Boundedness for multilinear commutator of Marcinkiewicz operator with variable kernels on Hardy and Herz-Hardy spaces

Abstract

In this paper, the $(H_{\overrightarrow{b}}^p,L^p)$- and $(H{\dot{K}}_{q,\overrightarrow{b}}^{\alpha ,p},{\dot{K}}_q^{\alpha ,p})$-type boundedness for the multilinear commutator related to the Marcinkiewicz operator with variable kernels is obtained.

MSC:42B20, 42B25.

1 Introduction and definitions

Let T be the Calderón-Zygmund operator and $b\in \mathit{BMO}(R^n)$. The commutator $[b,T]$ generated by T and b is defined by

$$[b,T](f)(x)=b(x)T(f)(x)-T(bf)(x).$$

A classical result of Coifman, Rochberg and Weiss (see [1, 2]) proved that the commutator $[b,T]$ is bounded on $L^p(R^n)$ ($1<p<\mathrm{\infty }$). However, it was observed that the $[b,T]$ is not bounded, in general, from $H^p(R^n)$ to $L^p(R^n)$. But if $H^p(R^n)$ is replaced by a suitable atomic space $H_b^P(R^n)$, then $[b,T]$ maps continuously $H_b^P(R^n)$ into $L^p(R^n)$ (see [3]). In addition, we easily know that $H_b^p(R^n)\subset H^p(R^n)$. In recent years, the theory of Herz-type Hardy spaces have been developed (see [4–7]). The main purpose of this paper is to consider the continuity of multilinear commutators related to Marcinkiewicz operators with variable kernels and $\mathit{BMO}(R^n)$ functions on certain Hardy and Herz-Hardy spaces. Let us first introduce some definitions (see [3–14]).

Given a positive integer m and $1\le j\le m$, we denote by $C_j^m$ the family of all finite subsets $\sigma =\{\sigma (1),\dots ,\sigma (j)\}$ of $\{1,\dots ,m\}$ of j different elements. For $\sigma \in C_j^m$, set ${\sigma }^c=\{1,\dots ,m\}\setminus \sigma$. For $\overrightarrow{b}=(b_1,\dots ,b_m)$ and $\sigma =\{\sigma (1),\dots ,\sigma (j)\}\in C_j^m$, set ${\overrightarrow{b}}_{\sigma }=(b_{\sigma (1)},\dots ,b_{\sigma (j)})$, $b_{\sigma }=b_{\sigma (1)}\cdots b_{\sigma (j)}$ and ${\parallel {\overrightarrow{b}}_{\sigma }\parallel }_{\mathit{BMO}}={\parallel b_{\sigma (1)}\parallel }_{\mathit{BMO}}\cdots {\parallel b_{\sigma (j)}\parallel }_{\mathit{BMO}}$.

Definition 1 Let $b_i$ ($i=1,\dots ,m$) be a locally integrable function and $0<p\le 1$. A bounded measurable function a on $R^n$ is said to be a $(p,\overrightarrow{b})$ atom if

• (1) $suppa\subset B=B(x_0,r)$,

• (2) ${\parallel a\parallel }_{L^{\mathrm{\infty }}}\le {|B|}^{-1/p}$,

• (3) ${\int }_{B}a(y)dy={\int }_{B}a(y){\prod }_{l\in \sigma }{b}_l(y)dy=0$ for any $\sigma \in C_j^m$, $1\le j\le m$.

A temperate distribution f is said to belong to $H_{\overrightarrow{b}}^p(R^n)$ if, in the Schwartz distribution sense, it can be written as

$$f(x)=\sum _{j=1}^{\mathrm{\infty }}{\lambda }_j{a}_j(x),$$

where every $a_j$ is $(p,\overrightarrow{b})$ atom, $\lambda \in C$ and ${\sum }_{j=1}^{\mathrm{\infty }}{|\lambda |}^p<\mathrm{\infty }$. Moreover, ${\parallel f\parallel }_{H_{\overrightarrow{b}}^p(R^n)}\approx {({\sum }_{j=1}^{\mathrm{\infty }}{|{\lambda }_j|}^p)}^{1/p}$.

Given a set $E\subset R^n$, the characteristic function of E is defined by ${\chi }_E$. Let $B_k=\{x\in R^n:|x|\le 2^k\}$ and $C_k=B_k\setminus B_{k-1}$and ${\chi }_k={\chi }_{B_k}$, $k\in Z$.

