}}2^{(k_0+1)q}\big \Vert \chi _{\{T_1+T_2>2^{k_0+1}\}}\big \Vert _{p(\cdot )}^q \\ {}&\lesssim \sum _{k_0\in {\mathbb{Z}}}2^{(k_0+1)q}\Big (\big \Vert \chi _{\{T_1>2^{k_0}\}}\big \Vert _{p(\cdot )}^q +\big \Vert \chi _{\{T_2>2^{k_0}\}}\big \Vert _{p(\cdot )}^q\Big ) \\ {}&\lesssim \sum _{k_0\in {\mathbb{Z}}}2^{k_0q}\big \Vert \chi _{\{T_1>2^{k_0}\}}\big \Vert _{p(\cdot )}^q +\sum _{k_0\in {\mathbb{Z}}}2^{k_0q}\big \Vert \chi _{\{T_2>2^{k_0}\}}\big \Vert _{p(\cdot )}^q \\ {}&\lesssim \sum _{k\in {\mathbb{Z}}}\bigg \Vert \sum _{m=0}^\infty \sum _i\Phi \bigg [\mu _{k,m,i}\Phi ^{-1}\Big ( \frac{1}{\Vert \chi _{I_{k,m,i}}\Vert _{p(\cdot )}}\Big )\bigg ] \chi _{I_{k,m,i}}\bigg \Vert ^q_{p(\cdot )}. \end{aligned}$$

Taking over all the admissible representations of (3.1) for f,  we obtain the desired result.

For \(q=\infty .\) On one hand, since \(0<\sigma <1-\frac{1}{\lambda },\) it follows from (3.7) that

$$\begin{aligned}&2^{k_0}\Vert \chi _{\{T_1>2^{k_0}\}}\Vert _{p(\cdot )} \\ &\lesssim 2^{k_0(\lambda \sigma -\lambda +1)} \Bigg \{\sum _{ k =-\infty }^{k_0-1}2^{k\lambda \theta (1-\sigma )} \Big \Vert \sum _{m=0}^\infty \sum _i\chi _{I_{k,m,i}}\Big \Vert ^\theta _{p(\cdot )} \Bigg \}^{1/\theta } \\ &= 2^{k_0(\lambda \sigma -\lambda +1)} \Bigg \{\sum _{ k =-\infty }^{k_0-1}2^{k\theta (\lambda -\lambda \sigma -1)} 2^{k\theta }\Big \Vert \sum _{m=0}^\infty \sum _i\chi _{I_{k,m,i}}\Big \Vert ^\theta _{p(\cdot )} \Bigg \}^{1/\theta } \\ {}&\le 2^{k_0(\lambda \sigma -\lambda +1)} \Bigg \{\sum _{ k =-\infty }^{k_0-1}2^{k\theta (\lambda -\lambda \sigma -1)} \Bigg \}^{1/\theta }\sup _{k\in {\mathbb{Z}}}2^{k}\Big \Vert \sum _{m=0}^\infty \sum _i\chi _{I_{k,m,i}}\Big \Vert _{p(\cdot )} \\ &\lesssim \sup _{k\in {\mathbb{Z}}}2^{k}\Big \Vert \sum _{m=0}^\infty \sum _i\chi _{I_{k,m,i}}\Big \Vert _{p(\cdot )}. \end{aligned}$$
(3.11)

On the other hand, since \(0<\beta <1,\) it follows from (3.9) that

$$\begin{aligned}&2^{k_0}\Vert \chi _{\{T_2>2^{k_0}\}}\Vert _{p(\cdot )} \\ &\lesssim 2^{k_0(1-\beta )} \Bigg \{\sum _{ k =k_0 }^{\infty }2^{(k+2)\beta \theta }\Big \Vert \sum _{m=0}^\infty \sum _i\chi _{I_{k,m,i}}\Big \Vert ^\theta _{p(\cdot )} \Bigg \}^{1/\theta } \\ &\lesssim 2^{k_0(1-\beta )} \Bigg \{\sum _{ k =k_0 }^{\infty }2^{k\theta (\beta -1)}2^{k\theta }\Big \Vert \sum _{m=0}^\infty \sum _i\chi _{I_{k,m,i}}\Big \Vert ^\theta _{p(\cdot )} \Bigg \}^{1/\theta } \\ &\le 2^{k_0(1-\beta )} \Bigg \{\sum _{ k =k_0 }^{\infty }2^{k\theta (\beta -1)} \Bigg \}^{1/\theta }\sup _{k\in {\mathbb{Z}}}2^{k}\Big \Vert \sum _{m=0}^\infty \sum _i\chi _{I_{k,m,i}}\Big \Vert _{p(\cdot )} \\ &\lesssim \sup _{k\in {\mathbb{Z}}}2^{k}\Big \Vert \sum _{m=0}^\infty \sum _i\chi _{I_{k,m,i}}\Big \Vert _{p(\cdot )}. \end{aligned}$$
(3.12)

