Bloom Type Inequality: The Off-diagonal Case

TL;DR

This paper proves a Bloom type inequality for off-diagonal fractional integral operators acting on product spaces, establishing bounds in weighted mixed-norm spaces and introducing a representation formula for fractional integrals.

Contribution

It introduces a new Bloom type inequality for off-diagonal fractional integrals on product spaces, with a novel representation formula for these operators.

Findings

Established a representation formula for fractional integrals.

Proved a Bloom type inequality for off-diagonal fractional integrals.

Derived bounds in weighted mixed-norm spaces.

Abstract

In this paper, we establish a representation formula for fractional integrals. As a consequence, for two fractional integral operators $I_{\lambda_1}$ and $I_{\lambda_2}$, we prove a Bloom type inequality \begin{align*} \mbox{\hbox to 8em{}}& \hskip -8em \left\|\big[I_{\lambda_1}^1,\big[b,I_{\lambda_2}^2\big]\big] \right\|_{L^{p_2}(L^{p_1})(\mu_2^{p_2}\times\mu_1^{p_1})\rightarrow L^{q_2}(L^{q_1})(\sigma_2^{q_2}\times\sigma_1^{q_1})} % \\ %& \lesssim_{\substack{[\mu_1]_{A_{p_1,q_1}(\mathbb R^n)},[\mu_2]_{A_{p_2,q_2}(\mathbb R^m)} \\ [\sigma_1]_{A_{p_1,q_1}(\mathbb R^n)},[\sigma_2]_{A_{p_2,q_2}(\mathbb R^m)}}} \|b\|_{\BMO_{\pro}(\nu)}, \end{align*} where the indices satisfy $1<p_1<q_1<\infty$, $1<p_2<q_2<\infty$, $1/q_1+1/p_1'=\lambda_1/n$ and $1/q_2+1/p_2'=\lambda_2/m$, the weights $\mu_1,\sigma_1 \in A_{p_1,q_1}(\mathbb R^n)$, $\mu_2,\sigma_2 \in A_{p_2,q_2}(\mathbb R^m)$ and $\nu:=\mu_1\sigma_1^{-1}\otimes \mu_2\sigma_2^{-1}$, $I_{\lambda_1}^1$ stands for $I_{\lambda_1}$ acting on the first variable and $I_{\lambda_2}^2$ stands for $I_{\lambda_2}$ acting on the second variable, $\BMO_{\rm{prod}}(\nu)$ is a weighted product $\BMO$ space and $L^{p_2}(L^{p_1})(\mu_2^{p_2}\times\mu_1^{p_1})$ and $ L^{q_2}(L^{q_1})(\sigma_2^{q_2}\times\sigma_1^{q_1}) $ are mixed-norm spaces.

Equations

$$\displaystyle\hskip-80.00012pt\left\|\big{[}I_{\lambda_{1}}^{1},\big{[}b,I_{\lambda_{2}}^{2}\big{]}\big{]}\right\|_{L^{p_{2}}(L^{p_{1}})(\mu_{2}^{p_{2}}\times\mu_{1}^{p_{1}})\rightarrow L^{q_{2}}(L^{q_{1}})(\sigma_{2}^{q_{2}}\times\sigma_{1}^{q_{1}})}\lesssim_{\begin{subarray}{c}[\mu_{1}]_{A_{p_{1},q_{1}}(\mathbb{R}^{n})},[\mu_{2}]_{A_{p_{2},q_{2}}(\mathbb{R}^{m})}\\ [\sigma_{1}]_{A_{p_{1},q_{1}}(\mathbb{R}^{n})},[\sigma_{2}]_{A_{p_{2},q_{2}}(\mathbb{R}^{m})}\end{subarray}}\|b\|_{\operatorname{BMO}_{\operatorname{prod}}(\nu)},$$

$$I_{\lambda }f(x):=\int \frac{f(y)}{|x-y{|}^{\lambda }}dy$$

$$I_{\lambda_{1}}^{1}f(x_{1},x_{2}):=I_{\lambda_{1}}\big{(}f(\cdot,x_{2})\big{)}(x_{1}),\quad I_{\lambda_{2}}^{2}f(x_{1},x_{2}):=I_{\lambda_{2}}\big{(}f(x_{1},\cdot)\big{)}(x_{2}).$$

$$\mathscr{D}^{0}:=\bigcup_{k\in\mathbb{Z}}\mathscr{D}^{0}_{k},\quad\mathscr{D}^{0}_{k}:=\Big{\{}2^{-k}\big{(}[0,1)^{n}+m\big{)}:m\in\mathbb{Z}^{n}\Big{\}}.$$

$$I\dotplus \omega :=I+\sum _{j:2^{-j}<\ell (I)}{2}^{-j}{\omega }_j.$$

$$\mathscr{D}^{\omega}:=\big{\{}I\dotplus w:I\in\mathscr{D}^{0}\big{\}}=\bigcup_{k\in\mathbb{Z}}\mathscr{D}^{\omega}_{k}.$$

$$S_{\lambda ,\omega }^{i,j}f:=\sum _{K\in {\mathcal{D}}^{\omega }}{A}_{\lambda ,K}^{i,j}f,$$

$$A_{\lambda ,K}^{i,j}f=\sum _{\begin{array}{c}I,J\in {\mathcal{D}}^{\omega }\\ I^{(i)}=K,J^{(j)}=K\end{array}}{a}_{\lambda ,I,J,K}⟨f,h_I⟩h_J$$

$$⟨g,I_{\lambda }f⟩=C\cdot {\mathbb{E}}_{\omega }\sum _{i.j=0}^{\infty }{2}^{-\max (i,j)/2}⟨g,S_{\lambda ,\omega }^{i,j}f⟩,$$

$$[w]_{A_{p}(\mathbb{R}^{n})}:=\sup_{Q}\big{\langle}w\big{\rangle}_{Q}\big{\langle}w^{1-p^{\prime}}\big{\rangle}_{Q}^{p-1}<\infty.$$

$$[w]_{A_{p,q}(\mathbb{R}^{n})}:=\sup_{Q}\big{\langle}w^{q}\big{\rangle}_{Q}\big{\langle}w^{-p^{\prime}}\big{\rangle}_{Q}^{q/p^{\prime}}<\infty,$$

