# Endpoint estimates for vector-valued multilinear commutator of fractional area integral operator

## Abstract

In this paper, we prove the endpoint estimates for vector-valued multilinear commutator of fractional area integral operator.

MSC:42B20, 42B25.

## 1 Introduction

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

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

A classical result of Coifman, Rochberb and Weiss (see ) proved that the commutator $\left[b,T\right]$ is bounded on ${L}^{p}\left({R}^{n}\right)$ ($1). In , the boundedness properties of the commutators for the extreme values of p are obtained. In this paper, we introduce vector-valued multilinear commutator of fractional area integral operator and prove the endpoint estimates for the commutator ${|{S}_{\psi ,\delta }^{\stackrel{\to }{b}}|}_{r}$ generated by the fractional area integral operator ${S}_{\psi ,\delta }$ and BMO functions.

## 2 Notations and results

We give the following definitions (see [2, 3, 57]).

Definition 1 Let $0<\delta , a function ψ satisfies:

1. (1)

${\int }_{{R}^{n}}\psi \left(x\right)\phantom{\rule{0.2em}{0ex}}dx=0$;

2. (2)

$|\psi \left(x\right)|\le C{\left(1+|x|\right)}^{-\left(n+1-\delta \right)}$;

3. (3)

$|\psi \left(x+y\right)-\psi \left(x\right)|\le C{|y|}^{\epsilon }{\left(1+|x|\right)}^{-\left(n+2-\delta \right)}$, $2|y|<|x|$.

Suppose that $1, ${b}_{j}$ ($j=1,\dots ,m$) are the fixed locally integrable functions on ${R}^{n}$. Set $\mathrm{\Gamma }\left(x\right)=\left\{\left(y,t\right)\in {R}_{+}^{n+1}:|x-y|\le t\right\}$ and the eigenfunction by ${\chi }_{\mathrm{\Gamma }\left(x\right)}$. We define the vector-valued multilinear commutator of fractional area integral operator by

$|{S}_{\psi ,\delta }^{\stackrel{\to }{b}}\left(f\right)\left(x\right){|}_{r}={\left(\sum _{i=1}^{\mathrm{\infty }}{\left({S}_{\psi ,\delta }^{\stackrel{\to }{b}}\left({f}_{i}\right)\left(x\right)\right)}^{r}\right)}^{1/r},$

where

${S}_{\psi ,\delta }^{\stackrel{\to }{b}}\left(f\right)\left(x\right)={\left({\int }_{\mathrm{\Gamma }\left(x\right)}|{F}_{t}^{\stackrel{\to }{b}}\left(f\right)\left(x,y\right){|}^{2}\frac{dy\phantom{\rule{0.2em}{0ex}}dt}{{t}^{n+1}}\right)}^{1/2}$

and

${F}_{t}^{\stackrel{˜}{b}}\left(f\right)\left(x\right)={\int }_{{R}^{n}}\left[\prod _{j=1}^{m}\left({b}_{j}\left(x\right)-{b}_{j}\left(z\right)\right)\right]{\psi }_{t}\left(y-z\right)f\left(z\right)\phantom{\rule{0.2em}{0ex}}dz.$

Definition 2 We call a locally integrable function b in the central BMO space, namely $\mathit{CMO}\left({R}^{n}\right)$, if the function b satisfies

${\parallel b\parallel }_{\mathit{CMO}}=\underset{r>1}{sup}|Q\left(0,r\right){|}^{-1}{\int }_{Q}|b\left(y\right)-{b}_{Q}|\phantom{\rule{0.2em}{0ex}}dy<\mathrm{\infty }.$

We have

${\parallel b\parallel }_{\mathit{CMO}}\approx \underset{r>1}{sup}\underset{c\in C}{inf}|Q\left(0,r\right){|}^{-1}{\int }_{Q}|b\left(y\right)-c|\phantom{\rule{0.2em}{0ex}}dy.$

Definition 3 Let $0<\delta , $1. We call a locally integrable function b in ${B}_{p}^{\delta }\left({R}^{n}\right)$, if the function b satisfies

${\parallel b\parallel }_{{B}_{p}^{\delta }}=\underset{r>1}{sup}{r}^{-n\left(1/p-\delta /n\right)}{\parallel b{\chi }_{Q\left(0,r\right)}\parallel }_{{L}^{p}}<\mathrm{\infty }.$

Now we state our theorems as follows.

