Martingale Operators and Hardy Spaces with Continuous Time Generated by Them

Abstract

In this paper, we introduce the martingale Hardy spaces and $BMO$ spaces generated by an operator T in continuous time and establish the atomic decomposition theorem of the space $H_p^T$ under the condition that T is predictable. We show that the $BM{O}_q$ spaces generated by the operator T are all equivalent and consider the sharp operator. Using the real interpolation method, we identify the interpolation spaces between the Hardy spaces and the $BMO$ spaces. With the aid of atomic decomposition, we establish some martingale inequalities between the Hardy spaces generated by two different operators.

1. Introduction

The history of continuous-time martingale theory dates back to the 1930s. Lévy [1] investigated the properties and convergence conditions of integrals composed of independent random variables. This work laid a foundation for the research on continuous-time martingale theory. Later, Ville [2] introduced the term martingale and extended its definition to continuous-time martingales. Doob [3] systematically explored continuous-time martingales and their connections to subharmonic functions. Burkholder [4] showed the connection between the $H^p$ spaces and harmonic functions. For a local continuous martingale X, Weisz [5] considered the maximal function $M^{*}\left(X\right)$, the quadratic variation $\left[X\right]$ and the conditional quadratic variation $⟨X⟩$. He established corresponding atomic decomposition theorems, key martingale inequalities and duality theorems. Afterward, Weisz [6] researched a new decomposition theorem for local martingales and studied the interpolation spaces between Hardy and $BMO$ spaces and between martingale Hardy spaces. The $BMO$ spaces $BM{O}_r^{⟨⟩}$ and $BM{O}_r^{\left[\right]}$ were introduced, and their relationships with $BMO$, $L_p$ and Hardy spaces were investigated in [7]. Hong et al. [8] studied asymmetric Doob inequalities in continuous time. Lu and Peng [9] introduced several martingale Orlicz–Hardy spaces and established some martingale inequalities. We refer the readers to [10,11,12,13,14,15] for additional contributions to the theory of continuous martingales.

In this paper, we investigate martingale operators as studied by Burkholder and Gundy in [16]. They introduced quasi-operators and proved some integral inequalities in discrete time. Weisz [17] further explored the martingale Hardy space generated by an operator T and established a corresponding atomic decomposition theorem, interpolation theorems, some martingale inequalities and dual theorems based on the work of [16]. The operator T maps the set of martingales into the set of non-negative $\mathcal{A}$-measurable functions. In particular, he pointed out that in discrete time, the maximal function M, the p-variation operator $S_p$ and the conditional p-variation operator $s_p$ belong to the class of the preceding operators. Subsequently, Fan [18] proved some inequalities concerning $S_p$ and $s_p$. Hou and Ren [19] established some inequalities about sublinear operators in weak Hardy spaces. Liu [20] introduced various types of Hardy martingale spaces and Lorentz Hardy martingale spaces and their corresponding results. Bakas et al. [21] investigated continuous bilinear decompositions of the multiplication between elements in martingale Hardy spaces and their dual spaces. Besides, for a local continuous martingale X, Pisier and Xu [22] studied the p-variation $S_p\left(X\right)$ and derived some important inequalities associated with $S_p\left(X\right)$. Peter and Victoir [23] further established quantitative bounds of the p-variation norm in the form of a $BDG$ inequality. Motivated by these works, we naturally consider the general form of the T operator in continuous time. Here, the operators M, $[·]$, $⟨·⟩$ and $S_p$ also belong to our class of operators. We focus on the Hardy and $BMO$ spaces generated by an arbitrary operator T. More precisely, our results generalize the related results in [5,6,7]. Several new results are proved, and many known results are generalized for arbitrary operators T in continuous time.

This paper is organized as follows. Section 2 provides essential definitions and preliminary concepts. In Section 3, we provide the definition of the atoms and establish the atomic decomposition theorem for the $H_p^T$ space generated by a predictable operator T. If $T=⟨·⟩$ in Theorem 1, our result can lead back to Theorem 1 in [5].

Section 4 introduces the sharp operator $T^{♯}$ of an operator T and considers the $BM{O}_q$ spaces in continuous time. We study the $BM{O}_q^T$ space, which is defined by the $L_{\infty }$-norm of $T_q^{♯}$ and prove that all $BM{O}_q^T$ spaces are equivalent for $0<q<\infty$ (see Theorem 2). Additionally, we demonstrate the equivalence between the $L_p$-norm of $T_q^{♯}\left(X\right)$ and the $H_p^T$-norm of X for $0<q<p<\infty$ (see Theorem 3).

In Section 5, we investigate the interpolation spaces between the Hardy and $BMO$ spaces in continuous time by the real method. To be precise, we prove that if T is predictable, $0<\theta <1$, $0<q\le \infty$ and $0<p_0\le 1$, then

$$\begin{array}{c}\hfill {\left(H_{p_0}^T,H_{\infty }^T\right)}_{\theta ,q}=H_{p,q}^T,\mathrm{where}{\displaystyle \frac{1}{p}}={\displaystyle \frac{1-\theta }{p_0}}.\end{array}$$

(1)

If T is predictable, $0<\theta <1$, $0<q\le \infty$ and $0<r<\infty$, then

$${\left(H_r^T,BM{O}^T\right)}_{\theta ,q}=H_{p,q}^T,\mathrm{where}{\displaystyle \frac{1}{p}}={\displaystyle \frac{1-\theta }{r}}.$$

(2)

In particular, setting $T=⟨·⟩$ in (1) and (2), the above statements can reduce to Theorem 4 and Colollary 2 in [6].

In the last section, with the help of atomic decomposition and interpolation theorems, we prove that if a continuous-time martingale inequality holds for a parameter p, then it also holds for all parameters less than p.

Before we proceed further, let us fix some notations. We use $\mathbb{N}$, $\mathbb{Z}$ and ${\mathbb{R}}^{+}$ to denote the set of nonnegative integers, the set of integers and the set of positive real numbers, respectively. The letter C denotes a positive constant, which may vary from line to line. The symbol $f\lesssim g$ stands for the inequality $f\le Cg$. If we write $f\approx g$, then it means $f\lesssim g\lesssim f$. We denote by ${\chi }_I$ the indicator function of a measurable set I.

2. Preliminaries and Notations

Let $(\mathsf{\Omega },\mathcal{F},\mathbb{P})$ be a complete probability space endowed with a right-continuous filtration $\mathcal{F}={\left({\mathcal{F}}_t\right)}_{t\in {\mathbb{R}}^{+}}$ satisfying the usual conditions, where each ${\mathcal{F}}_t$$(t\in {\mathbb{R}}^{+})$ is a sub-$\sigma$-algebra of $\mathcal{F}$ and ${\mathcal{F}}_s\subseteq {\mathcal{F}}_t$ for $s\le t$. Define ${\mathcal{F}}_{\infty }=\sigma \left({\bigcup }_{t\in {\mathbb{R}}^{+}}{\mathcal{F}}_t\right)$ with ${\mathcal{F}}_{\infty }=\mathcal{F}$. For $t\in {\mathbb{R}}^{+}$, we define

$${\mathcal{F}}_{t^{+}}=\bigcap _{s>t}{\mathcal{F}}_s,{\mathcal{F}}_{t^{-}}=\underset{s<t}{\bigvee }{\mathcal{F}}_s,$$

where ${\mathcal{F}}_{0^{-}}={\mathcal{F}}_0$. Throughout, we assume ${\mathcal{F}}_t={\mathcal{F}}_{t^{+}}$ for all $t\in {\mathbb{R}}^{+}$.

A stochastic process is a mapping X: ${\mathbb{R}}^{+}\times \Omega \to \mathbb{R}$ such that for every $t\in {\mathbb{R}}^{+}$, the function $\omega ⟼X_t\left(\omega \right)=X(t,\omega )$ is $\mathcal{F}$-measurable. A stochastic process X is adapted if $X(t,·)$ is ${\mathcal{F}}_t$-measurable for all $t\in {\mathbb{R}}^{+}$. X is called a regular process if X is adapted and all the functions $t\mapsto X(t,\omega )$ have a left and a right limit for every $\omega \in \mathsf{\Omega }$. Denote left and right limits by $X_{t^{-}}$ and $X_{t^{+}}$, respectively. The process is right-continuous (resp. left-continuous) if $X_{t^{+}}=X_t$ (resp. $X_{t^{-}}=X_t$) for all $t\in {\mathbb{R}}^{+}$.

The optional $\sigma$-algebra $\mathcal{G}$ is defined as the $\sigma$-algebra on ${\mathbb{R}}^{+}\times \mathsf{\Omega }$ generated by real-valued, regular and right-continuous processes. A stochastic process X is optional if it is $\mathcal{G}$-measurable. Similarly, the predictable $\sigma$-algebra $\mathcal{P}$ is generated by adapted continuous real processes. A process X is predictable if it is $\mathcal{P}$-measurable.

