Orlicz-Lorentz-Karamata Hardy martingale spaces: inequalities and fractional integral operators

Abstract

Let $0<q\le \infty$, b be a slowly varying function and $\Phi : [0,\infty ) \longrightarrow [0,\infty )$ be an increasing function with $\Phi (0)=0$ and $\lim \limits _{r \rightarrow \infty }\Phi (r)=\infty$. In this paper, we introduce a new class of function spaces $L_{\Phi ,q,b}$ which unify and generalize the Lorentz-Karamata spaces with $\Phi (t)=t^p$ and the Orlicz-Lorentz spaces with $b\equiv 1$. Based on the new spaces, we introduce five new Hardy spaces containing martingales, the so-called Orlicz-Lorentz-Karamata Hardy martingale spaces and then develop a theory of these martingale Hardy spaces. To be precise, we first investigate several properties of Orlicz-Lorentz-Karamata spaces and then present Doob’s maximal inequalities by using Hardy’s inequalities. The characterization of these Hardy martingale spaces are constructed via the atomic decompositions. As applications of the atomic decompositions, martingale inequalities and the relation of the different martingale Hardy spaces are presented. The dual theorems and a new John-Nirenberg type inequality for the new framework are also established. Moreover, we study the boundedness of fractional integral operators on Orlicz-Lorentz-Karamata Hardy martingale spaces. The results obtained here generalize the previous results for Lorentz-Karamata Hardy martingale spaces as well as for Orlicz-Lorentz Hardy martingales spaces. Especially, we remove the condition that b is non-decreasing as in [38, 39] and the condition $q_{\Phi ^{-1}}<1/q$ in [24], respectively.

1 Introduction

Martingale theory is widely studied in the field of mathematical physics, harmonic analysis and probability theory. The study of martingale theory can be traced back to Doob [11] and Burkholder and Gundy [8] who established the well-known inequality named after them. Since then, several significant results had been developed such as the duality between the martingale space and BMO and the atomic decompositions. The classical martingale theory is built on Lebesgue space $L_p$ (see [19, 59, 60]). Now the theory had been extended to other important function spaces in analysis.

Another interesting topic we shall touch in our work is Lorentz-Karamata space $L_{p,q,b}$. These spaces are defined by means of a slowly varying function b and generalized Lebesgue spaces, Lorentz spaces, Lorentz-Zygmund spaces and even generalized Lorentz-Zygmund spaces. They were introduced by Edmunds et al. [13] in 2000 and studied by Neves [48], Gogatishvili et al. [20]. This class offers not only a more general and unified insight for these families of spaces but also provides a framework in which it is easier to appreciate the central issues of different results, see [12, 17, 21, 30] and the references therein.

In probability theory, the study of Lorentz-Karamata martingale spaces has attracted more and more attention in martingale theory. By using Karamata theory, Ho [28] combined the martingale theory with Lorentz-Karamata spaces. He introduced the Lorentz-Karamata Hardy martingale spaces and discussed the atomic decompositions, dual theorems and some martingale inequalities of these spaces. Motivated by [37], Jiao et al. [38] further studied the Lorentz-Karamata Hardy martingale spaces and partially extended the results of [28] to a wider range. Inspired by [28, 38], Liu et al. [40] investigated the Lorentz-Karamata Hardy spaces for Banach space-valued martingales; Zhou et al. [69] and Wu et al. [64] studied the weak Orlicz-Karamata-Hardy martingale spaces and Orlicz-Karamata martingale spaces, respectively. In 2017, Liu and Zhou [39] investigated the boundedness of fractional integral operators on the Lorentz-Karamata martingale spaces.

In this paper, we introduce a class of Orlicz-Lorentz-Karamata spaces, which are much more wider than the Lorentz-Karamata spaces or the Orlicz-Lorentz spaces and other important function spaces, and then develop a theory of the martingale Hardy spaces in the new framework. The results obtained in this paper greatly broaden the scope of the previous results in Lorentz-Karamata Hardy martingale spaces and Orlicz-Lorentz Hardy martingale spaces. Besides the introduction, the paper is organized in eight sections as follows.

Section 2 gives the fundamental properties of the Orlicz-Lorentz-Karamata spaces. These properties are very useful to prove later the atomic characterization of the new Hardy spaces. Based on these Orlicz-Lorentz-Karamata spaces, we introduce a new kind of Hardy spaces, the so-called Orlicz-Lorentz-Karamata Hardy martingale spaces, which generalize the Lorentz-Karamata Hardy martingale spaces as well as the Orlicz-Lorentz Hardy martingale spaces introduced very recently by Ho [28] and Hao and Li [24].

Section 3 is devoted to Doob’s maximal inequalities on Orlicz-Lorentz-Karamata spaces. In [25], the authors presented Doob’s inequality on Orlicz-Lorentz-Karamata spaces under the following conditions: $\Phi$ is an Orlicz function with $p_{\Phi }>1$, $1\le q\le \infty$ and b is a slowly varying function. Here $p_{\Phi }$ and $q_\Phi$ is the lower and upper Simonenko index defined as

$$\begin{aligned} p_\Phi =\sup \Big \{p>0:\,t^{-p}\Phi (t)\;\text {is non-decreasing for all}\;t > 0\Big \} \end{aligned}$$

and

$$\begin{aligned} q_\Phi =\inf \Big \{q>0:\,t^{-q}\Phi (t)\;\text {is non-increasing for all}\;t > 0\Big \}. \end{aligned}$$

However, Doob’s inequality on Orlicz-Lorentz-Karamata spaces for $0<q< 1$ was still an open problem. We solve this problem in Theroem 3 by using a new characterization of Hardy-type inequality. More exactly, we generalize Doob’s maximal inequality on Orlicz-Lorentz-Karamata spaces for $0< q\le \infty$ and for $\Phi$ being an Orlicz function with lower type $p>1$ (see Section 2 for the definition of lower type). Note that the condition of lower type is weaker than the lower Simonenko index.

Section 4 presents the atomic decompositions of the five Orlicz-Lorentz-Karamata Hardy martingale spaces. Roughly speaking, under some weak conditions, we prove that a martingale $f=(f_n)_{ n \ge 0 }\in H_{\Phi ,q,b}^s$ (resp. $\mathcal {Q}_{\Phi ,q,b}$, $\mathcal {P}_{\Phi ,q,b}$) if and only if there exist a sequence $(a^k)_{ k \in \mathbb {Z} }$ of $(1,\Phi ,\infty )$ (resp. $(2,\Phi ,\infty )$, $(3,\Phi ,\infty )$) atoms and a sequence $(\mu _k)_{k\in \mathbb {Z}}$ of real numbers

$$\begin{aligned} \mu _k=\frac{\kappa \cdot 2^k}{\Phi ^{-1}\big (1/\mathbb {P}(\tau _k<\infty )\big )} \end{aligned}$$

($\kappa$ is a positive constant and $\tau _k$ is the stopping time associated with $a^k$) such that

$$\begin{aligned} f_n= \sum _{ k \in \mathbb {Z} } \mu _k \mathbb {E}_n a^k \quad a.e. \end{aligned}$$

and

$$\begin{aligned} \inf \bigg (\sum _{k\in \mathbb {Z}}\gamma _b^q\big (\mathbb {P}(\tau _k<\infty )\big )\mu _k^q\bigg )^{1/q} \approx \Vert f\Vert _{H_{\Phi ,q,b}^s}\;\;\; \big (\text {resp.}\; \Vert f\Vert _{\mathcal {Q}_{\Phi ,q,b}},\; \Vert f\Vert _{\mathcal {P}_{\Phi ,q,b}}\big ), \end{aligned}$$

where the infimum is taken over all decompositions of f as above. When the stochastic basis is regular, we also establish atomic decompositions for the martingale Hardy spaces $H_{\Phi ,q,b}^M$ and $H_{\Phi ,q,b}^S$. As corollaries of our theorems in this section, we generalize several known theorems from Lorentz-Karamata Hardy spaces and Orlicz-Lorentz Hardy spaces to Orlicz-Lorentz-Karamata Hardy spaces. Amongst others, we remove the restricting condition $0< q\le p\le 1$ of Theorem 5.4 in [28]; the restricting condition that b is non-decreasing in Theorem 3.1 of [38]; the restricting conditions $0<q\le 1$ and $q_{\Phi ^{-1}}<1/q$ of Theorems 3.1 and 3.2 in [24].

Section 5 shows the fundamental martingale inequalities and the relation of the different Orlicz-Lorentz-Karamata Hardy martingale spaces. The method we used here is to establish a sufficient condition for the $\sigma$-sublinear operator to be bounded from martingale Hardy spaces to Orlicz-Lorentz-Karamata spaces.

Section 6 applies the atomic decompositions of Orlicz-Lorentz-Karamata Hardy martingale spaces above to obtain that if $0<q\le 1$, then the dual space of $H_{\Phi ,q,b}^s$ is the generalized $BMO_2$ (usually denoted by BMO) martingale space $BMO_{2,\Phi ,b}$; if $1<q<\infty$, then the dual space of $H_{\Phi ,q,b}^s$ is $BMO_{2,q,\Phi ,b}$. The conclusions improve the recent results in [27, 37]. As a consequence, Theorems 1.2 and 1.5 in [38] are also true without the assumption that the slowly varying function b is non-decreasing. Moreover, we remove the restricting condition $q_{\Phi ^{-1}}<1/q$ of Theorem 4.3 in [24].

Section 7 describes the John-Nirenberg inequality for the martingale spaces $BMO_{r,q,\Phi ,b}$. Based mainly on the duality and the John-Nirenberg inequality, the space BMO plays an important role in classical analysis and martingale theory, see [38, 56, 60] and the references therein. The main result of this section (see Theorem 13) completes and generalizes the recent conclusion from Jiao et al. [37] and [38].

The last part of the paper, Section 8, extends the boundedness of the fractional integrals to a much more general setting. The fractional integrals are useful tools to analyze the function spaces in harmonic analysis, see [9, 14, 32, 52]. In probability theory, the fractional integrals were first introduced by Chao and Ombe [10] for dyadic martingales. The fractional integrals as special cases of martingale transforms (see [7]) have recently attracted widely attention. Now the fractional integrals have been defined for more general martingales than in [2], [23], [37] and [46, 47]. The goal of this section is to study the boundedness of the fractional integrals for Orlicz-Lorentz-Karamata Hardy martingale spaces. The results are very interesting and improve and generalize several known results, amongst others Theorem 4.4 in [39].

At the end of this section, we make some conventions. Throughout this paper, we denote by C the absolute positive constant that is independent of the main parameters involved, but whose value may differ from line to line. The symbol $f \preceq g$ stands for the inequality $f \le C g$. When we write $f \approx g$, this stands for $f \preceq g \preceq f$.

2 Orlicz-Lorentz-Karamata spaces

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

2.1 Orlicz spaces

Let us first recall some basic properties on Orlicz functions and Orlicz spaces in this subsection. Let $(\Omega ,\mathcal {F},\mathbb {P})$ be a complete probability space and f be an $\mathcal {F}$-measurable function defined on $\Omega$. The distribution function of f is the function $\lambda _s(f)$ defined by

$$\begin{aligned} \lambda _s(f)=\mathbb {P}\big (\{\omega \in \Omega : |f(\omega )|>s\}\big ), \ \ (s\ge 0). \end{aligned}$$

Denote by $f^*$ the decreasing rearrangement of f, defined by

$$\begin{aligned} f^*(t)=\inf \{s\ge 0: \lambda _s(f)\le t\}, \ \ (t\ge 0), \end{aligned}$$

with the convention that $\inf \emptyset =\infty$.

Recall that an Orlicz function $\Phi$ is an increasing function such that $\Phi (0)=0$ and $\lim \limits _{r \rightarrow \infty }\Phi (r)=\infty$. The Orlicz space $L_\Phi =L_\Phi (\Omega ,\mathcal {F},\mathbb {P})$ is the set of all $\mathcal {F}$-measurable functions f satisfying $\mathbb {E}\big (\Phi (c|f|)\big ) < \infty$ for some $c > 0$ and

$$\begin{aligned} \Vert f\Vert _{L_\Phi }=\inf \Big \{ c> 0 : \mathbb {E}\Big (\Phi \big ( {|f|}/{ c} \big )\Big )\le 1 \Big \}, \end{aligned}$$

where $\mathbb {E}$ denotes the expectation operator or the integral with respect to $\mathbb {P}$.

Remark 1

• (1) If $\Phi (t)=t^p \;(0<p<\infty )$, then $L_{\Phi }$ is the Lebesgue space $L_p$. In this case we denote $\Vert \cdot \Vert _{L_\Phi }$ by $\Vert \cdot \Vert _{L_p}$.

• (2) If $\Phi$ is a convex Orlicz function, then the functional $\Vert \cdot \Vert _{L_\Phi }$ is a norm and thereby $L_\Phi$ is a Banach space.

• (3) By a simple calculation, one can check that for any $A\in \mathcal {F}$, $\mathbb {P}(A)>0$,

  $$\begin{aligned} \Vert \chi _A\Vert _{L_\Phi } = \frac{1}{\Phi ^{-1}(\frac{1}{\mathbb {P}(A)})}, \end{aligned}$$

  where $\Phi ^{-1}$ denotes the left inverse function of $\Phi$, i.e. $\Phi ^{-1}(t)=\inf \{u\ge 0: \Phi (u)>t\}$.

Definition 1

Let $0<p<\infty$. One says that $\Phi$ is lower (resp. upper) type p if there exists a positive constant $C_{p}$ depending on p such that, for any $t \in [0,\infty )$ and $s \in (0, 1)$ (resp. $s \in [1, \infty )$),

$$\begin{aligned} \Phi (st)\le C_{p}s^p\Phi (t). \end{aligned}$$

Example 1

If $\Phi (t) = t^p$ for all $t \in (0,\infty )$ and $p \in (0, \infty )$, then the lower and upper types of $\Phi$ are equal to p.

For $0< p< r < \infty$, let

$$\begin{aligned} \Phi (t)=t^p+t^r\quad \quad \text {for all}\;t \in (0,\infty ). \end{aligned}$$

Then $\Phi$ is of lower type p and upper type r.

For $p\in (0,\infty )$ and $r\in \mathbb {R}$, let

$$\begin{aligned} \Phi (t)=t^p\big [\log (c_{p,r}+t)\big ]^r\quad \quad \text {for all}\;t \in (0,\infty ), \end{aligned}$$

where $c_{p,r}$ is a positive constant that ensures the increasing property of $\Phi$ on $(0,\infty )$. Then the lower and upper types of $\Phi$ are equal to p.

For $0< p< r < \infty$, consider the increasing function

$$\begin{aligned} \Phi (t)=\frac{t^r}{(1+t)^p}\quad \quad \text {for all}\;t \in (0,\infty ). \end{aligned}$$

Then $\Phi$ is of lower type $r-p$ and upper type r. The last two examples can be found in [15].

For a function $\varphi :[0,\infty )\rightarrow (0,\infty )$, if there is a non-decreasing (resp. non-increasing) function $\phi$ such that $\varphi \approx \phi$, then we say that $\varphi$ is equivalent to a non-decreasing (resp. non-increasing) function on $[0,\infty )$. By the definitions of lower and upper type, it is easy to see the following two lemmas.

Lemma 1

Let $\Phi$ be an Orlicz function of lower type $p \in (0,\infty )$. Then we have

• (i) the function $t \longmapsto t^{-p} \Phi (t)$ is equivalent to a non-decreasing function;

• (ii) the function $t \longmapsto t^{-1/{p}} \Phi ^{-1}(t)$ is equivalent to a non-increasing function;

• (iii) $\Phi ^{-1}(2t)\le C_p\Phi ^{-1}(t)$ for any $t\in [0,\infty )$, where $C_p$ is depending only on p.

Lemma 2

Let $\Phi$ be an Orlicz function of upper type $p \in (0,\infty )$. Then we have

• (iv) the function $t \longmapsto t^{-p} \Phi (t)$ is equivalent to a non-increasing function;

• (v) the function $t \longmapsto t^{-1/{p}} \Phi ^{-1}(t)$ is equivalent to a non-decreasing function;

• (vi) $\Phi ^{-1}(2t)\ge C_p\Phi ^{-1}(t)$ for any $t\in [0,\infty )$, where $C_p$ is depending only on p.

The following properties of Orlicz spaces will be used in the sequel.

Lemma 3

([68]) Let $\Phi$ be an Orlicz function with lower type $p\in (0,\infty )$. Then $\Vert \cdot \Vert _{L_\Phi }$ is a quasi-norm.

Remark 2

Let $\Phi$ be an Orlicz function with lower type $p\in (0,\infty )$. It follows from Lemma 3 and the Aoki-Rolewicz theorem (see [1]) that there exists $\varrho \in (0,1]$ such that

$$\begin{aligned} \Vert h_1+h_2+\cdots +h_n\Vert _{L_\Phi }^\varrho \le 4\big (\Vert h_1\Vert _{L_\Phi }^\varrho +\Vert h_2\Vert _{L_\Phi }^\varrho +\cdots +\Vert h_n\Vert _{L_\Phi }^\varrho \big ), \end{aligned}$$

for all $h_1,h_2,\cdots ,h_n$ (for any $n \ge 1$) in $L_\Phi$.

Lemma 4

Let $\Phi$ be an Orlicz function of lower type $p_-$ and upper type $p_+$ satisfying $0<p_- \le p_+<\infty$. For $A\in \mathcal {F}$, we have

$$\begin{aligned} \Vert \chi _A\Vert _{L_{p_-}} \preceq \Vert \chi _A\Vert _{L_\Phi }\preceq \Vert \chi _A\Vert _{L_{p_+}}. \end{aligned}$$

Proof

It is sufficient to consider the case of $\mathbb {P}(A)\ne 0$. It follows from Lemmas 1 (ii) and 2 (v) that

$$\begin{aligned} \Phi ^{-1}\bigg (\frac{1}{\mathbb {P}(A)}\bigg )\preceq \frac{1}{\mathbb {P}(A)^{1/p_-}} \cdot \Phi ^{-1}(1) \end{aligned}$$

and

$$\begin{aligned} \Phi ^{-1}(1)= \Phi ^{-1}\bigg (\mathbb {P}(A)\cdot \frac{1}{\mathbb {P}(A)}\bigg ) \preceq \mathbb {P}(A)^{1/p_+} \cdot \Phi ^{-1}\bigg (\frac{1}{\mathbb {P}(A)}\bigg ). \end{aligned}$$

According to Remark 1 (3), we have

$$\begin{aligned} \Vert \chi _A\Vert _{L_{p_-}}=\mathbb {P}(A)^{1/p_-}\preceq \Phi ^{-1}(1)\cdot \frac{1}{\Phi ^{-1}(\frac{1}{\mathbb {P}(A)})}\preceq \Vert \chi _A\Vert _{L_\Phi } \end{aligned}$$

and

$$\begin{aligned} \Vert \chi _A\Vert _{L_\Phi } = \frac{1}{\Phi ^{-1}(\frac{1}{\mathbb {P}(A)})}\preceq \frac{1}{\Phi ^{-1}(1)}\cdot \mathbb {P}(A)^{1/p_+} \preceq \Vert \chi _A\Vert _{L_{p_+}}. \end{aligned}$$

This completes the proof. $\square$

See [18, 43, 53] for more information on Orlicz functions and Orlicz spaces.

2.2 Orlicz-Lorentz-Karamata spaces

We recall the definition of slowly varying function in order to define the Orlicz-Lorentz-Karamata spaces.

Definition 2

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

The detailed study of Karamata theory, properties and examples of slowly varying functions can be found in [6, 13].

Proposition 1

Let $\Phi$ be an Orlicz function of lower type $p_-$ and upper type $p_+$ satisfying $0<p_- \le p_+<\infty$ and b be a slowly varying function. Then the function $b_1$ defined on $[1,\infty )$ by $b_1(t)=b(\Phi (t))$ is a slowly varying function.

Proof

We firstly prove that for any $\varepsilon >0$, $t^\varepsilon b_1(t)$ is equivalent to a non-decreasing function. Indeed, for $t\ge 1$, there is

$$\begin{aligned} t^\varepsilon b_1(t)=t^\varepsilon b(\Phi (t))= & {} \frac{t^\varepsilon }{\Phi (t)^{\varepsilon /p_+}}\Phi (t)^{\varepsilon /p_+}b(\Phi (t))\\= & {} \bigg (\frac{t^{p_+}}{\Phi (t)}\bigg )^{^{\varepsilon /p_+}}\Phi (t)^{\varepsilon /p_+}b(\Phi (t)). \end{aligned}$$

Since $\Phi$ is of upper type $p_+\in (0,\infty )$, it is easily to see that $\frac{t^{p_+}}{\Phi (t)}$ is equivalent to a non-decreasing function. Moreover, b is a slowly varying function, which implies that $\Phi (t)^{\varepsilon /p_+}b(\Phi (t))$ is equivalent to a non-decreasing function. Hence we obtain that $t^\varepsilon b_1(t)$ is equivalent to a non-decreasing function for any $\varepsilon >0$.

Similarly, we can prove that $t^{-\varepsilon } b_1(t)$ is equivalent to a non-increasing function for any $\varepsilon >0$. Thus $b_1$ is a slowly varying function. $\square$

Given a slowly varying function b on $[1,\infty )$, we denote by $\gamma _b$ the positive function defined by

$$\begin{aligned} \gamma _b(t)=b\big ( 1/t \big ), \quad \quad \quad \quad \text {for all}\;0<t\le 1. \end{aligned}$$

Proposition 2

[12]) Let b be a slowly varying function. For any given $\epsilon >0$, the function $t^\epsilon \gamma _b(t)$ is equivalent to a non-decreasing function and $t^{-\epsilon }\gamma _b(t)$ is equivalent to a non-increasing function on (0, 1].

Based on this, we now present the definition of the Orlicz-Lorentz-Karamata spaces.

Definition 3

Let $\Phi$ be an Orlicz function, $0<q\le \infty$ and let b be a slowly varying function. The Orlicz-Lorentz-Karamata space, denoted by $L_{\Phi , q, b}$, consists of those measurable functions f with $\Vert f\Vert _{L_{\Phi , q, b}}<\infty$, where

$$\begin{aligned} \big \Vert f\big \Vert _{L_{\Phi ,q,b}}= \left\{ \begin{array}{l} \Big (q\int _0^\infty \big (t\gamma _b(\lambda _t(f))\Vert \chi _{\{|f|>t\}}\Vert _{L_\Phi }\big )^q\frac{dt}{t}\Big )^{ {1}/{q}} \ \ \text {if}\;\;0<q<\infty ,\\ \sup \limits _{t>0}t\gamma _b\big (\lambda _t(f)\big ) \Vert \chi _{\{|f|>t\}}\Vert _{L_\Phi } \ \quad \quad \quad \quad \quad \;\; \text {if}\;\; q=\infty . \end{array} \right. \end{aligned}$$

These spaces are generalizations of classical Lorentz spaces, Lorentz-Karamata spaces and Orlicz-Lorentz spaces.

Remark 3

• (1) If $b\equiv 1$, $L_{\Phi ,q,b}$ gives to the Orlicz-Lorentz space $L_{\Phi ,q}$ introduced and studied in [24]. The Orlicz-Lorentz space $L_{\Phi ,q}$ coincides with the classical Lorentz space $L_{p,q}$ when $\Phi (t) = t^p$ for $0<p<\infty$. If $\Phi (t)=t^q$ for $0<q<\infty$, then $L_{\Phi ,q}$ are the usual Lebesgue spaces $L_{q}$. Moreover, if $q=\infty$, the space $L_{\Phi ,q}$ reduces to the weak Orlicz space.

• (2) If $\Phi (t) = t^p$ for $0<p<\infty$, $L_{\Phi ,q,b}$ becomes the Lorentz-Karamata space $L_{p,q,b}$. In this situation, when $b(t)=1+\log t$, the Lorentz-Karamata space reduces to the Lorentz-Zygmund space introduced and studied in [4].

• (3) Let $0<p<\infty$, $m \in \mathbb {N}$ and $\alpha =(\alpha _1,\alpha _2,\ldots ,\alpha _m) \in \mathbb {R}^m$. Define the family of the positive functions $\{\ell _k\}^m_{k=0}$ on $(0,\infty )$ by

  $$\begin{aligned} \ell _0(t) = 1/t,\quad \ell _k(t)=1+\log \big (\ell _{k-1}(t)\big ),\quad 0<t\le 1,\;1\le k\le m. \end{aligned}$$

  Moreover, define

  $$\begin{aligned} \Theta _\alpha ^m(t)=\prod _{k=1}^m\ell _k^{\alpha _k}(t). \end{aligned}$$

  The generalized Lorentz-Zygmund space consists of all $\mathcal {F}$-measurable functions f such that $\Vert f\Vert _{L_{p,q,\alpha }}<\infty$ where

  $$\begin{aligned} \big \Vert f\big \Vert _{L_{p,q,\alpha }}= \left\{ \begin{array}{l} \Big (\int _0^1\big (t^{1/p}\Theta _\alpha ^m(t)f^*(t)\big )^q \frac{dt}{t}\Big )^{1/q} \ \ \text {if}\;0<q<\infty ,\\ \sup \limits _{t>0} t^{1/p}\Theta _\alpha ^m(t)f^*(t) \ \quad \quad \quad \ \quad \;\; \text {if}\; q=\infty . \end{array} \right. \end{aligned}$$

It is easy to see that the generalized Lorentz-Zygmund space is a member of Lorentz-Karamata spaces (see also [12, 28]).

