Weighted estimates for the bilinear maximal operator on filtered measure spaces

TL;DR

This paper establishes weighted inequalities for the bilinear maximal operator on filtered measure spaces under the bilinear reverse Hölder condition, introducing new techniques like bilinear principal sets and a Carleson embedding theorem.

Contribution

It introduces a novel construction of bilinear principal sets and a new Carleson embedding theorem tailored for filtered measure spaces, advancing weighted inequality theory.

Findings

Weighted inequalities are established under the bilinear reverse Hölder condition.

Hytonen-Perez type weighted estimates are obtained for the bilinear maximal operator.

A new property called conditional sparsity of principal sets is identified.

Abstract

Assuming the bilinear reverse Holder's condition, we character weighted inequalities for the bilinear maximal operator on filtered measure spaces. We also obtain Hytonen-Perez type weighted estimates for the bilinear maximal operator. Our approaches are mainly based on the new construction of bilinear versions of principal sets and the new Carleson embedding theorem on filtered measure spaces. In particular, we find a new property of the construction and we call it the conditional sparsity of principal sets.

Equations

$$Mf(x)=\underset{x\in Q}{\sup }\frac{1}{|Q|}{\int }_Q|f(y)|dy,$$

$${\lambda }^p{\int }_{\{Mf>\lambda \}}u(x)dx\le C{\int }_{{\mathbb{R}}^n}|f(x){|}^{p}v(x)dx,\lambda >0,f\in L^p(v)$$

$$\Big{(}\frac{1}{|Q|}\int_{Q}u(x)dx\Big{)}\Big{(}\frac{1}{|Q|}\int_{Q}v(x)^{-\frac{1}{p-1}}dx\Big{)}^{p-1}<C,~{}p>1;$$

$$\frac{1}{|Q|}{\int }_{Q}u(x)dx\le C\underset{Q}{ess\; inf}v(x),p=1.$$

$$\int_{\mathbb{R}^{n}}\big{(}Mf(x)\big{)}^{p}v(x)dx\leq C\int_{\mathbb{R}^{n}}|f(x)|^{p}v(x)dx,~{}\forall f\in{L^{p}(v)}$$

$$\Big{(}\frac{1}{|Q|}\int_{Q}v(x)dx\Big{)}\Big{(}\frac{1}{|Q|}\int_{Q}v(x)^{-\frac{1}{p-1}}dx\Big{)}^{p-1}<C,~{}\forall Q\subset\mathbb{R}^{n}.$$

$$\int_{\mathbb{R}^{n}}\big{(}Mf(x)\big{)}^{p}u(x)dx\leq C\int_{\mathbb{R}^{n}}|f(x)|^{p}v(x)dx,~{}\forall f\in{L^{p}(v)}$$

$$\Big{(}\int_{Q}\big{(}M(\chi_{Q}v^{1-p^{\prime}})(x)\big{)}^{q}u(x)dx\Big{)}^{\frac{1}{q}}\leq C\Big{(}\int_{Q}v(x)^{1-p^{\prime}}dx\Big{)}^{\frac{1}{p}},~{}\forall Q\subset\mathbb{R}^{n},$$

$$\Big{(}\int_{\mathbb{R}^{n}}\big{(}Mf(x)\big{)}^{q}u(x)dx\Big{)}^{\frac{1}{q}}\leq C\Big{(}\int_{\mathbb{R}^{n}}|f(x)|^{p}v(x)dx\Big{)}^{\frac{1}{p}},~{}\forall f\in{L^{p}(v)}$$

$$\mathcal{M}(f_1,...,f_m)(x):=\underset{x\in Q}{\sup }\prod _{i=1}^m\frac{1}{|Q|}{\int }_Q|f_i(y_i)|d{y}_i,x\in {\mathbb{R}}^n$$

$$\sup_{Q}\Big{(}\frac{1}{|Q|}\int_{Q}v(x)dx\Big{)}\prod\limits^{m}_{i=1}\Big{(}\frac{1}{|Q|}\int_{Q}\omega_{i}(x)^{-\frac{1}{p_{i}-1}}dx\Big{)}^{\frac{p}{p^{\prime}_{i}}}<\infty,$$

$$v={\omega }_1^{\frac{p}{p_1}}{\omega }_2^{\frac{p}{p_2}}\mathrm{and}{\sigma }_i={\omega }_i^{-\frac{1}{p_i-1}},i=1,2,1/p=1/p_1+1/p_2.$$

$$\Vert \mathcal{M}(f_1{\sigma }_1,f_2{\sigma }_2){\Vert }_{L^p(v)}\le C\Vert f_1{\Vert }_{L^{p_1}({\sigma }_1)}\Vert f_2{\Vert }_{L^{p_2}({\sigma }_2)}.$$

$$\Vert \mathcal{M}(f_1{\sigma }_1,f_2{\sigma }_2){\Vert }_{L^p(v)}\le C\Vert f_1{\Vert }_{L^{p_1}({\sigma }_1)}\Vert f_2{\Vert }_{L^{p_2}({\sigma }_2)},$$

$$\Vert \mathcal{M}(f_1{\sigma }_1,f_2{\sigma }_2){\Vert }_{L^p(v)}\le C\Vert f_1{\Vert }_{L^{p_1}({\sigma }_1)}\Vert f_2{\Vert }_{L^{p_2}({\sigma }_2)}$$

$$[v,\overrightarrow{\omega }{]}_{S_{\overrightarrow{p}}}\le \Vert \mathcal{M}\Vert \le 32p_1^{\prime }{p}_2^{\prime }[v,\overrightarrow{\omega }{]}_{S_{\overrightarrow{p}}}[{\omega }_1,{\omega }_2{]}_{R{H}_{\overrightarrow{p}}}^{\frac{1}{p}}.$$

$$\Vert \mathcal{M}(f_1,f_2){\Vert }_{L^p(v)}\le C\Vert f_1{\Vert }_{L^{p_1}({\omega }_1)}\Vert f_2{\Vert }_{L^{p_2}({\omega }_2)}.$$

$$\Vert \mathcal{M}(f_1{\sigma }_1,f_2{\sigma }_2){\Vert }_{L^p(v)}\le C\Vert f_1{\Vert }_{L^{p_1}({\sigma }_1)}\Vert f_2{\Vert }_{L^{p_2}({\sigma }_2)}.$$

$$\Vert \mathcal{M}(f_1,f_2){\Vert }_{L^p(v)}\le C\Vert f_1{\Vert }_{L^{p_1}({\omega }_1)}\Vert f_2{\Vert }_{L^{p_2}({\omega }_2)}.$$

$$\Vert \mathcal{M}(f_1{\sigma }_1,f_2{\sigma }_2){\Vert }_{L^p(v)}\le C\Vert f_1{\Vert }_{L^{p_1}({\sigma }_1)}\Vert f_2{\Vert }_{L^{p_2}({\sigma }_2)}.$$