A map $\tau :\mathsf{\Omega }\to {\mathbb{R}}^{+}\cup \{\infty \}$ is said to be a stopping time with respect to ${\mathcal{F}}_t$ if $\{\omega :\tau \left(\omega \right)\le t\}\in {\mathcal{F}}_t$ for all $t\in {\mathbb{R}}^{+}$. The set of stopping times is denoted by $\mathcal{T}$. According to [24], $\tau$ is a stopping time if and only if ${\chi }_{[0,\tau ]}$ is predictable or, equivalently, if $\{\tau <t\}\in {\mathcal{F}}_t$ for all t. The graph of $\tau$ is defined as

$$\left[\left[\tau \right]\right]=\left\{(t,\omega )\in {\mathbb{R}}^{+}\times \Omega :t=\tau \left(\omega \right)<\infty \right\}.$$

A stopping time $\tau$ is predictable if $\left[\right[\tau \left]\right]\in \mathcal{P}$ and the set of all predictable stopping times is denoted by $\mathcal{P}\mathcal{T}$.

For any stopping time $\tau$, the $\sigma$-algebras ${\mathcal{F}}_{\tau }$ and ${\mathcal{F}}_{{\tau }^{-}}$ are defined as

$${\mathcal{F}}_{\tau }=\left\{F\in {\mathcal{F}}_{\infty }:F\cap \{\tau \le t\}\in {\mathcal{F}}_t,t\in {\mathbb{R}}^{+}\right\}$$

and

$${\mathcal{F}}_{{\tau }^{-}}=\left\{F\cap \{\tau <t\}:F\in {\mathcal{F}}_t,t\in {\mathbb{R}}^{+}\right\}\cup {\mathcal{F}}_0.$$

For a constant stopping time $\tau \left(\omega \right)\equiv t$, we have ${\mathcal{F}}_{\tau }={\mathcal{F}}_t$ and ${\mathcal{F}}_{{\tau }^{-}}={\mathcal{F}}_{t^{-}}$.

Denote the expectation operator by $\mathbb{E}$, the conditional expectation operator with respect to ${\mathcal{F}}_t$, ${\mathcal{F}}_{t^{-}}$, ${\mathcal{F}}_{\tau }$ and ${\mathcal{F}}_{{\tau }^{-}}$ by ${\mathbb{E}}_t$, ${\mathbb{E}}_{t^{-}}$, ${\mathbb{E}}_{\tau }$ and ${\mathbb{E}}_{{\tau }^{-}}$, respectively. We assume ${\mathbb{E}}_{0}f=0$ for all $f\in L_1$.

A stochastic process X is called a martingale if X is adapted, satisfies $\mathbb{E}\left(|X_t|\right)<\infty$ for all $t\in {\mathbb{R}}^{+}$ and ${\mathbb{E}}_s{X}_t=X_s$ whenever $s<t$. For simplicity, we assume $X_0=0$ for all martingales X. Denote by $\mathcal{M}$ the set of all martingales. ${\mathcal{M}}_p$ denotes the set of all $L_p$-bounded martingales, i.e., those for which

$${\parallel X\parallel }_{{\mathcal{M}}_p}=\underset{t\in {\mathbb{R}}^{+}}{sup}{\parallel X_t\parallel }_p<\infty .$$

In martingale theory, it is a classical result that if the martingale X is uniformly integrable (resp. $X\in {\mathcal{M}}_p$ with $p>1$), then there exists $X_{\infty }$ such that $X_t\to X_{\infty }$ a.e. and in $L_1$ (resp. $L_p$) as $t\to \infty$. Consequently, the map $X\mapsto X_{\infty }$ establishes an isomorphism between the space of uniformly integrable martingales, which is a subspace of ${\mathcal{M}}_1$ and $L_1$, and additionally between ${\mathcal{M}}_p$ and $L_p$ for $p>1$ (see [25]).

Let X be a real stochastic process and $\tau$ be a stopping time. The stopped process $X^{\tau }={\left(X_t^{\tau }\right)}_{t\in {\mathbb{R}}^{+}}$ is defined as

$$X_t^{\tau }=X_{t\wedge \tau }=X_t{\chi }_{\{t<\tau \}}+X_{\tau }{\chi }_{\{t\ge \tau \}}.$$

An adapted process X is said to be a local martingale when there exists a non-decreasing sequence of stopping times ${\left\{{\tau }_n\right\}}_{n=1}^{\infty }$ satisfying ${lim}_{n\to \infty }{\tau }_n=\infty$ a.e. and such that each stopped process $X^{{\tau }_n}$ is a uniformly integrable martingale. ${\left\{{\tau }_n\right\}}_{n=1}^{\infty }$ is said to be a fundamental sequence of X. Indeed, every martingale is a local martingale (take ${\tau }_n=n$), but the converse is not valid. Let ${\mathcal{M}}^{\mathrm{loc}}$ denote the set of all local martingales. A local martingale X is called locally $L_p$-bounded if there exists a non-decreasing sequence of stopping times ${\left\{{\tau }_n\right\}}_{n=1}^{\infty }$ with ${lim}_{n\to \infty }{\tau }_n=\infty$ a.e. such that each stopped process $X^{{\tau }_n}$ is an $L_p$-bounded martingale. Let ${\mathcal{M}}_p^{\mathrm{loc}}$ denote the set of such local martingales.

A stochastic process X is of class (D) if the set $\{X_{\tau }{\chi }_{\{\tau <\infty \}}:\tau \in \mathcal{T}\}$ is uniformly integrable. From [25], local martingales of class (D) are martingales. We further define,

$$\Delta X_t=X_t-X_{t^{-}},(X_0=0),$$

$$X_s^{*}=\underset{t\le s}{sup}|X_t|,X^{*}=\underset{t\in {\mathbb{R}}^{+}}{sup}\left|X_t\right|.$$

For $X\in {\mathcal{M}}_2^{\mathrm{loc}}$, there exists a unique predictable, right-continuous and increasing process $⟨X⟩$, called conditional quadratic variation (or sharp bracket) such that $X^2-⟨X⟩$ is a local martingale (see [25]). Similarly, if X is a local martingale, then there exists a unique right-continuous and increasing process $\left[X\right]$, called quadratic variation (or square bracket) such that $X^2-\left[X\right]$ is a local martingale and $\Delta {\left[X\right]}_t={(\Delta X_t)}^2$ (${\left[X\right]}_0=0$).

A martingale operator T is defined as an operator that maps the set of martingales to the set of non-negative $\mathcal{F}$-measurable functions. Throughout this paper, we assume that T satisfies the following conditions:

(B1)

Subadditivity: For any sequence of martingales ${\left(X_n\right)}_{n\in \mathbb{N}}$ and $X={\displaystyle \sum _{n=0}^{\infty }}{X}_n$ (in the sense of $X_t={\displaystyle \sum _{n=0}^{\infty }}{X}_{n,t}$ a.e. for all $t\in {\mathbb{R}}^{+}$), we have

$$T\left(X\right)\le \sum _{n=0}^{\infty }T\left(X_n\right),\mathrm{where}{\left(X_n\right)}_{n\in \mathbb{N}}\mathrm{are}\mathrm{martingales}.$$

(B2)

Homogeneity: $T\left(cX\right)=\left|c\right|T\left(X\right)$ for $c\in \mathbb{R}$.

(B3)

Locality: $T\left(X\right)=0$ on $\{⟨X⟩=0\}$.

(B4)

Symmetry: $T\left(X\right)=T(-X)$.

       

Define $T_s\left(X\right)=T\left(X^s\right)$ and $T^{*}\left(X\right)=\underset{s\in {\mathbb{R}}^{+}}{sup}{T}_s\left(X\right)$ with $T_0\left(X\right)=0$. Then,

(1)

$T(X-Y)\le T\left(X\right)+T\left(Y\right)$;

(2)

$T(X^{\mu }-X^{\nu })=0$ on $\{\mu =\nu \}$.

Furthermore, for a finite stopping time $\nu \in \mathcal{T}$, if $T_{\nu }\left(X\right)=T_t\left(X\right)$ on $\{\nu =t\}$, then $T_{\nu }\left(X\right)=T\left(X^{\nu }\right)$. Notably, the operator $T^{*}$ also satisfies (B1)–(B4). An operator T is adapted (resp. predictable) if $T_t\left(X\right)$ is ${\mathcal{F}}_t$-measurable (resp. ${\mathcal{F}}_{t^{-}}$-measurable). If $M\left(X^s\right)=\left|X_s\right|$, then $M_s^{*}\left(X\right)=X_s^{*}$ and $M^{*}\left(X\right)=X^{*}$$(s\in {\mathbb{R}}^{+})$. It is easy to see that the operators such as M, $⟨·⟩$, $[·]$ and $S_p$ satisfy (B1)–(B4), where $S_p\left(X\right)$ denotes the p-variation of X, i.e.,

$$S_p\left(X\right)=\underset{\begin{array}{c}0=t_0<t_1<\dots <t_n<\infty \\ n\in \mathbb{N}\end{array}}{sup}{\left(\sum _{i=1}^n{|X_{t_i}-X_{t_{i-1}}|}^p\right)}^{1/p}.$$

Now, we introduce the operator $T^{-}$. Let $\lambda ={\left({\lambda }_t\right)}_{t\in {\mathbb{R}}^{+}}$ be a non-negative, non-decreasing and predictable sequence of functions for which

$$T_t\left(X\right)\le {\lambda }_t(t\in {\mathbb{R}}^{+}).$$

Define

$$T_t^{-}\left(X\right)=\underset{\lambda }{inf}{\lambda }_t(t\in {\mathbb{R}}^{+}),T^{-}\left(X\right)=\underset{t\in {\mathbb{R}}^{+}}{sup}{T}_t^{-}\left(X\right).$$

Clearly, $T^{-}$ satisfies (B1)–(B4), and $T_t^{-}\left(X\right)$ is predictable and non-decreasing in t.