$$\|b\|_{\operatorname{BMO}_{\operatorname{prod}}^{\mathscr{D}^{\omega},\mathscr{D}^{\beta}}(w)}:=\sup_{\Omega}\Big{(}\frac{1}{w(\Omega)}\sum_{\begin{subarray}{c}I\in\mathscr{D}^{\omega},J\in\mathscr{D}^{\beta}\\ I\times J\subset\Omega\end{subarray}}|\langle b,h_{I}\otimes h_{J}\rangle|^{2}\langle w\rangle_{I\times J}^{-1}\Big{)}^{1/2}<\infty,$$

$$\Vert b{\Vert }_{{\mathrm{BMO}}_{\mathrm{prod}}(w)}:=\underset{{\mathcal{D}}^{\omega },{\mathcal{D}}^{\beta }}{\sup }\Vert b{\Vert }_{{\mathrm{BMO}}_{\mathrm{prod}}^{{\mathcal{D}}^{\omega },{\mathcal{D}}^{\beta }}(w)}.$$

$$\displaystyle\hskip-80.00012pt\big{\|}\big{[}I_{\lambda_{1}}^{1},\big{[}b,I_{\lambda_{2}}^{2}\big{]}\big{]}\big{\|}_{L^{p_{2}}(L^{p_{1}})(\mu_{2}^{p_{2}}\times\mu_{1}^{p_{1}})\rightarrow L^{q_{2}}(L^{q_{1}})(\sigma_{2}^{q_{2}}\times\sigma_{1}^{q_{1}})}$$

$${\lesssim }_{\begin{array}{c}[{\mu }_1{]}_{A_{p_1,q_1}({\mathbb{R}}^n)},[{\mu }_2{]}_{A_{p_2,q_2}({\mathbb{R}}^m)}\\ [{\sigma }_1{]}_{A_{p_1,q_1}({\mathbb{R}}^n)},[{\sigma }_2{]}_{A_{p_2,q_2}({\mathbb{R}}^m)}\end{array}}\Vert b{\Vert }_{{\mathrm{BMO}}_{\mathrm{prod}}(\nu )}.$$

$$\Vert f{\Vert }_{L^{p_2}(L^{p_1})({\mu }_2\times {\mu }_1)}:=$$

$$\displaystyle=\big{\|}\|f(\cdot,x_{2})\|_{L^{p_{1}}(\mu_{1})}\big{\|}_{L^{p_{2}}(\mu_{2})}.$$

$$\operatorname{dist}(I,\partial J)\leq\ell(J)\cdot\Big{(}\frac{\ell(I)}{\ell(J)}\Big{)}^{\gamma},$$

$${\mathbb{P}}_{\mathrm{good}}:=\mathbb{P}(\{\omega :I\dotplus \omega \text{is good}\})>0.$$

$$h_I^{\eta }(x)=h_{I_1\times \cdots \times I_n}^{({\eta }_1,\cdots ,{\eta }_n)}(x_1,\cdots ,x_n):={\otimes }_{i=1}^n{h}_{I_i}^{{\eta }_i}(x_i),$$

$${\Delta }_{I}f:=\sum _{\eta \in \{0,1{\}}^n\setminus \{0\}}⟨f,h_I^{\eta }⟩h_I^{\eta }.$$

$$f=\sum _{I\in {\mathcal{D}}^{\omega }}{\Delta }_{I}f=\sum _{I\in {\mathcal{D}}^{\omega }}\sum _{\eta \in \{0,1{\}}^n\setminus \{0\}}⟨f,h_I^{\eta }⟩h_I^{\eta }.$$

$$f=\sum _{I\in {\mathcal{D}}^{\omega }}⟨f,h_I⟩h_I.$$

$${\Delta }_{K,i}f=\sum _{\begin{array}{c}I\in {\mathcal{D}}^{\omega }\\ I^{(i)}=K\end{array}}{\Delta }_{I}f,K\in {\mathcal{D}}^{\omega },i\ge 0,$$

$$\big{|}S^{i,j}_{\lambda,\omega}f(x)\big{|}\leq\sum_{K\in\mathscr{D}^{\omega}}\frac{1}{|K|^{\lambda/n}}\int_{K}|f(y)|dy\cdot 1_{K}(x).$$

$$\sum _{K\in {\mathcal{D}}^{\omega }}\frac{1}{|K{|}^{\lambda /n}}{\int }_K|f(y)|dy\cdot 1_K(x)\lesssim I_{\lambda }|f|(x).$$

$$\big{|}S^{i,j}_{\lambda,\omega}f(x)\big{|}\lesssim I_{\lambda}|f|(x).$$

$$\Vert I_{\lambda }f{\Vert }_{L^q(w^q)}{\lesssim }_{[w{]}_{A_{p,q}({\mathbb{R}}^n)}}\Vert f{\Vert }_{L^p(w^p)}$$

$$\Vert S_{\lambda ,\omega }^{i,j}f{\Vert }_{L^q(w^q)}{\lesssim }_{[w{]}_{A_{p,q}({\mathbb{R}}^n)}}\Vert f{\Vert }_{L^p(w^p)}.$$

$$S_{\mathscr{D}^{\omega}}f:=\Big{(}\sum_{I\in\mathscr{D}^{\omega}}|\Delta_{I}f|^{2}\Big{)}^{1/2}.$$

$$\Vert f{\Vert }_{L^p(w)}{\sim }_{[w{]}_{A_p({\mathbb{R}}^n)}}\Vert S_{{\mathcal{D}}^{\omega }}{\Vert }_{L^p(w)}.$$

Full text

[TABLE]

where the indices satisfy $1<p_{1}<q_{1}<\infty$, $1<p_{2}<q_{2}<\infty$, $1/q_{1}+1/p_{1}^{\prime}=\lambda_{1}/n$ and $1/q_{2}+1/p_{2}^{\prime}=\lambda_{2}/m$, the weights $\mu_{1},\sigma_{1}\in A_{p_{1},q_{1}}(\mathbb{R}^{n})$, $\mu_{2},\sigma_{2}\in A_{p_{2},q_{2}}(\mathbb{R}^{m})$ and $\nu:=\mu_{1}\sigma_{1}^{-1}\otimes\mu_{2}\sigma_{2}^{-1}$, $I_{\lambda_{1}}^{1}$ stands for $I_{\lambda_{1}}$ acting on the first variable and $I_{\lambda_{2}}^{2}$ stands for $I_{\lambda_{2}}$ acting on the second variable, $\operatorname{BMO}_{\rm{prod}}(\nu)$ is a weighted product $\operatorname{BMO}$ space and $L^{p_{2}}(L^{p_{1}})(\mu_{2}^{p_{2}}\times\mu_{1}^{p_{1}})$ and $L^{q_{2}}(L^{q_{1}})(\sigma_{2}^{q_{2}}\times\sigma_{1}^{q_{1}})$ are mixed-norm spaces.

