B-Valued Martingale Hardy–Lorentz–Karamata Spaces

Abstract

In this paper, we investigate the Hardy–Lorentz–Karamata spaces for Banach space-valued martingales. Relying on the geometrical properties of the underlying Banach spaces, we establish the atomic decompositions and characterize the dual spaces of these spaces. We also obtain some martingale inequalities in the setting of Hardy–Lorentz–Karamata spaces.

1 Introduction

As a generalization of classical Lebesgue spaces, Lorentz spaces were introduced in [31]. In virtue of its wide applications in various fields, now it has become an important topic in modern analysis; see, e.g.,[1,2,3,4, 8, 38]. Weisz [38] introduced the martingale Hardy–Lorentz spaces $H_{p,q}^s$ and discussed the real interpolation between them. Moreover, it is proved that the dual of $H_{p,q}^s$ can be characterized as $H_{p',q'}^s$ where $1<p<\infty$, $1\le q<\infty$, $\frac{1}{p}+\frac{1}{p'}=1$ and $\frac{1}{q}+\frac{1}{q'}=1$. Jiao et al. [17] established the atomic decompositions of martingale Hardy–Lorentz spaces and obtained some continuous embeddings among these spaces, which improves some results of [38]. Now, one natural question arises, that is, what is the duality of $H_{p,q}^s$ for $0<p\le 1$ and $1<q<\infty$. This problem was recently resolved by Jiao et al. [21]. In [21], by aptly introducing a new generalized BMO martingale space, the authors successfully characterized the dual of Hardy–Lorentz spaces $H_{p,q}^s$ for $0<p\le 1$ and $1<q<\infty$. The technique developed in [21] has also been used to deal with some other problems (see, e.g., [22, 40, 46]).

The family of Lorentz–Karamata spaces, a generalization of the Lorentz spaces and the Lorentz–Zygmund spaces, has acquired considerable attentions over the past decades; see [8, 33] for some important results on Lorentz–Karamata spaces and their applications to Bessel and Riesz potentials. By using interpolation, some important results in analysis, such as the mapping properties for the Fourier transform, the Fourier integral operators, the oscillatory integral operators and the Fourier restriction theorem had been extended to Lorentz–Karamata spaces, see [10, 11]. Ho [12] firstly combined the martingale theory with the Lorentz–Karamata spaces, who introduced the martingale Hardy–Lorentz–Karamata spaces and discussed the atomic decomposition, dual space and interpolations of these spaces. Following the same idea used in [21], Jiao et al. [19] further studied the martingale Hardy–Lorentz–Karamata spaces and improved the main results of [12]. Very recently, inspired by [12, 19], Zhou et al. [46] defined the martingale weak Orlicz–Karamata–Hardy spaces and obtained some interesting results on these spaces; Wu et al. [40] considered the modular inequalities in the frame of martingale Orlicz–Karamata spaces.

The main purpose of this paper is to study Hardy–Lorentz–Karamata spaces for Banach space-valued (B-valued) martingales and extend the results in [12, 19] to the $\mathbf{B}$-valued martingale setting.

It is well known that the study of the B-valued martingale can be tracked back to Pisier’s fundamental paper [34]. Since then, the B-valued martingale theory has attracted a lot of attentions. For instance, martingale transforms and differential subordinations for B-valued martingales were discussed by Burkholder in [5] and [6]; Liu [26, 27] introduced the p-variation operator and discussed various B-valued martingale inequalities; Yu [44, 45] recently investigated the dual spaces of Orlicz–Hardy spaces and weak Orlicz–Hardy spaces for B-valued martingales; Liu et al. [24] studied B-valued martingale Hardy–Lorentz spaces. We refer to the very recent monograph [35] by Pisier for more information on martingales and Fourier analysis in Banach spaces.

The results mentioned above, and also many other B-valued martingale results (see, e.g., [15, 16, 18, 34, 35, 41]), are closely connected with the geometrical properties of the underlying spaces. Our conclusions in this paper have no exception. We should also mention that our proofs heavily depend on the establishment of atomic decompositions of Hardy–Lorentz–Karamata spaces for B-valued martingales. As an important tool in martingale theory, the atomic decompositions were first introduced by Herz in [9] for scalar-valued martingales. Afterward, this method was generalized by Weisz [37,38,39] and developed by many other authors (see, e.g., [14, 17, 28, 32]). As for $\mathbf B$-valued martingales, Liu et al. [29, 30] investigated the atomic decompositions of B-valued martingale Hardy spaces and studied the continuous embeddings between these spaces with small index; Yu [42, 43] established the duals of B-valued martingale Hardy spaces with the help of atomic decompositions.

This article is divided into five sections. Some notations and basic knowledge will be introduced in Sect. 2. In Sect. 3, we formulate atomic decompositions for $\mathbf B$-valued martingale Hardy–Lorentz–Karamata spaces.

 As usual, these theorems rely on the geometrical properties of the underlying Banach space B. Applying atomic decompositions established in Sect. 3, we prove three dualities in Sect. 4. In the last section, we discuss the continuous embeddings among martingale Hardy–Lorentz–Karamata spaces.

Throughout this paper, the sets of integers, nonnegative integers and complex numbers are always denoted by $\mathbb {Z}$, $\mathbb {N}$ and $\mathbb {C}$, respectively. We use C to denote a positive constant which may vary from line to line. The symbol $\subset$ means the continuous embedding and $f\approx g$ stands for $C^{-1}g\le f\le Cg$. We say f is equivalent to g if $f\approx g$.

2 Notation and Preliminaries

2.1 Lorentz–Karamata Spaces

In this section, we recall the definition of Lorentz–Karamata spaces and state some properties of these function spaces. The reader is referred to [8, 33] for more information about the Lorentz–Karamata spaces.

Definition 2.1

([8], Definition 3.4.32) A Lebesgue measurable function $b:[1,\infty )\rightarrow (0,\infty )$ is said to be a slowly varying function if for any given $\varepsilon >0$, the function $t^{\varepsilon }b(t)$ is equivalent to a nondecreasing function and the function $t^{-\varepsilon }b(t)$ is equivalent to a nonincreasing function on $[1,\infty )$.

According to the definition above, we define

$$\begin{aligned} \gamma _b(t)=b(\max (t,t^{-1})),\quad t>0. \end{aligned}$$

Remark 2.2

(i)It is clear that $\gamma _b$ is nonincreasing on (0, 1] if b is a nondecreasing function. (ii) For any given $\varepsilon >0$, the function $t^{\varepsilon }\gamma _b(t)$ is equivalent to a nondecreasing function and the function $t^{-\varepsilon }\gamma _b(t)$ is equivalent to a nonincreasing function on $(0,\infty )$.

Let $(\Omega ,{\mathcal {F}},\mathbb {P})$ be a complete probability space and B denote a Banach space with the norm $\Vert \cdot \Vert$. For any measurable function f defined on $(\Omega ,{\mathcal {F}},\mathbb {P})$ and taking values in B, write

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

and

$$\begin{aligned} f^*(t)=\inf \{s\ge 0: \lambda _f(s)\le t\}, \ \ (t\ge 0), \quad (\inf \emptyset =\infty ). \end{aligned}$$

Definition 2.3

([8]) Let $0<p<\infty$, $0<q \le \infty$ and b be a slowly varying function. The Lorentz–Karamata space $L_{p,q,b}$ consists of those measurable functions that satisfy $\Vert f\Vert _{{p,q,b}}<\infty$, where

$$\begin{aligned} \Vert f\Vert _{{p,q,b}}= \Bigg (\int _0^1 \big (t^{\frac{1}{p}}\gamma _b(t) f^*(t)\big )^q \frac{dt}{t} \Bigg )^{1/q},\quad 0<q<\infty , \end{aligned}$$

and

$$\begin{aligned} \Vert f\Vert _{{p,\infty ,b}}= \sup _{0\le t\le 1} t^{\frac{1}{p}}\gamma _b(t) f^*(t),\quad q=\infty . \end{aligned}$$

Remark 2.4

(i) If we take $b\equiv 1$ in the definition above, $L_{p,q,b}$ becomes the Lorentz space $L_{p,q}$. (ii) The quasi-norm $\Vert \cdot \Vert _{p,q ,b}$ has another equivalent characterization reads as follows ([12, Lemma 2.4]).

$$\begin{aligned} \Vert f\Vert _{{p,q,b}} \approx \Bigg (\int _0^\infty \Big (\lambda _f(s)^{\frac{1}{p}}\gamma _b\big (\lambda _f(s)\big )s\Big )^q \frac{ds}{s} \Bigg )^{1/q},\quad 0<q<\infty , \end{aligned}$$

and

$$\begin{aligned} \Vert f\Vert _{{p,\infty ,b}}\approx \sup _{s>0}\lambda _f(s)^{\frac{1}{p}}\gamma _b\big (\lambda _f(s)\big )s,\quad q=\infty . \end{aligned}$$

The following embedding relationship is useful in this paper, the reader is referred to Theorem 3.4.48 of [8].

Lemma 2.5

Let $(\Omega , \mathcal {F}, \mathbb {P})$ be a finite measure space, $p_1,p_2,q_1,q_2 \in (0,\infty )$ with $p_1>p_2$, and let $b_1,b_2$ be slowly varying functions. Then $L_{p_1,q_1,b_1} \subset L_{p_2,q_2,b_2}$.

2.2 B-valued Martingale Hardy–Lorentz–Karamata Spaces

Let $\{\mathcal {F}_n\}_{n\ge 0}$ be a nondecreasing sequence of sub-$\sigma$-algebras of $\mathcal {F}$ with $\mathcal {F}=\sigma (\bigcup _n\mathcal {F}_n)$. The expectation operator and the conditional expectation operator relative to $\mathcal {F}_n$ are denoted by $\mathbb {E}$ and $\mathbb {E}_n$, respectively. A sequence of B-valued random variables $f=(f_n)_{n\ge 0}$ is called a B-valued martingale if $\mathbb E_n (f_{n+1})=f_n$ for arbitrary $n\ge 0.$ Let $f=(f_n)_{n\ge 0}$ be a B-valued martingale adapted to $\{\mathcal {F}_n\}_{n\ge 0}$ such that $f_0=0$. For $n\ge 0$, write $df_n=f_n-f_{n-1}$ (with convention that $f_{-1}=0)$. For $1\le p<\infty$, we define the maximal function, the p-variation and the conditional p-variation of a B-valued martingale f as follows.

$$\begin{aligned} M_n(f)= & {} \sup \limits _{1\le i\le n}\Vert f_i\Vert , \quad M(f)=\sup \limits _{i\ge 1}\Vert f_i\Vert ;\\ S_n^p(f)= & {} \left( \sum \limits _{i=1}^n\Vert df_i\Vert ^p\right) ^{\frac{1}{p}},\quad S^p(f)=\left( \sum \limits _{i=1}^\infty \Vert df_i\Vert ^p\right) ^{\frac{1}{p}};\\ s_n^p(f)= & {} \left( \sum \limits _{i=1}^n\mathbb {E}_{i-1}\Vert df_i\Vert ^p\right) ^{\frac{1}{p}},\quad s^p(f)=\left( \sum \limits _{i=1}^\infty \mathbb {E}_{i-1}\Vert df_i\Vert ^p\right) ^{\frac{1}{p}}. \end{aligned}$$

Let $0<r_1<\infty , 0<r_2\le \infty$ and b be a slowly varying function. Let $\Gamma$ be the set of all sequences $(\lambda _n)_{n\ge 0}$ of nondecreasing, nonnegative and adapted functions with $\lambda _{\infty }=\lim _{n\rightarrow \infty }\lambda _n\in L_{r_1,r_2,b}$. We now give the definition of $\mathbf B$-valued martingale Hardy–Lorentz–Karamata space.

Definition 2.6

Let $0<r_1<\infty$, $0<r_2\le \infty$, $1\le p<\infty$ and b be a slowly varying function. Define

$$\begin{aligned} H_{r_1,r_2,b}(\mathbf{B})= & {} \big \{f=(f_n)_{n\ge 0}:\Vert f\Vert _{H_{r_1,r_2,b}(\mathbf{B})}=\Vert Mf\Vert _{r_1,r_2,b}<\infty \big \}.\\ H_{r_1,r_2,b}^{s^p}(\mathbf{B})= & {} \big \{f=(f_n)_{n\ge 0}:\Vert f\Vert _{H_{r_1,r_2,b}^{s^p}(\mathbf{B})}=\Vert s^p(f)\Vert _{r_1,r_2,b}<\infty \big \}.\\ H_{r_1,r_2,b}^{S^p}(\mathbf{B})= & {} \big \{f=(f_n)_{n\ge 0}:\Vert f\Vert _{H_{r_1,r_2,b}^{S^p}(\mathbf{B})}=\Vert S^p(f)\Vert _{r_1,r_2,b}<\infty \big \}.\\ Q_{r_1,r_2,b}^{S^p}(\mathbf{B})= & {} \big \{f=(f_n)_{n\ge 0}:\exists (\lambda _n)\in \Gamma , \ s.t.\ S_n^p(f)\le \lambda _{n-1}, n\ge 1, \lambda _{\infty }\in L_{r_1,r_2,b}\big \}, \end{aligned}$$

where

$$\begin{aligned} \Vert f\Vert _{Q_{r_1,r_2,b}^{S^p}(\mathbf{B})}= & {} \inf \limits _{(\lambda _n)\in \Gamma }\Vert \lambda _{\infty }\Vert _{r_1,r_2,b}<\infty .\\ D_{r_1,r_2,b}(\mathbf{B})= & {} \big \{f=(f_n)_{n\ge 0}:\exists (\lambda _n)\in \Gamma , \ s.t.\ \Vert f_n\Vert \le \lambda _{n-1}, n\ge 1, \lambda _{\infty }\in L_{r_1,r_2,b}\big \}, \end{aligned}$$

where

$$\begin{aligned} \Vert f\Vert _{D_{r_1,r_2,b}(\mathbf{B})}=\inf \limits _{(\lambda _n)\in \Gamma }\Vert \lambda _{\infty }\Vert _{r_1,r_2,b}<\infty . \end{aligned}$$

Remark 2.7

If we take $b\equiv 1$ in the definitions above, we obtain the $\mathbf{B}$-valued martingale Hardy–Lorentz spaces $H_{r_1,r_2}(\mathbf{B})$, $H_{r_1,r_2}^{s^p}(\mathbf{B})$ et al. (see [24]); and take $r_1=r_2=r$ and $b\equiv 1$ , the definitions come back to $\mathbf{B}$-valued martingale Hardy spaces $H_{r}(\mathbf{B})$, $H_{r}^{s^p}(\mathbf{B})$ et al. (see [28]).

