Atomic Decomposition of Weighted Multi-parameter Mixed Hardy Spaces

Abstract

It is well-known that atomic decomposition is an important tool to study the boundedness of some singular integral operators on Hardy spaces. Moreover, to study the boundedness of an operator in the Journé class, Fefferman R. builded a criterion by considering its action on rectangle atoms only. In this paper, we mainly establish atomic decomposition of multi-parameter mixed Hardy space which has been developed recently.

1 Introduction

Multi-parameter harmonic analysis containing multi-parameter function spaces and boundedness of operators have been extensively studied over the past decades. We refer readers to the work in [1, 2, 7,8,9,10, 12, 13, 15, 21,22,23, 25, 27, 29,30,32, 34,35,37, 39,40,47, 49,50,52].

The product Hardy space was first introduced in [25, 38]. Immediately after, Chang and Fefferman R. developed this theory in [4,5,6]. At the same time, Fefferman and Stein studied product convolution singular integral operators which satisfy analogous conditions of double Hilbert transform [18]. In [33], Journé generalized this result to product non-convolution singular integral operators and proved the $L^{\infty }\rightarrow BMO$ boundedness for such operators, which opened the door to prove the product $H^{p}$ boundedness of operators in the Journé’s class. Besides that, authors in [11, 37, 48] studied weighted multi-parameter Hardy spaces. For more results about multi-parameter Hardy spaces, we refer readers to [1,2,3, 11, 12, 20, 21, 25, 26, 28,29,30,31, 35,36,37, 46, 48].

Recently, the theory of multi-parameter mixed Hardy space has been developed in [14]. To be more precise, let $\psi _{0}^{(1)},\psi ^{(1)}\in {{\mathcal {S}}}({\mathbb {R}}^{n_{1}})$ with

$$\begin{aligned} \hbox {supp} \widehat{\psi _{0}^{(1)}}\subseteq \{\xi \in \mathbb R^{n_{1}}: |\xi |\le 2\};\ \widehat{\psi _{0}^{(1)}}(\xi )=1,\ \hbox {if}\ |\xi |\le 1, \end{aligned}$$

(1.1)

and

$$\begin{aligned} \hbox {supp} \widehat{\psi ^{(1)}}\subseteq \{\xi \in \mathbb R^{n_{1}}: \frac{1}{2}\le |\xi |\le 2\}, \end{aligned}$$

(1.2)

and

$$\begin{aligned} |\widehat{\psi _{0}^{(1)}}(\xi )|^{2}+\sum \limits ^{\infty }_{j =1}|\widehat{\psi ^{(1)}}(2^{-j}\xi )|^2=1, \ \hbox { for all } \xi \in {\mathbb {R}}^{n_{1}}. \end{aligned}$$

(1.3)

Let $\psi ^{(2)}\in {{\mathcal {S}}}({\mathbb {R}}^{n_{2}})$ with

$$\begin{aligned} \hbox {supp} \widehat{\psi ^{(2)}}\subseteq \{\xi \in \mathbb R^{n_{2}}: \frac{1}{2}\le |\xi |\le 2\}, \end{aligned}$$

(1.4)

and

$$\begin{aligned} \sum \limits _{j \in \mathbb Z}|\widehat{\psi ^{(2)}}(2^{-j}\xi )|^2=1, \ \hbox { for all } \xi \in {\mathbb {R}}^{n_{2}}\setminus \{0\}. \end{aligned}$$

(1.5)

Then, for $j,k\in {\mathbb {Z}}, j\ge 1$, set that $\psi _j^{(1)}(x)= 2^{jn_{1}} \psi ^{(1)}(2^jx)$, $\psi _k^{(2)}(x)= 2^{kn_{2}} \psi ^{(2)}(2^kx)$ and that $\psi _{j,k}(x,y)=\psi _j^{(1)}(x)\psi _k^{(2)}(y)$, $\psi _{0,k}(x,y)=\psi _0^{(1)}(x)\psi _k^{(2)}(y)$.

Denote that ${{\mathcal {S}}_{0}}({\mathbb {R}}^{n_{1}+n_{2}})=\{f\in {{\mathcal {S}}}({\mathbb {R}}^{n_{1}+n_{2}}): \int _{{\mathbb {R}}^{n_{2}}} f(x_{1},x_{2}) x_{2}^\alpha \textrm{d}x_{2}=0, \ \forall \ |\alpha |\ge 0, \ \forall \ x_{1}\in {\mathbb {R}}^{n_{1}}\}$. For $i=1,2$ and any $j\in {\mathbb {Z}}$, denote that $\Pi _{j}^{n_{i}}=\{I:$ I are dyadic cubes in ${\mathbb {R}}^{n_{i}}$ with the side length $l(I)=2^{-j}$, and the left lower corners of I are $x_{I}=2^{-j}\ell$, $\ell \in {\mathbb {Z}}^{n_{i}}\}$, $\Pi _{j,k}=\Pi _{j}^{n_{1}}\times \Pi _{k}^{n_{2}},$ and that $\Pi =\cup _{j,k\in {\mathbb {Z}}}\Pi _{j,k}.$

The following discrete multi-parameter Calderón’s reproducing formula was obtained in [16]:

Theorem A Suppose that $\psi _{0}^{(1)}$,$\psi ^{(1)}\in {{\mathcal {S}}}({\mathbb {R}}^{n_{1}})$ and $\psi ^{(2)}\in {{\mathcal {S}}}({\mathbb {R}}^{n_{2}})$ satisfy conditions in (1.1)–(1.3) and (1.4)–(1.5), respectively. Then

$$\begin{aligned} \qquad f(x_{1},x_{2}) =\sum \limits _{j\in {\mathbb {N}},k\in {\mathbb {Z}}}\sum \limits _{I\times J\in \Pi _{j,k}}|I||J|(\psi _{j,k}*f)(x_{I},x_{J})\times \psi _{j,k}(x_{1}-x_{I},x_{2}-x_{J}), \end{aligned}$$

(1.6)

where the series converges in $L^{2}({\mathbb {R}}^{n_{1}+n_{2}})$, $\mathcal S_{0} ({\mathbb {R}}^{{n_{1}+n_{2}}})$ and ${\mathcal {S}}'_{0}({\mathbb {R}}^{{n_{1}+n_{2}}})$, the dual space of ${\mathcal {S}}_{0} ({\mathbb {R}}^{{n_{1}+n_{2}}})$.

We recall some definitions of product weights in two parameter setting [24]. For $1<p<\infty ,$ a nonnegative locally integrable function $\omega \in A_p({\mathbb {R}}^{n_{1}}\times {\mathbb {R}}^{n_{2}})$ if there exists a constant $C>0$ such that

$$\begin{aligned} \Big (\frac{1}{|R|}\int _R \omega (x)\textrm{d}x \Big )\Big (\frac{1}{|R|}\int _R \omega (x)^{-1/(p-1)}\textrm{d}x \Big )^{p-1}<C \end{aligned}$$

for any dyadic rectangle R, that is $R\in \Pi$. We say that $\omega \in A_1({\mathbb {R}}^{n_{1}}\times {\mathbb {R}}^{n_{2}})$ if there exists a constant $C>0$ such that

$$\begin{aligned} M_s \omega (x)\le C \omega (x),\ \hbox {a.e.}\ x\in {\mathbb {R}}^{n_{1}+n_{2}}, \end{aligned}$$

where $M_s$ is the strong maximal function defined by

$$\begin{aligned} M_s f(x)= \sup \limits _{x \in R\in \Pi }\frac{1}{|R|}\int _{R} |f(y)| \textrm{d}y. \end{aligned}$$

Finally, define that $\omega \in A_\infty ({\mathbb {R}}^{n_{1}}\times {\mathbb {R}}^{n_{2}})$ by

$$\begin{aligned} A_\infty ({\mathbb {R}}^{n_{1}}\times {\mathbb {R}}^{n_{2}})=\bigcup \limits _{1\le p<\infty } A_p({\mathbb {R}}^{n_{1}}\times {\mathbb {R}}^{n_{2}}). \end{aligned}$$

In this paper, the classical Muckenhoupt’s weights on ${\mathbb {R}}^{n}$ is denoted by $A_p({\mathbb {R}}^{n})$.

Given a weight $\omega$ on ${\mathbb {R}}^{n}$, for $0<r<\infty$, $L_{\omega }^{r}({\mathbb {R}}^{n})$ is defined by

$$\begin{aligned} L_{\omega }^{r}({\mathbb {R}}^{n})=\{f:\int _{{\mathbb {R}}^{n}} |f(x)|^{r}\omega (x)\textrm{d}x<\infty \}. \end{aligned}$$

Based on the discrete Calderón’s identity (1.6), weighted mixed Hardy spaces are introduced in [14].

Definition 1.1

Let $0<p<\infty$ and $\omega \in A_\infty ({\mathbb {R}}^{n_{1}}\times {\mathbb {R}}^{n_{2}})$. Suppose that $\psi _{0}^{(1)}$,$\psi ^{(1)}\in {{\mathcal {S}}}({\mathbb {R}}^{n_{1}})$ and $\psi ^{(2)}\in {{\mathcal {S}}}({\mathbb {R}}^{n_{2}})$ satisfy conditions in (1.1)–(1.3) and (1.4)–(1.5), respectively. The weighted multi-parameter mixed Hardy space $H^{p}_{mix}(\omega ,{\mathbb {R}}^{n_{1}}\times {\mathbb {R}}^{n_{2}})$ is defined to be the set of $f\in {\mathcal {S}}'_0({\mathbb {R}}^{n_{1}+n_{2}})$ such that $\Vert f\Vert _{H^{p}_{mix}(\omega ,{\mathbb {R}}^{n_{1}}\times {\mathbb {R}}^{n_{2}})}=\Vert S(f)(x)\Vert _{L_{\omega }^p({\mathbb {R}}^{n_{1}+n_{2}})}<\infty ,$ where

$$\begin{aligned} S(f)(x)=\Big (\sum _{j\in {\mathbb {N}},k\in {\mathbb {Z}}} \sum \limits _{I\times J\in \Pi _{j}^{{n_{1}}}\times \Pi _{k}^{{n_{2}}}}|\psi _{j,k}*f(x_{I},x_{J})|^2\chi _{I}(x_{1})\chi _{J}(x_{2})\Big )^{\frac{1}{2}}. \end{aligned}$$

(1.7)

It is well-known that atomic decomposition plays an important role in studying the boundedness of singular operators, and it is much more complicated in multi-parameter setting. In the present paper, we consider the atomic decomposition of $H^{p}_{mix}(\omega ,{\mathbb {R}}^{n_{1}}\times {\mathbb {R}}^{n_{2}})$. The atoms of $H^{p}_{mix}(\omega ,{\mathbb {R}}^{n_{1}}\times {\mathbb {R}}^{n_{2}})$ are defined as follows.