$$\Vert \mathcal{M}\Vert =\Vert \mathcal{M}{\Vert }^{\prime }\le 32\cdot 2^{\frac{1}{p}}{p}_1^{\prime }{p}_2^{\prime }[v,{\omega }_1,{\omega }_2{]}_{A_{\overrightarrow{p}}}^{\frac{1}{p}}[{\omega }_1,{\omega }_2{]}_{W_{\overrightarrow{p}}^{\infty }}^{\frac{1}{p}}.$$

$${\int }_{G}fd\mu ={\int }_{G}gd\mu ,\forall G\in {\mathcal{G}}^0.$$

$$\mathcal{L}:=\bigcap _{i\in \mathbb{Z}}{L}_{{\mathcal{F}}_i^0}^1(\mathcal{F},\mu ).$$

$$L_{{\mathcal{F}}_i^0}^1(\mathcal{F},\mu )\supset L_{{\mathcal{F}}_j^0}^1(\mathcal{F},\mu )$$

$$Mf=\underset{i\in \mathbb{Z}}{\sup }|{\mathbb{E}}_i(f)|\text{ and }\mathcal{M}(f,g)=\underset{i\in \mathbb{Z}}{\sup }|{\mathbb{E}}_i(f)||{\mathbb{E}}_i(g)|,$$

$${{}^{\ast }M}_{i}f=\underset{j\ge i}{\sup }|{\mathbb{E}}_j(f)|\text{ and }{}^{\ast }{\mathcal{M}}_i(f,g)=\underset{j\ge i}{\sup }|{\mathbb{E}}_j(f)||{\mathbb{E}}_j(g)|,$$

$$\Big{(}\int_{\{\tau<+\infty\}}\sigma_{1}d\mu\Big{)}^{\frac{p}{p_{1}}}\Big{(}\int_{\{\tau<+\infty\}}\sigma_{2}d\mu\Big{)}^{\frac{p}{p_{2}}}\leq C\int_{\{\tau<+\infty\}}\sigma_{1}^{\frac{p}{p_{1}}}\sigma_{2}^{\frac{p}{p_{2}}}d\mu.$$

$$\Vert f{\Vert }_p\Vert g{\Vert }_q\le C\Vert fg{\Vert }_1,$$

$$\underset{j\in \mathbb{Z}}{\sup }{\mathbb{E}}_j(v){\mathbb{E}}_j({\omega }_1^{1-p_1^{\prime }}{)}^{\frac{p}{p_1^{\prime }}}{\mathbb{E}}_j({\omega }_2^{1-p_2^{\prime }}{)}^{\frac{p}{p_2^{\prime }}}\le C,$$

$$[v,\overrightarrow{\omega}]_{S_{\overrightarrow{p}}}:=\sup\limits_{i\in\mathbb{Z},\tau\in\mathcal{T}_{i}}\Big{(}\frac{\int_{\{\tau<+\infty\}}{\mathcal{M}}(\sigma_{1}\chi_{\{\tau<+\infty\}},\sigma_{2}\chi_{\{\tau<+\infty\}})^{p}vd\mu}{\sigma_{1}({\{\tau<+\infty\}})^{\frac{p}{p_{1}}}\sigma_{2}({\{\tau<+\infty\}})^{\frac{p}{p_{2}}}}\Big{)}^{\frac{1}{p}}<\infty,$$

Full text

The research of Wei Chen is supported by NSFC (11971419, 11771379) and the School Foundation of Yangzhou University (2019CXJ001). The research of Yong Jiao is supported by NSFC (11722114,11961131003).

1. 1 Introduction

2. 2 Preliminaries

3. 2.1 bilinear maximal operator on filtered measure spaces

4. 2.2 bilinear weights

5. 3 The construction of principal sets

6. 4 Carleson embedding theorem associated with the collection of principal sets

7. 5 Main results and their proofs

8. 5.1 Bilinear Version of one-weight Inequalities

9. 5.2 Bilinear Version of Two-weight Inequalities

1. Introduction

Let $\mathbb{R}^{n}$ be the $n\hbox{-dimensional}$ real Euclidean space and $f$ a real valued measurable function, the classical Hardy-littlewood maximal operator is defined by

[TABLE]

where $Q$ is a non-degenerate cube with its sides parallel to the coordinate axes and $|Q|$ is the Lebesgue measure of $Q.$

Let $u,~{}v$ be two weights, i.e. positive measurable functions. As is well known, for $p\geq 1,$ Muckenhoupt [22] showed that the inequality

[TABLE]

holds if and only if $(u,v)\in A_{p},$ i.e., for any cube $Q$ in $\mathbb{R}^{n}$ with sides parallel to the coordinates

[TABLE]

[TABLE]

Suppose that $u=v$ and $p>1,$ Muckenhoupt [22] also proved that

[TABLE]

holds if and only if $v$ satisfies

[TABLE]

The crucial step is to show that if $v$ satisfies $A_{p}$, then there is an $\varepsilon>0$ such that $v$ also satisfies $A_{p-\varepsilon}.$ However, the problem of finding all $u$ and $v$ such that

[TABLE]

is much hard and complicated. In order to solve the problem, Sawyer [26] established the testing condition $S_{p,q},$ i.e. for any cube $Q$ in $\mathbb{R}^{n}$ with sides parallel to the coordinates

[TABLE]

where $1<p\leq q<\infty$ and $\chi_{Q}$ is the characteristic function of $Q.$ The condition $S_{p,q}$ is a sufficient and necessary condition such that the weighted inequality

[TABLE]

holds. In this case, the method of proof is very interesting. Motivated by [22, 26], the theory of weights developed so rapidly that it is difficult to give its history a full account here (see [8] and [6] for more information).

Weighted estimates for the maximal operator $\prod_{j=1}^{m}Mf_{j}$ ($m$-fold product of $M$) in the multilinear setting were studied in [11] and [25]. Recently, the new multilinear maximal function

[TABLE]

associated with cubes with sides parallel to the coordinate axes was first defined and the corresponding weight theory was studied in [18]. The importance of this operator is that it is strictly smaller than the $m$-fold product of $M$. Moreover, it generalizes the Hardy–Littlewood maximal function (case $m=1$) and in several ways it controls the class of multilinear Calderón–Zygmund operators as shown in [18]. The relevant class of multiple weights for $\mathcal{M}$ is given by the condition $A_{\overrightarrow{p}}$ [18, Definition 3.5]: for $\overrightarrow{p}=(p_{1},p_{2},\cdot\cdot\cdot,~{}p_{m}),$ $\overrightarrow{\omega}=(\omega_{1},~{}\omega_{2},\cdot\cdot\cdot,~{}\omega_{m})$ and a weight $v,$ the weight vector $(v,\overrightarrow{\omega})\in A_{\overrightarrow{p}}$ if

