The John–Nirenberg Theorems for Martingales on Variable Lorentz–Karamata Spaces

Abstract

Let E be a rearrangement-invariant Banach function space. Let $\mathcal{P}(\Omega )$ denote the collection of all measurable variable exponents $p(·):\Omega \to (0,\infty )$ such that $0<\mathrm{ess}{\inf }_{w\in \Omega }p\left(w\right)\le \mathrm{ess}{\sup }_{w\in \Omega }p\left(w\right)<\infty$. In this paper, with the help of a new atomic decomposition of the variable Hardy–Lorentz–Karamata space $H_{p(·),q,b}^M$ via ${(s,p(·),E)}^M$-atoms, we characterize the dual space of $H_{p(·),q,b}^M$ for the two cases $0<q\le 1$ and $1<q\le \infty$, respectively. Using this, some new John–Nirenberg theorems associated with variable exponents are also established.

1. Introduction

Let $0<p<\infty$, $0<q\le \infty$, and b be a slowly varying function. Recall that the Lorentz–Karamata space $L_{p,q,b}(\Omega ,\mathcal{F},\mathbb{P})$ (abbreviated by $L_{p,q,b}$) is the collection of all measurable functions f in the probability space $(\Omega ,\mathcal{F},\mathbb{P})$ satisfying ${\parallel f\parallel }_{p,q,b}<\infty$. The norm ${\parallel f\parallel }_{p,q,b}$ is given by

$${\parallel f\parallel }_{p,q,b}:=\left\{\begin{array}{cc}{\left[{\int }_0^{\infty }{\left(\mathbb{P}{\left(\right|f|>t)}^{1/p}{\gamma }_b\left(\mathbb{P}\left(\right|f|>t)\right)t\right)}^q{\displaystyle \frac{dt}{t}}\right]}^{1/q},\hfill & 0<q<\infty ,\hfill \\ \underset{t>0}{sup}t\mathbb{P}{\left(\right|f|>t)}^{1/p}{\gamma }_b\left(\mathbb{P}\left(\right|f|>t)\right),\hfill & q=\infty ,\hfill \end{array}\right.$$

where ${\gamma }_b\left(t\right)=b\left(t^{-1}\right)$ for $0<t<1$. First proposed by Edmunds et al. in 2000 [1], the Lorentz–Karamata space provides a unified framework that generalizes several classical function spaces, such as Lebesgue spaces, Lorentz spaces, Zygmund spaces, and Lorentz–Zygmund spaces, by adjusting the parameters p, q, and b. More significant results of these spaces can be found in [2,3,4,5,6,7,8,9,10]; notably, they were first applied to characterize optimal Sobolev embeddings on regular Euclidean domains.

The development of variable Lebesgue spaces in harmonic analysis (see, e.g., [11]) has naturally led to increased research interest in martingale theory associated with variable exponents. Specifically, assume that $p(·)$ is ${\mathcal{F}}_n$-measurable for all $n\ge 0$; the Doob maximal inequality was established by Aoyama [12] in 2009. Subsequently, Nakai and Sadasue [13] demonstrated that Aoyama’s measurability assumption is not a necessary prerequisite for the Doob maximal inequality. Recently, Jiao et al. [14,15] gave the condition (4) as a replacement for the log-Hölder continuity and obtained the Doob maximal inequality in weak-type and strong-type. Under the condition (4), Jiao et al. [16] investigated real interpolation for variable martingale Hardy spaces. Note that Nakai and Sawano [17] first introduced the variable Hardy spaces based on Euclidean spaces ${\mathbb{R}}^n$. In this paper, we study the Hardy space in a probability space—a setting that diverges from the Euclidean case due to the absence of a natural metric structure. Weisz proved Doob’s and Burkholder–Davis–Gundy inequalities with variable exponents in [18] and explored equivalent characterizations of martingale Hardy–Lorentz spaces with variable exponents in [19]. For more information about variable martingale Hardy–Lorentz–Karamata spaces $H_{p(·),q,b}$, we refer readers to [20,21,22,23].

In this paper, we consider the John–Nirenberg theorems associated with variable Lorentz–Karamata spaces in martingale settings. Now, we consider martingales adapted to a non-decreasing stochastic basis ${\left\{{\mathcal{F}}_n\right\}}_{n\ge 0}$. Recall the classical John–Nirenberg theorem: if the stochastic basis ${\left\{{\mathcal{F}}_n\right\}}_{n\ge 0}$ is regular, then

$$BM{O}_p=BM{O}_1,1\le p<\infty ,$$

(1)

where ${\parallel f\parallel }_{BM{O}_p}={\parallel f\parallel }_{BM{O}_{L_p}}$ (the definition of this norm is given below). For details on the result (1), we refer the reader to [24]. Moreover, Yi et al. [25] extended this result to the setting of rearrangement-invariant Banach function space, i.e., if the stochastic basis is regular and E is a rearrangement-invariant Banach function space (see, e.g., [26]), then

$$BM{O}_E=BM{O}_1.$$

In 2015, Wu et al. [27] studied John–Nirenberg inequalities with variable exponents in probability spaces. For a more general case, the version of variable Lipschitz spaces was investigated in [28] (Theorem 5.14); i.e., for any $1\le q<\infty$, if $\alpha (·)$ satisfies (4), $0<{\alpha }_{-}\le {\alpha }_{+}<\infty$, and ${\left\{{\mathcal{F}}_n\right\}}_{n\ge 0}$ is regular, then

$$BM{O}_{E,q}\left(\alpha (·)\right)=BM{O}_{1,q}\left(\alpha (·)\right).$$

(2)

More results on the John–Nirenberg theorem can be found in [29,30,31].

Motivated by the above-established findings, we proceed to investigate a variable exponent generalization of the John–Nirenberg theorem. In this paper, we show that if ${\left\{{\mathcal{F}}_n\right\}}_{n\ge 0}$ is regular, then

$$BM{O}_{E,b}\left(\alpha (·)\right)=BM{O}_{1,b}\left(\alpha (·)\right)$$

and

$$BM{O}_{E,q,b}\left(\alpha (·)\right)=BM{O}_{1,q,b}\left(\alpha (·)\right)$$

with equivalent norms (see Theorems 3 and 4 for any unexplained terminology). As a special case of the above result, when $E=L_r$ ($1\le r<\infty$), we obtain [23] (Theorem 6.5).

Throughout this paper, we use $\mathbb{Z}$ to denote the set of integers and $\mathbb{N}$ for the set of non-negative integers. The symbol C represents a positive constant, whose value may differ from line to line. The symbol $A\lesssim B$ stands for the inequality $A\le CB$. If we write $A\approx B$, then it means $A\lesssim B\lesssim A$.

2. Preliminaries

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

2.1. Rearrangement-Invariant Banach Function Spaces

Let E be a rearrangement-invariant Banach function space defined in the probability space $(\Omega ,\mathcal{F},\mathbb{P})$; for the standard definitions of Banach function spaces and rearrangement-invariant Banach function spaces, we refer the reader to [26] (Chapters 1 and 2). Throughout this section, we assume that the probability space $(\Omega ,\mathcal{F},\mathbb{P})$ is non-atomic.

Now, we recall the Luxemburg representation theorem (see, for instance, [26] (Page 62)). If $(E,\parallel ·{\parallel }_E)$ is a rearrangement-invariant Banach function space, then there exists a rearrangement-invariant Banach function space $\widehat{E}$ over $[0,1]$ equipped with the norm ${\parallel ·\parallel }_{\widehat{E}}$ such that

$$\begin{array}{c}\hfill {\parallel f\parallel }_E={\parallel \mu (·,f)\parallel }_{\widehat{E}},\forall f\in E,\end{array}$$

where $\mu (·,f)$ is the non-increasing rearrangement function of f defined by

$$\begin{array}{c}\hfill \mu (t,f)=\underset{s>0}{inf}\{s:\mathbb{P}(\left|f\right|>s)\le t\},t>0.\end{array}$$

We call $(\widehat{E},\parallel ·{\parallel }_{\widehat{E}})$ the Luxemburg representation space of $(E,\parallel ·{\parallel }_E)$.

To proceed, we use the Boyd indices of E, first introduced by Boyd in [32]. We define the dilation operator $D_s$ (for $0<s<\infty$) in the space of measurable functions over $(0,1)$:

$$\begin{array}{c}\hfill D_{s}f\left(t\right)=\left\{\begin{array}{cc}f(t/s),\hfill & \mathrm{if}0<t<min(1,s),\hfill \\ 0,\hfill & \mathrm{if}s<t<1.\hfill \end{array}\right.\end{array}$$

Let $(\widehat{E},\parallel ·{\parallel }_{\widehat{E}})$ denote the Luxemburg representation space of E. The upper Boyd index and lower Boyd index of E are defined as

$$\begin{array}{c}\hfill q_E:=\underset{s>1}{inf}{\displaystyle \frac{logs}{log\parallel D_s\parallel }}\end{array}$$

and

$$\begin{array}{c}\hfill p_E:=\underset{0<s<1}{sup}{\displaystyle \frac{logs}{log\parallel D_s\parallel }}.\end{array}$$

Here, $\parallel D_s\parallel$ denotes the operator norm of $D_s$ in the Luxemburg representation space $\widehat{E}$. Note that the Boyd’s indices defined in [32] are the reciprocals of the ones defined in this paper. For any rearrangement-invariant Banach function space E, the indices satisfy

$$\begin{array}{c}\hfill 1\le p_E\le q_E\le \infty .\end{array}$$

Next, we introduce the associate space (also called Köthe dual) $E^{\prime }$ of E, which is defined as

$$\begin{array}{c}\hfill E^{\prime }=\left\{{f:\parallel f\parallel }_{E^{\prime }}<\infty \right\},\end{array}$$

where

$$\begin{array}{c}\hfill {\parallel f\parallel }_{E^{\prime }}=\underset{\begin{array}{c}g\in E,\\ {\parallel g\parallel }_E\le 1\end{array}}{sup}{\int }_{\Omega }\left|fg\right|d\mathbb{P}.\end{array}$$

For rearrangement-invariant Banach function spaces, the existence of a Fatou norm is equivalent to isometric embedding into the second Köthe dual. Specifically, E admits a Fatou norm if and only if $E^{\prime \prime }={\left(E^{\prime }\right)}^{\prime }$ holds isometrically. We utilize the following duality relation for Boyd indices (see Theorem II.4.11 [33]): if E is a rearrangement-invariant Banach function space equipped with a Fatou norm, then

$${\displaystyle \frac{1}{p_E}}+{\displaystyle \frac{1}{q_{E^{\prime }}}}=1,{\displaystyle \frac{1}{p_{E^{\prime }}}}+{\displaystyle \frac{1}{q_E}}=1.$$

(3)

As a standard example, the Lebesgue spaces $L_p$ (for $1\le p\le \infty$) are rearrangement-invariant Banach function spaces that possess Fatou norms.