Definition 1.2

Let $0<p\le 1$. A function $a(x_{1},x_{2})$ is said to be an atom for $H^{p}_{mix}(\omega ,{\mathbb {R}}^{n_{1}}\times {\mathbb {R}}^{n_{2}})$ if it satisfies the following properties:

• (1) $a(x_{1},x_{2})$ is supported in an open set $\Omega \subseteq {\mathbb {R}}^{n_{1}+n_{2}}$ with finite measure.

• (2) $\Vert a\Vert _{L_{\omega }^{2}({\mathbb {R}}^{n_{1}+n_{2}})}\le \omega (\Omega )^{\frac{1}{2}-\frac{1}{p}}.$

• (3) a can be further decomposed as

  $$\begin{aligned} a=\sum _{R\subset {\mathcal {M}}(\Omega )}a_{R} \end{aligned}$$

  with

  $$\begin{aligned} \sum _{R\subset {\mathcal {M}}(\Omega )}\Vert a_{R}\Vert _{L_{\omega }^{2}({\mathbb {R}}^{n_{1}+n_{2}})}^{2}\lesssim \omega (\Omega )^{1-\frac{2}{p}}, \end{aligned}$$

  (1.8)

where $a_{R}$ are named as rectangle-atoms associated with the dyadic rectangles $R=I\times J$, and supported in $\tau R$ for some positive integer $\tau > 1$ independent of a and $a_{R}$, and ${\mathcal {M}}(\Omega )$ is the set of all maximal dyadic rectangles in $\Omega$. Furthermore, $a_{R}$ has the following vanishing moment conditions:

• (i) in the $x_{2}$ direction,

  $$\begin{aligned} \ \hbox {for}\ \hbox {a.e.}\ x_{1}\in \tau I, \int a_{R}(x_{1},x_{2})x_{2}^{\alpha }\textrm{d}x_{2}=0,0\le |\alpha |\le N^{2}_{p}=\left[ \frac{2n_{2}}{p}-n_{2}\right] . \end{aligned}$$

• (ii) in the $x_{1}$ direction, there exists a positive constant $\varrho$, when $\ell (I)<\varrho$,

  $$\begin{aligned} \ \hbox {for}\ \hbox {a.e.}\ x_{2}\in \tau J, \int a_{R}(x_{1},x_{2})x_{1}^{\beta }\textrm{d}x_{1}=0, 0\le |\beta |\le N^{1}_{p}=\left[ \frac{2n_{1}}{p}-n_{1}\right] . \end{aligned}$$

Theorem 1.1

Let $0<p\le 1$, $\omega \in A_2({\mathbb {R}}^{n_{1}}\times {\mathbb {R}}^{n_{2}})$. Then there exists a constant $c_{p,n_{1},n_{2},\omega }$, such that for all $H^{p}_{mix}(\omega ,{\mathbb {R}}^{n_{1}}\times {\mathbb {R}}^{n_{2}})$ atoms a,

$$\begin{aligned} \Vert a\Vert _{H^{p}_{mix}(\omega ,{\mathbb {R}}^{n_{1}}\times {\mathbb {R}}^{n_{2}})}\le c_{p,n_{1},n_{2},\omega }. \end{aligned}$$

It has been proved in [14] that $L_{\omega }^{2}({\mathbb {R}}^{n_{1}+n_{2}})\cap H^{p}_{mix}(\omega ,{\mathbb {R}}^{n_{1}}\times {\mathbb {R}}^{n_{2}})$ is dense in $H^{p}_{mix}(\omega ,{\mathbb {R}}^{n_{1}}\times {\mathbb {R}}^{n_{2}})$ for all $0<p<\infty$. Once we obtain the atomic decomposition in this dense set, it is easy to generalize to whole space.

Theorem 1.2

Let $0<p\le 1$ and $\omega \in A_2({\mathbb {R}}^{n_{1}}\times {\mathbb {R}}^{n_{2}})$. Then $f\in H^{p}_{mix}(\omega ,{\mathbb {R}}^{n_{1}}\times {\mathbb {R}}^{n_{2}})$ if and only if there exist a sequence $\{a_{k}\}$ of $H^{p}_{mix}(\omega ,{\mathbb {R}}^{n_{1}}\times {\mathbb {R}}^{n_{2}})$ atoms and a sequence $\{\lambda _{k}\}$ of real numbers satisfying $\sum _{k\in {\mathbb {Z}}}|\lambda _{k}|^{p}\le C\Vert f\Vert ^{p}_{H^{p}_{mix}(\omega ,{\mathbb {R}}^{n_{1}}\times {\mathbb {R}}^{n_{2}})}$ such that

$$\begin{aligned} f(x)=\sum _{k\in {\mathbb {Z}}}\lambda _{k}a_{k}(x), \hbox {in}\ H^{p}_{mix}(\omega ,{\mathbb {R}}^{n_{1}}\times {\mathbb {R}}^{n_{2}}), \end{aligned}$$

where the series also converges to f in $L_{\omega }^{2}({\mathbb {R}}^{n_{1}+n_{2}})$.

One will see that the above atomic decomposition theorem is very useful to prove the boundedness of some kinds of singular operators, such as multi-pseudodifferential operators, and those in mixed Journé class. It will be exhibited in our following papers. As a direct application of Theorem  and Theorem 1.2, we have the following results.

Theorem 1.3

Let $0<p\le 1$ and $\omega \in A_2({\mathbb {R}}^{n_{1}}\times {\mathbb {R}}^{n_{2}})$. Assume that T is a linear singular integral operator bounded on $L_{\omega }^2({\mathbb {R}}^{n_{1}+n_{2}})$. Then T is bounded on $H^{p}_{mix}(\omega ,{\mathbb {R}}^{n_{1}}\times {\mathbb {R}}^{n_{2}})$ if and only if

$$\begin{aligned} \sup \{\Vert T(a)\Vert _{H^{p}_{mix}(\omega ,{\mathbb {R}}^{n_{1}}\times {\mathbb {R}}^{n_{2}} )}:a\ \ is\ \ any \ atom\ of\ H^{p}_{mix}(\omega ,{\mathbb {R}}^{n_{1}}\times {\mathbb {R}}^{n_{2}})\}<\infty . \end{aligned}$$

(1.9)

We now describe the strategy of this paper. In Section 2, we mainly prove the uniform boundedness of atoms on $H^{p}_{mix}(\omega ,{\mathbb {R}}^{n_{1}}\times {\mathbb {R}}^{n_{2}})$. Comparison with the unweighted case, the uniform boundedness of atoms on weighted mixed Hardy spaces is much more involved. In Section 3, we establish atomic decomposition on $H^{p}_{mix}(\omega ,{\mathbb {R}}^{n_{1}}\times {\mathbb {R}}^{n_{2}})$. The proof of Theorem 1.3 is also placed in this section.

2 The Uniform Boundedness of Atoms

In this section, we mainly discuss the uniform boundedness of atoms on $H^{p}_{mix}(\omega ,{\mathbb {R}}^{n_{1}}\times {\mathbb {R}}^{n_{2}})$ when $0<p\le 1$. First of all, let’s recall a key theorem discovered by Journé. For this, given any open set $\Omega \subseteq {\mathbb {R}}^{n_{1}+n_{2}}$, denote that ${\mathcal {M}}_{1}(\Omega )$ the collection of all dyadic rectangles $R=I\times J\subseteq \Omega , R\in \Pi ,$ which are maximal in the $x_{1}$ direction. Define ${\mathcal {M}}_{2}(\Omega )$ similarly. It is easy to see that ${\mathcal {M}}(\Omega )\subseteq {\mathcal {M}}_{1}(\Omega )$ and ${\mathcal {M}}(\Omega )\subseteq {\mathcal {M}}_{2}(\Omega )$.

Define that

$$\begin{aligned} {\widetilde{\Omega }}=\left\{ x \in {\mathbb {R}}^{n_{1}+n_{2}}: M^{\omega }_{s}(\chi _{\Omega })(x) > \frac{1}{(10\tau )^{n_{1}+n_{2}}}\right\} , \end{aligned}$$

and similarly for $\widetilde{{\widetilde{\Omega }}}$, $\widetilde{\widetilde{{\widetilde{\Omega }}}}$. Here, $M^{\omega }_{s}(g)(x)$ is a weighted strong Maximal function defined by

$$\begin{aligned} M^{\omega }_{s}(g)(x)=\sup _{x\in R}\frac{1}{\omega (R)}\int _{R}|g(y)|\omega (y)\textrm{d}y. \end{aligned}$$

Obviously, $\Omega \subseteq {\widetilde{\Omega }}\subseteq \widetilde{{\widetilde{\Omega }}}\subseteq \widetilde{\widetilde{{\widetilde{\Omega }}}}$. By the weighted strong maximal theorem (see [17]), one has $\omega (\widetilde{\widetilde{{\widetilde{\Omega }}}})\lesssim \omega (\Omega )$.