[TABLE]

where $\frac{1}{p}=\sum\limits^{m}_{i=1}\frac{1}{p_{i}}$ and $1\leq p_{1},p_{2},...,p_{m}<\infty.$ The more general case was extensively discussed in [10, 9]. Using a dyadic discretization technique, Damián, Lerner and Pérez [7] and Li, Moen and Sun [19] proved some sharp weighted norm inequalities for the multilinear maximal operator $\mathcal{M}.$ In order to establish the generalization of Sawyer’s theorem to the multilinear setting, a kind of monotone property and a reverse Hölder’s condition on the weights were introduced in [20] and [3], respectively. Note that if $v=\prod\limits_{i=1}^{m}\omega_{i}^{\frac{p}{p_{i}}},$ then the condition $(v,\overrightarrow{\omega})\in A_{\overrightarrow{p}}$ implies the reverse Hölder’s condition $\overrightarrow{\omega}\in RH_{\overrightarrow{p}}$ [1, Proposition 2.3]. In addition, Chen and Damián investigated a bound $B_{\overrightarrow{p}}$ [3, Theorem 2] and a mixed bound $A_{\overrightarrow{p}}-W^{\infty}_{\overrightarrow{p}}$ [3, Theorem 3] for the multilinear maximal operator, which are the multilinear versions of one-weight norm estimates [15, Theorem 4.3]. Still more recently, the multilinear fractional maximal operator and the multilinear fractional strong maximal operator associated with rectangles were studied in [1] and [2], respectively.

On the other hand, Tanaka and Terasawa [30] very recently developed a theory of weights for positive (linear) operators and the generalized Doob’s maximal operators on a filtered measure space. In particular, two-weight norm inequalities [30, Theorem 4.1] and one-weight norm estimates of Hytönen-Pérez type [30, Theorem 5.1] for Doob’s maximal operator were established by the use of the Carleson embedding theorem and the construction of principal set, respectively. Note that a filtered measure space naturally contains a filtered probability space with a filtration indexed by $\mathbb{N}$ and a Euclidean space with a dyadic filtration. It also contains a doubling metric measure space with dyadic lattice constructed by Hytönen and Kairema [16]. From this perspective, Dyadic Harmonic Analysis on the Euclidean space and Martingale Harmonic Analysis on a probability space can be unified on a filtered (infinite) measure space, as treated in [14, 27, 29]. We also mention that the Haar shift operators were studied by Lacey, Petermichl and Reguera in [17] and played an important role in the resolution of the so-called $A_{2}$ conjecture in [13]. On a filtered measure space, these operators could be seen from the point-of-view of martingale theory.

Motivated by the works above, the purpose of this paper is to develop a theory of weights for multilinear Doob’s maximal operator on a filtered measure space. For simplicity of notations, we only consider the bilinear case and all results can be extended to the multilinear case without essential difficulty.

The following theorem is our first main result, which gives the weights for which the bilinear maximal operator $\mathcal{M}$ is bounded from $L^{p_{1}}(\omega_{1})\times L^{p_{2}}(\omega_{2})$ to $L^{p}(\omega_{1}^{\frac{p}{p_{1}}}\omega_{2}^{\frac{p}{p_{2}}}).$ All unexplained notations can be found in Section 2.

Theorem 1.3**.**

Let $\omega_{1},~{}\omega_{2}$ be weights and $1<p_{1},~{}p_{2}<\infty.$ Suppose that

[TABLE]

1. (1)

If $(v,~{}\omega_{1},~{}\omega_{2})\in A_{\overrightarrow{p}}$, then there exists a positive constant $C$ such that for all $f_{1}\in L^{p_{1}}(\sigma_{1}),~{}f_{2}\in L^{p_{2}}(\sigma_{2})$ we have

[TABLE]

We denote the smallest constant $C$ in (1.4) by $\|\mathcal{M}\|.$ Then it follows that $\|\mathcal{M}\|\leq 16\cdot 4^{(q^{\prime}-1)}p^{\prime}_{1}p^{\prime}_{2}[v,\omega_{1},\omega_{2}]^{\frac{q^{\prime}}{p}}_{A_{\overrightarrow{P}}},$ where $q=\min\{p_{1},p_{2}\}.$ 2. (2)

Let $(\omega_{1},\omega_{2})\in RH_{\overrightarrow{p}}.$ If there exists a positive constant $C$ such that for all $f_{1}\in L^{p_{1}}(\sigma_{1}),$ $f_{2}\in L^{p_{2}}(\sigma_{2})$ we have

[TABLE]

then $(v,~{}\omega_{1},~{}\omega_{2})\in A_{\overrightarrow{p}}.$ We denote the smallest constant $C$ in (1.5) by $\|\mathcal{M}\|.$ Then it follows that $[v,\omega_{1},\omega_{2}]_{A_{\overrightarrow{P}}}\leq\|\mathcal{M}\|^{p}[\omega_{1},\omega_{2}]_{RH_{\overrightarrow{p}}}.$

First, Theorem 1.3 (more precisely, Corollary 5.3 below) is a bilinear analogue of [30, Corollary 4.5]. We remark that the reverse Hölder’s condition $RH_{\vec{p}}$ is automatically true in the linear case. Second, our theorem is an extension of [18, Theorem 3.7] and [19, Theorem 1.2] to a filtered measure space. In [18] they showed that the multilinear $A_{\vec{p}}$ condition has interesting characterization in terms of the linear $A_{p}$ classes [18, Theorem 3.6]. Then their proof was based on the Reverse Hölder’s inequality for linear $A_{p}$ classes. However, they are invalid on a filtered measure space (even on a filtered probability spaces without regularity condition [21, p.262]). Li, Moen and Sun [19, Theorem 1.2] found the optimal power on $[v,\omega_{1},\omega_{2}]_{A_{\overrightarrow{P}}}$ and their proof depends very much on the properties of the sparse family on Euclidean spaces. Because a filtered measure space contains a Euclidean space with a dyadic filtration, our theorem gives sharp bound for the bilinear maximal operator $\mathcal{M}.$ Motivated by [19, Theorem 1.2], our proof is mainly based on the bilinear construction of principal sets on filtered measure spaces. The germ of principal set appeared as the sparse family on $\mathbb{R}^{n}$ (see [15, 7] for more information) and was successfully constructed on the filtered measure space in [30, pp.942-943]. We find a new property (Section 3, P.(P. 3)) of the construction and we call it the conditional sparsity of principal sets.

Theorem 1.3 also completes the bilinear version of one-weight theory in the martingale setting [4, Proposition 1.15]. In fact, in [4] only the second part of Theorem 1.3 on a filtered probability space was proved. In addition, it is clear that (1.4) implies the condition $S_{\overrightarrow{p}}.$ Then it follows from Theorem 1.3 that the condition $A_{\overrightarrow{p}}$ implies the condition $S_{\overrightarrow{p}}.$ Hence, Theorem 1.3 is a bilinear analogue of the equivalence between $A_{p}$ and $S_{p}$ in [12].