Lemma 1

([26]). Let E be a Banach function space with associate space $E^{\prime }$. If $f\in E$ and $g\in E^{\prime }$, then $fg$ is integrable and

$$\begin{array}{c}\hfill \left|{\int }_{\Omega }fgd\mathbb{P}\right|\le {\parallel f\parallel }_E{\parallel g\parallel }_{E^{\prime }}.\end{array}$$

2.2. Variable Lebesgue Spaces

Throughout this paper, for any $p(·)\in \mathcal{P}(\Omega )$, define the conjugate variable exponent $p^{\prime }(·)$ of $p(·)$ by

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{p\left(\omega \right)}}+{\displaystyle \frac{1}{p^{\prime }\left(\omega \right)}}=1,\omega \in \Omega .\end{array}$$

The variable Lebesgue space $L_{p(·)}:=L_{p(·)}(\Omega )$ consists of all measurable functions f on $(\Omega ,\mathcal{F},\mathbb{P})$ such that there exists $\lambda >0$ with

$$\begin{array}{c}\hfill \rho (f/\lambda )={\int }_{\Omega }{\left({\displaystyle \frac{\left|f\right(·\left)\right|}{\lambda }}\right)}^{p(·)}d\mathbb{P}<\infty .\end{array}$$

Equipping with the (quasi)-norm

$$\begin{array}{c}\hfill {\parallel f\parallel }_{p(·)}:=inf\left\{\lambda >0:\rho (f/\lambda )\le 1\right\},\end{array}$$

$\left(L_{p(·)},{\parallel ·\parallel }_{p(·)}\right)$ becomes a (quasi)-Banach function space. Basic properties of the functional ${\parallel ·\parallel }_{p(·)}$ can be found in [11,17,34]. Moreover, the following lemma for variable Lebesgue spaces will be useful in the sequel.

Lemma 2

(Lemma 2.2 [28]). Let $p(·)\in \mathcal{P}(\Omega )$ satisfy $p_{+}\le 1$. For any $f,g\in L_{p(·)}$, we have

$$\begin{array}{c}\hfill {\parallel f\parallel }_{p(·)}+{\parallel g\parallel }_{p(·)}\le {∥\left|f\right|+\left|g\right|∥}_{p(·)}.\end{array}$$

2.3. Slowly Varying Functions

Let $f:[1,\infty )\to (0,\infty )$ be a function. We say that f is equivalent to a non-decreasing (resp. non-increasing) function g if $f\approx g$. To introduce the Lorentz–Karamata spaces with variable exponents, we recall the definition of slowly varying functions.

Definition 1

([2]). A Lebesgue measurable function $b:[1,\infty )\to (0,\infty )$ is said to be a slowly varying function if for any given $\epsilon >0$, the function $t^{\epsilon }b\left(t\right)$ is equivalent to a non-decreasing function and the function $t^{-\epsilon }b\left(t\right)$ is equivalent to a non-increasing function on $[1,\infty )$.

Example 1.

It is obvious that $\eta \left(t\right)\equiv 1$ and $1+logt$ are slowly varying functions (see [8]). Let $0<p<\infty$, $m\in \mathbb{N}$, and $\alpha =({\alpha }_1,{\alpha }_2,\dots ,{\alpha }_m)\in {\mathbb{R}}^m$. Define the family of positive functions ${\left\{{\ell }_k\right\}}_{k=0}^m$ on $(0,\infty )$ by

$${\ell }_0\left(t\right)=1/t\mathit{and}{\ell }_k\left(t\right)=1+log\left({\ell }_{k-1}\left(t\right)\right),0<t\le 1,1\le k\le m.$$

Moreover, define

$${\Theta }_{\alpha }^m\left(t\right)=\prod _{k=1}^m{\ell }_k^{{\alpha }_k}\left(t\right).$$

Clearly, ${\Theta }_{\alpha }^m$ is a slowly varying function (see [2]). Furthermore, it follows from [1] that ${(e+logt)}^{\alpha }{(log(e+logt))}^{\beta }(\alpha ,\beta \in \mathbb{R})$ and $exp\left(\sqrt{logt}\right)$ are also slowly varying functions.

Let b be a slowly varying function on $[1,\infty )$. We denote by ${\gamma }_b$ the positive function defined by

$${\gamma }_b\left(t\right)=b(max\{t,1/t\}),t\in (0,\infty ),$$

as introduced in [2]. The following proposition shows some properties of the slowly varying functions; for further details on such functions, we refer the reader to [2,5,6,8,9].

Proposition 1.

Let b be a slowly varying function on $[1,\infty )$.

(i) 

If b is a non-decreasing function, then ${\gamma }_b$ is non-increasing on $(0,1]$.

(ii) 

For any given $\epsilon >0$, the function $t^{\epsilon }{\gamma }_b\left(t\right)$ is equivalent to a non-decreasing function and the function $t^{-\epsilon }{\gamma }_b\left(t\right)$ is equivalent to a non-increasing function on $(0,\infty )$.

(iii) 

If ε and r are positive numbers, then there exists positive constants $c_{\epsilon }$ and $C_{\epsilon }$ such that

$$c_{\epsilon }min\{r^{\epsilon },r^{-\epsilon }\}b\left(t\right)\le b\left(rt\right)\le C_{\epsilon }max\{r^{\epsilon },r^{-\epsilon }\}b\left(t\right),t>0.$$

(iv) 

For any $a>0$, denote $b_1\left(t\right)=b\left(t^a\right)$ on $[1,\infty )$. Then, $b_1$ is also a slowly varying function.

(v) 

For any given $r\in \mathbb{R}$, the function $b^r$ is a slowly varying function and ${\gamma }_{b^r}={\gamma }_b^r$.

(vi) 

Let $0<p\le \infty$. For any positive constants α and β, we have

$$\begin{array}{c}\hfill {(\alpha +\beta )}^p{\gamma }_b(\alpha +\beta )\lesssim {\alpha }^p{\gamma }_b\left(\alpha \right)+{\beta }^p{\gamma }_b\left(\beta \right).\end{array}$$

2.4. Variable Lorentz–Karamata Spaces

The definition of variable Lorentz–Karamata spaces can be found in [23] as follows.

Definition 2.

Suppose that $p(·)\in \mathcal{P}(\Omega )$, $0<q\le \infty$, and b is a slowly varying function. The variable Lorentz–Karamata space $L_{p(·),q,b}:=L_{p(·),q,b}(\Omega )$ is the collection of all measurable functions f with ${\parallel f\parallel }_{p(·),q,b}<\infty$, where

$${\parallel f\parallel }_{p(·),q,b}=\left\{\begin{array}{cc}& {\left[{\int }_0^{\infty }{\left(t\parallel {\chi }_{\left\{\right|f|>t\}}{\parallel }_{p(·)}{\gamma }_b(\parallel {\chi }_{\left\{\right|f|>t\}}{\parallel }_{p(·)})\right)}^q{\displaystyle \frac{dt}{t}}\right]}^{{\displaystyle \frac{1}{q}}},\mathrm{if}0<q<\infty ,\hfill \\ & \underset{t>0}{sup}t\parallel {\chi }_{\left\{\right|f|>t\}}{\parallel }_{p(·)}{\gamma }_b(\parallel {\chi }_{\left\{\right|f|>t\}}{\parallel }_{p(·)}),\mathrm{if}q=\infty .\hfill \end{array}\right.$$

Remark 1.

These spaces coincide with the Lorentz–Karamata spaces $L_{p,q,b_1}$ in the case where $p(·)\equiv p$ is a positive constant and $b_1\left(t\right)=b\left(t^{1/p}\right)$ for $t\in [1,\infty )$. For comprehensive details on Lorentz–Karamata spaces, the reader is referred to [2,8,35]. Moreover, variable Lorentz–Karamata spaces reduce, in particular, to variable Lorentz spaces when $b\equiv 1$. For an introduction to variable Lorentz spaces, we refer the reader to [11,34,36,37].

In the case of $q=\infty$, we define a subspace of $L_{p(·),\infty ,b}$ as below.

Definition 3.

Let $p(·)\in \mathcal{P}(\Omega )$ and let b be a slowly varying function. We denote by ${\mathcal{L}}_{p(·),\infty ,b}$ the subset of $L_{p(·),\infty ,b}$ consisting of functions with absolutely continuous quasi-norm, i.e.,

$${\mathcal{L}}_{p(·),\infty ,b}=\left\{f\in L_{p(·),\infty ,b}:\underset{n\to \infty }{lim}{\parallel f{\chi }_{A_n}\parallel }_{p(·),q,b}=0\right\},$$

where ${\left(A_n\right)}_{n\ge 0}$ is any sequence of sets satisfying ${lim}_{n\to \infty }\mathbb{P}\left(A_n\right)=0$.