Now, for $0<p\le \infty$, the martingale Hardy space $H_p^T$ generated by T is defined as

$$H_p^T=\left\{X\in {\mathcal{M}}^{loc}{:\parallel X\parallel }_{H_p^T}={\parallel T^{*}\left(X\right)\parallel }_p<\infty \right\}.$$

Especially, if $T=M$, $T=⟨·⟩$ and $T=[·]$ in the definition above, respectively, we get the martingale Hardy spaces associated with the maximal function M, the conditional quadratic variation $⟨·⟩$ and the quadratic variation $[·]$, respectively. For the definitions of these martingale Hardy spaces, we refer the reader to [5].

3. Atomic Decompositions

In this section, we characterize the martingale space $H_p^T$. Firstly, let us review the definitions for atoms.

Definition 1.

A martingale a is called a $(T,p)$-atom if there exists a predictable stopping time ν such that

(1)

$a_t=0$ if $t<\nu$;

(2)

$\parallel T^{*}{\left(a\right)\parallel }_{\infty }\le \mathbb{P}{(\nu <\infty )}^{-1/p}$.

Theorem 1.

Let T be a predictable operator and $0<p<\infty$. If $X={\left(X_t\right)}_{t\in {\mathbb{R}}^{+}}\in H_p^T$ is a local martingale, then there exist a sequence ${\left(a^k\right)}_{k\in \mathbb{Z}}$ of $(T,p)$-atoms and a sequence ${\left({\mu }_k\right)}_{k\in \mathbb{Z}}$ of real numbers such that

$$\begin{array}{c}\hfill X_t=\sum _{k=-\infty }^{\infty }{\mu }_k{a}_t^{k}a.e.,t\in {\mathbb{R}}^{+},\end{array}$$

(3)

and

$$\begin{array}{c}\hfill {\left(\sum _{k=-\infty }^{\infty }{\left|{\mu }_k\right|}^p\right)}^{1/p}\lesssim {\parallel X\parallel }_{H_p^T}.\end{array}$$

(4)

Conversely, if $0<p\le 1$ and X has a decomposition of (3), then $X\in H_p^T$ and

$$\begin{array}{c}\hfill {\parallel X\parallel }_{H_p^T}\approx inf{\left(\sum _{k=-\infty }^{\infty }{\left|{\mu }_k\right|}^p\right)}^{1/p},\end{array}$$

(5)

where the infimum is taken over all the decompositions of the form (3).

Proof. 

Let $X={\left(X_t\right)}_{t\in {\mathbb{R}}^{+}}\in H_p^T$ be a local martingale. For each $k\in \mathbb{Z}$, we define the stopping times

$${\nu }_k=inf\left\{t\in {\mathbb{R}}^{+}:T_t^{*}\left(X\right)\ge 2^k\right\},(inf\varnothing =\infty ).$$

Obviously, ${\nu }_k$ is predictable (see [25]). We first prove the following decomposition:

$$\begin{array}{cc}\hfill X_t& =\sum _{k=-\infty }^{\infty }\left(X_t^{{\nu }_{k+1}^{-}}-X_t^{{\nu }_k^{-}}\right)\mathrm{a}.\mathrm{e}..\hfill \end{array}$$

(6)

It is easy to check that

$$\sum _{k=l}^m\left(X_t^{{\nu }_{k+1}^{-}}-X_t^{{\nu }_k^{-}}\right)=X_t^{{\nu }_{m+1}^{-}}-X_t^{{\nu }_l^{-}}$$

and ${lim}_{m\to \infty }{\nu }_m=\infty$. Thus, we get that $X_t^{{\nu }_{m+1}^{-}}\to X_t$. To derive (6), it suffices to prove that

$$\begin{array}{c}\hfill X_t^{{\nu }_l^{-}}\to 0\mathrm{as}l\to -\infty .\end{array}$$

(7)

Let

$$\nu =\underset{k\to -\infty }{lim}{\nu }_k=inf\left\{t\in {\mathbb{R}}^{+}:T_t^{*}\left(X\right)\ne 0\right\}.$$

Clearly, $\nu$ is not necessarily predictable. It follows from $\{T_{\nu }^{*}\left(X\right)=0\}\in {\mathcal{F}}_{\nu }$ that

$$\tau \left(\omega \right)={\nu }_{\{T_{\nu }^{*}\left(X\right)=0\}}\left(\omega \right)=\left\{\begin{array}{cc}\nu \left(\omega \right)\hfill & \mathrm{if}\omega \in \{T_{\nu }^{*}\left(X\right)=0\}\hfill \\ \infty \hfill & \mathrm{if}\omega \notin \{T_{\nu }^{*}\left(X\right)=0\}\hfill \end{array}\right.$$

is a stopping time (see [24]). Define

$${\tau }^{\prime }=inf\left\{t\in {\mathbb{R}}^{+}:T_{\nu }^{*}\left(X\right){\chi }_{[0,\nu ]}\left(t\right)\ne 0\right\}.$$

By the definition of ${\tau }^{\prime }$, we have that ${\tau }^{\prime }=\nu$ on the set $\{T_{\nu }^{*}\left(X\right)\ne 0\}$ and ${\tau }^{\prime }=\infty$ on the set $\{T_{\nu }^{*}\left(X\right)=0\}$. Thus, $\nu =\tau \wedge {\tau }^{\prime }$. Obviously, $T^{*}\left(X\right)$ and ${\chi }_{[0,\nu ]}$ are predictable. Hence, ${\tau }^{\prime }$ is a predictable stopping time. Therefore, there exists a non-decreasing sequence ${\tau }_n^{\prime }$ of stopping times such that ${lim}_{n\to \infty }{\tau }_n^{\prime }={\tau }^{\prime }$ and ${\tau }_n^{\prime }<{\tau }^{\prime }$ on $\{{\tau }^{\prime }>0\}$ (see [24]). If ${\tau }_m$ is a fundamental sequence for X, then we obtain $\tau \wedge {\tau }_n^{\prime }\wedge {\tau }_m\nearrow \nu$ as $n,m\to \infty$. According to the definition of $\tau$, we get

$$\mathbb{E}\left(X_{t\wedge \tau \wedge {\tau }_n^{\prime }\wedge {\tau }_m}^2-T_{t\wedge \tau \wedge {\tau }_n^{\prime }\wedge {\tau }_m}^{*}\left(X\right)\right)=0.$$

Apparently, $\mathbb{E}\left(X_{t\wedge \tau \wedge {\tau }_n^{\prime }\wedge {\tau }_m}^2\right)=0$. Hence, $X_{t\wedge \tau \wedge {\tau }_n^{\prime }}=0$. It follows from this fact that for $t\le \nu$, $X_t=0$ on the set $\{T_{\nu }^{*}\left(X\right)=0\}$ and $X_{t^{-}}=0$ on the set $\{T_{\nu }^{*}\left(X\right)\ne 0\}$. Now return to the proof of (7). If $t<\nu$, then (7) is valid. Let $t\ge \nu$. On the set $\{T_{\nu }^{*}\left(X\right)=0\}$, ${\nu }_l\searrow \nu$ (${\nu }_l\ne \nu$) and $X_t^{{\nu }_l^{-}}\to X_{\nu }$= 0 ($l\to -\infty$). On the set $\{T_{\nu }^{*}\left(X\right)\ne 0\}$, ${\nu }_l=\nu$ if $-l$ is sufficiently large. Thus, $X_t^{{\nu }_l^{-}}\to X_{\nu -}=0$ ($l\to -\infty$) which proves (7).

Assume that

$$a_t^k={\displaystyle \frac{X_t^{{\nu }_{k+1}^{-}}-X_t^{{\nu }_k^{-}}}{3·2^k\mathbb{P}{({\nu }_k<\infty )}^{1/p}}}\mathrm{and}{\mu }_k=3·2^k\mathbb{P}{({\nu }_k<\infty )}^{1/p}$$

$(\mathrm{let}{a}_t^k=0\mathrm{if}{\mu }_k=0)$. Then, we obtain

$$X_t=\sum _{k=-\infty }^{\infty }{\mu }_k{a}_t^k\mathrm{a}.\mathrm{e}..$$

Since X is a local martingale and the stopping times ${\left({\nu }_k\right)}_{k\in \mathbb{Z}}$ are predictable, then the stopped processes ${\left(X^{{\nu }_k^{-}}\right)}_{k\in \mathbb{Z}}$ are local martingales (see [25]). Consequently, for every fixed $k\in \mathbb{Z}$, $a^k={\left(a_t^k\right)}_{t\in {\mathbb{R}}^{+}}$ is a local martingale. For each $k\in \mathbb{Z}$, $a_t^k=0$ when $t<{\nu }_k$. Since T is local, we get

$$\begin{array}{cc}\hfill T^{*}\left(a^k\right)& \le {\displaystyle \frac{T^{*}\left(X^{{\nu }_{k+1}^{-}}\right)+T^{*}\left(X^{{\nu }_k^{-}}\right)}{{\mu }_k}}\hfill \\ \hfill & \le {\displaystyle \frac{2^{k+1}+2^k}{3·2^k\mathbb{P}{({\nu }_k<\infty )}^{1/p}}}\hfill \\ \hfill & =\mathbb{P}{({\nu }_k<\infty )}^{-1/p}.\hfill \end{array}$$

Hence, $a^k$ is really a $(T,p)$-atom and (3) holds.