The following two lemmas play an important role in B-valued martingale theory, which were firstly studied by Hoffmann-Jørgensen and Pisier [15] (see also [35, Corollary 10.23 and Theorem 10.59]). These lemmas state the facts that the usual results of B-valued martingale are connected closely with the geometrical properties of the underlying spaces. The reader is referred to [23, 35] for the definitions of uniform smoothness, uniform convexity and Radon Nikodým property (briefly by RNP).

Lemma 2.8

Let $\mathbf{B}$ be a Banach space and $1<p\le 2$. Then the following are equivalent:

• (i) $\mathbf{B}$ is isomorphic to a p-uniformly smooth space;

• (ii) There exists a constant $C_p>0$ only depending on p such that

  $$\begin{aligned} \mathbb E_n(\Vert f_m-f_n\Vert ^p)\le C_p\mathbb E_n\left( \sum \limits _{i=n+1}^{m}\Vert df_i\Vert ^p\right) , \ \quad \forall \ 0\le n\le m, \end{aligned}$$

  for every $\mathbf{B}$-valued martingale $f=(f_n)_{n\ge 0}$;

• (iii) For every $\mathbf{B}$-valued martingale $f=(f_n)_{n\ge 0}$, if  $\mathbb E\big (\sum \limits _{n=1}^{\infty }\Vert df_n\Vert ^p\big )< \infty$, then $f=(f_n)_{n\ge 0}$ converges in probability;

• (iv) There exists a constant C such that for every $\mathbf{B}$-valued martingale $f=(f_n)_{n\ge 0}$ and $1\le r<\infty$,

  $$\begin{aligned} \Vert M(f)\Vert _r\le C \Vert S^p(f)\Vert _r. \end{aligned}$$

Lemma 2.9

Let $\mathbf{B}$ be a Banach space, $2\le q<\infty$. Then the following are equivalent:

• (i) $\mathbf{B}$ is isomorphic to a q-uniformly convex space;

• (ii) There exists a constant $C_q>0$ only depending on q such that

  $$\begin{aligned} \mathbb E_n\Big (\sum \limits _{i=n+1}^{m}\Vert df_i\Vert ^q\Big )\le C_q\mathbb E_n(\Vert f_m-f_n\Vert ^q), \ \quad \forall \ 0\le n\le m, \end{aligned}$$

  for every $\mathbf{B}$-valued martingale $f=(f_n)_{n\ge 0}$;

• (iii) For every uniformly bounded $\mathbf{B}$-valued martingale $f=(f_n)_{n\ge 0}$, $S^q(f)<\infty , a.e..$

For the convenience of Sect. 5, we introduce the definition of dyadic martingales. Let $((0,1],{\mathcal {F}}, \mu )$ be a probability space, where $\mu$ is Lebesgue measure and sub-algebras $\{{\mathcal {F}}_n\}_{n\ge 0}$ are generated by:

$$\begin{aligned}{\mathcal {F}}_n=\sigma \text{-algebra } \text{ generated } \text{ by } \text{ atoms } \big (\frac{i}{2^n},\frac{i+1}{2^n}\big ],i=0,\cdots ,2^n-1. \end{aligned}$$

Remind that all martingales adapted to $\{\mathcal {F}_n\}_{n\ge 0}$ are called dyadic martingales.

Remark 2.10

([35], Corollaries 10.7 and 10.23) In (ii) of the two lemmas above, if we replace $\mathbf{B}$-valued martingale by $\mathbf{B}$-valued dyadic martingale, the equivalences still hold.

In the end of this section, we give two lemmas which are useful in this paper to verify that a function is in Lorentz–Karamata spaces $L_{p,q,b}$. Note that the original idea is from [1].

Lemma 2.11

([19], Lemma 2.12) Let $0<p<\infty$, $0<q\le \infty$ and b be a slowly varying function. Assume that the nonnegative sequence $\{2^k\mu _k\}\in l_q$. Further suppose that the nonnegative function $\varphi$ verifies the following property: there exists $0<\varepsilon <1$ such that, given an arbitrary integer $k_0$, we have $\varphi \le \psi _{k_0}+\eta _{k_0}$, where $\psi _{k_0}$ is essentially bounded and satisfies $\Vert \psi _{k_0}\Vert _\infty \le C2^{k_0}$, and

$$\begin{aligned} 2^{k_0\varepsilon p}\mathbb {P}(\eta _{k_0}>2^{k_0})\gamma _b^p\big (\mathbb {P}(\eta _{k_0}>2^{k_0})\big )\le C\sum \limits _{k=k_0}^\infty (2^{k\varepsilon }\mu _k)^p. \end{aligned}$$

Then $\varphi \in L_{p,q,b}$ and $\Vert \varphi \Vert _{p,q,b} \le C\Vert \{2^k\mu _k\}\Vert _{l_q}$.

Lemma 2.12

Let $0<p<\infty$, $0<q\le \infty$ and b be a slowly varying function and let the nonnegative sequence $\{\mu _k\}$ be such that $\{2^k\mu _k\}\in l_q$. Further suppose that the nonnegative function $\varphi$ satisfies the following property: there exists $0<\varepsilon <1$ such that, given an arbitrary integer $k_0$, we have $\varphi \le \psi _{k_0}+\eta _{k_0}$, where $\psi _{k_0}$ and $\eta _{k_0}$ satisfy

$$\begin{aligned} 2^{k_0p}\mathbb {P}^\varepsilon (\psi _{k_0}>2^{k_0})\gamma _b^{\varepsilon p}\big (\mathbb {P}(\psi _{k_0}>2^{k_0})\big )\le & {} C\sum \limits _{k=-\infty }^{k_0}(2^k\mu _k^\varepsilon )^p, \ \ \ 0<\varepsilon <\min (1,\frac{q}{p}),\\ 2^{k_0\varepsilon p}\mathbb {P}(\eta _{k_0}>2^{k_0})\gamma _b^{p}\big (\mathbb {P}(\eta _{k_0}>2^{k_0})\big )\le & {} C\sum \limits _{k=k_0}^\infty (2^{k\varepsilon }\mu _k)^p. \end{aligned}$$

Then $\Vert \psi _{k_0}\Vert _{p,q,b}\le C\Vert \{2^k\mu _k\}\Vert _{l_q}$ and $\Vert \eta _{k_0}\Vert _{p,q,b}\le C\Vert \{2^k\mu _k\}\Vert _{l_q}$. Moreover, $\varphi \in L_{p,q,b}$ and $\Vert \varphi \Vert _{p,q,b}\le C\Vert \{2^k\mu _k\}\Vert _{l_q}$.

Remark 2.13

Lemma 2.12 was firstly proved for $q=\infty$ in [25, Lemma 2.16]. The case $0<q<\infty$ can be treated similarly and we skip the details.

3 Atomic Decompositions

We formulate atomic decompositions for $\mathbf{B}$-valued martingale Hardy–Lorentz–Karamata spaces in this section. Let ${\mathcal {T}}$ be the set of all stopping times adapted to $\{{\mathcal {F}}_n\}_{n\ge 0}$. For any stopping time $\nu \in \mathcal {T}$ and a B -valued martingale $f=(f_n)_{n\ge 0}$ . Denote stopped martingale by $f^\nu =(f_n^\nu )_{n\ge 0}$, where $f^{\nu }_n=\sum _{i=0}^n\chi _{\{i\le \nu \}}df_i$.

Definition 3.1

Let $1\le p<\infty , 0<r<\infty$. A $\mathbf{B}$-valued measurable function a is called an atom of the type $(1,r,\infty ;p)$ ($(2,r,\infty ;p)$  or  $(3,r,\infty )$, respectively), if there exists a stopping time $\nu \in \mathcal {T}$($\nu$is called the stopping time associated witha) such that

• (i) $a_n=\mathbb {E}_na=0$, $\forall \, n\le \nu$;

• (ii) $\Vert s^p(a)\Vert _{\infty }$ ($\Vert S^p(a)\Vert _\infty$or$\Vert M(a)\Vert _\infty$, respectively)$\le \mathbb {P}(\nu <\infty )^{-\frac{1}{r}}$.

Theorem 3.2

Let $\mathbf{B}$ be a Banach space, $1<p\le 2$, $0<r_1\le p$, $0<r_2\le \infty$ and b be a nondecreasing slowly varying function. Then the following statements are equivalent:

• (i) $\mathbf{B}$ is isomorphic to a p-uniformly smooth space;

• (ii) For every $f=(f_n)_{n\ge 0}\in H_{r_1,r_2,b}^{s^p}(\mathbf{B})$, there exist a sequence $(a^k)_{k\in \mathbb Z}$ of $(1,r_1,\infty ;p)$-atoms and a sequence $(\mu _k)_{k\in \mathbb Z}\in l_{r_2}$ of positive numbers satisfying $\mu _k=A\cdot 2^k\mathbb {P}(\nu _k<\infty )^{\frac{1}{r_1}}$ (where A is a positive constant and $\nu _k$ is the stopping time associated with $a^k$) such that

  $$\begin{aligned} f_n=\sum \limits _{k\in \mathbb Z}\mu _k a_n^k, \ \quad a.e., \ \quad \forall n\ge 0, \end{aligned}$$

  (3.1)

  and

  $$\begin{aligned} \Vert f\Vert _{H_{r_1,r_2,b}^{s^p}(\mathbf{B})}\approx \inf \Vert \{\gamma _b\big (\mathbb P(\nu _k< \infty )\big )\mu _k\}_{k\in \mathbb Z}\Vert _{l_{r_2}},\ \quad \sup \limits _{k\in \mathbb {Z}}\Vert M(a^k)\Vert _{r_1}<\infty , \end{aligned}$$

  where $a_n^k=\mathbb {E}_n a^k$ and the infimum is taken over all the decompositions of (3.1).

Proof

$\mathrm{(i)}\Rightarrow \mathrm{(ii)}$. Let $f=(f_n)_{n\ge 0}\in H_{r_1,r_2,b}^{s^p}(\mathbf B )$. For every $k\in \mathbb {Z}$, we define the stopping time by

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

It is easy to check that $\nu _k$ is nondecreasing and for every $n\in \mathbb N$,

$$\begin{aligned} \sum \limits _{k\in \mathbb {Z}}(f_n^{\nu _{k+1}}-f_n^{\nu _k})=\sum _{m=0}^n\Big (\sum _{k\in \mathbb {Z}}\chi _{\{\nu _k<m\le \nu _{k+1}\}}df_m\Big )=f_n. \end{aligned}$$

For each $k\in \mathbb Z$ and $n\in \mathbb {N}$, let

$$\begin{aligned} \mu _k=3\cdot 2^k\mathbb {P}(\nu _k<\infty )^{\frac{1}{r_1}},\ \ \ a_n^k=\frac{f_n^{\nu _{k+1}}-f_n^{\nu _k}}{\mu _k}. \end{aligned}$$

Set $a_n^k=0$ if $\mu _k=0$. Then $a^k=(a_n^k)_{n\ge 0}$ is a B-valued martingale for any fixed $k\in \mathbb {Z}$. Since $s^p(f^{\nu _k})=s^p_{\nu _k}(f)\le 2^k$ and $s^p(a^k)=0$ on the set $\{\nu _k=\infty \}$, by the sublinearity of $s^p(\cdot )$, we deduce

$$\begin{aligned} s^p(a^k)&=s^p(a^k\chi _{\{\nu _k<\infty \}})\le \frac{1}{\mu _k}(s^p(f^{\nu _{k+1}})+s^p(f^{\nu _k}))\chi _{\{\nu _k<\infty \}}\\&\le \frac{1}{\mu _k}(2^{k+1}+2^k)\chi _{\{\nu _k<\infty \}}=\mathbb {P}(\nu _k<\infty )^{-\frac{1}{r_1}} \chi _{\{\nu _k<\infty \}}. \end{aligned}$$