Key words.   Fractional integrals, representation formula, mixed-norm spaces, Bloom type inequality, product BMO.

Mathematics Subject Classification: Primary 42B20

1 Introduction and The Main Results

Let $\mu$ and $\sigma$ be two weights in $\mathbb{R}^{n+m}$. A two-weight problem asks for a characterization of the boundedness of an operator $T:L^{p}(\mu)\rightarrow L^{p}(\sigma)$. In a Bloom type variant of this problem, $\mu$ and $\sigma$ are Muckenhoupt $A_{p}$ weights and a function $b$, which is taken from some appropriate weighted $\operatorname{BMO}$ space $\operatorname{BMO}(\nu)$ for some Bloom type weight $\nu:=\mu^{1/p}\sigma^{-1/p}$, is invoked. This leads us naturally to the commutator setting.

In the one-parameter case, Bloom [4] obtained such a two-weight estimate for $[b,H]$, where $H$ is the Hilbert transform. Holmes, Lacey and Wick [17] extended Bloom’s result to general Calderón-Zygmund operators. The iterated case is by Holmes and Wick [19] (see also Hytönen [26]). An improved iterated case is by Lerner, Ombrosi and Rivera-Ríos [29].

In the bi-parameter case, there are two types of commutators: one is involved with little BMO spaces, and the other is associated with the more complicated product BMO spaces. For the first type, Holmes, Petermichl and Wick [18] initiated the study of $[b,T]$ with $T$ being any bi-parameter Calderón-Zygmund operator and $b$ being some weighted little BMO function. Then Li, Martikainen and Vuorinen [30] extended Holmes-Petermichl-Wick’s result to higher order commutators, and provided a simpler proof for the first order case. In [5] Cao and Gu studied the related question for fractional integrals. For the second type, Li, Martikainen and Vuorinen [31] first studied the question for $[T_{n},[b,T_{m}]]$ (known as the Ferguson-Lacey type commutator [12]), where $T_{n}$ and $T_{m}$ are one-parameter Calderón-Zygmund operators in $\mathbb{R}^{n}$ and $\mathbb{R}^{m}$, respectively. Recently, Airta [1] generalized this result to the multi-parameter case.

In this paper, we focus on the Bloom type inequality for the Ferguson-Lacey type commutator involved with fractional integrals. Specifically, we prove a Bloom type inequality for $[I_{\lambda_{1}}^{1},[b,I_{\lambda_{2}}^{2}]]$, where

[TABLE]

is the fractional integral operator, and for a measurable function $f$ defined on $\mathbb{R}^{n+m}$, $I_{\lambda_{1}}^{1}f$ stands for $I_{\lambda_{1}}$ acting on the first variable of $f$ and $I_{\lambda_{2}}^{2}f$ stands for $I_{\lambda_{2}}$ acting on the second variable of $f$, i.e.,

[TABLE]

Our results extend similar results for singular integral operators. The main difference is that mixed-norm spaces [2, 3] are invoked when we study the off-diagonal case of Bloom type inequalities for fractional integral operators. Note that there are many results on mixed-norm spaces, e.g., see [6, 13, 16, 20, 35, 36, 37] for some recent advances on mixed-norm Lebesgue spaces, [8] for Besov spaces, [14, 27] for Triebel-Lizorkin spaces and [7, 21, 22] for Hardy spaces.

To get a Bloom type inequality for singular integral operators, a basic tool is the representation theorem (see Hytönen [25] for the one-parameter case, and Martikainen [32], Ou [34] for the bi-parameter and multi-parameter cases, respectively). To deal with fractional integral operators, we need to establish a representation formula. Before stating our main results, we introduce some notations.

Let $\mathscr{D}^{0}$ be the standard dyadic system in $\mathbb{R}^{n}$, i.e.,

[TABLE]

Given some $\omega=(\omega_{j})_{j\in\mathbb{Z}}\in(\{0,1\}^{n})^{\mathbb{Z}}$ and a cube $I$, denote

[TABLE]

We define the random dyadic system $\mathscr{D}^{\omega}$ by

[TABLE]

Definition 1.1** (Fractional dyadic shifts).**

Given two integers $i,j\geq 0$, the fractional dyadic shift associated with $(i,j)$ is defined by

[TABLE]

where each $A^{i,j}_{\lambda,K}$ has the form

[TABLE]

with the coefficients $a_{\lambda,I,J,K}$ satisfying $|a_{\lambda,I,J,K}|\leq|I|^{1/2}|J|^{1/2}/|K|^{\lambda/n}$.

We are now ready to state our first main result, a representation formula of fractional integral operators.

Theorem 1.2**.**

Let $0<\lambda<n$ be a constant and $I_{\lambda}$ be a fractional integral operator defined by (1.1). Then we have

[TABLE]

where $f,g\in L^{\infty}_{c}(\mathbb{R}^{n})$ and the constant $C$ depends only on $\lambda$ and the dimension $n$.

Recall that for $1<p<\infty$, the Muckenhoupt $A_{p}$ class consists of all locally integrable positive functions $w(x)$ for which

[TABLE]

And for $1<p<q<\infty$, $A_{p,q}(\mathbb{R}^{n})$ consists of all weight functions $w(x)$ for which

[TABLE]

where $\langle w\rangle_{Q}:=\frac{1}{Q}\int_{Q}w$ and the supremum is taken over all cubes $Q\subset\mathbb{R}^{n}$ with sides parallel to the axes.