2.5. Martingale and Variable Martingale Hardy–Lorentz–Karamata Spaces

Let ${\left\{{\mathcal{F}}_n\right\}}_{n\ge 0}$ be a non-decreasing sequence of sub-$\sigma$-algebras of $\mathcal{F}$, satisfying $\mathcal{F}=\sigma \left({\bigcup }_{n\ge 0}{\mathcal{F}}_n\right)$. We denote the expectation operator by $\mathbb{E}$ and the conditional expectation operator relative to ${\mathcal{F}}_n$ by ${\mathbb{E}}_n$. A sequence $f={\left(f_n\right)}_{n\ge 0}\subset L_1$ of measurable functions is called a martingale with respect to ${\left\{{\mathcal{F}}_n\right\}}_{n\ge 0}$ if it satisfies

$$\begin{array}{c}\hfill {\mathbb{E}}_n\left(f_{n+1}\right)=f_n\mathrm{for}\mathrm{all}n\ge 0.\end{array}$$

Denote by $\mathcal{M}$ the collection of all martingales $f={\left(f_n\right)}_{n\ge 0}$ with respect to ${\left\{{\mathcal{F}}_n\right\}}_{n\ge 0}$ such that $f_0=0$. For $f\in \mathcal{M}$, the martingale differences are defined by

$$d_{0}f:=0,d_{n}f:=f_n-f_{n-1}(n>0).$$

Let $\mathcal{T}$ be the set of all stopping times associated with ${\left\{{\mathcal{F}}_n\right\}}_{n\ge 0}$. For each $f\in \mathcal{M}$ and $\tau \in \mathcal{T}$, we define the stopped martingale $f^{\tau }={\left(f_n^{\tau }\right)}_{n\ge 0}$ by

$$f_n^{\tau }:=\sum _{m=0}^n{\chi }_{\{\tau \ge m\}}{d}_{m}f.$$

For a given martingale f, the maximal function and the conditional square function of f are defined, respectively, as follows:

$$M_m\left(f\right)=\underset{n\le m}{sup}|f_n|,M\left(f\right)=\underset{n\ge 0}{sup}\left|f_n\right|;$$

$$s_m\left(f\right)={\left(\sum _{n=0}^m{\mathbb{E}}_{n-1}{\left|d_{n}f\right|}^2\right)}^{{\displaystyle \frac{1}{2}}},s\left(f\right)={\left(\sum _{n=0}^{\infty }{\mathbb{E}}_{n-1}{\left|d_{n}f\right|}^2\right)}^{{\displaystyle \frac{1}{2}}}.$$

Definition 4.

Let $p(·)\in \mathcal{P}(\Omega )$, $0<q\le \infty$, and b be a slowly varying function. The variable martingale Hardy spaces associated with variable Lorentz–Karamata spaces are defined by

$$H_{p(·),q,b}^M=\left\{f\in {\mathcal{M}:\parallel f\parallel }_{H_{p(·),q,b}^M}={\parallel M\left(f\right)\parallel }_{p(·),q,b}<\infty \right\};$$

$$H_{p(·),q,b}^s=\left\{f\in {\mathcal{M}:\parallel f\parallel }_{H_{p(·),q,b}^s}={\parallel s\left(f\right)\parallel }_{p(·),q,b}<\infty \right\};$$

$${\mathcal{P}}_{p(·),q,b}=\left\{f\in \mathcal{M}:\exists \lambda ={\left({\lambda }_n\right)}_{n\ge 0}\in {\Lambda }_{p(·),q,b}\mathrm{s}.\mathrm{t}.\left|f_n\right|\le {\lambda }_{n-1}\right\}$$

$${with\parallel f\parallel }_{{\mathcal{P}}_{p(·),q,b}}=\underset{\lambda \in {\Lambda }_{p(·),q,b}}{inf}{\parallel {\lambda }_{\infty }\parallel }_{{\mathcal{L}}_{p(·),q,b}},$$

where ${\Lambda }_{p(·),q,b}$ is the class of all sequences $\lambda ={\left({\lambda }_n\right)}_{n\ge 0}$ of non-decreasing, non-negative, and adapted functions with ${\lambda }_{\infty }:=\underset{n\to \infty }{lim}{\lambda }_n\in {\mathcal{L}}_{p(·),q,b}(\Omega )$. We define ${\mathcal{H}}_{p(·),\infty ,b}^M$ as the space of all martingales such that $M\left(f\right)\in {\mathcal{L}}_{p(·),\infty ,b}$.

Remark 2.

When we set $p(·)\equiv p$ in Definition 4, this reduces to the martingale Hardy–Lorentz–Karamata spaces that were first introduced in [5,6]. Additionally, the variable martingale Hardy–Lorentz space (as defined in [28,36]) constitutes a special instance of the variable martingale Hardy–Lorentz–Karamata space when $b\equiv 1$.

2.6. Some More Notations

Recall that a set $B\in {\mathcal{F}}_n$ is called an atom (with respect to ${\mathcal{F}}_n$), if for any $A\subset B$ satisfying $A\in {\mathcal{F}}_n$ and $\mathbb{P}\left(A\right)<\mathbb{P}\left(B\right)$, we have $\mathbb{P}\left(A\right)=0$. In the theory of variable spaces, the log-Hölder continuity of $p(·)$ is commonly used. Throughout this paper, we adopt the following assumptions that each $\sigma$-algebra ${\mathcal{F}}_n$ is generated by countably many atoms. For $n\ge 0$, denote by $A\left({\mathcal{F}}_n\right)$ the set of all atoms in ${\mathcal{F}}_n$. We suppose that there exists an absolute constant $K_{p(·)}\ge 1$ (depending only on $p(·)$) such that

$$\begin{array}{c}\hfill \mathbb{P}{\left(A\right)}^{p_{-}\left(A\right)-p_{+}\left(A\right)}\le K_{p(·)},\forall A\in \bigcup _{n}A\left({\mathcal{F}}_n\right).\end{array}$$

(4)

Here, $p_{+}\left(A\right)$ and $p_{-}\left(A\right)$ are defined as follows:

$$\begin{array}{c}\hfill p_{+}\left(A\right):=\underset{\omega \in A}{\mathrm{ess}\sup }p\left(\omega \right)\mathrm{and}{p}_{-}\left(A\right):=\underset{\omega \in A}{\mathrm{ess}\inf }p\left(\omega \right).\end{array}$$

Note that, under condition (4) in this paper, each $\sigma$-algebra ${\mathcal{F}}_n$ is assumed to be countably generated by atoms.

For any $f\in L_1(\Omega )$, we can easily find that

$$\begin{array}{c}\hfill {\mathbb{E}}_n\left(f\right)=\sum _{A\in A\left({\mathcal{F}}_n\right)}\left({\displaystyle \frac{1}{\mathbb{P}\left(A\right)}}{\int }_{A}f\left(x\right)d\mathbb{P}\right){\chi }_A,\forall n\in \mathbb{N}.\end{array}$$

We now recall the regularity of a stochastic basis. A stochastic basis ${\left\{{\mathcal{F}}_n\right\}}_{n\ge 0}$ is regular if there exists a constant $R>0$ (independent of n) such that for any non-negative martingales ${\left(f_n\right)}_{n\ge 0}$,

$$\begin{array}{c}\hfill f_n\le R{f}_{n-1}.\end{array}$$

(5)

We refer the reader to [38] (Chapter 7) for further details.

The following lemmas will be used in the sequel.

Lemma 3

(Theorem 1.2 [21]). Let $p(·)\in \mathcal{P}(\Omega )$ satisfy (4), $0<q\le \infty$, and let b be a slowly varying function. If ${\left\{{\mathcal{F}}_n\right\}}_{n\ge 0}$ is regular, then

$$H_{p(·),q,b}^M=H_{p(·),q,b}^s={\mathcal{P}}_{p(·),q,b}.$$

Lemma 4.

Let $p(·)\in \mathcal{P}(\Omega )$ satisfy (4), $0<q<\infty$, and let b be a slowly varying function. If ${\left\{{\mathcal{F}}_n\right\}}_{n\ge 0}$ is regular, then $L_{\infty }$ is dense in $H_{p(·),q,b}^M$.

Proof. 

For each $f\in {\mathcal{P}}_{p(·),q,b}=H_{p(·),q,b}^M$ (by Lemma 3), it follows from [20] (Theorem 3.4) that

$$\begin{array}{c}\hfill f=\sum _{k\in \mathbb{Z}}{\mu }_k{a}^k,\end{array}$$

where $a^k$ and ${\mu }_k$ are defined in [20] (Theorem 3.4). Similarly to [21] (Remark 3.7), we easily find that ${\sum }_{k=l}^m{\mu }_k{a}^k$ converges to f in ${\mathcal{P}}_{p(·),q,b}$ as $l\to -\infty ,m\to +\infty$. Moreover, $M\left(a^k\right)\in L_{\infty }$ for each $k\in \mathbb{Z}$. Hence, we find that $L_{\infty }$ is dense in $H_{p(·),q,b}^M$. The proof is complete. □

3. Atomic Decompositions

In this section, we establish the atomic decompositions of the spaces $H_{p(·),q,b}^s$ and $H_{p(·),q,b}^M$ via ${(s,p(·),E)}^s$-atoms and ${(s,p(·),E)}^M$-atoms, respectively.

Definition 5.