Next, we verify (4). By the definitions of ${\mu }_k$ and ${\nu }_k$, we get

$$\begin{array}{cc}\hfill \sum _{k=-\infty }^{\infty }{\left|{\mu }_k\right|}^p& =3^p\sum _{k=-\infty }^{\infty }{2}^{kp}\mathbb{P}({\nu }_k<\infty )\hfill \\ \hfill & \le 3^p\sum _{k=-\infty }^{\infty }{2}^{kp}\mathbb{P}\left(T^{*}\left(X\right)\ge 2^k\right).\hfill \end{array}$$

Applying the Abel rearrangement, we find that

$$\begin{array}{cc}\hfill \sum _{k=-\infty }^{\infty }{\left|{\mu }_k\right|}^p& \lesssim \sum _{k=-\infty }^{\infty }\left({\left(2^p\right)}^{k+1}-{\left(2^p\right)}^k\right)\mathbb{P}\left(T^{*}{\left(X\right)}^p\ge {\left(2^p\right)}^k\right)\hfill \\ & \lesssim \sum _{k=-\infty }^{\infty }{\left(2^p\right)}^k\mathbb{P}\left({\left(2^p\right)}^{k-1}\le T^{*}{\left(X\right)}^p<{\left(2^p\right)}^k\right)\hfill \\ \hfill & \lesssim \mathbb{E}\left(T^{*}{\left(X\right)}^p\right),\hfill \end{array}$$

which means that (4) holds.

Conversely, suppose that the local martingale X has a decomposition of the form (3). For the case of $0<p\le 1$, by subadditivity (B1) and homogeneity (B2), we get

$$T^{*}{\left(X\right)}^p\lesssim \sum _{k=-\infty }^{\infty }{\left|{\mu }_k\right|}^p{T}^{*}{\left(a^k\right)}^p.$$

It follows from $T^{*}\left(a^k\right)=0$ on $\{{\nu }_k=\infty \}$ that

$$\begin{array}{cc}\hfill E\left[T^{*}{\left(X\right)}^p\right]& \lesssim \sum _{k=-\infty }^{\infty }{\left|{\mu }_k\right|}^p\mathbb{E}\left[T^{*}{\left(a^k\right)}^p\right]\hfill \\ & \lesssim \sum _{k=-\infty }^{\infty }{\left|{\mu }_k\right|}^p\mathbb{P}{({\nu }_k<\infty )}^{-1}\mathbb{P}({\nu }_k<\infty )\hfill \\ & =\sum _{k=-\infty }^{\infty }{\left|{\mu }_k\right|}^p.\hfill \end{array}$$

Hence,

$${\parallel X\parallel }_{H_p^T}\lesssim {\left(\sum _{k=-\infty }^{\infty }{\left|{\mu }_k\right|}^p\right)}^{1/p}.$$

Taking the infimum over all the preceding decomposition of the form (3), we get (5). The proof is complete. □

Remark 1.

If $T=⟨·⟩$ in Theorem 1, then this result can recover Theorem 1 in [5].

4. BMO Spaces and Sharp Operators

In this section, we introduce $BMO$ spaces associated with operator T satisfying (B1)–(B4), which are essential for establishing our main theorems. The classical $BM{O}_p$ spaces (see [7]) contain all martingales $X\in L_p$ for which

$$\begin{array}{cc}\hfill {\parallel X\parallel }_{BM{O}_p^{-}}& =\underset{t\in {\mathbb{R}}^{+}}{sup}\parallel {\left(E_t\left(|X_{\infty }-X_{t^{-}}{|}^p\right)\right)}^{1/p}{\parallel }_{\infty }<\infty (1\le p<\infty ),\hfill \\ \hfill {\parallel X\parallel }_{BM{O}_p}& =\underset{t\in {\mathbb{R}}^{+}}{sup}\parallel {\left(E_t\left(|X_{\infty }-X_t{|}^p\right)\right)}^{1/p}{\parallel }_{\infty }<\infty (1\le p<\infty ).\hfill \end{array}$$

These spaces satisfy the following inequalities:

$$\begin{array}{c}\hfill {\parallel X\parallel }_{BM{O}_p}\le 2\parallel X_{\infty }{\parallel }_{\infty }{,\parallel X\parallel }_{BM{O}_p^{-}}\le \parallel X_{\infty }{\parallel }_{\infty }{,\parallel X\parallel }_{BM{O}_2}\le {\parallel {⟨X⟩}_{\infty }^{1/2}\parallel }_{\infty }.\end{array}$$

Weisz [7] proved that all $BM{O}_p^{-}$ spaces $(1\le p<\infty )$ are equivalent and

$$L_{\infty }\subset BM{O}_p^{-}\subset BM{O}_p\subset L_p(1\le p<\infty ).$$

For an operator T satisfying (B1)–(B4), we define the sharp operator $T_q^{♯}$ as

$$\begin{array}{c}\hfill T_q^{♯}\left(X\right)=\underset{t\in {\mathbb{R}}^{+}}{sup}{\left[{\mathbb{E}}_t\left({\left(T^{*}\left(X-X^{t^{-}}\right)\right)}^q\right)\right]}^{1/q}(0<q<\infty ).\end{array}$$

Building on these foundations, we define the $BM{O}_q^T$ space generated by T as

$${\parallel X\parallel }_{BM{O}_q^T}={\parallel T_q^{♯}\left(X\right)\parallel }_{\infty }(0<q<\infty ).$$

In order to better characterize the $BM{O}_q^T$ space, we assume that T satisfies the following condition

$$\begin{array}{c}\hfill T\left(\underset{n\in \mathbb{N}}{lim\; inf}{X}_n\right)\le \underset{n\in \mathbb{N}}{lim\; inf}T\left(X_n\right),\end{array}$$

(8)

where ${\left(X_n\right)}_{n\in \mathbb{N}}$ are martingales. Obviously, $M^{*}$, the quadratic variation $[·]$ and the conditional quadratic variation $⟨·⟩$ also satisfy (8) (see [7] (pp. 70–72)).

To obtain the main conclusion of this section, we give the following lemmas.

Lemma 1

([5]). Let $0<q\le 1$ and A be a non-negative, non-decreasing process. If A is predictable with $A_0=0$ and satisfies

$$\begin{array}{c}\hfill {\left[{\mathbb{E}}_s\left({(A_t-A_s)}^q\right)\right]}^{1/q}\le C(t\ge s\ge 0),\end{array}$$

then

$$\begin{array}{c}\hfill {\mathbb{E}}_0\left(e^{t{A}_{\infty }^q}\right)\le {\displaystyle \frac{1}{1-t{C}^q}}(t{C}^q<1,t>0).\end{array}$$

Lemma 2.

Let $X={\left(X_t\right)}_{t\in {\mathbb{R}}^{+}}$ be a local martingale, T be a predictable operator satisfying $\left(B1\right)$–$\left(B4\right)$ and (8). Suppose that $0<q<\infty$ and there exists a function γ such that

$$\begin{array}{c}\hfill {\mathbb{E}}_t\left[{\left(T^{*}\left(X-X^t\right)\right)}^q\right]\le {\mathbb{E}}_t\left({\gamma }^q\right)\forall t\in {\mathbb{R}}^{+}.\end{array}$$

(9)

Then, for any $\beta >\alpha >0$,

$$\begin{array}{c}\hfill {(\beta -\alpha )}^q{\mathbb{E}}_0\left({\chi }_{\{T^{*}\left(X\right)>\beta \}}\right)\le {\mathbb{E}}_0\left({\chi }_{\{T^{*}\left(X\right)>\alpha \}}{\gamma }^q\right).\end{array}$$

Proof. 