For any $R\in {\mathcal {M}}(\Omega )$, let ${\tilde{I}}\subseteq {\mathbb {R}}^{n_{1}}$ be the largest dyadic cube containing I such that ${\tilde{R}}={\tilde{I}}\times J\subseteq {\widetilde{\Omega }}$, and ${\tilde{J}}$ be the largest dyadic cube containing J such that $\tilde{{\tilde{R}}} = {\tilde{I}}\times {\tilde{J}} \subseteq \widetilde{{\widetilde{\Omega }}}$. Define $\gamma =\gamma (R)=\frac{\ell ({\tilde{I}})}{\ell (I)}$ and $\gamma '=\gamma '(R)=\frac{\ell ({\tilde{J}})}{\ell (J)}$. The following weighted version of Journé’s lemma is in [19].

Lemma 2.1

Let $\Omega$ be an open set in ${\mathbb {R}}^{n_{1}+n_{2}}$. If $\omega \in A_\infty ({\mathbb {R}}^{n_{1}}\times {\mathbb {R}}^{n_{2}})$, then for any $\eta >0$

$$\begin{aligned} \sum _{R\in {\mathcal {M}}_{2}(\Omega )} \omega (R)\gamma (R)^{-\eta }\le C_{\eta }\omega (\Omega ). \end{aligned}$$

Proof of Theorem 1.1:

Let a be any $H^{p}_{mix}(\omega ,{\mathbb {R}}^{n_{1}}\times {\mathbb {R}}^{n_{2}})$ atom supported on an open set $\Omega \subseteq {\mathbb {R}}^{n_{1}+n_{2}}$ with $\omega (\Omega )<\infty$.

Firstly, by $L_{\omega }^{2}$ boundedness of operator S and condition 2 in the Definition 1.2, it is easy to have that

$$\begin{aligned} \int _{\widetilde{\widetilde{{\widetilde{\Omega }}}}}S(a)(x)^{p}\omega (x)\textrm{d}x\le & {} \omega (\widetilde{\widetilde{{\widetilde{\Omega }}}})^{1-p/2} \Vert S(a)(\cdot )\Vert ^{p}_{L_{\omega }^{2}({\mathbb {R}}^{n_{1}+n_{2}})}\\\le & {} C\omega (\Omega )^{1-p/2} \Vert a\Vert ^{p}_{L_{\omega }^{2}({\mathbb {R}}^{n_{1}+n_{2}})}\le C. \end{aligned}$$

Then this theorem will be proved if we obtain that

$$\begin{aligned} \int _{\widetilde{\widetilde{{\widetilde{\Omega }}}}^{c}}S(a)(x)^{p}\omega (x)\textrm{d}x<C, \end{aligned}$$

(2.1)

where C is a positive constant independent of a. According to the Definition 1.2, there are some rectangle atoms $a_{R}$ such that $a=\sum _{R\in {\mathcal {M}}(\Omega )}a_{R}.$ Then (2.1) follows from

$$\begin{aligned} \sum _{R\subset {\mathcal {M}}(\Omega )}\int _{\widetilde{\widetilde{{\widetilde{\Omega }}}}^{c}} S(a_{R})(x)^{p}\omega (x)\textrm{d}x\le C. \end{aligned}$$

(2.2)

For any $R\in {\mathcal {M}}(\Omega )$ centered at $(x_{0},y_{0})$, we now estimate $\int _{\widetilde{\widetilde{{\widetilde{\Omega }}}}^{c}}S(a_{R})(x)^{p}\omega (x)\textrm{d}x$. Note that ${\tilde{R}}\in {\mathcal {M}}_{1}({\widetilde{\Omega }})$, $\tilde{{\tilde{R}}}\in {\mathcal {M}}_{2}(\widetilde{{\widetilde{\Omega }}})$ and $10\tau \tilde{{\tilde{R}}}\subseteq \widetilde{\widetilde{{\widetilde{\Omega }}}}$. Hence one has that

$$\begin{aligned} \int _{\widetilde{\widetilde{{\widetilde{\Omega }}}}^{c}}S(a_{R})(x)^{p}\omega (x)\textrm{d}x\le & {} \int _{(10\tau {\tilde{I}})^{c}\times {\mathbb {R}}^{n_{2}}}S(a_{R})(x)^{p}\omega (x)\textrm{d}x\\{} & {} + \int _{{\mathbb {R}}^{n_{1}}\times (10\tau {\tilde{J}})^{c}}S(a_{R})(x)^{p}\omega (x)\textrm{d}x\\= & {} I+II. \end{aligned}$$

To estimate the first term, we split it into two terms as follows:

$$\begin{aligned} I= & {} \int _{(10\tau {\tilde{I}})^{c}\times (10\tau J)^{c}}S(a_{R})(x)^{p}\omega (x)\textrm{d}x+\int _{(10\tau {\tilde{I}})^{c}\times (10\tau J)}S(a_{R})(x)^{p}\omega (x)\textrm{d}x\\= & {} U(R)+V(R). \end{aligned}$$

Recall that $S(a_{R})(x_{1},x_{2})=\Big (\sum _{j\in {\mathbb {N}},k\in {\mathbb {Z}}} \sum \limits _{I'\times J'\in \Pi _{j}^{{n_{1}}}\times \Pi _{k}^{{n_{2}}}}|\psi _{j,k}*a_{R}(x_{I'},x_{J'})|^2\chi _{I'}(x_{1})\chi _{J'}(x_{2})\Big )^{\frac{1}{2}}$. Note that, for any fixed j, k, the rectangles in $\Pi _{j}^{{n_{1}}}\times \Pi _{k}^{{n_{2}}}$ are disjoint. Hence one can rewrite $S(a_{R})$ as follows:

$$\begin{aligned} S(a_{R})(x_{1},x_{2})= & {} \left( \sum _{j\in {\mathbb {N}},k\in {\mathbb {Z}}} |\sum \limits _{I'\times J'\in \Pi _{j}^{{n_{1}}}\times \Pi _{k}^{{n_{2}}}}\psi _{j,k}*a_{R}(x_{I'},x_{J'})\chi _{I'}(x_{1})\chi _{J'}(x_{2})|^2\right) ^{\frac{1}{2}}. \end{aligned}$$

According to side length of I, there are two cases: $\ell (I)\ge \varrho$ and $\ell (I)< \varrho .$

We now discuss the first case: $\ell (I)\ge \varrho$. For U(R), using the cancellation condition of $a_{R}$ in the $x_{2}$ direction and the Taylor’s Theorem, one has that

$$\begin{aligned} |\psi _{j,k}*a_{R}(x_{I'},x_{J'})|= & {} |\int \psi ^{(1)}_{j}(x_{I'}-y_{1})[\psi _{k}(x_{J'}-y_{2})\\{} & {} \quad -\sum _{|\alpha |\le N^{2}_{p}}2^{k|\alpha |}(y_{0}-y_{2})^{\alpha }(D^{\alpha }\psi ^{(2)})_{k}\\{} & {} \qquad (x_{J'}-y_{0})] a_{R}(y)\textrm{d}y|\\= & {} |\int \psi ^{(1)}_{j}(x_{I'}-y_{1})\\{} & {} \quad \sum _{|\alpha |= N^{2}_{p}+1}2^{k|\alpha |}(y_{0}-y_{2})^{\alpha }(D^{\alpha } \psi ^{(2)})_{k}\\{} & {} \quad (x_{J'}-y_{0}-\theta (y_{2}-y_{0})] a_{R}(y)\textrm{d}y| \end{aligned}$$

for some $\theta \in (0,1)$. Set $N=N^{2}_{p}+1$. It implies that

$$\begin{aligned}{} & {} |\psi _{j,k}*a_{R}(x_{I'},x_{J'})|\lesssim \\{} & {} \quad \int \frac{2^{jn_{1}}}{(1+2^{j}|x_{I'}-y_{1}|)^{n_{1}+L}} \frac{2^{k(n_{2}+N)}\ell (J)^{N}}{(1+2^{k}|x_{J'}-y_{0}-\theta (y_{2}-y_{0})|)^{n_{2}+L}} |a_{R}(y)|\textrm{d}y \end{aligned}$$

for any positive integer L. Note that $|x_{J'}-x_{2}|\le 2^{-k}$ if $x_{2}\in J'$, which yields that $1+2^{k}|x_{J'}-y_{0}-\theta (y_{2}-y_{0})|\thickapprox 1+2^{k}|x_{2}-y_{0}-\theta (y_{2}-y_{0})| \thickapprox 1+2^{k}|x_{2}-y_{0}|$ since $x_{2}\in (10\tau J)^{c}$. Similarly, $1+2^{j}|x_{I'}-y_{1}|\thickapprox 1+2^{j}|x_{1}-x_{0}|$. Hence when $(x_{1},x_{2})\in (10\tau {\tilde{I}})^{c}\times (10\tau J)^{c}$,

$$\begin{aligned}{} & {} |\psi _{j,k}*a_{R}(x_{I'},x_{J'})|\nonumber \\{} & {} \quad \lesssim \frac{2^{jn_{1}}}{(1+2^{j}|x_{1}-x_{0}|)^{n_{1}+L}} \frac{2^{k(n_{2}+N)}\ell (J)^{N}}{(1+2^{k}|x_{2}-y_{0}|)^{n_{2}+L}} \int |a_{R}(y)|\textrm{d}y\\{} & {} \quad \le \frac{2^{-jL}}{|x_{1}-x_{0}|^{n_{1}+L}} \frac{2^{k(n_{2}+N)}\ell (J)^{N}}{(1+2^{k} |x_{2}-y_{0}|)^{n_{2}+L}}\Vert a_{R}\Vert _{L_{\omega }^{2} ({\mathbb {R}}^{n_{1}+n_{2}})}(\omega ^{-1}(R))^{\frac{1}{2}}\nonumber \end{aligned}$$

(2.3)

by Hölder’s inequality. Moreover, by a standard estimate, if $L>N$, one has that

$$\begin{aligned} \sum _{k\in {\mathbb {Z}}} \frac{2^{k(n_{2}+N)}}{(1+2^{k}|x_{2}-y_{0}|)^{n_{2}+L}}\lesssim \frac{1}{|x_{2}-y_{0}|^{n_{2}+N}}. \end{aligned}$$