Theorem 1 Suppose $1, $0<\delta , and $\stackrel{\to }{b}=\left({b}_{1},\dots ,{b}_{m}\right)$ for ${b}_{j}\in \mathit{BMO}$, $1\le j\le m$. Then $|{S}_{\psi ,\delta }^{\stackrel{\to }{b}}{|}_{r}$ is bounded from ${L}^{n/\delta }$ to $\mathit{BMO}\left({R}^{n}\right)$.

Theorem 2 Let $1, $0<\delta , $1, and $\stackrel{\to }{b}=\left({b}_{1},\dots ,{b}_{m}\right)$, with ${b}_{j}\in \mathit{BMO}\left({R}^{n}\right)$, for $1\le j\le m$. Then $|{S}_{\psi ,\delta }^{\stackrel{\to }{b}}{|}_{r}$ is bounded from ${B}_{p}^{\delta }\left({R}^{n}\right)$ to $\mathit{CMO}\left({R}^{n}\right)$.

## 3 Proofs of theorems

We begin with a preliminaries lemma.

Lemma 1 (see [3, 4])

Let $1, $0<\delta , $1, $1/q=1/p-\delta /n$. Then $|{S}_{\psi ,\delta }{|}_{r}$ is bounded from ${L}^{p}\left({R}^{n}\right)$ to ${L}^{q}\left({R}^{n}\right)$.

Proof of Theorem 1 It is only to prove that there exists a constant ${C}_{Q}$, the following inequality holds:

$\frac{1}{|Q|}{\int }_{Q}||{S}_{\psi ,\delta }^{\stackrel{\to }{b}}\left(f\right)\left(x\right){|}_{r}-{C}_{Q}|\phantom{\rule{0.2em}{0ex}}dx\le C{\parallel |f{|}_{r}\parallel }_{{L}^{n/\delta }}.$

Fix a cube $Q=Q\left({x}_{0},r\right)$, let $f=g+h=\left\{{g}_{i}\right\}+\left\{{h}_{i}\right\}$ for ${g}_{i}={f}_{i}{\chi }_{Q}$, ${h}_{i}={f}_{i}{\chi }_{{\left(Q\right)}^{c}}$.

When $m=1$, set ${\left({b}_{1}\right)}_{Q}={|Q|}^{-1}{\int }_{Q}{b}_{1}\left(y\right)\phantom{\rule{0.2em}{0ex}}dy$, then

${F}_{t}^{{b}_{1}}\left({f}_{i}\right)\left(x,y\right)=\left({b}_{1}\left(x\right)-{\left({b}_{1}\right)}_{Q}\right){F}_{t}\left({f}_{i}\right)\left(y\right)-{F}_{t}\left(\left({b}_{1}-{\left({b}_{1}\right)}_{Q}\right){g}_{i}\right)\left(y\right)-{F}_{t}\left(\left({b}_{1}-{\left({b}_{1}\right)}_{Q}\right){h}_{i}\right)\left(y\right),$