Our second main purpose is to character two-weight inequalities for the bilinear maximal operator on the filtered measure space. Assuming the reverse Hölder’s condition, Theorem 1.6 below is a bilinear version of Sawyer’s result [26, Theorem A] on filtered measure spaces.

Theorem 1.6**.**

Let $v,\omega_{1},\omega_{2}$ be weights and $1<p_{1},p_{2}<\infty.$ Suppose that $1/p=1/p_{1}+1/p_{2}$ and $(\omega_{1},\omega_{2})\in RH_{\overrightarrow{p}},$ then the following statements are equivalent

1. $\rm(1)$

There exists a positive constant $C$ such that for all $f_{1}\in L^{p_{1}}(\sigma_{1}),~{}f_{2}\in L^{p_{2}}(\sigma_{2})$ we have

[TABLE]

where $\sigma_{i}=\omega_{i}^{-\frac{1}{p_{i}-1}},~{}i=1,~{}2.$ 2. $\rm(2)$

The triple of weights $(v,~{}\omega_{1},~{}\omega_{2})$ satisfies the condition $S_{\overrightarrow{p}}.$

Moreover, we denote the smallest constant $C$ in (1.7) by $\|\mathcal{M}\|.$ Then it follows that

[TABLE]

We also obtain Hytönen-Pérez type weighted estimates [15, theorem 4.3] for the bilinear maximal operator on filtered measure spaces. To be precise, we prove the following Theorems 1.8 and 1.11. Their linear cases were studied in [5, Theorem 1.7].

Theorem 1.8**.**

If $(v,\overrightarrow{\omega})\in B_{\overrightarrow{p}},$ then the following statements are valid:

1. (1)

There exists a positive constant $C$ such that for all $f\in L^{p_{1}}(\omega_{1}),~{}g\in L^{p_{2}}(\omega_{2})$ we have

[TABLE] 2. (2)

There exists a positive constant $C$ such that for all $f\in L^{p_{1}}(\sigma_{1}),~{}g\in L^{p_{2}}(\sigma_{2})$ we have

[TABLE]

Moreover, we denote the smallest constants $C$ in (1.9) and (1.10) by $\|\mathcal{M}\|$ and $\|\mathcal{M}\|^{\prime},$ respectively. Then it follows that $\|\mathcal{M}\|=\|\mathcal{M}\|^{\prime}\leq 32(2e)^{\frac{1}{p}}p^{\prime}_{1}p^{\prime}_{2}[v,\overrightarrow{\omega}]_{B_{\overrightarrow{p}}}^{\frac{1}{p}}.$

Theorem 1.11**.**

If $(v,\omega_{1},\omega_{2})\in A_{\overrightarrow{p}}$ and $(\omega_{1},\omega_{2})\in W^{\infty}_{\overrightarrow{p}},$ then the following statements are valid:

1. (1)

There exists a positive constant $C$ such that for all $f\in L^{p_{1}}(\omega_{1}),~{}g\in L^{p_{2}}(\omega_{2})$ we have

[TABLE] 2. (2)

There exists a positive constant $C$ such that for all $f\in L^{p_{1}}(\sigma_{1}),~{}g\in L^{p_{2}}(\sigma_{2}),$ we have

[TABLE]

Moreover, we denote the smallest constants $C$ in (1.12) and (1.13) by $\|\mathcal{M}\|$ and $\|\mathcal{M}\|^{\prime},$ respectively. Then it follows that

[TABLE]

Remark 1.14*.*

Using a dyadic discretization technique, Chen and Damián [3] investigated Theorems 1.6, 1.8 and 1.11 on $\mathbb{R}^{n}.$ In addition, Cao and Xue [1] and Sehba [28] gave the similar theorems for bilinear fractional maximal function on $\mathbb{R}^{n},$ respectively.

To prove Theorems 1.6, 1.8 and 1.11, the key ingredient is the bilinear version of Carleson embedding theorem associated with the collection of principal sets developed in Section 4. Note that Hytönen and Pérez gave the dyadic Carleson embedding theorem [15, Theorem 4.5] (see [24] for more information), and Chen and Damián obtained its multilinear analogue [3, Lemma 3] on $\mathbb{R}^{n}.$ In order to provide some two-weight norm estimates for multilinear fractional maximal function, Sehba [28] extensively discussed the more general Carleson embedding theorem. Tanaka and Terasawa [30, Section 3] introduced a refinement of the Carleson embedding theorem on a filtered measure space. In the present paper, our Carleson embedding theorem associated with the collection of principal sets is very different from [30, Theorem 3.1]; see Theorem 4.1 in Section 4.

Remark 1.15*.*

As treated on filtered probability spaces [4] and on Euclidean spaces [1, 3, 28], we do not know if the reverse Hölder condition $RH_{\vec{p}}$ in the theorems above is essential.

The article is organized as follows. In Section 2, we state some preliminaries. We construct bilinear versions of principal sets and Carleson embedding theorem in Section 3 and Section 4, respectively. In Section 5, we provide the proofs of the above theorems.

The letter $C$ will be used for constants that may change from one occurrence to another.

2. Preliminaries

This section consists of the preliminaries for this paper.

2.1. bilinear maximal operator on filtered measure spaces

In this subsection we introduce the bilinear maximal operator on filtered measure spaces, which are standard [30]. Let a triplet $(\Omega,\mathcal{F},\mu)$ be a measure space. Denote by $\mathcal{F}^{0}$ the collection of sets in $\mathcal{F}$ with finite measure. The measure space $(\Omega,\mathcal{F},\mu)$ is called $\sigma$-finite if there exist sets $E_{i}\in\mathcal{F}^{0}$ such that $\Omega=\bigcup\limits_{i=0}^{\infty}E_{i}$. In this paper all measure spaces are assumed to be $\sigma$-finite. Let $\mathcal{A}\subset\mathcal{F}^{0}$ be an arbitrary subset of $\mathcal{F}^{0}$. An $\mathcal{F}$-measurable function $f:\Omega\rightarrow\mathbb{R}$ is called $\mathcal{A}$-integrable if it is integrable on all sets of $\mathcal{A}$, i.e., $\chi_{E}f\in L^{1}(\mathcal{F},\mu)$ for all $E\in\mathcal{A}$. Denote the collection of all such functions by $L^{1}_{\mathcal{A}}(\mathcal{F},\mu).$

If $\mathcal{G}\subset\mathcal{F}$ is another $\sigma$-algebra, it is called a sub-$\sigma$-algebra of $\mathcal{F}$. A function $g\in L^{1}_{\mathcal{G}^{0}}(\mathcal{G},\mu)$ is called the conditional expectation of $f\in L^{1}_{\mathcal{G}^{0}}(\mathcal{F},\mu)$ with respect to $\mathcal{G}$ if there holds

[TABLE]

The conditional expectation of $f$ with respect to $\mathcal{G}$ will be denoted by $\mathbb{E}(f|\mathcal{G})$, which exists uniquely in $L^{1}_{\mathcal{G}^{0}}(\mathcal{G},\mu)$ due to $\sigma$-finiteness of $(\Omega,\mathcal{G},\mu).$

A family of sub-$\sigma$-algebras $(\mathcal{F}_{i})_{i\in\mathbb{Z}}$ is called a filtration of $\mathcal{F}$ if $\mathcal{F}_{i}\subset\mathcal{F}_{j}\subset\mathcal{F}$ whenever $i,j\in\mathbb{Z}$ and $i<j.$ We call a quadruplet $(\Omega,\mathcal{F},\mu;(\mathcal{F}_{i})_{i\in\mathbb{Z}})$ a $\sigma$-finite filtered measure space. As remarked in Section 1, it contains a filtered probability space with a filtration indexed by $\mathbb{N},$ a Euclidean space with a dyadic filtration and doubling metric space with dyadic lattice.