Define the stopping times:

$$S=inf\{t:T_t^{*}\left(X\right)\ge \alpha \}\mathrm{and}R=inf\{t:T_t^{*}\left(X\right)\ge \beta \}.$$

On the set $\{T_t^{*}\left(X\right)>\beta \}$, by the monotonicity of $T_t^{*}\left(X\right)$, we get

$$T_{S^{-}}^{*}\left(X\right)\le \alpha \le T_S^{*}\left(X\right)\mathrm{and}{T}_R^{*}\left(X\right)\ge \beta .$$

Thus,

$$T_R^{*}\left(X\right)-T_{S^{-}}^{*}\left(X\right)\ge \beta -\alpha .$$

In virtue of (B1), we obtain

$$\begin{array}{cc}\hfill {(\beta -\alpha )}^q{\mathbb{E}}_0\left({\chi }_{\{T_{\infty }^{*}\left(X\right)\ge \beta \}}\right)& \le {\mathbb{E}}_0\left[{\chi }_{\{T_{\infty }^{*}\left(X\right)\ge \beta \}}{\left(T_R^{*}\left(X\right)-T_{S^{-}}^{*}\left(X\right)\right)}^q\right]\hfill \\ \hfill & \le {\mathbb{E}}_0\left[{\chi }_{\{T_S^{*}\left(X\right)\ge \alpha \}}{\left(T^{*}\left(X\right)-T_{S^{-}}^{*}\left(X\right)\right)}^q\right]\hfill \\ \hfill & \le {\mathbb{E}}_0\left[{\chi }_{\{T_S^{*}\left(X\right)\ge \alpha \}}{\left(T^{*}\left(X-X^{S^{-}}\right)\right)}^q\right].\hfill \end{array}$$

(10)

Next, we show that for any predictable stopping time W,

$$\begin{array}{c}\hfill {\mathbb{E}}_{W^{-}}\left[{\left(T^{*}\left(X-X^{W^{-}}\right)\right)}^q\right]\le {\mathbb{E}}_{W^{-}}\left({\gamma }^q\right).\end{array}$$

(11)

Let ${\left(W_n\right)}_{n\in \mathbb{N}}$ be a sequence of predictable stopping times with $W_n\nearrow W$ on $\{W>0\}$. By Fatou’s lemma and (8), we get

$$\begin{array}{cc}\hfill {\mathbb{E}}_{W_N}\left[{\left(T^{*}\left(X-X^{W^{-}}\right)\right)}^q\right]& ={\mathbb{E}}_{W_N}\left[\underset{n\ge N}{lim\; inf}{\left(T^{*}\left(X-X^{W_n}\right)\right)}^q\right]\hfill \\ \hfill & \le \underset{n\ge N}{lim\; inf}{\mathbb{E}}_{W_N}\left[{\left(T^{*}\left(X-X^{W_n}\right)\right)}^q\right].\hfill \end{array}$$

From (9), for any predictable stopping time U, we have

$$\begin{array}{c}\hfill {\mathbb{E}}_U\left[{\left(T^{*}\left(X-X^U\right)\right)}^q\right]\le {\mathbb{E}}_U\left({\gamma }^q\right).\end{array}$$

Assume that

$$U_n=\sum _{k=0}^{\infty }{\displaystyle \frac{k+1}{2^n}}{\chi }_{\{k/2^n<U\le (k+1)/2^n\}}.$$

Obviously, $U_n$ is an elementary stopping time, $U_n\ge U$ and $U_n\searrow U$. As a consequence of Fatou’s lemma, (8) and ${\mathcal{F}}_U\subset {\mathcal{F}}_{U_n}$, we derive

$$\begin{array}{cc}\hfill {\mathbb{E}}_U\left[{\left(T^{*}\left(X-X^U\right)\right)}^q\right]& \le {\mathbb{E}}_U\left[\underset{n\to \infty }{lim\; inf}{\left(T^{*}\left(X-X^{U_n}\right)\right)}^q\right]\hfill \\ \hfill & \le \underset{n\to \infty }{lim\; inf}{\mathbb{E}}_U\left[{\left(T^{*}\left(X-X^{U_n}\right)\right)}^q\right]\hfill \\ \hfill & \le {\mathbb{E}}_U\left({\gamma }^q\right).\hfill \end{array}$$

Applying this to $W_n$, we obtain (11). Finally, since S is predictable, and $T_S^{*}\left(X\right)$ is ${\mathcal{F}}_{S^{-}}$-measurable (see [24,25]), from (10) and (11), we obtain

$${(\beta -\alpha )}^q{\mathbb{E}}_0\left({\chi }_{\{T^{*}\left(X\right)\ge \beta \}}\right)\le {\mathbb{E}}_0\left({\chi }_{\{T^{*}\left(X\right)\ge \alpha \}}{\gamma }^q\right).$$

The proof is complete. □

Theorem 2.

Let T be a predictable operator satisfying $\left(B1\right)$–$\left(B4\right)$ and (8). For any $0<q<\infty$, the spaces $BM{O}_q^T$ generated by T are all equivalent.

Proof. 

Let $X={\left(X_t\right)}_{t\in {\mathbb{R}}^{+}}\in BM{O}_q^T$ and denote $C={\parallel X\parallel }_{BM{O}_q^T}$. Define $A_t=T_t\left(X\right)$. By conditions (B1)–(B4), we know that $A_t$ is a non-decreasing, predictable process (see [26]). Let $0<q\le 1$ be fixed and $t\ge s\ge 0$. By $a^q-b^q\le {(a-b)}^q$ and (B1), we get that

$$\begin{array}{c}\hfill A_{\infty }-A_{t^{-}}\le T\left(X-X^{t^{-}}\right).\end{array}$$

(12)

From (12) and (B3), we conclude that

$$\begin{array}{cc}\hfill {\left({\mathbb{E}}_s\left({(A_t-A_s)}^q\right)\right)}^{1/q}& \le {\left({\mathbb{E}}_s{\left(\left[T\left(X-X^{s^{-}}\right)\right]\right)}^q\right)}^{1/q}\hfill \\ \hfill & \le {\parallel X\parallel }_{BM{O}_q^T}=C.\hfill \end{array}$$

Thus, by Lemma 1, we obtain

$$\begin{array}{c}\hfill {\mathbb{E}}_0\left(e^{t{A}_{\infty }^q}\right)\le {\mathbb{E}}_0\left(e^{t{\left(T^{*}\left(X\right)\right)}^q}\right)\le {\displaystyle \frac{1}{1-8tC}}(8tC<1).\end{array}$$

(13)

Consider the martingale $X-X^{s^{-}}$, substituting $X-X^{s^{-}}$ into (13), we have

$$\begin{array}{cc}\hfill {\mathbb{E}}_s\left[e^{t{\left(T^{*}\left(X-X^{s^{-}}\right)\right)}^q}\right]& \le {\mathbb{E}}_s\left[e^{t\left(\underset{s\le t}{sup}{\left(T^{*}\left(X-X^{s^{-}}\right)\right)}^q\right)}\right]\hfill \\ \hfill & \le {\displaystyle \frac{1}{1-8tC}}(8tC<1).\hfill \end{array}$$

The case $1<q<\infty$ follows from standard techniques in Garsia [26]. The proof is finished. □

Theorem 3.

Let $X={\left(X_t\right)}_{t\in {\mathbb{R}}^{+}}$ be a local martingale, T be a predictable operator satisfying $\left(B1\right)$–$\left(B4\right)$ and (8). Then,

$${\parallel X\parallel }_{H_p^T}\approx {\parallel T_q^{♯}\left(X\right)\parallel }_p(0<q<p<\infty ).$$

Proof. 

Set $\beta =2\alpha$. Lemma 2 yields that

$$\begin{array}{c}\hfill {\alpha }^q\mathbb{E}\left[{\chi }_{\{T^{*}\left(X\right)/2>\alpha \}}\right]\lesssim \mathbb{E}\left[{\chi }_{\{T^{*}\left(X\right)>\alpha \}}{\gamma }^q\right].\end{array}$$

(14)

Multiplying (14) by $p{\alpha }^{p-q-1}$ and integrating over $\alpha \in (0,\infty )$, we get

$$\begin{array}{cc}\hfill \mathbb{E}\left[|T^{*}{\left(X\right)/2|}^p\right]& \lesssim \mathbb{E}\left[{\gamma }^q{\int }_0^{\infty }p{\alpha }^{p-q-1}{\chi }_{\{T^{*}\left(X\right)>\alpha \}}d\alpha \right]\hfill \\ \hfill & =\mathbb{E}\left[{\gamma }^q{T}^{*}{\left(X\right)}^{p-q}\right].\hfill \end{array}$$

By Hölder’s inequality, we obtain

$$\parallel T^{*}{\left(X\right)\parallel }_p\lesssim {\parallel \gamma \parallel }_p(p>q).$$

Let $\gamma =T_q^{♯}\left(X\right)$. Therefore,

$${\parallel X\parallel }_{H_p^T}\lesssim {\parallel T_q^{♯}\left(X\right)\parallel }_p.$$

On the other hand, estimating $T_q^{♯}\left(X\right)$ by

$$T_q^{♯}\left(X\right)\le \underset{t\in {\mathbb{R}}^{+}}{sup}{\left[{\mathbb{E}}_t\left(T^{*}{\left(X\right)}^q\right)\right]}^{1/q}.$$

Applying Doob’s inequality, we have

$$\begin{array}{cc}\hfill \parallel T_q^{♯}{\left(X\right)\parallel }_p& \lesssim {\left(\mathbb{E}\left[\underset{t\in {\mathbb{R}}^{+}}{sup}{\mathbb{E}}_t\left({\left(T^{*}{\left(X\right)}^q\right)}^{p/q}\right)\right)\right]}^{1/p}\hfill \\ & \lesssim \parallel T^{*}{\left(X\right)\parallel }_p(p>q).\hfill \end{array}$$

This completes the proof. □

5. Interpolation of Martingale Spaces

In this section, we explore the interpolation spaces between $BMO$ spaces and martingale Hardy spaces generated by $H_p^T$. Fundamental definitions of interpolation theory are introduced below; we refer to Weisz [6] for further details.