By (3.11), (3.12) and the monotonicity of \(\Phi ^{-1},\) we obtain

$$\begin{aligned} \big \Vert \Phi (s(f))\big \Vert _{{p(\cdot ),\infty }}&\approx \sup _{k_0\in {\mathbb{Z}}}2^{k_0+1}\big \Vert \chi _{\{\Phi (s(f))>2^{k_0+1}\}}\big \Vert _{p(\cdot )} \\ {}&\lesssim \sup _{k_0\in {\mathbb{Z}}}2^{k_0}\big \Vert \chi _{\{T_1>2^{k_0}\}}\big \Vert _{p(\cdot )} +\sup _{k_0\in {\mathbb{Z}}}2^{k_0}\big \Vert \chi _{\{T_2>2^{k_0}\}}\big \Vert _{p(\cdot )} \\ {}&\lesssim \sup _{k\in {\mathbb{Z}}}2^{k}\Big \Vert \sum _{m=0}^\infty \sum _i\chi _{I_{k,m,i}}\Big \Vert _{p(\cdot )} \\ {}&\lesssim \sup _{k\in {\mathbb{Z}}}\bigg \Vert \sum _{m=0}^\infty \sum _i\Phi \bigg [\mu _{k,m,i}\Phi ^{-1}\Big ( \frac{1}{\Vert \chi _{I_{k,m,i}}\Vert _{p(\cdot )}}\Big )\bigg ] \chi _{I_{k,m,i}}\bigg \Vert _{p(\cdot )}. \end{aligned}$$

Taking over all the admissible representations of (3.1) for f,  we obtain the desired result. This completes the proof of the theorem. \(\square\)

Obviously, we can obtain that the converse part of Theorem 3.3 also holds for \(\big \Vert \Phi \big (M(f)\big )\big \Vert _{{p(\cdot ),q}}\) and \(\big \Vert \Phi \big (S(f)\big )\big \Vert _{{p(\cdot ),q}}\) as follows.

Theorem 3.4

Let \(p(\cdot )\in {\mathcal{P}}\) satisfy (2.2), \(\Phi \in {\mathcal{G}}\) be a concave function, \(0 < q \le \infty\) and \(\max \{p_{+},1\} <r\le \infty .\) If the martingale \(f=(f_n)_{ n \ge 0 }\) has a decomposition of type (3.1) with \(( a^{k,m,i}, I_{k,m,i}, \mu _{k,m,i})_{k\in {\mathbb{Z}}, m,i\in {\mathbb{N}}}\subset {\mathcal{A}}^M(\Phi ,p(\cdot ),q ,r)\) \(\big (\text{resp.}\; {\mathcal{A}}^S(\Phi ,p(\cdot ),q ,r)\big ),\) then for \(0<q<\infty ,\)

$$\begin{aligned}&\big \Vert \Phi \big (M(f)\big )\big \Vert _{{p(\cdot ),q}}\\ &\quad \lesssim \inf \left\{ \sum _{k\in {\mathbb{Z}}} \left\| \sum _{m=0}^\infty \sum _i\Phi \left[ \mu _{k,m,i}\Phi ^{-1}\left( \frac{1}{\Vert \chi _{I_{k,m,i}}\Vert _{p(\cdot )}}\right) \right] \chi _{I_{k,m,i}}\right\| ^q_{p(\cdot )}\right\} ^{1/q} \\ &\Bigg (\text{resp.}\ \big \Vert \Phi \big (S(f)\big )\big \Vert _{{p(\cdot ),q}}\\ &\quad \lesssim \inf \left\{ \sum _{k\in {\mathbb{Z}}} \left\| \sum _{m=0}^\infty \sum _i\Phi \left[ \mu _{k,m,i}\Phi ^{-1} \left( \frac{1}{\Vert \chi _{I_{k,m,i}}\Vert _{p(\cdot )}}\right) \right] \chi _{I_{k,m,i}}\right\| ^q_{p(\cdot )}\right\} ^{1/q}\Bigg ), \end{aligned}$$

(with the usual modification for \(q=\infty ),\) where the infimum is taken over all the preceding decompositions of f of the form (3.1).