so

$\begin{array}{r}|{S}_{\psi ,\delta }^{{b}_{1}}\left(f\right)\left(x\right){|}_{r}-|{S}_{\psi ,\delta }\left(\left({\left({b}_{1}\right)}_{2Q}-{b}_{1}\right)h\right){\left({x}_{0}\right)}_{r}|\\ \phantom{\rule{1em}{0ex}}\le {\left(\sum _{i=1}^{\mathrm{\infty }}{\parallel {\chi }_{\mathrm{\Gamma }\left(x\right)}\left({b}_{1}\left(x\right)-{\left({b}_{1}\right)}_{Q}\right){F}_{t}\left({f}_{i}\right)\left(y\right)\parallel }^{r}\right)}^{1/r}\\ \phantom{\rule{2em}{0ex}}+{\left(\sum _{i=1}^{\mathrm{\infty }}{\parallel {\chi }_{\mathrm{\Gamma }\left(x\right)}{F}_{t}\left(\left({\left({b}_{1}\right)}_{Q}-{b}_{1}\right){g}_{i}\right)\left(y\right)\parallel }^{r}\right)}^{1/r}\\ \phantom{\rule{2em}{0ex}}+{\parallel {\chi }_{\mathrm{\Gamma }\left(x\right)}{F}_{t}\left(\left({b}_{1}-{\left({b}_{1}\right)}_{Q}\right){f}_{2}\right)\left(y\right)-{\chi }_{\mathrm{\Gamma }\left({x}_{0}\right)}{F}_{t}\left(\left({b}_{1}-{\left({b}_{1}\right)}_{Q}\right)h\right)\left(y\right)\parallel }_{r}\\ \phantom{\rule{1em}{0ex}}=A\left(x\right)+B\left(x\right)+C\left(x\right).\end{array}$

For $A\left(x\right)$, suppose $1, $1/q=1/p-\delta /n$ and $1/q+1/{q}^{\prime }=1$, by the Hölder inequality, then

$\begin{array}{rcl}\frac{1}{|Q|}{\int }_{Q}|A\left(x\right)|\phantom{\rule{0.2em}{0ex}}dx& =& \frac{1}{|Q|}{\int }_{Q}|{b}_{1}\left(x\right)-{\left({b}_{1}\right)}_{Q}||{S}_{\psi ,\delta }\left(f\right)\left(x\right){|}_{r}\phantom{\rule{0.2em}{0ex}}dx\\ \le & {\left(\frac{1}{|Q|}{\int }_{Q}|{b}_{1}\left(x\right)-{\left({b}_{1}\right)}_{Q}{|}^{{q}^{\prime }}\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/{q}^{\prime }}\\ ×{\left(\frac{1}{|Q|}{\int }_{{R}^{n}}|{S}_{\psi ,\delta }\left(f\right)\left(x\right){|}_{r}^{q}{\chi }_{Q}\left(x\right)\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/q}\\ \le & C{\parallel {b}_{1}\parallel }_{\mathit{BMO}}|Q{|}^{-1/q}{\left({\int }_{{R}^{n}}|f\left(x\right){|}_{r}^{p}{\chi }_{Q}\left(x\right)\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/p}\\ \le & C{\parallel {b}_{1}\parallel }_{\mathit{BMO}}|Q{|}^{-1/q}\\ ×{\left[{\left({\int }_{{R}^{n}}|f\left(x\right){|}_{r}^{n/\delta }\phantom{\rule{0.2em}{0ex}}dx\right)}^{\delta p/n}{\left({\int }_{Q}{\chi }_{Q}\left(x\right)\phantom{\rule{0.2em}{0ex}}dx\right)}^{1-\delta p/n}\right]}^{1/p}\\ \le & C{\parallel {b}_{1}\parallel }_{\mathit{BMO}}|Q{|}^{-1/q}{\parallel |f{|}_{r}\parallel }_{{L}^{n/\delta }}|Q{|}^{\left(1-\delta p/n\right)/p}\\ \le & C{\parallel {b}_{1}\parallel }_{\mathit{BMO}}{\parallel |f{|}_{r}\parallel }_{{L}^{n/\delta }}.\end{array}$

For $B\left(x\right)$, fix $1, $1/v=1/u-\delta /n$, by the Hölder inequality, then