Now we introduce the weighted product BMO space. Let $\mathscr{D}^{\omega}$ be a dyadic system in $\mathbb{R}^{n}$, where $\omega=(\omega_{j})_{j\in\mathbb{Z}}\in(\{0,1\}^{n})^{\mathbb{Z}}$. And let $\mathscr{D}^{\beta}$ be a dyadic system in $\mathbb{R}^{m}$, where $\beta=(\beta_{j})_{j\in\mathbb{Z}}\in(\{0,1\}^{m})^{\mathbb{Z}}$. Given $w\in A_{2}(\mathbb{R}^{n+m})$, we say that a locally integrable function $b:\mathbb{R}^{n+m}\rightarrow\mathbb{C}$ belongs to the weighted product BMO space $\operatorname{BMO}_{\operatorname{prod}}^{\mathscr{D}^{\omega},\mathscr{D}^{\beta}}(w)$ if

[TABLE]

where $w(\Omega):=\int_{\Omega}w(x_{1},x_{2})dx_{1}dx_{2}$ and the supremum is taken over all subsets $\Omega\subset\mathbb{R}^{n+m}$ such that $|\Omega|<\infty$ and for every $x\in\Omega$, there exist some $I\in\mathscr{D}^{\omega}$ and $J\in\mathscr{D}^{\beta}$ satisfying $x\in I\times J\subset\Omega$. The non-dyadic product BMO norm can be defined by taking the supremum over all dyadic systems $\mathscr{D}^{\omega}$ and $\mathscr{D}^{\beta}$, i.e.,

[TABLE]

Now we state the second main result, a Bloom type inequality for fractional integral operators.

Theorem 1.3**.**

Let $I_{\lambda_{1}}$ and $I_{\lambda_{2}}$ be two fractional integral operators acting on functions defined on $\mathbb{R}^{n}$ and $\mathbb{R}^{m}$, respectively. Suppose that $1<p_{1}<q_{1}<\infty$, $1<p_{2}<q_{2}<\infty$, $1/q_{1}+1/p_{1}^{\prime}=\lambda_{1}/n$ and $1/q_{2}+1/p_{2}^{\prime}=\lambda_{2}/m$. Let $\mu_{1},\sigma_{1}\in A_{p_{1},q_{1}}(\mathbb{R}^{n})$ and $\mu_{2},\sigma_{2}\in A_{p_{2},q_{2}}(\mathbb{R}^{m})$. Set $\nu:=\mu_{1}\sigma_{1}^{-1}\otimes\mu_{2}\sigma_{2}^{-1}$. Then we have the quantitative estimate

[TABLE]

Here the weight is of tensor product type. We do not know how to relax this restriction. This is mainly due to the natural appearance of the mixed-norm spaces, in which even the boundedness of the strong maximal function with non-tensor product type weights is still open.

The paper is organized as follows. In Section 2, we collect some preliminary results. And in Section 3, we give a proof of Theorem 1.2. In Section 4, we present some results on mixed-norm spaces and then give a proof of the Bloom type inequality for fractional integral operators.

2 Notations and Preliminary Results

We denote $A\lesssim B$ if $A\leq C\cdot B$ for some constant $C$ that can depend on the dimension of the underlying spaces, on integration exponents, and on various other constants appearing in the assumptions.

We denote the product space $\mathbb{R}^{n+m}=\mathbb{R}^{n}\times\mathbb{R}^{m}$. For any $x\in\mathbb{R}^{n+m}$, we write $x=(x_{1},x_{2})$ with $x_{1}\in\mathbb{R}^{n}$ and $x_{2}\in\mathbb{R}^{m}$.

Let $\mu_{1}$ and $\mu_{2}$ be measures on $\mathbb{R}^{n}$ and $\mathbb{R}^{m}$, respectively. For $p_{1},p_{2}$ with $1\leq p_{1},p_{2}<\infty$ and a measurable function $f:\mathbb{R}^{n+m}\rightarrow\mathbb{C}$, we define the mixed-norm $\|f\|_{L^{p_{2}}(L^{p_{1}})(\mu_{2}\times\mu_{1})}$ by

[TABLE]

2.1 Random dyadic systems

Let $\mathscr{D}^{\omega}$ be defined by (1.2). For parameters $r\in\mathbb{Z}_{+}$ and $\gamma\in(0,1/2)$, we call a cube $I\in\mathscr{D}^{\omega}$ bad if there exists some $J\in\mathscr{D}^{\omega}$ such that $\ell(J)\geq 2^{r}\ell(I)$ and

[TABLE]

where $\ell(I)$ stands for the side length. In the treatment of a fractional integral operator $I_{\lambda}$ with exponent $\lambda$, the choice $\gamma=\frac{1}{2(\lambda+1)}$ is useful.

We say that a cube $I\in\mathscr{D}^{\omega}$ is good if it is not bad. It is well-known that $\mathbb{P}(\{\omega:I\dotplus\omega\ \ \text{is good}\})$ is independent of the choice of $I\in\mathscr{D}^{0}$ and if $r$ is sufficiently large, then

[TABLE]

Given two cubes $I,J\in\mathscr{D}^{\omega}$, we denote by $I\vee J$ the smallest cube in $\mathscr{D}^{w}$ that contains both $I$ and $J$. If it does not exist, we denote $I\vee J=\emptyset$.

2.2 Haar functions and martingale differences

In one dimension, for an interval $I$, the Haar functions are defined by $h^{0}_{I}=|I|^{-1/2}1_{I}$ and $h^{1}_{I}=|I|^{-1/2}(1_{I_{\ell}}-1_{I_{r}})$, here $I_{\ell}$ and $I_{r}$ are the left and right halves of the interval $I$, respectively. In higher dimensions, for a cube $I=I_{1}\times\cdots\times I_{n}\subset\mathbb{R}^{n}$, we define the Haar functions $h^{\eta}_{I}$ as

[TABLE]

where $\eta\in\{0,1\}^{n}\setminus\{0\}$.

For a locally integrable function $f:\mathbb{R}^{n}\rightarrow\mathbb{C}$, the martingale difference $\Delta_{I}$ associated with a dyadic cube $I\in\mathscr{D}^{\omega}$ is defined by

[TABLE]

Set \big{\langle}f\big{\rangle}_{I}:=\frac{1}{|I|}\int_{I}f. We also write E_{I}f:=\big{\langle}f\big{\rangle}_{I}1_{I}. We have the usual martingale decomposition

[TABLE]

Since the $\eta$’s do not play any major role, in the sequel we just simply write

[TABLE]

A martingale block is denoted by

[TABLE]

where $I^{(i)}$ denotes the unique dyadic cube $P\in\mathscr{D}^{\omega}$ such that $I\subset P$ and $\ell(P)=2^{i}\ell(I)$.