Therefore,

$$\begin{aligned} U(R)= & {} \int _{(10\tau {\tilde{I}})^{c}\times (10\tau J)^{c}} \Big (\sum _{j\in {\mathbb {N}},k\in {\mathbb {Z}}} \sum \limits _{I'\times J'\in \Pi _{j}^{{n_{1}}}\times \Pi _{k}^{{n_{2}}}}|\psi _{j,k}*a_{R}(x_{I'},x_{J'})|^2\chi _{I'}(x_{1})\chi _{J'}(x_{2})\Big )^{\frac{p}{2}}\omega (x)\textrm{d}x\\\lesssim & {} \int _{(\frac{\ell ({\tilde{I}})}{\ell (I)}I)^{c}\times (10\tau J)^{c}} \frac{1}{|x_{1}-x_{0}|^{(n_{1}+L)p}} \frac{\ell (J)^{Np}}{|x_{2}-y_{0}|^{(n_{2}+N)p}}\omega (x_{1},x_{2})\textrm{d}x\\{} & {} \cdot \Vert a_{R}\Vert ^{p}_{L_{\omega }^{2}({\mathbb {R}}^{n_{1}+n_{2}})} (\omega ^{-1}(R))^{\frac{p}{2}}\\\le & {} \sum ^{\infty }_{i=0}\sum ^{\infty }_{s=0}\int _{|x_{1}-x_{0}|\approx 2^{i}\ell ( {\tilde{I}})}\int _{|x_{2}-y_{0}|\approx 2^{s}\ell ( J)}\frac{1}{|x_{1}-x_{0}|^{(n_{1}+L)p}}\\{} & {} \frac{\ell (J)^{Np}}{|x_{2}-y_{0}|^{(n_{2}+N)p}}\omega (x_{1},x_{2})\textrm{d}x \cdot \Vert a_{R}\Vert ^{p}_{L_{\omega }^{2}({\mathbb {R}}^{n_{1}+n_{2}})}(\omega ^{-1}(R))^{\frac{p}{2}}\\\approx & {} \sum ^{\infty }_{i=0}\sum ^{\infty }_{s=0}(2^{i}\ell ({\tilde{I}}))^{-(n_{1}+L)p}(2^{s}\ell ( J))^{-(n_{2}+N)p}\ell (J)^{Np} \Vert a_{R}\Vert ^{p}_{L_{\omega }^{2}({\mathbb {R}}^{n_{1}+n_{2}})}(\omega ^{-1}(R))^{\frac{p}{2}}\\{} & {} \cdot \omega (2^{i}\frac{\ell ({\tilde{I}})}{\ell (I)}I\times 2^{s} J)\\\lesssim & {} \sum ^{\infty }_{i=0}\sum ^{\infty }_{s=0} (2^{i}\ell ({\tilde{I}}))^{-(n_{1}+L)p}(2^{s}\ell ( J))^{-(n_{2}+N)p}\ell (J)^{Np} \Vert a_{R}\Vert ^{p}_{L_{\omega }^{2}({\mathbb {R}}^{n_{1}+n_{2}})}(\omega ^{-1}(R))^{\frac{p}{2}}\\{} & {} \cdot 2^{2in_{1}}2^{2sn_{2}}(\frac{\ell ({\tilde{I}})}{\ell (I)})^{2n_{1}}\omega (I\times J)\\\lesssim & {} \ell ({\tilde{I}})^{-(n_{1}+L)p}\ell (J)^{-n_{2}p} \Vert a_{R}\Vert ^{p}_{L_{\omega }^{2}({\mathbb {R}}^{n_{1}+n_{2}})}(\omega ^{-1}(R))^{\frac{p}{2}} \left( \frac{\ell ({\tilde{I}})}{\ell (I)}\right) ^{2n_{1}}\omega (I\times J) \end{aligned}$$

by choosing L such that $(n_{1}+L)p-2n_{1}>0$. Hence,

$$\begin{aligned}{} & {} U(R)\lesssim (\frac{\ell ({\tilde{I}})}{\ell (I)})^{-[(n_{1}+L)p-2n_{1}]}|R|^{-p} \Vert a_{R}\Vert ^{p}_{L_{\omega }^{2}({\mathbb {R}}^{n_{1}+n_{2}})}(\omega ^{-1}(R))^{\frac{p}{2}} \omega (I\times J) \end{aligned}$$

since $\ell (I)\ge \varrho$. At last, since $\omega ^{-1}(R)\lesssim |R|^{2}\omega (R)^{-1}$ when $\omega \in A_{2}({\mathbb {R}}^{n_{1}}\times {\mathbb {R}}^{n_{2}})$, we have that

$$\begin{aligned}{} & {} U(R)\lesssim \gamma (R)^{-[(n_{1}+L)p-2n_{1}]} \omega (R)^{1-\frac{p}{2}}\Vert a_{R}\Vert ^{p}_{L_{\omega }^{2}({\mathbb {R}}^{n_{1}+n_{2}})}, \end{aligned}$$

where $\gamma (R)=\frac{\ell ({\tilde{I}})}{\ell (I)}$.

We now discuss V(R). In a fashion similar to obtain (2.3), one has that

$$\begin{aligned}{} & {} |\psi _{j,k}*a_{R}(x_{I'},x_{J'})| \le \int |\psi _{j}(x_{I'}-y_{1})||\int \psi _{k}(x_{J'}-y_{2}) a_{R}(y_{1},y_{2})\textrm{d}y_{2}|\textrm{d}y_{1}\\{} & {} \quad \lesssim \int \frac{2^{jn_{1}}}{(1+2^{j}|x_{I'}-y_{1}|)^{n_{1}+L}} |\int \psi _{k}(x_{J'}-y_{2}) a_{R}(y_{1},y_{2})\textrm{d}y_{2}|\textrm{d}y_{1}\\{} & {} \quad \lesssim \frac{2^{jn_{1}}}{(1+2^{j}|x_{1}-x_{0}|)^{n_{1}+L}}\int _{\tau I}|\int \psi _{k}(x_{J'}-y_{2}) a_{R}(y_{1},y_{2})\textrm{d}y_{2}|\textrm{d}y_{1}. \end{aligned}$$

It yields that

$$\begin{aligned}{} & {} |\sum \limits _{I'\times J'\in \Pi _{j}^{{n_{1}}}\times \Pi _{k}^{{n_{2}}}}\psi _{j,k}*a(x_{I'},x_{J'})\chi _{I'}(x_{1})\chi _{J'}(x_{2})|\\{} & {} \quad \lesssim \frac{2^{jn_{1}}}{(1+2^{j}|x_{1}-x_{0}|)^{n_{1}+L}} \int _{\tau I}\sum \limits _{ J'\in \Pi _{k}^{{n_{2}}}}|\int \psi _{k}(x_{J'}-y_{2})a_{R}(y_{1},y_{2})\textrm{d}y_{2}|\chi _{J'}(x_{2})\textrm{d}y_{1}. \end{aligned}$$

Hence

$$\begin{aligned}{} & {} \sum _{j\in {\mathbb {N}},k\in {\mathbb {Z}}} |\sum \limits _{I'\times J'\in \Pi _{j}^{{n_{1}}}\times \Pi _{k}^{{n_{2}}}}\psi _{j,k}*f(x_{I'},x_{J'})\chi _{I'}(x_{1})\chi _{J'}(x_{2})|^2\\{} & {} \quad \lesssim \sum \limits _{j\in {\mathbb {N}},k\in {\mathbb {Z}}}\\{} & {} \quad \left( \frac{2^{jn_{1}}}{(1{+}2^{j}|x_{1}{-}x_{0}|)^{n_{1}{+}L}}\right) ^{2} \left( \int _{\tau I}\sum \limits _{ J'\in \Pi _{k}^{{n_{2}}}}|\int \psi _{k}(x_{J'}-y_{2})a_{R}(y_{1},y_{2})\textrm{d}y_{2}|\chi _{J'}(x_{2})\textrm{d}y_{1}\right) ^{2}. \end{aligned}$$

Then,

$$\begin{aligned} V(R)\lesssim & {} \int _{(10\tau {\tilde{I}})^{c}\times (10\tau J)}\bigg (\sum \limits _{j\in {\mathbb {N}},k\in {\mathbb {Z}}}(\frac{2^{jn_{1}} }{(1+2^{j}|x_{1}-x_{0}|)^{n_{1}+L}})^{2} \\{} & {} \times \left( \int _{\tau I}\sum \limits _{ J'\in \Pi _{k}^{{n_{2}}}}|\int \psi _{k}(x_{J'}-y_{2})a_{R}(y_{1},y_{2})\textrm{d}y_{2} |\chi _{J'}(x_{2})\textrm{d}y_{1})^{2}\right) ^{p/2}\omega (x_{1},x_{2})\textrm{d}x\\\lesssim & {} \sum ^{\infty }_{i=0}\int _{|x_{1}-x_{0}|\approx 2^{i}\ell ( {\tilde{I}})}\int _{10\tau J}\left( \frac{1}{|x_{1}-x_{0}|^{(n_{1}+L)}}\right) ^{p}\\{} & {} \times \left( \sum \limits _{k\in {\mathbb {Z}}} \left( \int _{\tau I}\sum \limits _{ J'\in \Pi _{k}^{{n_{2}}}}|\int \psi _{k}(x_{J'}-y_{2})a_{R}(y_{1},y_{2})\textrm{d}y_{2}| \chi _{J'}(x_{2})\textrm{d}y_{1}\right) ^{2}\right) ^{p/2}\omega (x_{1},x_{2})\textrm{d}x\\\le & {} \frac{1}{\ell ( {\tilde{I}})^{Lp}} \sum ^{\infty }_{i=0}2^{-Lpi}\int _{|x_{1}-x_{0}|\approx 2^{i}\ell ( {\tilde{I}})}\int _{10\tau J}\\{} & {} \times \bigg (\sum \limits _{k\in {\mathbb {Z}}} (\frac{1}{|x_{1}-x_{0}|^{n_{1}}}\int _{\tau I}\sum \limits _{ J'\in \Pi _{k}^{{n_{2}}}}|\int \psi _{k}(x_{J'}-y_{2})a_{R}(y_{1},y_{2})\textrm{d}y_{2}|\chi _{J'}(x_{2})\textrm{d}y_{1})^{2}\bigg )^{p/2}\\{} & {} \omega (x_{1},x_{2})\textrm{d}x\\\le & {} \frac{1}{\ell ( {\tilde{I}})^{Lp}} \sum ^{\infty }_{i=0}2^{-Lpi}\int _{|x_{1}-x_{0}|\approx 2^{i}\ell ( {\tilde{I}})}\int _{10\tau J}\\{} & {} \times \bigg (\sum \limits _{k\in {\mathbb {Z}}} \big ({\mathcal {M}}^{(1)}(\sum \limits _{ J'\in \Pi _{k}^{{n_{2}}}}|\int \psi _{k}(x_{J'}{-}y_{2})a_{R} (\cdot ,y_{2})\textrm{d}y_{2}|\chi _{J'}(x_{2})\big )^{2}(x_{1})\bigg )^{p/2}\omega (x_{1},x_{2})\textrm{d}x, \end{aligned}$$