Then $\Vert s^p(a^k)\Vert _\infty \le \mathbb {P}(\nu _k<\infty )^{-\frac{1}{r_1}}$. According to condition $\mathrm{(i)}$ and Lemma 2.8(iv), using Hölder’s inequality, we get

$$\begin{aligned} \Vert M(a^k)\Vert _p\le C \Vert S^p(a^k)\Vert _p= C \Vert s^p(a^k)\Vert _p\le C\mathbb {P}(\nu _k<\infty )^{\frac{1}{p}-\frac{1}{r_1}}, \end{aligned}$$

which means $a^k$ is $L_p$-bounded for each $k\in \mathbb {Z}$. Furthermore, condition (i) implies B has the RNP (see [28, 35]), then $a_n^k$ converges almost everywhere as $n\rightarrow \infty$. We still denote its limit by $a^k$. Then for $n\le \nu _k$, $\mathbb {E}_na^k=a_n^k=0$. So $a^k$ is a $(1,r_1,\infty ; p)$-atom and (3.1) holds. Since $r_1\le {p}$, applying Hölder’s inequality, for any fixed $k\in \mathbb Z$, we have

$$\begin{aligned} \mathbb {E}(M(a^k)^{r_1})=\mathbb {E}\big (M(a^k)^{r_1}\chi _{\{\nu _k<\infty \}}\big )\le \big (\mathbb {E}(M(a^k)^p)\big )^{\frac{r_1}{p}} \mathbb {P}(\nu _k<\infty )^{1-\frac{r_1}{p}}\le C. \end{aligned}$$

Then $\sup _{k\in \mathbb {Z}}\Vert M(a^k)\Vert _{r_1}<\infty .$

We now estimate $\Vert \{\gamma _b\big (\mathbb {P}(\nu _k<\infty )\big )\mu _k\}_{k\in \mathbb {Z}}\Vert _{l_{r_2}}$ for $0<r_2\le {\infty }$. Firstly, we deal with $r_2=\infty$.

$$\begin{aligned} \Vert \{\gamma _b\big (\mathbb {P}(\nu _k<\infty )\big )\mu _k\}_{k\in \mathbb {Z}}\Vert _\infty= & {} \sup _{k\in \mathbb {Z}}|\big (\gamma _b\big (\mathbb {P}(\nu _k<\infty )\big )\mu _k|\\= & {} 3\cdot \sup \limits _{k\in \mathbb {Z}}2^k \mathbb {P}(\nu _k<\infty )^{\frac{1}{r_1}}\gamma _b\big (\mathbb {P}(\nu _k<\infty )\big )\\= & {} 3\cdot \sup \limits _{k\in \mathbb {Z}}2^k \mathbb {P}(s^p(f)>2^k)^{\frac{1}{r_1}}\gamma _b\big (\mathbb {P}(s^p(f)>2^k\big )\\\le & {} C\Vert s^p(f)\Vert _{r_1,\infty ,b}=C\Vert f\Vert _{H_{r_1,\infty ,b}^{s^p}(\mathbf{B})}. \end{aligned}$$

For $0<r_2<\infty$, since $\{\nu _k<\infty \}=\{s^p(f)>2^k\}$, we obtain

$$\begin{aligned}&\left( \sum \limits _{k\in \mathbb {Z}}\gamma _b^{r_2}\big (\mathbb {P}(\nu _k<\infty )\big )|\mu _k|^{r_2}\right) ^{\frac{1}{r_2}}\\&\quad =3\left( \sum \limits _{k\in \mathbb {Z}}2^{kr_2}\mathbb {P}(\nu _k<\infty )^{\frac{r_2}{r_1}}\gamma _b^{r_2}\big (\mathbb {P}(\nu _k<\infty )\big )\right) ^{\frac{1}{r_2}}\\&\quad \le C\left( \sum \limits _{k\in \mathbb {Z}}\int _{2^{k-1}}^{2^k}y^{r_2-1}dy \mathbb {P}(s^p(f)>2^k)^{\frac{r_2}{r_1}}\gamma _b^{r_2}\big (\mathbb {P}(s^p(f)>2^k)\big )\right) ^{\frac{1}{r_2}}\\&\quad \le C\left( \sum \limits _{k\in \mathbb {Z}}\int _{2^{k-1}}^{2^k}y^{r_2-1}\mathbb {P}(s^p(f)>y)^{\frac{r_2}{r_1}}\gamma _b^{r_2}\big (\mathbb {P}(s^p(f)>y)\big )dy\right) ^{\frac{1}{r_2}}\\&\quad \le C\left( \int _0^\infty y^{r_2-1}\mathbb {P}(s^p(f)>y)^{\frac{r_2}{r_1}}\gamma _b^{r_2}\big (\mathbb {P}(s^p(f)>y)\big )dy\right) ^{\frac{1}{r_2}}\\&\quad \le C\Vert f\Vert _{H_{r_1,r_2,b}^{s^p}(\mathbf{B})}. \end{aligned}$$

Then $\Vert \{\gamma _b\big (\mathbb {P}(\nu _k<\infty )\big )\mu _k\}_{k\in \mathbb {Z}}\Vert _{l_{r_2}}\le C\Vert f\Vert _{H_{r_1,r_2,b}^{s^p}(\mathbf{B})}$.

On the other hand, for any integer $k_0$, set

$$\begin{aligned} f=\sum \limits _{k\in \mathbb {Z}}\mu _ka^k=g+h, \end{aligned}$$

where $g=\sum _{k=-\infty }^{k_0-1}\mu _ka^k$ and $h=\sum _{k=k_0}^\infty \mu _ka^k$. In view of the sublinearity of $s^p(\cdot )$, we get

$$\begin{aligned} \Vert s^p(g)\Vert _\infty\le & {} \sum \limits _{k=-\infty }^{k_0-1}\mu _k\Vert s^p(a^k)\Vert _\infty \le \sum \limits _{k=-\infty }^{k_0-1}\mu _k\mathbb {P}(\nu _k<\infty )^{-\frac{1}{r_1}}\\\le & {} A\sum \limits _{k=-\infty }^{k_0-1}2^k=A\cdot 2^{k_0}. \end{aligned}$$

Since $s^p(h)\le \sum _{k=k_0}^\infty |\mu _k|s^p(a^k)$ and $\{s^p(a^k)>0\}\subset \{\nu _k<\infty \}$. We deduce

$$\begin{aligned} \{s^p(h)>0\}\subset \bigcup \limits _{k=k_0}^\infty \{s^p(a^k)>0\}\subset \bigcup \limits _{k=k_0}^\infty \{\nu _k<\infty \}. \end{aligned}$$

By Remark 2.2 (ii) and (i), for any $0<\varepsilon <1$, we have

$$\begin{aligned}&2^{k_0\varepsilon r_1}\mathbb {P}(s^p(h)>2^{k_0})\gamma _b^{r_1}\big (\mathbb {P}(s^p(h)>2^{k_0})\big )\nonumber \\&\quad \le C2^{k_0\varepsilon r_1}\mathbb {P}\left( \bigcup _{k=k_0}^{\infty }\{\nu _k<\infty \}\right) \gamma _b^{r_1}\left( \mathbb {P}\left( \bigcup _{k=k_0}^{\infty }\{\nu _k<\infty \}\right) \right) \nonumber \\&\quad \le C2^{k_0\varepsilon r_1}\sum \limits _{k=k_0}^\infty \mathbb {P}(\nu _k<\infty )\gamma _b^{r_1}\big (\mathbb {P}(\nu _k<\infty )\big )\nonumber \\&\quad \le C\sum \limits _{k=k_0}^\infty \left( 2^{k\varepsilon }\mathbb {P}(\nu _k<\infty )^{\frac{1}{r_1}}\gamma _b\big (\mathbb {P}(\nu _k<\infty )\big )\right) ^{r_1}. \end{aligned}$$

(3.2)

By Lemma 2.11, we obtain $s^p(f)\in L_{r_1,r_2,b}$ and

$$\begin{aligned} \quad \Vert s^p(f)\Vert _{r_1,r_2,b}\le C\Vert \{2^k\mathbb {P}(\nu _k<\infty )^{\frac{1}{r_1}}\gamma _b\big (\mathbb {P}(\nu _k<\infty )\big )\}_{k\in \mathbb {Z}}\Vert _{l_{r_2}}. \end{aligned}$$

Thus,

$$\begin{aligned} \Vert f\Vert _{H_{r_1,r_2,b}^{s^p}(\mathbf{B})}\approx \inf \Vert \{\gamma _b\big (\mathbb {P}(\nu _k<\infty )\big ){\mu _k}\}_{k\in \mathbb {Z}}\Vert _{l_{r_2}}, \end{aligned}$$

where the infimum is taken over all the decompositions of (3.1).

$\mathrm{(ii)}\Rightarrow \mathrm{(i)}$. Suppose that $f=(f_n)_{n\ge 0}$ is a $\mathbf{B}$-valued martingale with $\sum \limits _{n=0}^{\infty }\mathbb {E}\Vert df_n\Vert ^p<\infty$. Because of

$$\begin{aligned} \Vert s^p(f)\Vert _1^p\le \Vert s^p(f)\Vert _p^p=\Vert S^p(f)\Vert _p^p=\sum \limits _{n=0}^{\infty }\mathbb {E}\Vert df_n\Vert ^p<\infty , \end{aligned}$$

namely, $f\in {H_{1,1,1}^{s^p}(\mathbf{B})}$. Then f has the decomposition as (3.1), where $(\mu _k)_{k\in \mathbb {Z}}\in l_1$ and $\sup _{k\in \mathbb {Z}}\Vert M(a^k)\Vert _1<\infty .$ The latter property assures that $(a_n^k)_{n\ge 0}$ is uniformly integrable for every $k\in \mathbb Z$. Then $(a_n^k)_{n\ge 0}$ converges to $a^k$ in $L_1(\mathbf{B})$ as $n\rightarrow \infty$ ([35], Theorem 2.9). Thus, for any $\varepsilon >0$ and a fixed $k\in \mathbb {Z}$, there exists $N_k\in \mathbb N$ such that $\mathbb E \Vert a_m^k-a_n^k\Vert <\varepsilon$ as $m,n\ge N_k.$ Since $(\mu _k)_{k\in \mathbb {Z}}\in l_1$, for the $\varepsilon$ above, there exists $k_0\in \mathbb Z$ such that $\sum _{|k|>k_0}\mu _k <\varepsilon$. Set $N=\max _{|k|\le k_0}\{N_k\}$. For $k_0\in \mathbb Z$ mentioned above and $n,m\ge N$, we get

$$\begin{aligned} \mathbb {E}&\Vert f_m-f_n\Vert \le \sum \limits _{k\in \mathbb {Z}}\mu _k\mathbb {E}\Vert a_m^k-a_n^k\Vert \\&=\sum \limits _{|k|>k_0}\mu _k\mathbb {E}\Vert a_m^k-a_n^k\Vert +\sum \limits _{|k|\le k_0}\mu _k\mathbb {E}\Vert a_m^k-a_n^k\Vert \\&\le C\sum \limits _{|k|>k_0}\mu _k+\varepsilon \sum \limits _{|k|\le k_0}\mu _k, \end{aligned}$$

which means $(f_n)_{n\ge 0}$ is a Cauchy sequence in $L_1(\mathbf{B})$. Then $(f_n)_{n\ge 0}$ is convergent in probability. According to Lemma 2.8, we get the desired result. $\square$

Remark 3.3

If $r_2\ne \infty$ in the Theorem above, then $\sum _{k=l}^m \mu _ka_n^k$ converges to $f_n$ in $H_{r_1,r_2,b}^{s^p}(\mathbf{B})$ as $m\rightarrow \infty$, $l\rightarrow -\infty$. In fact, $\sum _{k=l}^m \mu _ka_n^k=f_n^{\nu _{m+1}}-f_n^{\nu _l}$, by the sublinearity of $s^p(\cdot )$, we get

$$\begin{aligned} \big \Vert f_n-\sum _{k=l}^m\mu _k a_n^k\big \Vert _{H_{r_1,r_2,b}^{s^p}(\mathbf{B})}&\le \Vert s^p(f_n-f_n^{\nu _{m+1}})+s^p(f_n^{\nu _l})\Vert _{r_1,r_2,b}\\&\le C\big (\Vert s^p(f_n-f_n^{\nu _{m+1}})\Vert _{r_1,r_2,b}+\Vert s^p(f_n^{\nu _l})\Vert _{r_1,r_2,b}\big ). \end{aligned}$$

Apparently, $s^p(f_n-f_n^{\nu _{m+1}}),\;s^p(f_n^{\nu _{l}})\le s^p(f_n)$ and $s^p(f_n-f_n^{\nu _{m+1}}),\; s^p(f_n^{\nu _{l}})\rightarrow 0$ a.e. as $m\rightarrow \infty$, $l\rightarrow -\infty$. By using the dominated convergence theorem, we have

$$\begin{aligned} \Vert s^p(f_n-f_n^{\nu _{m+1}})\Vert _{r_1,r_2,b},\;\Vert s^p(f_n^{\nu _l})\Vert _{r_1,r_2,b} \rightarrow 0\;\;\;\; as\; m\rightarrow \infty ,\;l\rightarrow -\infty . \end{aligned}$$

Then the conclusion holds.

