Lobachevskii Journal of Mathematics http://ljm.ksu.ru ISSN 1818-9962 Vol. 22, 2006, 7–18

© Changhong Wu and Lanzhe Liu

Abstract. In this paper, the $\left(H_{\overrightarrow{b}}^p,L^p\right)$ and $\left(H{\dot{K}}_{q,\overrightarrow{b}}^{\alpha ,p},{\dot{K}}_q^{\alpha ,p}\right)$ type boundedness for the multilinear commutator associated with the Littlewood-Paley operator are obtained.

1. Introduction and definition

Let $L_{loc}^q\left(R^n\right)=\left\{f^q\text{is locally integrable on}{R}^n\right\}$. Suppose $f\in L_{loc}^1\left(R^n\right)$, $B=B\left(x_0,r\right)=\left\{x\in R^n:\mid x-x_0\mid <r\right\}$ denotes a ball of $R^n$ centered at $x_0$ and having radius $r$, write $f_B=\mid B{\mid }^{-1}{\int <!--nolimits-->}_{B}f\left(x\right)dx$ and $f^{\#}\left(x\right)={sup}_{x\in B}\mid B{\mid }^{-1}{\int <!--nolimits-->}_B\mid f\left(x\right)-f_B\mid dx<\infty$. The function $f$ is said to belong to $BMO\left(R^n\right)$ if $f^{\#}\in L^{\infty }\left(R^n\right)$, and define $\mid \mid f\mid {\mid }_{BMO}=\mid \mid f^{\#}\mid {\mid }_{L^{\infty }}.$

Let $T$ be a linear operator and $K$ be a function on $R^n\times R^n$,

where $K$ satisfies:

(1) $\mid K\left(x,y+h\right)-K\left(x,y\right)\mid \le C\cdot \mid h{\mid }^{\alpha }\cdot \mid x-y{\mid }^{-n-\alpha }$ for $2\mid h\mid <\mid x-y\mid$, $0<\alpha \le 1$;

(2) $\mid \mid T\left(f\right)\mid {\mid }_{L^{p_0}}\le C\mid \mid f\mid {\mid }_{L^{p_0}}$ for some $1<p_0\le \infty$;

Then we call $T$ is the Calderón-Zygmund singular integral operator.

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

A classical result of Coifman, Rochberg and Weiss (see[2]) proved that the commutator $\left[b,T\right]$ is bounded on $L^p\left(R^n\right)\left(1<p<\infty \right)$. However, it was observed that the $\left[b,T\right]$ is not bounded, in general, from $H^p\left(R^n\right)$ to $L^p\left(R^n\right)$. But if $H^p\left(R^n\right)$ is replaced by a suitable atomic space $H_{\overrightarrow{b}}^p\left(R^n\right)$ and $H{\dot{K}}_{q,\overrightarrow{b}}^{\alpha ,p}\left(R^n\right)$(see[1][7][12]), then $\left[b,T\right]$ maps continuously $H_{\overrightarrow{b}}^p\left(R^n\right)$ into $L^p\left(R^n\right)$ and $H{\dot{K}}_{q,\overrightarrow{b}}^{\alpha ,p}\left(R^n\right)$ into ${\dot{K}}_q^{\alpha ,p}$. In addition we have easily known that $H_{\overrightarrow{b}}^p\left(R^n\right)\subset H^p\left(R^n\right),{\dot{K}}_{q,\overrightarrow{b}}^{\alpha ,p}\left(R^n\right)\subset H{\dot{K}}_q^{\alpha ,p}\left(R^n\right)$. The main purpose of this paper is to consider the continuity of the multilinear commutators related to the Littlewood-Paley operators and $BMO\left(R^n\right)$ functions on certain Hardy and Herz-Hardy spaces. Let us first introduce some definitions(see [1][3-10][12][13]).

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

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

(1)   $supp$ $a\subset B=B\left(x_0,r\right)$

(2)   $\mid \mid a\mid {\mid }_{L^{\infty }}\le \mid B\left(x_0,r\right){\mid }^{-1∕p}$

(3)   ${\int \; <!--nolimits-->}_{B}a\left(y\right)dy={\int \; <!--nolimits-->}_{B}a\left(y\right){\prod }_{l\in \sigma }{b}_l\left(y\right)dy=0\text{for any}\sigma \in C_j^m,1\le j\le m$.

A temperate distribution(see[14][15]) $f$ is said to belong to $H_{\overrightarrow{b}}^p\left(R^n\right)$, if, in the Schwartz distribution sense, it can be written as

where $a_j^{\prime }s$ are $\left(p,\overrightarrow{b}\right)$ atoms, ${\lambda }_j\in C$ and ${\sum }_{j=1}^{\infty }\mid {\lambda }_j{\mid }^p<\infty$. Moreover, $\mid \mid f\mid {\mid }_{H_{\overrightarrow{b}}^p}\approx {\left({\sum }_{j=1}^{\infty }\mid {\lambda }_j{\mid }^p\right)}^{1∕p}$.