Let $p(·)\in \mathcal{P}(\Omega )$ and E be a rearrangement-invariant Banach function space. A measurable function a is called a simple $\left(p\right(·),E)$-atom (briefly ${(s,p(·),E)}^s$-atom) if there exist $i\in \mathbb{N}$, $I\in A\left({\mathcal{F}}_i\right)$ such that

(i) 

The support of a is contained in I;

(ii) 

${\parallel s\left(a\right)\parallel }_E\le {\displaystyle \frac{\parallel {\chi }_I{\parallel }_E}{\parallel {\chi }_I{\parallel }_{p(·)}}}$;

(iii) 

${\mathbb{E}}_i\left(a\right)=0$.

If $s\left(a\right)$ in (ii) is substituted with $M\left(a\right)$, then the function a is called an ${(s,p(·),E)}^M$-atom.

Remark 3.

If we consider the special case $p(·)\equiv p$ and $E=L_{\infty }$, the ${(s,p(·),E)}^s$-atom (resp. ${(s,p(·),E)}^M$-atom) is $(1,p,\infty )$(resp. $(3,p,\infty )$) is a simple atom in [39] (Definition 2.4).

Definition 6.

Let $p(·)\in \mathcal{P}(\Omega )$, E be a rearrangement-invariant Banach function space, $0<q\le \infty$, and b be a slowly varying function. The atomic martingale Lorentz–Karamata space with variable exponent $H_{p(·),q,b}^{sat,*,E}(*=s\mathit{or}M)$ is defined to be the set of all $f={\left(f_n\right)}_{n\ge 0}\in \mathcal{M}$ such that for each $n\ge 0$,

$$\begin{array}{c}\hfill f_n=\sum _{k\in \mathbb{Z}}\sum _{i=0}^{n-1}\sum _j{\mu }_{k,i,j}{\mathbb{E}}_n\left(a^{k,i,j}\right).\end{array}$$

(6)

Here, ${\left(a^{k,i,j}\right)}_{k\in \mathbb{Z},i\in \mathbb{N},j}$ is a sequence of simple ${(s,p(·),E)}^{*}$-atoms associated with ${\left(I_{k,i,j}\right)}_{k\in \mathbb{Z},i\in \mathbb{N},j}$, and the sets $I_{k,i,j}$, are disjoint for any fixed k, ${\mu }_{k,i,j}=3·2^k{\parallel {\chi }_{I_{k,i,j}}\parallel }_{p(·)}$ and

$$\parallel {\left\{\parallel \sum _{i=0}^{\infty }\sum _j{\mu }_{k,i,j}{\parallel {\chi }_{I_{k,i,j}}\parallel }_{p(·)}^{-1}{\chi }_{I_{k,i,j}}{\parallel }_{p(·)}{\gamma }_b\left(\parallel \sum _{i=0}^{\infty }\sum _j{\chi }_{I_{k,i,j}}{\parallel }_{p(·)}\right)\right\}}_{k\in \mathbb{Z}}{\parallel }_{l_q}<\infty .$$

We equip $H_{p(·),q,b}^{sat,*,E}$ with the (quasi)-norm

$$\begin{array}{c}\hfill {\parallel f\parallel }_{H_{p(·),q,b}^{sat,*,E}}=inf\parallel {\left\{\parallel \sum _{i=0}^{\infty }\sum _j{\mu }_{k,i,j}{\parallel {\chi }_{I_{k,i,j}}\parallel }_{p(·)}^{-1}{\chi }_{I_{k,i,j}}{\parallel }_{p(·)}{\gamma }_b\left(\parallel \sum _{i=0}^{\infty }\sum _j{\chi }_{I_{k,i,j}}{\parallel }_{p(·)}\right)\right\}}_{k\in \mathbb{Z}}{\parallel }_{l_q},\end{array}$$

where the infimum is taken over all decompositions of f as above.

Lemma 5

(Lemma 5.5 [28]). Let $p(·)\in \mathcal{P}(\Omega )$ satisfy (4), $p_{+}<1$, and E be a rearrangement-invariant Banach function space. Take $0<\epsilon <p_{-}$ and $\gamma \in \left(1,{\displaystyle \frac{1}{p_{+}}}\wedge {\displaystyle \frac{1}{\epsilon }}\right)$. If $a^{k,i,j}$ is an ${(s,p(·),E)}^s$-atom for every $k,i,j$ associated with $I_{k,i,j}\in A\left({\mathcal{F}}_i\right)$, then

$$\begin{array}{c}\hfill Z:={∥\sum _{i=0}^{\infty }\sum _j{\left[\parallel {\chi }_{I_{k,i,j}}{\parallel }_{p(·)}s\left(a^{k,i,j}\right){\chi }_{I_{k,i,j}}\right]}^{\gamma \epsilon }∥}_{p(·)/\epsilon }\lesssim {∥\sum _{i=0}^{\infty }\sum _j{\chi }_{I_{k,i,j}}∥}_{p(·)/\epsilon }.\end{array}$$

Theorem 1.

Let E be a rearrangement-invariant Banach function space with $1\le p_E\le q_E<\infty$. If $p(·)\in \mathcal{P}(\Omega )$ satisfies (4) with $p_{+}<1$, $0<q\le \infty$, and b is a slowly varying function, then

$$\begin{array}{c}\hfill H_{p(·),q,b}^s=H_{p(·),q,b}^{sat,s,E}.\end{array}$$

Proof. 

Let $f\in H_{p(·),q,b}^s$. For each $k\in \mathbb{Z}$ and $n\in \mathbb{N}$, define

$$\begin{array}{c}\hfill {\tau }_k:=inf\left\{n\in \mathbb{N}:s_{n+1}\left(f\right)>2^k\right\}.\end{array}$$

It is easy to verify that ${\left\{{\tau }_k\right\}}_{k\in \mathbb{Z}}$ is a non-decreasing sequence of stopping times. For fixed $k\in \mathbb{Z}$, there exist disjoint atoms ${\left\{I_{k,i,j}\right\}}_j\subset A\left({\mathcal{F}}_i\right)$ such that

$$\begin{array}{c}\hfill \bigcup _j{I}_{k,i,j}=\{{\tau }_k=i\}.\end{array}$$

This means

$$\begin{array}{c}\hfill \{{\tau }_k<\infty \}=\bigcup _{i=0}^{\infty }\{{\tau }_k=i\}=\bigcup _{i=0}^{\infty }\bigcup _j{I}_{k,i,j}.\end{array}$$

Hence, we find that

$$\begin{array}{c}\hfill f_n=\sum _{k\in \mathbb{Z}}\sum _{i=0}^{n-1}\sum _j(f_n^{{\tau }_{k+1}}-f_n^{{\tau }_k}){\chi }_{I_{k,i,j}}.\end{array}$$

Let

$${\mu }_{k,i,j}:=3·2^k{\parallel {\chi }_{I_{k,i,j}}\parallel }_{p(·)}\mathrm{and}{a}_n^{k,i,j}:={\displaystyle \frac{f_n^{{\tau }_{k+1}}-f_n^{{\tau }_k}}{{\mu }_{k,i,j}}}{\chi }_{I_{k,i,j}}$$

(if ${\mu }_{k,i,j}=0$, then let $a_n^{k,i,j}=0$). By [21], we know that $a^{k,i,j}\in L_2$, ${\mathbb{E}}_n\left(a^{k,i,j}\right)=a_n^{k,i,j}$, and

$$s\left(a^{k,i,j}\right)\le {\displaystyle \frac{1}{\parallel {\chi }_{I_{k,i,j}}{\parallel }_{p(·)}}}.$$

This implies that

$$\parallel s\left(a^{k,i,j}\right){\parallel }_E\le {\displaystyle \frac{\parallel {\chi }_{I_{k,i,j}}{\parallel }_E}{\parallel {\chi }_{I_{k,i,j}}{\parallel }_{p(·)}}}.$$

Hence, $a^{k,i,j}$ is an ${(s,p(·),E)}^s$-atom. Similarly to the proof of [21] (Theorem 1.1), we find that

$$\begin{array}{c}\hfill {∥{\left\{\parallel \sum _{i=0}^{\infty }\sum _j{\mu }_{k,i,j}{\parallel {\chi }_{I_{k,i,j}}\parallel }_{p(·)}^{-1}{\chi }_{I_{k,i,j}}{\parallel }_{p(·)}{\gamma }_b\left(\parallel \sum _{i=0}^{\infty }\sum _j{\chi }_{I_{k,i,j}}{\parallel }_{p(·)}\right)\right\}}_{k\in \mathbb{Z}}∥}_{l_q}\lesssim {\parallel f\parallel }_{H_{p(·),q,b}^s}.\end{array}$$

For the converse part, let f have a decomposition (6). Using Lemma 5 instead of [21] (Lemma 3.5) and following the same argument as in [21] (Theorem 1.1), it is easy to see that

$$\begin{array}{c}\hfill {\parallel f\parallel }_{H_{p(·),q,b}^s}\lesssim {∥{\left\{\parallel \sum _{i=0}^{\infty }\sum _j{\mu }_{k,i,j}{\parallel {\chi }_{I_{k,i,j}}\parallel }_{p(·)}^{-1}{\chi }_{I_{k,i,j}}{\parallel }_{p(·)}{\gamma }_b\left(\parallel \sum _{i=0}^{\infty }\sum _j{\chi }_{I_{k,i,j}}{\parallel }_{p(·)}\right)\right\}}_{k\in \mathbb{Z}}∥}_{l_q}.\end{array}$$

The proof is complete. □

Theorem 2.