Lemma 5

Let $\Phi$ be an Orlicz function, $0<q\le \infty$ and let b be a slowly varying function. Then the following properties hold:

• (1) $\Vert f\Vert _{L_{\Phi ,q,b}} \ge 0$ and $\Vert f\Vert _{L_{\Phi ,q,b}} = 0$ if and only if $f=0$;

• (2) $\Vert \lambda \cdot f\Vert _{L_{\Phi ,q,b}}=|\lambda |\cdot \Vert f\Vert _{L_{\Phi ,q,b}}$ for any $\lambda \in \mathbb {C}$;

• (3) For any $A\in \mathcal {F}$ with $\mathbb {P}(A)>0$, we have

  $$\begin{aligned} \Vert \chi _A\Vert _{L_{\Phi ,q,b}} = \gamma _b\big (\mathbb {P}(A)\big )\Vert \chi _A\Vert _{L_\Phi }. \end{aligned}$$

  (2.1)

Proof

The properties (1) and (2) can be easily checked by direct calculation. For (3), it follows from the definition of $\Vert \cdot \Vert _{L_{\Phi ,q,b}}$ that

$$\begin{aligned} \Vert \chi _A\Vert _{L_{\Phi ,q,b}}= & {} \bigg (q\int _0^\infty \Big (t\gamma _b\big (\lambda _t(\chi _A)\big )\Vert \chi _{\{|\chi _A|>t\}}\Vert _{L_\Phi }\Big )^q\frac{dt}{t}\bigg )^{ {1}/{q}}\\= & {} \bigg (q\int _0^1 \Big (t\gamma _b\big (\mathbb {P}(A)\big )\Vert \chi _{A}\Vert _{L_\Phi }\Big )^q\frac{dt}{t}\bigg )^{ {1}/{q}} \\ {}= & {} \gamma _b\big (\mathbb {P}(A)\big )\Vert \chi _A\Vert _{L_\Phi }, \end{aligned}$$

where $0<q<\infty$. The case of $q=\infty$ can be proved analogously. $\square$

Lemma 6

Let $\Phi$ be an Orlicz function of upper type $p\in (0,\infty )$, $0<q\le \infty$ and let b be a slowly varying function. For $0<\kappa <\infty$, the functional $\Vert \cdot \Vert _{L_{\Phi ,q,b}}$ can be discretized as follows

$$\begin{aligned} \quad \quad \Vert f\Vert _{L_{\Phi ,q,b}}\approx \bigg (\sum _{k\in \mathbb {Z}} \kappa ^q\cdot 2^{kq}\gamma ^q_b\big (\lambda _{\kappa \cdot 2^k}(f)\big )\big \Vert \chi _{\{|f|>\kappa \cdot 2^k\}}\big \Vert _{L_\Phi }^q\bigg )^{ {1}/{q}}, \end{aligned}$$

(2.2)

when $0<q<\infty$, and

$$\begin{aligned} \Vert f\Vert _{L_{\Phi ,\infty ,b}}\approx \sup _{k\in \mathbb {Z}} 2^k \gamma _b\big (\lambda _{\kappa \cdot 2^k}(f)\big )\big \Vert \chi _{\{|f|>\kappa \cdot 2^k\}}\big \Vert _{L_\Phi }. \end{aligned}$$

Proof

We only prove the case of $0<q<\infty$:

$$\begin{aligned} \Vert f\Vert ^q_{L_{\Phi ,q,b}}= & {} q\int _0^\infty \big (t\gamma _b(\lambda _t(f))\Vert \chi _{\{|f|>t\}} \Vert _{L_\Phi }\big )^q\frac{dt}{t}\\= & {} q\sum _{k\in \mathbb {Z}} \int _{\kappa \cdot 2^k}^{\kappa \cdot 2^{k+1}}\big (t\gamma _b(\lambda _t(f))\Vert \chi _{\{|f|>t\}} \Vert _{L_\Phi }\big )^q\frac{dt}{t}\\= & {} q\sum _{k\in \mathbb {Z}} \int _{\kappa \cdot 2^k}^{\kappa \cdot 2^{k+1}}t^q\bigg ( \gamma _b(\lambda _t(f))\big (\lambda _t(f)\big )^{\frac{1}{p}} \frac{\big (1/\lambda _t(f)\big )^{\frac{1}{p}}}{\Phi ^{-1}\big (1/\lambda _t(f)\big )}\bigg )^q\frac{dt}{t}. \end{aligned}$$

Since the functions

$$\begin{aligned} {\gamma _b(\lambda _t(f))\big (\lambda _t(f)\big )^{\frac{1}{p}}} \qquad \text{ and } \qquad {\frac{\big (1/\lambda _t(f)\big )^{\frac{1}{p}}}{\Phi ^{-1}\big (1/\lambda _t(f)\big )}} \end{aligned}$$

are equivalent to non-increasing functions by Lemma 2 and Proposition 2, we get

$$\begin{aligned} \Vert f\Vert ^q_{L_{\Phi ,q,b}}\preceq & {} q\sum _{k\in \mathbb {Z}}\bigg (\gamma _b(\lambda _{\kappa \cdot 2^k}(f))\big (\lambda _{\kappa \cdot 2^k}(f)\big )^{\frac{1}{p}} \frac{\big (1/\lambda _{\kappa \cdot 2^k}(f)\big )^{\frac{1}{p}}}{\Phi ^{-1} \big (1/\lambda _{\kappa \cdot 2^k}(f)\big )}\bigg )^q \int _{\kappa \cdot 2^k}^{\kappa \cdot 2^{k+1}}t^q\frac{dt}{t}\\= & {} \sum _{k\in \mathbb {Z}} (2^q-1)\Big (\kappa \cdot 2^{k}\gamma _b\big (\lambda _{\kappa \cdot 2^k}(f)\big )\big \Vert \chi _{\{|f|>\kappa \cdot 2^k\}} \big \Vert _{L_\Phi }\Big )^q, \end{aligned}$$

which means

$$\begin{aligned} \Vert f\Vert _{L_{\Phi ,q,b}} \preceq \bigg (\sum _{k\in \mathbb {Z}} \Big (\kappa \cdot 2^{k}\gamma _b\big (\lambda _{\kappa \cdot 2^k}(f)\big )\big \Vert \chi _{\{|f|>\kappa \cdot 2^k\}} \big \Vert _{L_\Phi }\Big )^q\bigg )^{1/q}. \end{aligned}$$

Conversely, according to Lemma 2, we have

$$\begin{aligned}&\Vert f\Vert ^q_{L_{\Phi ,q,b}} \\&\quad = q\sum _{k\in \mathbb {Z}} \int _{\kappa \cdot 2^k}^{\kappa \cdot 2^{k+1}}t^q\bigg ( \gamma _b(\lambda _t(f))\big (\lambda _t(f)\big )^{\frac{1}{p}} \frac{\big (1/\lambda _t(f)\big )^{\frac{1}{p}}}{\Phi ^{-1}\big (1/\lambda _t(f)\big )}\bigg )^q\frac{dt}{t}\\&\quad \succeq q\sum _{k\in \mathbb {Z}}\bigg (\gamma _b(\lambda _{\kappa \cdot 2^{k+1}}(f)) \big (\lambda _{\kappa \cdot 2^{k+1}}(f)\big )^{\frac{1}{p}} \frac{1/\big (\lambda _{\kappa \cdot 2^{k+1}}(f)\big )^{\frac{1}{p}}}{\Phi ^{-1}\big (1/\lambda _{\kappa \cdot 2^{k+1}}(f)\big )}\bigg )^q \int _{\kappa \cdot 2^k}^{\kappa \cdot 2^{k+1}}t^q\frac{dt}{t}\\&\quad = \sum _{k\in \mathbb {Z}} \frac{2^q-1}{2^q}\Big (\kappa \cdot 2^{k+1}\gamma _b\big (\lambda _{\kappa \cdot 2^{k+1}}(f)\big ) \big \Vert \chi _{\{|f|>\kappa \cdot 2^{k+1}\}} \big \Vert _{L_\Phi }\Big )^q. \end{aligned}$$

Consequently, we conclude

$$\begin{aligned} \Vert f\Vert _{L_{\Phi ,q,b}} \succeq \bigg (\sum _{k\in \mathbb {Z}} \Big (\kappa \cdot 2^{k}\gamma _b\big (\lambda _{\kappa \cdot 2^k}(f)\big )\big \Vert \chi _{\{|f|>\kappa \cdot 2^k\}} \big \Vert _{L_\Phi }\Big )^q\bigg )^{1/q}. \end{aligned}$$

Thus the estimation (2.2) is true. $\square$

Remark 4

Let $\Phi$ be an Orlicz function of upper type $p\in (0,\infty )$ and let b be a slowly varying function. Especially for $\kappa =1$, it follows from the formula (2.2) that

$$\begin{aligned} \Vert f\Vert _{L_{\Phi ,q,b}}\approx \bigg (\sum _{k\in \mathbb {Z}} 2^{kq}\gamma ^q_b\big (\lambda _{2^k}(f)\big )\big \Vert \chi _{\{|f|>2^k\}}\big \Vert _{L_\Phi }^q\bigg )^{ {1}/{q}}, \ \ (0<q<\infty ). \end{aligned}$$

If we take $b\equiv 1$ for the above estimation, then we get

$$\begin{aligned} \Vert f\Vert _{L_{\Phi ,q}}\approx \bigg (\sum _{k\in \mathbb {Z}} 2^{kq}\big \Vert \chi _{\{|f|>2^k\}}\big \Vert _{L_\Phi }^q\bigg )^{ {1}/{q}}, \ \ (0<q<\infty ). \end{aligned}$$

Moreover, if we consider the case $\Phi (t)=t^p$ for $0<p<\infty$, we obtain

$$\begin{aligned} \Vert f\Vert _{L_{p,q,b}}\approx \bigg (\sum _{k\in \mathbb {Z}} 2^{kq}\gamma ^q_b(\lambda _{2^k}(f))\big (\lambda _{2^k}(f)\big )^{q/p}\bigg )^{ {1}/{q}}, \ \ (0<q<\infty ). \end{aligned}$$

Taking $b\equiv 1$ and $\Phi (t)=t^p$ for $0<p< \infty$, we also obtain

$$\begin{aligned} \Vert f\Vert _{L_{p,q}}\approx \bigg (\sum _{k\in \mathbb {Z}} 2^{kq}\big (\lambda _{2^k}(f)\big )^{q/p}\bigg )^{ {1}/{q}}, \ \ (0<q<\infty ). \end{aligned}$$

Next we prove the following useful properties of Orlicz-Lorentz-Karamata spaces.

Proposition 3

Let $\Phi$ be an Orlicz function of lower type $p_-$ and upper type $p_+$ satisfying $0<p_- \le p_+<\infty$, $0<q\le \infty$ and let b be a slowly varying function. Then we have $L_{p_+,q,b}\subseteq L_{\Phi ,q,b} \subseteq L_{p_-,q,b}$, i.e.,

$$\begin{aligned} \Vert f\Vert _{L_{p_-,q,b}} \preceq \Vert f\Vert _{L_{\Phi ,q,b}} \preceq \Vert f\Vert _{L_{p_+,q,b}}. \end{aligned}$$

(2.3)

Proof

Apparently, it follows from Lemma 4 and the definition of $\Vert \cdot \Vert _{L_{\Phi ,q,b}}$ that (2.3) holds. $\square$

Proposition 4

Let $\Phi$ be an Orlicz function of lower type $p_-$ and upper type $p_+$ satisfying $0<p_- \le p_+<\infty$, $0<q\le \infty$ and let b be a slowly varying function. Then $\Vert f\Vert _{L_{\Phi ,q,b}}$ and

$$\begin{aligned} |||f |||_{L_{\Phi ,q,b}} = \left\{ \begin{array}{l} \Big (q\int _0^1\big (\frac{1}{\Phi ^{-1}(1/t)}\gamma _b(t) f^*(t)\big )^q\frac{dt}{t}\Big )^{1/q} \ \ \;\,\, \text {if} \;\; 0<q<\infty ,\\ \sup \limits _{t>0} \frac{1}{\Phi ^{-1}(1/t)}\gamma _b(t) f^*(t) \ \quad \quad \quad \quad \ \quad \;\; \text {if} \;\; q=\infty . \end{array} \right. \end{aligned}$$

(2.4)

are equivalent functionals.

Proof

For any measurable function f, there exists a sequence of non-negative simple functions $\{f_n\}_{n \in \mathbb {N}}$ such that $f_n \uparrow |f|$ a.e.. Moreover, $d_{f_n} \uparrow d_f$ and $f^*_n \uparrow f^*$. Therefore, by using Lebesgue monotone convergence theorem, it suffices to establish that the quasi-norm defined in (2.4) is equivalent to $\Vert f\Vert _{L_{\Phi ,q,b}}$ for non-negative simple functions.

Now let

$$\begin{aligned} f(\omega )=\sum _{i=1}^N\alpha _i\chi _{A_i}(\omega ), \end{aligned}$$

where $\{A_i \}_{i=1}^N$ is a family of disjoint measurable sets and $\{\alpha _j\}_{j=1}^N \subseteq \mathbb {R}$ satisfying $0 \le \alpha _j \le \alpha _i$ as $1 \le i\le j \le N$. For any $t\ge 0$, we have

$$\begin{aligned} \lambda _t(f)=\sum _{j=1}^N \beta _j \chi _{[\alpha _{j+1},\alpha _j) }(t), \end{aligned}$$

where $\alpha _{N+1}=0$ and $\beta _j=\sum \limits _{i=1}^j \mathbb {P}(A_i)$ for $1\le j\le N$. Also, one can see that

$$\begin{aligned} f^*(t)=\sum _{j=1}^N \alpha _j \chi _{[\beta _{j-1},\beta _j)}(t), \end{aligned}$$

where $\beta _0=0$.

We first consider the case of $q=\infty$. Since $\Phi ^{-1}(t)$ is increasing on $(0,\infty )$ and $\frac{\Phi ^{-1}(t)}{t^{1/p_+}}$ is equivalent to a non-decreasing function on $(0,\infty )$ (see Lemma 2), then we get

$$\begin{aligned} \big \Vert f\big \Vert _{L_{\Phi ,\infty ,b}}= & {} \sup _{t>0}t\Vert \chi _{\{|f|>t\}}\Vert L_{\Phi }\gamma _b\big (\lambda _t(f)\big ) =\sup _{t>0}\frac{t\gamma _b\big (\lambda _t(f)\big )}{\Phi ^{-1}\big (1/\lambda _t(f)\big )}\\= & {} \sup _{t>0}\sum _{j=1}^N\frac{t\gamma _b(\beta _j)}{\Phi ^{-1}(1/\beta _j)}\chi _{[\alpha _{j+1},\alpha _j)}(t) =\max _{1\le j\le N}\frac{\alpha _j\gamma _b(\beta _j)}{\Phi ^{-1}(1/\beta _j)} \end{aligned}$$

and

$$\begin{aligned} |||f |||_{L_{\Phi ,\infty ,b}}= & {} \sup _{t>0}\frac{\gamma _b(t)}{\Phi ^{-1}(1/t)}f^*(t) =\sup _{t>0}\sum _{j=1}^N\frac{\alpha _j\gamma _b(t)}{\Phi ^{-1}(1/t)}\chi _{[\beta _{j-1},\beta _j)}(t)\\= & {} \sup _{t>0}\sum _{j=1}^N\alpha _j\frac{(1/t)^{1/p_+}}{\Phi ^{-1}(1/t)} t^{1/p_+}\gamma _b(t)\chi _{[\beta _{j-1},\beta _j)}(t)\\\approx & {} \max _{1\le j\le N}\frac{\alpha _j\gamma _b(\beta _j)}{\Phi ^{-1}(1/\beta _j)}, \end{aligned}$$

which implies

$$\begin{aligned} |||f |||_{L_{\Phi ,\infty ,b}} \approx \Vert f\Vert _{L_{\Phi ,\infty ,b}}. \end{aligned}$$

Now we consider the case of $0<q<\infty$. Set

$$\begin{aligned} K_b(t)=q\int _0^t\Big (\frac{\gamma _b(s)}{\Phi ^{-1}(1/s)}\Big )^q\frac{ds}{s}. \end{aligned}$$

Then it follows from the Abel transformation that

$$\begin{aligned} |||f |||^q_{L_{\Phi ,q,b}}= & {} q\sum _{i=1}^N\alpha ^q_i \int _{\beta _{i-1}}^{\beta _i}\Big (\frac{\gamma _b(t)}{\Phi ^{-1}(1/t)}\Big )^q\frac{dt}{t} =\sum _{i=1}^N\alpha ^q_i\big (K_b(\beta _i)-K_b(\beta _{i-1})\big )\\= & {} \sum _{i=1}^N\big (\alpha ^q_{i}-\alpha ^q_{i+1}\big )K_b(\beta _i). \end{aligned}$$

Since $\frac{\Phi ^{-1}(t)}{t^{1/p_+}}$ is equivalent to a non-decreasing function on $(0,\infty )$, and $\frac{\Phi ^{-1}(t)}{t^{1/p_-}}$ is equivalent to a non-increasing function on $(0,\infty )$ (see Lemmas 1 and 2), then we have

$$\begin{aligned} K_b(\beta _i)= & {} q \int _0^{\beta _i}\Big (\frac{\gamma _b(t)}{\Phi ^{-1}(1/t)}\Big )^q\frac{dt}{t}= q \int _0^{\beta _i}\Big (\frac{(1/t)^{1/p_+}}{\Phi ^{-1}(1/t)}\Big )^q t^{q/p_+}\big (\gamma _b(t)\big )^q \frac{dt}{t}\\\preceq & {} q\Big (\frac{(1/\beta _i)^{1/p_+}}{\Phi ^{-1}(1/\beta _i)}\Big )^q\int _0^{\beta _i} \big (t^{\frac{ 1}{2p_+}}\gamma _b(t)\big )^q\;t^{\frac{q}{2 l_+}-1}dt\\\preceq & {} q\Big (\frac{(1/\beta _i)^{1/p_+}}{\Phi ^{-1}(1/\beta _i)}\Big )^q \Big (\beta _i^{\frac{1}{2p_+}}\gamma _b(\beta _i)\Big )^q \int _0^{\beta _i} t^{\frac{q}{2 p_+}-1}dt\\= & {} 2p_+ \Big (\frac{(1/\beta _i)^{1/p_+}}{\Phi ^{-1}(1/\beta _i)}\Big )^q \beta _i^{q/ p_+}\big (\gamma _b(\beta _i)\big )^q\\= & {} 2p_+ \Big (\frac{\gamma _b(\beta _i)}{\Phi ^{-1}(1/\beta _i)}\Big )^q. \end{aligned}$$

Similarly, by Lemma 1,

$$\begin{aligned} K_b(\beta _i)= & {} q \int _0^{\beta _i}\Big (\frac{\gamma _b(t)}{\Phi ^{-1}(1/t)}\Big )^q\frac{dt}{t}= q \int _0^{\beta _i} \Big (\frac{(1/t)^{1/p_-}}{\Phi ^{-1}(1/t)}\Big )^q t^{q/p_{-}}\big (\gamma _b(t)\big )^q \frac{dt}{t}\\\succeq & {} q\Big (\frac{(1/\beta _i)^{1/p_-}}{\Phi ^{-1}(1/\beta _i)}\Big )^q\int _0^{\beta _i} \big (t^{- 1/p_{-}}\gamma _b(t)\big )^q\;t^{2q/p_{-}-1}dt\\\succeq & {} q\Big (\frac{(1/\beta _i)^{1/p_-}}{\Phi ^{-1}(1/\beta _i)}\Big )^q \Big (\beta _i^{-1/p_-}\gamma _b(\beta _i)\Big )^q \int _0^{\beta _i} t^{2q/ p_--1}dt\\= & {} \frac{p_-}{2} \Big (\frac{(1/\beta _i)^{1/p_-}}{\Phi ^{-1}(1/\beta _i)}\Big )^q \beta _i^{q/p_{-}}\big (\gamma _b(\beta _i)\big )^q\\= & {} \frac{p_-}{2} \Big (\frac{\gamma _b(\beta _i)}{\Phi ^{-1}(1/\beta _i)}\Big )^q. \end{aligned}$$

Hence we get

$$\begin{aligned}&\frac{p_-}{2}\sum _{i=1}^N \big (\alpha ^q_{i}-\alpha ^q_{i+1}\big )\Big (\frac{\gamma _b(\beta _i)}{\Phi ^{-1}(1/\beta _i)}\Big )^q \preceq |||f |||^q_{L_{\Phi ,q,b}}\\&\quad \preceq 2 p_+ \sum _{i=1}^N\big (\alpha ^q_{i}-\alpha ^q_{i+1}\big )\Big (\frac{\gamma _b(\beta _i)}{\Phi ^{-1}(1/\beta _i)}\Big )^q. \end{aligned}$$

Moreover, we have

$$\begin{aligned} \big \Vert f\big \Vert ^q_{L_{\Phi ,q,b}}= & {} q\int _0^\infty \big (t\gamma _b(\lambda _t(f))\Vert \chi _{\{|f|>t\}}\Vert _{L_\Phi }\big )^q\frac{dt}{t}\\= & {} q\int _0^\infty \Big ( \frac{t\gamma _b(\lambda _t(f))}{\Phi ^{-1}(1/\lambda _t(f))} \Big )^q\frac{dt}{t}\\= & {} q\sum ^N_{i=1}\int _{\alpha _{i+1}}^{\alpha _i}\Big ( \frac{t\gamma _b(\beta _i)}{\Phi ^{-1}(1/\beta _i)} \Big )^q\frac{dt}{t}\\= & {} \sum _{i=1}^N\big (\alpha ^q_{i}-\alpha ^q_{i+1}\big )\Big (\frac{\gamma _b(\beta _i)}{\Phi ^{-1}(1/\beta _i)}\Big )^q. \end{aligned}$$

Thus we get

$$\begin{aligned} |||f |||_{L_{\Phi ,q,b}}\approx \Vert f\Vert _{L_{\Phi ,q,b}}, \end{aligned}$$

which completes the proof. $\square$

Lemma 7

If $\Phi$ is an Orlicz function of lower type $p_-$ and upper type $p_+$ satisfying $0<p_- \le p_+<\infty$, $0<q\le \infty$ and b is a slowly varying function, then the Orlicz-Lorentz-Karamata spaces $L_{\Phi ,q,b}$ are (quasi)-Banach spaces.

Proof

We only have to prove that

$$\begin{aligned} \Vert f+g\Vert _{L_{\Phi ,q,b}} \preceq \Vert f\Vert _{L_{\Phi ,q,b}}+\Vert g\Vert _{L_{\Phi ,q,b}}. \end{aligned}$$

(2.5)

Indeed, it follows from Proposition 4 and $(f+g)^*(t_1+t_2)\le f^*(t_1)+g^*(t_2)$ for $0\le t_1,t_2<\infty$ that

$$\begin{aligned}&\Vert f+g\Vert _{L_{\Phi ,q,b}} \approx |||f+g |||_{L_{\Phi ,q,b}}\\&\quad = \bigg (q\int _0^1\Big (\frac{\gamma _b(t)}{\Phi ^{-1}\big (1/t\big )} (f+g)^*(t)\Big )^q\frac{dt}{t}\bigg )^{1/q}\\&\quad = \bigg (q\int _0^{\frac{1}{2}}\Big (\frac{\gamma _b(2t)}{\Phi ^{-1}\big (1/(2t)\big )} (f+g)^*(2t)\Big )^q\frac{dt}{t}\bigg )^{1/q}\\&\quad \le \bigg (q\int _0^{\frac{1}{2}}\Big (\frac{\gamma _b(2t)}{\Phi ^{-1}\big (1/(2t)\big )} \big (f^*(t)+g^*(t)\big )\Big )^q\frac{dt}{t}\bigg )^{1/q}\\&\quad \preceq \bigg (q\int _0^{\frac{1}{2}}\Big (\frac{\gamma _b(2t)}{\Phi ^{-1}\big (1/(2t)\big )} f^*(t)\Big )^q\frac{dt}{t}\bigg )^{1/q} \\&\qquad + \bigg (q\int _0^{\frac{1}{2}}\Big (\frac{\gamma _b(2t)}{\Phi ^{-1}\big (1/(2t)\big )} g^*(t)\Big )^q\frac{dt}{t}\bigg )^{1/q}. \end{aligned}$$

By applying Proposition 2, we have

$$\begin{aligned}&\Vert f+g\Vert _{L_{\Phi ,q,b}}\\&\quad \preceq \bigg (q\int _0^{\frac{1}{2}}\Big (\frac{(2t)^{-1}\gamma _b(2t)2t}{\Phi ^{-1}\big (1/(2t)\big )} f^*(t)\Big )^q\frac{dt}{t}\bigg )^{1/q} \\&\qquad + \bigg (q\int _0^{\frac{1}{2}}\Big (\frac{(2t)^{-1}\gamma _b(2t)2t}{\Phi ^{-1}\big (1/(2t)\big )} g^*(t)\Big )^q\frac{dt}{t}\bigg )^{1/q}\\&\quad \preceq \bigg (q\int _0^{\frac{1}{2}}\Big (\frac{t^{-1}\gamma _b(t)2t}{\Phi ^{-1}\big (1/(2t)\big )} f^*(t)\Big )^q\frac{dt}{t}\bigg )^{1/q} + \bigg (q\int _0^{\frac{1}{2}}\Big (\frac{t^{-1}\gamma _b(t)2t}{\Phi ^{-1}\big (1/(2t)\big )} g^*(t)\Big )^q\frac{dt}{t}\bigg )^{1/q} \\&\quad = \bigg (q\int _0^{\frac{1}{2}}\Big (\frac{2\gamma _b(t)}{\Phi ^{-1}\big (1/(2t)\big )} f^*(t)\Big )^q\frac{dt}{t}\bigg )^{1/q} + \bigg (q\int _0^{\frac{1}{2}}\Big (\frac{2\gamma _b(t)}{\Phi ^{-1}\big (1/(2t)\big )} g^*(t)\Big )^q\frac{dt}{t}\bigg )^{1/q}. \end{aligned}$$

According to $\frac{1}{\Phi ^{-1}\big (1/(2t)\big )}\preceq \frac{1}{\Phi ^{-1}\big (1/t\big )}$ from Lemma 1 (iii), we obtain that

$$\begin{aligned}{} & {} \Vert f+g\Vert _{L_{\Phi ,q,b}}\\{} & {} \quad \preceq \bigg (q\int _0^{\frac{1}{2}}\Big (\frac{\gamma _b(t)}{\Phi ^{-1}\big (1/t\big )} f^*(t)\Big )^q\frac{dt}{t}\bigg )^{1/q} + \bigg (q\int _0^{\frac{1}{2}}\Big (\frac{\gamma _b(t)}{\Phi ^{-1}\big (1/t\big )} g^*(t)\Big )^q\frac{dt}{t}\bigg )^{1/q}\\{} & {} \quad \le |||f |||_{L_{\Phi ,q,b}}+|||g |||_{L_{\Phi ,q,b}}. \end{aligned}$$

Thus we get that

$$\begin{aligned} \Vert f+g\Vert _{L_{\Phi ,q,b}}\preceq |||f |||_{L_{\Phi ,q,b}}+|||g |||_{L_{\Phi ,q,b}}. \end{aligned}$$

This proves (2.5) as well as the lemma. $\square$

Let $(X,\Vert \cdot \Vert _X)$ be a (quasi)-normed space. A function $f\in X$ is said to have absolutely continuous (quasi)-norm in X, if

$$\begin{aligned} \lim \limits _{n\rightarrow \infty }\Vert f\chi _{A_n}\Vert _X=0 \end{aligned}$$

for every sequence $(A_n)_{n\ge 0}$ satisfying $\lim \limits _{n\rightarrow \infty }\mathbb {P}(A_n)=0$.

Proposition 5

Let $\Phi$ be an Orlicz function of lower type $p_-$ and upper type $p_+$ satisfying $0<p_-\le p_+<\infty$, $0<q<\infty$ and let b be a slowly varying function. Then every $f\in L_{\Phi ,q,b}$ has absolutely continuous quasi-norm.