We now establish the \(\infty\)-atom decompositions for \(\Phi (f)\) in \({\mathcal{D}}_{p(\cdot ), q}\) and \({\mathcal{Q}}_{p(\cdot ), q}.\)

Theorem 3.5

Let \(p(\cdot )\in {\mathcal{P}}\) satisfy (2.2), \(\Phi \in {\mathcal{G}}\) be a concave function and \(0 < q \le \infty .\) If the martingale \(f=(f_n)_{ n \ge 0 }\) satisfies \(\Vert \Phi (f)\Vert _{{\mathcal{D}}_{p(\cdot ),q}}<\infty ,\) then there exists a sequence of triples \(( a^{k,m,i}, I_{k,m,i}, \mu _{k,m,i})_{k\in {\mathbb{Z}},m,i\in {\mathbb{N}}}{} \subset {\mathcal{A}}^M(\Phi ,p(\cdot ),q ,\infty )\) such that for each \(n \ge 0 ,\) (3.1) holds and

$$\begin{aligned}&\left\{ \sum _{k\in {\mathbb{Z}}}\left\| \sum _{m=0}^\infty \sum _i\Phi \left[ \mu _{k,m,i}\Phi ^{-1}\left( \frac{1}{\Vert \chi _{I_{k,m,i}} \Vert _{p(\cdot )}}\right) \right] \chi _{I_{k,m,i}}\right\| ^q_{p(\cdot )}\right\} ^{1/q}\\ &\quad \lesssim \Vert \Phi (f)\Vert _{{\mathcal{D}}_{p(\cdot ),q}}, \quad (0<q<\infty ), \end{aligned}$$
(3.13)

(with the usual modification for \(q=\infty ).\)

Conversely, if the martingale \(f=(f_n)_{ n \ge 0 }\) has a decomposition of type (3.1), then for \(0<q<\infty ,\)

$$\begin{aligned}&\Vert \Phi (f)\Vert _{{\mathcal{D}}_{p(\cdot ),q}}\\ &\quad \lesssim \inf \left\{ \sum _{k\in {\mathbb{Z}}} \left\| \sum _{m=0}^\infty \sum _i\Phi \left[ \mu _{k,m,i}\Phi ^{-1} \left( \frac{1}{\Vert \chi _{I_{k,m,i}}\Vert _{p(\cdot )}}\right) \right] \chi _{I_{k,m,i}}\right\| ^q_{p(\cdot )}\right\} ^{1/q}, \end{aligned}$$

(with the usual modification for \(q=\infty ),\) where the infimum is taken over all the preceding decompositions of f of the form (3.1).

Proof

The proof follows the ideas in Theorem 3.3, so we omit some details. For \(0<q<\infty .\) Suppose that \(f=(f_n)_{ n \ge 0 }\) satisfies \(\Vert \Phi (f)\Vert _{{\mathcal{D}}_{p(\cdot ),q}}<\infty .\) We define stopping times as follows

$$\begin{aligned} \tau _k= \inf \big \{n \in {\mathbb{N}}: \lambda _n > \Phi ^{-1}(2^k)\big \} ,~~~~~\inf \emptyset = \infty , \end{aligned}$$

where \((\lambda _n)_{n\ge 0}\) is a sequence of nondecreasing, nonnegative and adapted functions such that \(|f_n| \le \lambda _{n-1}\) and \(\Phi (\lambda _{\infty })\in L_{p(\cdot ),q}.\)

Let \((a^{k,m,i})_{k\in {\mathbb{Z}},m,i\in {\mathbb{N}}}\) and \((\mu _{k,m,i})_{k\in {\mathbb{Z}},m,i\in {\mathbb{N}}}\) be defined as in the proof of the Theorem 3.3. Obviously, \(a^{k,m,i}\) is a simple \((\Phi ,p(\cdot ), \infty )^M\)-atom for fixed \(k\in {\mathbb{Z}},m,i\in {\mathbb{N}}.\) Now we present \(( a^{k,m,i}, I_{k,m,i}, \mu _{k,m,i})_{k\in {\mathbb{Z}}, m,i\in {\mathbb{N}}} \subset {\mathcal{A}}^M(\Phi ,p(\cdot ),q ,\infty ) .\) Since \(\{\tau _k<\infty \}=\{\Phi (\lambda _{\infty }) >2^k \},\) then we have

$$\begin{aligned}&\sum _{k\in {\mathbb{Z}}}\bigg \Vert \sum _{m=0}^\infty \sum _i\Phi \bigg [\mu _{k,m,i}\Phi ^{-1}\Big ( \frac{1}{\Vert \chi _{I_{k,m,i}}\Vert _{p(\cdot )}}\Big )\bigg ] \chi _{I_{k,m,i}}\bigg \Vert ^q_{p(\cdot )}\\ &\quad \le \sum _{ k \in {\mathbb{Z}} }2^{(k+2)q}\Vert \chi _{\{\tau _k<\infty \}}\Vert ^q_{p(\cdot )} = \sum _{ k \in {\mathbb{Z}} }2^{(k+2)q}\Vert \chi _{\{\Phi (\lambda _{\infty }) >2^k\}}\Vert ^q_{p(\cdot )}\\ &\quad \lesssim \Vert \Phi (\lambda _{\infty })\Vert _{{p(\cdot ),q}}^q. \end{aligned}$$

Taking the infimum over \((\lambda _n)_{n\ge 0}\in \Lambda [{\mathcal{D}}_{p(\cdot ),q, \Phi }](f),\) we obtain the desired result.