Let E be a rearrangement-invariant Banach function space with $1\le p_E\le q_E<\infty$. Suppose that $p(·)\in \mathcal{P}(\Omega )$ satisfies (4) with $p_{+}<1$, $0<q\le \infty$, and b is a slowly varying function. If ${\left\{{\mathcal{F}}_n\right\}}_{n\ge 0}$ is regular, then

$$\begin{array}{c}\hfill H_{p(·),q,b}^M=H_{p(·),q,b}^{sat,M,E}.\end{array}$$

Proof. 

Let $f\in H_{p(·),q,b}^M$. Define the stopping times with respect to ${\left\{{\mathcal{F}}_n\right\}}_{n\ge 0}$ by

$$v_k:=inf\left\{n\in \mathbb{N}:\left|f_n\right|>2^k\right\},k\in \mathbb{Z}.$$

Moreover,

$$F_i^k:=\left\{{\mathbb{E}}_{i-1}\left({\chi }_{\left\{v_k=i\right\}}\right)\ge {\displaystyle \frac{1}{R}}\right\}\in {\mathcal{F}}_{i-1},i\in \mathbb{N}.$$

Since ${\left\{{\mathcal{F}}_i\right\}}_{i\ge 0}$ is regular, we find that

$$\left\{v_k=i\right\}\subset F_i^k\mathrm{and}\mathbb{P}\left(F_i^k\right)\le R\mathbb{P}\left(v_k=i\right),$$

where R is from (5). Therefore, there exist atoms $\overline{I_{k,i,j}}\in A\left({\mathcal{F}}_i\right)$ such that ${\bigcup }_j\overline{I_{k,i,j}}=\left\{v_k=i\right\}$. Denote by $I_{k,i,j}\in A\left({\mathcal{F}}_{i-1}\right)$ that which contains $\overline{I_{k,i,j}}$. Now, we define another set of stopping times as follows:

$${\tau }_k\left(\omega \right):=inf\left\{i\in \mathbb{N}:\omega \in F_{i+1}^k\right\}.$$

It is easy to see that ${\left\{{\tau }_k\right\}}_{k\in \mathbb{Z}}$ is a non-decreasing sequence of stopping times and ${\tau }_k\left(\omega \right)\le i-1$ when $v_k\left(\omega \right)=i$. More precisely, ${\tau }_k<v_k$ on the set $\left\{v_k\ne \infty \right\}$. Moreover, by [21] (Lemma 3.6),

$${∥{\chi }_{\left\{{\tau }_k<\infty \right\}}∥}_{p(·)}\lesssim {∥{\chi }_{\left\{v_k<\infty \right\}}∥}_{p(·)}={∥{\chi }_{\left\{M\left(f\right)>2^k\right\}}∥}_{p(·)}\le 2^{-k}{\parallel M\left(f\right)\parallel }_{p(·)}.$$

Then, ${∥{\chi }_{\left\{{\tau }_k<\infty \right\}}∥}_{p(·)}\to 0$ as $k\to \infty$, which implies that

$$\underset{k\to \infty }{lim}\mathrm{P}\left({\tau }_k=\infty \right)=1.$$

So, ${lim}_{k\to \infty }{\tau }_k=\infty$ a.e. and ${lim}_{k\to \infty }{f}_n^{{\tau }_k}=f_n$ a.e. for $n\in \mathbb{N}$. Let ${\mu }_{k,i,j}$ and $a_n^{k,i,j}$ be the same as in the proof of Theorem 1. We have

$$M\left({\left(a_n^{k,i,j}\right)}_{n\ge 0}\right)\le {\displaystyle \frac{M\left(f_n^{{\tau }_{k+1}}\right)+M\left(f_n^{{\tau }_k}\right)}{{\mu }_{k,i,j}}}\le {\displaystyle \frac{2^{k+1}+2^k}{{\mu }_{k,i,j}}}={\displaystyle \frac{1}{{∥{\chi }_{I_{k,i,j}}∥}_{p(·)}}}.$$

Thus, ${\left(a_n^{k,i,j}\right)}_{n\ge 0}$ is an $L_2$-bounded martingale. Moreover, there exists $a^{k,i,j}\in L_2$ and

$${∥M\left(a^{k,i,j}\right)∥}_E\le {\displaystyle \frac{\parallel {\chi }_{I_{k,i,j}}{\parallel }_E}{{∥{\chi }_{I_{k,i,j}}∥}_{p(·)}}},$$

which implies $a^{k,i,j}$ is an ${(s,p(·),E)}^M$-atom. Similarly to the proof of [21] (Theorem 1.1), we get

$$\begin{array}{c}\hfill {∥{\left\{\parallel \sum _{i=0}^{\infty }\sum _j{\mu }_{k,i,j}{\parallel {\chi }_{I_{k,i,j}}\parallel }_{p(·)}^{-1}{\chi }_{I_{k,i,j}}{\parallel }_{p(·)}{\gamma }_b\left(\parallel \sum _{i=0}^{\infty }\sum _j{\chi }_{I_{k,i,j}}{\parallel }_{p(·)}\right)\right\}}_{k\in \mathbb{Z}}∥}_{l_q}\lesssim {\parallel f\parallel }_{H_{p(·),q,b}^M}.\end{array}$$

For the converse part, the proof is similar to Theorem 1. The proof is complete. □

4. John–Nirenberg Theorems

In this section, we prove the John–Nirenberg theorems. To this end, we define new bounded mean oscillation martingale spaces associated with a rearrangement-invariant Banach function space.

Definition 7.

Let $\alpha (·)+1\in \mathcal{P}(\Omega )$, E be a Banach function space with its associate space $E^{\prime }$, and b be a slowly varying function. We define the space $BM{O}_{E,b}\left(\alpha (·)\right)$ as the set of all functions $f\in E$ for which

$$\begin{array}{c}\hfill {\parallel f\parallel }_{{\mathrm{BMO}}_{E,b}\left(\alpha (·)\right)}=\underset{n\ge 0}{sup}\underset{I\in A\left({\mathcal{F}}_n\right)}{sup}{\displaystyle \frac{{∥{\chi }_I∥}_{E^{\prime }}{∥(f-f_n){\chi }_I∥}_E}{{∥{\chi }_I∥}_{\frac{1}{\alpha (·)+1}}{\gamma }_b\left({∥{\chi }_I∥}_{\frac{1}{\alpha (·)+1}}\right)}}\end{array}$$

is finite.

Remark 4.

When $\alpha (·)=0$ and $b\equiv 1$, this definition reduces to the classical martingale $BMO$ space. For a positive constant $\alpha (·)={\alpha }_0>0$, the definition coincides with the classical martingale Lipschitz space. Further details on these special cases can be found in [39].

Proposition 2.

Let $p(·)\in \mathcal{P}(\Omega )$ with $0<p_{+}<1$, $0<q\le 1$, b be a slowly varying function, and E be a Banach function space. If $f={\left(f_n\right)}_{n\ge 0}\in H_{p(·),q,b}^{sat,M,E}$ with a decomposition as in (6), then

$$\begin{array}{c}\hfill \sum _{k\in \mathbb{Z}}\left(\sum _{i=0}^{\infty }\sum _j{\mu }_{k,i,j}\right){\gamma }_b\left(\parallel \sum _{i=0}^{\infty }\sum _j{\chi }_{I_{k,i,j}}{\parallel }_{p(·)}\right)\lesssim {\parallel f\parallel }_{H_{p(·),q,b}^{sat,M,E}},\end{array}$$

(7)

where ${\mu }_{k,i,j}=3·2^k{\parallel {\chi }_{I_{k,i,j}}\parallel }_{p(·)}$ and ${\left(I_{k,i,j}\right)}_j\subset A\left({\mathcal{F}}_i\right)$ are as in Definition 6.

Proof. 

Applying Lemma 2, we find that

$$\begin{array}{cc}\hfill & \sum _{k\in \mathbb{Z}}\left(\sum _{i=0}^{\infty }\sum _j{\mu }_{k,i,j}\right){\gamma }_b\left(\parallel \sum _{i=0}^{\infty }\sum _j{\chi }_{I_{k,i,j}}{\parallel }_{p(·)}\right)\hfill \\ \hfill & \lesssim \sum _{k\in \mathbb{Z}}\left(\sum _{i=0}^{\infty }\sum _j{2}^k{\parallel {\chi }_{I_{k,i,j}}\parallel }_{p(·)}\right){\gamma }_b\left(\parallel \sum _{i=0}^{\infty }\sum _j{\chi }_{I_{k,i,j}}{\parallel }_{p(·)}\right)\hfill \\ \hfill & \lesssim \sum _{k\in \mathbb{Z}}{2}^k{∥\sum _{i=0}^{\infty }\sum _j{\chi }_{I_{k,i,j}}∥}_{p(·)}{\gamma }_b\left(\parallel \sum _{i=0}^{\infty }\sum _j{\chi }_{I_{k,i,j}}{\parallel }_{p(·)}\right)\hfill \\ \hfill & \le {\left\{\sum _{k\in \mathbb{Z}}{2}^{kq}{∥\sum _{i=0}^{\infty }\sum _j{\chi }_{I_{k,i,j}}∥}_{p(·)}^q{\gamma }_b^q\left(\parallel \sum _{i=0}^{\infty }\sum _j{\chi }_{I_{k,i,j}}{\parallel }_{p(·)}\right)\right\}}^{{\displaystyle \frac{1}{q}}},\hfill \end{array}$$

which implies (7). This complete the proof. □

Lemma 6.

Let $p(·)\in \mathcal{P}(\Omega )$ satisfy condition (4) with $0<p_{+}<1$ and $0<q\le 1$. Let E be a rearrangement-invariant Banach function space equipped with a Fatou norm such that $1\le p_E\le q_E<\infty$ and b be a slowly varying function. If ${\left\{{\mathcal{F}}_n\right\}}_{n\ge 0}$ is regular, then

$$\begin{array}{c}\hfill {\left(H_{p(·),q,b}^M\right)}^{*}={\mathrm{BMO}}_{E,b}\left(\alpha (·)\right),\end{array}$$

where $\alpha (·)={\displaystyle \frac{1}{p(·)}}-1$ and the norms of these spaces are equivalent.