Proof

Since $f\in L_{\Phi ,q,b}$, for any $\varepsilon >0$, there exists $N_1\in \mathbb {N}$ such that

$$\begin{aligned} \bigg (\sum _{k=N_1}^{\infty }2^{kq}\gamma _{b}^q\big (\lambda _{2^k}(f)\big )\Vert \chi _{\{|f|>2^k\}}\Vert _{L_\Phi }^q\bigg )^{\frac{1}{q}}<\varepsilon . \end{aligned}$$

Let the sequence $(A_n)_{n\ge 0}$ satisfy $\lim \limits _{n\rightarrow \infty }\mathbb {P}(A_n)=0$. There exists $N_2\in \mathbb {N}$ such that for $n\ge N_2$, $\mathbb {P}(A_n)<\big (\frac{\varepsilon }{2^{N_1}\gamma _{b}(1)}\big )^{2p_+}$. Now let $n\ge N_2$. By using Lemmas 6 and 4, we have

$$\begin{aligned} \Vert f\chi _{A_n}\Vert _{L_{\Phi ,q,b}}^q{} & {} \approx \sum _{k=N_1}^{\infty }2^{kq}\gamma _{b}^q\big (\lambda _{2^k}(f\chi _{A_n})\big )\Vert \chi _{\{|f\chi _{A_n}|>2^k\}}\Vert _{L_\Phi }^q\\{} & {} \quad +\sum _{k=-\infty }^{N_1-1}2^{kq}\gamma _{b}^q\big (\lambda _{2^k}(f\chi _{A_n})\big )\Vert \chi _{\{|f\chi _{A_n}|>2^k\}}\Vert _{L_\Phi }^q\\{} & {} \preceq \sum _{k=N_1}^{\infty }2^{kq}\gamma _{b}^q\big (\lambda _{2^k}(f)\big )\Vert \chi _{\{|f|>2^k\}}\Vert _{L_\Phi }^q \\{} & {} \quad +\sum _{k=-\infty }^{N_1-1}2^{kq}\gamma _{b}^q\big (\mathbb {P}(A_n)\big )\Vert \chi _{A_n}\Vert _{L_\Phi }^q\\{} & {} < \varepsilon ^q+\sum _{k=-\infty }^{N_1-1}2^{kq}\gamma _{b}^q\big (\mathbb {P}(A_n)\big )\Vert \chi _{A_n}\Vert _{L_\Phi }^{q}\\{} & {} \preceq \varepsilon ^q+\sum _{k=-\infty }^{N_1-1}2^{kq}\gamma _{b}^q\big (\mathbb {P}(A_n)\big )\mathbb {P}(A_n)^{q/p_+}\\{} & {} = \varepsilon ^q+\sum _{k=-\infty }^{N_1-1}2^{kq}\gamma _{b}^q\big (\mathbb {P}(A_n)\big )\mathbb {P}(A_n)^{q/2p_+}\mathbb {P}(A_n)^{q/2p_+}\\{} & {} \preceq \varepsilon ^q+\sum _{k=-\infty }^{N_1-1}2^{kq}\gamma _{b}^q(1)\mathbb {P}(A_n)^{q/2p_+}\\{} & {} < \varepsilon ^q+\gamma _{b}^q(1)\Big (\frac{\varepsilon }{2^{N_1}\gamma _{b}(1)}\Big )^q\sum _{k=-\infty }^{N_1-1}2^{kq} =\Big (1+\frac{1}{2^q-1}\Big )\varepsilon ^q, \end{aligned}$$

which yields that $\Vert f\chi _{A_n}\Vert _{L_{\Phi ,q,b}} \preceq \varepsilon$. Hence, f has absolutely continuous quasi-norm. $\square$

Proposition 6

Let $0 < s \le q \le \infty$, $\Phi$ be an Orlicz function of lower type $p\in (s,\infty )$ and let b be a slowly varying function. Then we obtain that $|||f |||_{L_{\Phi ,q,b}}$ and

$$\begin{aligned} \big \Vert f\big \Vert _{L_{\Phi ,q,b,s}}= \left\{ \begin{array}{ll} \Big (q\int _0^1\big (\frac{\gamma _b(t)}{\Phi ^{-1}(1/t)} f_s^{**}(t)\big )^q\frac{dt}{t}\Big )^{1/q} &{} \text {if} \;\; 0<q<\infty ,\\ \sup \limits _{t>0} \frac{\gamma _b(t)}{\Phi ^{-1}(1/t)} f_s^{**}(t) \ \quad \quad &{}\text {if} \;\; q=\infty , \end{array} \right. \end{aligned}$$

are equivalent (quasi)-norms, where

$$\begin{aligned} f_s^{**}(t)=\Big (\frac{1}{t}\int _0^t\big (f^*(x)\big )^sdx\Big )^{1/s}, \quad \quad \quad \quad t>0. \end{aligned}$$

Proof

Obviously, $f^*(t)\le f_s^{**}(t)$ for $t>0$. Thus it is sufficient to prove

$$\begin{aligned} \Vert f\Vert _{L_{\Phi ,q,b,s}}\preceq |||f |||_{L_{\Phi ,q,b}}. \end{aligned}$$

(2.6)

From the definitions of $\Vert \cdot \Vert _{L_{\Phi ,q,b,s}}$ and $|||\cdot |||_{L_{\Phi ,q,b}}$, we see that

$$\begin{aligned} \big \Vert f\big \Vert ^s_{L_{\Phi ,q,b,s}}= & {} \Bigg (q\int _0^\infty \bigg [\frac{\gamma _b(t)}{\Phi ^{-1}(1/t)} \Big (\frac{1}{t}\int _0^t\big (f^*(x)\big )^sdx\Big )^{1/s}\bigg ]^q\frac{dt}{t}\Bigg )^{s/q}\\= & {} q^{s/q}\Bigg (\int _0^\infty \bigg [U(t)\int _0^tg(x)dx\bigg ]^{\alpha }dt \Bigg )^{1/\alpha } \end{aligned}$$

and

$$\begin{aligned} |||f |||^s_{L_{\Phi ,q,b}}= & {} \bigg (q\int _0^1\Big (\frac{\gamma _b(t)}{\Phi ^{-1}(1/t)} f^*(t)\Big )^q\frac{dt}{t}\bigg )^{s/q}\\= & {} q^{s/q}\bigg (\int _0^\infty \Big (V(t)g(t)\Big )^{\alpha }dt \bigg )^{1/\alpha }, \end{aligned}$$

where

$$\begin{aligned} U(t)&{:}{=}\Big (\frac{\gamma _b(t)}{\Phi ^{-1}\big (1/t\big )}\Big )^st^{-1-s/q}, \quad V(t){:}{=}\Big (\frac{\gamma _b(t)}{\Phi ^{-1}\big (1/t\big )}\Big )^st^{-s/q},\\ g(t)&{:}{=}\big (f^*(t)\big )^s\quad \;\;\text {and} \quad \;\;\alpha {:}{=}\frac{q}{s}\ge 1. \end{aligned}$$

Applying Theorem 1.1 in [45], we know that, if the following estimation

$$\begin{aligned} \sup _{r>0}\bigg [\int _r^\infty \big |U(t) \big |^\alpha dt\bigg ]^{1/\alpha } \bigg [\int _0^r\big |V(t) \big |^{-\alpha '}dt\bigg ]^{1/\alpha '} \end{aligned}$$

(2.7)

is finite, then

$$\begin{aligned} \bigg [\int _0^\infty \Big |U(t)\int _0^tg(x)dx\Big |^\alpha dt\bigg ]^{1/\alpha } \le C \bigg [\int _0^\infty \big |V(t)g(t)\big |^\alpha dt\bigg ]^{1/\alpha }, \end{aligned}$$

where $\alpha '=\alpha /(\alpha -1).$

Hence, in order to get the inequality (2.6), we just compute the formula (2.7). Now let’s estimate the

$$\begin{aligned} \bigg [\int _r^\infty \big |U(t) \big |^\alpha dt\bigg ]^{1/\alpha }\quad \text {and}\quad \bigg [\int _0^r\big |V(t) \big |^{-\alpha '}dt\bigg ]^{1/\alpha '} \end{aligned}$$

respectively. The estimations are divided into the following two steps:

Step 1. We first consider the case of $s<q$. That is

$$\begin{aligned} 0<s<\min \big \{p,q\big \}\quad \text {and}\quad p-s>0. \end{aligned}$$

Since $\frac{\Phi ^{-1}(t)}{t^{1/p}}$ is equivalent to non-increasing on $(0,\infty )$ via Lemma 1 (ii), we have

$$\begin{aligned} \bigg [\int _r^\infty \big |U(t) \big |^\alpha dt\bigg ]^{1/\alpha }= & {} \bigg [\int _r^\infty \Big [\Big (\frac{\gamma _b(t)}{\Phi ^{-1}\big (1/t\big )}\Big )^st^{-1-s/q}\Big ]^{q/s}dt\bigg ]^{1/\alpha }\\= & {} \bigg [\int _r^\infty \Big (\frac{(1/t)^{1/p}}{\Phi ^{-1}\big (1/t\big )}\Big )^q \big (t^{\frac{s-p}{2ps}}\gamma _b(t)\big )^q t^{\frac{q(s-p)}{2ps}-1} dt\bigg ]^{s/q}\\\preceq & {} \Big (\frac{(1/r)^{1/p}}{\Phi ^{-1}\big (1/r\big )}\Big )^s \big (r^{\frac{s-p}{2ps}}\gamma _b(r)\big )^s \bigg [\int _r^\infty t^{\frac{q(s-p)}{2ps}-1} dt\bigg ]^{s/q}\\= & {} \Big (\frac{2ps}{q(p-s)}\Big )^{s/q} \Big (\frac{\gamma _b(r)}{\Phi ^{-1}\big (1/r\big )}\Big )^s\frac{1}{r} \end{aligned}$$

and

$$\begin{aligned}{} & {} \bigg [\int _0^r\big |V(t) \big |^{-\alpha '}dt\bigg ]^{1/\alpha '} = \bigg [\int _0^r\Big (\frac{\Phi ^{-1}(1/t)}{\gamma _b(t) }\Big )^{\frac{qs}{q-s}}t^{\frac{s}{q-s}} dt\bigg ]^{(q-s)/q} \\{} & {} \quad = \bigg [\int _0^r\Big (\frac{\Phi ^{-1}(1/t)}{ (1/t)^{1/p}}\Big )^{\frac{qs}{q-s}} \Big (t^{\frac{s-p}{2ps}}\gamma _b(t)\Big )^{-\frac{qs}{q-s}} t^{\frac{q(p-s)}{2(q-s)p}-1} dt\bigg ]^{(q-s)/q}\\{} & {} \quad \preceq \Big (\frac{\Phi ^{-1}(1/r)}{ (1/r)^{1/p}}\Big )^s \Big (r^{\frac{s-p}{2ps}}\gamma _b(r)\Big )^{-s} \bigg [\int _0^r t^{\frac{q(p-s)}{2(q-s)p}-1} dt\bigg ]^{(q-s)/q}\\{} & {} \quad = \Big (\frac{ 2(q-s)p }{ q(p-s) }\Big )^{(q-s)/q} \Big (\frac{\Phi ^{-1}(1/r)}{ (1/r)^{1/p}}\Big )^s \big (\gamma _b(r)\big )^{-s}r^{{(p-s)}/{p}}\\{} & {} \quad = \Big (\frac{ 2(q-s)p }{ q(p-s) }\Big )^{(q-s)/q}\Big (\frac{\Phi ^{-1}(1/r)}{ \gamma _b(r)}\Big )^sr . \end{aligned}$$

This means

$$\begin{aligned} \sup _{r>0}\bigg [\int _r^\infty \big |U(t) \big |^\alpha dt\bigg ]^{1/\alpha } \bigg [\int _0^r\big |V(t) \big |^{-\alpha '}dt\bigg ]^{1/\alpha '}<C. \end{aligned}$$

Thus we see that (2.6) holds for $s<q$.

Step 2. Now we consider the case $s=q$. Then $0<q<p$ and define

$$\begin{aligned} U(t){:}{=}\Big (\frac{\gamma _b(t)}{\Phi ^{-1}\big (1/t\big )}\Big )^qt^{-2}, \ V(t){:}{=}\Big (\frac{\gamma _b(t)}{\Phi ^{-1}\big (1/t\big )}\Big )^qt^{-1}\ \text {and}\ g(t){:}{=}\big (f^*(t)\big )^q. \end{aligned}$$

Similarly to Step 1, we obtain that

$$\begin{aligned} \int _r^\infty \big |U(t) \big | dt= & {} \int _r^\infty \Big (\frac{\gamma _b(t)}{\Phi ^{-1}\big (1/t\big )}\Big )^qt^{-2}dt\\= & {} \int _r^\infty \Big (\frac{(1/t)^{1/p}}{\Phi ^{-1}\big (1/t\big )}\Big )^q \big (t^{\frac{q-p}{2pq}}\gamma _b(t)\big )^q t^{\frac{q-p}{2p}-1} dt\\\preceq & {} \frac{2p}{p-q}\Big (\frac{\gamma _b(r)}{\Phi ^{-1}\big (1/r\big )}\Big )^q\frac{1}{r} \end{aligned}$$

and

$$\begin{aligned} \sup _{0<t<r}\big |V(t)\big |^{-1}=\sup _{0<t<r} \Big (\frac{\Phi ^{-1}\big (1/t\big )}{\gamma _b(t)}\Big )^qt =\Big (\frac{\Phi ^{-1}\big (1/r\big )}{\gamma _b(r)}\Big )^qr . \end{aligned}$$

Thus we have

$$\begin{aligned} \sup _{r>0}\bigg \{\int _r^\infty \big |U(t) \big | dt \cdot \sup _{0<t<r}\big |V(t) \big |^{-1}\bigg \}\preceq \frac{2p}{p-q}, \end{aligned}$$

which implies that (2.6) holds for $s=q$. This completes the proof. $\square$

Theorem 1

If $0 < q \le \infty$, $0<s \le \min \{1,q\}$, $\Phi$ is an Orlicz function of lower type $p\in (s,\infty )$ and b is a slowly varying function, then $|||\cdot |||_{L_{\Phi ,q,b}}$ is equivalent to an s-norm. Namely, there exists a finite C such that for any $h_k$, $k=1,2,\ldots ,n$,

$$\begin{aligned} |||h_1+h_2+\cdots +h_n |||^s_{L_{\Phi ,q,b}}\le C \big ( |||h_1 |||^s_{L_{\Phi ,q,b}}+|||h_2|||^s_{L_{\Phi ,q,b}}+\cdots +|||h_n|||^s_{L_{\Phi ,q,b}}\big ). \end{aligned}$$

Proof

It follows from p. 64 of [22] that for $0 < s \le 1$,

$$\begin{aligned} \big ((f+g)^{**}_s(t)\big )^s \le \big (f^{**}_s(t)\big )^s+\big (g^{**}_s(t)\big )^s, \quad \quad t>0. \end{aligned}$$

(2.8)

According to Proposition 6, inequality (2.8) and Minkowski’s inequality, we have

$$\begin{aligned}&|||h_1+h_2+\cdots +h_n |||^s_{L_{\Phi ,q,b}}\approx \big \Vert h_1+h_2+\cdots +h_n\big \Vert ^s_{L_{\Phi ,q,b,s}} \\&\quad = \Bigg (q\int _0^1\bigg [\Big (\frac{\gamma _b(t)}{\Phi ^{-1}(1/t)}\Big )^s \big ((h_1+h_2+\cdots +h_n)^{**}_s(t)\big )^s\bigg ]^{q/s}\frac{dt}{t}\Bigg )^{s/q}\\&\quad \le \Bigg (q\int _0^1\bigg [\Big (\frac{\gamma _b(t)}{\Phi ^{-1}(1/t)}\Big )^s \\&\qquad \big ((h_1)^{**}_s(t)\big )^s+\big ((h_2)^{**}_s(t)\big )^s +\cdots +\big ((h_n)^{**}_s(t)\big )^s\bigg ]^{q/s}\frac{dt}{t}\Bigg )^{s/q} \\&\quad \le \big \Vert h_1\big \Vert ^s_{L_{\Phi ,q,b,s}}+\big \Vert h_2\big \Vert ^s_{L_{\Phi ,q,b,s}} +\cdots +\big \Vert h_n\big \Vert ^s_{L_{\Phi ,q,b,s}} \\&\quad \approx |||h_1 |||^s_{L_{\Phi ,q,b}}+ |||h_2 |||^s_{L_{\Phi ,q,b}} +\cdots + |||h_n |||^s_{L_{\Phi ,q,b}}. \end{aligned}$$

Thus there exists a constant C independently on n such that

$$\begin{aligned}&|||h_1+h_2+\cdots +h_n|||^s_{L_{\Phi ,q,b}} \\&\quad \le C \big (|||h_1|||^s_{L_{\Phi ,q,b}}+|||h_2|||^s_{L_{\Phi ,q,b}}+\cdots +|||h_n|||^s_{L_{\Phi ,q,b}}\big ). \end{aligned}$$

This completes the proof. $\square$

Combining this with Proposition 4 and Theorem 1, we get the following result.

Theorem 2

Let $\Phi$ be an Orlicz function of lower type $p_-$ and upper type $p_+$ satisfying $0<s<p_- \le p_+<\infty$, $0<q\le \infty$, $0<s \le \min \{1,q\}$ and let b be a slowly varying function. $\Vert \cdot \Vert _{L_{\Phi ,q,b}}$ is equivalent to an s-norm.

2.3 Orlicz-Lorentz-Karamata Hardy martingale spaces

We recall some notations for martingales and define Orlicz-Lorentz-Karamata Hardy martingale spaces. Let $(\Omega ,\mathcal {F},\mathbb {P})$ be a probability space and $\{\mathcal {F}_n\}_{n\ge 0}$ be a non-decreasing sequence of sub-$\sigma$-algebras of $\mathcal {F}$ such that $\mathcal {F}=\mathcal {\sigma }\big (\bigcup _{n\ge 0}\mathcal {F}_n\big )$. Recall that the expectation operator and the conditional expectation operator with respect to ${\mathcal {F}_n}$ are denoted by $\mathbb {E}$ and $\mathbb {E}_n$, i.e. $\mathbb {E}_n f=\mathbb {E}(f|{\mathcal {F}_n})$ for any $\mathcal {F}$-measurable function f, respectively. A sequence of measurable functions $f=(f_n)_{n\ge 0} \subseteq L_1$ is called a martingale with respect to $\{\mathcal {F}_n\}_{n\ge 0}$ if $\mathbb {E}_nf_{n+1}=f_n$ for every $n\ge 0.$ Denote by $\mathcal {M}$ the set of all martingales $f=(f_n)_{n\ge 0}$ 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

$$\begin{aligned} d_nf=f_n-f_{n-1} \qquad (n\ge 0), \qquad f_{-1}=0. \end{aligned}$$

We shall consider the following special martingale operators. For any martingale $f\in \mathcal {M}$, we define the maximal functions, the square functions and the conditional square functions of f, respectively, as follows

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

Let $\Lambda$ be the class of non-negative, non-decreasing and adapted sequences $\lambda =(\lambda _n)_{n\ge 0}$ with respect to $\{\mathcal {F}_n\}_{n\ge 0}$, i.e., $\lambda _n$ is $\mathcal {F}_n$ measurable for all $n \in \mathbb {N}$. Define

$$\begin{aligned} \lambda _\infty = \lim _{n\rightarrow \infty }{\lambda _n}. \end{aligned}$$

Let $\Phi$ be an Orlicz function, $0<q\le \infty$ and let b be a slowly varying function. For $f\in \mathcal {M}$, let

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

We define the Orlicz-Lorentz-Karamata Hardy martingale spaces via the martingale operators above in the following way:

$$\begin{aligned} H^M_{\Phi ,q,b}= & {} \big \{f\in \mathcal {M}: M(f)\in L_{\Phi ,q,b}\big \},~~~~~~\Vert f\Vert _{H^M_{\Phi ,q,b}}={\Vert M(f)\Vert }_{L_{\Phi ,q,b}};\\ H^S_{\Phi ,q,b}= & {} \big \{f\in \mathcal {M}: S(f)\in L_{\Phi ,q,b}\big \},~~~~~~\Vert f\Vert _{H^S_{{\Phi ,q,b}}}={\Vert S(f)\Vert }_{L_{\Phi ,q,b}};\\ H^s_{{\Phi ,q,b}}= & {} \big \{f\in \mathcal {M}: s(f)\in L_{{\Phi ,q,b}}\big \},~~~~~~\Vert f\Vert _{H^s_{{\Phi ,q,b}}}={\Vert s(f)\Vert }_{L_{\Phi ,q,b}};\\ \mathcal {Q}_{\Phi ,q,b}= & {} \big \{f\in \mathcal {M}: \Vert f\Vert _{Q_{{\Phi ,q,b}}}<\infty \big \},\\ \Vert f\Vert _{\mathcal {Q}_{\Phi ,q,b}}= & {} \inf \limits _{(\lambda _n)_{n\ge 0} \in \Lambda [\mathcal {Q}_{\Phi ,q,b}](f)}\Vert \lambda _\infty \Vert _{L_{\Phi ,q,b}};\\ \mathcal {P}_{\Phi ,q,b}= & {} \big \{f\in \mathcal {M}: \Vert f\Vert _{\mathcal {P}_{\Phi ,q,b}}<\infty \big \},\\ \Vert f\Vert _{\mathcal {P}_{\Phi ,q,b}}= & {} \inf \limits _{(\lambda _n)_{n\ge 0}\in \Lambda [\mathcal {P}_{\Phi ,q,b}](f)}\Vert \lambda _\infty \Vert _{L_{\Phi ,q,b}}. \end{aligned}$$

Remark 5

If we consider the special case $\Phi (t)=t^p$ for $0<p<\infty$ with the notations above, then we obtain the definition of $H_{p,q,b}^M$, $H^S_{p,q,b}$, $H^s_{p,q,b}$, ${\mathcal {Q}}_{p,q,b}$ and ${\mathcal {P}}_{p,q,b}$, respectively; see Ho [28]. If $b\equiv 1$, we obtain the definition of $H_{\Phi ,q}^M$, $H^S_{\Phi ,q}$, $H^s_{\Phi ,q}$, ${\mathcal {Q}}_{\Phi ,q}$ and ${\mathcal {P}}_{\Phi ,q}$, respectively; see Hao and Li [24]. In addition, if $\Phi (t)=t^p$ for $0<p<\infty$, then we obtain the definition of $H_{p,q}^M$, $H^S_{p,q}$, $H^s_{p,q}$, ${\mathcal {Q}}_{p,q}$ and ${\mathcal {P}}_{p,q}$, respectively; see Jiao et al. [33, 37, 60]. Moreover, if $p=q$, we obtain the martingale Hardy spaces $H_q^M$, $H^S_q$, $H^s_q$, ${\mathcal {Q}}_q$ and ${\mathcal {P}}_q$, respectively; see Weisz [59, 60].

The bounded $L_{\Phi ,q,b}$-martingale spaces are defined by

$$\begin{aligned} L_{{\Phi ,q,b}}=\Big \{f=(f_n)_{n\ge 0}: \sup _{n\ge 0}\Vert f_n\Vert _{L_{\Phi ,q,b}}<\infty \Big \},~~~~~~\Vert f\Vert _{L_{\Phi ,q,b}}=\sup _{n\ge 0}\Vert f_n\Vert _{L_{\Phi ,q,b}}. \end{aligned}$$

In particular, for $\Phi (t)=t^q$ and $b\equiv 1$, we call these spaces bounded $L_{q}$-martingale spaces.

The stochastic basis $\{\mathcal {F}_n\}_{n\ge 0}$ is said to be regular, if for $n\ge 0$ and $A\in \mathcal {F}_n$, there exists a $B\in \mathcal {F}_{n-1}$ such that

$$\begin{aligned} A\subseteq B \qquad \text{ and } \qquad \mathbb {P}(B)\le \mathcal {R} \mathbb {P}(A), \end{aligned}$$

where $\mathcal {R}$ is a positive constant independent of n. Equivalently, $\{\mathcal {F}_n\}_{n\ge 0}$ is regular if there exists a constant $\mathcal {R}>0$ such that

$$\begin{aligned} f_n\le \mathcal {R} f_{n-1} \end{aligned}$$

(2.9)

for all non-negative martingales $(f_n)_{n\ge 0}$. We refer to [42, 50, 60] for the theory of martingale Hardy spaces.

3 Doob’s maximal inequalities

Doob’s maximal inequality plays a central role in harmonic analysis, probability and ergodic theory. The classical Doob maximal inequality (see Doob [11], and also [19, 60]) states that

$$\begin{aligned} \Vert M(f)\Vert _{L_p}\le \frac{p}{p-1}\Vert f\Vert _{L_p},\quad \quad \quad p>1. \end{aligned}$$

Doob used this inequality to prove the basic, almost sure convergence properties of martingales. Over the course of the past few decades, Doob’s maximal inequality has been rapidly developed to various function spaces. In this section, we study the Doob maximal inequalities on Orlicz-Lorentz-Karamata spaces.

Theorem 3

Let $\Phi$ be an Orlicz function of lower type $p\in (1,\infty )$, $0<q\le \infty$ and let b be a slowly varying function. Then we have

$$\begin{aligned} |||f |||_{L_{\Phi ,q,b}}\le |||M(f) |||_{L_{\Phi ,q,b}} \preceq |||f |||_{L_{\Phi ,q,b}},\quad \quad \forall \;f=(f_n)_{n\ge 0}\in L_{\Phi ,q,b}. \end{aligned}$$

As an application of Theorem 3, by using Proposition 4, we obtain the following result.

Theorem 4

Let $\Phi$ be an Orlicz function of lower type $p_-$ and upper type $p_+$ satisfying $1<p_- \le p_+<\infty$, $0<q\le \infty$ and let b be a slowly varying function. Then we have

$$\begin{aligned} \Vert f\Vert _{L_{\Phi ,q,b}}\le \Vert M(f)\Vert _{L_{\Phi ,q,b}} \preceq \Vert f\Vert _{L_{\Phi ,q,b}},\quad \quad \forall \;f=(f_n)_{n\ge 0}\in L_{\Phi ,q,b}. \end{aligned}$$

In order to prove Theorem 3, we firstly present another characterization of the functional $|||\cdot |||_{L_{\Phi ,q,b}}$ under suitable conditions.

Proposition 7

Let $\Phi$ be an Orlicz function of lower type $p\in (1,\infty )$, $0< q\le \infty$ and let b be a slowly varying function. Then $|||f |||_{L_{\Phi ,q,b}}$ and

$$\begin{aligned} \big \Vert f\big \Vert _{L_{\Phi ,q,b,1}}= \left\{ \begin{array}{l} \Big (q\int _0^\infty \Big (\frac{\gamma _b(t)}{\Phi ^{-1}(1/t)} f^{**}(t)\Big )^q\frac{dt}{t}\Big )^{1/q} \ \ \;\,\, \text {if} \;\; 0<q<\infty ,\\ \sup \limits _{t>0} \frac{\gamma _b(t)}{\Phi ^{-1}(1/t)} f^{**}(t) \ \quad \quad \quad \quad \ \quad \;\; \;\;\text {if} \;\; q=\infty , \end{array} \right. \end{aligned}$$

are equivalent (quasi)-norms. Here $f^{**}$ denotes the maximal non-increasing rearrangement of f, which is defined as

$$\begin{aligned} f^{**}(t)=\frac{1}{t}\int _0^t f^*(x)dx, \quad \quad \quad \quad t>0. \end{aligned}$$

To prove Proposition 7, we need the following characterization of Hardy-type inequalities.