where ${\mathcal {M}}^{(1)}$ is the Hardy-Littlewood maximal operator associated with the first direction $x_{1}$. By Hölder’s inequality and $L^{2}$ boundedness of ${\mathcal {M}}^{(1)}$, one has that

$$\begin{aligned}{} & {} \int _{|x_{1}-x_{0}|\approx 2^{i}\ell ( {\tilde{I}})}\int _{10\tau J} \bigg (\sum \limits _{k\in {\mathbb {Z}}} \big ({\mathcal {M}}^{(1)}(\sum \limits _{ J'\in \Pi _{k}^{{n_{2}}}}|\int \psi _{k}(x_{J'}-y_{2})a_{R}\\{} & {} \quad (\cdot ,y_{2})\textrm{d}y_{2}|\chi _{J'}(x_{2})\big )^{2}(x_{1})\bigg )^{p/2} \cdot \omega (x_{1},x_{2})\textrm{d}x\\{} & {} \quad \le \omega \bigg (\frac{2^{i}\ell ( {\tilde{I}})}{\ell (I)}I,10\tau J \bigg )^{1-\frac{p}{2}}\\{} & {} \quad \bigg (\int \int \sum \limits _{k\in {\mathbb {Z}}} ({\mathcal {M}}^{(1)}\bigg (\sum \limits _{ J'\in \Pi _{k}^{{n_{2}}}}|\int \psi _{k}(x_{J'}-y_{2})a_{R}\\{} & {} \quad (\cdot ,y_{2})\textrm{d}y_{2}|\chi _{J'}(x_{2})\bigg )^{2}(x_{1}) \omega (x_{1},x_{2})\textrm{d}x\bigg )^{p/2}\\{} & {} \quad \lesssim \omega (\frac{2^{i}\ell ( {\tilde{I}})}{\ell (I)}I,10\tau J)^{1-\frac{p}{2}}\\{} & {} \quad \bigg (\int \int \sum \limits _{k\in {\mathbb {Z}}} \bigg (\sum \limits _{ J'\in \Pi _{k}^{{n_{2}}}}|\int \psi _{k}(x_{J'}-y_{2})a_{R}(x_{1},y_{2})\textrm{d}y_{2} |\chi _{J'}(x_{2})\bigg )^{2}\omega (x_{1},x_{2})\textrm{d}x\bigg )^{p/2}\\{} & {} \quad \lesssim \omega \left( \frac{2^{i}\ell ( {\tilde{I}})}{\ell (I)}I,10\tau J \right) ^{1-\frac{p}{2}}\Vert a_{R}\Vert ^{p}_{L_{\omega }^{2}({\mathbb {R}}^{n_{1}+n_{2}})}\\{} & {} \quad \lesssim \left( \frac{2^{i}\ell ( {\tilde{I}})}{\ell (I)}\right) ^{2n_{1}(1-\frac{p}{2})}\omega (R)^{1-\frac{p}{2}} \Vert a_{R}\Vert ^{p}_{L_{\omega }^{2}({\mathbb {R}}^{n_{1}+n_{2}})}. \end{aligned}$$

It gives that

$$\begin{aligned} V(R)\lesssim & {} \frac{1}{\ell ( {\tilde{I}})^{Lp}} \sum ^{\infty }_{i=0}2^{-Lpi}(\frac{2^{i}\ell \bigg ( {\tilde{I}})}{\ell (I)}\bigg )^{2n_{1}(1-\frac{p}{2})}\omega (R)^{1-\frac{p}{2}} \Vert a_{R}\Vert ^{p}_{L_{\omega }^{2}({\mathbb {R}}^{n_{1}+n_{2}})}\\\lesssim & {} \gamma (R)^{-(n_{1}+L)p+2n_{1}} \omega (R)^{1-\frac{p}{2}}\Vert a_{R}\Vert ^{p}_{L_{\omega }^{2}({\mathbb {R}}^{n_{1}+n_{2}})}, \end{aligned}$$

since $\ell (I)\ge \varrho$.

For the second case $\ell (I)< \varrho ,$ in a fashion similar to case one, using the cancellation condition of $a_{R}$ in two directions and Taylor’s Theorem, one has that

$$\begin{aligned}{} & {} |\psi _{j,k}*a_{R}(x_{I'},x_{J'})|\\{} & {} \quad \lesssim \frac{2^{j(n_{1}+N')}\ell (I)^{N'}}{(1+2^{j}|x_{1}-x_{0}|)^{n_{1}+L}} \frac{2^{k(n_{2}+N)}\ell (J)^{N}}{(1+2^{k}|x_{2}-y_{0}|)^{n_{2}+L}} \Vert a_{R}\Vert _{L_{\omega }^{2} ({\mathbb {R}}^{n_{1}+n_{2}})}(\omega ^{-1}(R))^{\frac{1}{2}}, \end{aligned}$$

where $N'=N^{1}_{p}+1$. It follows that

$$\begin{aligned}{} & {} U(R)\lesssim \int _{(\frac{\ell ({\tilde{I}})}{\ell (I)}I)^{c}\times (10\tau J)^{c}} \frac{\ell (I)^{N'p}}{|x_{1}-x_{0}|^{(n_{1}+N')p}} \frac{\ell (J)^{Np}}{|x_{2}-y_{0}|^{(n_{2}+N)p}}\omega (x_{1},x_{2})\textrm{d}x \\{} & {} \qquad \cdot \Vert a_{R}\Vert ^{p}_{L_{\omega }^{2}({\mathbb {R}}^{n_{1}+n_{2}})}(\omega ^{-1}(R))^{\frac{p}{2}}\\{} & {} \quad \le \sum ^{\infty }_{i=0}\sum ^{\infty }_{s=0}\int _{|x_{1}-x_{0}|\approx 2^{i}\ell ( {\tilde{I}})}\int _{|x_{2}-y_{0}|\approx 2^{s}\ell ( J)}\frac{\ell (I)^{N'p}}{|x_{1}-x_{0}|^{(n_{1}+N')p}} \frac{\ell (J)^{Np}}{|x_{2}-y_{0}|^{(n_{2}+N)p}}\omega (x_{1},x_{2})\textrm{d}x\\{} & {} \qquad \cdot \Vert a_{R}\Vert ^{p}_{L_{\omega }^{2}({\mathbb {R}}^{n_{1}+n_{2}})} (\omega ^{-1}(R))^{\frac{p}{2}}\\ \\{} & {} \quad \lesssim \sum ^{\infty }_{i=0}\sum ^{\infty }_{s=0} (2^{i}\ell ({\tilde{I}}))^{-(n_{1}+N')p}\ell (I)^{N'p}(2^{s}\ell ( J))^{-(n_{2}+N)p}\ell (J)^{Np} \Vert a_{R}\Vert ^{p}_{L_{\omega }^{2}({\mathbb {R}}^{n_{1}+n_{2}})}(\omega ^{-1}(R))^{\frac{p}{2}}\\{} & {} \qquad \cdot 2^{2in_{1}}2^{2sn_{2}}(\frac{\ell ({\tilde{I}})}{\ell (I)})^{2n_{1}}\omega (I\times J)\\{} & {} \quad \lesssim \gamma (R)^{-[(n_{1}+N')p-2n_{1}]} \omega (R)^{1-\frac{p}{2}}\Vert a_{R}\Vert ^{p}_{L_{\omega }^{2}({\mathbb {R}}^{n_{1}+n_{2}})}, \end{aligned}$$

since $(n_{1}+N')p-2n_{1}>0$.

For V(R), let $x_{1}\in (10\tau {\tilde{I}})^{c}$. Similarly, for any positive integer L,

$$\begin{aligned} |\psi _{j,k}*a_{R}(x_{I'},x_{J'})| \lesssim \frac{2^{j(n_{1}+N)}\ell (I)^{N}}{(1+2^{j}|x_{1}-x_{0}|)^{n_{1}+L}} \int _{\tau I}|\int \psi _{k}(x_{J'}-y_{2})a_{R}(y_{1},y_{2})\textrm{d}y_{2}|\textrm{d}y_{1}. \end{aligned}$$

It yields that

$$\begin{aligned}{} & {} |\sum \limits _{I'\times J'\in \Pi _{j}^{{n_{1}}}\times \Pi _{k}^{{n_{2}}}}\psi _{j,k}*a(x_{I'},x_{J'})\chi _{I'}(x_{1})\chi _{J'}(x_{2})|\\{} & {} \quad \lesssim \frac{2^{j(n_{1}+N)}\ell (I)^{N}}{(1+2^{j}|x_{1}-x_{0}|)^{n_{1}+L}} \int _{\tau I}\sum \limits _{ J'\in \Pi _{k}^{{n_{2}}}}|\int \psi _{k}(x_{J'}-y_{2})a_{R}(y_{1},y_{2})\textrm{d}y_{2}|\chi _{J'}(x_{2})\textrm{d}y_{1}. \end{aligned}$$