Proof. 

Let $\phi \in BM{O}_{E,b}\left(\alpha (·)\right)$. By Lemma 1, we have

$$\begin{array}{c}\hfill \parallel (\phi -{\phi }_n){\chi }_I{\parallel }_1\le \parallel (\phi -{\phi }_n){\chi }_I{\parallel }_E{\parallel {\chi }_I\parallel }_{E^{\prime }}.\end{array}$$

This means that

$$\begin{array}{c}\hfill {\parallel \phi \parallel }_{BM{O}_{1,b}\left(\alpha (·)\right)}\le {\parallel \phi \parallel }_{BM{O}_{E,b}\left(\alpha (·)\right)}.\end{array}$$

Then, $\phi \in BM{O}_{1,b}\left(\alpha (·)\right)$. In light of [23] (Theorems 6.1 and 6.5), we easily find that $BM{O}_{1,b}\left(\alpha (·)\right)\subset L_1$. Define the functional as

$$\begin{array}{c}\hfill l_{\phi }\left(f\right)=\mathbb{E}\left(f\phi \right),\forall f\in L_{\infty }.\end{array}$$

Similarly to the proof of [23] (Theorem 5.6), using Theorem 2, Proposition 2, and Lemma 3, we obtain

$$\begin{array}{c}\hfill |l_{\phi }{\left(f\right)|\lesssim \parallel \phi \parallel }_{BM{O}_{1,b}\left(\alpha (·)\right)}{\parallel f\parallel }_{H_{p(·),q,b}^M}.\end{array}$$

On the other hand, Lemma 4 implies that $L_{\infty }$ is dense in $H_{p(·),q,b}^M$. This guarantees that $l_{\phi }$ extends uniquely to a continuous linear functional on $H_{p(·),q,b}^M$.

For the converse part, take any $l\in {\left(H_{p(·),q,b}^M\right)}^{*}$. Since $L_2$ is contained in $H_{p(·),q,b}^M$, there exists some $\phi \in L_2$ such that

$$\begin{array}{c}\hfill l\left(f\right)=\mathbb{E}\left(f\phi \right),\forall f\in L_2.\end{array}$$

It remains to show that $\phi \in BM{O}_{E,b}\left(\alpha (·)\right)$. By the proof of [23] (Theorem 5.6), we have ${\parallel \phi \parallel }_{BM{O}_{2,b}\left(\alpha (·)\right)}\le \parallel l\parallel$. Thus, to establish $\phi \in BM{O}_{E,b}\left(\alpha (·)\right)$, it is sufficient to verify

$$\begin{array}{c}\hfill {\parallel \phi \parallel }_{BM{O}_{E,b}\left(\alpha (·)\right)}\lesssim {\parallel \phi \parallel }_{BM{O}_{2,b}\left(\alpha (·)\right)}.\end{array}$$

For any given $n\in \mathbb{N}$ and each $I\in A\left(\mathcal{F}\right)$, the associate space $E^{\prime }$ of E guarantees the existence of some $h\in E^{\prime }$ with ${\parallel h\parallel }_{E^{\prime }}\le 1$ such that

$$\begin{array}{c}\hfill {∥(\phi -{\phi }_n){\chi }_I∥}_E\le 2\left|{\int }_I(\phi -{\phi }_n)hd\mathbb{P}\right|.\end{array}$$

We now introduce the following definitions:

$$\begin{array}{c}\hfill a={\displaystyle \frac{\parallel {\chi }_I{\parallel }_{E^{\prime }}(h-h_n){\chi }_I}{2c\parallel {\chi }_I{\parallel }_{p(·)}{\gamma }_b\left(\parallel {\chi }_I{\parallel }_{p(·)}\right)}},\alpha (·)={\displaystyle \frac{1}{p(·)}}-1,\end{array}$$

where c denotes the constant appearing in [28] (Lemma 5.3). From (3), it is easy to check that $1<p_{E^{\prime }}\le q_{E^{\prime }}\le \infty$. Thus, it follows from [28] (Lemma 5.3) that

$$\begin{array}{c}\hfill {\parallel M\left(a\right)\parallel }_{E^{\prime }}\le c{\parallel a\parallel }_{E^{\prime }}\le {\displaystyle \frac{\parallel {\chi }_I{\parallel }_{E^{\prime }}}{\parallel {\chi }_I{\parallel }_{p(·)}}}.\end{array}$$

This means that a is an ${(s,p(·),E^{\prime })}^M$-atom. Then, by Lemma 3 and Theorem 2, we get

$$\begin{array}{c}\hfill (h-h_n){\chi }_I={\displaystyle \frac{2c\parallel {\chi }_I{\parallel }_{p(·)}·{\gamma }_b(\parallel {\chi }_I{\parallel }_{p(·)})}{\parallel {\chi }_I{\parallel }_{E^{\prime }}}}a\in H_{p(·),q,b}^M\end{array}$$

along with the estimate

$$\begin{array}{c}\hfill \parallel (h-h_n){\chi }_I{\parallel }_{H_{p(·),q,b}^M}\le {\displaystyle \frac{2c\parallel {\chi }_I{\parallel }_{p(·)}{\gamma }_b(\parallel {\chi }_I{\parallel }_{p(·)})}{\parallel {\chi }_I{\parallel }_{E^{\prime }}}}.\end{array}$$

It follows from [23] (Theorem 5.6) that ${\left(H_{p(·),q,b}^s\right)}^{*}=BM{O}_{2,q,b}\left(\alpha (·)\right),\alpha (·)={\displaystyle \frac{1}{p(·)}}-1.$ Given that ${\left\{{\mathcal{F}}_n\right\}}_{n\ge 0}$ is regular, together with Lemma 3, we conclude that

$${\left(H_{p(·),q,b}^M\right)}^{*}=BM{O}_{2,q,b}\left(\alpha (·)\right),\alpha (·)={\displaystyle \frac{1}{p(·)}}-1.$$

By [23] (Proposition 5.5), we have ${\parallel f\parallel }_{BM{O}_{2,b}\left(\alpha (·)\right)}\approx {\parallel f\parallel }_{BM{O}_{2,q,b}\left(\alpha (·)\right)}$ for any $0<q\le 1$. Therefore, it is clear that ${\left(H_{p(·),q,b}^M\right)}^{*}=BM{O}_{2,b}\left(\alpha (·)\right)$ if ${\left\{{\mathcal{F}}_n\right\}}_{n\ge 0}$. Then,

$$\begin{array}{cc}\hfill {\displaystyle \frac{\parallel {\chi }_I{\parallel }_{E^{\prime }}{\parallel (\phi -{\phi }_n){\chi }_I\parallel }_E}{\parallel {\chi }_I{\parallel }_{\frac{1}{\alpha (·)+1}}{\gamma }_b\left(\parallel {\chi }_I{\parallel }_{\frac{1}{\alpha (·)+1}}\right)}}& \le {\displaystyle \frac{2\parallel {\chi }_I{\parallel }_{E^{\prime }}\left|{\int }_I(\phi -{\phi }_n)hd\mathbb{P}\right|}{\parallel {\chi }_I{\parallel }_{\frac{1}{\alpha (·)+1}}{\gamma }_b\left(\parallel {\chi }_I{\parallel }_{\frac{1}{\alpha (·)+1}}\right)}}\hfill \\ \hfill & ={\displaystyle \frac{2\parallel {\chi }_I{\parallel }_{E^{\prime }}\left|{\int }_I\phi (h-h_n)d\mathbb{P}\right|}{\parallel {\chi }_I{\parallel }_{\frac{1}{\alpha (·)+1}}{\gamma }_b\left(\parallel {\chi }_I{\parallel }_{\frac{1}{\alpha (·)+1}}\right)}}\hfill \\ \hfill & \le {\displaystyle \frac{2\parallel {\chi }_I{\parallel }_{E^{\prime }}{\parallel \phi \parallel }_{BM{O}_{2,b}\left(\alpha (·)\right)}{\parallel (h-h_n){\chi }_I\parallel }_{H_{p(·),q,b}^M}}{\parallel {\chi }_I{\parallel }_{\frac{1}{\alpha (·)+1}}{\gamma }_b\left(\parallel {\chi }_I{\parallel }_{\frac{1}{\alpha (·)+1}}\right)}}\hfill \\ \hfill & \le {2c\parallel \phi \parallel }_{BM{O}_{2,b}\left(\alpha (·)\right)}.\hfill \end{array}$$

As a consequence, we obtain

$$\begin{array}{c}\hfill {\parallel f\parallel }_{BM{O}_{E,b}\left(\alpha (·)\right)}\lesssim {\parallel f\parallel }_{BM{O}_{2,b}\left(\alpha (·)\right)},\end{array}$$

which finishes the proof. □

An immediate consequence of the above lemma is the following John–Nirenberg inequality.

Theorem 3.

Let $\alpha (·)+1\in \mathcal{P}(\Omega )$ satisfy (4) and $0<{\alpha }_{-}\le {\alpha }_{+}<\infty$. Let E be a rearrangement-invariant Banach function space with a Fatou norm such that $1\le p_E\le q_E<\infty$ and b be a slowly varying function. If ${\left\{{\mathcal{F}}_n\right\}}_{n\ge 0}$ is regular, then

$$\begin{array}{c}\hfill BM{O}_{E,b}\left(\alpha (·)\right)=BM{O}_{1,b}\left(\alpha (·)\right)\end{array}$$

with equivalent norms.