$\begin{array}{r}\frac{1}{|Q|}{\int }_{Q}|B\left(x\right)|\phantom{\rule{0.2em}{0ex}}dx\\ \phantom{\rule{1em}{0ex}}=\frac{1}{|Q|}{\int }_{Q}|{S}_{\psi ,\delta }\left(\left({b}_{1}-{\left({b}_{1}\right)}_{Q}\right)g\right)\left(x\right){|}_{r}\phantom{\rule{0.2em}{0ex}}dx\\ \phantom{\rule{1em}{0ex}}\le {\left(\frac{1}{|Q|}{\int }_{{R}^{n}}|{S}_{\psi ,\delta }{\left(\left({b}_{1}-{\left({b}_{1}\right)}_{Q}\right)g\right)\left(x\right)\right)}^{v}\phantom{\rule{0.2em}{0ex}}dx{|}_{r}\right)}^{1/v}\\ \phantom{\rule{1em}{0ex}}\le C|Q{|}^{-1/v}{\left({\int }_{{R}^{n}}|{b}_{1}\left(x\right)-{\left({b}_{1}\right)}_{Q}{|}^{u}|f\left(x\right){|}_{r}^{u}{\chi }_{Q}\left(x\right)\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/u}\\ \phantom{\rule{1em}{0ex}}\le C{\left(\frac{1}{|Q|}{\int }_{Q}|{b}_{1}\left(x\right)-{\left({b}_{1}\right)}_{Q}{|}^{u}\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/u}{\parallel |f{|}_{r}\parallel }_{{L}^{n/\delta }}\\ \phantom{\rule{1em}{0ex}}\le C{\parallel {b}_{1}\parallel }_{\mathit{BMO}}{\parallel |f{|}_{r}\parallel }_{{L}^{n/\delta }}.\end{array}$

For $C\left(x\right)$, we have

$\begin{array}{rcl}C\left(x\right)& =& {\parallel {\chi }_{\mathrm{\Gamma }\left(x\right)}{F}_{t}\left(\left({b}_{1}-{\left({b}_{1}\right)}_{Q}\right){f}_{2}\right)\left(y\right)-{\chi }_{\mathrm{\Gamma }\left({x}_{0}\right)}{F}_{t}\left(\left({b}_{1}-{\left({b}_{1}\right)}_{Q}\right)h\right)\left(y\right)\parallel }_{r}\\ \le & {\left[\int {\int }_{{R}_{+}^{n+1}}{\left({\int }_{{Q}^{c}}|{\chi }_{\mathrm{\Gamma }\left(x\right)}-{\chi }_{\mathrm{\Gamma }\left({x}_{0}\right)}\parallel {b}_{1}\left(z\right)-{\left({b}_{1}\right)}_{Q}\parallel {\psi }_{t}\left(y-z\right)||f\left(z\right){|}_{r}\phantom{\rule{0.2em}{0ex}}dz\right)}^{2}\frac{dy\phantom{\rule{0.2em}{0ex}}dt}{{t}^{n+1}}\right]}^{1/2}\\ \le & C{\int }_{{Q}^{c}}|{b}_{1}\left(z\right)-{\left({b}_{1}\right)}_{Q}||f\left(z\right){|}_{r}\\ ×|\int {\int }_{|x-y|\le t}\frac{{t}^{1-n}\phantom{\rule{0.2em}{0ex}}dy\phantom{\rule{0.2em}{0ex}}dt}{{\left(t+|y-z|\right)}^{2n+2-2\delta }}-\int {\int }_{|{x}_{0}-y|\le t}\frac{{t}^{1-n}\phantom{\rule{0.2em}{0ex}}dy\phantom{\rule{0.2em}{0ex}}dt}{{\left(t+|y-z|\right)}^{2n+2-2\delta }}{|}^{1/2}\phantom{\rule{0.2em}{0ex}}dz\\ \le & C{\int }_{{Q}^{c}}|{b}_{1}\left(z\right)-{\left({b}_{1}\right)}_{Q}||f\left(z\right){|}_{r}\\ ×\left(\int {\int }_{|y|\le t,|x+y-z|\le t}|\frac{1}{{\left(t+|x+y-z|\right)}^{2n+2-2\delta }}\\ {-\frac{1}{{\left(t+|{x}_{0}+y-z|\right)}^{2n+2-2\delta }}|\frac{dy\phantom{\rule{0.2em}{0ex}}dt}{{t}^{n-1}}\right)}^{1/2}\phantom{\rule{0.2em}{0ex}}dz\\ \le & {\int }_{{Q}^{c}}|{b}_{1}\left(z\right)-{\left({b}_{1}\right)}_{Q}||f\left(z\right){|}_{r}{\left(\int {\int }_{|y|\le t,|x+y-z|\le t}\frac{|x-{x}_{0}|{t}^{1-n}}{{\left(t+|x+y-z|\right)}^{2n+3-2\delta }}\phantom{\rule{0.2em}{0ex}}dy\phantom{\rule{0.2em}{0ex}}dt\right)}^{1/2}\phantom{\rule{0.2em}{0ex}}dz.\end{array}$