Then

$$\begin{aligned}{} & {} \sum _{j\in {\mathbb {N}},k\in {\mathbb {Z}}} |\sum \limits _{I'\times J'\in \Pi _{j}^{{n_{1}}}\times \Pi _{k}^{{n_{2}}}}\psi _{j,k}*f(x_{I'},x_{J'})\chi _{I'}(x_{1})\chi _{J'}(x_{2})|^2\\{} & {} \quad \lesssim \sum \limits _{j\in {\mathbb {N}},k\in {\mathbb {Z}}}(\frac{2^{j(n_{1}+N)}\ell (I)^{N}}{(1+2^{j}|x_{1}-x_{0}|)^{n_{1}+L}})^{2} (\int _{\tau I}\sum \limits _{ J'\in \Pi _{k}^{{n_{2}}}}|\int \psi _{k}(x_{J'}-y_{2})a_{R}(y_{1},y_{2})\textrm{d}y_{2}|\chi _{J'}(x_{2})\textrm{d}y_{1})^{2}. \end{aligned}$$

Thus

$$\begin{aligned}{} & {} V(R)\\{} & {} \quad \lesssim \int _{(10\tau {\tilde{I}})^{c}\times (10\tau J)}\bigg (\sum \limits _{j\in {\mathbb {N}},k\in {\mathbb {Z}}}\bigg (\frac{2^{j(n_{1}+N)} \ell (I)^{N}}{(1+2^{j}|x_{1}-x_{0}|)^{n_{1}+L}}\bigg )^{2} \\{} & {} \qquad \bigg (\int _{\tau I}\sum \limits _{ J'\in \Pi _{k}^{{n_{2}}}}|\int \psi _{k}(x_{J'}-y_{2})a_{R}(y_{1},y_{2})\textrm{d}y_{2} |\chi _{J'}(x_{2})\textrm{d}y_{1}\bigg )^{2}\bigg )^{p/2}\omega (x_{1},x_{2})\textrm{d}x\\{} & {} \quad \lesssim \sum ^{\infty }_{i=0}\int _{|x_{1}-x_{0}|\approx 2^{i}\ell ( {\tilde{I}})}\int _{10\tau J}\bigg (\frac{\ell (I)^{N}}{|x_{1}-x_{0}|^{(n_{1}+N)}}\bigg )^{p}\\{} & {} \qquad \bigg (\sum \limits _{k\in {\mathbb {Z}}} \bigg (\int _{\tau I}\sum \limits _{ J'\in \Pi _{k}^{{n_{2}}}}|\int \psi _{k}(x_{J'}-y_{2})a_{R}(y_{1},y_{2})\textrm{d}y_{2}| \chi _{J'}(x_{2})\textrm{d}y_{1}\bigg )^{2}\bigg )^{p/2}\omega (x_{1},x_{2})\textrm{d}x\\ \\{} & {} \quad \le \bigg (\frac{\ell (I)^{N}}{\ell ( {\tilde{I}})^{N}}\bigg )^{p} \sum ^{\infty }_{i=0}2^{-Npi}\int _{|x_{1}-x_{0}|\approx 2^{i}\ell ( {\tilde{I}})}\int _{10\tau J}\\{} & {} \quad \bigg (\sum \limits _{k\in {\mathbb {Z}}} ({\mathcal {M}}^{(1)}\bigg (\sum \limits _{ J'\in \Pi _{k}^{{n_{2}}}}|\int \psi _{k}(x_{J'}-y_{2})a_{R}(\cdot ,y_{2})\textrm{d}y_{2}| \chi _{J'}(x_{2})\bigg )^{2}(x_{1})\bigg )^{p/2}\omega (x_{1},x_{2})\textrm{d}x\\{} & {} \quad \lesssim \bigg (\frac{\ell (I)^{N}}{\ell ( {\tilde{I}})^{N}}\bigg )^{p} \sum ^{\infty }_{i=0}2^{-Npi}\omega \bigg (\frac{2^{i}\ell ( {\tilde{I}})}{\ell (I)}I,10\tau J \bigg )^{1-\frac{p}{2}}\Vert a_{R}\Vert ^{p}_{L_{\omega }^{2}({\mathbb {R}}^{n_{1}+n_{2}})}\\{} & {} \quad \lesssim \gamma (R)^{-(n_{1}+N)p+2n_{1}} \omega (R)^{1-\frac{p}{2}}\Vert a_{R}\Vert ^{p}_{L_{\omega }^{2}({\mathbb {R}}^{n_{1}+n_{2}})}. \end{aligned}$$

Hence, we obtain that

$$\begin{aligned} I\lesssim (\frac{\ell ({\tilde{I}})}{\ell (I)})^{-(n_{1}+N')p+2n_{1}} \omega (R)^{1-\frac{p}{2}}\Vert a_{R}\Vert ^{p}_{L_{\omega }^{2}({\mathbb {R}}^{n_{1}+n_{2}})}. \end{aligned}$$

Similarly,

$$\begin{aligned} II\lesssim & {} \gamma '(R)^{-(n_{2}+N)p+2n_{2}} \omega (R)^{1-\frac{p}{2}}\Vert a_{R}\Vert ^{p}_{L_{\omega }^{2}({\mathbb {R}}^{n_{1}+n_{2}})}. \end{aligned}$$

By weighted version of Journé’s Lemma 2.1, one has that

$$\begin{aligned}{} & {} \sum _{R\in {\mathcal {M}}(\Omega )}\int _{\widetilde{\widetilde{{\widetilde{\Omega }}}}^{c}}|S(a_{R})(x)|^{p}\omega (x)\textrm{d}x\\{} & {} \lesssim \sum _{R\in {\mathcal {M}}(\Omega )}\gamma (R)^{-(n_{1}p+N' p-2n_{1})}\omega (R)^{1-\frac{p}{2}}\Vert a_{R}\Vert ^{p}_{L_{\omega }^{2}({\mathbb {R}}^{n_{1}+n_{2}})}\nonumber \\{} & {} \qquad +\sum _{R\in {\mathcal {M}}(\Omega )}\gamma '(R)^{-(n_{2}p+N p-2n_{2})}\omega (R)^{1-\frac{p}{2}}\Vert a_{R}\Vert ^{p}_{L_{\omega }^{2}({\mathbb {R}}^{n_{1}+n_{2}})} \\{} & {} \quad \le \big (\sum _{R\in {\mathcal {M}}_{1}(\Omega )}\gamma (R)^{-(n_{1}p+N' p-2n_{1})p'}\omega (R)\big )^{1/p'}\left( \sum _{R\in {\mathcal {M}}(\Omega )} \Vert a_{R}\Vert ^{2}_{L_{\omega }^{2}({\mathbb {R}}^{n_{1}+n_{2}})}\right) ^{p/2}\\{} & {} \quad +\big (\sum _{R\in {\mathcal {M}}_{2}(\Omega )}\gamma '(R)^{-(n_{2}p+N p-2n_{2})}\omega (R)\big )^{1/p'}\left( \sum _{R\in {\mathcal {M}}(\Omega )}\Vert a_{R}\Vert ^{2}_{L_{\omega }^{2}({\mathbb {R}}^{n_{1}+n_{2}})}\right) ^{p/2}\\{} & {} \quad \lesssim \omega (\Omega )^{1-p/2}\left( \sum _{R\in {\mathcal {M}}(\Omega )}\Vert a_{R}\Vert ^{2}_{L_{\omega }^{2}({\mathbb {R}}^{n_{1}+n_{2}})}\right) ^{p/2} \le C, \end{aligned}$$

where $p'=\frac{2}{2-p}$ satisfying $\frac{p}{2}+\frac{1}{p'}=1$. Thus we obtain (2.2) and then finish the proof. $\square$

3 Atomic Decomposition

To discuss the atomic decomposition of $H^{p}_{mix}(\omega ,{\mathbb {R}}^{n_{1}}\times {\mathbb {R}}^{n_{2}})$, we need a new discrete Calderón-type identity composed by some test functions with compact supports which was obtained in [14]. To do this, given a positive integer M large enough, let $\phi _{0}^{(1)},\phi ^{(1)}\in {{\mathcal {S}}}({\mathbb {R}}^{n_{1}})$ satisfy that

$$\begin{aligned} \hbox {supp}\phi _{0}^{(1)} \subseteq \{x\in \mathbb R^{n_{1}}: |x|\le 1\};\ \int \phi _{0}^{(1)}=1, \end{aligned}$$

(3.1)

and

$$\begin{aligned} \hbox {supp}\phi ^{(1)} \subseteq \{x\in \mathbb R^{n_{1}}: |x|\le 1\};\ \int \phi ^{(1)}(x)x^{\alpha }\textrm{d}x=0,\ \hbox {for\ all}\ |\alpha | \le M, \end{aligned}$$

(3.2)

and

$$\begin{aligned} |\widehat{\phi _{0}^{(1)}}(\xi )|^{2}+\sum \limits ^{\infty }_{j =1}|\widehat{\phi ^{(1)}}(2^{-j}\xi )|^2=1, \ \hbox { for all } \xi \in {\mathbb {R}}^{n_{1}}. \end{aligned}$$

(3.3)

Let $\phi ^{(2)}\in {{\mathcal {S}}}({\mathbb {R}}^{n_{2}})$ with

$$\begin{aligned} \hbox {supp} \phi ^{(2)}\subseteq \{x\in \mathbb R^{n_{2}}: |x|\le 1\};\ \int \phi ^{(2)}(x)x^{\alpha }\textrm{d}x=0,\ \hbox {for\ all}\ |\alpha | \le M, \end{aligned}$$

(3.4)

and

$$\begin{aligned} \sum \limits _{j \in \mathbb Z}|\widehat{\phi ^{(2)}}(2^{-j}\xi )|^2=1, \ \hbox { for all } \xi \in {\mathbb {R}}^{n_{2}}\setminus \{0\}. \end{aligned}$$

(3.5)