For the converse part, let

$$\begin{aligned} \lambda _n=\sum _{k\in {\mathbb{Z}}}\sum _{m=0}^n\sum _i\mu _{k,m,i} \Vert M(a^{k,m,i})\Vert _{\infty }\chi _{I_{k,m,i}}. \end{aligned}$$

Clearly, \((\lambda _n)_{n\ge 0}\) is a nondecreasing, nonnegative and adapted sequence with \(|f_n|\le \lambda _{n-1}.\) For an arbitrary \(k_0,\) set

$$\begin{aligned} J_1=\sum _{k=-\infty }^{k_0-1}\sum _{m=0}^\infty \sum _i \Phi \big (\mu _{k,m,i} \Vert M(a^{k,m,i})\Vert _{\infty }\big )\chi _{I_{k,m,i}} \end{aligned}$$

and

$$\begin{aligned} J_2=\sum _{k=k_0}^{\infty }\sum _{m=0}^\infty \sum _i \Phi \big (\mu _{k,m,i} \Vert M(a^{k,m,i})\Vert _{\infty }\big )\chi _{I_{k,m,i}}. \end{aligned}$$

By replacing \(T_1\) (resp. \(T_2\)) in the proof of Theorem 3.3 with \(J_1\) (resp. \(J_2\)), we get

$$\begin{aligned} \Vert \Phi (f)\Vert ^q_{{\mathcal{D}}_{p(\cdot ),q}}&\le \Vert \Phi (\lambda _{\infty })\Vert ^q_{{p(\cdot ),q}} \lesssim \Vert J_1\Vert ^q_{{p(\cdot ),q}}+\Vert J_2\Vert ^q_{{p(\cdot ),q}} \\ {}&\lesssim \sum _{k\in {\mathbb{Z}}}\bigg \Vert \sum _{m=0}^\infty \sum _i\Phi \bigg [\mu _{k,m,i}\Phi ^{-1}\Big ( \frac{1}{\Vert \chi _{I_{k,m,i}}\Vert _{p(\cdot )}}\Big )\bigg ] \chi _{I_{k,m,i}}\bigg \Vert ^q_{p(\cdot )}. \end{aligned}$$

Taking over all the admissible representations of (3.1) for f,  we obtain the desired result. The proof of the case \(q=\infty\) is analogous. This completes the proof of the theorem. \(\square\)

Similar to the method of the proof above, we can present the atomic decomposition for \(\Phi (f)\) under the functional of \(\Vert \cdot \Vert _{{\mathcal{Q}}_{p(\cdot ),q}}.\)

Theorem 3.6

Let \(p(\cdot )\in {\mathcal{P}}\) satisfy (2.2), \(\Phi \in {\mathcal{G}}\) be a concave function and \(0 < q \le \infty .\) If the martingale \(f=(f_n)_{ n \ge 0 }\) satisfies \(\Vert \Phi (f)\Vert _{{\mathcal{Q}}_{p(\cdot ),q}}<\infty ,\) then there exists a sequence of triples \((a^{k,m,i}, I_{k,m,i}, \mu _{k,m,i})_{k\in {\mathbb{Z}},m,i\in {\mathbb{N}}}\subset {\mathcal{A}}^S(\Phi ,p(\cdot ),q ,\infty )\) such that for each \(n \ge 0 ,\) (3.1) holds and

$$\begin{aligned}&\left\{ \sum _{k\in {\mathbb{Z}}}\left\| \sum _{m=0}^\infty \sum _i\Phi \left[ \mu _{k,m,i}\Phi ^{-1}\left( \frac{1}{\Vert \chi _{I_{k,m,i}} \Vert _{p(\cdot )}}\right) \right] \chi _{I_{k,m,i}}\right\| ^q_{p(\cdot )}\right\} ^{1/q}\\ &\quad \lesssim \Vert \Phi (f)\Vert _{{\mathcal{Q}}_{p(\cdot ),q}}, \quad (0<q<\infty ), \end{aligned}$$

(with the usual modification for \(q=\infty ).\)

Conversely, if the martingale \(f=(f_n)_{ n \ge 0 }\) has a decomposition of type (3.1), then for \(0<q<\infty ,\)

$$\begin{aligned}&\Vert \Phi (f)\Vert _{{\mathcal{Q}}_{p(\cdot ),q}}\\ &\quad \lesssim \inf \left\{ \sum _{k\in {\mathbb{Z}}} \left\| \sum _{m=0}^\infty \sum _i\Phi \left[ \mu _{k,m,i}\Phi ^{-1} \left( \frac{1}{\Vert \chi _{I_{k,m,i}}\Vert _{p(\cdot )}}\right) \right] \chi _{I_{k,m,i}}\right\| ^q_{p(\cdot )}\right\} ^{1/q}, \end{aligned}$$

(with the usual modification for \(q=\infty ),\) where the infimum is taken over all the preceding decompositions of f of the form (3.1).