Lemma 8

([3], Theorem 7) Let $0<q\le 1$, v and w be non-negative measurable functions where v is assumed to be locally integrable on $(0,\infty )$. Then the weighted Hardy inequality

$$\begin{aligned} \bigg [\int _0^\infty w(x)\Big (\int _0^xg(t)dt\Big )^q dx\bigg ]^{1/q} \le C \bigg [\int _0^\infty v(x)g(x)^q dx\bigg ]^{1/q}, \end{aligned}$$

holds for all non-negative, non-increasing functions g on $(0,\infty )$ if and only if the following conditions hold

$(a)\quad \quad \quad \quad \quad \quad \quad \quad \sup \limits _{x>0}\frac{1}{V(x)^{1/q}}\Big (\int _{0}^{x}t^{q}w(t)dt\Big )^{1/q}<\infty$

and

$(b)\quad \quad \quad \quad \quad \quad \quad \quad \sup \limits _{x>0}\frac{x}{V(x)^{1/q}}\Big (\int _{x}^\infty w(t)dt\Big )^{1/q}<\infty ,$

where

$$\begin{aligned} {V}(x)=\int _{0}^{x}v(t)dt,\quad \quad x>0. \end{aligned}$$

Now we prove Proposition 7:

Proof

Obviously, $f^*(t)\le f^{**}(t)$ for $t>0$. Thus it is enough to prove that

$$\begin{aligned} \Vert f\Vert _{L_{\Phi ,q,b,1}}\lesssim |||f |||_{L_{\Phi ,q,b}}. \end{aligned}$$

(3.1)

Taking $s=1$ in Proposition 6, it is easy to see that (3.1) holds when $1\le q \le \infty$. Now we verify (3.1) for $0<q<1$. From the definitions of $\Vert \cdot \Vert _{L_{\Phi ,q,b,1}}$ and $|||\cdot |||_{L_{\Phi ,q,b}}$, we see that

$$\begin{aligned} \big \Vert f\big \Vert _{L_{\Phi ,q,b,1}}= & {} \Bigg \{q\int _0^\infty \bigg [\bigg (\frac{\gamma _b(x)}{x^{1+1/q}\Phi ^{-1}(1/x)} \bigg )^q\bigg (\int _0^x f^*(t)dt\bigg )^q \bigg ]dx\Bigg \}^{1/q}\\= & {} q^{1/q}\bigg [\int _0^\infty w(x)\Big (\int _0^x g(t)dt\Big )^q dt\bigg ]^{1/q} \end{aligned}$$

and

$$\begin{aligned} |||f |||_{L_{\Phi ,q,b}}= & {} \bigg (q\int _0^1\Big (\frac{\gamma _b(x)}{x^{1/q}\Phi ^{-1}(1/x)}\Big )^q \big [f^*(x)\big ]^q dx\bigg )^{1/q}\\= & {} q^{1/q}\bigg (\int _0^\infty v(x)\big [g(x)\big ]^q dx \bigg )^{1/q}, \end{aligned}$$

where

$$\begin{aligned} w(x){:}{=} \bigg (\frac{\gamma _b(x)}{x^{1+1/q}\Phi ^{-1}(1/x)} \bigg )^q, \; v(x){:}{=} \bigg (\frac{\gamma _b(x)}{x^{1/q}\Phi ^{-1}(1/x)}\bigg )^q \;\text {and}\; g(x){:}{=}f^*(x). \end{aligned}$$

Clearly, g(x) is a non-negative and non-increasing function, and

$$\begin{aligned} \sup _{x>0}\frac{1}{V(x)^{1/q}}\Big (\int _{0}^{x}t^{q}w(t)dt\Big )^{1/q}=1<\infty . \end{aligned}$$

(3.2)

Since $\frac{\Phi ^{-1}(t)}{t^{1/{p}}}$ is equivalent to a non-increasing function on $(0,\infty )$ by Lemma 1, we have that $\frac{(1/t)^{1/p}}{\Phi ^{-1}(1/t)}$ is also equivalent to a non-increasing function on $(0,\infty )$. Hence we obtain that, for $x>0$,

$$\begin{aligned} V(x)= & {} \int _0^x\bigg (\frac{\gamma _b(t)}{t^{1/q}\Phi ^{-1}(1/t)}\bigg )^qdt\\= & {} \int _0^x \Big (\frac{(1/t)^{1/p}}{\Phi ^{-1}\big (1/t\big )}\Big )^q \big (t^{1/p -1/q}\gamma _b(t)\big )^qdt\\\succeq & {} x\bigg (\frac{\gamma _b(x)}{x^{1/q}\Phi ^{-1}(1/x)}\bigg )^q=\bigg (\frac{\gamma _b(x)}{\Phi ^{-1}(1/x)}\bigg )^q \end{aligned}$$

and

$$\begin{aligned} \int _{x}^\infty w(t)dt= & {} \int _{x}^\infty \bigg (\frac{\gamma _b(t)}{t^{1+1/q}\Phi ^{-1}(1/t)} \bigg )^qdt\\= & {} \int _{x}^\infty \Big (\frac{(1/t)^{1/p}}{\Phi ^{-1}\big (1/t\big )}\Big )^q \big (t^{-\varepsilon _0}\gamma _b(t)\big )^qt^{(1/p -1+\varepsilon _0)q-1} dt\\\preceq & {} \bigg (\frac{\gamma _b(x)}{x\Phi ^{-1}(1/x)}\bigg )^q, \end{aligned}$$

where $0<\varepsilon _0<1-\frac{1}{p}$. Thus we get the following estimation

$$\begin{aligned} \sup \limits _{x>0}\frac{x}{V(x)^{1/q}}\Big (\int _{x}^\infty w(t)dt\Big )^{1/q}\le C_{p,q,b}<\infty . \end{aligned}$$

(3.3)

Inequalities (3.2) and (3.3) and Lemma 8 imply

$$\begin{aligned} \big \Vert f\big \Vert _{L_{\Phi ,q,b,1}}= & {} q^{1/q}\bigg [\int _0^\infty w(x)\Big (\int _0^xg(t)dt\Big )^q dt\bigg ]^{1/q}\\\le & {} C q^{1/q}\bigg (\int _0^\infty v(x)\big [g(x)\big ]^q dx \bigg )^{1/q}\\= & {} C|||f |||_{L_{\Phi ,q,b}}. \end{aligned}$$

This completes the proof. $\square$

Lemma 9

([42], Theorem 3.6.3) Let $f=(f_n)_{n\ge 0}\in L_1$. Then

$$\begin{aligned} \big (M(f)\big )^*(t)\le f^{**}(t),\quad \quad \quad \forall \;t>0. \end{aligned}$$

Now we prove Theorem 3:

Proof

Obviously, $|||f |||_{L_{\Phi ,q,b}}\le |||M(f) |||_{L_{\Phi ,q,b}}$. It follows from Proposition 7 and Lemma 9 that

$$\begin{aligned} |||M(f) |||_{L_{\Phi ,q,b}}= & {} \bigg (q\int _0^1\Big (\frac{1}{\Phi ^{-1}(1/t)}\gamma _b(t) \big (M(f)\big )^*(t)\Big )^q\frac{dt}{t}\bigg )^{1/q}\\\le & {} \bigg (q\int _0^1\Big (\frac{1}{\Phi ^{-1}(1/t)}\gamma _b(t) f^{**}(t)\Big )^q\frac{dt}{t}\bigg )^{1/q}\\= & {} \big \Vert f\big \Vert _{L_{\Phi ,q,b,1}} \preceq |||f |||_{L_{\Phi ,q,b}}. \end{aligned}$$

This completes the proof of the theorem. $\square$

4 Atomic decomposition

Atomic decomposition is a powerful tool for dealing with interpolation theorems, duality theorems and some fundamental inequalities both in martingale theory and harmonic analysis. We refer to [5, 26, 34, 36, 41, 44, 51, 57, 60,61,62,63, 65,66,67] for more information on atomic decomposition. In this section we construct the atomic decomposition of Orlicz-Lorentz-Karamata Hardy martingale spaces. First, we give the definition of the atom.

Definition 4

Let $\Phi$ be an Orlicz function and $0<r\le \infty$. A measurable function, a, is called a $(1,\Phi ,r)$-atom (resp. $(2,\Phi ,r)$-atom, $(3,\Phi ,r)$-atom) if there exists a stopping time $\tau \in \mathcal {T}$ such that

• 1. $a_n=\mathbb {E}_n a =0$ if $\tau \ge n$;

• 2. $\Vert s(a)\Vert _{L_r}$ (resp. $\Vert S(a)\Vert _{L_r}$, $\Vert M(a)\Vert _{L_r}$ ) $\le \mathbb {P}(\tau<\infty )^{1/r}\Phi ^{-1}\big ( {1}/{\mathbb {P}(\tau <\infty )}\big )$.

Notice that if we consider the special case $\Phi (t)=t^p$ $(t\ge 0, \, 0<p<\infty )$, the $(i,\Phi ,\infty )$-atom ($i=1,2,3$) is the same as Definition 2.1 in Weisz [60]. Denote by $\mathcal {A}_i{(\Phi ,q,b,r)}$ the set of all sequences of triples $(a^k,\tau _k,\mu _k)_{k\in \mathbb {Z}}$ satisfying

$$\begin{aligned} \bigg (\sum _{k\in \mathbb {Z}}\gamma _b^q\big (\mathbb {P}(\tau _k<\infty )\big )\mu _k^q\bigg )^{1/q}<\infty \end{aligned}$$

(with the usual modification for $q = \infty$), where $(a^k)_{k\in \mathbb {Z}}$ are $(i,\Phi ,r)$-atoms ($i=1,2,3$), $(\tau _k)_{k\in \mathbb {Z}}\subseteq \mathcal {T}$ are associated with $(a^k)_{k\in \mathbb {Z}}$, and there exists a positive constant $\kappa$ such that

$$\begin{aligned} (\mu _k)_{k\in \mathbb {Z}}=\bigg (\frac{\kappa \cdot 2^k}{\Phi ^{-1}\big (1/\mathbb {P}(\tau _k<\infty )\big )}\bigg )_{k\in \mathbb {Z}}. \end{aligned}$$

Definition 5

Let $\Phi$ be an Orlicz function, $0<q,r\le \infty$ and let b be a slowly varying function. The atomic martingale Orlicz-Lorentz-Karamata space $H_{\Phi ,q,b}^{at,r,1}$ $\big ($resp. $H_{\Phi ,q,b}^{at,r,2}$ or $H_{\Phi ,q,b}^{at,r,3}$ $\big )$ is defined to be the set of all $f \in \mathcal {M}$ satisfying that there exists a sequence of triples, $(a^k,\tau _k,\mu _k)_{k\in \mathbb {Z}}\subseteq \mathcal {A}_1{(\Phi ,q,b,r)}$ $\big ($resp. $\mathcal {A}_2{(\Phi ,q,b,r)}$ or $\mathcal {A}_3{(\Phi ,q,b,r)}$ $\big )$ such that for each $n\ge 0$ and almost every $\omega \in \Omega$,

$$\begin{aligned} f_n(\omega )= \sum _{ k \in \mathbb {Z} } \mu _k \big (\mathbb {E}_n a^k\big )(\omega ). \end{aligned}$$

(4.1)

Endow $H_{\Phi ,q,b}^{at,r,1}$ $\big ($resp. $H_{\Phi ,q,b}^{at,r,2}$ or $H_{\Phi ,q,b}^{at,r,3}$ $\big )$ with the (quasi)-norm

$$\begin{aligned} \Vert f\Vert _{H_{\Phi ,q,b}^{at,r,1}}\; \Big (\text {resp.}\; \Vert f\Vert _{H_{\Phi ,q,b}^{at,r,2}}\; \text {or}\; \Vert f\Vert _{H_{\Phi ,q,b}^{at,r,3}}\Big )=\inf \bigg (\sum _{k\in \mathbb {Z}}\gamma _b^q\big (\mathbb {P}(\tau _k<\infty )\big )\mu _k^q\bigg )^{1/q}, \end{aligned}$$

where the infimum is taken over all decompositions of f as above (with the usual modification for $q = \infty$ $)$.

Remark 6

We claim that every $(1,\Phi ,\infty )$-atom a is a $(1,\Phi ,r)$-atom for $0<r<\infty$. Indeed, from the definition of $(1,\Phi ,\infty )$-atom, there is

$$\begin{aligned} \Vert s(a)\Vert _{L_r}&=\Vert s(a)\chi _{\{\tau<\infty \}}\Vert _{L_r}\le \Vert s(a)\Vert _{L_\infty }\Vert \chi _{\{\tau<\infty \}}\Vert _{L_r} \\&\le \mathbb {P}(\tau< \infty )^{\frac{1}{r}}\Phi ^{-1}\Big (\frac{1}{\mathbb {P}(\tau < \infty )}\Big ). \end{aligned}$$

This implies that

$$\begin{aligned} H_{\Phi ,q,b}^{at,\infty ,i}\subseteq H_{\Phi ,q,b}^{at,r,i},\quad \quad \quad \quad \quad (i=1,2,3) \end{aligned}$$

with respect to their (quasi)-norms respectively.

Theorem 5

Let $\Phi$ be an Orlicz function of lower type $p_-$ and upper type $p_+$ satisfying $0<p_- \le p_+<\infty$, $0<q\le \infty$ and let b be a slowly varying function. Then

$$\begin{aligned} H_{\Phi ,q,b}^s=H_{\Phi ,q,b}^{at,\infty ,1}, \end{aligned}$$

with equivalent (quasi)-norms.

Proof

Let $f \in H_{\Phi ,q,b}^s$. For any $k\in \mathbb {Z}$, define

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

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 $n\in \mathbb {N}$,

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

For every $k\in \mathbb {Z}$ and $n\in \mathbb {N}$, set

$$\begin{aligned} \mu _k=\frac{3\cdot 2^k }{\Phi ^{-1}\big ( 1/\mathbb {P}(\tau _k<\infty )\big ) }\quad \text{ and }\quad a_n^k= \frac{ f_n^{\tau _{k+1}}-f_n^{\tau _{k}} }{\mu _k}. \end{aligned}$$

If $\mu _k=0$ then let $a_n^k=0$ for all $k\in \mathbb {Z}$ and $n\in \mathbb {N}$. Obviously, $(a_n^k)_{n \ge 0}$ is a martingale for each fixed $k\in \mathbb {Z}$. Moreover, in view of the definition of $\tau _k$, we have $s(f^{\tau _k}) = s_{\tau _k}(f)\le 2^k$. Hence, by using the definition of $\mu _k$, we obtain

$$\begin{aligned} s\big ((a_n^k)_{n\ge 0}\big )\le \frac{s(f_n^{\tau _{k+1}})+s(f_n^{\tau _{k}})}{\mu _k} \le \Phi ^{-1}\big (1/\mathbb {P}(\tau _k<\infty )\big ),\quad \;n\in \mathbb {N}. \end{aligned}$$

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

$$\begin{aligned} \Vert s(a^k)\Vert _{L_\infty } \le \Phi ^{-1}\big (1/\mathbb {P}(\tau _k<\infty )\big ). \end{aligned}$$

Furthermore, for $\tau _k\ge n$,

$$\begin{aligned} a_n^k= \frac{ f_n^{\tau _{k+1}}-f_n^{\tau _{k}} }{\mu _k}=\frac{ f_n-f_n }{\mu _k}=0. \end{aligned}$$

Thus we conclude that $a^k$ is really a $(1,\Phi ,\infty )$-atom.

Suppose that $0<q<\infty$. Since $\{\tau _k<\infty \}=\{s(f)>2^k\}$, then it follows from Remark 4 that

$$\begin{aligned} \sum _{k\in \mathbb {Z}}\gamma _b^q\big (\mathbb {P}(\tau _k<\infty )\big )\mu _k^q\le & {} 3^q \sum _{ k \in \mathbb {Z} }\gamma _b^q\big (\mathbb {P}(\tau _k<\infty )\big )\Big (\frac{2^k }{\Phi ^{-1}\big ( 1/\mathbb {P}(\tau _k<\infty )\big ) }\Big )^q\\= & {} 3^q \sum _{ k \in \mathbb {Z} } 2^{kq}\gamma _b^q\big (\lambda _{2^k}(s(f))\big )\big \Vert \chi _{\{s(f)>2^k\}}\big \Vert _{L_\Phi }^q\\\preceq & {} \big \Vert s(f)\big \Vert ^q_{L_{\Phi ,q,b}}. \end{aligned}$$

Thus we deduce that

$$\begin{aligned} \bigg (\sum _{k\in \mathbb {Z}}\gamma _b^q\big (\mathbb {P}(\tau _k<\infty )\big )\mu _k^q\bigg )^{1/q} \preceq \Vert f\Vert _{H_{\Phi ,q,b}^s}. \end{aligned}$$

For the converse part, assume that the martingale f has the decomposition of (4.1). For an arbitrary integer $k_0$, let

$$\begin{aligned} f=\sum _{ k \in \mathbb {Z} } \mu _k a^k=T_1 + T_2,\quad \quad (n\in \mathbb {N}), \end{aligned}$$

where

$$\begin{aligned} T_{1}=\sum _{ k =-\infty }^{k_0-1} \mu _k a^k\quad \text {and}\quad T_{2}=\sum _{ k =k_0 }^{\infty } \mu _k a^k. \end{aligned}$$

It follows from the sublinearity of the conditional square operator s that

$$\begin{aligned} s(T_{ 1})\le & {} \big \Vert s(T_{ 1})\big \Vert _{L_\infty }\\= & {} \Big \Vert s\Big (\sum _{ k =-\infty }^{k_0-1} \mu _k a^k\Big )\Big \Vert _{L_\infty }\\\le & {} \Big \Vert \sum _{ k =-\infty }^{k_0-1}\mu _ks(a^k)\Big \Vert _{L_\infty } \le \sum _{ k =-\infty }^{k_0-1}\mu _k\big \Vert s(a^k)\big \Vert _{L_\infty }. \end{aligned}$$

Since $a^k$ is a $(1,\Phi ,\infty )$-atom for each $k\in \mathbb {Z}$, we obtain

$$\begin{aligned} s(T_{ 1})\le & {} \sum _{ k =-\infty }^{k_0-1}\mu _k\Phi ^{-1}\big (1/\mathbb {P}(\tau _k<\infty )\big ) \nonumber \\= & {} \sum _{ k =-\infty }^{k_0-1}\kappa \cdot 2^k=\kappa \cdot 2^{k_0}. \end{aligned}$$

(4.2)

Combining this with formulas (2.2), (2.1) and (4.2), we can deduce that

$$\begin{aligned} \Vert f\Vert ^q_{H^s_{\Phi ,q,b}}&= \Vert s(f)\Vert ^q_{L_{\Phi ,q,b}} \nonumber \\&= \sum _{k\in \mathbb {Z}} \kappa ^q\cdot 2^{kq}\gamma _b \big (\lambda _{\kappa \cdot 2^k}(s(f))\big )\big \Vert \chi _{\{s(f)>\kappa \cdot 2^k\}}\big \Vert _{L_\Phi }^q\nonumber \\&= \sum _{k\in \mathbb {Z}} \kappa ^q\cdot 2^{kq}\big \Vert \chi _{\{s(f)>\kappa \cdot 2^k\}}\big \Vert _{L_{\Phi ,q,b}}^q\nonumber \\&= \sum _{k\in \mathbb {Z}} \kappa ^q\cdot 2^{(k+1)q}\big \Vert \chi _{\{s(f)>\kappa \cdot 2^{k+1}\}}\big \Vert _{L_{\Phi ,q,b}}^q\nonumber \\&= (2\kappa )^q\sum _{k_0\in \mathbb {Z}} 2^{k_0q}\big \Vert \chi _{\{s(f)>2\kappa \cdot 2^{k_0}\}}\big \Vert _{L_{\Phi ,q,b}}^q\nonumber \\&\le (2\kappa )^q\sum _{k_0\in \mathbb {Z}} 2^{k_0q}\big \Vert \chi _{\{s(T_{ 1})>\kappa \cdot 2^{k_0}\}} +\chi _{\{s(T_{ 2})>\kappa \cdot 2^{k_0}\}}\big \Vert _{L_{\Phi ,q,b}}^q\nonumber \\&= (2\kappa )^q\sum _{k_0\in \mathbb {Z}} 2^{k_0q}\big \Vert \chi _{\{s(T_{2})>\kappa \cdot 2^{k_0}\}}\big \Vert _{L_{\Phi ,q,b}}^q. \end{aligned}$$

(4.3)

On the other hand, since $a^k$ is a $(1,\Phi ,\infty )$-atom for each $k\in \mathbb {Z}$, then we have $s(a^k)=0$ on the set $\{\tau _k=\infty \}$. This means

$$\begin{aligned} \{s(a^k)>0\}\subseteq \{\tau _k<\infty \}, \end{aligned}$$

which implies

$$\begin{aligned} \{s(T_{ 2})> \kappa \cdot 2^{k_0}\}\subseteq & {} \{s(T_{ 2})>0\}\nonumber \\\subseteq & {} \bigcup _{k=k_0}^\infty \{s(a^k)>0\}\nonumber \\\subseteq & {} \bigcup _{k=k_0}^\infty \{\tau _k<\infty \}. \end{aligned}$$

(4.4)

Moreover, applying with Theorem 2, there exists $0<\alpha <\min \{1,p_-, q\}$ such that

$$\begin{aligned} \big \Vert \chi _{\{s(T_{ 2})>\kappa \cdot 2^{k_0}\}}\big \Vert _{L_{\Phi ,q,b}}^\alpha&\le \bigg \Vert \sum _{k=k_0}^\infty \chi _{\{\tau _k<\infty \}}\bigg \Vert _{L_{\Phi ,q,b}}^\alpha \nonumber \\&\preceq \sum _{k=k_0}^\infty \big \Vert \chi _{\{\tau _k<\infty \}}\big \Vert _{L_{\Phi ,q,b}}^\alpha . \end{aligned}$$

(4.5)

Let $0<\eta <1$. For $1=\frac{\alpha }{q}+\frac{q-\alpha }{q}$, it follows from Hölder’s inequality and (4.5) that

$$\begin{aligned}&\Vert \chi _{\{s(T_{2})>\kappa \cdot 2^{k_0}\}}\Vert _{L_{\Phi ,q,b}} \\&\quad \preceq \bigg (\sum _{k=k_0}^\infty \big \Vert \chi _{\{\tau _k<\infty \}}\big \Vert _{L_{\Phi ,q,b}}^\alpha \bigg )^{1/\alpha }\\&\quad = \bigg (\sum _{k=k_0}^\infty 2^{-k\eta \alpha }2^{k\eta \alpha }\big \Vert \chi _{\{\tau _k<\infty \}}\big \Vert ^\alpha _{L_{\Phi ,q,b}}\bigg )^{1/\alpha }\\&\quad \le \bigg (\sum _{k=k_0}^\infty 2^{-k\eta \alpha \frac{q}{q-\alpha }}\bigg )^{\frac{q-\alpha }{q \alpha }} \bigg (\sum _{k=k_0}^\infty 2^{k\eta q}\big \Vert \chi _{\{\tau _k<\infty \}}\big \Vert ^q_{L_{\Phi ,q,b}}\bigg )^{1/q}\\&\quad \preceq 2^{-k_0\eta }\bigg (\sum _{k=k_0}^\infty 2^{k\eta q}\big \Vert \chi _{\{\tau _k<\infty \}}\big \Vert ^q_{L_{\Phi ,q,b}}\bigg )^{1/q}. \end{aligned}$$

Combining this with (4.3), we obtain that

$$\begin{aligned} \Vert f\Vert ^q_{H^s_{\Phi ,q,b}}\preceq & {} (2\kappa )^q\sum _{k_0\in \mathbb {Z}}2^{k_0(1-\eta )q}\sum _{k=k_0}^\infty 2^{k\eta q}\big \Vert \chi _{\{\tau _k<\infty \}}\big \Vert _{L_{\Phi ,q,b}}^q\\= & {} (2\kappa )^q\sum _{k\in \mathbb {Z}} 2^{k\eta q}\big \Vert \chi _{\{\tau _k<\infty \}}\big \Vert _{L_{\Phi ,q,b}}^q \sum _{k_0=-\infty }^k2^{k_0(1-\eta )q}\\\preceq & {} (2\kappa )^q\sum _{k\in \mathbb {Z}} 2^{k \eta q}\big \Vert \chi _{\{\tau _k<\infty \}}\big \Vert _{L_{\Phi ,q,b}}^q 2^{k(1-\eta )q}\\= & {} (2\kappa )^q\sum _{k\in \mathbb {Z}} 2^{kq}\big \Vert \chi _{\{\tau _k<\infty \}}\big \Vert _{L_{\Phi ,q,b}}^q\\= & {} (2\kappa )^q\sum _{k\in \mathbb {Z}} 2^{kq} \bigg (\frac{\gamma _b\big (\mathbb {P}(\tau _k<\infty )\big ) }{\Phi ^{-1}\big (1/\mathbb {P}(\tau _k<\infty )\big )}\bigg )^q\\\preceq & {} \sum _{k\in \mathbb {Z}}\gamma _b^q\big (\mathbb {P}(\tau _k<\infty )\big ) \bigg (\frac{\kappa \cdot 2^k }{\Phi ^{-1}\big (1/\mathbb {P}(\tau _k<\infty )\big )}\bigg )^q\\= & {} \sum _{k\in \mathbb {Z}}\gamma _b^q\big (\mathbb {P}(\tau _k<\infty )\big ) \mu _k^q . \end{aligned}$$

This means

$$\begin{aligned} \Vert f\Vert _{H^s_{\Phi ,q,b}} \preceq \bigg (\sum _{k\in \mathbb {Z}}\gamma _b^q\big (\mathbb {P}(\tau _k<\infty )\big )\mu _k^q\bigg )^{1/q}. \end{aligned}$$

Taking the infimum over all the preceding decompositions of f of the form (4.1), we can conclude the result. The proof is similar for $q=\infty$. This completes the proof of the theorem. $\square$

Remark 7

If $\Phi$ is an Orlicz function of lower type $p_-$ and upper type $p_+$ satisfying $0<p_- \le p_+<\infty$, $0<q< \infty$ and b is a slowly varying function, then the sum $\sum ^N_{k=M} \mu _k a^k$ converges to f in $H_{\Phi ,q,b}^s$ as $M \rightarrow -\infty$, $N\rightarrow \infty$. Indeed, since

$$\begin{aligned} \sum ^N_{k=M} \mu _k a^k = \sum ^N_{k=M} (f^{\tau _{k+1}}-f^{\tau _{k}})=f^{\tau _{N+1}}-f^{\tau _{M}}, \end{aligned}$$

(4.6)

then, by using (2.5), we get

$$\begin{aligned} \big \Vert f-\sum _{ k \in \mathbb {Z} }\mu _k a^k \big \Vert _{H_{\Phi ,q,b}^s}= & {} \big \Vert s\big ((f-f^{\tau _{N+1}})+f^{\tau _{M}}\big ) \big \Vert _{ L_{\Phi ,q,b} }\\\le & {} \big \Vert s\big ((f-f^{\tau _{N+1}}) + s\big (f^{\tau _{M}}\big )\big \Vert _{ L_{\Phi ,q,b} }\\\le & {} C\Big (\big \Vert s\big (f-f^{\tau _{N+1}}\big )\big \Vert _{L_{\Phi ,q,b}} +\big \Vert s\big (f^{\tau _{M}}\big )\big \Vert _{ L_{\Phi ,q,b} }\Big ). \end{aligned}$$

It follows from $s\big (f-f^{\tau _{N+1}}\big )^2=s(f)^2-s\big (f^{\tau _{N+1}})^2$ that $s\big (f-f^{\tau _{N+1}}\big )\le s(f)$, $s\big (f^{\tau _{M}}\big )\le s(f)$ and $s\big (f-f^{\tau _{N+1}}\big )\longrightarrow 0$, $s\big (f^{\tau _{M}}\big )\longrightarrow 0$ as $M \longrightarrow -\infty$, $N \longrightarrow \infty$. Then, by Proposition 5, we obtain

$$\begin{aligned} \big \Vert s\big (f-f^{\tau _{N+1}}\big )\big \Vert _{L_{\Phi ,q,b}}\longrightarrow 0, \quad \quad \big \Vert s\big (f^{\tau _{M}}\big )\big \Vert _{ L_{\Phi ,q,b} }\longrightarrow 0, \end{aligned}$$

as $M \longrightarrow -\infty$, $N \longrightarrow \infty$. This means

$$\begin{aligned} \big \Vert f-\sum _{ k \in \mathbb {Z} }\mu _k a^k \big \Vert _{H_{\Phi ,q,b}^s}\longrightarrow 0,\quad \quad \text {as}\;\; M \longrightarrow -\infty ,\; N \longrightarrow \infty . \end{aligned}$$

Obviously, for $k\in \mathbb {Z}$, $a^k = (a^k_n)_{n\ge 0}$ (here $a^k$ is a $(1, \Phi ,\infty )$-atom) is $L_2$-bounded. Hence $L_2 = H^s_2$ is dense in $H^s_{\Phi ,q,b}$.

Theorem 6

Let $\Phi$ be an Orlicz function of lower type $p_-$ and upper type $p_+$ satisfying $0<p_- \le p_+<\infty$, $0<q\le \infty$ and let b be a slowly varying function. If $\{\mathcal {F}_n\}_{n\ge 0}$ is regular, then

$$\begin{aligned} H_{\Phi ,q,b}^M=H_{\Phi ,q,b}^{at,\infty ,3}, \end{aligned}$$

with equivalent (quasi)-norms.