We write

[TABLE]

Notice that

[TABLE]

whenever $i<j.$ For a function $f\in\mathcal{L}$ we will denote $\mathbb{E}(f|\mathcal{F}_{i})$ by $\mathbb{E}_{i}(f).$ By the tower rule of conditional expectations, a family of functions $\mathbb{E}_{i}(f)\in L^{1}_{\mathcal{F}^{0}_{i}}(\mathcal{F},\mu)$ becomes a martingale.

By a weight we mean a nonnegative function which belongs to $\mathcal{L}$ and, by a convention, we will denote the set of all weights by $\mathcal{L}^{+}.$

Let $(\Omega,\mathcal{F},\mu;(\mathcal{F}_{i})_{i\in\mathbb{Z}})$ be a $\sigma$-finite filtered measure space. Then a function $\tau:~{}\Omega\rightarrow\{-\infty\}\cup\mathbb{Z}\cup\{+\infty\}$ is called a stopping time if for any $i\in\mathbb{Z},$ we have $\{\tau=i\}\in\mathcal{F}_{i}.$ The family of all stopping times is denoted by $\mathcal{T}.$ Fixing $i\in\mathbb{Z},$ we denote $\mathcal{T}_{i}=\{\tau\in\mathcal{T}:~{}\tau\geq i\}.$

Suppose that functions $f\in\mathcal{L}$ and $g\in\mathcal{L},$ the maximal operator and bilinear maximal operator are defined by

[TABLE]

respectively. Fix $i\in\mathbb{Z},$ we define the tailed maximal operator and tailed bilinear maximal operator by

[TABLE]

respectively.

Let $B\in\mathcal{F},$ $w\in\mathcal{L}^{+},$ we always denote $\int_{\Omega}\chi_{B}d\mu$ and $\int_{\Omega}\chi_{B}\omega d\mu$ by $|B|$ and $|B|_{\omega},$ respectively.

2.2. bilinear weights

In this subsection we define several kinds of bilinear weights.

Definition 2.1**.**

Let $\omega_{1},~{}\omega_{2}$ be weights and $1<p_{1},~{}p_{2}<\infty.$ Suppose that $\frac{1}{p}=\frac{1}{p_{1}}+\frac{1}{p_{2}}.$ Denote that $\overrightarrow{p}=(p_{1},p_{2})$ and $\sigma_{s}=\omega_{s}^{-\frac{1}{p_{s}-1}}\in\mathcal{L}^{+},~{}s=1,~{}2.$ We say that the couple of weights $(\omega_{1},~{}\omega_{2})$ satisfies the reverse Hölder’s condition $RH_{\overrightarrow{p}},$ if there exists a positive constant $C$ such that for all $i\in\mathbb{Z}$ and $\tau\in\mathcal{T}_{i},$ we have

[TABLE]

We denote by $[\omega_{1},\omega_{2}]_{RH_{\overrightarrow{p}}}$ the smallest constant $C$ in (2.2).

Remark 2.3*.*

In the literature there exist many reverse Hölder’s inequalities of the type

[TABLE]

where $\frac{1}{p}+\frac{1}{q}=1,$ $C$ is a constant and the functions $f$ and $g$ are subjected to suitable restrictions. The suitable restrictions can be found in [23, 31]. In our paper, we find that the reverse Hölder’s condition is useful for bilinear weighted theory.

Definition 2.4**.**

Let $v,~{}\omega_{1}\hbox{ and }\omega_{2}$ be weights and $1<p_{1},~{}p_{2}<\infty.$ Suppose that $\frac{1}{p}=\frac{1}{p_{1}}+\frac{1}{p_{2}}.$ Denote that $\overrightarrow{p}=(p_{1},p_{2})$ and $\sigma_{s}=\omega_{s}^{-\frac{1}{p_{s}-1}}\in\mathcal{L}^{+},~{}s=1,~{}2.$ We say that the triple of weights $(v,~{}\omega_{1},~{}\omega_{2})$ satisfies the condition $A_{\overrightarrow{p}},$ if there exists a positive constant $C$ such that

[TABLE]

where $\frac{1}{p_{s}}+\frac{1}{p^{\prime}_{s}}=1,~{}s=1,~{}2.$ We denote by $[v,\omega_{1},\omega_{2}]_{A_{\overrightarrow{p}}}$ the smallest constant $C$ in (2.5).

Definition 2.6**.**

Let $v,~{}\omega_{1}\hbox{ and }\omega_{2}$ be weights and $1<p_{1},~{}p_{2}<\infty.$ Suppose that $\frac{1}{p}=\frac{1}{p_{1}}+\frac{1}{p_{2}}.$ Denote that $\overrightarrow{p}=(p_{1},p_{2})$ and $\sigma_{s}=\omega_{s}^{-\frac{1}{p_{s}-1}}\in\mathcal{L}^{+},~{}s=1,~{}2.$ We say that the triple of weights $(v,~{}\omega_{1},~{}\omega_{2})$ satisfies the condition $S_{\overrightarrow{p}},$ if

[TABLE]

where $\frac{1}{p_{s}}+\frac{1}{p^{\prime}_{s}}=1,~{}s=1,~{}2.$

Definition 2.8**.**

Let $v,~{}\omega_{1}\hbox{ and }\omega_{2}$ be weights and $1<p_{1},~{}p_{2}<\infty.$ Suppose that $\frac{1}{p}=\frac{1}{p_{1}}+\frac{1}{p_{2}}.$ Denote that $\overrightarrow{p}=(p_{1},p_{2})$ and $\sigma_{s}=\omega_{s}^{-\frac{1}{p_{s}-1}}\in\mathcal{L}^{+},~{}s=1,~{}2.$ We say that the couple of weights $(\omega_{1},~{}\omega_{2})$ satisfies the condition $B_{\overrightarrow{p}},$ if there exists a positive constant $C$ such that for all $i\in\mathbb{Z}$ we have

[TABLE]

We denote by $[v,\omega_{1},\omega_{2}]_{B_{\overrightarrow{p}}}$ the smallest constant $C$ in (2.9).