Theorem 3.4

Let $\mathbf{B}$ be a Banach space, $1<p\le 2$, $0<r_1\le p$, $0<r_2\le \infty$ and b be a nondecreasing slowly varying function. Then the following statements are equivalent:

• (i) $\mathbf{B}$ is isomorphic to a p-uniformly smooth space;

• (ii) For every $f=(f_n)_{n\ge 0}\in Q_{r_1,r_2,b}^{S^p}(\mathbf{B})$, there exist a sequence $(a^k)_{k\in \mathbb Z}$ of $(2,r_1,\infty ;p)$-atoms and a sequence $(\mu _k)_{k\in \mathbb Z}\in l_{r_2}$ of positive numbers satisfying $\mu _k=A\cdot 2^k\mathbb {P}(\nu _k<\infty )^{\frac{1}{r_1}}$ (where A is a positive constant and $\nu _k$ is the stopping time associated with $a^k$) such that

  $$\begin{aligned} f_n=\sum \limits _{k\in \mathbb Z}\mu _k a_n^k, \ \quad a.e., \ \quad \forall n\ge 0, \end{aligned}$$

  (3.3)
  $$\begin{aligned} \Vert f\Vert _{Q_{r_1,r_2,b}^{S^p}(\mathbf{B})}\approx \inf \Vert \{\gamma _b\big (\mathbb P(\nu _k< \infty )\big )\mu _k\}_{k\in \mathbb Z}\Vert _{l_{r_2}},\ \quad \sup \limits _{k\in \mathbb {Z}}\Vert M(a^k)\Vert _{r_1}<\infty , \end{aligned}$$

  where $a_n^k=\mathbb {E}_n a^k$ and the infimum is taken over all the decompositions of (3.3).

Proof

$\mathrm{(i)}\Rightarrow \mathrm{(ii)}$. The proof is similar to that of Theorem 3.2, then we give it in sketch. Let $f=(f_n)_{n\ge 0}\in Q_{r_1,r_2,b}^{S^p}(\mathbf{B})$. For each $k\in \mathbb Z$, define the stopping time

$$\begin{aligned} \nu _k=\inf \{n\in \mathbb {N}: \lambda _n>2^k\}, \ \ \ (\inf \emptyset =\infty ), \end{aligned}$$

(3.4)

where $(\lambda _n)_{n\ge 0}$ is the sequence in the definition of $Q_{r_1,r_2,b}^{S^p}(\mathbf{B})$. Let $\mu _k$ and $a_n^k$$(k\in \mathbb {Z})$ be the same as in the proof of Theorem 3.2. Then (3.3) holds, where $(a^k)_{k\in \mathbb Z}$ is a sequence of $(2,r_1,\infty ;p)$-atoms. Moreover, $\sup _{k\in \mathbb {Z}}\Vert M(a^k)\Vert _{r_1}<\infty$ and $\Vert \{\gamma _b\big (\mathbb P(\nu _k< \infty )\big )\mu _k\}_{k\in \mathbb Z}\Vert _{l_{r_2}}\le C\Vert f\Vert _{Q_{r_1,r_2,b}^{S^p}(\mathbf{B})}$ still hold (Note that $\{\nu _k<\infty \}=\{\lambda _{\infty }>2^k\}$ in this case). On the other hand, let

$$\begin{aligned} \lambda _n=\sum \limits _{k\in \mathbb {Z}}\mu _k\chi _{\{\nu _k\le n\}}\Vert S^p(a^k)\Vert _\infty . \end{aligned}$$

Then $(\lambda _n)_{n\ge 0}\in \Gamma$ with $S_{n+1}^p(f)\le \lambda _n$ for every $n\ge 0$. Given an integer $k_0$, let

$$\begin{aligned} \lambda _\infty =\lambda _\infty ^{(1)}+\lambda _\infty ^{(2)}, \end{aligned}$$

where

$$\begin{aligned} \lambda _\infty ^{(1)}=\sum \limits _{k=-\infty }^{k_0-1}\mu _k\chi _{\{\nu _k<\infty \}}\Vert S^p(a^k)\Vert _\infty , \ \ \lambda _\infty ^{(2)}=\sum \limits _{k=k_0}^\infty \mu _k\chi _{\{\nu _k<\infty \}}\Vert S^p(a^k)\Vert _\infty . \end{aligned}$$

Replacing $s^p(g)$ and $s^p(h)$ by $\lambda _\infty ^{(1)}$ and $\lambda _\infty ^{(2)}$ in Theorem 3.2, respectively. It follows from the similar argument that $f\in Q_{r_1,r_2,b}^{S^p}(\mathbf{B})$ and $\Vert f\Vert _{Q_{r_1,r_2,b}^{S^p}(\mathbf{B})}\approx \inf \Vert \{\gamma _b\big (\mathbb P(\nu _k< \infty )\big )\mu _k\}_{k\in \mathbb Z}\Vert _{l_{r_2}}$, where the infimum is taken over all the decompositions of (3.3).

$\mathrm{(ii)}\Rightarrow \mathrm{(i)}$. Let $f=(f_n)_{n\ge 0}$ be a $\mathbf{B}$-valued martingale with $S^p(f)\in L_{\infty }$. We have $\Vert S^p(f)\Vert _p^p\le \Vert S^p(f)\Vert _{\infty }^p<\infty$. For every $n\ge 0$, let $\lambda _n=\Vert S^p(f)\Vert _{\infty }$. It is obvious that $(\lambda _n)_{n\ge 0}\in \Gamma$ and $S_n^p(f)\le \lambda _{n-1}$. Then $\Vert f\Vert _{Q_{1,1,1}^{S^p}(\mathbf{B})}\le \Vert S^p(f)\Vert _{\infty }<\infty$. Thus, $f_n$ has the decomposition as (3.3) with $\sup _{k\in \mathbb {Z}}\Vert M(a^k)\Vert _1<\infty$. The rest of the proof is similar to that of $\mathrm{(ii)}\Rightarrow \mathrm{(i)}$ in Theorem 3.2. We conclude the proof of the theorem. $\square$

Theorem 3.5

Let $\mathbf{B}$ be a Banach space, $0<r_1<\infty$, $0<r_2\le \infty$ and b be a nondecreasing slowly varying function. Then the following statements are equivalent:

• (i) $\mathbf{B}$ has the RNP;

• (ii) For every $f=(f_n)_{n\ge 0}\in D_{r_1,r_2,b}(\mathbf{B})$, there exist a sequence $(a^k)_{k\in \mathbb Z}$ of $(3,r_1,\infty )$-atoms and a sequence $(\mu _k)_{k\in \mathbb Z}\in l_{r_2}$ of positive numbers satisfying $\mu _k=A\cdot 2^k\mathbb {P}(\nu _k<\infty )^{\frac{1}{r_1}}$ (where A is a positive constant and $\nu _k$ is the stopping time associated with $a^k$) such that

  $$\begin{aligned}&f_n=\sum \limits _{k\in \mathbb Z}\mu _k a_n^k, \ \quad a.e., \ \quad \forall n\ge 0,\nonumber \\&\quad \Vert f\Vert _{D_{r_1,r_2,b}(\mathbf{B})}\approx \inf \Vert \{\gamma _b\big (\mathbb P(\nu _k< \infty )\big )\mu _k\}_{k\in \mathbb Z}\Vert _{l_{r_2}},\ \quad \sup \limits _{k\in \mathbb {Z}}\Vert M(a^k)\Vert _{r_1}<\infty , \end{aligned}$$

  (3.5)

  where $a_n^k=\mathbb {E}_n a^k$, and the infimum is taken over all the decompositions of (3.5).

Proof

$\mathrm{(i)}\Rightarrow \mathrm{(ii)}$. The proof is similar to the one of Theorem 3.4. Let $f=(f_n)_{n\ge 0}\in D_{r_1,r_2,b}(\mathbf{B})$. In this case, for any $k\in \mathbb {Z}$, the stopping time $\nu _k$ is also defined as (3.4), where $(\lambda _n)_{n\ge 0}$ is the sequence in the definition of $D_{r_1,r_2,b}(\mathbf{B})$. Let $\mu _k$ and $a_n^k$ be the same as in the proof of Theorem 3.2. Obviously, $\lambda _{\nu _k-1}\le 2^k$. For every $n\ge 0$, we get

$$\begin{aligned} \Vert a_n^k\Vert&=\frac{1}{\mu _k}(\Vert f_n^{\nu _{k+1}}\Vert +\Vert f_n^{\nu _k}\Vert )\le \frac{1}{\mu _k}(\lambda _{\nu _{k+1}-1}+\lambda _{\nu _k-1})\chi _{\{\nu _k<\infty \}}\\&\le \mathbb {P}(\nu _k<\infty )^{-\frac{1}{r_1}}\chi _{\{\nu _k<\infty \}}. \end{aligned}$$

Therefore,

$$\begin{aligned} \Vert M(a^k)\Vert _{\infty }\le \mathbb {P}(\nu _k<\infty )^{-\frac{1}{r_1}}{{,}} \end{aligned}$$

and $a_n^k=0$ if $\nu _k\ge n$. Then $a^k$ is a $(3,r_1,\infty )$-atom for any $k\in \mathbb {Z}$. The condition $\mathrm{(i)}$ implies that there exists a $\mathbf{B}$-valued integrable function (still denoted by $a^k$) such that $a_n^k$ converges to $a^k$ as $n\rightarrow \infty$ in $L_1(\mathbf{{B}})$ (see [35, Theorem 2.9]). Then $a_n^k=\mathbb {E}_n a^k$ and (3.5) hold. Moreover, we have $\sup _{k\in \mathbb {Z}}\Vert M(a^k)\Vert _{r_1}<\infty$ and $\Vert \{\gamma _b\big (\mathbb P(\nu _k< \infty )\big )\mu _k\}_{k\in \mathbb Z}\Vert _{l_{r_2}}\le C\Vert f\Vert _{D_{r_1,r_2,b}(\mathbf{B})}$. On the other hand, let

$$\begin{aligned} \lambda _n=\sum _{k\in \mathbb {Z}}\mu _k\chi _{\{\nu _k\le n\}}\Vert M(a^k)\Vert _\infty . \end{aligned}$$

Then $(\lambda _n)_{n\ge 0}\in \Gamma$ with $\Vert f_{n+1}\Vert \le \lambda _n$. For any $k_0\in \mathbb {Z}$, set

$$\begin{aligned} \lambda _\infty =\lambda _\infty ^{(1)}+\lambda _\infty ^{(2)}, \end{aligned}$$

where

$$\begin{aligned} \lambda _\infty ^{(1)}=\sum \limits _{k=-\infty }^{k_0-1}\mu _k\chi _{\{\nu _k<\infty \}}\Vert M(a^k)\Vert _\infty , \ \ \lambda _\infty ^{(2)}=\sum \limits _{k=k_0}^\infty \mu _k\chi _{\{\nu _k<\infty \}}\Vert M(a^k)\Vert _\infty . \end{aligned}$$

Replacing $s^p(g)$ and $s^p(h)$ in the proof of Theorem 3.2 by $\lambda _\infty ^{(1)}$ and $\lambda _\infty ^{(2)}$, respectively. It follows from a similar discussion and Lemma 2.11 that $f\in D_{r_1,r_2,b}(\mathbf{B})$ and $\Vert f\Vert _{D_{r_1,r_2,b}(\mathbf{B})}\approx \inf \Vert \{\gamma _b\big (\mathbb P(\nu _k< \infty )\big )\mu _k\}_{k\in \mathbb Z}\Vert _{l_{r_2}}$, where the infimum is taken over all the decompositions of (3.5).

$\mathrm{(ii)}\Rightarrow \mathrm{(i)}$. Let $f=(f_n)_{n\ge 0}$ be a $\mathbf{B}$-valued martingale with $\sup _{n\ge 0}\Vert f_n\Vert _{\infty }<\infty$. Furthermore, let $\lambda _n=\sup _{n\ge 0}\Vert f_n\Vert _{\infty }$ for all $n\ge 0$. It is obvious $(\lambda _n)_{n\ge 0}\in \Gamma$ with $\Vert f_{n}\Vert \le \lambda _{n-1}$. Then $\Vert f\Vert _{D_{1,1,1}(\mathbf{B})}\le \sup _{n\ge 0}\Vert f_n\Vert _{\infty }$, which means $f\in D_{1,1,1}(\mathbf{B})$. The rest of the proof is similar to that of $\mathrm{(ii)}\Rightarrow \mathrm{(i)}$ in Theorem 3.2. Then we obtain $(f_n)_{n\ge 0}$ converges in probability and $(f_n)_{n\ge 0}$ is uniformly integrable. Then $(f_n)_{n\ge 0}$ converges in $L_1(\mathbf{B})$. Therefore, the space $\mathbf{B}$ has the RNP (see [7] or [28, p.31]). The proof is complete. $\square$

4 Duality Theorems

In this section, by the atomic decompositions discussed in last section we establish several dual results for $\mathbf{B}$-valued martingale Hardy–Lorentz–Karamata space ${H_{r_1,r_2,b}^{s^p}(\mathbf{B})}$ . Some necessary definitions and lemmas have to be presented or proved in preparation.

