An extension of the classical John-Nirenberg inequality of martingales

1.   Introduction

Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space and $(\mathcal{F}_n)_{n\geq0}$ an increasing sequence of sub-$\sigma$-algebras of $\mathcal{F}$ with the associated conditional expectations $(\mathbb{E}_n)_{n\geq0}$. A sequence $f = (f_n)_{n\geq0}$ adapted to $(\mathcal{F}_n)_{n\geq0}$ is said to be a martingale if $\mathbb{E}(|f_n|) < \infty$ and $E_n(f_{n+1}) = f_n$ for every $n\geq0$. For the sake of simplicity, we assume $f_0 = 0$. Let $1\leq p < \infty$. The quasi-Banach spaces $bmo_p$ are defined as follows:

Here, the notation $f$ in $|f-f_{n}|^p$ stands for $f_\infty$. It follows from ^[7] that

Before describing our main results, we recall the classical John-Nirenberg inequality in the martingale theory (see ^[6,7]).

Theorem 1. If the stochastic basis $\{\mathcal{F}_n\}_{n\geq0}$ is regular, then for $1\leq p < \infty$ we have that

with equivalent norms.

In 2014, Yi et al. ^[8] proved the John-Nirenberg inequality on the rearrangement-invariant Banach function space $E$ with $1\leq p_E\leq q_E < \infty$. In 2019, Li ^[4] considered the John-Nirenberg theorem on Lorentz space $bmo_{p, q}$ with $1 < p < \infty$ and $0 < q < \infty$.

In this paper, we first prove the John-Nirenberg inequality of $bmo_p$ martingale spaces for $0 < p < \infty$, extending Theorem 1 via a new interpolation method. Then, we extend this result to a wider class of the symmetric quasi-Banach function space $E$ with $0 < p_E\leq q_E < \infty$.

2.   Preliminaries and notations

Let us first recall some basic facts on the symmetric quasi-Banach function spaces. Let $((0, \infty), \mathcal{F}, P)$ be the Lesbegue measure space and $L_0(0, \infty)$ be the space of all Lesbegue measurable real-valued functions defined on $(0, \infty)$. Let $E$ be a quasi-Banach subspace of $L_0(0, \infty)$, simply called a quasi-Banach function space on $(0, \infty)$ in the sequel. A quasi-Banach function space $E$ on $(0, \infty)$ is called symmetric if for any $g\in E$ and any measurable function $f$ with $\mu_t(f)\leq \mu_t(g)$ ($\mu_t(f)$ and $\mu_t(g)$ respectively represent the non-increasing rearrangement of $f$ and $g$) for all $t\geq0$, $f\in E$ and $\|f\|_E\leq \|g\|_E$. $E$ is said to have the Fatou property if for every net $(x_i)_{i\in I}$ in $E$ satisfying $0\leq x_i \uparrow$ and $\sup_{i\in I} \|x_i\|_E < \infty$ the supremum $x = \sup_{i\in I}x_i$ exists in $E$ and $\|x_i\|_E\uparrow \|x\|_E$.

The K${\ddot{\mbox{o}}}$the dual of a symmetric Banach function space $E$ on $(0, \infty)$ is given by

with the norm $\|f\|_{E^\times}: = \sup\{\int^\infty_0|f(t)g(t)|dt:\|g\|_E\leq1\}.$ The space $E^{\times}$ is symmetric and has the Fatou property. Refer to ^[1,5] for more details.

For a quasi-Banach function space $E$ on $(0, \infty)$, the lower and upper Boyd indices $p_E$ and $q_E$ of $E$ are respectively defined by

where the dilation operator $D_s$ on $L_0(0, \infty)$ is defined by $(D_sf)(t) = f(t/s)$ for all $t\in (0, \infty)$. For a symmetric quasi-Banach function space $E$ on $(0, \infty)$, $D_s$ is a bounded linear operator on $E$ for every $s > 0$ and $0\leq p_E\leq q_E\leq\infty$ (see [2, Lemma 2.2]).