Proof

Let $f\in H_{\Phi ,q,b}^M$. For each fixed $k\in \mathbb {Z}$, the stopping times $v_k$ are defined by

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

Define

$$\begin{aligned} F_n^k {:}{=} \left\{ \mathbb {E}_{n-1} \chi _{\{v_k=n\}} \ge \frac{1}{\mathcal {R}}\right\} \in \mathcal {F}_{n-1}, \qquad n \in \mathbb {N}. \end{aligned}$$

Since $\{\mathcal {F}_n\}_{n\ge 0}$ is regular, we can see that

$$\begin{aligned} \{v_k=n\}\subseteq F^k_n \qquad \text{ and } \qquad \mathbb {P}(F_n^k)\le \mathcal {R}\mathbb {P}(v_k=n). \end{aligned}$$

Define a new family of stopping times by

$$\begin{aligned} \tau _k(\omega ){:}{=}\inf \{n\in \mathbb {N}: \omega \in F_{n+1}^k\}. \end{aligned}$$

Then $v_k(\omega )=n$ means $\omega \in F_n^k$, which deduces $\tau _k(\omega )\le n-1$. In other words, $\tau _k< v_k$ on the set $\{v_k\ne \infty \}$. Clearly, the sequence of these stopping times $(\tau _k)_{k\in \mathbb {Z}}$ is non-decreasing. Moreover, by applying Proposition 3, we get that

$$\begin{aligned} \Vert \chi _{\{\tau _k<\infty \}}\Vert _{L_{p_-}}&\preceq \Vert \chi _{\{v_k<\infty \}}\Vert _{L_{p_-}} =\Vert \chi _{\{M(f)>2^k\}}\Vert _{L_{p_-}}\\&\le 2^{-k}\Vert M(f)\Vert _{L_{p_-}}\le 2^{-k}\Vert M(f)\Vert _{L_{\Phi ,q,b}}, \end{aligned}$$

where the first inequality was proved in Lemma 4 of [57]. Then

$$\begin{aligned} \Vert \chi _{\{\tau _k<\infty \}}\Vert _{L_{p_-}}\rightarrow 0 \qquad \hbox { as}\ k\rightarrow \infty , \end{aligned}$$

which means that

$$\begin{aligned} \lim \limits _{k\rightarrow \infty }\mathbb {P}(\tau _k=\infty )=1. \end{aligned}$$

Hence $\lim \limits _{k\rightarrow \infty }\tau _k=\infty$ a.e. and for $n\in \mathbb {N}$, $\lim \limits _{k\rightarrow \infty }f_n^{\tau _k}=f_n$ a.e. Let $(a^k)_{k\in \mathbb {Z}}$ and $(\mu _k)_{k\in \mathbb {Z}}$ be defined as in the proof of the Theorem 5, then the conclusions $f_n=\sum _{k\in \mathbb {Z}}\mu _k \mathbb {E}_{n}a^k$ and

$$\begin{aligned} \bigg (\sum _{k\in \mathbb {Z}}\gamma _b^q\big (\mathbb {P}(\tau _k<\infty )\big )\mu _k^q\bigg )^{1/q} \preceq \Vert f\Vert _{H_{\Phi ,q,b}^M} \end{aligned}$$

still hold.

We omit the proof of the converse part, since it is similar to the proof of Theorem 5. The proof is completed. $\square$

Similarly to the method of the proof above, it is easy to get the atomic decomposition of $H^S_{\Phi ,q ,b}$ as follows.

Theorem 7

Let $\Phi$ be an Orlicz function of lower type $p_-$ and upper type $p_+$ satisfying $0<p_- \le p_+<\infty$, $0<q\le \infty$ and let b be a slowly varying function. If $\{\mathcal {F}_n\}_{n\ge 0}$ is regular, then

$$\begin{aligned} H_{\Phi ,q,b}^S=H_{\Phi ,q,b}^{at,\infty ,2}, \end{aligned}$$

with equivalent (quasi)-norms.

Next, we consider the atomic decompositions of $\mathcal {Q}_{\Phi ,q,b}$ and $\mathcal {P}_{\Phi ,q,b}$.

Theorem 8

Let $\Phi$ be an Orlicz function of lower type $p_-$ and upper type $p_+$ satisfying $0<p_- \le p_+<\infty$, $0<q\le \infty$ and let b be a slowly varying function. Then

$$\begin{aligned} \mathcal {Q}_{\Phi ,q,b}=H_{\Phi ,q,b}^{at,\infty ,2}\quad \text {and} \quad \mathcal {P}_{\Phi ,q,b}=H_{\Phi ,q,b}^{at,\infty ,3}, \end{aligned}$$

with equivalent (quasi)-norms.

Proof

The proof is similar to that of Theorem 5, so we only sketch it. If $f=(f_n)_{n \ge 0} \in \mathcal {Q}_{\Phi ,q,b}$ (resp. $\mathcal {P}_{\Phi ,q,b}$), then the stopping times $\tau _k$ are defined by

$$\begin{aligned} \tau _k= \inf \{n \in \mathbb {N}: \lambda _n > 2^k\} , \end{aligned}$$

where $(\lambda _n)_{n\ge 0}$ is an adapted, non-decreasing sequence such that almost everywhere $|S_n(f)| \le \lambda _{n-1}$ (resp. $|f_n| \le \lambda _{n-1}$) and $\lambda _{\infty }\in L_{\Phi ,q,b}.$ Let $(a^k)_{k\in \mathbb {Z}}$ and $(\mu _k)_{k\in \mathbb {Z}}$ be defined as in the proof of Theorem 5. Then the conclusions

$$\begin{aligned} f_n=\sum _{k\in \mathbb {Z}}\mu _k \mathbb {E}_{n}a^k \quad \text{ a.e. } \end{aligned}$$

and

$$\begin{aligned} \bigg (\sum _{k\in \mathbb {Z}}\gamma _b^q\big (\mathbb {P}(\tau _k<\infty )\big )\mu _k^q\bigg )^{1/q} \preceq \Vert f\Vert _{\mathcal {Q}_{\Phi ,q,b}} \quad (\text{ resp. } \ \Vert f\Vert _{\mathcal {P}_{\Phi ,q,b}}) \end{aligned}$$

still hold.

To prove the converse part, let

$$\begin{aligned} \lambda _n=\sum _{k\in \mathbb {Z}}\mu _k \Vert S(a^k)\Vert _{L_\infty }\chi _{\{\tau _k \le n\}} ~~ \;\; \Big (\text {resp.} \;\lambda _n=\sum _{k\in \mathbb {Z}}\mu _k \Vert M(a^k)\Vert _{L_\infty }\chi _{\{\tau _k \le n\}}\Big ). \end{aligned}$$

Then $(\lambda _n)_{n\ge 0}$ is a non-negative, non-decreasing and adapted sequence. It follows from the first condition of the atom that

$$\begin{aligned} S_{n+1}(f) \le \lambda _n \qquad (\text{ resp. } |f_{n+1}|\le \lambda _n). \end{aligned}$$

(4.7)

For any given integer $k_0$, set

$$\begin{aligned} \lambda _\infty =\lambda _\infty ^{\top }+\lambda _\infty ^{\bot }, \end{aligned}$$

where

$$\begin{aligned} \lambda _\infty ^{\top }= & {} \sum _{k=-\infty }^{k_0-1}\mu _k \Vert S(a^k)\Vert _{L_\infty }\chi _{\{\tau _k< \infty \}}\\ \Big (\text {resp.} \;\lambda _\infty ^{\top }= & {} \sum _{k=-\infty }^{k_0-1}\mu _k \Vert M(a^k)\Vert _{L_\infty }\chi _{\{\tau _k < \infty \}}\Big ), \end{aligned}$$

and

$$\begin{aligned} \lambda _\infty ^{\bot }= & {} \sum ^{\infty }_{k=k_0}\mu _k \Vert S(a^k)\Vert _{L_\infty }\chi _{\{\tau _k< \infty \}}\\ \Big (\text {resp.} \;\lambda _\infty ^{\bot }= & {} \sum ^{\infty }_{k=k_0}\mu _k \Vert M(a^k)\Vert _{L_\infty }\chi _{\{\tau _k < \infty \}}\Big ). \end{aligned}$$

By replacing $s(T_1)$ and $s(T_2)$ in the proof of Theorem 5 with $\lambda _\infty ^{\top }$ and $\lambda _\infty ^{\bot }$, respectively, we get that $f\in \mathcal {Q}_{\Phi ,q,b}$ (resp. $f\in \mathcal {P}_{\Phi ,q,b}$) and

$$\begin{aligned} \Vert f\Vert _{\mathcal {Q}_{\Phi ,q,b}}\;\; (\text {resp.} \;\,\Vert f\Vert _{\mathcal {P}_{\Phi ,q,b}})\preceq \inf \bigg (\sum _{k\in \mathbb {Z}}\gamma _b^q\big (\mathbb {P}(\tau _k<\infty )\big )\mu _k^q\bigg )^{1/q}, \end{aligned}$$

where the infimum is taken over all decompositions of f of the form (4.1). This completes the proof of the theorem. $\square$

In particular, if we consider the case $\Phi (t)=t^p$, then the following result holds:

Corollary 1

Let $0<p<\infty$, $0<q \le \infty$ and b be a slowly varying function. We get the atomic decomposition of Lorentz-Karamata Hardy martingale spaces $H_{p,q,b}^s$, $\mathcal {Q}_{p,q,b}$ and $\mathcal {P}_{p,q,b}$, respectively.

Remark 8

We refer the reader to [38] for the atomic decomposition of Lorentz-Karamata Hardy martingale spaces in the case of $0<p<\infty$, $0<q\le \infty$ and b is a non-decreasing slowly varying function. It is noteworthy that the slowly varying function b does not need to satisfy the non-decreasing condition in Corollary 1. Hence our results improve the atomic decomposition of Lorentz-Karamata Hardy martingale spaces in [38].

Especially for $b\equiv 1$, we obtain the following conclusion:

Corollary 2

Let $\Phi$ be an Orlicz function of lower type $p_-$ and upper type $p_+$ satisfying $0<p_- \le p_+<\infty$ and $0<q\le \infty$. Then we obtain the atomic decomposition of Orlicz-Lorentz Hardy martingale spaces $H_{\Phi ,q}^s$, $\mathcal {Q}_{\Phi ,q}$ and $\mathcal {P}_{\Phi ,q}$, respectively.

Remark 9

We refer the reader to [24] for the atomic decomposition of Orlicz-Lorentz Hardy martingale spaces in the case of $0<q\le 1$ and $\Phi$ is an Orlicz function with $q_{\Phi ^{-1}} <1/q$. Notice that, there is no need for the condition $q_{\Phi ^{-1}} <1/q$ in Corollary 2. Hence we also extend and improve the atomic decomposition of Orlicz-Lorentz Hardy martingale spaces in [24].

5 Martingale inequalities

In this section, we show some martingale inequalities between different Orlicz-Lorentz-Karamata Hardy martingale spaces. The method we used here is to establish a sufficient condition for a $\sigma$-sublinear operator to be bounded from martingale Orlicz-Lorentz-Karamata Hardy spaces to Orlicz-Lorentz-Karamata spaces.

Let us recall that an operator $T:X\rightarrow Y$ is said to be $\sigma$-sublinear if for any constant c,

$$\begin{aligned} \bigg |T\bigg (\sum _{k=1}^{\infty }f_k\bigg )\bigg |\le \sum _{k=1}^{\infty }|T(f_k)|\ \ \text {and}\ \ |T(cf)|=|c||T(f)|, \end{aligned}$$

where X is a martingale space and Y is a measurable function space.

Lemma 10

Let $\Phi$ be an Orlicz function of lower type $p_-$ and upper type $p_+$ satisfying $0<p_- \le p_+<\infty$, $0<q\le \infty$ and let b be a non-decreasing slowly varying function. Suppose that $\max \{p_+,1\}<r<\infty$. If $T:H_r^s\rightarrow L_r$ is a bounded $\sigma$-sublinear operator and

$$\begin{aligned} \{|T(a)|>0\}\subseteq \{\nu <\infty \}, \end{aligned}$$

for every $(1,\Phi ,\infty )$-atom a associated with the stopping time $\nu$, then for $f\in H_{\Phi ,q,b}^s$,

$$\begin{aligned} \Vert T(f)\Vert _{L_{\Phi ,q,b}}\preceq \Vert f\Vert _{H_{\Phi ,q,b}^s}. \end{aligned}$$

Remark 10

Let $0<L<\infty$ and $\widetilde{\Phi }(t)=\Phi \big (t^{L}\big )$ for $t\in [0,\infty )$. If $\Phi$ is an Orlicz function of lower type $p_-$ and upper type $p_+$, then $\widetilde{\Phi }$ is an Orlicz function of lower type $Lp_-$ and upper type $Lp_+$. Under the conditions of Lemma 10, we have

$$\begin{aligned} \big \Vert |T(a)|^L\big \Vert _{L_\Phi } \preceq \Vert s(a)^L\Vert _{L_\Phi }, \end{aligned}$$

where $0<L<r/p_+$. Indeed, it was proved in [66] that T is bounded from $H_{\widetilde{\Phi }}^s$ to $L_{\widetilde{\Phi }}$. Hence we obtain that

$$\begin{aligned} \big \Vert |T(a)|^L\big \Vert _{L_\Phi }= \Vert T(a)\Vert _{L_{\widetilde{\Phi }}}^L \preceq \Vert s(a)\Vert _{L_{\widetilde{\Phi }}}^L=\big \Vert s(a)^L\big \Vert _{L_\Phi }. \end{aligned}$$

Moreover, since a is a $(1,\Phi ,\infty )$-atom, we get

$$\begin{aligned} \big \Vert |T(a)|^{L}\big \Vert _{L_\Phi }&\preceq \big \Vert s(a)^L\big \Vert _{L_\Phi }\le \big \Vert s(a)^L\big \Vert _{L_\infty }\Vert \chi _{\{\nu<\infty \}}\Vert _{L_\Phi }\\&= \Vert s(a)\Vert _{L_\infty }^L\Vert \chi _{\{\nu<\infty \}}\Vert _{L_\Phi }\le \Vert \chi _{\{\nu <\infty \}}\Vert _{L_\Phi }^{1-L}. \end{aligned}$$

Now we turn to prove Lemma 10:

Proof

Let $f\in H_{\Phi ,q,b}^s$. According to Theorem 5, there exist a sequence $(a^k)_{k\in \mathbb {Z}}$ of $(1,\Phi ,\infty )$-atoms associated with stopping times $(\tau _k)_{k\in \mathbb {Z}}$ and a sequence of real numbers

$$\begin{aligned} \mu _k=\frac{3\cdot 2^k}{\Phi ^{-1}\big (1/\mathbb {P}(\tau _k<\infty )\big )}\quad \quad \;k\in \mathbb {Z} \end{aligned}$$

such that

$$\begin{aligned} \sum _{k\in \mathbb {Z}} \mu _k a^k=f \quad \text{ a.e. } \end{aligned}$$

and

$$\begin{aligned} \Big (\sum _{k\in \mathbb {Z}}\gamma _b^q\big (\mathbb {P}(\tau _k<\infty )\big )|\mu _k|^q\Big )^{1/q} \preceq \Vert f\Vert _{H^s_{\Phi ,s,b}}. \end{aligned}$$

For an arbitrary integer $k_0$, set

$$\begin{aligned} D_1{:}{=}\sum _{k=-\infty }^{k_0-1}\mu _{k}T(a^{k}) \quad \text {and} \quad D_2{:}{=}\sum _{k=k_0}^{\infty }\mu _{k}T(a^{k}). \end{aligned}$$

Then by the $\sigma$-sublinearity of T, there is

$$\begin{aligned} T(f)\le \sum _{k\in \mathbb {Z}}\mu _{k}T(a^{k})=D_1+D_2. \end{aligned}$$

Let $0<\varepsilon <\min \{\varrho ,p_-,q\}$, $1<L<\min \{\frac{r}{p_+},\frac{1}{\varepsilon }\}$ and $0<\sigma <1-\frac{1}{L}$, where $\varrho$ is the same constant as in Remark 2. By using Hölder’s inequality, we conclude

$$\begin{aligned} D_1&= \sum _{k=-\infty }^{k_0-1} \mu _{k }T(a^{k })\\&\le \bigg (\sum _{k=-\infty }^{k_0-1}2^{k\sigma L'}\bigg )^\frac{1}{L'}\bigg [\sum _{k=-\infty }^{k_0-1}2^{-k\sigma L}\big (\mu _{k}T(a^{k})\big )^L\bigg ]^{\frac{1}{L}}\\&\approx 2^{k_0\sigma }\bigg [\sum _{k=-\infty }^{k_0-1}2^{-k\sigma L}\big (\mu _{k}T(a^{k})\big )^L\bigg ]^{\frac{1}{L}}, \end{aligned}$$

where $\frac{1}{L'}+\frac{1}{L}=1$. According to Remark 2 and Remark 10, we have

$$\begin{aligned} \Vert \chi _{\{D_1>2^{k_0}\}}\Vert _{L_\Phi }&\le \ 2^{-k_0L}\Vert D_1^L\Vert _{L_\Phi }\preceq 2^{-k_0L}\Bigg \Vert 2^{k_0\sigma L}\sum _{k=-\infty }^{k_0-1}2^{-kL \sigma }\big (\mu _{k}T(a^{k})\big )^L\Bigg \Vert _{L_\Phi }\\&= 2^{k_0L(\sigma -1)}\Bigg \Vert \sum _{k=-\infty }^{k_0-1}2^{-kL\sigma }\big ( \mu _kT(a^{k})\big )^L\Bigg \Vert _{L_\Phi }^{\varepsilon \cdot \frac{1}{\varepsilon }}\\&\le 2^{k_0L(\sigma -1)}\Bigg (\sum _{k=-\infty }^{k_0-1}2^{-kL\sigma \varepsilon }\mu _k^{L\varepsilon } \big \Vert \big ( T(a^{k})\big )^{L}\big \Vert ^\varepsilon _{L_\Phi } \Bigg )^{\frac{1}{\varepsilon }}\\&\preceq 2^{k_0L(\sigma -1)}\Bigg (\sum _{k=-\infty }^{k_0-1}2^{-kL\sigma \varepsilon }\mu _k^{L\varepsilon } \Vert \chi _{\{\tau _k<\infty \}}\Vert ^{(1-L)\varepsilon }_{L_\Phi } \Bigg )^{\frac{1}{\varepsilon }} . \end{aligned}$$

Hence, we can choose $\delta >0$ such that $1<\delta <L(1-\sigma )$. By Hölder’s inequality, we have

$$\begin{aligned} \Vert \chi _{\{D_1>2^{k_0}\}}\Vert _{L_\Phi }&\preceq 2^{k_0L(\sigma -1)}\Bigg (\sum _{k=-\infty }^{k_0-1}2^{-kL\sigma \varepsilon }\mu _k^{L\varepsilon } \Vert \chi _{\{\tau _k<\infty \}}\Vert ^{(1-L)\varepsilon }_{L_\Phi } \Bigg )^{\frac{1}{\varepsilon }} \\&\preceq 2^{k_0L(\sigma -1)}\Bigg (\sum _{k=-\infty }^{k_0-1}2^{kL\varepsilon (1-\sigma )} \Vert \chi _{\{\tau _k<\infty \}}\Vert ^{\varepsilon }_{L_\Phi } \Bigg )^{\frac{1}{\varepsilon }}\\&= 2^{k_0L(\sigma -1)}\Bigg (\sum _{k=-\infty }^{k_0-1}2^{k\varepsilon (L(1-\sigma )-\delta )} 2^{k\varepsilon \delta }\big \Vert \chi _{\{\tau _k<\infty \}}\big \Vert _{L_\Phi }^{\varepsilon }\Bigg )^{\frac{1}{\varepsilon }}\\&\le 2^{k_0L(\sigma -1)}\Bigg (\sum _{k=-\infty }^{k_0-1}2^{\frac{qk\varepsilon (L(1-\sigma )-\delta )}{q-\varepsilon }} \Bigg )^{\frac{q-\varepsilon }{q\varepsilon }}\Bigg (\sum _{k=-\infty }^{k_0-1}2^{kq\delta } \big \Vert \chi _{\{\tau _k<\infty \}}\big \Vert _{L_\Phi }^{q}\Bigg )^{\frac{1}{q}}\\&\preceq 2^{-k_0\delta }\Bigg (\sum _{k=-\infty }^{k_0-1}2^{kq\delta }\big \Vert \chi _{\{\tau _k<\infty \}}\big \Vert _{L_\Phi }^{q}\Bigg )^{\frac{1}{q}}. \end{aligned}$$

Let $b_1(t)=b\big (\Phi (t)\big )$ for $t\in [1,\infty )$. From Proposition 1, $b_1$ is a slowly varying function. Clearly, $b_1$ is non-decreasing. Hence we get

$$\begin{aligned}&\sum _{k_0\in \mathbb {Z}}2^{k_0q}\Vert \chi _{\{D_1>2^{k_0}\}}\Vert _{L_\Phi }^{q}\gamma _b^{q}\big (\mathbb {P}(D_1>2^{k_0})\big )\\&\quad = \sum _{k_0\in \mathbb {Z}}2^{k_0q}\Vert \chi _{\{D_1>2^{k_0}\}}\Vert _{L_\Phi }^{q}\gamma _{b_1}^{q}\big (\Vert \chi _{\{D_1>2^{k_0}\}}\Vert _{L_\Phi }\big )\\&\quad \preceq \sum _{k_0\in \mathbb {Z}}2^{k_0q}2^{-k_0\delta q}\sum _{m=-\infty }^{k_0-1}2^{mq\delta }\big \Vert \chi _{\{\tau _m<\infty \}}\big \Vert _{L_\Phi }^{q} \\&\qquad \gamma _{b_1}^{q}\Bigg [2^{-k_0\delta } \Bigg (\sum _{l=-\infty }^{k_0-1}2^{lq\delta }\big \Vert \chi _{\{\tau _l<\infty \}}\big \Vert _{L_\Phi }^{q}\Bigg )^{\frac{1}{q}}\Bigg ]\\&\quad \preceq \sum _{k_0\in \mathbb {Z}}2^{k_0q(1-\delta )}\sum _{m=-\infty }^{k_0-1}2^{mq\delta }\big \Vert \chi _{\{\tau _m<\infty \}}\big \Vert _{L_\Phi }^{q}\gamma _{b_1}^{q}\Big (2^{(m-k_0)\delta } \Vert \chi _{\{\tau _m<\infty \}}\Vert _{L_\Phi }\Big ). \end{aligned}$$

Set $0<z<\frac{\delta -1}{\delta }$. Since $t^z\gamma _{b_1}(t)$ is equivalent to a non-decreasing function, then

$$\begin{aligned}&\gamma _{b_1}^{q}\Big (2^{(m-k_0)\delta } \Vert \chi _{\{\tau _m<\infty \}}\Vert _{L_\Phi }\Big )\\&\quad = \Big (2^{(m-k_0)\delta } \Vert \chi _{\{\tau _m<\infty \}}\Vert _{L_\Phi }\Big )^{-z}\Big (2^{(m-k_0)\delta } \Vert \chi _{\{\tau _m<\infty \}}\Vert _{L_\Phi }\Big )^{z} \\&\qquad \gamma _{b_1}^{q}\Big (2^{(m-k_0)\delta } \Vert \chi _{\{\tau _m<\infty \}}\Vert _{L_\Phi }\Big )\\&\quad \preceq \Big (2^{(m-k_0)\delta } \Vert \chi _{\{\tau _m<\infty \}}\Vert _{L_\Phi }\Big )^{-z}\big ( \Vert \chi _{\{\tau _m<\infty \}}\Vert _{L_\Phi }\big )^{z}\gamma _{b_1}^{q}\big ( \Vert \chi _{\{\tau _m<\infty \}}\Vert _{L_\Phi }\big )\\&\quad = 2^{-(m-k_0)\delta z}\gamma _{b_1}^{q}\big ( \Vert \chi _{\{\tau _m<\infty \}}\Vert _{L_\Phi }\big ) , \end{aligned}$$

where $m<k_0$. Combining this with Remark 4 and the Abel transformation, we have

$$\begin{aligned}&\big \Vert D_1\big \Vert _{L_{\Phi ,q,b}}^q \\&\quad \approx \sum _{k_0\in \mathbb {Z}}2^{k_0q}\Vert \chi _{\{D_1>2^{k_0}\}}\Vert _{L_\Phi }^{q}\gamma _b^{q}\big (\mathbb {P}(D_1>2^{k_0})\big )\\&\quad \preceq \sum _{k_0\in \mathbb {Z}}2^{k_0q(1-\delta )}\sum _{k=-\infty }^{k_0-1}2^{k\delta q}\big \Vert \chi _{\{\tau _k<\infty \}}\big \Vert _{L_\Phi }^{q}\gamma _{b_1}^{q}\Big (2^{(k-k_0)\delta } \Vert \chi _{\{\tau _k<\infty \}}\Vert _{L_\Phi }\Big )\\&\quad \preceq \sum _{k_0\in \mathbb {Z}}2^{k_0q(1-\delta )}\sum _{k=-\infty }^{k_0-1}2^{k\delta q}\big \Vert \chi _{\{\tau _k<\infty \}}\big \Vert _{L_\Phi }^{q}2^{-(k-k_0)\delta zq}\gamma _{b_1}^q\Big (\Vert \chi _{\{\tau _k<\infty \}}\Vert _{L_\Phi }\Big )\\&\quad =\sum _{k_0\in \mathbb {Z}}2^{k_0q(1-\delta +\delta z)}\sum _{k=-\infty }^{k_0-1}2^{k\delta q(1-z)}\big \Vert \chi _{\{\tau _k<\infty \}}\big \Vert _{L_\Phi }^{q} \gamma _{b_1}^q\Big (\Vert \chi _{\{\tau _k<\infty \}}\Vert _{L_\Phi }\Big ) \end{aligned}$$

and further

$$\begin{aligned}&\big \Vert D_1\big \Vert _{L_{\Phi ,q,b}}^q \nonumber \\&\quad \preceq \sum _{k\in \mathbb {Z}}2^{kq\delta (1-z)}\big \Vert \chi _{\{\tau _k<\infty \}}\big \Vert _{L_\Phi }^{q}\gamma _{b_1}^q\big (\Vert \chi _{\{\tau _k<\infty \}}\Vert _{L_\Phi }\big ) \sum _{k_0=k+1}^{\infty }2^{k_0q(1-\delta +\delta z)} \nonumber \\&\quad \preceq \sum _{k\in \mathbb {Z}}2^{kq}\big \Vert \chi _{\{\tau _k<\infty \}}\big \Vert _{L_\Phi }^{q}\gamma _{b_1}^q\big (\Vert \chi _{\{\tau _k<\infty \}}\Vert _{L_\Phi }\big ) \nonumber \\&= \sum _{k\in \mathbb {Z}}2^{kq}\big \Vert \chi _{\{\tau _k<\infty \}}\big \Vert _{L_\Phi }^{q}\gamma _b^q\big (\mathbb {P}(\tau _k<\infty )\big ) . \end{aligned}$$

(5.1)

Now, we estimate $D_2$. Since $\{|T(a^k)|>0\}\subseteq \{\tau _k<\infty \}$ for each $k\in \mathbb {Z}$, we have

$$\begin{aligned} \{D_2>2^{k_0}\}\subseteq \{D_2>0\}\subseteq \bigcup _{k=k_0}^{\infty }\{ \tau _k<\infty \}. \end{aligned}$$

(5.2)

Moreover, by using Theorem 2, there exists $0<s<\min \{1,p_-, q\}$ such that

$$\begin{aligned} \big \Vert \chi _{\{D_2>2^{k_0}\}}\big \Vert _{L_{\Phi ,q,b}}^s \le \bigg \Vert \sum _{k=k_0}^\infty \chi _{\{\tau _k<\infty \}}\bigg \Vert _{L_{\Phi ,q,b}}^s \preceq \sum _{k=k_0}^\infty \big \Vert \chi _{\{\tau _k<\infty \}}\big \Vert _{L_{\Phi ,q,b}}^s. \end{aligned}$$

(5.3)