Definition 4.1

Let $1\le p<\infty$, $\alpha \ge 0$ and b be a slowly varying function. The generalized $\mathbf{B}$-valued martingale BMO space is defined by

$$\begin{aligned} BMO_{\alpha ,b}^{s^p}(\mathbf{B})=\big \{f=(f_n)_{n\ge 0}\in H_p^{s^p}(\mathbf{B}) :\Vert f\Vert _{BMO_{\alpha ,b}^{s^p}(\mathbf{B})}<\infty \big \}, \end{aligned}$$

where

$$\begin{aligned} \Vert f\Vert _{BMO_{\alpha ,b}^{s^p}(\mathbf{B})}=\sup \limits _{{\nu }\in {\mathcal {T}}} \mathbb P(\nu<\infty )^{-\frac{1}{p}-\alpha }\gamma _b^{-1}\big (\mathbb {P}(\nu <\infty )\big )\Vert s^p(f-f^{\nu })\Vert _p. \end{aligned}$$

Motivated by [21, Definition 1.1], we introduce the new generalized $\mathbf{B}$-valued martingale BMO space as follows.

Definition 4.2

Let $1\le p<\infty$, $1\le r<\infty$, $\alpha \ge 0$ and b be a slowly varying function. The new generalized $\mathbf{B}$-valued martingale BMO space is defined by

$$\begin{aligned} BMO_{r,\alpha ,b}^{s^p}(\mathbf{B})=\big \{f=(f_n)_{n\ge 0}\in H_p^{s^p}(\mathbf{B}):\Vert f\Vert _{BMO_{r,\alpha ,b}^{s^p}(\mathbf{B})}<\infty \big \}, \end{aligned}$$

where

$$\begin{aligned} \Vert f\Vert _{BMO_{r,\alpha ,b}^{s^p}(\mathbf{B})} =\sup \frac{\sum \limits _{k\in \mathbb {Z}}2^k\mathbb {P}(\nu _k<\infty )^{1-\frac{1}{p}}\Vert s^p(f-f^{\nu _k})\Vert _p}{\bigg (\sum \limits _{k\in \mathbb {Z}}\Big (2^k\gamma _b \big (\mathbb {P}(\nu _k<\infty )\big ) \mathbb {P}(\nu _k<\infty )^{1+\alpha }\Big )^r\bigg )^{\frac{1}{r}}}, \end{aligned}$$

where the supremum is taken over all stopping time sequences $\{\nu _k\}_{k\in \mathbb {Z}}$ such that $\big \{2^k\gamma _b\big (\mathbb {P}(\nu _k<\infty )\big )\mathbb {P}(\nu _k<\infty )^{1+\alpha }\big \}_{k\in \mathbb {Z}}\in {l_r}$.

The following lemma plays a key role to prove the dual result of Hardy–Lorentz–Karamata spaces for B-valued martingale, which is similar to ([24], Lemma 4.1). So we omit the proof.

Lemma 4.3

Let $1<p\le 2$, $0<r_1\le p$, $0<r_2<\infty$ and b be a slowly varying function. If $\mathbf{B}$ is isomorphic to a p-uniformly smooth space, then $H_p^{s^p}(\mathbf{B})$ is dense in $H_{r_1,r_2,b}^{s^p}(\mathbf{B})$.

Lemma 4.4

([43], Lemma 4.5) Let $\mathbf{B}$ be a reflexive Banach space and $1<p<\infty$. Then $\big (H_p^{s^p}(\mathbf{B})\big )^*=H_q^{s^q}(\mathbf{B}^*)$, where $\frac{1}{p}+\frac{1}{q}=1$.

Theorem 4.5

Let $0<r_1, r_2\le {1}$, $1<p\le 2$ and b be a slowly varying function. If $\mathbf{B }$ is isomorphic to a p-uniformly smooth space, then

$$\begin{aligned} \big (H_{r_1,r_2,b}^{s^p}(\mathbf{B})\big )^*= BMO_{\alpha ,b}^{s^q}(\mathbf{B}^*), \quad q=\frac{p}{p-1},\, \alpha =\frac{1}{r_1}-1. \end{aligned}$$

Proof

Let $g\in BMO_{\alpha ,b}^{s^q}(\mathbf{B}^*)\subset H_q^{s^q}(\mathbf{B}^*)$. Since $r_1<p$, by Lemma 2.5, we have $\Vert f\Vert _{H_{r_1,r_2,b}^{s^p}(\mathbf{B})}=\Vert s^p(f)\Vert _{r_1,r_2,b}\le \Vert s^p(f)\Vert _{p}=\Vert f\Vert _{H_p^{s^p}(\mathbf{B})}$. Then $H_p^{s^p}(\mathbf{B})\subset {H_{r_1,r_2,b}^{s^p}(\mathbf{B})}$. Since $\mathbf{B}$ is isomorphic to a p-uniformly smooth space, then $\mathbf{B}$ is reflexive ([23], Proposition 1.e.3). According to Lemma 4.4, we define

$$\begin{aligned} \phi _g(f)=\mathbb {E}(fg),\quad \forall f\in H_p^{s^p}(\mathbf{B}). \end{aligned}$$

It follows from Theorem 3.2 and Hölder’s inequality that

$$\begin{aligned} |\mathbb {E}(fg)|&=\sum _{k\in \mathbb {Z}}\mu _k\mathbb {E}\left( \sum _{i=1}^{\infty }da_i^k d(g_i-g_i^{\nu _k})\right) \nonumber \\&\le \sum _{k\in \mathbb {Z}}\mu _k\mathbb {E}\left( \left( \sum _{i=1}^{\infty }\mathbb {E}_{i-1}\Vert da_i^k\Vert ^p \right) ^{\frac{1}{p}} \left( \sum _{i=1}^{\infty }\mathbb {E}_{i-1}\Vert d(g_i-g_i^{\nu _k})\Vert ^q \right) ^{\frac{1}{q}}\right) \nonumber \\&\le \sum _{k\in \mathbb {Z}}\mu _k\mathbb {E}\Big (s^p(a^k)^p \chi _{\{\nu _k<\infty \}}\Big )^{\frac{1}{p}}\Vert s^q(g-g^{\nu _k})\Vert _q\nonumber \\&\le \sum _{k\in \mathbb {Z}}\mu _k\gamma _b\big (\mathbb {P}(\nu _k<\infty )\big )\mathbb {P}(\nu _k<\infty )^{-\frac{1}{q}-\alpha }\gamma _b^{-1}\big (\mathbb {P}(\nu _k<\infty )\big )\Vert s^q(g-g^{\nu _k})\Vert _q. \end{aligned}$$

(4.1)

Since $0<r_2\le 1$. Applying Theorem 3.2 again, we have

$$\begin{aligned} |\phi _g(f)|\le C \Vert \{\gamma _b\big (\mathbb {P}(\nu _k<\infty )\big )\mu _k\}\Vert _{l_{r_2}} \Vert g\Vert _{BMO_{\alpha ,b}^{s^q}(\mathbf{B}^*)}\le C \Vert f\Vert _{H_{r_1,r_2,b}^{s^p}(\mathbf{B})} \Vert g\Vert _{BMO_{\alpha ,b}^{s^q}(\mathbf{B}^*)}. \end{aligned}$$

By Lemma 4.3, $\phi _g$ can be uniquely extended to a continuous functional on $H_{r_1,r_2,b}^{s^p}(\mathbf{B})$.

To prove the converse part, let $\phi \in \big (H_{r_1,r_2,b}^{s^p}(\mathbf{B})\big )^*$. Since $H_p^{s^p}(\mathbf{B})\subset {H_{r_1,r_2,b}^{s^p}(\mathbf{B})}$, by Lemma 4.4, there exists $g\in H_q^{s^q}(\mathbf{B}^*)$ satisfying

$$\begin{aligned} \phi (f)=\mathbb {E}(fg), \quad \forall f\in H_p^{s^p}(\mathbf{B}). \end{aligned}$$

For any stopping time $\nu \in \mathcal {T}$, set $b=g-g^{\nu }$. Then $b\in H_q^{s^q}(\mathbf{B}^*)$. Since $\mathbf{B}$ is reflexive, using Lemma 4.4 again, we know that $\big (H_q^{s^q}(\mathbf{B}^*)\big )^*=H_p^{s^p}(\mathbf{B})$. Then there exists $a\in H_p^{s^p}(\mathbf{B})$ with $\Vert a\Vert _{H_p^{s^p}(\mathbf{B})}\le 1$ such that $\Vert b\Vert _{H_q^{s^q}(\mathbf{B}^*)}=|\mathbb {E}\big (g(a-a^{\nu })\big )|$. Set

$$\begin{aligned} h=\frac{a-a^{\nu }}{2\mathbb {P}(\nu<\infty )^{\frac{1}{r_1}-\frac{1}{p}}\gamma _b\big (\mathbb {P}(\nu <\infty )\big )}. \end{aligned}$$

Note that $s(h)=0$ on the set $\{\nu =\infty \}$. Then $s(h)=s(h)\chi _{\{\nu <\infty \}}$. It follows from Hölder’s inequality that

$$\begin{aligned} \Vert h\Vert ^{r_2}_{H_{r_1,r_2,b}^{s^p}(\mathbf{B})}= & {} \int _0^{\mathbb {P}(\nu<\infty )}\Big (t^{1/{r_1}}\gamma _b(t)\big (s^p(h)\big )^*(t)\Big )^{r_2}\frac{dt}{t}\\\le & {} \left( \int _0^{\mathbb {P}(\nu<\infty )}\Big (\big (s^p(h)\big )^*(t)\Big )^pdt\right) ^{\frac{r_2}{p}}\\&\quad \cdot \left( \int _0^{\mathbb {P}(\nu <\infty )} \big (t^{\frac{r_2}{r_1}-1}\gamma _b^{r_2}(t)\big )^{\frac{p}{p-r_2}}dt\right) ^{\frac{p-r_2}{p}}. \end{aligned}$$

By Remark 2.2 (ii), we know that $t^{\frac{r_2}{pr_1}}\gamma _b(t)$ is equivalent to a nondecreasing function, then

$$\begin{aligned}&\int _0^{\mathbb {P}(\nu<\infty )}\left( t^{\frac{r_2}{r_1}-1}\gamma _b^{r_2}(t)\right) ^{\frac{p}{p-r_2}}dt\\&\quad =\int _0^{\mathbb {P}(\nu<\infty )} \left( t^{\frac{r_2}{pr_1}}\gamma _b^{r_2}(t)\right) ^{\frac{p}{p-r_2}}\cdot t^{\frac{p(r_2-r_1)-r_2}{r_1(p-r_2)}}dt\\&\quad \le C\mathbb {P}(\nu<\infty )^{\frac{r_2}{r_1(p-r_2)}}\cdot \gamma _b\big (\mathbb {P}(\nu<\infty )\big )^{\frac{pr_2}{p-r_2}}\cdot \int _0^{\mathbb {P}(\nu<\infty )}t^{\frac{p(r_2-r_1)-r_2}{r_1(p-r_2)}}dt\\&\quad = C\mathbb {P}(\nu<\infty )^{\frac{r_2(p-r_1)}{r_1(p-r_2)}}\cdot \gamma _b\big (\mathbb {P}(\nu <\infty )\big )^{\frac{pr_2}{p-r_2}} \end{aligned}$$

for some positive constant C. Then

$$\begin{aligned} \Vert h\Vert _{H_{r_1,r_2,b}^{s^p}(\mathbf{B})}\le & {} C \Vert s^p(h)\Vert _p\mathbb {P}(\nu<\infty )^{\frac{1}{r_1}-\frac{1}{p}}\gamma _b\big (\mathbb {P}(\nu<\infty )\big )\\= & {} C\frac{\Vert s^p(a-a^{\nu })\Vert _p}{2\mathbb {P}(\nu<\infty )^{\frac{1}{r_1}-\frac{1}{p}}\gamma _b\big (\mathbb {P}(\nu<\infty )\big )}\mathbb {P}(\nu<\infty )^{\frac{1}{r_1}-\frac{1}{p}} \gamma _b\big (\mathbb {P}(\nu<\infty )\big )\\= & {} C\frac{\Vert a-a^{\nu })\Vert _{H^{s^p}_p(\mathbf{B})}}{2\mathbb {P}(\nu<\infty )^{\frac{1}{r_1}-\frac{1}{p}}\gamma _b\big (\mathbb {P}(\nu<\infty )\big )}\mathbb {P}(\nu<\infty )^{\frac{1}{r_1}-\frac{1}{p}} \gamma _b\big (\mathbb {P}(\nu <\infty )\big ) \\\le & {} C \end{aligned}$$

Thus,

$$\begin{aligned} \Vert \phi \Vert&\ge C^{-1}|\phi (h)|=C^{-1}|\mathbb {E}(gh)|\\&=C^{-1}\mathbb {P}(\nu<\infty )^{\frac{1}{p}-\frac{1}{r_1}}\gamma _b^{-1}\big (\mathbb {P}(\nu<\infty )\big )|\mathbb {E}\big (g(a-a^{\nu })\big )|\\&=C^{-1}\mathbb {P}(\nu<\infty )^{\frac{1}{p}-\frac{1}{r_1}}\gamma _b^{-1}\big (\mathbb {P}(\nu<\infty )\big )\Vert b\Vert _{H_q^{s^q}(\mathbf{B}^*)}\\&=C^{-1}\mathbb {P}(\nu<\infty )^{-\frac{1}{q}-\alpha }\gamma _b^{-1}\big (\mathbb {P}(\nu <\infty )\big )\Vert s^q(g-g^{\nu })\Vert _q. \end{aligned}$$

Taking the supremum over all stopping times, we get $\Vert g\Vert _{BMO_{\alpha ,b}^{s^q}(\mathbf{B}^*)}\le C \Vert \phi \Vert$. The proof is complete. $\square$

Remark 4.6

If we take $r_1=r_2=r,b\equiv 1$, the conclusion above arrives at Theorem 4.6 in [43] .