4 \(\Phi\)-moment martingale inequalities

In this section, using the atomic decompositions we presented above, we deduce some fundamental \(\Phi\)-moment martingale inequalities on variable Lorentz spaces. We need the following lemma.

Lemma 4.1

(See [14, 34]) Let h be a martingale. Then

$$\begin{aligned}&\Vert M(h)\Vert _2\le 2 \Vert S(h)\Vert _2=2\Vert s(h)\Vert _2\le 2\Vert M(h)\Vert _2;\\ &\Vert s(h)\Vert _r\le \sqrt{\frac{r}{2}} \; \big \Vert M(h)\big \Vert _r,\quad (r\ge 2);\\ &\Vert s(h)\Vert _r\le \sqrt{\frac{r}{2}} \; \big \Vert S(h)\big \Vert _r,\quad (r\ge 2). \end{aligned}$$

Moreover, if the stochastic basis \(\{{{\mathcal{F}}}_n\}_{n\ge 0}\) is regular, then

$$\begin{aligned} \Vert M(h)\Vert _{r}\approx \Vert S(h)\Vert _{r}\approx \Vert s(h)\Vert _{r},\quad (0<r<\infty ). \end{aligned}$$
(4.1)

Theorem 4.2

Let \(p(\cdot )\in {\mathcal{P}}\) satisfy (2.2) and \(\Phi \in {\mathcal{G}}\) be a concave function. Then

  1. (1)

    For \(0<p_+<2\) and \(0<q\le \infty ,\)

    $$\begin{aligned} \big \Vert \Phi (M(f))\big \Vert _{{p(\cdot ),q}} \lesssim \big \Vert \Phi (s(f))\big \Vert _{{p(\cdot ),q}},\quad \big \Vert \Phi (S(f))\big \Vert _{{p(\cdot ),q}} \lesssim \big \Vert \Phi (s(f))\big \Vert _{{p(\cdot ),q}}. \end{aligned}$$
  2. (2)

    For \(0<q \le \infty ,\)

    $$\begin{aligned} \big \Vert \Phi (s(f))\big \Vert _{{p(\cdot ),q}} \lesssim \big \Vert \Phi (f)\big \Vert _{{\mathcal{D}}_{p(\cdot ),q}},\quad \big \Vert \Phi (s(f))\big \Vert _{{p(\cdot ),q}} \lesssim \big \Vert \Phi (f)\big \Vert _{{\mathcal{Q}}_{p(\cdot ),q}}. \end{aligned}$$
  3. (3)

    For \(0<p_+<2\) and \(0<q\le \infty ,\)

    $$\begin{aligned} \big \Vert \Phi (f)\big \Vert _{{\mathcal{Q}}_{p(\cdot ),q}} \lesssim \big \Vert \Phi (f)\big \Vert _{{\mathcal{D}}_{p(\cdot ),q}} \lesssim \big \Vert \Phi (f)\big \Vert _{{\mathcal{Q}}_{p(\cdot ),q}}. \end{aligned}$$

Proof

  1. (1)

    Let \(0<p_+<2\) and \(0<q\le \infty .\) Suppose that \(f=(f_n)_{ n \ge 0 }\) is a martingale with \(\big \Vert \Phi \big (s(f)\big )\big \Vert _{p(\cdot ),q}<\infty .\) By Theorem 3.3, there exists a sequence of triples \((a^{k,m,i}, I_{k,m,i}, \mu _{k,m,i})_{k\in {\mathbb{Z}},m,i\in {\mathbb{N}}}{} \subset {\mathcal{A}}^s(\Phi ,p(\cdot ),q ,2)\) such that (3.1) and (3.2) hold. Using Lemma 4.1, we have

    $$\begin{aligned} \Big \Vert M\Big (\frac{1}{2}a^{k,m,i}\Big )\Big \Vert _2\le \big \Vert s(a^{k,m,i})\big \Vert _2\le {\mathbb{P}}(I_{k,m,i})^{1/2}\Phi ^{-1}\Big ( \frac{1}{\Vert \chi _{I_{k,m,i}}\Vert _{p(\cdot )}}\Big ). \end{aligned}$$