Definition 2.

For a measurable function f, the non-increasing rearrangement function of f is defined as

$$\tilde{f}\left(t\right)=inf\left\{y>0:\mathbb{P}\left(\{\omega :|f\left(\omega \right)|>y\}\right)\le t\right\},t\in {\mathbb{R}}^{+}.$$

Definition 3.

Let $0<p<\infty$ and $0<q\le \infty$. The Lorentz space $L_{p,q}$ consists of those measurable functions f with ${\parallel f\parallel }_{p,q}<\infty$, where

$${\parallel f\parallel }_{p,q}=\left\{\begin{array}{cc}{\left({\displaystyle \underset{0}{\overset{\infty }{\mathit{\int }}}\tilde{f}{\left(t\right)}^q{t}^{q/p}\frac{dt}{t}}\right)}^{1/q},\hfill & 0<p,q<\infty ;\hfill \\ \underset{t\in {\mathbb{R}}^{+}}{sup}{t}^{1/p}\tilde{f}\left(t\right),\hfill & 0<p<\infty ,q=\infty .\hfill \end{array}\right.$$

Definition 4.

The martingale Hardy–Lorentz space $H_{p,q}^T$ ($0<p<\infty$, $0<q\le \infty$) consists of all local martingales X satisfying

$${\parallel X\parallel }_{H_{p,q}^T}={\parallel T^{*}\left(X\right)\parallel }_{p,q}<\infty .$$

If $p=q$ in Definition 4, we obtain martingale Hardy spaces $H_p^T$. Moreover, taking $T=⟨·⟩$ in the definition above, we obtain the martingale Hardy–Lorentz space associated with the conditional quadratic variation $⟨·⟩$.

Let $(A_0,A_1)$ be a compatible couple of quasi-normed spaces continuously embedded in a topological vector space A. The following definitions are standard in interpolation theory (see [6,27]).

Definition 5.

For $f\in A_0+A_1$, the K-functional is defined by

$$K(t,f,A_0,A_1)=\underset{\begin{array}{c}f=f_0+f_1\\ f_i\in A_i\end{array}}{inf}\left(\parallel f_0{\parallel }_{A_0}+t{\parallel f_1\parallel }_{A_1}\right),t\in {\mathbb{R}}^{+}.$$

Definition 6.

For $0<\theta <1$ and $0<q\le \infty$, the real interpolation space ${(A_0,A_1)}_{\theta ,q}$ is defined as

$${\parallel f\parallel }_{{(A_0,A_1)}_{\theta ,q}}=\left\{\begin{array}{cc}{\left(\underset{0}{\overset{\infty }{\int }}{\left[t^{-\theta }K(t,f,A_0,A_1)\right]}^q{\displaystyle \frac{dt}{t}}\right)}^{1/q},\hfill & 0<q<\infty ,\hfill \\ \underset{t\in {\mathbb{R}}^{+}}{sup}{t}^{-\theta }K(t,f,A_0,A_1),\hfill & q=\infty .\hfill \end{array}\right.$$

By convention, ${(A_0,A_1)}_{0,q}=A_0$ and ${(A_0,A_1)}_{1,q}=A_1$. From [17], the K-functional for $(L_{p_0},L_{\infty })$ satisfies

$$K(t,f,L_{p_0},L_{\infty })\approx {\left(\underset{0}{\overset{t^{p_0}}{\int }}\tilde{f}{\left(x\right)}^{p_0}dx\right)}^{1/p_0},0<p_0<\infty .$$

Let $(B_0,B_1)$ be another compatible couple in a topological vector space B. A map $U:A_0+A_1\to B_0+B_1$ is called quasi-linear if for any $a\in A_0+A_1$ and decomposition $a=a_0+a_1$, there exists $b_i\in B_i$ such that

$$U\left(a\right)=b_0+b_1\mathrm{and}\parallel b_i{\parallel }_{B_i}\le K_i{\parallel a_i\parallel }_{A_i}(K_i>0,i=0,1).$$

The following well-known lemmas are useful to prove our main results in this section.

Lemma 3

([28]). Let $0\le {\theta }_0<{\theta }_1\le 1$, $0<q_0,q_1\le \infty$ and $X_i={(A_0,A_1)}_{{\theta }_i,q_i}$$(i=0,1)$. If $0<\eta <1$ and $0<q\le \infty$, then

$${(X_0,X_1)}_{\eta ,q}={(A_0,A_1)}_{\theta ,q},\theta =(1-\eta ){\theta }_0+\eta {\theta }_1.$$

If $A_0$ and $A_1$ are complete and $0<{\theta }_0={\theta }_1=\rho <1$, then

$${\left({(A_0,A_1)}_{\rho ,q_0},{(A_0,A_1)}_{\rho ,q_1}\right)}_{\eta ,q}={(A_0,A_1)}_{\rho ,q},$$

where

$${\displaystyle \frac{1}{q}}={\displaystyle \frac{1-\eta }{q_0}}+{\displaystyle \frac{\eta }{q_1}}.$$

Lemma 4

([28]). Let $0<\eta <1$, $0<p_0,p_1,q_0,q_1,q\le \infty$ and $p_0\ne p_1$. Then,

$${(L_{p_0,q_0},L_{p_1,q_1})}_{\eta ,q}=L_{p,q},{\displaystyle \frac{1}{p}}={\displaystyle \frac{1-\eta }{p_0}}+{\displaystyle \frac{\eta }{p_1}}.$$

In particular,

$${(L_{p_0},L_{p_1})}_{\eta ,p}=L_p,{\displaystyle \frac{1}{p}}={\displaystyle \frac{1-\eta }{p_0}}+{\displaystyle \frac{\eta }{p_1}}.$$

Furthermore, for $0<p<\infty$,

$${(L_{p_0,q_0},L_{p_1,q_1})}_{\eta ,q}=L_{p,q},{\displaystyle \frac{1}{q}}={\displaystyle \frac{1-\eta }{q_0}}+{\displaystyle \frac{\eta }{q_1}}.$$

Lemma 5

([17,29]). Suppose $B_i$$(i=0,1)$ has the lattice property, i.e., $\left|X\right|\le \left|Y\right|$ a.e. implies ${\parallel X\parallel }_{B_i}\le {\parallel Y\parallel }_{B_i}$. If U is a subadditive operator bounded from $A_i$ to $B_i$ with norms $K_i$$(i=0,1)$, then for $0<q\le \infty$ and $0\le \theta \le 1$, we have

$${\parallel U\left(a\right)\parallel }_{{(B_0,B_1)}_{\theta ,q}}\le K_0^{1-\theta }{K}_1^{\theta }{\parallel a\parallel }_{{(A_0,A_1)}_{\theta ,q}}.$$

Next, a new decomposition theorem for the interpolation spaces generated by an operator T is given.

Theorem 4.

Let T be a predictable operator satisfying $\left(B1\right)$–$\left(B4\right)$, $y>0$, $0<p\le 1$ and $X={\left(X_t\right)}_{t\in {\mathbb{R}}^{+}}\in H_p^T$ be a local martingale. Then, X can be decomposed as $X=R+Q$, where R and Q satisfy

$$\begin{array}{c}\hfill {\parallel R\parallel }_{H_{\infty }^T}\le 4y\end{array}$$

(15)

and

$$\begin{array}{c}\hfill {\parallel Q\parallel }_{H_p^T}\lesssim {\left(\underset{\{T^{*}\left(X\right)>y\}}{\int }{T}^{*}{\left(X\right)}^{p}d\mathbb{P}\right)}^{1/p}.\end{array}$$

(16)

Proof. 