By the definition of fractional dyadic shifts, it is is easy to see that

[TABLE]

Cruz-Uribe and Moen [10] proved that

[TABLE]

Thus we have

[TABLE]

2.3 Weights

For any $1<p<q<\infty$ and $w\in A_{p,q}(\mathbb{R}^{n})$, a simple calculation gives that $w^{q}\in A_{q}(\mathbb{R}^{n})$, $w^{-p^{\prime}}\in A_{p^{\prime}}(\mathbb{R}^{n})$ and $w^{-q^{\prime}}\in A_{q^{\prime}}(\mathbb{R}^{n})$.

In [33], Muckenhoupt and Wheeden proved the weighted bound for fractional integral operators. Specifically, they showed that if $0<\lambda<n$ and $1/q+1/p^{\prime}=\lambda/n$, then

[TABLE]

if and only if $w(x)\in A_{p,q}(\mathbb{R}^{n})$. Combining with the inequality (2.4), we have

[TABLE]

2.4 Maximal functions and martingale difference square functions

Let $\mathscr{D}^{\omega}$ be a dyadic system in $\mathbb{R}^{n}$, where $\omega=(\omega_{j})_{j\in\mathbb{Z}}\in(\{0,1\}^{n})^{\mathbb{Z}}$. For a measurable function defined on $\mathbb{R}^{n}$, we define the martingale difference square function $S_{\mathscr{D}^{\omega}}f$ by

[TABLE]

Wilson [38] proved the following weighted estimates for martingale difference square functions.

Proposition 2.1** ([38, Theorem 2.1]).**

For any $1<p<\infty$, $w\in A_{p}(\mathbb{R}^{n})$ and a measurable function $f$, we have the following norm equivalence,

[TABLE]

And Cruz-Uribe, Martell and Pérez [9] gave the following sharp estimate.

Proposition 2.2** ([9, Theorem 1.8]).**

Given $p,1<p<\infty$, then for any measurable function $f$ and $w\in A_{p}(\mathbb{R}^{n})$,

[TABLE]

Further, the exponent $\max(\frac{1}{2},\frac{1}{p-1})$ is the best possible.

Next we introduce some notations on the product space $\mathbb{R}^{n+m}$. Given a measurable function $f$ defined on $\mathbb{R}^{n+m}$, we define the strong maximal function $M_{S}$ by

[TABLE]

where $R=Q\times Q^{\prime}$ and $Q\subset\mathbb{R}^{n}$ and $Q^{\prime}\subset\mathbb{R}^{m}$ are cubes with sides parallel to the axes.

Let $\mathscr{D}^{\omega}$ and $\mathscr{D}^{\beta}$ be dyadic systems in $\mathbb{R}^{n}$ and $\mathbb{R}^{m}$, respectively, where $\omega=(\omega_{j})_{j\in\mathbb{Z}}\in(\{0,1\}^{n})^{\mathbb{Z}}$ and $\beta=(\beta_{j})_{j\in\mathbb{Z}}\in(\{0,1\}^{m})^{\mathbb{Z}}$. We define the dyadic maximal functions by

[TABLE]

[TABLE]

[TABLE]

It is obvious that $M^{1}_{\mathscr{D}^{\omega}}f\leq M_{S}f$, $M^{2}_{\mathscr{D}^{\beta}}f\leq M_{S}f$ and $M_{\mathscr{D}^{\omega},\mathscr{D}^{\beta}}f\leq M_{S}f$, thanks to the Lebesgue differentiation theorem.

Let $0<\lambda_{1}<n$ and $0<\lambda_{2}<m$. We define the partial fractional maximal functions by

[TABLE]

[TABLE]

By (2.3), it is easy to see that

[TABLE]

Similarly,

[TABLE]

And we define the martingale difference square functions on the product space by

[TABLE]

where $\Delta_{I}^{1}f(x)=\Delta_{I}(f(\cdot,x_{2}))(x_{1})$, $\Delta_{J}^{2}f(x)=\Delta_{J}(f(x_{1},\cdot))(x_{2})$ and $\Delta_{I\times J}f(x)=\Delta_{J}^{2}(\Delta_{I}^{1}f)(x)$.

Similarly we set $E_{I}^{1}:=E_{I}(f(\cdot,x_{2}))(x_{1})$ and $E_{J}^{2}:=E_{J}(f(x_{1},\cdot))(x_{2})$.

Notice that $\Delta_{I}^{1}f=h_{I}\otimes\langle f,h_{I}\rangle_{1},\Delta_{J}^{2}=\langle f,h_{J}\rangle_{2}\otimes h_{J}$ and $\Delta_{I\times J}f=\langle f,h_{I}\otimes h_{J}\rangle h_{I}\otimes h_{J}$, where $\langle f,h_{I}\rangle_{1}$ is defined by

[TABLE]

and $\langle f,h_{J}\rangle_{2}$ is defined similarly.

The Martingale blocks are defined in the natural way,

[TABLE]

3 Proof of Theorem 1.2

In this section, we give a proof of Theorem 1.2. First, we introduce a result by Hytönen [24].

Lemma 3.1** ([24, Lemma 3.7]).**

Let $I,J\in\mathscr{D}^{\omega}$ be such that $I$ is good, $I\cap J=\emptyset$ and $\ell(I)\leq\ell(J)$. Then there exists some $K\supset I\cup J$ which satisfies

[TABLE]

To prove the main result, we also need the following lemma.

Lemma 3.2**.**

Let $I$ and $J$ be cubes in $\mathbb{R}^{n}$. If $I$ and $J$ share the same center, then $\langle 1_{J},I_{\lambda}h_{I}\rangle=0$.

Proof.

For $\eta\in\{0,1\}^{n}\setminus\{0\}$, let the Haar functions $h^{\eta}_{I}$ be defined by (2.1). Observe that there exists some $i$ such that $\eta_{i}=1$. Without loss of generality, we assume that $i=1$. We denote the center of the interval $I_{1}$ by $c$. Moreover, we further divide cubes $I$ and $J$ into the following parts,

[TABLE]

[TABLE]

Then

[TABLE]

where

[TABLE]

and

[TABLE]

So we have $A_{1}+A_{2}=0$. Similarly we can show that $A_{3}+A_{4}=0$. Therefore,

[TABLE]

This completes the proof. ∎

We are now ready to give a proof of Theorem 1.2.

Proof of Theorem 1.2.

Using (2.2) to expand $f$ and $g$, we have

[TABLE]

Since the goodness of a cube is independent of its position, we have

[TABLE]

First, we estimate the first term I. We have

[TABLE]

Now we estimate the four terms separately.