Theorem 4.7

Let $1<p\le 2$, $0<r_1\le {1}, 1<r_2<\infty$ and b be a nondecreasing slowly varying function. If $\mathbf{B }$ is isomorphic to a p-uniformly smooth space, then

$$\begin{aligned} \big (H_{r_1,r_2,b}^{s^p}(\mathbf{B})\big )^*=BMO_{r_2,\alpha ,b}^{s^q}(\mathbf{B}^*), \quad q=\frac{p}{p-1}, \alpha =\frac{1}{r_1}-1. \end{aligned}$$

Proof

At first, let $g\in BMO_{r_2,\alpha ,b}^{s^q}(\mathbf{B}^*)\subset H_q^{s^q}(\mathbf{B}^*)$. Since $H_p^{s^p}(\mathbf{B})\subset {H_{r_1,r_2,b}^{s^p}(\mathbf{B})}$. Define

$$\begin{aligned} \phi _g(f)=\mathbb {E}(fg),\quad \forall f\in H_p^{s^p}(\mathbf{B}). \end{aligned}$$

Similar to the discussion of (4.1), we get

$$\begin{aligned} |\mathbb {E}(fg)| =\sum _{k\in \mathbb {Z}}\mu _k\mathbb {E}\left( \sum \limits _{i=1}^{\infty }da_i^k d(g_i-g_i^{\nu _k})\right) \le A\sum _{k\in \mathbb {Z}}2^k\mathbb {P}(\nu _k<\infty )^{1-\frac{1}{q}}\Vert s^q(g-g^{\nu _k})\Vert _q. \end{aligned}$$

It follows from the definition of $\Vert \cdot \Vert _{BMO_{r_2,\alpha ,b}^{s^q}(\mathbf{B}^*)}$ and Theorem 3.2 that

$$\begin{aligned} |\mathbb {E}(fg)|&\le A \left( \sum _{k\in \mathbb {Z}}\left( 2^k\gamma _b\big (\mathbb {P}(\nu _k<\infty )\big )\mathbb {P}(\nu _k<\infty )^{1+\alpha }\right) ^{r_2}\right) ^{\frac{1}{r_2}} \Vert g\Vert _{BMO_{r_2,\alpha ,b}^{s^q}(\mathbf{B}^*)}\\&\le A \left( \sum _{k\in \mathbb {Z}}\left( 2^k\gamma _b\big (\mathbb {P}(\nu _k<\infty )\big )\mathbb {P}(\nu _k<\infty )^{\frac{1}{r_1}}\right) ^{r_2}\right) ^{\frac{1}{r_2}}\Vert g\Vert _{BMO_{r_2,\alpha ,b}^{s^q}(\mathbf{B}^*)}\\&\le C \Vert f\Vert _{H_{r_1,r_2,b}^{s^p}(\mathbf{B})}\cdot \Vert g\Vert _{BMO_{r_2,\alpha ,b}^{s^q}(\mathbf{B}^*)}. \end{aligned}$$

Therefore, we get $|\phi _g(f)|\le C \Vert f\Vert _{H_{r_1,r_2,b}^{s^p}(\mathbf{B})}\cdot \Vert g\Vert _{BMO_{r_2,\alpha ,b}^{s^q}(\mathbf{B}^*)}$. According to Lemma 4.3, $\phi _g$ can be uniquely extended to a continuous functional on $H_{r_1,r_2,b}^{s^p}(\mathbf{B})$.

Conversely, let $\phi \in \big (H_{r_1,r_2,b}^{s^p}(\mathbf{B})\big )^*$. Since $H_p^{s^p}(\mathbf{B})\subset {H_{r_1,r_2,b}^{s^p}(\mathbf{B})}$, using Lemma 4.4, there exists $g\in H_q^{s^q}(\mathbf{B}^*)$ such that

$$\begin{aligned} \phi (f)=\mathbb {E}(fg), \quad \forall f\in H_p^{s^p}(\mathbf{B}). \end{aligned}$$

Let $\{\nu _k\}_{k\in \mathbb {Z}}$ be any stopping time sequence satisfying $\{2^k\gamma _b\big (\mathbb {P}(\nu _k<\infty )\big )\mathbb {P}(\nu _k<\infty )^{\frac{1}{r_1}}\}_{k\in \mathbb {Z}}\in l_{r_2}$. Similar to Theorem 4.5, there exists some $a_k\in H_p^{s^p}(\mathbf{B})$ with $\Vert a_k\Vert _{H_p^{s^p}(\mathbf{B})}\le 1$ such that $\Vert g-g^{\nu _k}\Vert _{H_q^{s^q}(\mathbf{B}^*)}=\mathbb {E}\big (a_k(g-g^{\nu _k})\big )$ for each $k\in \mathbb Z$. For an arbitrary $N\in \mathbb N$, set

$$\begin{aligned} h=\sum _{k=-N}^{N}2^k\mathbb {P}(\nu _k<\infty )^{1-\frac{1}{q}}(a_k-a_k^{\nu _k}). \end{aligned}$$

Let $h=G+H,$ where

$$\begin{aligned} G=\sum _{k=-N}^{k_0-1}2^k\mathbb {P}(\nu _k<\infty )^{1-\frac{1}{q}}(a_k-a_k^{\nu _k})\quad \text {and}\quad H=\sum _{k=k_0}^N2^k\mathbb {P}(\nu _k<\infty )^{1-\frac{1}{q}}(a_k-a_k^{\nu _k}), \end{aligned}$$

if $-N\le k_0\le N$; $G=0$ if $k_0\le -N$; $H=0$ if $k_0>N$. Since $\frac{r_2}{r_1}\in (1,\infty )$, for some $0<\lambda <1$, by Hölder’s inequality, we have

$$\begin{aligned} |s^p(G)|&\le \sum _{k=-N}^{k_0-1} \mu _{k}|s^p(a^{k})| \chi _{\{\nu _k<\infty \}}\\&\le \left( \sum _{k=-N}^{k_0-1} 2^{k\lambda q}\right) ^{1/{q}}\left( \sum _{k=-N}^{k_0-1} 2^{-k\lambda p} \left( \mu _{k}|s^p(a^{k}) | \chi _{\{\nu _k<\infty \}}\right) ^p\right) ^{1/{p}}\\&\le C 2^{k_0\lambda } \left( \sum _{k=-N}^{k_0-1} 2^{-k\lambda p} \left( \mu _{k}|s^p(a^{k}) | \chi _{\{\nu _k<\infty \}}\right) ^p\right) ^{1/{p}}. \end{aligned}$$

Since $a^k$ is a $(1,r_1,\infty ,p)$-atom for each $k\in \mathbb Z$, we have $\Vert s^p(a^k)\Vert _\infty \le \mathbb P(\nu _k<\infty )^{-\frac{1}{r_1}}$. It follows from Chebyshev’s and Hölder’s inequalities that

$$\begin{aligned} \mathbb P (s^p(G)>2^{k_0})&\le \Big \Vert \frac{|s^p(G)|^p}{2^{k_0p}}\Big \Vert _1 \le C2^{k_0p(\lambda -1)} \sum _{k=-N}^{k_0-1} 2^{-k\lambda p}\mu _k^p \Vert s^p(a^k)\chi _{\{\nu _k<\infty \}}\Vert _p^p\\&\le C2^{k_0p(\lambda -1)} \sum _{k=-N}^{k_0-1} 2^{-k\lambda p}\mu _k^p \big (\Vert s^p(a^k)\Vert _{\infty }\Vert \chi _{\{\nu _k<\infty \}}\Vert _p\big )^p\\&\le C2^{k_0p(\lambda -1)} \sum _{k=-\infty }^{k_0-1} 2^{k(1-\lambda ) p} \mathbb P(\nu _k<\infty )\\&\le C \sum _{k=-\infty }^{k_0-1} 2^{(k-k_0)(1-\lambda ) p} =: C\times C_1. \end{aligned}$$

By Remark 2.2 (ii), since $t^{\frac{1}{r_1}}\gamma _b(t)$ is a nondecreasing function, then for any $0<\varepsilon <1$, we have

$$\begin{aligned} \mathbb P&(s^p(G)>2^{k_0})^\varepsilon \gamma _b^{\varepsilon r_1} \big (\mathbb P(s^p(G)>2^{k_0})\big )\\&\le C \left( 2^{k_0p(\lambda -1)} \sum _{k=-\infty }^{k_0-1} 2^{k(1-\lambda ) p} \mathbb P(\nu _k<\infty )\right) ^{\varepsilon }\\&\qquad \times \gamma _b^{\varepsilon r_1}\left( \sum _{k=-\infty }^{k_0-1} C_1^{-1}2^{(k-k_0)(1-\lambda ) p} \mathbb P(\nu _k<\infty )\right) \\&\le C\sum _{k=-\infty }^{k_0-1}2^{(k-k_0)(1-\lambda ) p\varepsilon } \mathbb P(\nu _k<\infty )^\varepsilon \cdot \gamma _b^{\varepsilon r_1}\big ( C_1^{-1}2^{(k-k_0)(1-\lambda ) p} \mathbb P(\nu _k<\infty )\big ), \end{aligned}$$

where the last inequality is valid because $\gamma _b(t)$ is nonincreasing on (0, 1], see Remark 2.2(i).

Set

$$\begin{aligned} z:=\varepsilon -\frac{r_1}{(1-\lambda )p}>0, \end{aligned}$$

where $\varepsilon \in (\frac{r_1}{p},1)$ (this implies $\varepsilon <\frac{r_2}{r_1}$) and $\lambda \in (0,1-\frac{r_1}{\varepsilon p})$. Then $(1-\lambda )p(\varepsilon -z)=r_1$. Since $t^z\gamma _b(t)$ is equivalent to a nondecreasing function,

$$\begin{aligned}&2^{k_0r_1}\mathbb P(s^p(G)>2^{k_0})^\varepsilon \gamma _b^{\varepsilon r_1} \big (\mathbb P(s^p(G)>2^{k_0})\big ) \nonumber \\&\quad \le C 2^{k_0r_1}\sum _{k=-\infty }^{k_0-1}2^{(k-k_0)(1-\lambda ) p\varepsilon } \mathbb P(\nu _k<\infty )^\varepsilon \cdot 2^{-(k-k_0)(1-\lambda ) pz} \cdot \gamma _b^{\varepsilon r_1}\big (\mathbb P(\nu _k<\infty )\big ) \nonumber \\&\quad =C \sum _{k=-\infty }^{k_0-1} \left( 2^{k}\mathbb P (\nu _k<\infty )^{\frac{\varepsilon }{r_1}}\gamma _b^{\varepsilon } \big (\mathbb P (\nu _k<\infty )\big )\right) ^{r_1}. \end{aligned}$$

(4.2)

On the other hand, $\{s^p(H)>0\}\subset \bigcup \limits _{k=k_0}^N\{\nu _k<\infty \}$. Similar to that of (3.2), for each $0<\varepsilon <1$, we obtain

$$\begin{aligned}&2^{k_0\varepsilon r_1}\mathbb {P}(s^p(H)>2^{k_0})\gamma _b^{r_1}\big (\mathbb {P}(s^p(H)>2^{k_0})\big )\\&\quad \le C\sum \limits _{k=k_0}^{\infty }\Big (2^{k\varepsilon }\mathbb {P}(\nu _k<\infty )^{\frac{1}{r_1}}\gamma _b\big (\mathbb {P}(\nu _k<\infty )\big )\Big )^{r_1}. \end{aligned}$$

By Lemma 2.12, we get $s^p(h)\in L_{r_1,r_2,b}$, and

$$\begin{aligned} \Vert s^p(h)\Vert _{r_1,r_2,b}\le C \Vert \{2^k\mathbb {P}(\nu _k<\infty )^{\frac{1}{r_1}}\gamma _b\big (\mathbb {P}(\nu _k<\infty )\big )\}_{k\in \mathbb {Z}}\Vert _{l_{r_2}}. \end{aligned}$$

Consequently, $h\in H_{r_1,r_2,b}^{s^p}(\mathbf{B})$ and

$$\begin{aligned} \Vert h\Vert _{H_{r_1,r_2,b}^{s^p}(\mathbf{B})}\le C\left( \sum \limits _{k\in \mathbb {Z}}\Big (2^k\mathbb {P}(\nu _k<\infty )^{\frac{1}{r_1}}\gamma _b\big (\mathbb {P}(\nu _k<\infty )\big )\Big )^{r_2}\right) ^{\frac{1}{r_2}} . \end{aligned}$$