Definition 2 Let $0<p$, $q<\mathrm{\infty }$, $\alpha \in R$. For $k\in Z$, set $B_k=\{x\in R^n:|x|\le 2^k\}$ and $C_k=B_k\setminus B_{k-1}$. Denote by ${\chi }_k$ the characteristic function of $C_k$ and by ${\chi }_0$ the characteristic function of $B_0$.

• (1) The homogeneous Herz space is defined by

  $${\dot{K}}_q^{\alpha ,p}\left(R^n\right)=\{f\in L_{\mathrm{loc}}^q(R^n\setminus \{0\}):{\parallel f\parallel }_{{\dot{K}}_q^{\alpha ,p}}<\mathrm{\infty }\},$$

where

$${\parallel f\parallel }_{{\dot{K}}_q^{\alpha ,p}}={\left[\sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}{2}^{k\alpha p}{\parallel f{\chi }_k\parallel }_{L^q}^p\right]}^{1/p}.$$

• (2) The nonhomogeneous Herz space is defined by

  $$K_q^{\alpha ,p}\left(R^n\right)=\{f\in L_{\mathrm{loc}}^q\left(R^n\right):{\parallel f\parallel }_{K_q^{\alpha ,p}}<\mathrm{\infty }\},$$

where

$${\parallel f\parallel }_{K_q^{\alpha ,p}}={[\sum _{k=1}^{\mathrm{\infty }}{2}^{k\alpha p}{\parallel f{\chi }_k\parallel }_{L^q}^p+{\parallel f{\chi }_0\parallel }_{L^q}^p]}^{1/p}.$$

Definition 3 Let $\alpha \in R$, $1<q<\mathrm{\infty }$, $0<\alpha <n(1-1/q)$, $b_i\in \mathit{BMO}(R^n)$, $1\le i\le m$. A function a on $R^n$ is called a central $(\alpha ,q,\overrightarrow{b})$-atom (or a central $(\alpha ,q,\overrightarrow{b})$-atom of restrict type) if

• (1) $suppa\subset B=B(0,r)$ (or for some $r\ge 1$),

• (2) ${\parallel a\parallel }_{L^q}\le {|B|}^{-\alpha /n}$,

• (3) ${\int }_{B}a(x)dx={\int }_{B}a(x){\prod }_{l\in \sigma }{b}_l(x)dx=0$ for any $\sigma \in C_j^m$, $1\le j\le m$.

A temperate distribution f is said to belong to $H{\dot{K}}_{q,\overrightarrow{b}}^{\alpha ,p}(R^n)$ (or $H{K}_{q,\overrightarrow{b}}^{\alpha ,p}(R^n)$) if it can be written as $f={\sum }_{j=-\mathrm{\infty }}^{\mathrm{\infty }}{\lambda }_j{a}_j$ (or $f={\sum }_{j=0}^{\mathrm{\infty }}{\lambda }_j{a}_j$), in the Schwartz distribution sense, where $a_j$ is a central $(\alpha ,q,\overrightarrow{b})$-atom (or a central $(\alpha ,q,\overrightarrow{b})$-atom of restrict type) supported on $B(0,2^j)$ and ${\sum }_{-\mathrm{\infty }}^{\mathrm{\infty }}{|{\lambda }_j|}^p<\mathrm{\infty }$ (or ${\sum }_{j=0}^{\mathrm{\infty }}{|{\lambda }_j|}^p<\mathrm{\infty }$). Moreover, ${\parallel f\parallel }_{H{\dot{K}}_{q,\overrightarrow{b}}^{\alpha ,p}}(\text{or }{\parallel f\parallel }_{H{K}_{q,\overrightarrow{b}}^{\alpha ,p}})\approx {({\sum }_j{|{\lambda }_j|}^p)}^{1/p}$.

Definition 4 Let Ω be homogeneous of degree zero on $R^n$ such that ${\omega }_r(\delta )$ is defined

$${\omega }_r(\delta )=\underset{|\rho |<\delta }{sup}{({\int }_{S^{n-1}}{|\mathrm{\Omega }(\rho \acute{x})-\mathrm{\Omega }(\acute{x})|}^{r}d\sigma (\acute{x}))}^{1/r},$$

where $|\rho |={sup}_{\acute{x}\in S^{n-1}}|\rho \acute{x}-\acute{x}|$.