The term $\sigma_{\operatorname{out}}$. By Lemma 3.1, we know that $K=I\vee J$ exists and

[TABLE]

We write

[TABLE]

Denote the center of the cube $I$ by $y_{I}$. We have

[TABLE]

So we obtain

[TABLE]

The term $\sigma_{\operatorname{near}}$. Again by Lemma 3.1, with the condition $I\cap J=\emptyset$ and $\operatorname{dist}(I,J)\leq\ell(J)(\ell(I)/\ell(J))^{1/(2\lambda+1)}$, we know that $K=I\vee J$ exists and $\ell(K)\leq 2^{r}\ell(I)$. So $\sigma_{\operatorname{near}}$ can be written as

[TABLE]

Since $I_{\lambda}$ is bounded from $L^{p}$ to $L^{q}$, we have

[TABLE]

Observe that $2^{-r}\ell(K)\leq\ell(I)\leq\ell(J)\leq\ell(K)$ and $1/{p^{\prime}}+1/{q}={\lambda}/{n}$. We have

[TABLE]

So

[TABLE]

The term $\sigma_{\operatorname{shallow\ in}}$. It is easy to see that the term $\sigma_{\operatorname{shallow\ in}}$ can be written as

[TABLE]

With similar arguments as those for calculating the term $\sigma_{\operatorname{near}}$, we have

[TABLE]

Hence

[TABLE]

The term $\sigma_{\operatorname{deep\ in}}$. This term can be written as

[TABLE]

The cube $J$ has $2^{n}$ children in $\mathscr{D}^{w}$. $I\subsetneq J$ implies that $I$ must be contained in one of the children $J_{1}$. Since $\ell(I)\leq 2^{-r-1}\ell(J)$, we have $\ell(I)\leq 2^{-r}\ell(J_{1})$. Hence $I$ is good and

[TABLE]

We denote the center of the cube $I$ by $y_{I}$. Let $J^{\prime}$ be the cube with side length $2\ell(J)$ and the same center $y_{I}$. Then we have $J^{\prime}\supset J$. By Lemma 3.2,

[TABLE]

Now we see from the definition of Haar functions that $h_{J}$ is a constant on $J_{1}$. More precisely, $h_{J}\equiv\langle h_{J}\rangle_{J_{1}}$. It follows that

[TABLE]

So we obtain

[TABLE]

Hence

[TABLE]

Similarly we can calculate the term II and get the conclusion as desired. This completes the proof. ∎

4 Proof of Theorem 1.3

In this section, we give a proof of Theorem 1.3. First, we introduce some results on mixed-norm spaces. We begin with a result on the weighted estimates of the strong maximal function.

Proposition 4.1** ([28, Theorem 1]).**

Let $1<p,q<\infty$ and weights $w_{1}\in A_{p}(\mathbb{R}^{n})$ and $w_{2}\in A_{q}(\mathbb{R}^{m})$. Then

[TABLE]

Moreover, we have the Fefferman-Stein inequality

[TABLE]

It is easy to see that the above result remains true whenever $M_{S}$ is replaced by $M_{\mathscr{D}^{\omega}}^{1}$, $M_{\mathscr{D}^{\beta}}^{2}$ or $M_{\mathscr{D}^{\omega},\mathscr{D}^{\beta}}$.

Next we introduce the extrapolation theorem [11].

Proposition 4.2**.**

Assume that for a pair of nonnegative functions $(f,g)$, for some $p_{0}\in[1,\infty)$ and for all $w\in A_{p_{0}}$, we have

[TABLE]

where $N$ is an increasing function and the constant $C$ does not depend on $w$. Then for all $1<p<\infty$ and all $w\in A_{p}$, we have

[TABLE]

where K(w)\leq C_{1}N\big{(}C_{2}[w]_{A_{p}}^{\max(1,\frac{p_{0}-1}{p-1})}\big{)} for $w\in A_{p}$.

With the above extrapolation theorem, we can prove the following estimates of martingale difference square functions.

Lemma 4.3**.**

Under the same hypotheses as in Proposition 4.1, we have

[TABLE]

Proof.

First we prove (4.1). For any $f\in L^{q}(L^{p})(w_{2}\times w_{1})$, we have

[TABLE]

By Proposition 2.1, we have

[TABLE]

Hence

[TABLE]

Next we prove (4.2). Denote $\|f(\cdot,x_{2})\|_{L^{p}(w_{1})}$ and $\|S_{\mathscr{D}^{\beta}}^{2}f(\cdot,x_{2})\|_{L^{p}(w_{1})}$ by $F(x_{2})$ and $G(x_{2})$, respectively. We have

[TABLE]

Notice that for any weight $w\in A_{p}(\mathbb{R}^{m})$, we have

[TABLE]

By Proposition 2.2,

[TABLE]

This give us

[TABLE]

Note that $x^{\max(\frac{1}{2},\frac{1}{p-1})}$ is increasing on $(0,\infty)$. By Proposition 4.2, for any $w\in A_{q}(\mathbb{R}^{m})$, we have

[TABLE]

That is,

[TABLE]

Finally we prove (4.3). By the Kahane-Khintchine inequality [23],

[TABLE]

where $(\varepsilon_{I})_{I\in\mathscr{D}^{\omega}}$ is a Rademacher sequence defined on some probability space \big{(}\Omega,\mathbb{P}\big{)}.

Since $p,q>1$, by Minkowski’s integral inequality,

[TABLE]

It follows from (4.2) that

[TABLE]

On the other hand, we see from Proposition 2.1 that

[TABLE]

Combining the above inequalities, we get

[TABLE]

This completes the proof. ∎

The $\ell^{2}$-valued extension of linear operators on classical $L^{p}$ spaces is well known, e.g., see Grafakos [15, Chapter 4]. Here we give a similar result on mixed-norm spaces. Since we do not find a reference, we present a short proof.

Lemma 4.4**.**

Let $\mu_{1},\sigma_{1}$ be measures on $\mathbb{R}^{n}$ and $\mu_{2},\sigma_{2}$ be measures on $\mathbb{R}^{m}$. Given $0<p_{1},p_{2},q_{1},q_{2}<\infty$ and suppose that $T$ is a bounded linear operator from $L^{p_{2}}(L^{p_{1}})(\mu_{2}\times\mu_{1})$ to $L^{q_{2}}(L^{q_{1}})(\sigma_{2}\times\sigma_{1})$. Then $T$ has an $\ell^{2}$-valued extension, that is, for all complex-valued functions $f_{j}$ in $L^{p_{2}}(L^{p_{1}})(\mu_{2}\times\mu_{1})$ we have

[TABLE]

Proof.