Then

$$\begin{aligned}&\sum \limits _{k=-N}^N2^k\mathbb {P}(\nu _k<\infty )^{1-\frac{1}{q}}\Vert s^q(g-g^{\nu _k})\Vert _q\\&\quad =\sum \limits _{k=-N}^N2^k\mathbb {P}(\nu _k<\infty )^{1-\frac{1}{q}}\mathbb {E}\big (a_k(g-g^{\nu _k})\big ) \\&\quad =\sum \limits _{k=-N}^N2^k\mathbb {P}(\nu _k<\infty )^{1-\frac{1}{q}}\mathbb {E}\big ((a_k-a_k^{\nu _k})g\big ) \\&\quad =\mathbb {E}(hg)=\phi (h)\le \Vert h\Vert _{H_{r_1,r_2,b}^{s^p}(\mathbf{B})}\Vert \phi \Vert \\&\quad \le C\left( \sum \limits _{k\in \mathbb {Z}}\Big (2^k\mathbb {P}(\nu _k<\infty )^{\frac{1}{r_1}}\gamma _b\big (\mathbb {P}(\nu _k<\infty )\big )\Big )^{r_2}\right) ^{\frac{1}{r_2}}\Vert \phi \Vert . \\&\quad = C\left( \sum \limits _{k\in \mathbb {Z}}\Big (2^k\mathbb {P}(\nu _k<\infty )^{1+\alpha }\gamma _b\big (\mathbb {P}(\nu _k<\infty )\big )\Big )^{r_2}\right) ^{\frac{1}{r_2}}\Vert \phi \Vert . \end{aligned}$$

We have

$$\begin{aligned} \frac{\sum \limits _{k=-N}^{N}2^k\mathbb {P}(\nu _k<\infty )^{1-\frac{1}{q}} \Vert s^q(g-g^{\nu _k})\Vert _q}{\left( \sum \limits _{k\in \mathbb {Z}} \Big (2^k\mathbb {P}(\nu _k<\infty )^{1+\alpha }\gamma _b \big (\mathbb {P}(\nu _k<\infty )\big )\Big )^{r_2}\right) ^{\frac{1}{r_2}}}\le C \Vert \phi \Vert . \end{aligned}$$

Taking over all $N\in \mathbb {N}$ and the supremum over all of such stopping time sequences satisfying $\{2^k\mathbb {P}(\nu _k<\infty )^{1+\alpha } \gamma _b\big (\mathbb {P}(\nu _k<\infty )\big )\}_{k\in \mathbb {Z}}\in l_{r_2}$, we obtain $\Vert g\Vert _{BMO_{r_2,\alpha ,b}^{s^q}(\mathbf{B}^*)}\le C\Vert \phi \Vert$. The proof is complete. $\square$

We now consider another question what is the duality in the theorem above for $r_2=\infty$. To handle this problem, we define the generalized weak BMO martingale space associated with slowly varying function. Firstly, we introduce the following definition.

Definition 4.8

([20], Definition 1.4) Let $\mathbf{B}$ be a Banach space, $1\le p<\infty$, $0<r<\infty$ and b be a slowly varying function. Denote by ${\mathcal {L}}_{r,\infty ,b}$ the set of all $f\in {L}_{r,\infty ,b}$ having the absolute continuous quasi-norm defined by

$$\begin{aligned} {\mathcal {L}}_{r,\infty ,b} =\{f\in {L}_{r,\infty ,b} : \lim _{\mathbb P(A)\rightarrow 0}\Vert f\chi _A\Vert _{r,\infty ,b}=0 \}. \end{aligned}$$

Note that ${\mathcal {L}}_{r,\infty ,b}$ is a linear closed subspace of $L_{r,\infty ,b}$. We now define a closed subspace of $H_{r,\infty ,b}^{s^p}(\mathbf{B})$ as follows

$$\begin{aligned} {\mathcal {H}}_{r,\infty ,b}^{s^p}(\mathbf{B})=\{f=(f_n)_{n\ge 0}:s^p(f)\in {\mathcal {L}}_{r,\infty ,b}\}. \end{aligned}$$

Remark 4.9

If we take $r_2=\infty$ in Lemma 4.3 and replace $H_{r_1,r_2,b}^{s^p}(\mathbf{B})$ by ${\mathcal {H}}_{r_1,\infty ,b}^{s^p}(\mathbf{B})$, the conclusion still holds.

Definition 4.10

Let $1\le p<\infty$, $\alpha \ge 0$ and b be a slowly varying function. The generalized weak BMO martingale spaces $wBMO_{\alpha ,b}^{s^p}(\mathbf{B})$ are defined by

$$\begin{aligned} wBMO_{\alpha ,b}^{s^p}(\mathbf{B})=\Big \{f=(f_n)_{n\ge 0}\in H_p^{s^p}(\mathbf{B}): \Vert f\Vert _{wBMO_{\alpha ,b}^{s^p}(\mathbf{B})}<\infty \Big \}, \end{aligned}$$

where

$$\begin{aligned} \Vert f\Vert _{wBMO_{\alpha ,b}^{s^p}(\mathbf{B})}=\sup \frac{\sum \limits _{k\in \mathbb {Z}}2^k \mathbb {P}(\nu _k<\infty )^{1-\frac{1}{p}}\Vert s^p(f-f^{\nu _k})\Vert _p}{ \sup _k2^k\gamma _b\big (\mathbb {P}(\nu _k<\infty )\big )\mathbb {P}(\nu _k<\infty )^{1+\alpha }} \end{aligned}$$

and the supremum is taken over all stopping time sequences $\{\nu _k\}_{k\in \mathbb {Z}}$ such that $\big \{2^k\gamma _b\big (\mathbb {P}(\nu _k<\infty )\big )\mathbb {P} (\nu _k<\infty )^{1+\alpha }\big \}_{k\in \mathbb {Z}}\in l_\infty$.

Similar to the proof of Theorem 4.7, we give the following duality only without proof.

Theorem 4.11

Let $\mathbf{B}$ be a Banach space, $0<r\le 1$, $1<p\le 2$, $\alpha \ge 0$ and b be a nondecreasing slowly varying function. If $\mathbf{B }$ is isomorphic to a p-uniformly smooth space, then

$$\begin{aligned} \big (\mathcal {H}_{r,\infty ,b}^{s^p}(\mathbf{B})\big )^*=wBMO_{\alpha ,b}^{s^q}(\mathbf{B}^*), \quad q=\frac{p}{p-1}, \alpha =\frac{1}{r}-1. \end{aligned}$$

5 Martingale Inequalities

As another application of the atomic decomposition, in this section, we get a sufficient condition for a $\sigma$-sublinear operator to be bounded from $\mathbf{B}$-valued martingale Hardy–Lorentz–Karamata space to $L_{p,q,b}$. Firstly, we give the definition of $\sigma$-sublinear operator.

An operator $T:X\rightarrow Y$ is called a $\sigma$-sublinear operator if for any $\lambda \in \mathbb C$ it satisfies

$$\begin{aligned} \Big |T\Big (\sum _{i=1}^{\infty }f_i\Big )\Big |\le \sum _{i=1}^{\infty }|T(f_i)| \quad \text {and} \quad |T(\lambda f)|= |\lambda ||T(f)|, \end{aligned}$$

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

Lemma 5.1

Let $1<p\le 2$, b be a nondecreasing slowly varying function and let $T:H_p^{s^p}(\mathbf{B})\rightarrow L_p(\Omega )$ be a $\sigma$-sublinear bounded operator. If $\mathbf{B}$ is isomorphic to a p-uniformly smooth space and $\{|Ta|>0\}\subset \{\nu <\infty \}$ for all $(1, r_1, \infty ; p)$-atom a (where $\nu$ is the stopping time associated with a), then for $0<r_1<p$ and $0<r_2\le \infty$, we get

$$\begin{aligned} \Vert Tf\Vert _{r_1,r_2,b}\le C\Vert f\Vert _{H_{r_1,r_2,b}^{s^p}(\mathbf{B})}. \end{aligned}$$

Proof

For any $f\in H_{r_1,r_2,b}^{s^p}(\mathbf B )$. According to Theorem 3.2, f has a atomic decomposition as (3.1). For an arbitrary $k_0\in \mathbb {Z}$, set

$$\begin{aligned} f=\sum _k{\mu _k a^k}=\sum _{k=-\infty }^{k_0-1}{\mu _k a^k}+\sum _{k=k_0}^{\infty }{\mu _k a^k}=g+h, \end{aligned}$$

where

$$\begin{aligned} g=\sum \limits _{k=-\infty }^{k_0-1}{\mu _k a^k}, \quad h=\sum \limits _{k=k_0}^{\infty }{\mu _k a^k}. \end{aligned}$$

By the $\sigma$-sublinearity of the operator T, we get

$$\begin{aligned} |T(g)|\le \sum _{k=-\infty }^{k_0-1}\mu _k|T(a^k)| \quad \text {and} \quad |T(h)|\le \sum _{k=k_0}^{\infty }\mu _k|T(a^k)| \end{aligned}$$

Now we firstly prove $Tg\in L_{r_1,r_2,b}$. We consider two cases: $\frac{r_2}{r_1}\in [1,\infty ]$ and $\frac{r_2}{r_1}\in (0,1)$.

Case 1: $\frac{r_2}{r_1}\in [1,\infty ]$. In this situation, the proof is similar to the converse part of Theorem 4.7. Indeed, following the same argument used in (4.2), by Lemma 2.12, we get the desired result.

Case 2: $\frac{r_2}{r_1}\in (0,1)$. In this situation, set $z:=1-\frac{r_1}{(1-\lambda )p\varepsilon },$ where $\lambda \in (0,1-\frac{r_1}{p})$, $\varepsilon \in (\frac{r_1}{(1-\lambda )p},1)$. Similar to (4.2), we deduce

$$\begin{aligned}&2^{k_0r_1}\mathbb P(T(g)>2^{k_0})^\varepsilon \gamma _b^{\varepsilon r_1} \big (\mathbb P(T(g)>2^{k_0})\big ) \nonumber \\&\quad \le C \left( \, \sum _{k=-\infty }^{k_0-1} \left( 2^{\frac{k}{\varepsilon }}\mathbb P (\nu _k<\infty )^{ \frac{1}{r_1}} \gamma _b (\mathbb P (\nu _k<\infty ))\right) ^{r_1}\right) ^\varepsilon . \end{aligned}$$

Then

$$\begin{aligned} \Vert T(g)\Vert _{r_1,r_2,b}^{r_2}&=\sum _{k_0=-\infty }^\infty 2^{k_0 r_2}\mathbb P(|T(g)|>2^{k_0})^{\frac{r_2}{r_1}} \gamma _b^{r_2}\big (\mathbb P(|T(g)|>2^{k_0})\big )\\&\le C\sum _{k_0=-\infty }^\infty 2^{k_0(1-\frac{1}{\varepsilon })r_2} \left( \,\sum _{k=-\infty }^{k_0-1} \left( 2^{\frac{k}{\varepsilon }}\mathbb P (\nu _k<\infty )^{ \frac{1}{r_1}} \gamma _b \big (\mathbb P (\nu _k<\infty )\big )\right) ^{r_1}\right) ^{\frac{r_2}{r_1}}\\&\le C\sum _{k_0=-\infty }^\infty 2^{k_0(1-\frac{1}{\varepsilon })r_2 } \sum _{k=-\infty }^{k_0-1} 2^{k\frac{1}{\varepsilon } r_2} \mathbb P(\nu _k<\infty )^{\frac{r_2}{r_1}}\gamma _b^{r_2}\big (\mathbb P(\nu _k<\infty )\big )\\&= C\sum _{k=-\infty }^{\infty } 2^{k\frac{1}{\varepsilon } r_2} \mathbb P(\nu _k<\infty )^{\frac{r_2}{r_1}}\gamma _b^{r_2}\big (\mathbb P(\nu _k<\infty )\big ) \sum _{k_0=k+1}^\infty 2^{k_0(1-\frac{1}{\varepsilon })r_2 }\\&\le C \sum _{k=-\infty }^{\infty } 2^{k r_2} \mathbb P(\nu _k<\infty )^{\frac{r_2}{r_1}}\gamma _b^{r_2}\big (\mathbb P(\nu _k<\infty )\big ). \end{aligned}$$

Consequently, by Theorem 3.2, we have $\Vert T(g)\Vert _{r_1,r_2,b} \le C \Vert f\Vert _{H_{r_1,r_2,b}^{s^p}}$.

Secondly, we show that $Th\in L_{r_1,r_2,b}$. By Remark 2.2 (ii), $t^{\frac{1}{r_1}}\gamma _b(t)$ is a nondecreasing function. Since $\{T(h)>0\}\subset \bigcup \limits _{k=k_0}^\infty \{\nu _k<\infty \}$, for any $0<\varepsilon <1$, we obtain

$$\begin{aligned}&2^{k_0\varepsilon r_1}\mathbb {P}(T(h)>2^{k_0})\gamma _b^{r_1}\big (\mathbb {P}(T(h)>2^{k_0})\big )\\&\quad \le C2^{k_0\varepsilon r_1}\sum _{k=k_0}^{\infty }\mathbb {P}(\nu _k<\infty ) \gamma _b^{r_1}\big (\mathbb {P}(\nu _k<\infty )\big )\\&\quad \le C\sum _{k=k_0}^\infty \Big (2^{k\varepsilon }\mathbb {P}(\nu _k<\infty )^{\frac{1}{r_1}} \gamma _b\big (\mathbb {P}(\nu _k<\infty )\big )\Big )^{r_1}, \end{aligned}$$

where the first “$\le$” is because the function $\gamma _b(t)$ is nonincreasing on (0,1] (see Remark 2.2(i)). By Lemma 2.12, we get $T(h)\in L_{r_1,r_2,b}$. Then $T(f)\in L_{r_1,r_2,b}$. According to Theorem 3.2, we obtain

$$\begin{aligned} \Vert T(f)\Vert _{r_1,r_2,b}\le C\Vert \{2^k\mathbb {P}(\nu _k<\infty )^{\frac{1}{r_1}}\gamma _b\big (\mathbb {P}(\nu _k<\infty )\big )\}_{k\in \mathbb {Z}}\Vert _{l_{r_2}}\le C\Vert f\Vert _{H_{r_1,r_2,b}^{s^p}(\mathbf{B})}. \end{aligned}$$

$\square$

Similarly, we only give the following two lemmas without proofs. Note that the proofs rely on Theorems 3.4 and 3.5, respectively.