    Hence, \(\frac{1}{2}a^{k,m,i}\) is a simple \((\Phi ,p(\cdot ), 2)^M\)-atom and

    $$\begin{aligned} \Big (\frac{1}{2}a^{k,m,i}, I_{k,m,i}, \mu _{k,m,i}\Big )_{k\in {\mathbb{Z}},m,i\in {\mathbb{N}}}\subset {\mathcal{A}}^M(\Phi ,p(\cdot ),q ,2). \end{aligned}$$

    Moreover, applying (3.1), (3.2), Lemma 2.7 and Theorem 3.4, we obtain that for each \(n \ge 0 ,\)

    $$\begin{aligned} \frac{1}{2}f_n= \sum _{ k \in {\mathbb{Z}} }\sum _{m=0}^{n-1}\sum _i\mu _{k,m,i} {\mathbb{E}}_n \Big (\frac{1}{2}\,a^{k,m,i}\Big ) \quad \text{a.e.}, \end{aligned}$$

    and

    $$\begin{aligned}&\big \Vert \Phi (M(f))\big \Vert _{{p(\cdot ),q}}\le 2\,\Big \Vert \Phi \Big (M\Big (\frac{1}{2}f\Big )\Big )\Big \Vert _{{p(\cdot ),q}}\\ &\lesssim \left\{ \sum _{k\in {\mathbb{Z}}}\bigg \Vert \sum _{m=0}^\infty \sum _i\Phi \bigg [\mu _{k,m,i}\Phi ^{-1}\Big ( \frac{1}{\Vert \chi _{I_{k,m,i}}\Vert _{p(\cdot )}}\Big )\bigg ] \chi _{I_{k,m,i}}\bigg \Vert ^q_{p(\cdot )}\right\} ^{1/q}\\ &\lesssim \Vert \Phi (s(f))\Vert _{{p(\cdot ),q}}, \quad (0<q<\infty ), \end{aligned}$$

    (with the usual modification for \(q = \infty\)). The second inequality of (1) can be proved analogously.

  2. (2)

    Let \(\max \{p_+,2\}<r<\infty ,\) \(0 < q \le \infty\) and martingale \(f=(f_n)_{ n \ge 0 }\) satisfy \(\Vert \Phi (f)\Vert _{{\mathcal{D}}_{p(\cdot ),q}}<\infty .\) According to Theorem 3.5, there exists a sequence of triples \((a^{k,m,i}, I_{k,m,i}, \mu _{k,m,i})_{k\in {\mathbb{Z}},m,i\in {\mathbb{N}}}{} \subset {\mathcal{A}}^M(\Phi ,p(\cdot ),q ,\infty )\) such that for each \(n \ge 0 ,\) (3.1) and (3.13) hold. For fixed \(k\in {\mathbb{Z}},m,i\in {\mathbb{N}},\) we get

    $$\begin{aligned} \sqrt{\frac{2}{r}} \;\Vert s(a^{k,m,i})\Vert _r&\le \big \Vert M(a^{k,m,i})\big \Vert _r\le \Vert \chi _{I_{k,m,i}}\Vert _r\big \Vert M(a^{k,m,i})\big \Vert _{\infty }\\ &\le {\mathbb{P}}(I_{k,m,i})^{1/r}\Phi ^{-1}\Big ( \frac{1}{\Vert \chi _{I_{k,m,i}}\Vert _{p(\cdot )}}\Big ). \end{aligned}$$

    By Lemma 4.1 and \(a^{k,m,i}\) being a simple \((\Phi ,p(\cdot ), \infty )^M\)-atom. So \(\sqrt{\frac{2}{r}}a^{k,m,i}\) is a simple \((\Phi ,p(\cdot ), r)^s\)-atom and

    $$\begin{aligned} \left( \sqrt{\frac{2}{r}}a^{k,m,i}, I_{k,m,i}, \mu _{k,m,i}\right) _{k\in {\mathbb{Z}},m,i\in {\mathbb{N}}}\subset {\mathcal{A}}^s(\Phi ,p(\cdot ),q ,r). \end{aligned}$$