Suppose that $X={\left(X_t\right)}_{t\in {\mathbb{R}}^{+}}$. Let $N\in \mathbb{Z}$ satisfy $2^{N-1}<y\le 2^N$. For each $k\in \mathbb{Z}$, define the same predictable stopping time as in Theorem 1:

$${\nu }_k=inf\left\{t\in {\mathbb{R}}^{+}:T_t^{*}\left(X\right)\ge 2^k\right\}.$$

From Theorem 1, X has the following decomposition

$$X_t=\sum _{k=-\infty }^{\infty }{\mu }_k{a}_t^k\mathrm{a}.\mathrm{e}.,$$

where $a^k$ are $(T,p)$-atoms and ${\mu }_k=3·2^k\mathbb{P}{({\nu }_k<\infty )}^{1/p}$. Define

$$R_t=\sum _{k=-\infty }^N{\mu }_k{a}_t^k,Q_t=\sum _{k=N+1}^{\infty }{\mu }_k{a}_t^k.$$

Evidently, $X_t=R_t+Q_t$ for all $t\in {\mathbb{R}}^{+}$ and $R=X^{{\nu }_{N+1}^{-}}$. By the definition of the stopping time ${\nu }_k$, we have

$$T^{*}\left(R\right)=T^{*}\left(X^{{\nu }_{N+1}^{-}}\right)\le 2^{N+1}\le 4y.$$

Thus, (15) holds. For Q, by Theorem 1, we obtain that

$$\begin{array}{cc}\hfill {\parallel Q\parallel }_{H_p^T}^p& \le \sum _{k=N+1}^{\infty }{\left|{\mu }_k\right|}^p\hfill \\ \hfill & \lesssim \sum _{k=N+1}^{\infty }{\left(2^k\right)}^p\mathbb{P}\left(T^{*}\left(X\right)\ge 2^k\right).\hfill \end{array}$$

Applying the Abel rearrangement, we get that

$$\begin{array}{cc}\hfill {\parallel Q\parallel }_{H_p^T}^p& \lesssim \underset{\{T^{*}\left(X\right)>2^N\}}{\int }{T}^{*}{\left(X\right)}^{p}d\mathbb{P}\hfill \\ \hfill & \lesssim \underset{\{T^{*}\left(X\right)>y\}}{\int }{T}^{*}{\left(X\right)}^{p}d\mathbb{P},\hfill \end{array}$$

which means (16) holds. This completes the proof. □

Now, we characterize the interpolation spaces between the martingale Hardy spaces generated by the predictable operator T.

Theorem 5.

Let T be a predictable operator satisfying $\left(B1\right)$–$\left(B4\right)$. For $0<\theta <1$, $0<p_0\le 1$ and $0<q\le \infty$, we have

$${(H_{p_0}^T,H_{\infty }^T)}_{\theta ,q}=H_{p,q}^T,\mathit{where}{\displaystyle \frac{1}{p}}={\displaystyle \frac{1-\theta }{p_0}}.$$

In order to prove this theorem, we first need to present two lemmas as follows.

Lemma 6.

Let T be a predictable operator satisfying $\left(B1\right)$–$\left(B4\right)$. If $0<p_0\le 1$, then

$$K(t,X,H_{p_0}^T,H_{\infty }^T)\lesssim {\left(\underset{0}{\overset{t^{p_0}}{\int }}{\tilde{S}\left(x\right)}^{p_0}dx\right)}^{1/p_0},$$

where $S=T^{*}\left(X\right)$.

Proof. 

We pick $N\in \mathbb{Z}$ such that $2^{N-1}<y\le 2^N$, where $y=\tilde{S}\left(t^{p_0}\right)$ for fixed $t\in [0,1]$. Let $Q_t$ and $R_t$ denote the martingales from Theorem 4 associated with y. We apply the definition of K to obtain

$$K(t,X,H_{p_0}^T,H_{\infty }^T)\lesssim \parallel Q_t{\parallel }_{H_{p_0}^T}+t{\parallel R_t\parallel }_{H_{\infty }^T}.$$

In view of Theorem 4, we deduce

$$\begin{array}{cc}\hfill \parallel Q_t{\parallel }_{H_{p_0}^T}& \lesssim {\left(\underset{\{S\ge y\}}{\int }{S}^{p_0}d\mathbb{P}\right)}^{1/p_0}\hfill \\ \hfill & ={\left(\underset{0}{\overset{t^{p_0}}{\int }}{\left[\tilde{S}\left(x\right)\right]}^{p_0}dx\right)}^{1/p_0}.\hfill \end{array}$$

Moreover,

$$t\parallel R_t{\parallel }_{H_{\infty }^T}\lesssim t\tilde{S}\left(t^{p_0}\right)\lesssim {\left(\underset{0}{\overset{t^{p_0}}{\int }}{\tilde{S}\left(x\right)}^{p_0}dx\right)}^{1/p_0},$$

which completes the proof. □

Lemma 7

([29]). For a non-increasing function $f:(0,\infty )\to {\mathbb{R}}^{+}$ and $0<s<q\le \infty$, we have

$${\left(\underset{0}{\overset{\infty }{\int }}{\left({\displaystyle \frac{1}{t}}\underset{0}{\overset{t}{\int }}f\left(u\right)du\right)}^q{t}^s{\displaystyle \frac{dt}{t}}\right)}^{1/q}\lesssim {\left(\underset{0}{\overset{\infty }{\int }}f{\left(t\right)}^q{t}^s{\displaystyle \frac{dt}{t}}\right)}^{1/q}.$$

Proof of Theorem 5.

We deduce from Lemma 6 and the definition of real interpolation that

$$\begin{array}{cc}\hfill {\parallel X\parallel }_{{(H_{p_0}^T,H_{\infty }^T)}_{\theta ,q}}^q\lesssim & \underset{0}{\overset{1}{\int }}{t}^{-\theta q}{\left(\underset{0}{\overset{t^{p_0}}{\int }}{\left[\tilde{S}\left(x\right)\right]}^{p_0}dx\right)}^{q/p_0}{\displaystyle \frac{dt}{t}}\hfill \\ \hfill \lesssim & \underset{0}{\overset{1}{\int }}{t}^{(1-\theta q)/p_0}{\left({\displaystyle \frac{1}{t}}\underset{0}{\overset{t}{\int }}{\left[\tilde{S}\left(x\right)\right]}^{p_0}dx\right)}^{q/p_0}{\displaystyle \frac{dt}{t}}.\hfill \end{array}$$

Applying Lemma 7, we get that

$${\parallel X\parallel }_{{(H_{p_0}^T,H_{\infty }^T)}_{\theta ,q}}^q\lesssim \underset{0}{\overset{1}{\int }}{t}^{(1-\theta q)/p_0}{\left[\tilde{S}\left(t\right)\right]}^q{\displaystyle \frac{dt}{t}}={\parallel S\parallel }_{p,q}^q.$$

Clearly, $T^{*}:H_{\infty }^T\to L_{\infty }$ and $T^{*}:H_{p_0}^T\to L_{p_0}$ are bounded. Therefore, by Theorem 4 and Lemma 5, we get

$$\begin{array}{c}\hfill T^{*}:{(H_{p_0}^T,H_{\infty }^T)}_{\theta ,q}\to {(L_{p_0},L_{\infty })}_{\theta ,q}=L_{p,q},\end{array}$$

is bounded. Therefore,

$${\parallel X\parallel }_{H_{p,q}^T}=\parallel T^{*}{\left(X\right)\parallel }_{p,q}\lesssim {\parallel X\parallel }_{{(H_{p_0}^T,H_{\infty }^T)}_{\theta ,q}},$$

which finishes the proof of Theorem 5 for $0<q<\infty$. By appropriately modifying the previous proof, the theorem can also be proven in the case of $q=\infty$. □

Extending the application of Lemma 3 and Theorem 5, we establish the following result.

Corollary 1.

Let T be a predictable operator satisfying $\left(B1\right)$–$\left(B4\right)$. If $0<\eta <1$, $0<p_0\ne p_1\le \infty$ and $0<q_0,q_1,q\le \infty$, then

$${(H_{p_0,q_0}^T,H_{p_1,q_1}^T)}_{\eta ,q}=H_{p,q}^T,\mathit{where}{\displaystyle \frac{1}{p}}={\displaystyle \frac{1-\eta }{p_0}}+{\displaystyle \frac{\eta }{p_1}}.$$

We proceed to characterize the interpolation spaces between $H_p^T$ and $BM{O}^T$, where $BM{O}^T$ denotes one of the spaces $BM{O}_q^T$.

Theorem 6.

Let T be a predictable operator satisfying $\left(B1\right)$–$\left(B4\right)$ and (8). If $0<\theta <1$, $0<r\le 1$ and $0<q\le \infty$, then

$${(H_r^T,BM{O}^T)}_{\theta ,q}=H_{p,q}^T,\mathit{where}{\displaystyle \frac{1}{p}}={\displaystyle \frac{1-\theta }{r}}.$$

Proof. 

It is straightforward to verify that

$$\begin{array}{c}\hfill {\parallel X\parallel }_{BM{O}^T}\lesssim {\parallel X\parallel }_{H_{\infty }^T}.\end{array}$$

(17)

Corollary 1 and (17) show that

$${\parallel X\parallel }_{{(H_r^T,BM{O}^T)}_{\theta ,q}}\lesssim {\parallel X\parallel }_{{(H_r^T,H_{\infty }^T)}_{\theta ,q}}={\parallel X\parallel }_{H_{p,q}^T}.$$

To prove the converse part, fix $0<u<r$ for the operator $T_u^{♯}$. Theorem 3 establishes the boundedness: $T_u^{♯}:H_r^T\to L_r$. Moreover, the operator $T_u^{♯}:BM{O}^T\to L_{\infty }$ is also bounded. Applying Lemmas 4 and 5 with $p=q$, we obtain

$$\parallel T_u^{♯}{\left(X\right)\parallel }_p\lesssim {\parallel X\parallel }_{{(H_r^T,BM{O}^T)}_{\theta ,p}}.$$

By Theorem 3, we get

$${\parallel X\parallel }_{H_p^T}\lesssim \parallel T_u^{♯}{\left(X\right)\parallel }_p\lesssim {\parallel X\parallel }_{{(H_r^T,BM{O}^T)}_{\theta ,p}},$$

which establishes the result for $p=q$, namely,

$${(H_r^T,BM{O}^T)}_{\theta ,p}=H_p^T,\mathrm{where}{\displaystyle \frac{1}{p}}={\displaystyle \frac{1-\theta }{r}}.$$

Utilizing Lemma 3, the result could be proven by a standard argument (see [30]). This completes the proof. □

Applying Lemma 3 and Theorem 6, we establish the following interpolation result between the martingale Hardy–Lorentz spaces and $BMO$ spaces generated by T.