By the Khintchine inequality,

[TABLE]

where $(\varepsilon_{j})$ is a Rademacher sequence defined on a probability space \big{(}\Omega,\mathbb{P}\big{)}. This gives us

[TABLE]

By the Kahane-Khintchine inequality, we have

[TABLE]

Hence

[TABLE]

Since $T$ is bounded from $L^{p_{2}}(L^{p_{1}})(\mu_{2}\times\mu_{1})$ to $L^{q_{2}}(L^{q_{1}})(\sigma_{2}\times\sigma_{1})$, we see from the above arguments that

[TABLE]

When $q_{2}>p_{2}$, we see from Minkowski’s integral inequality that

[TABLE]

Using the Kahane-Khintchine inequality again, we get

[TABLE]

Hence for $q_{2}>p_{2}$,

[TABLE]

When $q_{2}\leq p_{2}$, since \big{(}\Omega,\mathbb{P}\big{)} is a probability space, by Hölder’s inequality, we get

[TABLE]

Similar arguments as those for the case of $q_{2}>p_{2}$ show that when $q_{2}\leq p_{2}$,

[TABLE]

This completes the proof. ∎

Next we give the weighted estimates of the partial fractional integral operators $I_{\lambda_{1}}^{1}$ and $I_{\lambda_{2}}^{2}$.

Lemma 4.5**.**

Let $I_{\lambda_{1}}$ and $I_{\lambda_{2}}$ be fractional integral operators in $\mathbb{R}^{n}$ and $\mathbb{R}^{m}$, respectively, where $0<\lambda_{1}<n$ and $0<\lambda_{2}<m$. Let $1<p_{1},q_{1},p_{2},q_{2}<\infty$ be constants such that $1/q_{1}+1/p_{1}^{\prime}=\lambda_{1}/n$ and $1/q_{2}+1/p_{2}^{\prime}=\lambda_{2}/m$. Suppose that $\mu_{1}\in A_{p_{1},q_{1}}(\mathbb{R}^{n})$ and $\mu_{2}\in A_{p_{2},q_{2}}(\mathbb{R}^{m})$. Then for any measurable function $f$ defined on $\mathbb{R}^{n+m}$, we have

[TABLE]

Proof.

Observe that

[TABLE]

Taking the $L^{p_{2}}(\mu_{2}^{p_{2}})$ norm and the $L^{q_{2}}(\mu_{2}^{q_{2}})$ norm respectively, we get the first two inequalities.

On the other hand, by Minkowski’s integral inequality, we get

[TABLE]

This proves (4.6). And (4.7) can be proved similarly. ∎

Li, Martikainen and Vuorinen [30, 31] showed that a product $bf$ can be expanded by paraproduct operators. Specifically, let $\mathscr{D}^{\omega}$ and $\mathscr{D}^{\beta}$ be dyadic systems in $\mathbb{R}^{n}$ and $\mathbb{R}^{m}$, respectively, where $\omega=(\omega_{j})_{j\in\mathbb{Z}}\in(\{0,1\}^{n})^{\mathbb{Z}}$ and $\beta=(\beta_{j})_{j\in\mathbb{Z}}\in(\{0,1\}^{m})^{\mathbb{Z}}$. The paraproduct operators are defined by

[TABLE]

[TABLE]

and

[TABLE]

[TABLE]

The ”illegal” biparameter paraproduct is

[TABLE]

We can decompose the multiplication $bf$ in the biparameter sense,

[TABLE]

In the following we give the weighted estimates of $A_{k}(b,\cdot)$ on mixed-norm spaces.

Lemma 4.6**.**

Suppose that $b\in\operatorname{BMO}_{\operatorname{prod}}(\nu)$, where $\nu:=\mu_{1}^{1/p}\sigma_{1}^{-1/p}\otimes\mu_{2}^{1/q}\sigma_{2}^{-1/q}$, $\mu_{1}$, $\sigma_{1}\in A_{p}(\mathbb{R}^{n})$, $\mu_{2}$, $\sigma_{2}\in A_{q}(\mathbb{R}^{m})$ and $1<p,q<\infty$. Then for $k=1,2,3,4$, we have

[TABLE]

The classical weighted $L^{p}$ norm estimates of $A_{k}(b,\cdot)$ were proved by Holmes, Petermichl and Wick in [18]. The main tool they used was the following weighted $H^{1}$-$\operatorname{BMO}_{\operatorname{prod}}$ duality estimate.

Proposition 4.7** ([18, Proposition 4.1]).**

For every $b\in\operatorname{BMO}_{\operatorname{prod}}^{\mathscr{D}^{\omega},\mathscr{D}^{\beta}}(w)$, we have

[TABLE]

By Proposition 4.7, we have the following estimate

[TABLE]

Using the same idea as in [18] we can prove Lemma 4.6. Here we only outline the general strategy and leave the details to interested readers.

Proof of Lemma 4.6.

By the dual property of mixed-norm spaces [3], it suffices to show that for any $f\in L^{q}(L^{p})(\mu_{2}\times\mu_{1})$ and $g\in L^{q^{\prime}}(L^{p^{\prime}})(\sigma_{2}^{\prime}\times\sigma_{1}^{\prime})$ with $\sigma_{1}^{\prime}=\sigma_{1}^{1-p^{\prime}}$ and $\sigma_{2}^{\prime}=\sigma_{2}^{1-q^{\prime}}$,

[TABLE]

First, we write $\langle A_{k}(b,f),g\rangle=\langle b,\phi\rangle$, where $\phi$ depends on $f$ and $g$. By Proposition 4.7, $|\langle A_{k}(b,f),g\rangle|\lesssim\|b\|_{\operatorname{BMO}_{\operatorname{prod}}^{\mathscr{D}^{\omega},\mathscr{D}^{\beta}}(\nu)}\|S_{\mathscr{D}^{\omega},\mathscr{D}^{\beta}}\phi\|_{L^{1}(\nu)}$.