Definition 2.10**.**

Let $\omega_{1}\hbox{ and }\omega_{2}$ be weights and $1<p_{1},~{}p_{2}<\infty.$ Suppose that $\frac{1}{p}=\frac{1}{p_{1}}+\frac{1}{p_{2}}.$ Denote that $\overrightarrow{p}=(p_{1},p_{2})$ and $\sigma_{s}=\omega_{s}^{-\frac{1}{p_{s}-1}}\in\mathcal{L}^{+},~{}s=1,~{}2.$ We say that the couple of weights $(\omega_{1},~{}\omega_{2})$ satisfies the condition $W^{\infty}_{\overrightarrow{p}},$ if there exists a positive constant $C$ such that for all $i\in\mathbb{Z}$ and $\tau\in\mathcal{T}_{i}$ we have

[TABLE]

We denote by $[\omega_{1},\omega_{2}]_{W^{\infty}_{\overrightarrow{p}}}$ the smallest constant $C$ in (2.11).

Remark 2.12*.*

If $p_{1}=p_{2}$ and $\omega_{1}=\omega_{2}$ in the above definitions, we obtain the linear ones.

3. The construction of principal sets

Let $i\in\mathbb{Z},$ $h_{1}\in\mathcal{L}^{+}$ and $h_{2}\in\mathcal{L}^{+}.$ Fixing $k\in\mathbb{Z},$ we define a stopping time

[TABLE]

For $\Omega_{0}\in\mathcal{F}^{0}_{i},$ we denote that

[TABLE]

and assume $\mu(P_{0})>0.$ It follows that $P_{0}\in\mathcal{F}^{0}_{i}.$ We write $\mathcal{K}_{1}(P_{0}):=i$ and $\mathcal{K}_{2}(P_{0}):=k.$ We let $\mathcal{P}_{1}:=\{P_{0}\}$ which we call the first generation of principal sets. To get the second generation of principal sets we define a stopping time

[TABLE]

where $P^{c}_{0}=\Omega\setminus P_{0}.$ We say that a set $P\subset P_{0}$ is a principal set with respect to $P_{0}$ if it satisfies $\mu(P)>0$ and there exists $j>i$ and $l>k+1$ such that

[TABLE]

Noticing that such $j$ and $l$ are unique, we write $\mathcal{K}_{1}(P):=j$ and $\mathcal{K}_{2}(P):=l.$ We let $\mathcal{P}(P_{0})$ be the set of all principal sets with respect to $P_{0}$ and let $\mathcal{P}_{2}:=\mathcal{P}(P_{0})$ which we call the second generalization of principal sets.

We now need to verify that

[TABLE]

where

[TABLE]

Indeed, we have

[TABLE]

It follows from the Hölder’s inequality for sum that

[TABLE]

Applying the Hölder’s inequality for conditional expectations, we have

[TABLE]

This clearly implies

[TABLE]

For any $P^{\prime}_{0}\in(P_{0}\cap\mathcal{F}^{0}_{i}),$ there exists a set $\Omega^{\prime\prime}_{0}\in\mathcal{F}^{0}_{i}$ such that

[TABLE]

Taking $\Omega^{\prime}_{0}=\Omega_{0}\cap\Omega^{\prime\prime}_{0},$ we have $P^{\prime}_{0}=\{4^{k-1}<\mathbb{E}(h_{1}|\mathcal{F}_{i})\mathbb{E}(h_{2}|\mathcal{F}_{i})\leq 4^{k}\}\cap\Omega^{\prime}_{0}.$ Using $\Omega^{\prime}_{0}$ instead of $\Omega_{0}$ in (3.1), we deduce that

[TABLE]

Moreover, we obtain that

[TABLE]

Since $P^{\prime}_{0}$ is arbitrary, we have $\chi_{P_{0}}\leq 2\mathbb{E}_{i}(\chi_{E(P_{0})})\chi_{P_{0}}.$

The next generalizations are defined inductively,

[TABLE]

and we define the collection of principal sets $\mathcal{P}$ by

[TABLE]

It is easy to see that the collection of principal sets $\mathcal{P}$ satisfied the following properties:

• (P. 1)

The set $E(P)$ where $P\in\mathcal{P},$ are disjoint and $P_{0}=\bigcup\limits_{P\in\mathcal{P}}E(P);$

• (P. 2)

$P\in\mathcal{F}_{{\mathcal{K}}_{1}(P)};$

• (P. 3)

$\chi_{P}\leq 2\mathbb{E}(\chi_{E(P)}|\mathcal{F}_{{\mathcal{K}}_{1}(P)})\chi_{P}$;

• (P. 4)

$4^{{\mathcal{K}}_{2}(P)-1}<\mathbb{E}(h_{1}|\mathcal{F}_{{\mathcal{K}}_{1}(P)})\mathbb{E}(h_{2}|\mathcal{F}_{{\mathcal{K}}_{1}(P)})\leq 4^{{\mathcal{K}}_{2}(P)}$ on $P;$

• (P. 5)

$\sup\limits_{j\geq i}\mathbb{E}_{j}(h_{1}\chi_{P})\mathbb{E}_{j}(h_{2}\chi_{P})\leq 4^{{\mathcal{K}}_{2}(P)+1}$ on $E(P).$

We call (P.(P. 3)) the conditional sparsity of principal sets. Then we use the principal sets to represent the tailed bilinear maximal operator and obtain the following Lemma $\ref{repre}.$

Lemma 3.2**.**

Let $i\in\mathbb{Z},$ $h_{1}\in\mathcal{L}^{+}$ and $h_{2}\in\mathcal{L}^{+}.$ Fixing $k\in\mathbb{Z}$ and $\Omega_{0}\in\mathcal{F}^{0}_{i},$ we denote

[TABLE]

If $\mu(P_{0})>0,$ then

[TABLE]

4. Carleson embedding theorem associated with the collection of principal sets

For $\omega_{1}\in\mathcal{L}^{+}$ and $\omega_{2}\in\mathcal{L}^{+},$ we set $\sigma_{1}:=\omega_{1}^{-\frac{1}{p_{1}-1}}\in\mathcal{L}^{+}$ and $\sigma_{2}=:\omega_{2}^{-\frac{1}{p_{2}-1}}\in\mathcal{L}^{+}.$ Suppose that $f^{p_{1}}_{1}\omega_{1}\in L_{1}^{+}$ and $f^{p_{2}}_{2}\omega_{2}\in L_{1}^{+}.$ It follows from Hölder’s inequality that $f_{1}\in\mathcal{L}^{+}$ and $f_{2}\in\mathcal{L}^{+}.$ Let $h_{1}=f_{1}$ and $h_{2}=f_{2}.$ Fixing $k\in\mathbb{Z},$ $i\in\mathbb{Z}$ and $\Omega_{0}\in\mathcal{F}^{0}_{i}$ such that $\mu(\{4^{k-1}<\mathbb{E}(h_{1}|\mathcal{F}_{i})\mathbb{E}(h_{2}|\mathcal{F}_{i})\leq 4^{k}\}\cap\Omega_{0})>0,$ we apply the construction of principal sets to give the following Carleson embedding theorem.