Let $0<\eta <1$. For $1=\frac{s}{q}+\frac{q-s}{q}$, it follows from Hölder’s inequality and (5.3) that

$$\begin{aligned} \big \Vert \chi _{\{D_2>2^{k_0}\}}\big \Vert _{L_{\Phi ,q,b}}\preceq & {} \bigg (\sum _{k=k_0}^\infty \big \Vert \chi _{\{\tau _k<\infty \}}\big \Vert _{L_{\Phi ,q,b}}^s\bigg )^{1/s}\\= & {} \bigg (\sum _{k=k_0}^\infty 2^{-k\eta s}2^{k\eta s}\big \Vert \chi _{\{\tau _k<\infty \}}\big \Vert ^s_{L_{\Phi ,q,b}}\bigg )^{1/s}\\\le & {} \bigg (\sum _{k=k_0}^\infty 2^{-k\eta s\frac{q}{q-s}}\bigg )^{\frac{q-s}{q s}} \bigg (\sum _{k=k_0}^\infty 2^{k\eta q}\big \Vert \chi _{\{\tau _k<\infty \}}\big \Vert ^q_{L_{\Phi ,q,b}}\bigg )^{1/q}\\\preceq & {} 2^{-k_0\eta }\bigg (\sum _{k=k_0}^\infty 2^{k\eta q}\big \Vert \chi _{\{\tau _k<\infty \}}\big \Vert ^q_{L_{\Phi ,q,b}}\bigg )^{1/q}. \end{aligned}$$

It follows from Remark 4 and Abel’s transformation that

$$\begin{aligned} \big \Vert D_2\big \Vert _{L_{\Phi ,q,b}}^q&\approx \sum _{k_0\in \mathbb {Z}}2^{k_0q}\Vert \chi _{\{D_2>2^{k_0}\}}\Vert _{L_\Phi }^{q}\gamma _b^{q}\big (\mathbb {P}(D_2>2^{k_0})\big )\nonumber \\&= \sum _{k_0\in \mathbb {Z}}2^{k_0q}\Vert \chi _{\{D_2>2^{k_0}\}}\Vert _{L_{\Phi ,q,b}}^q \nonumber \\&\preceq \sum _{k_0\in \mathbb {Z}}2^{k_0(1-\eta )q}\sum _{k=k_0}^\infty 2^{k\eta q}\big \Vert \chi _{\{\tau _k<\infty \}}\big \Vert _{L_{\Phi ,q,b}}^q\nonumber \\&= \sum _{k\in \mathbb {Z}} 2^{k\eta q}\big \Vert \chi _{\{\tau _k<\infty \}}\big \Vert _{L_{\Phi ,q,b}}^q \sum _{k_0=-\infty }^k2^{k_0(1-\eta )q}\nonumber \\&\preceq \sum _{k\in \mathbb {Z}} 2^{k \eta q}\big \Vert \chi _{\{\tau _k<\infty \}}\big \Vert _{L_{\Phi ,q,b}}^q 2^{k(1-\eta )q}\nonumber \\&= \sum _{k\in \mathbb {Z}} 2^{k q}\big \Vert \chi _{\{\tau _k<\infty \}}\big \Vert _{L_{\Phi ,q,b}}^q\nonumber \\&= \sum _{k\in \mathbb {Z}}2^{kq}\big \Vert \chi _{\{\tau _k<\infty \}}\big \Vert _{L_\Phi }^{q}\gamma _b^q\big (\mathbb {P}(\tau _k<\infty )\big ). \end{aligned}$$

(5.4)

Thus, combining with (5.1) and (5.4), we have

$$\begin{aligned} \Vert T(f)\Vert ^q_{L_{\Phi ,q,b}}&\le \Vert D_1+D_2\Vert _{L_{\Phi ,q,b}}^q\\&\preceq \Vert D_1\Vert ^q_{L_{\Phi ,q,b}}+\Vert D_2\Vert ^q_{L_{\Phi ,q,b}}\\&\preceq \sum _{k\in \mathbb {Z}}2^{kq}\big \Vert \chi _{\{\tau _k<\infty \}}\big \Vert _{L_\Phi }^{q}\gamma _b^q\big (\mathbb {P}(\tau _k<\infty )\big )\\&\approx \sum _{k\in \mathbb {Z}}\gamma _b^q\big (\mathbb {P}(\tau _k<\infty )\big )\mu _k^q \preceq \Vert f\Vert _{H_{\Phi ,q,b}^s}^q. \end{aligned}$$

The proof is completed now. $\square$

Similar to the method of the proof above, we can present the following results.

Lemma 11

Let $\Phi$ be an Orlicz function of lower type $p_-$ and upper type $p_+$ satisfying $0<p_- \le p_+<\infty$, $0<q\le \infty$ and let b be a non-decreasing slowly varying function. Suppose that $\{\mathcal {F}_n\}_{n\ge 0}$ is regular and $\max \{p_+,1\}<r<\infty$. If $T:H_r^S\rightarrow L_r \; (\text {resp.}\ H_r^M\rightarrow L_r)$ is a bounded $\sigma$-sublinear operator and

$$\begin{aligned} \{|T(a)|>0\}\subseteq \{\nu <\infty \}, \end{aligned}$$

for every $(2,\Phi ,\infty )$-atom (resp. $(3,\Phi ,\infty )$-atom) a associated with the stopping time $\nu$, then for $f\in H_{\Phi ,q,b}^S$ $(\text {resp.}\; f\in H_{\Phi ,q,b}^M)$,

$$\begin{aligned} \Vert T(f)\Vert _{L_{\Phi ,q,b}}\preceq \Vert f\Vert _{H_{\Phi ,q,b}^S}\ \ \big (\text {resp.}\ \Vert T(f)\Vert _{L_{\Phi ,q,b}}\preceq \Vert f\Vert _{H_{\Phi ,q,b}^M}\big ). \end{aligned}$$

Lemma 12

Let $\Phi$ be an Orlicz function of lower type $p_-$ and upper type $p_+$ satisfying $0<p_- \le p_+<\infty$, $0<q\le \infty$ and let b be a non-decreasing slowly varying function. Suppose that $\max \{p_+,1\}<r<\infty$. If $T:H_r^S\rightarrow L_r\;(\text {resp.}\ H_r^M\rightarrow L_r)$ is a bounded $\sigma$-sublinear operator and

$$\begin{aligned} \{|T(a)|>0\}\subseteq \{\nu <\infty \}, \end{aligned}$$

for every $(2,\Phi ,\infty )$-atom (resp. $(3,\Phi ,\infty )$-atom) a associated with the stopping time $\nu$, then for $f\in \mathcal {Q}_{\Phi ,q,b}$ $(\text {resp.}\ f\in \mathcal {P}_{\Phi ,q,b})$,

$$\begin{aligned} \Vert T(f)\Vert _{L_{\Phi ,q,b}}\preceq \Vert f\Vert _{\mathcal {Q}_{\Phi ,q,b}}\ \ \big (\text {resp.}\ \Vert T(f)\Vert _{L_{\Phi ,q,b}}\preceq \Vert f\Vert _{\mathcal {P}_{\Phi ,q,b}}\big ). \end{aligned}$$

To get the martingale inequalities for martingale Orlicz-Lorentz-Karamata Hardy spaces, we need the following lemma.

Lemma 13

([19, 42, 60]) Let f be a martingale. Then

$$\begin{aligned} \Vert M(f)\Vert _{L_2}\le & {} 2\Vert S(f)\Vert _{L_2}=2\Vert s(f)\Vert _{L_2}\le 2\Vert M(f)\Vert _{L_2};\\ \Vert s(f)\Vert _{L_r}\le & {} \sqrt{\frac{r}{2}}\Vert M(f)\Vert _{L_r},\ \ \ \ (r\ge 2);\\ \Vert s(f)\Vert _{L_r}\le & {} \sqrt{\frac{r}{2}}\Vert S(f)\Vert _{L_r},\ \ \ \ (r\ge 2). \end{aligned}$$

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

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

(5.5)

Theorem 9

Let $\Phi$ be an Orlicz function of lower type $p_-$ and upper type $p_+$ satisfying $0<p_- \le p_+<\infty$, $0<q\le \infty$ and let b be a non-decreasing slowly varying function. Then

$$\begin{aligned} \Vert f\Vert _{H_{\Phi ,q,b}^M}&\preceq \Vert f\Vert _{H_{\Phi ,q,b}^s},\ \ \Vert f\Vert _{H_{\Phi ,q,b}^S}\preceq \Vert f\Vert _{H_{\Phi ,q,b}^s}\ \ \ 0<p_-\le p_+<2; \end{aligned}$$

(5.6)

$$\begin{aligned} \Vert f\Vert _{H_{\Phi ,q,b}^M}&\preceq \Vert f\Vert _{\mathcal {P}_{\Phi ,q,b}},\ \ \Vert f\Vert _{H_{\Phi ,q,b}^S}\preceq \Vert f\Vert _{\mathcal {Q}_{\Phi ,q,b}}; \end{aligned}$$

(5.7)

$$\begin{aligned} \Vert f\Vert _{H_{\Phi ,q,b}^S}&\preceq \Vert f\Vert _{\mathcal {P}_{\Phi ,q,b}},\ \ \Vert f\Vert _{H_{\Phi ,q,b}^M}\preceq \Vert f\Vert _{\mathcal {Q}_{\Phi ,q,b}}; \end{aligned}$$

(5.8)

$$\begin{aligned} \Vert f\Vert _{H_{\Phi ,q,b}^s}&\preceq \Vert f\Vert _{\mathcal {P}_{\Phi ,q,b}},\ \ \Vert f\Vert _{H_{\Phi ,q,b}^s}\preceq \Vert f\Vert _{\mathcal {Q}_{\Phi ,q,b}}; \end{aligned}$$

(5.9)

$$\begin{aligned} \Vert f\Vert _{\mathcal {P}_{\Phi ,q,b}}&\preceq \Vert f\Vert _{\mathcal {Q}_{\Phi ,q,b}}\preceq \Vert f\Vert _{\mathcal {P}_{\Phi ,q,b}}. \end{aligned}$$

(5.10)

Moreover, if $\{\mathcal {F}_n\}_{n\ge 0}$ is regular, then

$$\begin{aligned} H_{\Phi ,q,b}^S=\mathcal {Q}_{\Phi ,q,b}=\mathcal {P}_{\Phi ,q,b}=H_{\Phi ,q,b}^M=H_{\Phi ,q,b}^s \end{aligned}$$

with equivalent norms.

Proof

It is clear that the operators s, S, and M are all $\sigma$-sublinear and

$$\begin{aligned} \{s(a_1)>0\}\subseteq \{\nu _1<\infty \}, \qquad \{S(a_2)>0\}\subseteq \{\nu _2<\infty \} \end{aligned}$$

and

$$\begin{aligned} \{M(a_3)>0\}\subseteq \{\nu _3<\infty \}, \end{aligned}$$

where $a_i\;(i=1,2,3)$ is and $(i,\Phi ,\infty )$-atom associated with the stopping time $\nu _i$, respectively.

Lemma 13 gives that the operators $M:H_2^s\rightarrow L_2$ and $S:H_2^s\rightarrow L_2$ are bounded. Hence, by using Lemma 10, we get that the inequalities in (5.6) hold if $0<p_-\le p_+<2$.

The inequalities in (5.7) can be directly get from the definitions of $\mathcal {P}_{\Phi ,q,b}$ and $\mathcal {Q}_{\Phi ,q,b}$.

According to Doob’s maximal inequality and Brukholder-Davis-Gundy’s inequality (see [60], Theorem 2.12), namely,

$$\begin{aligned} c_r\Vert f\Vert _{H_r^S}\le \Vert f\Vert _{H_r^M}\le C_r\Vert f\Vert _{H_r^S},\ \ \quad \quad (1\le r<\infty ), \end{aligned}$$

we obtain the operators $M:H_r^S\rightarrow L_r$ and $S:H_r^M\rightarrow L_r$ are bounded. Hence according to Lemma 12, the inequalities in (5.8) hold.

Let $\max \{p_+,2\}<r<\infty$. It follows from Lemma 13 that $s:H_r^M\rightarrow L_r$ and $s:H_r^S\rightarrow L_r$ are bounded. Then by Lemma 12, we obtain the inequalities in (5.9).

To prove (5.10), we take $f=(f_n)_{n\ge 0}\in \mathcal {Q}_{\Phi ,q,b}$. This means there exists $(\lambda _n^{(1)})_{n\ge 0}\in \Lambda [\mathcal {Q}_{\Phi ,q,b}](f)$ such that $S_n(f)\le \lambda _{n-1}^{(1)}$ with $\lambda _\infty ^{(1)}\in L_{\Phi ,q,b}$. Since

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

and $(\lambda _{n}^{(1)}+M_{n}(f))_{n\ge 0}\in \Lambda [\mathcal {Q}_{\Phi ,q,b}](f)$, by using the second inequality of (5.8), we get

$$\begin{aligned} \Vert f\Vert _{\mathcal {P}_{\Phi ,q,b}}\le \Vert \lambda _\infty ^{(1)}+M(f)\Vert _{L_{\Phi ,q,b}}\preceq \Vert M(f)\Vert _{H_{\Phi ,q,b}} +\Vert \lambda _\infty ^{(1)}\Vert _{L_{\Phi ,q,b}}\preceq \Vert f\Vert _{\mathcal {Q}_{\Phi ,q,b}}. \end{aligned}$$

If $f=(f_n)_{n\ge 0}\in \mathcal {P}_{\Phi ,q,b}$, then there exists $(\lambda _n^{(2)})_{n\ge 0}\in \Lambda [\mathcal {P}_{\Phi ,q,b}](f)$ such that $|f_n|\le \lambda _{n-1}^{(2)}$ with $\lambda _\infty ^{(2)}\in L_{\Phi ,q,b}$. Since

$$\begin{aligned} S_n(f)\le S_{n-1}(f)+|f_n-f_{n-1}|\le S_{n-1}(f)+2M_n(f)\le S_{n-1}(f)+2\lambda _n^{(2)} \end{aligned}$$

and $(\lambda _{n}^{(2)}+S_{n}(f))_{n\ge 0}\in \Lambda [\mathcal {P}_{\Phi ,q,b}](f)$, by using the first inequality of (5.8), we have

$$\begin{aligned} \Vert f\Vert _{\mathcal {Q}_{\Phi ,q,b}}\le \Vert \lambda _\infty ^{(2)}+S(f)\Vert _{L_{\Phi ,q,b}}\preceq \Vert S(f)\Vert _{H_{\Phi ,q,b}} +\Vert \lambda _\infty ^{(2)}\Vert _{L_{\Phi ,q,b}}\preceq \Vert f\Vert _{\mathcal {P}_{\Phi ,q,b}}. \end{aligned}$$

Hence, inequality (5.10) holds and it yields that $\mathcal {P}_{\Phi ,q,b}=\mathcal {Q}_{\Phi ,q,b}$.

Let $\{\mathcal {F}_n\}_{n\ge 0}$ be regular and $\max \{p_+,1\}<r<\infty$. Combining (5.5) with Lemma 10 and Lemma 11, we have

$$\begin{aligned} H_{\Phi ,q,b}^s=H_{\Phi ,q,b}^M=H_{\Phi ,q,b}^S. \end{aligned}$$

It follows from Theorem 6 and Theorem 8 that $H_{\Phi ,q,b}^M=\mathcal {P}_{\Phi ,q,b}$. This and (5.10) imply

$$\begin{aligned} H_{\Phi ,q,b}^S=\mathcal {Q}_{\Phi ,q,b}=\mathcal {P}_{\Phi ,q,b}=H_{\Phi ,q,b}^M=H_{\Phi ,q,b}^s. \end{aligned}$$

The proof of this theorem is completed. $\square$

The following corollary follows from Theorems 4 and 9.

Corollary 3

Let $\Phi$ be an Orlicz function of lower type $p_-$ and upper type $p_+$ satisfying $1<p_- \le p_+<\infty$, $0<q\le \infty$ and let b be a non-decreasing slowly varying function. Then $H_{\Phi ,q,b}^M$ is equivalent $L_{\Phi ,q,b}$. If $\{\mathcal {F}_n\}_{n\ge 0}$ is regular, then the spaces $H_{\Phi ,q,b}^S$, $\mathcal {Q}_{\Phi ,q,b}$, $\mathcal {P}_{\Phi ,q,b}$, $H_{\Phi ,q,b}^M$, $H_{\Phi ,q,b}^s$ and $L_{\Phi ,q,b}$ are all equivalent.

Taking $b\equiv 1$ in Theorem 9, we obtain the fundamental martingale inequalities on Orlicz-Lorentz Hardy spaces.

Corollary 4

Let $\Phi$ be an Orlicz function of lower type $p_-$ and upper type $p_+$ satisfying $0<p_- \le p_+<\infty$ and $0<q\le \infty$. Then

$$\begin{aligned} \Vert f\Vert _{H_{\Phi ,q}^M}&\preceq \Vert f\Vert _{H_{\Phi ,q}^s},\ \ \Vert f\Vert _{H_{\Phi ,q}^S}\preceq \Vert f\Vert _{H_{\Phi ,q}^s}\ \ \ 0<p_-\le p_+<2;\\ \Vert f\Vert _{H_{\Phi ,q}^M}&\preceq \Vert f\Vert _{\mathcal {P}_{\Phi ,q}},\ \ \Vert f\Vert _{H_{\Phi ,q}^S}\preceq \Vert f\Vert _{\mathcal {Q}_{\Phi ,q}};\\ \Vert f\Vert _{H_{\Phi ,q}^S}&\preceq \Vert f\Vert _{\mathcal {P}_{\Phi ,q}},\ \ \Vert f\Vert _{H_{\Phi ,q}^M}\preceq \Vert f\Vert _{\mathcal {Q}_{\Phi ,q}};\\ \Vert f\Vert _{H_{\Phi ,q}^s}&\preceq \Vert f\Vert _{\mathcal {P}_{\Phi ,q}},\ \ \Vert f\Vert _{H_{\Phi ,q}^s}\preceq \Vert f\Vert _{\mathcal {Q}_{\Phi ,q}};\\ \Vert f\Vert _{\mathcal {P}_{\Phi ,q}}&\preceq \Vert f\Vert _{\mathcal {Q}_{\Phi ,q}}\preceq \Vert f\Vert _{\mathcal {P}_{\Phi ,q}}. \end{aligned}$$

Moreover, if $\{\mathcal {F}_n\}_{n\ge 0}$ is regular, then the spaces $H_{\Phi ,q}^S$, $\mathcal {Q}_{\Phi ,q}$, $\mathcal {P}_{\Phi ,q}$, $H_{\Phi ,q}^M$ and $H_{\Phi ,q}^s$ are all equivalent. If $1<p_- \le p_+<\infty$, then these spaces are also equivalent to $L_{\Phi ,q}$.

Note that this result was proved for $\Phi (t)=t^p$, i.e., for Hardy and Hardy-Lorentz spaces, in [59, 60].

6 Duality theorems

The goal of this section is to establish the dual spaces of Orlicz-Lorentz-Karamata Hardy martingale spaces $H^s_{\Phi ,q,b}$. The dual spaces of $H_p^s$ and $H_{p,q}^s$ are known. More exactly, the dual spaces of $H_p^s$ ($0<p\le 1$) are the BMO spaces when $p=1$ and the Lipschitz spaces $\Lambda _2(1/p-1)$ where $0<p<1$. Moreover, the dual spaces of Lorentz Hardy martingale spaces $H_{p,q}^s$ ($0<p\le 1$, $1<q<\infty$) are the generalized BMO martingale spaces $BMO_{2,q}(1/p-1)$ (see [37, 59, 60]). In this section, we generalize these results for the $H^s_{\Phi ,q,b}$ spaces when $\Phi$ is an Orlicz function and $0<q<\infty$.

To state the duality, we need some more notations in this section. Let $1\le p < \infty$, and let $L_p^0$ be the set of all $f \in L_p$ such that $\mathbb {E}_0f = 0$. For any $f \in L_p^0$, let $f_n = \mathbb {E}_n f$. Then, it is known that $(f_n )_{n\ge 0}$ is an $L_p$-bounded martingale in $\mathcal {M}$ and converges to f in $L_p$ (see [49]). Now we define the generalized BMO martingale spaces as follows.

Definition 6

Let $\Phi$ be an Orlicz function, $1\le p < \infty$ and let b be a slowly varying function. The generalized BMO martingale space $BMO_{p,\Phi ,b}$ is defined by

$$\begin{aligned} BMO_{p,\Phi ,b} =\Big \{ f \in L_p^0: \Vert f\Vert _{BMO_{p,\Phi ,b}}< \infty \Big \}, \end{aligned}$$

where

$$\begin{aligned} \Vert f\Vert _{BMO_{p,\Phi ,b}}=\sup _{ \tau \in \mathcal {T} }\frac{\Phi ^{-1}\big (1/\mathbb {P}(\tau<\infty )\big ) {\mathbb {P}(\tau<\infty )^{1-1/p}}}{\gamma _b\big (\mathbb {P}(\tau <\infty )\big )}\big \Vert f-f^\tau \big \Vert _{L_p} . \end{aligned}$$

Note that if $\Phi (t) = t^{\frac{1}{\lambda +1}}$, $\lambda \in (-1,\infty )$, we have the martingale Campanato spaces $\mathcal {L}_{p,\lambda }$ introduced in [46]. Moreover, if $\Phi (t)=t^p$ for $0<p \le 1$, the spaces $\mathcal {L}_{2,\Phi }$ coincide with $BMO_{2,\alpha }$, where $\alpha =1/p-1$. See [37, 59, 60] for the notation $BMO_{2,\alpha }$. In particular, take $\Phi (t)=t$, the $\mathcal {L}_{2,\Phi }$ can be reduced to the Banach space BMO (Bounded Mean Oscillation, see [27, 59, 60]). It is well known that the dual space of ${H^s_1}$ is BMO in analogy with the result of Fefferman [16] for the classical case. We refer to [19] and [42, 59, 60] for the facts above.

Now we can identity the dual space of $H^s_{\Phi ,q,b}$ ($0<q\le 1$):

Theorem 10

If $\Phi$ is an Orlicz function of lower type $p_-$ and upper type $p_+$ satisfying $0<p_- \le p_+<2$, $0<q\le 1$ and b is a slowly varying function, then the dual space of $H_{\Phi ,q,b}^s$ is $BMO_{2,\Phi ,b}$. That is

$$\begin{aligned} \big ( H_{\Phi ,q,b}^s\big )^*=BMO_{2,\Phi ,b}. \end{aligned}$$

To prove the theorem above, we need a classical result from [12], Theorem 3.4.48] as follows.

Lemma 14

Let $p_1,p_2,q_1,q_2 \in (0,\infty )$ with $p_2 < p_1$ and let $b_1,b_2$ be slowly varying functions. Then

$$\begin{aligned} L_{p_1,q_1,b_1}\subseteq L_{p_2,q_2,b_2} \end{aligned}$$

with respect to (quasi)-norms.

Now we turn to prove Theorem 10:

Proof

Since $\Phi$ is an Orlicz function of lower type $p_-$ and upper type $p_+$ satisfying $0<p_- \le p_+<2$, by using Lemma 14 and Proposition 3, we obtain that

$$\begin{aligned} \Vert f\Vert _{H_{\Phi ,q,b}^s} =\Vert s(f)\Vert _{L_{\Phi ,q,b}} \preceq \Vert s(f)\Vert _{L_{p_+,q,b}} \preceq \Vert s(f)\Vert _{L_{2,2,1}}=\Vert s(f)\Vert _{L_2}. \end{aligned}$$

In addition, Remark 7 assures that $L_2$ is dense in $H_{\Phi ,q,b}^s$. We shall prove that

$$\begin{aligned} \ell (f) = \mathbb {E}[fg] , \quad \quad \; \forall \; f\in L_2 \end{aligned}$$

(6.1)

is a continuous linear functional on $H_{\Phi ,q,b}^s$, where $g\in BMO_{2,\Phi ,b}\subseteq L_2$ is arbitrary. We take the same stopping time $\tau _k$, atoms $a^k$ and real numbers $\mu _k$ ($k\in \mathbb {Z}$) as we did in Theorem 5. Modifying slightly the proof of that theorem, we obtain that

$$\begin{aligned} \sum _{k\in \mathbb {Z}} \mu _k a^k=f\quad \quad a.e. \end{aligned}$$

holds and also in $L_2$ norm if $f\in L_2$. Since $g\in L_2$, we have

$$\begin{aligned} \ell (f) = \mathbb {E}[fg]=\sum _{k\in \mathbb {Z}} \mu _k \mathbb {E}(a^k g). \end{aligned}$$

By the definition of atom $a^k$, we get that

$$\begin{aligned} \mathbb {E}[a^kg]=\mathbb {E}[a^k(g-g^k)]. \end{aligned}$$

Hence, applying Hölder’s inequality, we conclude that

$$\begin{aligned} |\ell (f)|&\le \sum _{k\in \mathbb {Z}}\mu _k\big |\mathbb {E}[a^k(g-g^{\tau _k})]\big | \nonumber \\&\le \sum _{k\in \mathbb {Z}}\mu _k\Vert a^k\Vert _{L_2}\Vert g-g^{\tau _k}\Vert _{L_2} \nonumber \\&\le \sum _{k\in \mathbb {Z}}\mu _k\Vert s(a^k)\Vert _{L_\infty } \Vert \chi _{\{\tau _k<\infty \}}\Vert _{L_2}\Vert g-g^{\tau _k}\Vert _{L_2} \nonumber \\&\le \sum _{k\in \mathbb {Z}}\mu _k \Phi ^{-1}\big (1/\mathbb {P}(\tau _k<\infty )\big )\Vert \chi _{\{\tau _k<\infty \}}\Vert _{L_2} \Vert g-g^{\tau _k}\Vert _{L_2} \nonumber \\&= \sum _{k\in \mathbb {Z}}\gamma _b\big (\mathbb {P}(\tau _k<\infty )\big )\mu _k \frac{\Phi ^{-1}\big (1/\mathbb {P}(\tau _k<\infty )\big ) {\mathbb {P}(\tau _k<\infty )^{1/2}}}{\gamma _b\big (\mathbb {P}(\tau _k<\infty )\big )}\Vert g-g^{\tau _k}\Vert _{L_2} \nonumber \\&\le \Vert g\Vert _{BMO_{2,\Phi ,b}} \sum _{k\in \mathbb {Z}}\gamma _b\big (\mathbb {P}(\tau _k<\infty )\big )\mu _k. \end{aligned}$$

(6.2)

Since $0 < q \le 1$, it follows from concavity that

$$\begin{aligned} |\ell (f)|^q\le \sum _{k\in \mathbb {Z}}\gamma _b^q\big (\mathbb {P}(\tau _k<\infty )\big )\mu _k^q\Vert g\Vert _{BMO_{2,\Phi ,b}}^q. \end{aligned}$$

Therefore, Theorem 5 gives that

$$\begin{aligned} |\ell (f)| \preceq \Vert f\Vert _{H^s_{\Phi ,q,b}}\Vert g\Vert _{BMO_{2,\Phi ,b}}. \end{aligned}$$

Conversely, let $\ell$ be any continuous functional on $H^s_{\Phi ,q,b}$. We show that there exists $g \in BMO_{2,\Phi ,b}$ such that

$$\begin{aligned} \ell (f) = \mathbb {E}[fg] \qquad (f\in L_2^0) \qquad \text{ and } \qquad \Vert g\Vert _{BMO_{2,\Phi ,b}}\le C \Vert \ell \Vert . \end{aligned}$$

(6.3)

First we note that, by

$$\begin{aligned} \Vert f\Vert _{H^s_{\Phi ,q,b}} \le C\Vert s(f)\Vert _{L_2}= C\Vert f\Vert _{L_2}, \quad \quad f\in L_2^0, \end{aligned}$$

the space $L_2$ can be embedded continuously in $H^s_{\Phi ,q,b}$. Consequently, there exists a $\widetilde{g}\in L_2$ such that

$$\begin{aligned} \ell (f) = \mathbb {E}[f \widetilde{g}], \quad \quad f\in L_2^0. \end{aligned}$$