Remark 5.

If $b\equiv 1$ in Theorem 3, then we obtain [28] (Theorem 5.10).

Now, we consider the duality for the case $1<q<\infty$. Unlike the situation where $0<q\le 1$, some novel insights are required here. A key contribution of this paper is the introduction of a generalized $BMO$ martingale space developed specifically for this setting.

Definition 8.

Let E be a Banach function space, $0<q\le \infty$, $\alpha (·)+1\in \mathcal{P}(\Omega )$, and b be a slowly varying function. The generalized martingale space $BM{O}_{E,q,b}\left(\alpha (·)\right)$ is defined by

$$\begin{array}{c}\hfill BM{O}_{E,q,b}\left(\alpha (·)\right)=\left\{f\in {E:\parallel f\parallel }_{BM{O}_{E,q,b}\left(\alpha (·)\right)}<\infty \right\},\end{array}$$

where the norm is given by

$$\begin{array}{c}\hfill {\parallel f\parallel }_{BM{O}_{E,q,b}\left(\alpha (·)\right)}=sup{\displaystyle \frac{{\sum }_{k\in \mathbb{Z}}{\sum }_{i\in \mathbb{N}}{\sum }_j{2}^k\parallel {\chi }_{I_{k,i,j}}{\parallel }_{E^{\prime }}{\parallel (f-f_i){\chi }_{I_{k,i,j}}\parallel }_E}{{\left({\sum }_{k\in \mathbb{Z}}{2}^{kq}{∥{\sum }_{i\in \mathbb{N}}{\sum }_j{\chi }_{I_{k,i,j}}∥}_{\frac{1}{\alpha (·)+1}}^q{\gamma }_b^q\left({∥{\sum }_{i\in \mathbb{N}}{\sum }_j{\chi }_{I_{k,i,j}}∥}_{\frac{1}{\alpha (·)+1}}\right)\right)}^{1/q}}}\end{array}$$

and the supremum is taken over all sequences of atoms ${\left\{I_{k,i,j}\right\}}_{k\in \mathbb{Z},i\in \mathbb{N},j}$ such that $I_{k,i,j}$ are disjoint if k is fixed, $I_{k,i,j}$ belong to ${\mathcal{F}}_i$, and

$$\begin{array}{c}\hfill {\left\{2^k{∥\sum _{i\in \mathbb{N}}\sum _j{\chi }_{I_{k,i,j}}∥}_{{\displaystyle \frac{1}{\alpha (·)+1}}}{\gamma }_b\left({∥\sum _{i\in \mathbb{N}}\sum _j{\chi }_{I_{k,i,j}}∥}_{{\displaystyle \frac{1}{\alpha (·)+1}}}\right)\right\}}_k\in {\ell }_q.\end{array}$$

$BM{O}_{r,\infty ,b}\left(\alpha (·)\right)$ can be defined in a similar manner.

Similarly to [23] (Proposition 5.5), we establish the following proposition.

Proposition 3.

Let E be a rearrangement-invariant Banach function space, $0<q<\infty$, $\alpha (·)\ge 0$ with $\alpha (·)+1\in \mathcal{P}(\Omega )$, and b be a slowly varying function. Then,

$$\begin{array}{c}\hfill {\parallel f\parallel }_{BM{O}_{E,b}\left(\alpha (·)\right)}\le {\parallel f\parallel }_{BM{O}_{E,q,b}\left(\alpha (·)\right)}.\end{array}$$

If $0<q\le 1$ and ${\alpha }_{-}>0$, then $BM{O}_{E,b}\left(\alpha (·)\right)=BM{O}_{E,q,b}\left(\alpha (·)\right)$.

Proof. 

Restricting the supremum in the definition of $BM{O}_{E,q,b}\left(\alpha (·)\right)$ to a single atom yields the $BM{O}_{E,b}\left(\alpha (·)\right)$-norm, which directly proves the first inequality. In the case where $0<q\le 1$, we have

$$\begin{array}{cc}\hfill & {\parallel f\parallel }_{BM{O}_{E,q,b}\left(\alpha (·)\right)}\hfill \\ \hfill & \le sup{\displaystyle \frac{{\sum }_{k\in \mathbb{Z}}{\sum }_{i\in \mathbb{N}}{\sum }_j{2}^k\parallel {\chi }_{I_{k,i,j}}{\parallel }_{E^{\prime }}{\parallel (f-f_i){\chi }_{I_{k,i,j}}\parallel }_E}{{\sum }_{k\in \mathbb{Z}}{2}^k{∥{\sum }_{i\in \mathbb{N}}{\sum }_j{\chi }_{I_{k,i,j}}∥}_{\frac{1}{\alpha (·)+1}}{\gamma }_b\left({∥{\sum }_{i\in \mathbb{N}}{\sum }_j{\chi }_{I_{k,i,j}}∥}_{\frac{1}{\alpha (·)+1}}\right)}}\hfill \\ & \le {\parallel f\parallel }_{BM{O}_{E,b}\left(\alpha (·)\right)}sup{\displaystyle \frac{{\sum }_{k\in \mathbb{Z}}{\sum }_{i\in \mathbb{N}}{\sum }_j{2}^k{\parallel {\chi }_{I_{k,i,j}}\parallel }_{\frac{1}{\alpha (·)+1}}{\gamma }_b\left(\parallel {\chi }_{I_{k,i,j}}{\parallel }_{\frac{1}{\alpha (·)+1}}\right)}{{\sum }_{k\in \mathbb{Z}}{∥{\sum }_{i\in \mathbb{N}}{\sum }_j{\chi }_{I_{k,i,j}}∥}_{\frac{1}{\alpha (·)+1}}{\gamma }_b\left({∥{\sum }_{i\in \mathbb{N}}{\sum }_j{\chi }_{I_{k,i,j}}∥}_{\frac{1}{\alpha (·)+1}}\right)}}.\hfill \end{array}$$

By [21] (Lemma 2.10), we find that

$$\begin{array}{c}\hfill BM{O}_{E,q,b}\left(\alpha (·)\right)\lesssim BM{O}_{E,b}\left(\alpha (·)\right).\end{array}$$

Hence, we find that $BM{O}_{E,b}\left(\alpha (·)\right)=BM{O}_{E,q,b}\left(\alpha (·)\right)$ when $0<q\le 1$ and ${\alpha }_{-}>0$. This complete the proof. □

Now, we establish the following result.

Lemma 7.

Let $p(·)\in \mathcal{P}(\Omega )$ satisfy condition (4) with $0<p_{+}<1$ and $1<q<\infty$. Let E be a rearrangement-invariant Banach function space equipped with a Fatou norm such that $1\le p_E\le q_E<\infty$ and b be a slowly varying function. If ${\left\{{\mathcal{F}}_n\right\}}_{n\ge 0}$ is regular, then

$$\begin{array}{c}\hfill {\left(H_{p(·),q,b}^M\right)}^{*}={\mathrm{BMO}}_{E,q,b}\left(\alpha (·)\right),\end{array}$$

where $\alpha (·)={\displaystyle \frac{1}{p(·)}}-1$ and the norms of these spaces are equivalent.

Proof. 

By Lemma 1, we have ${\parallel g\parallel }_{BM{O}_{1,q,b}\left(\alpha (·)\right)}\le {\parallel g\parallel }_{BM{O}_{E,q,b}\left(\alpha (·)\right)}$ for all $g\in BM{O}_{E,q,b}\left(\alpha (·)\right)$. For any $\phi \in BM{O}_{E,q,b}\left(\alpha (·)\right)$, it follows that $\phi \in BM{O}_{1,q,b}\left(\alpha (·)\right)\subset L_1$. We define the functional as

$$\begin{array}{c}\hfill l_{\phi }\left(f\right)=\mathbb{E}\left(f\phi \right),\forall f\in L_{\infty }.\end{array}$$

Combining the inclusion $L_{\infty }\subset H_{p(·),q,b}^M$ with [21] (Theorem 1.1) and Lemma 3, we deduce

$$\begin{array}{c}\hfill f=\sum _{k\in \mathbb{Z}}\sum _{i=0}^{\infty }\sum _j{\mu }_{k,i,j}{a}^{k,i,j},\forall f\in L_{\infty },\end{array}$$

where ${\mu }_{k,i,j}=3·2^k{\parallel {\chi }_{I_{k,i,j}}\parallel }_{p(·)}$ and $a^{k,i,j}$ are ${(s,p(·),\infty )}^M$-atoms associated with ${\left(I_{k,i,j}\right)}_j\subset A\left({\mathcal{F}}_i\right)$. It follows from (iii) of Definition 5 that $\mathbb{E}\left(a^{k,i,j}\phi \right)=\mathbb{E}\left(a^{k,i,j}(\phi -{\phi }_i)\right)$ holds for all $k,i,j$, where ${\phi }_i={\mathbb{E}}_i\left(\phi \right)$. Hence, we find that

$$\begin{array}{cc}\hfill |l_{\phi }\left(f\right)|& \le \sum _{k\in \mathbb{Z}}\sum _{i=0}^{\infty }\sum _j{\mu }_{k,i,j}\left|{\int }_{\Omega }{a}^{k,i,j}(\phi -{\phi }_i)d\mathbb{P}\right|\hfill \\ \hfill & \le \sum _{k\in \mathbb{Z}}\sum _{i=0}^{\infty }\sum _j{\mu }_{k,i,j}\parallel a^{k,i,j}{\parallel }_{\infty }{\parallel (\phi -{\phi }_i){\chi }_{I_{k,i,j}}\parallel }_1\hfill \\ \hfill & \lesssim \sum _{k\in \mathbb{Z}}\sum _{i=0}^{\infty }\sum _j{2}^k{\parallel (\phi -{\phi }_i){\chi }_{I_{k,i,j}}\parallel }_1.\hfill \end{array}$$

In light of the definition of the $BM{O}_{1,q,b}\left(\alpha (·)\right)$-norm and Lemma 3, we have

$$\begin{array}{cc}\hfill |l_{\phi }\left(f\right)|& \lesssim {\left(\sum _{k\in \mathbb{Z}}{2}^{kq}{∥\sum _{i\in \mathbb{N}}\sum _j{\chi }_{I_{k,i,j}}∥}_{p(·)}^q{\gamma }_b^q\left({∥\sum _{i\in \mathbb{N}}\sum _j{\chi }_{I_{k,i,j}}∥}_{p(·)}\right)\right)}^{1/q}{\parallel \phi \parallel }_{BM{O}_{1,q,b}\left(\alpha (·)\right)}\hfill \\ \hfill & \lesssim {\parallel f\parallel }_{H_{p(·),q,b}^M}{\parallel \phi \parallel }_{BM{O}_{E,q,b}\left(\alpha (·)\right)}.\hfill \end{array}$$

As $L_{\infty }$ is dense in $H_{p(·),q,b}^M$ (see Lemma 4), the functional $l_{\phi }$ admits a unique continuous extension to $H_{p(·),q,b}^M$.