    It follows from (3.1), (3.2), (3.13) and Lemma 2.7 that for each \(n \ge 0 ,\)

    $$\begin{aligned} \sqrt{\frac{2}{r}}\,f_n= \sum _{ k \in {\mathbb{Z}} }\sum _{m=0}^{n-1}\sum _i\mu _{k,m,i} {\mathbb{E}}_n \left( \sqrt{\frac{2}{r}}\,a^{k,m,i}\right) \quad \text{a.e.}, \end{aligned}$$

    and

    $$\begin{aligned}&\big \Vert \Phi (s(f))\big \Vert _{{p(\cdot ),q}}\le \sqrt{\frac{r}{2}}\,\left\| \Phi \left( s\left( \sqrt{\frac{2}{r}}f\right) \right) \right\| _{{p(\cdot ),q}}\\ &\lesssim \left\{ \sum _{k\in {\mathbb{Z}}}\left\| \sum _{m=0}^\infty \sum _i\Phi \left[ \mu _{k,m,i}\Phi ^{-1}\left( \frac{1}{\Vert \chi _{I_{k,m,i}}\Vert _{p(\cdot )}}\right) \right] \chi _{I_{k,m,i}}\right\| ^q_{p(\cdot )}\right\} ^{1/q}\\ &\lesssim \big \Vert \Phi (f)\big \Vert _{{\mathcal{D}}_{p(\cdot ),q}}, \quad (0<q<\infty ), \end{aligned}$$

    (with the usual modification for \(q = \infty\)). The second inequality of (2) can be proved analogously.

  3. (3)

    Let \(0<p_+<2,\) \(0<q\le \infty\) and \(f=(f_n)_{ n \ge 0 }\) be a martingale with \(\Vert \Phi (f)\Vert _{{\mathcal{Q}}_{p(\cdot ),q}}<\infty .\) For any \(\epsilon >0,\) there exists \((\lambda _n)_{n\ge 0}\in \Lambda [{\mathcal{Q}}_{p(\cdot ),q, \Phi }](f)\) such that \(\Vert \Phi (\lambda _{\infty })\Vert _{p(\cdot ),q}\le \Vert \Phi (f)\Vert _{{\mathcal{Q}}_{p(\cdot ),q}}+\epsilon .\) Then, there has

    $$\begin{aligned} |f_n|&=|f_n-f_{n-1}+f_{n-1}|\le |f_n-f_{n-1}|+M_{n-1}(f) \\ {}&\le S_n(f)+M_{n-1}(f)\le \lambda _{n-1}+M_{n-1}(f). \end{aligned}$$

    Applying (1) and (2), we have

    $$\begin{aligned} \big \Vert \Phi \big (\lambda _{\infty }+M(f)\big )\big \Vert _{p(\cdot ),q}&\le \big \Vert \Phi (\lambda _{\infty })+\Phi \big (M(f)\big )\big \Vert _{p(\cdot ),q} \\ &\lesssim \Vert \Phi (\lambda _{\infty })\Vert _{p(\cdot ),q}+\big \Vert \Phi \big (M(f)\big )\big \Vert _{{p(\cdot ),q}} \\ &\lesssim \Vert \Phi (f)\Vert _{{\mathcal{Q}}_{p(\cdot ),q}}+\epsilon +\big \Vert \Phi \big (s(f)\big )\big \Vert _{{p(\cdot ),q}} \\ &\lesssim \Vert \Phi (f)\Vert _{{\mathcal{Q}}_{p(\cdot ),q}}+\epsilon . \end{aligned}$$
    (4.2)

    Hence, \(\big (\lambda _n+M_{n}(f)\big )_{n\ge 0}\in \Lambda [{\mathcal{D}}_{p(\cdot ),q, \Phi }](f).\) From the definition of \(\Vert \cdot \Vert _{{\mathcal{D}}_{p(\cdot ),q}}\) and (4.2), we have

    $$\begin{aligned} \Vert \Phi (f)\Vert _{{\mathcal{D}}_{p(\cdot ),q}}\lesssim \Vert \Phi (f)\Vert _{{\mathcal{Q}}_{p(\cdot ),q}}+\epsilon . \end{aligned}$$

    Taking \(\epsilon \longrightarrow 0,\) we obtain the right inequality in (3).

We consider the rest part of (3) in the following. Let \(f=(f_n)_{ n \ge 0 }\) be a martingale with \(\Vert \Phi (f)\Vert _{{\mathcal{D}}_{p(\cdot ),q}}<\infty .\) For any \(\eta >0,\) there exists \((\gamma _n)_{n\ge 0}\in \Lambda [{\mathcal{D}}_{p(\cdot ),q, \Phi }](f)\) such that \(\Vert \Phi (\gamma _{\infty })\Vert _{p(\cdot ),q}\le \Vert \Phi (f)\Vert _{{\mathcal{D}}_{p(\cdot ),q}}+\eta .\) Then, this yields

$$\begin{aligned} S_n(f)&\le S_{n-1}(f)+|f_n-f_{n-1}|\le S_{n-1}(f)+2M_n(f) \\ {}&\le S_{n-1}(f)+2\gamma _{n-1}. \end{aligned}$$