Let $g =\widetilde{g}-\mathbb {E}_0[\widetilde{g}]$. Then $g\in L_2^0$ and $\mathbb {E}[f \widetilde{g}] = \mathbb {E}[fg]$ for $f\in L_2^0$, that is,

$$\begin{aligned} \ell (f) = \mathbb {E}[fg], \quad \quad f\in L_2^0. \end{aligned}$$

Next we show that $g \in BMO_{2,\Phi ,b}$ and that $\Vert g\Vert _{BMO_{2,\Phi ,b}}\le C \Vert \ell \Vert$. Indeed, for any $\tau \in \mathcal {T}$ with $\mathbb {P}(\tau <\infty ) \ne 0$, suppose that

$$\begin{aligned} \varphi {:}{=}\frac{\Phi ^{-1}\big (1/\mathbb {P}(\tau<\infty )\big ) \mathbb {P}(\tau<\infty )^{1/2}}{\Vert g-g^\tau \Vert _{L_2}\gamma _b\big (\mathbb {P}(\tau <\infty )\big )}(g-g^\tau ). \end{aligned}$$

The function $\varphi$ is not necessarily a $(1,\Phi ,\infty )$-atom, however, it satisfies the condition (1) of Definition 4, namely,

$$\begin{aligned} s(\varphi )=s(\varphi )\chi _{\{\tau < \infty \}}. \end{aligned}$$

Let $h=g-g^\tau$. Note that $s(h)=s(h)\chi _{\{\tau < \infty \}}$ also holds. Hölder’s inequality assures that

$$\begin{aligned} \big \Vert h\big \Vert ^q_{H^s_{\Phi ,q,b}}&\approx \int _0^{\mathbb {P}(\tau<\infty )}\left( \frac{\gamma _b(t)}{\Phi ^{-1}\big (1/t\big )}\big (s(h)\big )^*(t)\right) ^q\frac{dt}{t}\\&\le \left( \int _0^{\mathbb {P}(\tau<\infty )}\big [\big (s(h)\big )^*(t)\big ]^2dt\right) ^{\frac{q}{2}} \left( \int _0^{\mathbb {P}(\tau <\infty )} \left( \frac{\gamma _b^q(t)1/t}{\Phi ^{-1}\big (1/t\big )^q}\right) ^{\frac{2}{2-q}} dt\right) ^{\frac{2-q}{2}} . \end{aligned}$$

Since $\frac{\Phi ^{-1}(t)}{t^{1/p_+}}$ is equivalent to a non-decreasing function on $(0,\infty )$, we have

$$\begin{aligned}&\left( \int _0^{\mathbb {P}(\tau<\infty )} \left( \frac{\gamma _b^q(t)1/t}{\Phi ^{-1}\big (1/t\big )^q}\right) ^{\frac{2}{2-q}} dt\right) ^{\frac{2-q}{2}}\\&\quad = \Bigg (\int _0^{\mathbb {P}(\tau<\infty )} \left( \frac{(1/t)^{1/p_+}}{\Phi ^{-1}\big (1/t\big )}\right) ^{\frac{2q}{2-q}} \Big (t^{\frac{2-p_+}{3p_+}}\gamma _b(t)\Big )^{\frac{2q}{2-q}}t^{\frac{q(2-p_+)}{3(2-q)p_+}-1} dt\Bigg )^{\frac{2-q}{2}}\\&\quad \preceq \left( \frac{\mathbb {P}(\tau<\infty )^{\frac{-1 -p_+}{3p_+}}}{\Phi ^{-1}\big (1/\mathbb {P}(\tau<\infty )\big )}\right) ^q \gamma ^q_b\big (\mathbb {P}(\tau<\infty )\big ) \left( \int _{0}^{\mathbb {P}(\tau<\infty )} t^{\frac{q(2-p_+)}{3(2-q)p_+}-1} dt\right) ^{\frac{2-q}{2}}\\&\quad = \bigg (\frac{3(2-q)p_+}{q(2-p_+)}\bigg )^{\frac{2-q}{2}} \left( \frac{\gamma _b\big (\mathbb {P}(\tau<\infty )\big )\mathbb {P}(\tau<\infty )^{-1/2}}{\Phi ^{-1}\big (1/\mathbb {P}(\tau <\infty )\big )}\right) ^q . \end{aligned}$$

Thus there exists a positive constant c such that

$$\begin{aligned} \big \Vert h\big \Vert _{H^s_{\Phi ,q,b}}\le & {} c \Vert s(h)\Vert _{L_2} \frac{\gamma _b\big (\mathbb {P}(\tau<\infty )\big )\mathbb {P}(\tau<\infty )^{-1/2}}{\Phi ^{-1}\big (1/\mathbb {P}(\tau<\infty )\big )}\\= & {} c \Vert h\Vert _{L_2}\frac{\gamma _b\big (\mathbb {P}(\tau<\infty )\big )\mathbb {P}(\tau<\infty )^{-1/2}}{\Phi ^{-1}\big (1/\mathbb {P}(\tau <\infty )\big )}, \end{aligned}$$

which means

$$\begin{aligned} \big \Vert g-g^\tau \big \Vert _{H^s_{\Phi ,q,b}}\le c \big \Vert g-g^\tau \big \Vert _{L_2}\frac{\gamma _b\big (\mathbb {P}(\tau<\infty )\big ) \mathbb {P}(\tau<\infty )^{-1/2}}{\Phi ^{-1}\big (1/\mathbb {P}(\tau <\infty )\big )}. \end{aligned}$$

Hence, we get

$$\begin{aligned} \big \Vert \varphi \big \Vert _{H^s_{\Phi ,q,b}} =\frac{\Phi ^{-1}\big (1/\mathbb {P}(\tau<\infty )\big ) \mathbb {P}(\tau<\infty )^{1/2}}{\Vert g-g^\tau \Vert _{L_2}\gamma _b\big (\mathbb {P}(\tau <\infty )\big )} \big \Vert g-g^\tau \big \Vert _{H^s_{\Phi ,q,b}} \le c. \end{aligned}$$

Thus, we obtain

$$\begin{aligned} \Vert \ell \Vert\ge & {} \frac{1}{c} \ell (\varphi ) = \frac{1}{c} \mathbb {E}(\varphi g) = \frac{1}{c} \mathbb {E}\big (\varphi (g-g^\tau )\big )\\= & {} \frac{1}{c} \frac{\Phi ^{-1}\big (1/\mathbb {P}(\tau<\infty )\big ) \mathbb {P}(\tau<\infty )^{1/2}}{\gamma _b\big (\mathbb {P}(\tau <\infty )\big )} \Vert g-g^\tau \Vert _{L_2} . \end{aligned}$$

Therefore, $g \in BMO_{2,\Phi ,b}$. Taking the supremum over all stopping times, we proved (6.3). Hence, we establish the embedding $\big (H^s_{\Phi ,q}\big )^* \subseteq BMO_{2,\Phi ,b}$ with respect to (quasi)-norms. This completes the proof of the theorem. $\square$

As a consequence, Theorem 1.2 in [38] is also true without the assumption that the slowly varying function b is non-decreasing. More precisely,

Corollary 5

Let $0<p<2$, $0<q\le 1$ and let b be a slowly varying function. Then the dual space of $H^s_{p,q,b}$ is $BMO_{2,b}(\alpha )$ with $\alpha =1/p-1$.

Taking $\Phi (t)=t^p$ for $t\in [0,\infty )$ in Theorem 10, we can see that Corollary 5 holds. If we consider the special case $b\equiv 1$ in Theorem 10, we conclude the following result.

Corollary 6

Let $\Phi$ be an Orlicz function of lower type $p_-$ and upper type $p_+$ satisfying $0<p_- \le p_+<2$ and $0<q\le 1$. Then the dual space of $H_{\Phi ,q}^s$ is $BMO_{2,\Phi }$. That is

$$\begin{aligned} \big ( H_{\Phi ,q}^s\big )^*=BMO_{2,\Phi }. \end{aligned}$$

There is no restricted condition $q_{\Phi ^{-1}}<1/q$ in Corollary 6. Hence, it generalizes the very recent result in [24]. For another application of Theorem 10, let $\Phi (t)=t^p$ and $b\equiv 1$. Then the dual space of Lorentz Hardy martingale space is $BMO_2(\alpha )$, where $\alpha =1/p-1$.

Corollary 7

([37]) If $0< p <2$ and $0 <q\le 1$, then the dual space of $H_{p,q}^s$ is $BMO_2(\alpha )$, where $\alpha =1/p-1$.

Note that the conclusion of Corollary 7 requires not the restriction $q<p$. So this generalizes the result in [28].

Now, we consider the dual of Orlicz-Lorentz-Karamata Hardy martingale spaces $H^s_{\Phi ,q,b}$ for $1<q<\infty$.

Definition 7

Let $\Phi$ be an Orlicz function, $1\le p < \infty$, $0<q<\infty$ and let b be a slowly varying function. The generalized BMO martingale space $BMO_{p,q,\Phi ,b}$ is defined by

$$\begin{aligned} BMO_{p,q,\Phi ,b} =\Big \{ f \in L_p^0: \Vert f\Vert _{BMO_{p,q,\Phi ,b}}< \infty \Big \}, \end{aligned}$$

where

$$\begin{aligned} \Vert f\Vert _{BMO_{p,q,\Phi ,b}}=\sup \frac{\sum _{k\in \mathbb {Z}}2^k {\mathbb {P}(\nu _k<\infty )^{1-1/p}}\Vert f-f^{\nu _k}\Vert _{L_p}}{\Big (\sum _{k\in \mathbb {Z}}\big [2^k\gamma _b\big (\mathbb {P}(\nu _k<\infty )\big ) \Vert \chi _{\{\nu _k<\infty \}}\Vert _{L_\Phi }\big ]^q\Big )^{1/q}} \end{aligned}$$

and the supremum is taken over all stopping time sequences $\{\nu _k\}_{k\in \mathbb {Z}}$ such that $\big \{ 2^k\gamma _b\big (\mathbb {P}(\nu _k<\infty )\big )\Vert \chi _{\{\nu _k<\infty \}}\Vert _{L_\Phi }\big \}_{k\in \mathbb {Z}}\in l_q$.

It follows from the idea of Proposition 4.6 in [35] and Proposition 6.7 in [63] that $BMO_{p,q,\Phi ,b}$ and $BMO_{p,\Phi ,b}$ have the following connection.

Proposition 8

Let $\Phi$ be an Orlicz function, $1\le p < \infty$, $0<q<\infty$ and let b be a slowly varying function. Then

$$\begin{aligned} \Vert f\Vert _{BMO_{p,\Phi ,b}}\le \Vert f\Vert _{BMO_{p,q,\Phi ,b}}. \end{aligned}$$

Moreover, if $0<q \le 1$, then $\Vert f\Vert _{BMO_{p,\Phi ,b}} = \Vert f\Vert _{BMO_{p,q,\Phi ,b}}$.

Proof

Let $f\in BMO_{p,q,\Phi ,b}$. For any given $\tau \in \mathcal {T}$ and fixing $n_0\in \mathbb {Z}$, we set, for $k\in \mathbb {Z}$,

$$\begin{aligned} \nu _k= \left\{ \begin{array}{l} \tau \quad \quad \quad \;\;\text {if}\;k=n_0,\\ +\infty \ \quad \quad \text {if}\; k\ne n_0. \end{array} \right. \end{aligned}$$

It is easy to see that

$$\begin{aligned} \bigg (\sum _{k\in \mathbb {Z}}\big [2^k\gamma _b\big (\mathbb {P}(\nu _k<\infty )\big ) \Vert \chi _{\{\nu _k<\infty \}}\Vert _{L_\Phi }\big ]^q\bigg )^{1/q}<\infty . \end{aligned}$$

This implies $\Vert f\Vert _{BMO_{p,\Phi ,b}}\le \Vert f\Vert _{BMO_{p,q,\Phi ,b}}$.

For the converse, let $0<q\le 1$, $f\in BMO_{p,\Phi ,b}$ and

$$\begin{aligned} \big \{ 2^k\gamma _b\big (\mathbb {P}(\nu _k<\infty )\big )\Vert \chi _{\{\nu _k<\infty \}}\Vert _{L_\Phi }\big \}_{k\in \mathbb {Z}}\in l_q. \end{aligned}$$

Then we get that

$$\begin{aligned}&\frac{\sum _{k\in \mathbb {Z}}2^k {\mathbb {P}(\nu _k<\infty )^{1-1/p}}\Vert f-f^{\nu _k}\Vert _{L_p}}{\Big (\sum _{k\in \mathbb {Z}}\big [2^k\gamma _b\big (\mathbb {P}(\nu _k<\infty )\big ) \Vert \chi _{\{\nu _k<\infty \}}\Vert _{L_\Phi }\big ]^q\Big )^{1/q}}\\&\quad \le \frac{\sum _{k\in \mathbb {Z}}2^k {\mathbb {P}(\nu _k<\infty )^{1-1/p}}\Vert f-f^{\nu _k}\Vert _{L_p}}{ \sum _{k\in \mathbb {Z}} 2^k\gamma _b\big (\mathbb {P}(\nu _k<\infty )\big ) \Vert \chi _{\{\nu _k<\infty \}}\Vert _{L_\Phi } }\\&\quad \le \frac{\sum _{k\in \mathbb {Z}}2^k\gamma _b\big (\mathbb {P}(\nu _k<\infty )\big )\Vert \chi _{\{\nu _k<\infty \}}\Vert _{L_\Phi } \sup _{\nu \in \mathcal {T}}\frac{ {\mathbb {P}(\nu<\infty )^{1-1/p}}\Vert f-f^{\nu }\Vert _{L_p}}{\gamma _b (\mathbb {P}(\nu<\infty ) )\Vert \chi _{\{\nu<\infty \}}\Vert _{L_\Phi }}}{\sum _{k\in \mathbb {Z}} 2^k\gamma _b\big (\mathbb {P}(\nu _k<\infty )\big ) \Vert \chi _{\{\nu _k<\infty \}}\Vert _{L_\Phi }}\\&\quad = \Vert f\Vert _{BMO_{p,\Phi ,b}}. \end{aligned}$$

Taking over all stopping time sequences $\{\nu _k\}_{k\in \mathbb {Z}}$ associated with

$$\begin{aligned} \big \{ 2^k\gamma _b\big (\mathbb {P}(\nu _k<\infty )\big )\Vert \chi _{\{\nu _k<\infty \}}\Vert _{L_\Phi }\big \}_{k\in \mathbb {Z}}\in l_q, \end{aligned}$$

we have

$$\begin{aligned} \Vert f\Vert _{BMO_{p,\Phi ,b}}\le \Vert f\Vert _{BMO_{p,q,\Phi ,b}}. \end{aligned}$$

Thus, if $0<q \le 1$, then $\Vert f\Vert _{BMO_{p,\Phi ,b}} = \Vert f\Vert _{BMO_{p,q,\Phi ,b}}$. $\square$

Remark 11

In Definition 7, if $\Phi (t)=t^{1/(1+\alpha )}$ for $\alpha \ge 0$, then the generalized BMO martingale space $BMO_{p,q,\Phi ,b}$ goes back to $BMO_{p,q,b}(\alpha )$, which was introduced in [38]. In addition, if $b\equiv 1$, $BMO_{p,q,b}(\alpha )$ is $BMO_{p,q}(\alpha )$, which was introduced in [37].

Now we prove that the dual of $H_{\Phi ,q,b}^s$ is $BMO_{p,q,\Phi ,b}$ if $1<q<\infty$.

Theorem 11

If $\Phi$ is an Orlicz function of lower type $p_-$ and upper type $p_+$ satisfying $0<p_- \le p_+<2$, $1<q<\infty$ and b is a slowly varying function, then the dual space of $H_{\Phi ,q,b}^s$ is $BMO_{2,q,\Phi ,b}$. That is

$$\begin{aligned} \big ( H_{\Phi ,q,b}^s\big )^*=BMO_{2,q,\Phi ,b}. \end{aligned}$$

Proof

Let $g\in BMO_{2,q,\Phi ,b}\subseteq L_2$ and define the linear functional

$$\begin{aligned} \ell (f)=\mathbb {E}[fg],\quad \quad \forall \;f\in L_2. \end{aligned}$$

For any $f \in L_2$, Theorem 5 provides a sequence $(a^k )_{k\in \mathbb {Z}}$ of $(1, \Phi ,\infty )$-atoms and a sequence of real numbers $(\mu _k )_{k\in \mathbb {Z}}$ satisfying

$$\begin{aligned} \mu _k =\frac{ \kappa \cdot 2^k}{\Phi ^{-1}\big (1/\mathbb {P}(\tau _k < \infty )\big )}, \end{aligned}$$

where $\kappa$ is a positive constant and $(\tau _k)_{k\in \mathbb {Z}}$ is the corresponding stopping time sequence. Moreover, $f =\sum _{k\in \mathbb {Z}} \mu _ka^k$ and

$$\begin{aligned} \bigg (\sum _{k\in \mathbb {Z}}\gamma _b^q\big (\mathbb {P}(\tau _k<\infty )\big )\mu _k^q\bigg )^{1/q}\preceq \big \Vert f\big \Vert _{H^s_{\Phi ,q,b}}. \end{aligned}$$

Similarly to (6.2), we get

$$\begin{aligned} |\ell (f)|\le & {} \sum _{k\in \mathbb {Z}}\mu _k \Phi ^{-1}\big (1/\mathbb {P}(\tau _k<\infty )\big )\Vert \chi _{\{\tau _k<\infty \}}\Vert _{L_2} \Vert g-g^{\tau _k}\Vert _{L_2}\\= & {} \kappa \cdot \sum _{k\in \mathbb {Z}}2^k\mathbb {P}(\tau _k<\infty ) \Vert g-g^{\tau _k}\Vert _{L_2} \\\le & {} \kappa \cdot \Vert g\Vert _{BMO_{2,q,\Phi ,b}} \bigg (\sum _{k\in \mathbb {Z}}\Big [2^k\gamma _b\big (\mathbb {P}(\tau _k<\infty )\big ) \big \Vert \chi _{\{\tau _k<\infty \}}\big \Vert _{L_\Phi }\Big ]^q\bigg )^{1/q} \\= & {} \Vert g\Vert _{BMO_{2,q,\Phi ,b}}\bigg (\sum _{k\in \mathbb {Z}}\gamma _b^q\big (\mathbb {P}(\tau _k<\infty )\big )\mu _k^q\bigg )^{1/q} \\\preceq & {} \Vert g\Vert _{BMO_{2,q,\Phi ,b}}\Vert f\Vert _{H^s_{\Phi ,q,b}} . \end{aligned}$$

Since $L_2$ is dense in $H^s_{\Phi ,q,b}$ from Remark 7, the functional $\ell$ can be uniquely extended to a continuous linear functional on $H^s_{\Phi ,q,b}$.

Conversely, let $\ell \in ( H_{\Phi ,q,b}^s)^*$. According to Theorem 5 and Remark 6, we have

$$\begin{aligned} L_2\subseteq H_{\Phi ,q,b}^s = H_{\Phi ,q,b}^{at,\infty ,1} \subseteq H_{\Phi ,q,b}^{at,2,1}. \end{aligned}$$

This means that

$$\begin{aligned} (H_{\Phi ,q,b}^{at,2,1})^* \subseteq ( H_{\Phi ,q,b}^s)^*\subseteq L_2. \end{aligned}$$

Hence there exists a $g \in L_2$ such that

$$\begin{aligned} \ell (f)=\mathbb {E}[fg]\quad \quad \forall \;f\in L_2, \end{aligned}$$

and $\ell$ can be extended to a continuous functional $\tilde{\ell }$ on $H_{\Phi ,q,b}^{at,2,1}$ such that $\Vert \ell \Vert =\Vert \tilde{\ell }\Vert$. Let $\{\nu _k\}_{k\in \mathbb {Z}}$ be an arbitrary stopping time sequence such that

$$\begin{aligned} \bigg (\sum _{k\in \mathbb {Z}}\Big [2^k\gamma _b\big (\mathbb {P}(\nu _k<\infty )\big ) \big \Vert \chi _{\{\nu _k<\infty \}}\big \Vert _{L_\Phi }\Big ]^q\bigg )^{1/q}<\infty . \end{aligned}$$

(6.4)

Now we set

$$\begin{aligned} h_k=\frac{(g-g^{\nu _k})\mathbb {P}(\nu _k<\infty )^{1/2}}{\Vert g-g^{\nu _k} \Vert _{L_2} \Vert \chi _{\{\nu _k<\infty \}}\Vert _{L_\Phi }}. \end{aligned}$$

(6.5)

We claim that $h_k$ is a $(1,\Phi ,2)$-atom for each $k\in \mathbb {Z}$. Indeed, if $n\le \nu _k$, then clearly $\mathbb {E}_nh_k=0$. Moreover,

$$\begin{aligned} \Vert s(h_k)\Vert _{L_2}=\Vert h_k\Vert _{L_2}=\frac{\mathbb {P}(\nu _k<\infty )^{1/2}}{\Vert \chi _{\{\nu _k<\infty \}}\Vert _{L_\Phi }}=\mathbb {P}(\nu _k<\infty )^{1/2} \Phi ^{-1}\big (1/\mathbb {P}(\nu _k<\infty )\big ). \end{aligned}$$

It follows from formulae (6.4) and (6.5) that

$$\begin{aligned} f=\sum _{k\in \mathbb {Z}} 2^k \Vert \chi _{\{\nu _k<\infty \}} \Vert _{L_\Phi } h_k \in H_{\Phi ,q,b}^{at,2,1} \end{aligned}$$

and

$$\begin{aligned} \Vert f\Vert _{H_{\Phi ,q,b}^{at,2,1}}\le \bigg (\sum _{k\in \mathbb {Z}}\Big [2^k\gamma _b\big (\mathbb {P}(\nu _k<\infty )\big ) \big \Vert \chi _{\{\nu _k<\infty \}}\big \Vert _{L_\Phi }\Big ]^q\bigg )^{1/q}. \end{aligned}$$

Let

$$\begin{aligned} (f)^N=\sum _{k=-N}^N 2^k \Vert \chi _{\{\nu _k<\infty \}} \Vert _{L_\Phi } h_k. \end{aligned}$$

Consequently,

$$\begin{aligned}&\sum _{k=-N}^{N}2^k\mathbb {P}(\nu _k<\infty )^{\frac{1}{2}}\Vert g-g^{\nu _k}\Vert _{L_2}\\&\quad = \sum _{k=-N}^{N} 2^k \Vert \chi _{\{\nu _k<\infty \}}\Vert _{L_\Phi } \mathbb {E}[h_k(g-g^{\nu _k})]\\&\quad = \sum _{k=-N}^{N} 2^k \Vert \chi _{\{\nu _k<\infty \}}\Vert _{L_\Phi } \mathbb {E}[h_kg] = \mathbb {E}[(f)^Ng]=\tilde{\ell }((f)^N) \le \Vert \tilde{\ell }\Vert \big \Vert (f)^N\big \Vert _{H_{\Phi ,q,b}^{at,2,1}}\\&\quad \le \Vert \ell \Vert \big \Vert f\big \Vert _{H_{\Phi ,q,b}^{at,2,1}} \le \Vert \ell \Vert \bigg (\sum _{k\in \mathbb {Z}}\big [2^k\gamma _b\big (\mathbb {P}(\nu _k<\infty )\big ) \Vert \chi _{\{\nu _k<\infty \}}\Vert _{L_\Phi }\big ]^q\bigg )^{1/q}. \end{aligned}$$

Thus we obtain

$$\begin{aligned} \frac{\sum _{k=-N}^{N}2^k\mathbb {P}(\nu _k<\infty )^{\frac{1}{2}}\Vert g-g^{\nu _k}\Vert _{L_2}}{\Big (\sum _{k\in \mathbb {Z}}\big [2^k\gamma _b\big (\mathbb {P}(\nu _k<\infty )\big ) \Vert \chi _{\{\nu _k<\infty \}}\Vert _{L_\Phi }\big ]^q\Big )^{1/q}} \le \Vert \ell \Vert . \end{aligned}$$

Taking $N \longrightarrow \infty$ and the supremum over all stopping time sequences such that

$$\begin{aligned} \{ 2^k\gamma _b\big (\mathbb {P}(\nu _k<\infty )\big )\Vert \chi _{\{\nu _k<\infty \}}\Vert _{L_\Phi }\}_{k\in \mathbb {Z}}\in l_q, \end{aligned}$$

we conclude

$$\begin{aligned} \Vert g\Vert _{BMO_{2,q,\Phi ,b}}\le \Vert \ell \Vert . \end{aligned}$$

The proof is complete. $\square$

Theorem 11 improves the recent results [37, 38]. To be more specific, we obtain the conclusion of Theorem 1.5 in [38] without the condition that b is non-decreasing. That is, if we consider the case $\Phi (t)=t^p$ for $t\in [0,\infty )$ in Theorem 10, one get the following result:

Corollary 8

Let $0<p<2$, $1<q<\infty$ and let b be a slowly varying function. Then the dual space of $H^s_{p,q,b}$ is $BMO_{2,q,b}(\alpha )$ with $\alpha =1/p-1$.

In particular, if $\Phi (t)=t^p$ for $t\in [0,\infty )$ and $b\equiv 1$ in Theorem 11, we obtain the next corollary, which was proved in [37, 59, 60].

Corollary 9

If $0<p<2$ and $1<q<\infty$, then the dual space of $H_{p,q}^s$ is $BMO_{2,q}(\alpha )$, where $\alpha =1/p-1$.

7 The generalized John-Nirenberg theorem

In this section, we prove the generalized John-Nirenberg theorem when the stochastic basis $\{\mathcal {F}_n\}_{n\ge 0}$ is regular.