If ${\int }_0^1\frac{{\omega }_r(\delta )}{\delta }d\delta <\mathrm{\infty }$, we say $\mathrm{\Omega }(x)$ satisfied $L^r$-Dini condition.

Definition 5 Let $0<ϵ<n$, $0<\gamma \le 1$ and Ω be homogeneous of degree zero on $R^n$ such that ${\int }_{S^{n-1}}\mathrm{\Omega }(x^{\prime })d\sigma (x^{\prime })=0$. Assume that $\mathrm{\Omega }\in {Lip}_{\gamma }(S^{n-1})$, that is, there exists a constant $M>0$ such that for any $x,y\in S^{n-1}$, $|\mathrm{\Omega }(x)-\mathrm{\Omega }(y)|\le M{|x-y|}^{\gamma }$. The Marcinkiewicz multilinear commutator is defined by

$${\mu }_{ϵ}^{\tilde{b}}(f)(x)={\left({\int }_0^{\mathrm{\infty }}{|F_t^{\tilde{b}}(f)(x)|}^2\frac{dt}{t^3}\right)}^{1/2},$$

where

$$F_t^{\tilde{b}}(f)(x)={\int }_{|x-y|\le t}\frac{\mathrm{\Omega }(x-y)}{{|x-y|}^{n-1-ϵ}}\left[\prod _{j=1}^m(b_j(x)-b_j(y))\right]f(y)dy.$$

Set

$$F_t(f)(x)={\int }_{|x-y|\le t}\frac{\mathrm{\Omega }(x-y)}{{|x-y|}^{n-1-ϵ}}f(y)dy,$$

we also define that

$${\mu }_{ϵ}(f)(x)={\left({\int }_0^{\mathrm{\infty }}{|F_t(f)(x)|}^2\frac{dt}{t^3}\right)}^{1/2},$$

which is the Marcinkiewicz operator (see [9, 13, 14]).

2 Theorems and proofs

We begin with three preliminary lemmas.

Lemma 1 (see [12])

Let $1<r<\mathrm{\infty }$, $b_j\in \mathit{BMO}(R^n)$ for $j=1,\dots ,k$ and $k\in N$. Then we have

$$\frac{1}{|Q|}{\int }_Q\prod _{j=1}^k|b_j(y)-{(b_j)}_Q|dy\le C\prod _{j=1}^k{\parallel b_j\parallel }_{\mathit{BMO}(R^n)}$$

and

$${\left(\frac{1}{|Q|}{\int }_Q\prod _{j=1}^k{|b_j(y)-{(b_j)}_Q|}^{r}dy\right)}^{1/r}\le C\prod _{j=1}^k{\parallel b_j\parallel }_{\mathit{BMO}(R^n)}.$$

Lemma 2 (see [14])

Let $0<ϵ<n$, $1<s<n/ϵ$ and $1/r=1/s-ϵ/n$. Then ${\mu }_{ϵ}^{\overrightarrow{b}}$ is bounded from $L^s(R^n)$ to $L^r(R^n)$.

Lemma 3 (see [15])

Let $0<\mu <n$, $\mathrm{\Omega }(x,z)\in L^{\mathrm{\infty }}(R^n)$ satisfy $L^r(S^{n-1})$ ($r\ge 1$) conditions, that is, there exists a constant $0<a_0<1/2$ such that for $|y_0|<a_{0}R$,

$${\left({\int }_{R<|x|<2R}\right|\frac{\mathrm{\Omega }(x,x-y)}{{|x-y|}^{n-\mu }}-\frac{\mathrm{\Omega }(x,x)}{{|x-y|}^{n-\mu }}{|}^{r}dx)}^{1/r}\le C{R}^{n/r-(n-\mu )}(\frac{|y|}{R}+{\int }_{|y|/2R}^{|y|/R}\frac{{\omega }_r(\delta )}{\delta }d\delta ).$$

Theorem 1 Let $0<ϵ<n$, $n/(n+1/2-ϵ)<q\le 1$, $1/q=1/p-ϵ/n$, $\overrightarrow{b}=(b_1,\dots ,b_m)$, $b_i\in \mathit{BMO}$, $1\le i\le m$. Then ${\mu }_{ϵ}^{\overrightarrow{b}}$ is bounded from $H_{\overrightarrow{b}}^p(R^n)$ to $L^q(R^n)$.