Notice that when $|y|\le t$, $2t+|x+y-z|\ge 2t+|x-z|-|y|\ge t+|x-z|$, and

${\int }_{0}^{\mathrm{\infty }}\frac{t\phantom{\rule{0.2em}{0ex}}dt}{{\left(t+|x-z|\right)}^{2n+3-2\delta }}=C|x-z{|}^{-2n-1+2\delta },$

then, for $x\in Q$,

$\begin{array}{rcl}C\left(x\right)& \le & {\int }_{{Q}^{c}}|{b}_{1}\left(z\right)-{\left({b}_{1}\right)}_{Q}||f\left(z\right){|}_{r}{\left(\int {\int }_{|y|\le t}\frac{{2}^{2n+3-2\delta }|{x}_{0}-x|{t}^{1-n}\phantom{\rule{0.2em}{0ex}}dy\phantom{\rule{0.2em}{0ex}}dt}{{\left(2t+2|x+y-z|\right)}^{2n+3-2\delta }}\right)}^{1/2}\phantom{\rule{0.2em}{0ex}}dz\\ \le & C{\int }_{{Q}^{c}}|{b}_{1}\left(z\right)-{\left({b}_{1}\right)}_{Q}||f\left(z\right){|}_{r}|x-{x}_{0}{|}^{1/2}{\left(\int {\int }_{|y|\le t}\frac{{t}^{1-n}\phantom{\rule{0.2em}{0ex}}dy\phantom{\rule{0.2em}{0ex}}dt}{{\left(t+|x-z|\right)}^{2n+3-2\delta }}\right)}^{1/2}\phantom{\rule{0.2em}{0ex}}dz\\ \le & C{\int }_{{Q}^{c}}|{b}_{1}\left(z\right)-{\left({b}_{1}\right)}_{Q}||f\left(z\right){|}_{r}|x-{x}_{0}{|}^{1/2}{\left({\int }_{0}^{\mathrm{\infty }}\frac{t\phantom{\rule{0.2em}{0ex}}dt}{{\left(t+|x-z|\right)}^{2n+3-2\delta }}\right)}^{1/2}\phantom{\rule{0.2em}{0ex}}dz\\ \le & C{\int }_{{Q}^{c}}|{b}_{1}\left(z\right)-{\left({b}_{1}\right)}_{Q}||f\left(z\right){|}_{r}\frac{|{x}_{0}-x{|}^{1/2}}{|{x}_{0}-z{|}^{n+1/2-\delta }}\phantom{\rule{0.2em}{0ex}}dz\\ \le & C\sum _{k=1}^{\mathrm{\infty }}{\int }_{{2}^{k+1}Q\setminus {2}^{k}Q}|{b}_{1}\left(z\right)-{\left({b}_{1}\right)}_{Q}||f\left(z\right){|}_{r}\frac{|{x}_{0}-x{|}^{1/2}}{|{x}_{0}-z{|}^{n+1/2-\delta }}\phantom{\rule{0.2em}{0ex}}dz\\ \le & C\sum _{k=1}^{\mathrm{\infty }}{2}^{-k/2}|{2}^{k+1}Q{|}^{-1+\delta /n}{\int }_{{2}^{k+1}Q}|{b}_{1}\left(z\right)-{\left({b}_{1}\right)}_{Q}||f\left(z\right){|}_{r}\phantom{\rule{0.2em}{0ex}}dz\\ \le & C{\parallel {b}_{1}\parallel }_{\mathit{BMO}}\sum _{k=1}^{\mathrm{\infty }}k{2}^{-k/2}{\parallel |f{|}_{r}\parallel }_{{L}^{n/\delta }}\\ \le & C{\parallel {b}_{1}\parallel }_{\mathit{BMO}}{\parallel |f{|}_{r}\parallel }_{{L}^{n/\delta }},\end{array}$

so that

$\frac{1}{|Q|}{\int }_{Q}|C\left(x\right)|\phantom{\rule{0.2em}{0ex}}dx\le C{\parallel {b}_{1}\parallel }_{\mathit{BMO}}{\parallel |f{|}_{r}\parallel }_{{L}^{n/\delta }}.$