Corollary 2.

Let T be a predictable operator satisfying $\left(B1\right)$–$\left(B4\right)$ and (8). If $0<\theta <1$, $0<p_0<\infty$ and $0<q_0,q\le \infty$, then

$${(H_{p_0,q_0}^T,BM{O}^T)}_{\theta ,q}=H_{p,q}^T,\mathit{where}{\displaystyle \frac{1}{p}}={\displaystyle \frac{1-\theta }{p_0}}.$$

Setting $T=⟨·⟩$ in Corollaries 1 and 2, the following statements hold.

Corollary 3.

If $0<\eta <1$, $0<p_0\ne p_1\le \infty$ and $0<q_0,q_1,q\le \infty$, then

$${(H_{p_0,q_0}^{⟨⟩},H_{p_1,q_1}^{⟨⟩})}_{\eta ,q}=H_{p,q}^{⟨⟩},\mathit{where}{\displaystyle \frac{1}{p}}={\displaystyle \frac{1-\eta }{p_0}}+{\displaystyle \frac{\eta }{p_1}}.$$

Corollary 4.

If $0<\theta <1$, $0<p_0<\infty$ and $0<q_0,q\le \infty$, then

$${(H_{p_0,q_0}^{⟨⟩},BM{O}^{⟨⟩})}_{\theta ,q}=H_{p,q}^{⟨⟩},\mathit{where}{\displaystyle \frac{1}{p}}={\displaystyle \frac{1-\theta }{p_0}}.$$

6. Martingale Inequalities

In this section, we investigate martingale inequalities in the martingale space $H_p^T$. Our approach is based on the method, which was introduced by Weisz [17].

Theorem 7.

Let the operators T be predictable and U be adapted satisfying $\left(B1\right)$–$\left(B4\right)$. If there exists $0<p_1\le \infty$ such that

$$\parallel U^{*}{\left(X\right)\parallel }_{p_1}\lesssim {\parallel T^{*}\left(X\right)\parallel }_{p_1}$$

(18)

for all local martingales X, then

$${\parallel X\parallel }_{H_p^U}\lesssim {\parallel X\parallel }_{H_p^T}(0<p\le p_1).$$

Proof. 

For the case of $0<p\le 1\wedge p_1$, assume that $X\in H_p^T$ and $X={\displaystyle \sum _{k=-\infty }^{\infty }}{\mu }_k{a}^k$ is the same decomposition in Theorem 3.2, then

$${\left(\sum _{k=-\infty }^{\infty }{\left|{\mu }_k\right|}^p\right)}^{1/p}\lesssim {\parallel X\parallel }_{H_p^T}.$$

For the operator U, we have the following inequality

$$U^{*}{\left(X\right)}^p\lesssim \sum _{k=-\infty }^{\infty }{\left|{\mu }_k\right|}^p{U}^{*}{\left(a^k\right)}^p.$$

Now, it suffices to show that for every $(T,p)$-atom a,

$$\mathbb{E}\left[U^{*}{\left(a^k\right)}^p\right]\le C.$$

Using the fact that $T^{*}\left(a^k\right)=0$ on the set $\{{\nu }_k=\infty \}$, we have

$$\mathbb{E}\left[U^{*}{\left(a^k\right)}^p\right]=\mathbb{E}\left[U^{*}{\left(a^k\right)}^p{\chi }_{\{{\nu }_k<\infty \}}\right]={∥U^{*}{\left(a^k\right)}^p{\chi }_{\{{\nu }_k<\infty \}}∥}_1.$$

Applying (18) and Hölder’s inequality, we derive

$$\begin{array}{cc}\hfill \mathbb{E}\left[U^{*}{\left(a\right)}^p\right]& \le {\mathbb{E}}^{p/p_1}\left[U^{*}{\left(a\right)}^{p_1}\right]\mathbb{P}{({\nu }_k<\infty )}^{1-p/p_1}\hfill \\ \hfill & \le C{\mathbb{E}}^{p/p_1}\left[T^{*}{\left(a\right)}^{p_1}\right]\mathbb{P}{({\nu }_k<\infty )}^{1-p/p_1}\hfill \\ \hfill & \le C{\left[\mathbb{P}{({\nu }_k<\infty )}^{-p_1/p}\mathbb{P}({\nu }_k<\infty )\right]}^{p/p_1}\mathbb{P}{({\nu }_k<\infty )}^{1-p/p_1}\hfill \\ \hfill & =C.\hfill \end{array}$$

For the case of $p_1>1$, using the fact that

$$U^{*}:H_{p_1}^T\to L_{p_1}\mathrm{and}{U}^{*}:H_1^T\to L_1$$

are bounded. Combining this with Lemma 5, we may infer that

$$U^{*}:(H_{p_1}^T,H_1^T)\to (L_{p_1},L_1)$$

is bounded. By Corollary 1 and Lemma 4, we obtain that $U^{*}:H_p^T\to L_p$ is bounded for $1\le p\le p_1$. The proof is complete. □

It follows from the definition of the operator T that $H_{\infty }^T=H_{\infty }^{T^{-}}$ and

$${\parallel X\parallel }_{H_p^T}\lesssim {\parallel X\parallel }_{H_p^{T^{-}}}(0<p\le \infty ).$$

(19)

The following results are a consequence of Theorem 6.1 and (19).

Corollary 5.

Let U and T be adapted operators satisfying $\left(B1\right)$–$\left(B4\right)$. If there exists $0<p_1\le \infty$ such that (18) holds, then

$${\parallel X\parallel }_{H_p^U}\lesssim {\parallel X\parallel }_{H_p^{T^{-}}}(0<p\le p_1).$$

For a regular stochastic basis, the converse part of (19) also holds (see [17]).

Corollary 6.

Let ${\left({\mathcal{F}}_t\right)}_{t\in {\mathbb{R}}^{+}}$ be regular. If $0<p\le \infty$, then $H_p^T=H_p^{T^{-}}$.

The proof of this result is similar to that of Proposition 2 by Weisz [31].

Corollary 7.

Let ${\left({\mathcal{F}}_t\right)}_{t\in {\mathbb{R}}^{+}}$ be regular, T and U be adapted operators satisfying $\left(B1\right)$–$\left(B4\right)$. If there exists $0<p_1\le \infty$ such that for all local martingales X

$$\begin{array}{c}\hfill \parallel U^{*}{\left(X\right)\parallel }_{p_1}\approx {\parallel T^{*}\left(X\right)\parallel }_{p_1},\end{array}$$

then

$${\parallel X\parallel }_{H_p^U}\approx {\parallel X\parallel }_{H_p^T}(0<p\le p_1).$$

Proof. 

Applying Corollaries 5 and 6, we obtain

$$\parallel U^{*}{\left(X\right)\parallel }_p\lesssim \parallel T^{-}{\left(X\right)\parallel }_p\lesssim {\parallel T^{*}\left(X\right)\parallel }_p.$$

The reverse inequality follows similarly. This completes the proof. □

From Theorem 7 and the Burkholder–Davis–Gundy inequality (see [25]), we get the following result.

Corollary 8.

Let $0<p\le 2$. For any local martingale X, we have

$${\parallel X\parallel }_{H_p^M}\lesssim {\parallel X\parallel }_{H_p^{⟨⟩}}{\mathit{and}\parallel X\parallel }_{H_p^{\left[\right]}}\lesssim {\parallel X\parallel }_{H_p^{⟨⟩}}.$$

Applying Corollary 8, [31] (Proposition 2 (ii)), and the Burkholder–Davis–Gundy inequality, the following statement also holds.

Corollary 9.

Let $0<p\le 2$ and ${\left({\mathcal{F}}_t\right)}_{t\in {\mathbb{R}}^{+}}$ be regular. For any local martingale X, we have

$${\parallel X\parallel }_{H_p^M}\approx {\parallel X\parallel }_{H_p^{⟨⟩}}\approx {\parallel X\parallel }_{H_p^{\left[\right]}}.$$

7. Conclusions

This paper investigates the Hardy and $BMO$ spaces generated by arbitrary operators T in continuous time, aiming to generalize classical results on martingale inequalities, atomic decomposition theorems and interpolation theorems. The core contributions lie in establishing a unified framework for operators T, which encompasses key operators such as the maximal function M, quadratic variation $[·]$, conditional quadratic variation $⟨·⟩$ and p-variation $S_p$. These findings help the understanding of continuous martingale spaces under general operators. More precisely, some discrete-time results in [16,17] are extended to continuous time and the related results in [5,6,7] are generalized.

Future research directions may include extending the framework to option pricing under uncertainty and models of market observable interest rates in finance, in the same line, among others, of [32,33].