Theorem 12

Let $\Phi$ be an Orlicz function of lower type $p_-$ and upper type $p_+$ satisfying $0<p_- \le p_+\le 1$, $1< p,\,q < \infty$ and let b be a slowly varying function. If the stochastic basis $\{\mathcal {F}_n\}_{n\ge 0}$ is regular, then

$$\begin{aligned} (H^s_{\Phi ,q,b})^*=BMO_{p,q,\Phi ,b}. \end{aligned}$$

Proof

Let $p'$ be the conjugate number of p, that is, $1/p'+1/p=1$. Obviously, $p'>1$. Firstly we claim that $L_{p'}\subseteq H^s_{\Phi ,q,b}$. Indeed, for any $f\in L_{p'}$, it follows from Proposition 3, Lemma 14 and Corollary 4 that

$$\begin{aligned} \Vert f\Vert _{H^s_{\Phi ,q,b}}=\Vert s(f)\Vert _{L_{\Phi ,q,b}}\preceq \Vert s(f)\Vert _{L_{1,q,b}} \preceq \Vert s(f)\Vert _{L_{p',p',1}}=\Vert s(f)\Vert _{L_{p'}}\approx \Vert f\Vert _{L_{p'}} . \end{aligned}$$

On the basis of Theorem 5, there exist a sequence $(a^k )_{k\in \mathbb {Z}}$ of $(1, \Phi ,\infty )$-atoms and a sequence of real numbers $(\mu _k )_{k\in \mathbb {Z}}$ satisfying

$$\begin{aligned} \mu _k =\frac{ \kappa \cdot 2^k}{\Phi ^{-1}\big (1/\mathbb {P}(\tau _k < \infty )\big )}, \end{aligned}$$

where $\kappa$ is a positive constant and $(\tau _k)_{k\in \mathbb {Z}}$ is the corresponding stopping time sequence, such that

$$\begin{aligned} f =\sum _{k\in \mathbb {Z}} \mu _ka^k \qquad a.e. \end{aligned}$$

(7.1)

and

$$\begin{aligned} \bigg (\sum _{k\in \mathbb {Z}}\gamma _b^q\big (\mathbb {P}(\tau _k<\infty )\big )\mu _k^q\bigg )^{1/q}\preceq \big \Vert f\big \Vert _{H^s_{\Phi ,q,b}}. \end{aligned}$$

Similarly to the proof of Theorem 10, we conclude that (7.1) holds also in the $L_{p'}$-norm. Then for any given $g\in BMO_{p,q,\Phi ,b}\subseteq L_p$, we have

$$\begin{aligned} \ell _g(f){:}{=} \mathbb {E}[fg] = \sum _{k\in \mathbb {Z}}\mu _k\mathbb {E}[a^kg], \quad \quad \quad \quad \forall \;f\in L_{p'}. \end{aligned}$$

According to the Hölder inequality, we get

$$\begin{aligned} |\ell _g(f)|\le & {} \sum _{k\in \mathbb {Z}}\mu _k\big |\mathbb {E}[a^k(g-g^{\tau _k})]\big | \le \sum _{k\in \mathbb {Z}}\mu _k\Vert a^k\Vert _{L_{p'}}\Vert g-g^{\tau _k}\Vert _{L_p}\\\preceq & {} \sum _{k\in \mathbb {Z}}\mu _k\Vert s(a^k)\Vert _{L_{p'}} \Vert g-g^{\tau _k}\Vert _{L_p}\\\le & {} \sum _{k\in \mathbb {Z}}\mu _k\Vert s(a^k)\Vert _{L_\infty }\Vert \chi _{\{\tau _k<\infty \}}\Vert _{L_{p'}} \Vert g-g^{\tau _k}\Vert _{L_p}\\\le & {} \sum _{k\in \mathbb {Z}}\mu _k \Phi ^{-1}\big (1/\mathbb {P}(\tau _k<\infty )\big )\mathbb {P}(\tau _k<\infty )^{1-1/p} \Vert g-g^{\tau _k}\Vert _{L_p}\\= & {} \kappa \cdot \sum _{k\in \mathbb {Z}}2^k\mathbb {P}(\tau _k<\infty )^{1-1/p} \Vert g-g^{\tau _k}\Vert _{L_p}\\\le & {} \kappa \cdot \Vert g\Vert _{BMO_{2,q,\Phi ,b}} \bigg (\sum _{k\in \mathbb {Z}}\Big [2^k\gamma _b\big (\mathbb {P}(\tau _k<\infty )\big ) \big \Vert \chi _{\{\tau _k<\infty \}}\big \Vert _{L_\Phi }\Big ]^q\bigg )^{1/q}\\= & {} \Vert g\Vert _{BMO_{2,q,\Phi ,b}}\bigg (\sum _{k\in \mathbb {Z}}\gamma _b^q\big (\mathbb {P}(\tau _k<\infty )\big )\mu _k^q\bigg )^{1/q}\\\preceq & {} \Vert g\Vert _{BMO_{2,q,\Phi ,b}}\Vert f\Vert _{H^s_{\Phi ,q,b}} . \end{aligned}$$

Thus $\ell _g$ can be extended to a continuous functional on $H^s_{\Phi ,q,b}$.

To prove the converse, let $\ell \in ( H_{\Phi ,q,b}^s)^*$. It follows from Theorem 5, Remark 6 and Corollary 4 that

$$\begin{aligned} L_{p'}=H_{p'}^s =H^s_{p',p',1}\subseteq H^s_{1,q,b}\subseteq H_{\Phi ,q,b}^s = H_{\Phi ,q,b}^{at,\infty ,1} \subseteq H_{\Phi ,q,b}^{at, p', 1}. \end{aligned}$$

This means that

$$\begin{aligned} \big (H_{\Phi ,q,b}^{at, p', 1}\big )^* \subseteq ( H_{\Phi ,q,b}^s)^*\subseteq L_p. \end{aligned}$$

Hence there exists a $g \in L_p$ such that

$$\begin{aligned} \ell (f)=\ell _g(f)=\mathbb {E}[fg],\quad \quad \forall \;f\in L_{p'}, \end{aligned}$$

and $\ell$ can be extended to a continuous functional $\tilde{\ell }$ on $H_{\Phi ,q,b}^{at, p', 1}$ such that $\Vert \ell \Vert =\Vert \tilde{\ell }\Vert$. Let $\{\nu _k\}_{k\in \mathbb {Z}}$ be an arbitrary stopping time sequence such that

$$\begin{aligned} \bigg (\sum _{k\in \mathbb {Z}}\Big [2^k\gamma _b\big (\mathbb {P}(\nu _k<\infty )\big ) \big \Vert \chi _{\{\nu _k<\infty \}}\big \Vert _{L_\Phi }\Big ]^q\bigg )^{1/q}<\infty . \end{aligned}$$

(7.2)

Based on the sequence $\{\nu _k\}_{k\in \mathbb {Z}}$, we set

$$\begin{aligned} h_k=\frac{|g-g^{\nu _k}|^{p-1}\text {sign}(g-g^{\nu _k})\mathbb {P}(\nu _k<\infty )^{1/{p'}}}{\Vert g-g^{\nu _k} \Vert _{L_p}^{p-1} \Vert \chi _{\{\nu _k<\infty \}}\Vert _{L_\Phi }}. \end{aligned}$$

(7.3)

Now we claim that ${h_k}/{c_0}$ is a $(1,\Phi ,{p'})$-atom for each $k\in \mathbb {Z}$, where $c_0>1$ is the constant only dependents on p. Indeed, via Corollary 4, if $n\le \nu _k$, then $\mathbb {E}_nh_k=0$ and

$$\begin{aligned} \Vert s(h_k)\Vert _{L_{p'}}&\le c_0\Vert h_k\Vert _{L_{p'}}=c_0\frac{\mathbb {P}(\nu _k<\infty )^{1/{p'}}}{\Vert \chi _{\{\nu _k<\infty \}}\Vert _{L_\Phi }} \\&=c_0\mathbb {P}(\nu _k<\infty )^{1/{p'}} \Phi ^{-1}\big (1/\mathbb {P}(\nu _k<\infty )\big ). \end{aligned}$$

It follows from formulas (7.2) and (7.3) that

$$\begin{aligned} f=\sum _{k\in \mathbb {Z}} 2^k \Vert \chi _{\{\nu _k<\infty \}} \Vert _{L_\Phi } h_k \in H_{\Phi ,q,b}^{at, p', 1} \end{aligned}$$

and

$$\begin{aligned} \Vert f\Vert _{H_{\Phi ,q,b}^{at, p', 1}}\le \bigg (\sum _{k\in \mathbb {Z}}\Big [2^k\gamma _b\big (\mathbb {P}(\nu _k<\infty )\big ) \big \Vert \chi _{\{\nu _k<\infty \}}\big \Vert _{L_\Phi }\Big ]^q\bigg )^{1/q}. \end{aligned}$$

For an arbitrary positive integer N, set

$$\begin{aligned} (f)^N=\sum _{k=-N}^N 2^k \Vert \chi _{\{\nu _k<\infty \}} \Vert _{L_\Phi } h_k. \end{aligned}$$

Consequently,

$$\begin{aligned}&\sum _{k=-N}^{N}2^k\mathbb {P}(\nu _k<\infty )^{1-1/{p}}\Vert g-g^{\nu _k}\Vert _{L_p}\\&\quad =\sum _{k=-N}^{N}2^k\mathbb {P}(\nu _k<\infty )^{1/{p'}}\Vert g-g^{\nu _k}\Vert _{L_p} \\&\quad = \sum _{k=-N}^{N} 2^k \Vert \chi _{\{\nu _k<\infty \}}\Vert _{L_\Phi } \mathbb {E}\big [h_k(g-g^{\nu _k})\big ] \\&\quad = \sum _{k=-N}^{N} 2^k \Vert \chi _{\{\nu _k<\infty \}}\Vert _{L_\Phi } \mathbb {E}[h_kg]\\&\quad = \mathbb {E}[(f)^Ng]=\tilde{\ell }\big ((f)^N\big ) \le \big \Vert \tilde{\ell }\big \Vert \big \Vert (f)^N\big \Vert _{H_{\Phi ,q,b}^{at, p', 1}} \le \Vert \ell \big \Vert \big \Vert f\big \Vert _{H_{\Phi ,q,b}^{at, p', 1}}\\&\quad \le \big \Vert \ell \big \Vert \Big (\sum _{k\in \mathbb {Z}}\big [2^k\gamma _b\big (\mathbb {P}(\nu _k<\infty )\big ) \Vert \chi _{\{\nu _k<\infty \}}\Vert _{L_\Phi }\big ]^q\Big )^{1/q}. \end{aligned}$$

Thus we obtain

$$\begin{aligned} \frac{\sum _{k=-N}^{N}2^k\mathbb {P}(\nu _k<\infty )^{1-1/p}\Vert g-g^{\nu _k}\Vert _{L_p}}{\Big (\sum _{k\in \mathbb {Z}}\big [2^k\gamma _b\big (\mathbb {P}(\nu _k<\infty )\big ) \Vert \chi _{\{\nu _k<\infty \}}\Vert _{L_\Phi }\big ]^q\Big )^{1/q}} \le \big \Vert \ell \big \Vert . \end{aligned}$$

Taking $N \longrightarrow \infty$ and the supremum over all stopping time sequences such that $\{ 2^k\gamma _b\big (\mathbb {P}(\nu _k<\infty )\big )\Vert \chi _{\{\nu _k<\infty \}}\Vert _{L_\Phi }\}_{k\in \mathbb {Z}}\in l_q$, we conclude $\Vert g\Vert _{BMO_{p,q,\Phi ,b}}\le \Vert \ell \Vert$. The proof is complete. $\square$

Obviously, combining Theorem 12 with Theorem 11, we obtain the conclusion as follows.

Theorem 13

Let $\Phi$ be an Orlicz function of lower type $p_-$ and upper type $p_+$ satisfying $0<p_- \le p_+\le 1$, $1<p,\,q<\infty$ and let b be a slowly varying function. If the stochastic basis $\{\mathcal {F}_n\}_{n\ge 0}$ is regular, then

$$\begin{aligned} BMO_{p,q,\Phi ,b}=BMO_{2,q,\Phi ,b}. \end{aligned}$$

As an application of Theorem 13, combining with Remark 11, we get the following result:

Corollary 10

Let $\alpha \ge 0$, $1<p,\,q<\infty$ and b be a slowly varying function. If the stochastic basis $\{\mathcal {F}_n\}_{n\ge 0}$ is regular, then

$$\begin{aligned} BMO_{p,q,b}(\alpha )=BMO_{2,q,b}(\alpha ). \end{aligned}$$

Remark 12

Note that, in the corollary above, the slowly varying function b is not necessarily non-decreasing as in the result of [38].

Remark 13

Theorem 13 and Corollary 10 are similar to the vector-valued characterizations of BMO on Euclidean space in [29].

For another application of Theorem 13, let $\Phi (t)=t^p$ and $b\equiv 1$. It is clear that we get the generalized John-Nirenberg theorem for Lorentz Hardy martingale spaces $H^s_{p,q}$, which was proved in [37, 59, 60].

Corollary 11

Let $\alpha \ge 0$, $1<p,\,q<\infty$. If the stochastic basis $\{\mathcal {F}_n\}_{n\ge 0}$ is regular, then

$$\begin{aligned} BMO_{p,q}(\alpha )=BMO_{2,q}(\alpha ). \end{aligned}$$

If $b\equiv 1$ and the stopping time sequence $\{\nu _k\}_{k\in \mathbb {Z}}$ is a sequence whose one element is a stopping time $\nu$ and the others are $\infty$ in Definition 7, then Theorem 13 returns to the famous result, the John-Nirenberg characterization of BMO as

$$\begin{aligned} BMO_p=BMO. \end{aligned}$$

8 Boundedness of generalized fractional integral operators

We now extend the boundedness of generalized fractional integrals on Orlicz-Lorentz-Karamata Hardy martingale spaces. In this section, we always assume that $(\Omega ,\mathcal {F},\mathbb {P})$ is a complete and non-atomic probability space. Here, we suppose that every $\sigma$-algebra $\mathcal {F}_n$ is generated by countable many atoms. $B\in \mathcal {F}_n$ is called an atom if for any $A\subseteq B$ with $A\in \mathcal {F}_n$ satisfying $\mathbb {P}(A)<\mathbb {P}(B)$, we have $\mathbb {P}(A)=0$. Denote by $A(\mathcal {F}_n)$ the set of all atoms in $\mathcal {F}_n$. Without loss of generality, we always suppose that the constant in (2.9) satisfies $\mathcal {R}\ge 2$. Now we give the definition of generalized fractional integrals as follows.

Definition 8

Let $\phi$ be an Orlicz function. For $f=(f_n)_{n\ge 0}\in \mathcal {M}$, the generalized fractional integral $I_\phi f=\big ((I_\phi f)_n\big )_{n\ge 0}$ of f is defined by

$$\begin{aligned} (I_\phi f)_n=\sum \limits _{k=1}^n \phi (b_{k-1})d_kf, \end{aligned}$$

where

$$\begin{aligned} b_k(\omega )=\sum _{B\in A(\mathcal {F}_k)}\mathbb {P}(B)\chi _{B}(\omega ), \quad \quad \text {a.e.}\;\omega \in \Omega , \;\; \forall \; k\ge 1. \end{aligned}$$

Moreover, if $\phi (t)=t^\alpha$ ($\alpha >0$), then we say that $I_\phi f$ =$I_\alpha f$ is the fractional integral (of order $\alpha$) of martingale f which was introduced in [23, 37, 54].

Remark 14

Note that $I_\phi f=\big ((I_\phi f)_n\big )_{n\ge 0}$ is also a martingale, more exactly, a martingale transform introduced by Burkholder. Especially, if $\Omega = [0, 1]$ and the $\sigma$-algebra $\mathcal {F}_n$ is generated by the dyadic intervals of [0, 1], then $I_\alpha f$ is closely related to a class of multiplier transformations of Walsh-Fourier series [58]. We refer to [55] for the classical fractional integrals.

In order to prove the boundedness of generalized fractional integrals on Orlicz-Lorentz-Karamata Hardy martingale spaces, we need the following lemmas.

Lemma 15

Let $\{\mathcal {F}_n\}_{ n \ge 0}$ be regular and $\phi$ be an Orlicz function of upper type $p\in (0,1)$. If $f=(f_n)_{n\ge 0}\in \mathcal {M}$ and $B\in \mathcal {F}$ such that $S(f)\le \chi _B$, then there exists a positive constant $C_\phi$ independent of f and B such that

$$\begin{aligned} S{(I_\phi f)}\le C_\phi \phi \big (\mathbb {P}(B)\big )\chi _B. \end{aligned}$$

(8.1)

The proof of Lemma 15 is similar to the proof of Lemma 5.3 in [24]. Its proof is left to the reader.

If we consider the case of $\phi (t)=t^\alpha$ for $\alpha >0$ and $t\in [0,\infty )$, the fractional integral operator $I_\alpha$ has the following property.

Lemma 16

([23]) Let $\alpha >0$ and $\{\mathcal {F}_n\}_{ n \ge 0}$ be regular. If $f=(f_n)_{n\ge 0}\in \mathcal {M}$ and $B\in \mathcal {F}$ such that $S(f)\le \chi _B$, then there exists a positive constant $C_\alpha$ independent of f and B such that

$$\begin{aligned} S{(I_\alpha f)}\le C_\alpha \mathbb {P}(B)^\alpha \chi _B. \end{aligned}$$

(8.2)

It is noteworthy that the case of $\alpha \ge 1$ in Lemma 16 is not included in Lemma 15.

Lemma 17

Let $\{\mathcal {F}_n\}_{ n \ge 0}$ be regular, $0<q\le \infty$, $\phi$ be an Orlicz function of upper type $p\in (0,1)$ and let b be a slowly varying function. Suppose that a is a $(2,\Phi ,\infty )$-atom as in Definition 4. If there exists $\Psi$ such that $\Psi ^{-1}(t)\ge \Phi ^{-1}(t) \cdot \phi (1/t)$ for any $t\ge 1$, then we have

$$\begin{aligned} \Vert I_\phi a\Vert _{\mathcal {Q}_{{\Psi ,q,b}}}\le C_\phi \gamma _b\big (\mathbb {P}(\tau <\infty )\big ), \end{aligned}$$

where $\tau$ is the stopping time associated with $(2,\Phi ,\infty )$-atom a and $C_\phi$ is the same constant as in Lemma 15.

Proof

From the definition of the $(2,\Phi ,\infty )$-atom a, we have

$$\begin{aligned} S(a)\le \Phi ^{-1}\big (1/\mathbb {P}(\tau<\infty )\big )\chi _{\{\tau <\infty \}}. \end{aligned}$$

This means

$$\begin{aligned} S\Big (\frac{a}{\Phi ^{-1}\big (1/\mathbb {P}(\tau<\infty )\big )}\Big ) =\frac{S(a)}{\Phi ^{-1}\big (1/\mathbb {P}(\tau<\infty )\big )} \le \chi _{\{\tau <\infty \}}. \end{aligned}$$

By using Lemma 15, we can obtain that

$$\begin{aligned} S\left( I_{\phi }\left( \frac{a}{\Phi ^{-1}\big (1/\mathbb {P}(\tau<\infty )\big )}\right) \right) \le C_\phi \phi \big ( \mathbb {P}(\tau<\infty )\big )\chi _{\{\tau <\infty \}}. \end{aligned}$$

Then, by the condition $\Psi ^{-1}(t)\ge \Phi ^{-1}(t) \cdot \phi (1/t)$ for any $t\ge 1$, we have

$$\begin{aligned} S(I_\phi a)\le & {} C_\phi \phi \big ( \mathbb {P}(\tau<\infty )\big ) \Phi ^{-1}\big (1/\mathbb {P}(\tau<\infty )\big )\chi _{\{\tau<\infty \}}\\\le & {} C_\phi \Psi ^{-1}\big (1/\mathbb {P}(\tau<\infty )\big )\chi _{\{\tau <\infty \}}. \end{aligned}$$

Now, let

$$\begin{aligned} \lambda _n=\Vert S(I_\phi a)\Vert _{L_\infty }\chi _{\{\tau \le n\}}. \end{aligned}$$

Then $(\lambda _n)_{n\ge 0}$ is a non-negative, non-decreasing and adapted sequence and, similarly to (4.7), $S_{n+1}(I_\phi a) \le \lambda _n$. Hence, we obtain

$$\begin{aligned} \Vert I_\phi a\Vert _{\mathcal {Q}_{{\Psi ,q,b}}}\le & {} \Vert \lambda _\infty \Vert _{L_{\Psi ,q,b}}\\\le & {} C_\phi \Psi ^{-1}\big (1/\mathbb {P}(\tau<\infty )\big )\left\| \chi _{\{ \tau<\infty \}} \right\| _{L_{\Psi ,q,b}}\\= & {} C_\phi \gamma _b\big (\mathbb {P}(\tau <\infty )\big ), \end{aligned}$$

where $C_\phi$ is the same constant as in Lemma 15. The proof is complete. $\square$

Following the same proof as in Lemma 17, we obtain the following conclusion via Lemma 16.

Lemma 18

Let $\{\mathcal {F}_n\}_{ n \ge 0}$ be regular, $0<q\le \infty$, $\alpha >0$ and let b be a slowly varying function. Suppose that a is a $(2,\Phi ,\infty )$-atom as in Definition 4, where $\Phi$ is an Orlicz function. If there exists $\Psi$ such that $\Phi ^{-1}(t) \le \Psi ^{-1}(t) \cdot t^\alpha$ for any $t\ge 1$, then we have

$$\begin{aligned} \Vert I_\alpha a\Vert _{\mathcal {Q}_{{\Psi ,q,b}}}\le C_\alpha \gamma _b\big (\mathbb {P}(\tau <\infty )\big ), \end{aligned}$$

where $\tau$ is the stopping time associated with $(2,\Phi ,\infty )$-atom a and $C_\alpha$ is the same constant as in Lemma 16.

When we consider $\Phi (t)=t^p$ and $\Psi (t)=t^r$ in Lemma 18, one can see that

Corollary 12

Let $\{\mathcal {F}_n\}_{ n \ge 0}$ be regular, $0<q\le \infty$, $\alpha >0$ and let b be a slowly varying function. Suppose that a is a $(2,p,\infty )$-atom for $0<p<\infty$. If there exists r such that $\alpha \ge \frac{1}{p}- \frac{1}{r}$, then we have

$$\begin{aligned} \Vert I_\alpha a\Vert _{\mathcal {Q}_{{r,q,b}}}\le C_\alpha \gamma _b\big (\mathbb {P}(\tau <\infty )\big ), \end{aligned}$$

where $\tau$ is the stopping time associated with $(2,p,\infty )$-atom a and $C_\alpha$ is the same constant as in Lemma 16.

We now prove the boundedness of generalized fractional integrals on Orlicz-Lorentz-Karamata Hardy martingale spaces via atomic decomposition. Recall that all the spaces $H_{\Phi ,q,b}^S$, $\mathcal {Q}_{\Phi ,q,b}$, $\mathcal {P}_{\Phi ,q,b}$, $H_{\Phi ,q,b}^M$ and $H_{\Phi ,q,b}^s$ are equivalent if $\{\mathcal {F}_n\}_{n\ge 0}$ is regular (see Theorem 9).

Theorem 14

Let $\{\mathcal {F}_n\}_{ n \ge 0}$ be regular, $0 < s\le \min \{1,q\}$ and b be a slowly varying function. Let $\Psi$ be an Orlicz function of lower type $p_\Psi ^-$ and upper type $p_\Psi ^+$ satisfying $s<p_\Psi ^- \le p_\Psi ^+<\infty$ and let $\Phi$ be an Orlicz function of lower type $p_\Phi ^-$ and upper type $p_\Phi ^+$ satisfying $0<p_\Phi ^- \le p_\Phi ^+<\infty$. If there exists an Orlicz function $\phi$ with upper type $p\in (0,1)$ such that $\Phi ^{-1} (t) \cdot \phi (1/t) \le \Psi ^{-1} (t)$ for any $t \ge 1$, then there exists a constant C such that

$$\begin{aligned} \Vert I_\phi f\Vert _{\mathcal {Q}_{\Psi ,q,b}}\le C\Vert f\Vert _{\mathcal {Q}_{\Phi ,s,b}} \end{aligned}$$

for all $f\in \mathcal {Q}_{\Phi ,s,b}$.

Proof

Let $f\in \mathcal {Q}_{\Phi ,s,b}$. According to Theorem 8, there exist a sequence $(a^k)_{k\in \mathbb {Z}}$ of $(2,\Phi ,\infty )$-atoms and a sequence of real numbers

$$\begin{aligned} \mu _k=\frac{3\cdot 2^k}{\Phi ^{-1}\big (1/\mathbb {P}(\tau _k<\infty )\big )}\quad \quad \;k\in \mathbb {Z}, \end{aligned}$$

($\tau _k$ is the stopping time associated with $a^k$) such that for all $n\ge 0$,

$$\begin{aligned} \sum _{k\in \mathbb {Z}} \mu _k \mathbb {E}_n a^k=f_n \qquad \text{ a.e. } \end{aligned}$$

and

$$\begin{aligned} \Big (\sum _{k\in \mathbb {Z}}\gamma _b^s\big (\mathbb {P}(\tau _k<\infty )\big )|\mu _k|^s\Big )^{1/s} \preceq \Vert f\Vert _{\mathcal {Q}_{\Phi ,s,b}}. \end{aligned}$$

Combining this with Lemma 17 and Theorem 2, we can finish the proof of Theorem 14. Indeed, since $0 < s \le \min \{1,q\}$ and $s<p_{\Psi }^-$, we have

$$\begin{aligned} \big \Vert I_\phi f\big \Vert ^s_{\mathcal {Q}_{\Psi ,q,b}}&=\big \Vert \sum _{k\in \mathbb {Z}} \mu _k I_\phi a^k \big \Vert ^s_{\mathcal {Q}_{\Psi ,q,b}} \preceq \sum _{k\in \mathbb {Z}} |\mu _k|^s \big \Vert I_\phi a^k\big \Vert ^s_{\mathcal {Q}_{\Psi ,q,b}}\\&\le C^s_\phi \sum _{k\in \mathbb {Z}} \gamma _b^s\big (\mathbb {P}(\tau _k<\infty )\big )|\mu _k|^s \le C^s_\phi \big \Vert f\big \Vert ^s_{ \mathcal {Q}_{\Phi ,s,b} }. \end{aligned}$$

This completes the proof of the theorem. $\square$

Based on the argument similar to the proof of Theorem 14 and using Lemma 18, we obtain the following result.

Theorem 15

Let $\{\mathcal {F}_n\}_{ n \ge 0}$ be regular, $0 < s\le \min \{1,q\}$ and b be a slowly varying function. Let $\Psi$ be an Orlicz function of lower type $p_\Psi ^-$ and upper type $p_\Psi ^+$ satisfying $s<p_\Psi ^- \le p_\Psi ^+<\infty$ and let $\Phi$ be an Orlicz function of lower type $p_\Phi ^-$ and upper type $p_\Phi ^+$ satisfying $0<p_\Phi ^- \le p_\Phi ^+<\infty$. If there exists $\alpha >0$ such that $\Phi ^{-1} (t) \le \Psi ^{-1} (t) \cdot t^\alpha$ for any $t \ge 1$, then there exists a constant C such that

$$\begin{aligned} \Vert I_\alpha f\Vert _{\mathcal {Q}_{\Psi ,q,b}}\le C\Vert f\Vert _{\mathcal {Q}_{\Phi ,s,b}} \end{aligned}$$

for all $f\in \mathcal {Q}_{\Phi ,s,b}$.

As an application of Theorem 14, we obtain the following result:

Corollary 13

Let $\{\mathcal {F}_n\}_{ n \ge 0}$ be regular, $0 < s\le \min \{1,q\}$, $\Psi$ be an Orlicz function of lower type $p_\Psi ^-$ and upper type $p_\Psi ^+$ satisfying $s<p_\Psi ^- \le p_\Psi ^+<\infty$ and let $\Phi$ be an Orlicz function of lower type $p_\Phi ^-$ and upper type $p_\Phi ^+$ satisfying $0<p_\Phi ^- \le p_\Phi ^+<\infty$. If there exists an Orlicz function $\phi$ with upper type $p\in (0,1)$ such that $\Phi ^{-1} (t) \cdot \phi (1/t) \le \Psi ^{-1} (t)$ for any $t \ge 1$, then there exists a constant C such that

$$\begin{aligned} \Vert I_\phi f\Vert _{\mathcal {Q}_{\Psi ,q}}\le C\Vert f\Vert _{\mathcal {Q}_{\Phi ,s}} \end{aligned}$$

for all $f\in \mathcal {Q}_{\Phi ,s}$.

Proof

Let $b\equiv 1$ in Theorem 14. $\square$

The next corollary follows from Theorem 15.

Corollary 14

Suppose that $\{\mathcal {F}_n\}_{ n \ge 0}$ is regular, $0<q_1\le \min \{1,q_2, p_2\}$, $0<p_1<p_2<\infty$, $\alpha \ge {1}/{p_1}- {1}/{p_2}$ and b is a slowly varying function. Then there exists a constant C such that

$$\begin{aligned} \Vert I_\alpha f\Vert _{\mathcal {Q}_{p_2,q_2,b}} \le C \Vert f\Vert _{\mathcal {Q}_{p_1,q_1,b}} \end{aligned}$$

for all $f\in \mathcal {Q}_{p_1,q_1,b}$.

Remark 15

Note that this result was proved in [31] and [12] for $1<p_1<p_2<\infty$. Moreover, the slowly varying function b is not necessarily non-decreasing. Thus, Corollary 14 improves and completes Theorem 4.4 in [39].