Theorem 4.1**.**

For $P\in\mathcal{P}$ and $l\in\mathbb{Z},$ let

[TABLE]

We denote $\mathcal{Q}=:\bigcup\limits_{P\in\mathcal{P}}\bigcup\limits_{l\in\mathbb{Z}}A_{P}^{l}.$ If the nonnegative numbers $a_{Q}$ and non-negative function $\sigma_{1}^{\frac{p}{p_{1}}}\sigma_{2}^{\frac{p}{p_{2}}}$ satisfy

[TABLE]

where $A$ is an absolute constant, then

[TABLE]

where $\mathbb{E}^{\sigma_{s}}(\cdot|\mathcal{F}_{\mathcal{K}_{1}(P)})$ is the conditional expectation with respect to $\mathcal{F}_{\mathcal{K}_{1}(P)},$ $\sigma_{s}d\mu$ in place of $d\mu,$ $s=1,~{}2.$

Proof.

We view the sum

[TABLE]

as an integral on a measure space $(\mathcal{Q},~{}2^{\mathcal{Q}},~{}\nu)$ built over $\mathcal{Q},$ assigning to each $Q\in\mathcal{Q}$ the measure $a_{Q}.$ Thus

[TABLE]

where \mathcal{D}_{\lambda}=\big{\{}A_{P}^{l}\in\mathcal{Q}:~{}\mathop{\hbox{essinf}}\limits_{A_{P}^{l}}\big{(}\mathbb{E}^{\sigma_{1}}(h_{1}\sigma^{-1}_{1}|\mathcal{F}_{\mathcal{K}_{1}(P)})\mathbb{E}^{\sigma_{2}}(h_{2}\sigma^{-1}_{2}|\mathcal{F}_{\mathcal{K}_{1}(P)})\big{)}>\lambda\big{\}}. Let

[TABLE]

Then $\tau\in\mathcal{T}_{i}$ and $\bigcup\limits_{Q\in\mathcal{D}_{\lambda}}Q\subset\{{{}^{*}\mathcal{M}}_{i}^{\sigma_{1},\sigma_{2}}(h_{1}\sigma^{-1}_{1}\chi_{P_{0}},h_{2}\sigma^{-1}_{2}\chi_{P_{0}})>\lambda\}=\{\tau<+\infty\},$ where

[TABLE]

Thus

[TABLE]

which implies that

[TABLE]

where

[TABLE]

It follows from Fubini’s theorem that

[TABLE]

Applying Hölder’s inequality and Doob’s maximal inequality, we obtain that

[TABLE]

where

[TABLE]

Combining (4.4), (4.5) and the inequalities above, we conclude this proof. ∎

Remark 4.6*.*

Using $A_{P}^{l}:=E(P)\cap\{2^{l}<\mathbb{E}(\sigma_{1}|\mathcal{F}_{\mathcal{K}_{1}(P)})\mathbb{E}(\sigma_{2}|\mathcal{F}_{\mathcal{K}_{1}(P)})\leq 2^{l+1}\}$ instead of $\eqref{con1},$ we still have Theorem 4.3.

5. Main results and their proofs

5.1. Bilinear Version of one-weight Inequalities

Proof of Theorem 1.3..

(1) Let $i\in\mathbb{Z}$ be arbitrarily chosen and fixed. For $k\in\mathbb{Z}$ and $\Omega_{0}\in\mathcal{F}^{0}_{i},$ we denote

[TABLE]

We claim that

[TABLE]

where $q=\min\{p_{1},p_{2}\}.$ To see this, denote $h_{1}=f_{1}\sigma_{1}\chi_{P_{0}}$ and $h_{2}=f_{2}\sigma_{2}\chi_{P_{0}}.$ For the above $i,$ $P_{0},$ $h_{1}$ and $h_{2},$ we apply the construction of principal sets. It follows from Lemma 3.2 that

[TABLE]

Without loss of generality assume that $1<p_{1}\leq p_{2}<\infty.$ We now estimate $\int_{E(P)}vd\mu$ as follows:

[TABLE]

It follows from the definition of $A_{\overrightarrow{P}}$ and the conditional sparsity of principal sets that

[TABLE]

Applying Hölder’s inequality for the conditional expectation, we have

[TABLE]

It follows from $p^{\prime}_{1}\geq p^{\prime}_{2}$ that $\frac{1}{2p^{\prime}_{s}}2p(p^{\prime}_{1}-1)-\frac{p}{p_{s}}=\frac{pp^{\prime}_{1}}{p^{\prime}_{s}}-p\geq 0,~{}s=1,~{}2.$ Then

[TABLE]

Thus

[TABLE]

Noting that $E(P)\subset P$ and $4^{{\mathcal{K}}_{2}(P)-1}<\mathbb{E}(h_{1}|\mathcal{F}_{{\mathcal{K}}_{1}(P)})\mathbb{E}(h_{2}|\mathcal{F}_{{\mathcal{K}}_{1}(P)})$ on $P,$ we obtain that

[TABLE]

where the last equality uses a standard fact that

[TABLE]

Using Hölder’s inequality, we get

[TABLE]

It follows from (5.2) that

[TABLE]

Applying the Hölder’s inequality for sum and Doob’s maximal inequality, we obtain that

[TABLE]

Hence, the estimation (5.1) is proved. Consequently,

[TABLE]

It follows from the Hölder’s inequality for sum that

[TABLE]

Since the measure space $(\Omega,\mathcal{F},\mu)$ is $\sigma$-finite, we have

[TABLE]

Using the monotone convergence theorem, we obtain that

[TABLE]

(2) Fix $i\in\mathbb{Z}.$ For $B\in\mathcal{F}^{0}_{i},$ set $f_{1}=\chi_{B}$ and $f_{2}=\chi_{B}.$ Then

[TABLE]

It follows from the assumption that

[TABLE]

Since $(\omega_{1},\omega_{2})\in RH_{\overrightarrow{p}},$ we have

[TABLE]

Thus

[TABLE]

where we have used Hölder’s inequality for conditional expectations. Then we obtain

[TABLE]

∎

Corollary 5.3**.**

Let $\omega_{1},~{}\omega_{2}$ be weights and $1<p_{1},~{}p_{2}<\infty.$ Suppose that $\frac{1}{p}=\frac{1}{p_{1}}+\frac{1}{p_{2}}$ and $v=\omega_{1}^{\frac{p}{p_{1}}}\omega_{2}^{\frac{p}{p_{2}}}.$

1. (1)

If $(v,\omega_{1},\omega_{2})\in A_{\overrightarrow{p}}$, then there exists a positive constant $C$ such that for all $f_{1}\in L^{p_{1}}(\omega_{1}),~{}f_{2}\in L^{p_{2}}(\omega_{2})$ we have