Proof It suffices to show that there exists a constant $C>0$ such that for every $(p,\overrightarrow{b})$ atom a,

$${\parallel {\mu }_{ϵ}^{\overrightarrow{b}}(a)\parallel }_{L^q}\le C.$$

Let a be a $(p,\overrightarrow{b})$ atom supported on a ball $B=B(x_0,2d)$. We write

$${\int }_{R^n}{|{\mu }_{ϵ}^{\overrightarrow{b}}(a)(x)|}^{q}dx={\int }_{|x-x_0|\le 2d}{|{\mu }_{ϵ}^{\overrightarrow{b}}(a)(x)|}^{q}dx+{\int }_{|x-x_0|>2d}{|{\mu }_{ϵ}^{\overrightarrow{b}}(a)(x)|}^{q}dx=I+\mathit{II}.$$

For I, taking $r,s>1$ with $q<s<n/ϵ$ and $1/r=1/s-ϵ/n$, by Hölder’s inequality and the $(L^s,L^r)$-boundedness of ${\mu }_{ϵ}^{\overrightarrow{b}}$, we get

$$I\le C{\parallel {\mu }_{ϵ}^{\overrightarrow{b}}(a)\parallel }_{L^r}^q{|B(x_0,2d)|}^{1-q/r}\le C{\parallel a\parallel }_{L^s}^q{|B|}^{1-q/r}\le C{|B|}^{-q/p+q/s+1-q/r}\le C.$$

For II, denoting $\lambda =({\lambda }_1,\dots ,{\lambda }_m)$ with ${\lambda }_i={(b_i)}_B$, $1\le i\le m$, where ${(b_i)}_B={|B(x_0,2d)|}^{-1}\times {\int }_{B(x_0,2d)}{b}_i(x)dx$, by Hölder’s inequality and the vanishing moment of a, we get

$$\begin{array}{rcl}\mathit{II}& \le & {\left[{\int }_0^{|x-x_0|+2d}\right|{\int }_{|x-y|<t}\prod _{j=1}^m(b_j(x)-b_j(y))a(y)\frac{\mathrm{\Omega }(x,x-y)}{{|x-y|}^{n-1-ϵ}}dy{|}^2\frac{dt}{t^3}]}^{1/2}\\ +{\left[{\int }_{|x-x_0|+2d}^{\mathrm{\infty }}\right|{\int }_{|x-y|<t}\prod _{j=1}^m(b_j(x)-b_j(y))a(y)\frac{\mathrm{\Omega }(x,x-y)}{{|x-y|}^{n-1-ϵ}}dy{|}^2\frac{dt}{t^3}]}^{1/2}\\ =& {\mathit{II}}_1+{\mathit{II}}_2.\end{array}$$

Note that $|x-y|\sim |x-x_0|\sim |x-x_0|+2d$ for $|x-x_0|>2d$, $y\in B$. For $1/t+1/r=1$, we have