Combining (1) and (2), we get

$$\begin{aligned} \big \Vert \Phi \big (S(f)+2\gamma _{\infty }\big )\big \Vert _{p(\cdot ),q}&\le \big \Vert \Phi \big (S(f)\big )+\Phi \big (2\gamma _{\infty }\big )\big \Vert _{p(\cdot ),q} \\ &\lesssim \big \Vert \Phi \big (S(f)\big )\big \Vert _{{p(\cdot ),q}}+\big \Vert \Phi (2\gamma _{\infty })\big \Vert _{p(\cdot ),q} \\ &\le \big \Vert \Phi \big (S(f)\big )\big \Vert _{{p(\cdot ),q}}+2\big \Vert \Phi (\gamma _{\infty })\big \Vert _{p(\cdot ),q} \\ &\lesssim \big \Vert \Phi \big (s(f)\big )\big \Vert _{{p(\cdot ),q}}+\Vert \Phi (f)\Vert _{{\mathcal{D}}_{p(\cdot ),q}}+\eta \\ &\lesssim \Vert \Phi (f)\Vert _{{\mathcal{D}}_{p(\cdot ),q}}+\eta . \end{aligned}$$
(4.3)

This means \(\big (S_{n}(f)+2\gamma _{n}\big )_{n\ge 0}\in \Lambda [{\mathcal{Q}}_{p(\cdot ), q, \Phi }](f).\) By the definition of \(\Vert \cdot \Vert _{{\mathcal{Q}}_{p(\cdot ),q}}\) and (4.3), we have

$$\begin{aligned} \Vert \Phi (f)\Vert _{{\mathcal{Q}}_{p(\cdot ),q}}\lesssim \Vert \Phi (f)\Vert _{{\mathcal{D}}_{p(\cdot ),q}}+\eta . \end{aligned}$$

Taking \(\eta \longrightarrow 0,\) we obtain the left inequality in (3). We conclude that

$$\begin{aligned} \Vert \Phi (f)\Vert _{{\mathcal{D}}_{p(\cdot ),q}}\lesssim \Vert \Phi (f)\Vert _{{\mathcal{Q}}_{p(\cdot ),q}} \lesssim \Vert \Phi (f)\Vert _{{\mathcal{D}}_{p(\cdot ),q}}. \end{aligned}$$

\(\square\)

Remark 4.3

Let \(p(\cdot )\equiv p.\) Then, we obtain all the results in Peng and Li [40]. Moreover, if \(p(\cdot )\equiv 1=q,\) then we obtain all the results in Jiao and Yu [24].

Remark 4.4

Let \(\Phi (t)=t.\) Then, we obtain some results in [21]. Moreover, if \(\Phi (t)=t\) and \(p(\cdot )\equiv p,\) then we obtain many results in the classical martingale space theory as in [17, 44].

Furthermore, if the stochastic basis is regular, similarly to proof of Theorem 3.5 in [21], the other part of Theorem 3.4 also holds, then we can easily get the following further conclusion via (4.1) and the proof process of Theorems 4.2.

Theorem 4.5

Let \(p(\cdot )\in {\mathcal{P}}\) satisfy (2.2), \(\Phi \in {\mathcal{G}}\) be a concave function and \(0<q\le \infty .\) If the stochastic basis \(\{{{\mathcal{F}}}_n\}_{n\ge 0}\) is regular, then

$$\begin{aligned} \big \Vert \Phi (M(f))\big \Vert _{{p(\cdot ),q}}&\approx \big \Vert \Phi (S(f))\big \Vert _{{p(\cdot ),q}}\approx \big \Vert \Phi (s(f))\big \Vert _{{p(\cdot ),q}}\\ &\approx \big \Vert \Phi (f)\big \Vert _{{\mathcal{Q}}_{p(\cdot ),q}} \approx \big \Vert \Phi (f)\big \Vert _{{\mathcal{D}}_{p(\cdot ),q}}. \end{aligned}$$

If the stochastic basis \(\{{{\mathcal{F}}}_n\}_{n\ge 0}\) is regular, let \(\Phi (t)=t,\) then we obtain from Theorem 4.5 that the five variable martingale Lorentz–Hardy spaces are equivalent, see also Corollary 3.8 in [22].

Corollary 4.6

Let \(p(\cdot )\in {\mathcal{P}}\) satisfy (2.2) and \(0<q\le \infty .\) If the stochastic basis \(\{{{\mathcal{F}}}_n\}_{n\ge 0}\) is regular, then

$$\begin{aligned} H^M_{p(\cdot ),q} = H^S_{p(\cdot ),q}= H^s_{p(\cdot ),q} = {\mathcal{Q}}_{p(\cdot ),q} = {\mathcal{D}}_{p(\cdot ),q} \end{aligned}$$

with equivalent quasi-norms.