[TABLE]

We denote the smallest constant $C$ in (5.4) by $\|\mathcal{M}\|.$ Then it follows that

[TABLE]

where $q=\min\{p_{1},p_{2}\}.$ 2. (2)

Let $(\omega_{1},\omega_{2})\in RH_{\overrightarrow{p}}.$ If there exists a positive constant $C$ such that for all $f_{1}\in L^{p_{1}}(\omega_{1}),~{}f_{2}\in L^{p_{2}}(\omega_{2})$ we have

[TABLE]

then $(v,~{}\omega_{1},~{}\omega_{2})\in A_{\overrightarrow{p}}.$ We denote the smallest constant $C$ in (5.5) by $\|\mathcal{M}\|.$ Then it follows that $[v,\omega_{1},\omega_{2}]_{A_{\overrightarrow{P}}}\leq\|\mathcal{M}\|^{p}[\omega_{1},\omega_{2}]_{RH_{\overrightarrow{p}}}.$

5.2. Bilinear Version of Two-weight Inequalities

Proof of Theorem 1.6. .

To prove $\eqref{Spa}\Rightarrow\eqref{Spb}.$ Fix $i\in\mathbb{Z}.$ For $\tau\in\mathcal{T}_{i},$ set $f_{1}=\chi_{\{\tau<+\infty\}}$ and $f_{2}=\chi_{\{\tau<+\infty\}}.$ It follows from (1.7) that

[TABLE]

Thus $[v,\overrightarrow{\omega}]_{S_{\overrightarrow{p}}}\leq\|\mathcal{M}\|.$

To prove $\eqref{Spb}\Rightarrow\eqref{Spa},$ as we show in Theorem 1.3, we have

[TABLE]

This also implies that

[TABLE]

where $A^{l}_{P}:=E(P)\cap\{2^{l}<\mathbb{E}(\sigma_{1}|\mathcal{F}_{\mathcal{K}_{1}(P)})\mathbb{E}(\sigma_{2}|\mathcal{F}_{\mathcal{K}_{1}(P)})\leq 2^{l+1}\}.$ By the properties of principal sets, we have

[TABLE]

It follows that

[TABLE]

For simplicity, we denote

[TABLE]

Then

[TABLE]

Now we claim that

[TABLE]

To see this, we apply the Carleson embedding theorem to these $a_{A^{l}_{P}}.$ By Theorem 4.1, it suffices to prove

[TABLE]

For $\tau\in\mathcal{T}_{i},$ we have

[TABLE]

Therefore, (5.8) is proved and (5.7) immediately follows. Employing an argument similar to the one in the proof of Theorem 1.3, we have

[TABLE]

∎

Corollary 5.9**.**

Let $v,\omega_{1},\omega_{2}$ be weights and $1<p_{1},p_{2}<\infty.$ Suppose that $1/p=1/p_{1}+1/p_{2}$ and $(\omega_{1},\omega_{2})\in RH_{\overrightarrow{p}},$ then the following statements are equivalent$:$

1. $\rm(1)$

There exists a positive constant $C$ such that

[TABLE] 2. $\rm(2)$

The triple of weights $(v,~{}\omega_{1},~{}\omega_{2})$ satisfies the condition $S_{\overrightarrow{p}}.$

Moreover, we denote the smallest constant $C$ in (5.10) by $\|\mathcal{M}\|.$ Then it follows that

[TABLE]

Corollary 5.11**.**

Let $\omega_{1},~{}\omega_{2}$ be weights and $1<p_{1},~{}p_{2}<\infty.$ Suppose that $\frac{1}{p}=\frac{1}{p_{1}}+\frac{1}{p_{2}}$ and $v=\omega_{1}^{\frac{p}{p_{1}}}\omega_{2}^{\frac{p}{p_{2}}}.$ If $(v,\omega_{1},\omega_{2})\in A_{\overrightarrow{p}}$, then $(v,\omega_{1},\omega_{2})\in S_{\overrightarrow{p}}$ and $[v,\overrightarrow{\omega}]_{S_{\overrightarrow{p}}}\leq 16\cdot 4^{(q^{\prime}-1)}p^{\prime}_{1}p^{\prime}_{2}[v,\omega_{1},\omega_{2}]^{\frac{q^{\prime}}{p}}_{A_{\overrightarrow{P}}}.$

Proof of Theorem 1.8..

It is clear that $\eqref{thm B 1}\Leftrightarrow\eqref{thm B 2},$ so $\|\mathcal{M}\|=\|\mathcal{M}\|^{\prime}.$ To prove $\eqref{thm B 2},$ noting that (5.2) and $E(P)\subset P$, we have

[TABLE]

It follows that

[TABLE]

where $A^{l}_{P}=P\cap\{2^{l}<\mathbb{E}(\sigma_{1}|\mathcal{F}_{\mathcal{K}_{1}(P)})\mathbb{E}(\sigma_{2}|\mathcal{F}_{\mathcal{K}_{1}(P)})\leq 2^{l+1}\}.$ Following the arguments used in the proof of Theorem 1.6, we denote

[TABLE]

Then

[TABLE]

Applying the Carleson embedding theorem to these $a_{A^{l}_{P}},$ we claim that

[TABLE]

In fact, for $\tau\in\mathcal{T}_{i},$ noting that $A_{P}^{l}\subset P,$ we have

[TABLE]

It follows from the conditional sparsity of principal sets that ${\rm I}$ is controlled by

[TABLE]

which is equal to

[TABLE]

where $r$ is an arbitrary real number and bigger than $1.$ Using Jensen’s inequality, we have

[TABLE]

Since for all $P\in\mathcal{P}(P)$ and $l\in\mathbb{Z},$ $\mathbb{E}(P)\cap A_{p}^{l}$ are disjoint sets, it follows that

[TABLE]

Letting $r\rightarrow+\infty,$ we deduce that

[TABLE]

Therefore, the estimation (5.13) is proved. It follows from Theorem 4.1 and (5.12) that

[TABLE]

Then by similar arguments as Theorem 1.3, we get

[TABLE]

This completes the proof. ∎

Proof of Theorem 1.11..

This proof is similar to the proof of Theorem 1.8. For $A_{P}^{l}$ and $a_{A_{P}^{l}}$ defined in the proof of Theorem 1.8, it suffices to check the Carleson embedding condition,

[TABLE]

Indeed, for $\tau\in\mathcal{T}_{i},$ it follows from the definition of $A_{\overrightarrow{p}}$ that

[TABLE]

It follows from the conditional sparsity of principal sets that ${\rm II}$ is controlled by

[TABLE]

which is smaller than

[TABLE]

It follows from the definition of $W^{\infty}_{\overrightarrow{p}}$ that

[TABLE]

Therefore, by (5.12) and Theorem 4.1, we obtain that

[TABLE]

Then by similar arguments as Theorem 1.3, we obtain

[TABLE]

This finishes the proof. ∎