$$\begin{array}{rcl}{\mathit{II}}_1& \le & C{\int }_{R^n}{\left({\int }_{|x-y|}^{|x-x_0|+2d}\frac{dt}{t^3}\right)}^{1/2}\prod _{j=1}^m|b_j(x)-b_j(y)||a(y)|\frac{|\mathrm{\Omega }(x,x-y)|}{{|x-y|}^{n-1-ϵ}}dy\\ \le & C{\int }_{R^n}|\frac{1}{{|x-y|}^2}-\frac{1}{{(|x-x_0|+2d)}^2}{|}^{1/2}\prod _{j=1}^m|b_j(x)-b_j(y)||a(y)|\frac{|\mathrm{\Omega }(x,x-y)|}{{|x-y|}^{n-1-ϵ}}dy\\ \le & C{\int }_{R^n}\prod _{j=1}^m|b_j(x)-b_j(y)||a(y)|\frac{|\mathrm{\Omega }(x,x-y)|}{{|x-y|}^{n-1-\epsilon }}\frac{{|y-x_0|}^{1/2}}{{|x-x_0|}^{3/2}}dy\\ \le & C{\int }_{R^n}\prod _{j=1}^m|b_j(x)-b_j(y)||a(y)|\frac{|\mathrm{\Omega }(x,x-y)|}{{|x-x_0|}^{n+1/2-ϵ}}{|y-x_0|}^{1/2}dy\\ \le & C\sum _{j=1}^{m-1}\sum _{\sigma \in C_j^m}\frac{1}{{|x-x_0|}^{n+1/2-ϵ}}\left({\int }_B\left|{(\overrightarrow{b}(y)-\lambda )}_{{\sigma }^c}\right||a(y)||\mathrm{\Omega }(x,x-y)|{|y-x_0|}^{1/2}dy\right)\\ \times \left|{(\overrightarrow{b}(x)-\lambda )}_{\sigma }\right|\\ \le & C\sum _{j=1}^{m-1}\sum _{\sigma \in C_j^m}\frac{1}{{|x-x_0|}^{n+1/2-ϵ}}{\left({\int }_B{\left(\left|{(\overrightarrow{b}(y)-\lambda )}_{{\sigma }^c}\right||a(y)|{|y-x_0|}^{1/2}\right)}^{t}dy\right)}^{1/t}\\ \times {\left({\int }_B{\left(|\mathrm{\Omega }(x,x-y)|\right)}^{r}dy\right)}^{1/r}\left|{(\overrightarrow{b}(x)-\lambda )}_{\sigma }\right|\\ \le & C\sum _{j=1}^{m-1}\sum _{\sigma \in C_j^m}\frac{d^{n(1-1/p+1/t+1/r+1/2n)}}{{|x-x_0|}^{n+1/2-ϵ}}{\parallel {\overrightarrow{b}}_{{\sigma }^c}\parallel }_{\mathit{BMO}}\cdot {\parallel \mathrm{\Omega }\parallel }_{L^{\mathrm{\infty }}\times L^r}\left|{(\overrightarrow{b}(x)-\lambda )}_{\sigma }\right|\\ \le & C\sum _{j=1}^{m-1}\sum _{\sigma \in C_j^m}\frac{d^{n(-1/p+1/t+1/r+1/2n)}}{{|x-x_0|}^{n+1/2-ϵ}}{\parallel {\overrightarrow{b}}_{{\sigma }^c}\parallel }_{\mathit{BMO}}\left|{(\overrightarrow{b}(x)-\lambda )}_{\sigma }\right|,\end{array}$$

so we have

For ${\mathit{II}}_2$, we can obtain

$$\begin{array}{rcl}{\mathit{II}}_2& =& {\int }_{R^n}{\left({\int }_{|x-x_0|+2d}^{\mathrm{\infty }}\frac{dt}{t^3}\right)}^{1/2}|\prod _{j=1}^m(b_j(x)-b_j(y))a(y)\frac{\mathrm{\Omega }(x,x-y)}{{|x-y|}^{n-1-ϵ}}dy|\\ \le & C|{\int }_{R^n}\prod _{j=1}^m(b_j(x)-b_j(y))a(y)|\frac{\mathrm{\Omega }(x,x-y)}{{|x-y|}^{n-1-ϵ}}-\frac{\mathrm{\Omega }(x,x)}{{|x|}^{n-1-ϵ}}\left|dy\right|\frac{1}{|x-x_0|+2d}\\ \le & C{\int }_{R^n}\prod _{j=1}^m|b_j(x)-b_j(y)||a(y)||\frac{\mathrm{\Omega }(x,x-y)}{{|x-y|}^{n-ϵ}}-\frac{\mathrm{\Omega }(x,x)}{{|x|}^{n-ϵ}}|dy\\ \le & C\sum _{j=1}^{m-1}\sum _{\sigma \in C_j^m}\left|{(\overrightarrow{b}(x)-\lambda )}_{\sigma }\right|{\int }_B\left|{(\overrightarrow{b}(y)-\lambda )}_{{\sigma }^c}\right||a(y)||\frac{\mathrm{\Omega }(x,x-y)}{{|x-y|}^{n-ϵ}}-\frac{\mathrm{\Omega }(x,x)}{{|x|}^{n-ϵ}}|dy.\end{array}$$

Thus, by Lemma 3, we have

This finishes the proof of Theorem 1. □

Theorem 2 Let $0<ϵ<n$, $0<p<\mathrm{\infty }$, $1<q_1$, $q_2<\mathrm{\infty }$, $1/q_1-1/q_2=ϵ/n$, $n(1-1/q_1)\le \alpha <n(1-1/q_1)+1/2+ϵ$ and $b_i\in \mathit{BMO}(R^n)$, $1\le i\le m$, $\overrightarrow{b}=(b_1,\dots ,b_m)$. Then ${\mu }_{ϵ}^{\overrightarrow{b}}$ is bounded from $H{\dot{K}}_{q_1,\overrightarrow{b}}^{\alpha ,p}(R^n)$ to ${\dot{K}}_{q_2}^{\alpha ,p}(R^n)$.