Theorem 3.1

For $0<p\le 1$, let $\phi _{0}^{(1)}$, $\phi ^{(1)}\in {{\mathcal {S}}}({\mathbb {R}}^{n_{1}})$ and $\phi ^{(2)}\in {{\mathcal {S}}}({\mathbb {R}}^{n_{2}})$ satisfy conditions (3.1)–(3.3) and (3.4)–(3.5), respectively. Suppose that $\omega \in A_2({\mathbb {R}}^{n_{1}}\times {\mathbb {R}}^{n_{2}})$. Then for any $f \in L_{\omega }^{2}({\mathbb {R}}^{n_{1}+n_{2}})\cap H^{p}_{mix}(\omega ,{\mathbb {R}}^{n_{1}}\times {\mathbb {R}}^{n_{2}})$, there exists ${\tilde{f}}\in L_{\omega }^{2}({\mathbb {R}}^{n_{1}+n_{2}})\cap H^{p}_{mix}(\omega ,{\mathbb {R}}^{n_{1}}\times {\mathbb {R}}^{n_{2}})$ such that

$$\begin{aligned} \qquad f(x)= & {} \sum \limits _{j\in {\mathbb {N}},k\in {\mathbb {Z}}} \sum \limits _{I\times J\in \Pi _{j+N}^{{n_{1}}}\times \Pi _{k+N}^{{n_{2}}}}|I||J|(\phi _{j,k}*{\tilde{f}})(x_{I},x_{J})\times \phi _{j,k}(x_{1}-x_{I},x_{2}-x_{J}),\nonumber \\ \end{aligned}$$

(3.6)

where the series converges in $L_{\omega }^{2}({\mathbb {R}}^{n_{1}+n_{2}})$ and $H^{p}_{mix}(\omega ,{\mathbb {R}}^{n_{1}}\times {\mathbb {R}}^{n_{2}})$, and N is some large positive integer independent of f. Moreover,

$$\begin{aligned} \Vert {\tilde{f}}\Vert _{L_{\omega }^{2}({\mathbb {R}}^{n_{1}+n_{2}})}\approx \Vert f\Vert _{L_{\omega }^{2}({\mathbb {R}}^{n_{1}+n_{2}})} \end{aligned}$$

and

$$\begin{aligned} \Vert {\tilde{f}}\Vert _{H^{p}_{mix}(\omega ,{\mathbb {R}}^{n_{1}}\times {\mathbb {R}}^{n_{2}})}\approx \Vert f\Vert _{H^{p}_{mix}(\omega ,{\mathbb {R}}^{n_{1}}\times {\mathbb {R}}^{n_{2}})}. \end{aligned}$$

Proof of Theorem 1.2:

For $f\in L_{\omega }^{2}({\mathbb {R}}^{n_{1}+n_{2}})\cap H^{p}_{mix}(\omega ,{\mathbb {R}}^{n_{1}}\times {\mathbb {R}}^{n_{2}})$, let ${\tilde{S}}(f)(x)= \big (\sum \limits _{j\in {\mathbb {N}},k\in {\mathbb {Z}}} \sum \limits _{I\times J\in \Pi _{j+N}^{{n_{1}}}\times \Pi _{k+N}^{{n_{2}}}}|(\phi _{j,k}*{\tilde{f}})(x_{I},x_{J}|^{2}\chi _{I}\chi _{J}\big )^{1/2}$, where $\phi _{j,k}$ satisfies the conditions of Theorem 3.1. Firstly, rewrite (3.6)as follows:

$$\begin{aligned} \qquad f(x)= & {} \sum \limits _{j=0,k\in {\mathbb {Z}}} \sum \limits _{I\times J\in \Pi _{j+N}^{{n_{1}}}\times \Pi _{k+N}^{{n_{2}}}}|I||J|(\phi _{j,k}*{\tilde{f}})(x_{I},x_{J})\times \phi _{j,k}(x_{1}-x_{I},x_{2}-x_{J})\\{} & {} +\sum \limits _{j>1,k\in {\mathbb {Z}}} \sum \limits _{I\times J\in \Pi _{j+N}^{{n_{1}}}\times \Pi _{k+N}^{{n_{2}}}}|I||J|(\phi _{j,k}*{\tilde{f}})(x_{I},x_{J})\times \phi _{j,k}(x_{1}-x_{I},x_{2}-x_{J})\\= & {} f_{1}(x)+f_{2}(x). \end{aligned}$$

We now decompose $f_{1}$ into atoms. For any $i\in {\mathbb {Z}}$, set that

$$\begin{aligned}{} & {} \Omega _i =\{x \in {\mathbb {R}}^{n_{1}+n_{2}}: {\tilde{S}}(f)(x)> 2^i\}. \\{} & {} \quad {\mathcal {B}}_i=\{R: R\in \Pi _{j+N}^{{n_{1}}}\times \Pi _{k+N}^{{n_{2}}}, j=0,k\in {\mathbb {Z}},\\{} & {} \qquad \ \omega (R \cap \Omega _i) > \frac{1}{2} \omega (R), \ \ \omega (R \cap \Omega _{i+1})\le \frac{1}{2}\omega (R)\},\end{aligned}$$

and

$$\begin{aligned} {\tilde{\Omega }}_i=\{x \in {\mathbb {R}}^{n_{1}+n_{2}}: M^{\omega }_{s}(\chi _{\Omega _i}) > \frac{1}{10^{N(n_{1}+n_{2})}}\}. \end{aligned}$$

Obviously, $\cup _{R\in {\mathcal {B}}_i}R\subseteq {\tilde{\Omega }}_i$. For $R\in \Pi _{j+N}^{{n_{1}}}\times \Pi _{k+N}^{{n_{2}}}$, denote $\phi _{R}=\phi _{j,k}$. Then one can rewrite $f_{1}$ as following

$$\begin{aligned} f_{1}(x)= & {} \sum \limits _{i=-\infty }^{+\infty } \sum _{R\in {\mathcal {B}}_i}|R|(\phi _{R}*{\tilde{f}})(x_{I},x_{J})\times \phi _{R}(x_{1}-x_{I},x_{2}-x_{J}) =\sum \limits _{i}\lambda _{i}a_{i}(x). \end{aligned}$$

Set

$$\begin{aligned} {\tilde{\Omega }}_i^{'}=\left\{ x \in {\mathbb {R}}^{n_{1}+n_{2}}: M_{s}(\chi _{{\tilde{\Omega }}_i}) > \frac{1}{10^{N(n_{1}+n_{2})}}\right\} . \end{aligned}$$

Note that if $(x_{1},x_{2})\in {\text {supp}}\phi _{R}(\cdot -x_{I},\cdot -x_{J})$, one has that $|x_{1}-x_{I}|\le 2^{-j},|x_{2}-x_{J}|\le 2^{-k}$, which implies that ${\text {supp}}\phi _{R}(\cdot -x_{I},\cdot -x_{J})\subseteq 10^{N}R\subseteq {\tilde{\Omega }}_i^{'}.$ By the weighted boundedness of $M_{s}$, one has that $\omega ({\tilde{\Omega }}'_i)\thickapprox \omega ({\tilde{\Omega }}_i)$. Denote that

$$\begin{aligned} \lambda _{i}=C\omega ({\tilde{\Omega }}'_i)^{\frac{1}{p}-\frac{1}{2}} \Vert \Big (\sum _{R\in {\mathcal {B}}_i}|\phi _{R}*{\tilde{f}}(x_{I},x_{J})\chi _{R}(x)|^{2}\Big )^{\frac{1}{2}}\Vert _{L_{\omega }^2({\mathbb {R}}^{n_{1}+n_{2}})}, \end{aligned}$$

and

$$\begin{aligned} a_{i}(x)=\frac{1}{\lambda _{i}}\sum _{R\in {\mathcal {B}}_i}|R|(\phi _{R}*{\tilde{f}})(x_{I},x_{J})\phi _{R}(x_{1}-x_{I},x_{2}-x_{J}). \end{aligned}$$

Then one has that

$$\begin{aligned} f_{1}(x) =\sum \limits _{i}\lambda _{i}a_{i}(x). \end{aligned}$$

We now check that every $a_{i}$ is an atom $H^{p}_{mix}(\omega ,{\mathbb {R}}^{n_{1}}\times {\mathbb {R}}^{n_{2}})$. Firstly, ${\text {supp}}a_{i}\subseteq {\tilde{\Omega }}'_i$. Moreover, by the duality argument,

$$\begin{aligned}{} & {} \Vert a_{i}\Vert _{L_{\omega }^2({\mathbb {R}}^{n_{1}+n_{2}})}=\frac{1}{\lambda _{i}}\sup _{\Vert g\Vert _{L_{\omega ^{-1}}^2({\mathbb {R}}^{n_{1}+n_{2}})}\le 1}|\int (\sum _{R\in {\mathcal {B}}_i}|R|\left( \phi _{R}*{\tilde{f}}\right) \\{} & {} \quad (x_{I},x_{J})\phi _{R}(x_{1}-x_{I},x_{2}-x_{J}))g(x)\textrm{d}x|\\{} & {} \quad =\frac{1}{\lambda _{i}}\sup _{\Vert g\Vert _{L_{\omega ^{-1}}^2({\mathbb {R}}^{n_{1}+n_{2}})}\le 1}|\int \sum _{R\in {\mathcal {B}}_i}(\phi _{R}*{\tilde{f}})(x_{I},x_{J})\times {\tilde{\phi }}_{R}*g(x_{I},x_{J})\chi _{R}(x)\textrm{d}x|\\{} & {} \quad \le \frac{1}{\lambda _{i}}\sup _{\Vert g\Vert _{L_{\omega ^{-1}}^2({\mathbb {R}}^{n_{1}+n_{2}})}\le 1}\int \big (\sum _{R\in {\mathcal {B}}_i}|(\phi _{R}*{\tilde{f}})(x_{I},x_{J})|^{2}\chi _{R}(x)\omega (x)\big )^{1/2}\\{} & {} \qquad \big (\sum _{R\in {\mathcal {B}}_i}|({\tilde{\phi }}_{R}*g)(x_{I},x_{J})|^{2}\chi _{R}(x)\omega ^{-1}(x)\big )^{1/2}\textrm{d}x, \end{aligned}$$

which yields that $\Vert a_{i}\Vert _{L_{\omega }^2({\mathbb {R}}^{n_{1}+n_{2}})}\le \omega ({\tilde{\Omega }}'_i)^{\frac{1}{2}-\frac{1}{p}}$.