Next we show that $S_{\mathscr{D}^{\omega},\mathscr{D}^{\beta}}\phi\lesssim(\mathcal{O}_{1}f)(\mathcal{O}_{2}g)$, where $\mathcal{O}_{1}$ and $\mathcal{O}_{2}$ are operators that are combinations of maximal operators $M_{S}$, $M_{\mathscr{D}^{\omega}}$, $M_{\mathscr{D}^{\beta}}$ and square functions $S_{\mathscr{D}^{\omega},\mathscr{D}^{\beta}},S_{\mathscr{D}^{\omega}},S_{\mathscr{D}^{\beta}}$. By Proposition 4.1 and Lemma 4.3, we have

[TABLE]

and

[TABLE]

Finally, by Hölder’s inequality, the $L^{1}(\nu)$-norm of $S_{\mathscr{D}^{\omega},\mathscr{D}^{\beta}}\phi$ is controlled by the $L^{q}(L^{p})(\mu_{2}\times\mu_{1})$ norm of $\mathcal{O}_{1}$ and the $L^{q^{\prime}}(L^{p^{\prime}})(\sigma_{2}^{\prime}\times\sigma_{1}^{\prime})$ norm of $\mathcal{O}_{2}$, i.e.,

[TABLE]

Now we get the conclusion as desired. ∎

We are now ready to prove the main result.

Proof of Theorem 1.3.

By the representation formula in Theorem 1.2, it suffices to prove the Bloom type inequality for \big{[}S_{\lambda_{1},\omega}^{i,j,1},\big{[}b,S_{\lambda_{2},\beta}^{s,t,2}\big{]}\big{]}, where $\omega=(\omega_{j})_{j\in\mathbb{Z}}\in(\{0,1\}^{n})^{\mathbb{Z}}$, $\beta=(\beta_{j})_{j\in\mathbb{Z}}\in(\{0,1\}^{m})^{\mathbb{Z}}$, $i,j,s,t\in\mathbb{Z}^{+}\cup\{0\}$ and

[TABLE]

[TABLE]

and the coefficients satisfy

[TABLE]

By (4.8), we have

[TABLE]

where

[TABLE]

and

[TABLE]

We start looking at the sum over $k$. First of all, combining (2.4) and Lemmas 4.5 and 4.6, we conclude that for $k\leq 4$ all the individual terms are bounded.

Next we consider the case of $k\geq 5$. In this case the term S_{\lambda_{1},\omega}^{i,j,1}\big{(}A_{k}\big{(}b,S_{\lambda_{2},\beta}^{s,t,2}f\big{)}\big{)} should be paired with another term. For $k=5,6$, it is paired with -A_{k}\big{(}b,S_{\lambda_{2},\beta}^{s,t,2}S_{\lambda_{1},\omega}^{i,j,1}f\big{)} and for $k=7,8$, it is paired with -S_{\lambda_{1},\omega}^{i,j,1}S_{\lambda_{2},\beta}^{s,t,2}\big{(}A_{k}\big{(}b,f\big{)}\big{)}. We consider only the case of $k=5$. Other cases can be proved similarly.

Since $S_{\lambda_{2},\beta}^{s,t,2}S_{\lambda_{1},\omega}^{i,j,1}=S_{\lambda_{1},\omega}^{i,j,1}S_{\lambda_{2},\beta}^{s,t,2}$, by (2.4) and Lemma 4.5, we have

[TABLE]

By the dual property of mixed-norm spaces, it suffices to estimate \big{|}\big{\langle}S_{\lambda_{1},\omega}^{i,j,1}\big{(}A_{5}\big{(}b,f\big{)}\big{)} -A_{5}\big{(}b,S_{\lambda_{1},\omega}^{i,j,1}f\big{)},g\big{\rangle}\big{|} for $g\in L^{q_{2}^{\prime}}(L^{q_{1}^{\prime}})(\sigma_{2}^{-q_{2}^{\prime}}\times\sigma_{1}^{-q_{1}^{\prime}})$.

Since

[TABLE]

and

[TABLE]

we reduce the problem to estimate

[TABLE]

and

[TABLE]

First, we estimate (4.12). For $1\leq r\leq i$, we have

[TABLE]

where we use (4.10) in the last step. We see from (4.9) that

[TABLE]

where $\nu=\mu_{1}\sigma_{1}^{-1}\otimes\mu_{2}\sigma_{2}^{-1}$. By Hölder’s inequality,

[TABLE]

Since $\mu_{1}\in A_{p_{1},q_{1}}(\mathbb{R}^{n})$ implies $\mu_{1}^{p_{1}}\in A_{p_{1}}(\mathbb{R}^{n})$ and $\mu_{2}\in A_{p_{2},q_{2}}(\mathbb{R}^{m})$ implies $\mu_{2}^{q_{2}}\in A_{q_{2}}(\mathbb{R}^{m})$, by Proposition 4.1 and Lemma 4.3,

[TABLE]

By (2.6), we have

[TABLE]

Since $\sigma_{1}\in A_{p_{1},q_{1}}(\mathbb{R}^{n})$, we have $\sigma_{1}^{-1}\in A_{q_{1}^{\prime},p_{1}^{\prime}}$. It follows from Lemma 4.5 that

[TABLE]

On the other hand, we see from $\sigma_{2}\in A_{p_{2},q_{2}}(\mathbb{R}^{m})$ that $\sigma^{-p_{2}^{\prime}}_{2}\in A_{p_{2}^{\prime}}(\mathbb{R}^{m})$. By Proposition 4.1, we get

[TABLE]

So we conclude that

[TABLE]

Next, we estimate (4.13). Using the same method as that for estimating (4.12), we have

[TABLE]

Also by (2.6) we get

[TABLE]

By Lemma 4.5, $I_{\lambda_{1},\mathscr{D}^{\omega}}^{1}$ is bounded from $L^{q_{2}}(L^{p_{1}})(\mu_{2}^{q_{2}}\times\mu_{1}^{p_{1}})$ to $L^{q_{2}}(L^{q_{1}})(\mu_{2}^{q_{2}}\times\mu_{1}^{q_{1}})$. Note that $I_{\lambda_{1},\mathscr{D}^{\omega}}^{1}$ is linear. It follows from Lemma 4.4 that

[TABLE]

Now we see from Proposition 4.1 and Lemma 4.3 that

[TABLE]

By Proposition 4.1,

[TABLE]

It follows that

[TABLE]

Taking the summation over $r$ and $l$, we obtain

[TABLE]

Combining with (4.11), we get

[TABLE]

It remains to estimate

[TABLE]

where

[TABLE]

Noting that

[TABLE]

with similar arguments we get that

[TABLE]

This completes the proof. ∎