Proof Let $f\in H{\dot{K}}_{q,\overrightarrow{b}}^{\alpha ,p}(R^n)$ and $f(x)={\sum }_{j=-\mathrm{\infty }}^{\mathrm{\infty }}{\lambda }_j{a}_j(x)$ be the atomic decomposition for f as in Definition 3. We write

For JJ, by the $(L^{q_1},L^{q_2})$-boundedness of ${\mu }_{ϵ}^{\overrightarrow{b}}$, we get

$$\begin{array}{rcl}\mathit{JJ}& \le & C\sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}{2}^{k\alpha p}{(\sum _{j=k-2}^{\mathrm{\infty }}|{\lambda }_j|{\parallel a_j\parallel }_{L^{q_1}})}^p\\ \le & C\sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}{2}^{k\alpha p}{(\sum _{j=k-2}^{\mathrm{\infty }}|{\lambda }_j|2^{-j\alpha })}^p\\ \le & \{\begin{array}{cc}C{\sum }_{k=-\mathrm{\infty }}^{\mathrm{\infty }}{2}^{k\alpha p}({\sum }_{j=k-2}^{\mathrm{\infty }}{|{\lambda }_j|}^p{2}^{-j\alpha p}),\hfill & 0<p\le 1,\hfill \\ C{\sum }_{k=-\mathrm{\infty }}^{\mathrm{\infty }}{2}^{k\alpha p}({\sum }_{j=k-2}^{\mathrm{\infty }}{|{\lambda }_j|}^p{2}^{-j\alpha p/2}){({\sum }_{j=k-2}^{\mathrm{\infty }}{2}^{-j\alpha p^{\prime }/2})}^{p/p^{\prime }},\hfill & p>1,\hfill \end{array}\\ \le & \{\begin{array}{cc}C{\sum }_{j=-\mathrm{\infty }}^{\mathrm{\infty }}{|{\lambda }_j|}^p({\sum }_{k=-\mathrm{\infty }}^{j+2}{2}^{(k-j)\alpha p}),\hfill & 0<p\le 1,\hfill \\ C{\sum }_{j=-\mathrm{\infty }}^{\mathrm{\infty }}{|{\lambda }_j|}^p({\sum }_{k=-\mathrm{\infty }}^{j+2}{2}^{(k-j)\alpha p/2}){({\sum }_{k=-\mathrm{\infty }}^{j+2}{2}^{(k-j)\alpha p^{\prime }/2})}^{p/p^{\prime }},\hfill & p>1,\hfill \end{array}\\ \le & C\sum _{j=-\mathrm{\infty }}^{\mathrm{\infty }}{|{\lambda }_j|}^p\le C{\parallel f\parallel }_{H{\dot{K}}_{q_1,\overrightarrow{b}}^{\alpha ,p}}^p.\end{array}$$

For J, let $x\in B_k\setminus B_{k-1}$, $b_j^i={|B_j|}^{-1}{\int }_{B_j}{b}_i(x)dx$, $1\le i\le m$, $\overrightarrow{b^{\prime }}=(b_j^1,\dots ,b_j^m)$, we have

$$\begin{array}{rcl}{\mu }_{ϵ}^{\overrightarrow{b}}(a_j)(x)& =& {\left({\int }_0^{\mathrm{\infty }}\right|{\int }_{|x-y|<t}\prod _{i=1}^m(b_i(x)-b_i(y))\frac{\mathrm{\Omega }(x,x-y)}{{|x-y|}^{n-1-ϵ}}{a}_j(y)dy{|}^2\frac{dt}{t^3})}^{1/2}\\ =& {\left({\int }_0^{|x|+2^j}\right|{\int }_{|x-y|<t}\prod _{i=1}^m(b_i(x)-b_i(y))\frac{\mathrm{\Omega }(x,x-y)}{{|x-y|}^{n-1-ϵ}}{a}_j(y)dy{|}^2\frac{dt}{t^3})}^{1/2}\\ +{\left({\int }_{|x|+2^j}^{\mathrm{\infty }}\right|{\int }_{|x-y|<t}\prod _{i=1}^m(b_i(x)-b_i(y))\frac{\mathrm{\Omega }(x,x-y)}{{|x-y|}^{n-1-ϵ}}{a}_j(y)dy{|}^2\frac{dt}{t^3})}^{1/2}\\ =& G+H.\end{array}$$