Lemma 5.2

Let $1<p\le 2$ , b be a nondecreasing slowly varying function and let $T:H_p^{S^p}(\mathbf{B})\rightarrow L_p(\Omega )$ be a $\sigma$-sublinear bounded operator. If $\mathbf{B}$ is isomorphic to a p-uniformly smooth space and $\{|Ta|>0\}\subset \{\nu <\infty \}$ for all $(2, r_1, \infty ; p)$-atom a (where $\nu$ is the stopping time associated with a), then for $0<r_1<p$ and $0<r_2\le \infty$, we get

$$\begin{aligned} \Vert Tf\Vert _{r_1,r_2,b}\le C\Vert f\Vert _{Q_{r_1,r_2,b}^{S^p}(\mathbf{B})}. \end{aligned}$$

Lemma 5.3

Let $0< q<\infty$ , b be a nondecreasing slowly varying function and let $T:H_q(\mathbf{B})\rightarrow L_q(\Omega )$ be a $\sigma$-sublinear bounded operator. If $\mathbf{B}$ has the RNP and $\{|Ta|>0\}\subset \{\nu <\infty \}$ for all $(3, r_1, \infty )$-atom a (where $\nu$ is the stopping time associated with a), then for $0<r_1<q$ and $0< r_2\le \infty$, we get

$$\begin{aligned} \Vert Tf\Vert _{r_1,r_2,b}\le C\Vert f\Vert _{D_{r_1,r_2,b}(\mathbf{B})}. \end{aligned}$$

Theorem 5.4

Let $\mathbf{B}$ be a Banach space and b be a nondecreasing slowly varying function. For $1<p\le 2$, $0<r_1<p$ and $0<r_2\le \infty$. Then the following statements are equivalent:

• (i) $\mathbf{B}$ is isomorphic to a p-uniformly smooth space;

• (ii) For each $\mathbf{B}$-valued martingale $f=(f_n)_{n\ge 0}\in H_{r_1,r_2,b}^{s^p}(\mathbf{B})$, there exists a constant C such that

  $$\begin{aligned} \Vert Mf\Vert _{r_1,r_2,b}\le C \Vert f\Vert _{H_{r_1,r_2,b}^{s^p}(\mathbf{B})}; \end{aligned}$$

• (iii) For each $\mathbf{B}$-valued martingale $f=(f_n)_{n\ge 0}\in Q_{r_1,r_2,b}^{S^p}(\mathbf{B})$, there exists a constant C such that

  $$\begin{aligned} \Vert Mf\Vert _{r_1,r_2,b}\le C \Vert f\Vert _{Q_{r_1,r_2,b}^{S^p}(\mathbf{B})}. \end{aligned}$$

Proof

$\mathrm{(i)}\Rightarrow \mathrm{(ii)}$. Let $f=(f_n)_{n\ge 0}\in H_{r_1,r_2,b}^{s^p}(\mathbf{B})$. The maximal operator $T=M$ is $\sigma$-sublinear. By condition $\mathrm{(i)}$, it follows from Lemma 2.8 (iv) that

$$\begin{aligned} \Vert Mf\Vert _p\le C\Vert S^p(f)\Vert _p=C\Vert s^p(f)\Vert _p=C\Vert f\Vert _{H_p^{s^p}(\mathbf{B})}, \end{aligned}$$

which implies $M: H_p^{s^p}(\mathbf{B})\rightarrow L_p(\Omega )$ is bounded. For any $(1,r_1,\infty ; p)$-atom a and the corresponding stopping time $\nu$, we have $\{|Ta|>0\}=\{|Ma|>0\}\subset \{\nu <\infty \}$. Then, from Lemma 5.1, we get the desired inequality.

$\mathrm (i)\Rightarrow (iii)$. Note that the maximal operator $M:H_p^{S^p}(\mathbf{B})\rightarrow L_p(\Omega )$ is bounded. Similar to that of $\mathrm{(i)}\Rightarrow \mathrm{(ii)}$, it can be proved by Lemma 5.2.

$\mathrm{(ii)}\Rightarrow \mathrm{(i)}$. Given an arbitrary $\mathbf{B}$-valued martingale $f=(f_n)_{n\ge 0}$ with $\mathbb {E}s^p(f)^p=\mathbb {E}\big (\sum _{n=1}^{\infty }\Vert df_n\Vert ^p\big )<\infty$. Since $0<r_1<p$, we obtain $\Vert s^p(f)\Vert _{r_1,r_2,b}\le \Vert s^p(f)\Vert _{p,p}<\infty$. Then $f\in H_{r_1,r_2,b}^{s^p}(\mathbf{B})$. Consider a new martingale $g^{(n)}=(g_m^{(n)})_{m\ge 0}$, where $g_m^{(n)}=f_{m+n}-f_n, (m\ge 0)$. Obviously, $s^p(g^{(n)})^p=s^p(f)^p-s^p_n(f)^p\rightarrow 0$ as $n\rightarrow \infty$ and $s^p(g^{(n)})\le s^p(f)$. Furthermore, from the condition $\mathrm (ii)$ we get

$$\begin{aligned} \Vert f_{m+n}-f_n\Vert _{r_1,r_2,b}\le \sup _{m\ge 0}\Vert f_{m+n}-f_n\Vert _{r_1,r_2,b}\le \Vert Mg^{(n)}\Vert _{r_1,r_2,b}\le C\Vert s^p(g^{(n)})\Vert _{r_1,r_2,b}. \end{aligned}$$

Using the dominated convergence theorem, $(f_n)_{n\ge 1}$ is a Cauchy sequence in $L_{r_1,r_2,b}(\mathbf{B})$. So $f_n$ is convergent in probability. It follows from Lemma 2.8 that $\mathbf{B}$ is isomorphic to a p-uniformly smooth space.

$\mathrm{(iii)}\Rightarrow \mathrm{(i)}$. Given an arbitrary $\mathbf{B}$-valued dyadic martingale $f=(f_n)_{n\ge 0}$ such that $\mathbb {E}\big (\sum _{n=1}^{\infty }\Vert df_n\Vert ^p\big )<\infty$. It is similar to that of $\mathrm{(ii)}\Rightarrow \mathrm{(i)}$, we get $f\in H_{r_1,r_2,b}^{s^p}(\mathbf{B})$. For every $n\ge 0$, set $\lambda _n=s_{n+1}^p(f)$. Then $(\lambda _n)_{n\ge 0}$ is a nondecreasing, nonnegative, and adapted sequence. Since f is a $\mathbf{B}$-valued dyadic martingale, we obtain $S_n^p(f)\le Cs_n^p(f)$. Then

$$\begin{aligned} \Vert f\Vert _{Q_{r_1,r_2,b}^{S^p}(\mathbf{B})}\le \Vert s^p(f)\Vert _{r_1,r_2,b}<\infty . \end{aligned}$$

Namely, $f\in Q_{r_1,r_2,b}^{S^p}(\mathbf{B})$. Consider a new martingale $g^{(n)}=(g_m^{(n)})_{m\ge 0}$ as above. By condition $\mathrm (iii)$, we have

$$\begin{aligned} \Vert f_{m+n}-f_n\Vert _{r_1,r_2,b}\le \Vert Mg^{(n)}\Vert _{r_1,r_2,b}\le C\Vert g^{(n)}\Vert _{Q_{r_1,r_2,b}^{S^p}(\mathbf{B})}\le C\Vert s^p(g^{(n)})\Vert _{r_1,r_2,b}. \end{aligned}$$

Applying the dominated convergence theorem, $\{f_n\}_{n\ge 1}$ is a Cauchy sequence in $L_{r_1,r_2,b}(\mathbf{B})$. So $f_n$ is convergent in probability. Follows from Lemma 2.8 and Remark 2.10, $\mathbf{B}$ is isomorphic to a p-uniformly smooth space. We conclude the proof of the theorem. $\square$

Lemma 5.5

([35], Theorem 10.58) Let $2\le q\le r<\infty$. Then the following statements are equivalent:

• (i) $\mathbf{B}$ is isomorphic to a q-uniformly convex space;

• (ii) There exists a constant C such that for every martingale $f=(f_n)_{n\ge 0}\in H_r(\mathbf{B})$,

  $$\begin{aligned} \Vert f\Vert _{H_r^{S^q}(\mathbf{B})}\le C \Vert f\Vert _{H_r(\mathbf{B})}; \end{aligned}$$

• (iii) There exists a constant C such that for every martingale $f=(f_n)_{n\ge 0}\in H_r(\mathbf{B})$,

  $$\begin{aligned} \Vert f\Vert _{H_r^{s^q}(\mathbf{B})}\le C \Vert f\Vert _{H_r(\mathbf{B})}. \end{aligned}$$

Theorem 5.6

Let $\mathbf{B}$ be a Banach space and b be a nondecreasing slowly varying function. For $2\le q<\infty$, $0<r_1<q$ and $0< r_2\le \infty$. Then the following statements are equivalent:

• (i) $\mathbf{B}$ is isomorphic to a q-uniformly convex space;

• (ii) For each $\mathbf{B}$-valued martingale $f=(f_n)_{n\ge 0}\in D_{r_1,r_2,b}(\mathbf{B})$, there exists a constant C such that

  $$\begin{aligned} \Vert f\Vert _{H_{r_1,r_2,b}^{S^q}(\mathbf{B})}\le C \Vert f\Vert _{D_{r_1,r_2,b}(\mathbf{B})}; \end{aligned}$$

• (iii) For each $\mathbf{B}$-valued martingale $f=(f_n)_{n\ge 0}\in D_{r_1,r_2,b}(\mathbf{B})$, there exists a constant C such that

  $$\begin{aligned} \Vert f\Vert _{H_{r_1,r_2,b}^{s^q}(\mathbf{B})}\le C \Vert f\Vert _{D_{r_1,r_2,b}(\mathbf{B})}. \end{aligned}$$

Proof

$\mathrm{(i)}\Rightarrow \mathrm{(ii)}$. It is clear that $\{S^q(a)>0\}\subset \{\nu <\infty \}$, where a is a $(3, r_1, \infty )$-atom and $\nu$ is the corresponding stopping time. According to Lemma 5.5, we know that the $\sigma$-sublinear operator $S^q: H_q(\mathbf {B} )\rightarrow L_q(\Omega )$ is bounded. Condition $\mathrm (i)$ implies that $\mathbf{B}$ has the RNP. It follows from Lemma 5.3 that

$$\begin{aligned} \Vert S^q(f)\Vert _{r_1,r_2,b} \le C\Vert f\Vert _{D_{r_1,r_2,b}(\mathbf{B})},\quad \forall f=(f_n)_{n\ge 0}\in D_{r_1,r_2,b}(\mathbf{B}). \end{aligned}$$

Namely, $\Vert f\Vert _{H_{r_1,r_2,b}^{S^q}(\mathbf{B})}\le C \Vert f\Vert _{D_{r_1,r_2,b}(\mathbf{B})}$.

$\mathrm{(i)}\Rightarrow \mathrm{(iii)}$. It is similar to that of $\mathrm{(i)}\Rightarrow \mathrm{(ii)}$ above.

$\mathrm{(ii)}\Rightarrow \mathrm{(i)}$. Let $f=(f_n)_{n\ge 0}$ be an arbitrary $\mathbf{B}$-valued martingale with $\sup _{n\ge {0}}\Vert f_n\Vert _{\infty }<\infty$. Set $\lambda _n=\sup _{n\ge {0}}\Vert f_n\Vert _{\infty }$, then $\Vert f\Vert _{D_{r_1,r_2,b}(\mathbf{B})}<\infty$. Since $\Vert f\Vert _{H_{r_1,r_2,b}^{S^q}(\mathbf{B})}\le C \Vert f\Vert _{D_{r_1,r_2,b}(\mathbf{B})}$, we get $S^q(f)<\infty$. Using Lemma 2.9, we obtain $\mathbf{B}$ is isomorphic to a q-convex space.

$\mathrm{(iii)}\Rightarrow \mathrm{(i)}$. Consider a $\mathbf{B}$-valued dyadic martingale $f=(f_n)_{n\ge 0}$ such that $\sup _{n\ge {0}}\Vert f_n\Vert _{\infty }<\infty$. Then $S^q(f)\le Cs^q(f)<\infty$. Applying Lemma 2.9 and Remark 2.10, we get the desired result. $\square$