Given a quasi-Banach function space $E$ on $(0, \infty)$, for $0 < r < \infty$, $E^{(r)}$ will denote the quasi-Banach function space on $(0, \infty)$ defined by $E^{(r)} = \{x: |x|^r\in E\}$ and equipped with the quasi-norm $\|x\|_{E^{(r)}} = \big\||x|^r\big\|^{\frac{1}{r}}_E.$ Note that

Let $E_i$ be a quasi-Banach function space on $(0, \infty)$ for $i = 1, 2.$ The pointwise product space $E_1\odot E_2$ is defined by

with the functional $\|\cdot\|_{E_1\odot E_2}$ being defined by

We need the following lemmas (see Theorem 2.1 in ^[1]).

Lemma 1. Let $E$ and $F$ be two symmetric Banach function spaces on $(0, \infty)$.

(ⅰ) If $0 < p < \infty$, then $(E\odot F)^{(p)} = E^{(p)}\odot F^{(p)}.$

(ⅱ) $L_1(0, \infty) = E\odot E^\times$.

Lemma 2. Let $E$ be a symmetric quasi-Banach function space on $(0, \infty)$ with the Fatou property. If $p_E > p$, then $E^{(\frac{1}{p})}$ can be renormed as a symmetric Banach function space.

Proof. By (2.1), we have that $p_{E^{(\frac{1}{p})}} = \frac{1}{p}p_E > 1$. Thus $E^{(\frac{1}{p})}$ is an interpolation space for the couple $\big(L_1(0, \infty), L_\infty(0, \infty)\big)$ (see [3, Lemma 3.6]). Therefore, according to Lemma 2.2 in ^[1], we get that $E^{(\frac{1}{p})}$ can be renormed as a symmetric Banach function space.

Now we define the Hardy spaces and $BMO$ spaces of martingales. For a martingale $f = (f_n)_{n\geq0}$, we denote its martingale difference by $df_i = f_i-f_{i-1}$ (with convention $f_0 = 0$). Then the conditional quadratic variation and the square function are defined by

Let $0 < p < \infty$. Define

Here, the notation $f$ in $|f-f_{n-1}|^p$ stands for $f_\infty$.

A stochastic basis $(\mathcal{F}_n)_{n\geq0}$ is said to be regular if, for $n\geq0$ and $A\in \mathcal{F}_n$, there exists $B\in \mathcal{F}_{n-1}$ such that $A\subset B$ and $\mathbb{P}(B)\leq R \mathbb{P}(A)$, where $R$ is a positive constant independent of $n$. A martingale is said to be regular if it is adapted to a regular $\sigma$-algebra sequence. This means that there exists a constant $R > 0$ such that $f_n\leq R f_{n-1}$ for all nonnegative martingales $(f_n)_{n\geq0}$ adapted to the stochastic basis $(\mathcal{F}_n)_{n\geq0}$. We refer the reader to Long ^[6] and Weisz ^[7] for the theory of martingales.

In what follows, unless otherwise specified, for two nonnegative quantities $A$ and $B$, by $A\lesssim B$ we mean that there exists an absolute constant $C > 0$ such that $A\leq CB$, and by $A\thickapprox B$ that $A\lesssim B$ and $B\lesssim A$.

3.   Main results

In this section, we first establish the John-Nirenberg theorem of the $bmo_p$ spaces for $0 < p < 1$.