Conversely, let $l\in {\left(H_{p(·),q,b}^M\right)}^{*}$. Since $L_2\subset H_{p(·),q,b}^M$, the Riesz representation theorem implies the existence of some $\phi \in L_2\subset L_1$ such that

$$\begin{array}{c}\hfill l\left(f\right)=\mathbb{E}\left(f\phi \right),\forall f\in L_2.\end{array}$$

We still need to verify that $\phi \in BM{O}_{E,q,b}\left(\alpha (·)\right)$. By [23] (Theorem 5.6), we have

$${\parallel \phi \parallel }_{BM{O}_{2,q,b}\left(\alpha (·)\right)}\lesssim \parallel l\parallel .$$

Thus, it suffices to establish ${\parallel \phi \parallel }_{BM{O}_{E,q,b}\left(\alpha (·)\right)}\lesssim {\parallel \phi \parallel }_{BM{O}_{2,q,b}\left(\alpha (·)\right)}$.

Let ${\left\{I_{k,i,j}\right\}}_{k\in \mathbb{Z},i\in \mathbb{N},j}$ be an arbitrary sequence of atoms satisfying the fact that $I_{k,i,j}$ are disjoint for fixed k, $I_{k,i,j}\in A\left({\mathcal{F}}_i\right)$ for fixed $k,i$, and

$$\begin{array}{c}\hfill {\left\{2^k{∥\sum _{i\in \mathbb{N}}\sum _j{\chi }_{I_{k,i,j}}∥}_{p(·)}\right\}}_{k\in \mathbb{Z}}\in {\ell }_q.\end{array}$$

Applying duality, for each $k,i,j$, there exists $h^{k,i,j}\in E^{\prime }$ such that

$$\begin{array}{cc}\hfill \parallel (\phi -{\phi }_i){\chi }_{I_{k,i,j}}{\parallel }_E& \le 2\int (\phi -{\phi }_i){\chi }_{I_{k,i,j}}{h}^{k,i,j}d\mathbb{P}\hfill \\ \hfill & =2\int \left(h^{k,i,j}-{\mathbb{E}}_j\left(h^{k,i,j}\right)\right){\chi }_{I_{k,i,j}}\phi d\mathbb{P}.\hfill \end{array}$$

Following the proof strategy of Lemma 6, we define

$$\begin{array}{c}\hfill a^{k,i,j}={\displaystyle \frac{\parallel {\chi }_{I_{k,i,j}}{\parallel }_{E^{\prime }}\left(h^{k,i,j}-{\mathbb{E}}_i\left(h^{k,i,j}\right)\right){\chi }_{I_{k,i,j}}}{2c\parallel {\chi }_{I_{k,i,j}}{\parallel }_{p(·)}{\gamma }_b\left(\parallel {\chi }_{I_{k,i,j}}{\parallel }_{p(·)}\right)}},\alpha (·)={\displaystyle \frac{1}{p(·)}}-1,\end{array}$$

where c denotes the constant from the Doob maximal inequality (see [28] (Lemma 5.3)). Each $a^{k,i,j}$ is an ${(s,p(·),E)}^M$-atom. By Theorem 2, we observe that

$$\begin{array}{c}\hfill f=\sum _{k\in \mathbb{Z}}\sum _{i\in \mathbb{N}}\sum _j{2}^k{\parallel {\chi }_{I_{k,i,j}}\parallel }_{p(·)}{\gamma }_b\left(\parallel {\chi }_{I_{k,i,j}}{\parallel }_{p(·)}\right)a^{k,i,j}\in H_{p(·),q,b}^M\end{array}$$

and

$$\begin{array}{c}\hfill {\parallel f\parallel }_{H_{p(·),q,b}^M}\lesssim {\left(\sum _{k\in \mathbb{Z}}{2}^{kq}{∥\sum _{i\in \mathbb{N}}\sum _j{\chi }_{I_{k,i,j}}∥}_{p(·)}^q{\gamma }_b^q\left({∥\sum _{i\in \mathbb{N}}\sum _j{\chi }_{I_{k,i,j}}∥}_{p(·)}\right)\right)}^{1/q}.\end{array}$$

It follows from the above argument and [23] (Theorem 5.6) that

$$\begin{array}{cc}\hfill & \sum _{k\in \mathbb{Z}}\sum _{i\in \mathbb{N}}\sum _j{2}^k\parallel {\chi }_{I_{k,i,j}}{\parallel }_{E^{\prime }}{\parallel (\phi -{\phi }_j){\chi }_{I_{k,i,j}}\parallel }_E\hfill \\ \hfill & \le 2\sum _{k\in \mathbb{Z}}\sum _{i\in \mathbb{N}}\sum _j{2}^k{\parallel {\chi }_{I_{k,i,j}}\parallel }_{E^{\prime }}\int \left(h^{k,i,j}-{\mathbb{E}}_i\left(h^{k,i,j}\right)\right){\chi }_{I_{k,i,j}}\phi d\mathbb{P}\hfill \\ \hfill & \lesssim {\parallel \phi \parallel }_{BM{O}_{2,q,b}\left(\alpha (·)\right)}{\parallel f\parallel }_{H_{p(·),q,b}^M}.\hfill \end{array}$$

This implies that

$$\begin{array}{c}\hfill {\parallel \phi \parallel }_{BM{O}_{E,q,b}\left(\alpha (·)\right)}\lesssim {\parallel \phi \parallel }_{BM{O}_{2,q,b}\left(\alpha (·)\right)}.\end{array}$$

The proof is complete. □

Note that $L_2$ is dense in ${\mathcal{H}}_{p(·),\infty ,b}^M$ from [21] (Remark 3.7). By adapting a similar argument used in Lemma 7, we can establish

$$\begin{array}{c}\hfill {\left({\mathcal{H}}_{p(·),\infty ,b}^M\right)}^{*}=BM{O}_{E,\infty ,b}\left(\alpha (·)\right).\end{array}$$

Then, the John–Nirenberg theorem stated below follows immediately as a consequence of Lemma 7 and the above fact.

Theorem 4.

Let $\alpha (·)+1\in \mathcal{P}(\Omega )$ satisfy (4), $1<q\le \infty$, and $0<{\alpha }_{-}\le {\alpha }_{+}<\infty$. Let E be a rearrangement-invariant Banach function space with a Fatou norm such that $1\le p_E\le q_E<\infty$, and b be a slowly varying function. If ${\left\{{\mathcal{F}}_n\right\}}_{n\ge 0}$ is regular, then

$$\begin{array}{c}\hfill BM{O}_{E,q,b}\left(\alpha (·)\right)=BM{O}_{2,q,b}\left(\alpha (·)\right)\end{array}$$

with equivalent norms.

If $E=L_r(1\le r<\infty )$ in Theorems 3 and 4, the following result, which was proved in [23] (Theorem 6.5), follows from Proposition 3.

Corollary 1.

Let $\alpha (·)+1\in \mathcal{P}(\Omega )$ satisfy (4) with $0<{\alpha }_{-}\le {\alpha }_{+}<\infty$, $0<q\le \infty$, and b be a slowly varying function. If ${\left\{{\mathcal{F}}_n\right\}}_{n\ge 0}$ is regular, then

$$\begin{array}{c}\hfill BM{O}_{r,q,b}\left(\alpha (·)\right)=BM{O}_{2,q,b}\left(\alpha (·)\right)\end{array}$$

with equivalent norms for all $1\le r<\infty$.

Remark 6.

Moreover, if $b\equiv 1$, then Theorem 4 becomes [28] (Theorem 5.14) and Corollary 1 reduces to [23] (Corollary 6.8).

5. Conclusions

This paper established the atomic decompositions of the variable Hardy–Lorentz–Karamata spaces $H_{p(·),q,b}^s$ and $H_{p(·),q,b}^M$ via ${(s,p(·),E)}^s$-atoms and ${(s,p(·),E)}^M$-atoms, respectively. The key point lies in defining some new bounded mean oscillation martingale spaces associated with rearrangement-invariant Banach function space. Following this, the atomic decomposition was applied to investigate two key results: the dual theorems of $H_{p(·),q,b}^M$ and the John–Nirenberg theorems with variable exponents for the cases $0<q\le 1$ and $1<q\le \infty$. A central contribution of this paper lies in developing a unified framework that enables a proof of the John–Nirenberg theorem. Specifically, we extend the principal results of [28] to the Karamata setting and generalize relevant findings in [23,26].