For G, noting that $y\in B_j$, $x\in B(0,2^k)\setminus B(0,2^{k-1})$, $j\le k-3$, we know $|x-y|\sim |x|\sim |x|+2^j$. Then, similar to the proof of Theorem 1, we obtain

$$\begin{array}{rcl}G& \le & C{\int }_{B_j}|{\int }_{|x-y|}^{|x|+2^j}\frac{dt}{t^3}{|}^{1/2}\prod _{i=1}^m|b_i(x)-b_i(y)|\frac{|\mathrm{\Omega }(x,x-y)||a_j(y)|}{{|x-y|}^{n-1-ϵ}}dy\\ \le & {\int }_{B_j}|\frac{1}{{|x-y|}^2}-\frac{1}{{(|x|+2^j)}^2}{|}^{1/2}\prod _{i=1}^m|b_i(x)-b_i(y)|\frac{|\mathrm{\Omega }(x,x-y)||a_j(y)|}{{|x-y|}^{n-1-ϵ}}dy\\ \le & C{2}^{j(1/2+ϵ)}{\int }_{B_j}\frac{1}{{|x|}^{n+1/2}}\prod _{i=1}^m|b_i(x)-b_i(y)||\mathrm{\Omega }(x,x-y)||a_j(y)|dy\\ \le & C\frac{2^{j(1/2+ϵ)}}{{|x|}^{n+1/2}}\sum _{i=0}^m\sum _{\sigma \in C_i^m}\left|{(\overrightarrow{b}(x)-\overrightarrow{b^{\prime }})}_{\sigma }\right|{\int }_{B_j}|\mathrm{\Omega }(x,x-y)||a_j(y)|\left|{(\overrightarrow{b}(y)-\overrightarrow{b})}_{{\sigma }^c}\right|dy\\ \le & C\frac{2^{j(1/2+ϵ)}}{{|x|}^{n+1/2}}\sum _{j=1}^{m-1}\sum _{\sigma \in C_j^m}{\left({\int }_{B_j}{\left(\left|{(\overrightarrow{b}(y)-\overrightarrow{b})}_{{\sigma }^c}\right||a_j(y)|\right)}^{t}dy\right)}^{1/t}\\ \times {\left({\int }_B{\left(|\mathrm{\Omega }(x,x-y)|\right)}^{r}dy\right)}^{1/r}\left|{(\overrightarrow{b}(x)-\overrightarrow{b^{\prime }})}_{\sigma }\right|\\ \le & C\frac{2^{j(1/2+ϵ+n(1/t+1/r-1/q_1-\alpha ))}}{{|x|}^{n+1/2}}\sum _{i=0}^m\sum _{\sigma \in C_i^m}{\parallel {\overrightarrow{b}}_{{\sigma }^c}\parallel }_{\mathit{BMO}}\left|{(\overrightarrow{b}(x)-\overrightarrow{b^{\prime }})}_{\sigma }\right|,\\ H& \le & {\left({\int }_{|x|+2^j}^{\mathrm{\infty }}\right|{\int }_{|x-y|<t}\prod _{i=1}^m(b_i(x)-b_i(y))|\frac{\mathrm{\Omega }(x-y,x)}{{|x-y|}^{n-1-ϵ}}-\frac{\mathrm{\Omega }(x,x)}{{|x|}^{n-1-ϵ}}|a_j(y)dy{|}^2\frac{dt}{t^3})}^{1/2}\\ \le & {\int }_{B_j}\frac{1}{|x|+2^j}|\frac{\mathrm{\Omega }(x-y,x)}{{|x-y|}^{n-1-ϵ}}-\frac{\mathrm{\Omega }(x,x)}{{|x|}^{n-1-ϵ}}|\prod _{i=1}^m|b_i(x)-b_i(y)||a_j(y)|dy\\ \le & C\frac{1}{|x|+2^j}\sum _{i=0}^m\sum _{\sigma \in C_i^m}\left|{(\overrightarrow{b}(x)-\overrightarrow{b^{\prime }})}_{\sigma }\right|{\int }_{B_j}|\frac{\mathrm{\Omega }(x-y,x)}{{|x-y|}^{n-1-ϵ}}-\frac{\mathrm{\Omega }(x,x)}{{|x|}^{n-1-ϵ}}||a_j(y)|\left|{(\overrightarrow{b}(y)-\overrightarrow{b})}_{{\sigma }^c}\right|dy\\ \le & C\frac{1}{|x|+2^j}\sum _{i=0}^m\sum _{\sigma \in C_i^m}\left|{(\overrightarrow{b}(x)-\overrightarrow{b^{\prime }})}_{\sigma }\right|{\left({\int }_{B_j}\right|\frac{\mathrm{\Omega }(x-y,x)}{{|x-y|}^{n-1-ϵ}}-\frac{\mathrm{\Omega }(x,x)}{{|x|}^{n-1-ϵ}}{|}^{r}dy)}^{1/r}\\ \times {\left({\int }_{B_j}{\left(|a_j(y)|\right|{(\overrightarrow{b}(y)-\overrightarrow{b})}_{{\sigma }^c})}^t\right|dy)}^{1/t}\\ \le & C\frac{1}{|x|+2^j}\sum _{i=0}^m\sum _{\sigma \in C_i^m}{\parallel a\parallel }_{L^{q_1}}{|B_j|}^{1/t-1/q_1}{\parallel {\overrightarrow{b}}_{{\sigma }^c}\parallel }_{\mathit{BMO}}\left|{(\overrightarrow{b}(x)-\overrightarrow{b^{\prime }})}_{\sigma }\right|(\frac{1}{2^j}+{\int }_{B_j}\frac{{\omega }_r(\delta )}{\delta }d\delta )\\ \le & C\frac{2^{jn(-1/q_1+1/t-\alpha )}}{|x|+2^j}\sum _{i=0}^m\sum _{\sigma \in C_i^m}{\parallel {\overrightarrow{b}}_{{\sigma }^c}\parallel }_{\mathit{BMO}}\left|{(\overrightarrow{b}(x)-\overrightarrow{b^{\prime }})}_{\sigma }\right|,\end{array}$$