Furthermore, for $Q\in {\mathcal {M}}(\Omega _i)$, if we set that

$$\begin{aligned} a_{i,Q}(x)=\frac{1}{\lambda _{i}}\sum _{R\in {\mathcal {B}}_i,R\subseteq Q}|Q|(\phi _{R}*{\tilde{f}})(x_{I},x_{J})\phi _{R}(x_{1}-x_{I},x_{2}-x_{J}), \end{aligned}$$

then $a_{i}=\sum _{Q\in {\mathcal {M}}(\Omega _i)}a_{i,Q}$, and ${\text {supp}}a_{i,Q}\subseteq 2^{N+4}Q$. Moreover, the side length of Q in the first direction is $2^{-N}$ denoted by $\varrho$ since $Q\in {\mathcal {B}}_i$, and there is no any vanishing moment in this direction. While $a_{i,Q}$ satisfies vanishing moment in $x_{2}$ direction. On the other hand, by the duality argument again,

$$\begin{aligned} \Vert a_{i,Q}\Vert ^{2}_{L_{\omega }^2({\mathbb {R}}^{n_{1}+n_{2}})} \le \frac{1}{\lambda _{i}}\int \sum _{R\in {\mathcal {B}}_i,R\subseteq Q}|(\phi _{R}*{\tilde{f}})(x_{I},x_{J})|^{2}\chi _{R}(x)\omega (x)\textrm{d}x. \end{aligned}$$

It gives that

$$\begin{aligned}\sum _{Q\in {\mathcal {M}}(\Omega _i)}\Vert a_{i,Q}\Vert _{L_{\omega }^{2}({\mathbb {R}}^{n_{1}+n_{2}})}^{2}\lesssim & {} \frac{1}{\lambda ^{2}_{i}}\int \sum _{R\in {\mathcal {B}}_i}|(\phi _{R}*{\tilde{f}})(x_{I},x_{J})|^{2}\chi _{R}(x)\omega (x)\textrm{d}x\\\lesssim & {} \omega (\Omega '_i)^{1-\frac{2}{p}}. \end{aligned}$$

For $\sum _{i} \lambda ^{p}_{i}$, using $\omega (R\cap {\tilde{\Omega }}_{i}\backslash \Omega _{i+1})>\frac{1}{2}\omega (R)$ when $R\in B_{i}$, one has that

$$\begin{aligned}{} & {} \Vert \Big (\sum _{R\in {\mathcal {B}}_i}|\phi _{R}*{\tilde{f}}(x_{I},x_{J})\chi _{R}(x)|^{2}\Big )^{\frac{1}{2}}\Vert ^{2} _{L_{\omega }^2({\mathbb {R}}^{n_{1}+n_{2}})}\\= & {} \sum _{R\in {\mathcal {B}}_i}\omega (R)|(\phi _{R}*{\tilde{f}})(x_{I},x_{J})|^{2} \\\le & {} 2\sum _{R\in {\mathcal {B}}_i}\omega (R\cap {\tilde{\Omega }}_{i}\backslash \Omega _{i+1})|(\phi _{R}*{\tilde{f}})(x_{I},x_{J})|^{2}\\= & {} 2\int _{{\tilde{\Omega }}_{i}\backslash \Omega _{i+1}}\sum _{R\in {\mathcal {B}}_i}|(\phi _{R}*{\tilde{f}})(x_{I},x_{J})|^{2}\chi _{R}(x)\omega (x)\textrm{d}x \\\le & {} 2^{2(i+1)+1}\omega ({\tilde{\Omega }}_{i}). \end{aligned}$$

Hence

$$\begin{aligned} \sum _{i} \lambda _{i}^{p} \le C\sum _{i}2^{pi}\omega (\Omega _i)\le C \Vert {\tilde{S}}(f)\Vert ^{p}_{L_{\omega }^{p}({\mathbb {R}}^{n_{1}+n_{2}})} \le C\Vert f\Vert ^{p}_{H^{p}_{mix}(\omega ,{\mathbb {R}}^{n_{1}}\times {\mathbb {R}}^{n_{2}})} \end{aligned}$$

(3.7)

with a constant C independent of f.

Similarly, one can obtain the atomic decompositions of $f_{2}$. Different from the above, the rectangle atoms decomposed from $f_{2}$ have desired vanishing moment both in two directions. The results that $\sum \limits _{i}\lambda _{i}a_{i}(x)\rightarrow f$ in $H^{p}_{mix}(\omega ,{\mathbb {R}}^{n_{1}}\times {\mathbb {R}}^{n_{2}})$ and in $L_{\omega }^2({\mathbb {R}}^{n_{1}+n_{2}})$ are followed from (3.7) and duality argument, respectively.

For $f\in H^{p}_{mix}(\omega ,{\mathbb {R}}^{n_{1}}\times {\mathbb {R}}^{n_{2}})$, by the density, there are $\{f_{i}\}_{i\ge 0}\subseteq L_{\omega }^{2}({\mathbb {R}}^{n_{1}+n_{2}})\cap H^{p}_{mix}(\omega ,{\mathbb {R}}^{n_{1}}\times {\mathbb {R}}^{n_{2}})$ such that $\Vert f_{i}\Vert _{H^{p}_{mix}(\omega ,{\mathbb {R}}^{n_{1}}\times {\mathbb {R}}^{n_{2}})}\le 2^{-i+1}\Vert f\Vert _{H^{p}_{mix}(\omega ,{\mathbb {R}}^{n_{1}}\times {\mathbb {R}}^{n_{2}})}$, and $f(x)=\sum _{i\ge 0}f_{i}(x)$ in $H^{p}_{mix}(\omega ,{\mathbb {R}}^{n_{1}}\times {\mathbb {R}}^{n_{2}})$. Since $f_{i}\in L^{2}({\mathbb {R}}^{n_{1}+n_{2}})\cap H^{p}_{mix}(\omega ,{\mathbb {R}}^{n_{1}}\times {\mathbb {R}}^{n_{2}})$, we can decompose $f_{i}$ into atoms to obtain that

$$\begin{aligned}{} & {} f_{i}(x)=\sum _{k}\lambda ^{(i)}_{k}a^{(i)}_{k}(x),\ \hbox {in}\ H^{p}_{mix}(\omega ,{\mathbb {R}}^{n_{1}}\times {\mathbb {R}}^{n_{2}}),\ \ \hbox {and} \\{} & {} \quad \sum _{k}|\lambda ^{(i)}_{k}|^{p}\le C\Vert f_{i}\Vert ^{p}_{H^{p}_{mix}(\omega ,{\mathbb {R}}^{n_{1}}\times {\mathbb {R}}^{n_{2}})}, \end{aligned}$$

where C is a absolute constant, which yields that

$$\begin{aligned} f(x)=\sum _{i\ge 0}\sum _{k}\lambda ^{(i)}_{k}a^{(i)}_{k}(x). \end{aligned}$$

Moreover,

$$\begin{aligned}\sum \limits _{i\ge 0}\sum \limits _{k}|\lambda ^{(i)}_{k}|^{p}\le & {} \sum \limits _{i\ge 0}C\Vert f_{i}\Vert ^{p}_{H^{p}_{mix}(\omega ,{\mathbb {R}}^{n_{1}}\times {\mathbb {R}}^{n_{2}})}\\\le & {} C\sum \limits _{i\ge 0}2^{-i+1}\Vert f\Vert ^{p}_{H^{p}_{mix}(\omega ,{\mathbb {R}}^{n_{1}}\times {\mathbb {R}}^{n_{2}})}\lesssim \Vert f\Vert ^{p}_{H^{p}_{mix}(\omega ,{\mathbb {R}}^{n_{1}}\times {\mathbb {R}}^{n_{2}})}.\end{aligned}$$

The converse is obvious by Theorem .

This completes the proof. $\square$

Proof of Theorem 1.3:

If T is bounded on $H^{p}_{mix}(\omega ,{\mathbb {R}}^{n_{1}}\times {\mathbb {R}}^{n_{2}})$, then (1.9) is obtained by Theorem  directly. For the converse, let $f\in L_{\omega }^{2}({\mathbb {R}}^{n_{1}+n_{2}})\cap H^{p}_{mix}(\omega ,{\mathbb {R}}^{n_{1}}\times {\mathbb {R}}^{n_{2}})$, then by Theorem 1.2, there exists a sequence atoms $\{a_{k}\}$ such that $f(x)=\sum _{k\in {\mathbb {Z}}}\lambda _{k}a_{k}(x)$ in $L_{\omega }^{2}({\mathbb {R}}^{n_{1}+n_{2}})$, where the real numbers sequence $\{\lambda _{k}\}$ satisfies $\sum _{k\in {\mathbb {Z}}}|\lambda _{k}|^{p}\lesssim \Vert f\Vert ^{p}_{H^{p}_{mix}(\omega ,{\mathbb {R}}^{n_{1}}\times {\mathbb {R}}^{n_{2}})}$. Since T is bounded on $L_{\omega }^2({\mathbb {R}}^{n_{1}+n_{2}})$, one has $T(f)(x)=\sum \lambda _{k}T(a_{k})(x)$ in $L_{\omega }^2({\mathbb {R}}^{n_{1}+n_{2}})$, which implies that this series (subsequence) converges almost everywhere. Hence,

$$\begin{aligned} \Vert T(f)\Vert ^{p}_{H^{p}_{mix}(\omega ,{\mathbb {R}}^{n_{1}}\times {\mathbb {R}}^{n_{2}})}\le \sum |\lambda _{k}|^{p}\Vert T(a_{k})\Vert ^{p}_{H^{p}_{mix}(\omega ,{\mathbb {R}}^{n_{1}}\times {\mathbb {R}}^{n_{2}})}\lesssim \Vert f\Vert ^{p}_{H^{p}_{mix}(\omega ,{\mathbb {R}}^{n_{1}}\times {\mathbb {R}}^{n_{2}})}. \end{aligned}$$

by (1.9).

This completes the proof. $\square$