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

(1) The homogeneous Herz space is defined by

where

(2) The nonhomogeneous Herz space is defined by

where

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

(1)   $supp$ $a\subset B=B\left(x_0,r\right)$(or for some $r\ge 1$),

(2)   $\mid \mid a\mid {\mid }_{L^q}\le \mid B\left(x_0,r\right){\mid }^{-\alpha ∕n}$,

(3)   ${\int \; <!--nolimits-->}_{B}a\left(x\right)x^{\beta }dx={\int <!--nolimits-->}_{B}a\left(x\right)x^{\beta }{\prod }_{i\in \sigma }{b}_i\left(x\right)dx=0\text{for any}\sigma \in C_j^m,1\le j\le m,$ $0\le \mid \beta \mid \le \alpha ,$ where $\beta =\left({\beta }_1,...,{\beta }_n\right)$ is the multi-indices with ${\beta }_i\in N$ for $1\le i\le n$ and $\mid \beta \mid ={\sum }_{i=1}^n{\beta }_i$.

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

where the infimum are taken over all the decompositions of $f$ as above.

Definition 4. Let $\varepsilon >0$ and $\psi$ be a fixed function which satisfies the following properties:

(1)   ${\int \; <!--nolimits-->}_{R^n}\psi \left(x\right)dx=0$,

(2)   $\mid \psi \left(x\right)\mid \le C{\left(1+\mid x\mid \right)}^{-\left(n+1\right)}$,

(3)   $\mid \psi \left(x+y\right)-\psi \left(x\right)\mid \le C\mid y{\mid }^{\varepsilon }{\left(1+\mid x\mid \right)}^{-\left(n+1+\varepsilon \right)}$ when $2\mid y\mid <\mid x\mid$;

The Littlewood-Paley multilinear commutator is defined by

where

$\Gamma \left(x\right)=\left\{\left(y,t\right)\in R_{+}^{n+1}:\mid x-y\mid <t\right\}$ and ${\psi }_t\left(x\right)=t^{-n}\psi \left(x∕t\right)$ for $t>0$. Set $F_t\left(f\right)\left(y\right)={\int \; <!--nolimits-->}_{R^n}{\psi }_t\left(y-x\right)f\left(x\right)dx$. We also define that

which is the Littlewood-Paley operator(see [15]).

2.Theorems and Proofs

Theorem 1. Let $\varepsilon >0$,$b_i\in BMO,1\le i\le m,\overrightarrow{b}=\left(b_1,\cdots ,b_m\right),n∕\left(n+\varepsilon \right)<p\le 1$. Then the multilinear commutator $S_{\psi }^{\overrightarrow{b}}$ is bounded from $H_{\overrightarrow{b}}^p\left(R^n\right)$ to $L^p\left(R^n\right).$

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

Let $a$ be a $\left(p,\overrightarrow{b}\right)$ atom supported on a ball $B=B\left(x_0,r\right)$. When $m=1$ see [7], and now we prove $m>1$. Write

For $I$, taking $q>1$, by Hölder’s inequality and the $L^q$-boundedness of $S_{\psi }^{\overrightarrow{b}},$ we have

For $II$, denoting $\lambda =\left({\lambda }_1,\cdots ,{\lambda }_m\right)$ with ${\lambda }_i={\left(b_i\right)}_B,1\le i\le m$, where ${\left(b_i\right)}_B=\mid B\left(x_0,r\right){\mid }^{-1}{\int <!--nolimits-->}_{B\left(x_0,r\right)}{b}_i\left(x\right)dx$, by Hölder’s inequality and the vanishing moment of $a$, we get

noting that $z\in B,x\in B\left(x_0,2^{k+1}r\right)\setminus B\left(x_0,2^{k}r\right)$, then

Notice that $2t+\mid x_0-y\mid >2t+\mid x_0-x\mid -\mid x-y\mid >t+\mid x_0-x\mid$ when $\mid x-y\mid <t$, and it is easy to calculate that

then, we deduce

So

This finishes the proof of Theorem 1.

Theorem 2. Let $0<p<\infty$, $1<q<\infty ,n\left(1-1∕q\right)\le \alpha <n\left(1-1∕q\right)+\varepsilon$, $\varepsilon >0$,and $b_i\in BMO\left(R^n\right),1\le i\le m$, $\overrightarrow{b}=\left(b_1,\cdots ,b_m\right).$ Then $S_{\psi }^{\overrightarrow{b}}$ is bounded from $H{\dot{K}}_{q,\overrightarrow{b}}^{\alpha ,p}\left(R^n\right)$ to ${\dot{K}}_q^{\alpha ,p}\left(R^n\right).$

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

For $II$, by the boundedness of $S_{\psi }^{\overrightarrow{b}}$ on $L^q$ and the Hölder’s inequality, we have

For $I$, when m=1, let $b_j^1=\mid B_j{\mid }^{-1}{\int <!--nolimits-->}_{B_j}{b}_1\left(x\right)dx.$ We have