thus

For the sake of simplicity, we denote

then

$${\parallel {\mu }_{ϵ}^{\overrightarrow{b}}(a_j){\chi }_k\parallel }_{L^{q_2}}\le C{\parallel \overrightarrow{b}\parallel }_{\mathit{BMO}}W(j,k),$$

we obtain

$$\begin{array}{rcl}J& \le & C{\parallel \overrightarrow{b}\parallel }_{\mathit{BMO}}^p\sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}{2}^{k\alpha p}{(\sum _{j=-\mathrm{\infty }}^{k-3}|{\lambda }_j|W(j,k))}^p\\ \le & \{\begin{array}{cc}C{\parallel \overrightarrow{b}\parallel }_{\mathit{BMO}}^p{\sum }_{j=-\mathrm{\infty }}^{\mathrm{\infty }}{|{\lambda }_j|}^p{\sum }_{k=j+3}^{\mathrm{\infty }}W{(j,k)}^p,\hfill & 0<p\le 1,\hfill \\ C{\parallel \overrightarrow{b}\parallel }_{\mathit{BMO}}^p{\sum }_{j=-\mathrm{\infty }}^{\mathrm{\infty }}{|{\lambda }_j|}^p[{\sum }_{k=j+3}^{\mathrm{\infty }}W{(j,k)}^{p/2}]{[{\sum }_{k=j+3}^{\mathrm{\infty }}W{(j,k)}^{p^{\prime }/2}]}^{p/p^{\prime }},\hfill & p>1\hfill \end{array}\\ \le & C{\parallel \overrightarrow{b}\parallel }_{\mathit{BMO}}^p\sum _{j=-\mathrm{\infty }}^{\mathrm{\infty }}{|{\lambda }_j|}^p\le C{\parallel \overrightarrow{b}\parallel }_{\mathit{BMO}}^p{\parallel f\parallel }_{H{\dot{K}}_{q_1,\overrightarrow{b}}^{\alpha ,p}}^p.\end{array}$$

This completes the proof of Theorem 2. □

Remark Theorem 2 also holds for nonhomogeneous Herz-type spaces.