Theorem 2. If the stochastic basis $(\mathcal{F}_n)_{n\geq0}$ is regular, then, for any $f\in bmo_{1}$

Proof. From H$\rm{\ddot{o}}$lder's inequality it follows that

To prove the converse we choose $1 < p_1 < \infty$ and $0 < \theta < 1$ such that $1 = ({1-\theta})/{p}+{\theta}/{p_1}$. Fix $n$, and for any $0 < r < \infty$, let $T_n:L_{r}(\mathcal{F}_n)\rightarrow L_p(\mathcal{F})$ be a linear operator with $T_n(a) = (f-f_n)a.$ Then by the definition of $bmo_p$, we have the following inequalities:

Thus by interpolation, we have that

Noting that $(L_p, L_{p_1})_{\theta} = L_1$ with equal norms and using the inequality

(see [7, Corollory 2.51]) we reduce that

which implies that

Remark 1. (i) If, in the proof of Theorem 2, we replace $f-f_{n}$ with $f-f_{n-1}$ and $bmo_p$ and $bmo_1$ with $BMO_p$ and $BMO_1$ then

(ii) According to Theorem 1, $bmo_p$ coincides with $bmo_1$ for $1\leq p < \infty$. While for $0 < p < 1$, if a priori we assume that $f\in bmo_{1}$. Theorem 2 tells us the norms of $bmo_p$ and $bmo_1$ are also equivalent.

Recall that if ${(\mathcal{{F}}_n)}_{n\geq0}$ is regular, then $H_1^s = H_1^S$ which follows that their dual spaces $bmo_2$ and $BMO_2$ are equivalent. Hence, by Theorem 2, Theorem 1, (i) of Remark 1 and [7, Theorem 2.50], we obtain the following result.

corollary 1. Let $0 < p < \infty$. If the stochastic basis ${(\mathcal{F}_n)}_{n\geq0}$ is regular, then for any $f\in BMO_{1}$ and $f\in bmo_{1}$

Now we present the John-Nirenberg inequality of martingale spaces associated with symmetric quasi-Banach function spaces, generalizing the results obtained in ^[4,8].

Theorem 3. Let E be a symmetric quasi-Banach function space on $(0, \infty)$ with $0 < p_E\leq q_E < \infty$ that has the Fatou property. If ${(\mathcal{F}_n)}_{n\geq0}$ is regular, then for any $f\in bmo_{1}$,

where

Proof. Choose $p$ and $q$ such that $0 < p < p_E\leq q_E < q < \infty$. Then by Lemma 2, $E^{(\frac{1}{p})}$ can be renormed as a symmetric Banach function space; so, we assume that $E^{(\frac{1}{p})}$ is a symmetric Banach function space. By (ii) of Lemma 1, we have that $L_1(0, \infty) = E^{(\frac{1}{p})}\odot E^{(\frac{1}{p})\times}$. It follows that

where $F = (E^{(\frac{1}{p})\times})^{p}$ (see (i) of Lemma 1). Fix $n$. Take $a\in L_p(\mathcal{F}_n)$ with $\|a\|_p\leq1$. Then by (3.3), there exist $a_1\in{E}$ and $a_2\in{F}$ such that $a = a_1a_2$ and $\|a_1\|_E, \|a_2\|_F\leq1$. Thus we have that

which implies $\|f\|_{bmo_p}\leq\|f\|_{bmo_E}.$ Therefore, by Theorem 2, $\|f\|_{bmo_1}\leq\|f\|_{bmo_E}.$

Now we turn to the converse inequality. Fix $n$. Similar to the definition of the operator $T_n$ in Theorem 3.1, we can view $f-f_{n}$ as an operator from $L_p(\mathcal{F}_n)$ to $L_p(\mathcal{F})$ and from $L_{q}(\mathcal{F}_n)$ to $L_{q}(\mathcal{F})$; then, we get that

By Lemma 3.6 in ^[3], we have that $E$ is an interpolation space in $(L_p(0, \infty), L_q(0, \infty))$ which implies that

where $C > 0$ is a constant depending only on $p$ and $q$. Putting (3.4) and (3.5) together and using Corollary 1, we obtain that

It follows that $\|f\|_{bmo_E}\leq C\|f\|_{bmo_1}$. This completes the proof.

Remark 2. When $E = L_p(0, \infty)$ for $0 < p < \infty$, (3.2) implies that

Acknowledgments

The second author was supported in part by NSFC No.11801489.

Conflict of interest

The authors declare that they have no conflicts of interest.