When $m>1$, let ${\stackrel{\to }{b}}_{Q}=\left({\left({b}_{1}\right)}_{Q},\dots ,{\left({b}_{m}\right)}_{Q}\right)\in {R}^{n}$, where

${\left({b}_{j}\right)}_{Q}=|Q{|}^{-1}{\int }_{Q}{b}_{j}\left(y\right)\phantom{\rule{0.2em}{0ex}}dy,\phantom{\rule{1em}{0ex}}1\le j\le m,$

let $f=g+h=\left\{{g}_{i}\right\}+\left\{{h}_{i}\right\}$ for ${g}_{i}={f}_{i}{\chi }_{Q}$, ${h}_{i}={f}_{i}{\chi }_{{\left(Q\right)}^{c}}$. We have

$\begin{array}{r}{F}_{t}^{\stackrel{\to }{b}}\left({f}_{i}\right)\left(x,y\right)\\ \phantom{\rule{1em}{0ex}}={\int }_{{R}^{n}}\left[\prod _{j=1}^{m}\left({b}_{1}\left(x\right)-{b}_{1}\left(z\right)\right)\right]{\psi }_{t}\left(y-z\right){f}_{i}\left(z\right)\phantom{\rule{0.2em}{0ex}}dz\\ \phantom{\rule{1em}{0ex}}=\left({b}_{1}\left(x\right)-{\left({b}_{1}\right)}_{Q}\right)\cdots \left({b}_{m}\left(x\right)-{\left({b}_{m}\right)}_{Q}\right){F}_{t}\left({f}_{i}\right)\left(y\right)\\ \phantom{\rule{2em}{0ex}}+{\left(-1\right)}^{m}{F}_{t}\left(\left({b}_{1}-{\left({b}_{1}\right)}_{Q}\right)\cdots \left({b}_{m}-{\left({b}_{m}\right)}_{Q}\right){f}_{i}\right)\left(y\right)\\ \phantom{\rule{2em}{0ex}}+\sum _{j=1}^{m-1}\sum _{\sigma \in {C}_{j}^{m}}{\left(-1\right)}^{m-j}{\left(\stackrel{\to }{b}\left(x\right)-{\stackrel{\to }{b}}_{Q}\right)}_{\sigma }\\ \phantom{\rule{2em}{0ex}}×{\int }_{{R}^{n}}{\left(\stackrel{\to }{b}\left(z\right)-{\stackrel{\to }{b}}_{Q}\right)}_{{\sigma }^{c}}{\psi }_{t}\left(y-z\right){f}_{i}\left(z\right)\phantom{\rule{0.2em}{0ex}}dz\\ \phantom{\rule{1em}{0ex}}=\left({b}_{1}\left(x\right)-{\left({b}_{1}\right)}_{Q}\right)\cdots \left({b}_{m}\left(x\right)-{\left({b}_{m}\right)}_{Q}\right){F}_{t}\left({f}_{i}\right)\left(y\right)\\ \phantom{\rule{2em}{0ex}}+{\left(-1\right)}^{m}{F}_{t}\left(\left({b}_{1}-{\left({b}_{1}\right)}_{Q}\right)\cdots \left({b}_{m}-{\left({b}_{m}\right)}_{Q}\right){g}_{i}\right)\left(y\right)\\ \phantom{\rule{2em}{0ex}}+{\left(-1\right)}^{m}{F}_{t}\left(\left({b}_{1}-{\left({b}_{1}\right)}_{Q}\right)\cdots \left({b}_{m}-{\left({b}_{m}\right)}_{Q}\right){h}_{i}\right)\left(y\right)\\ \phantom{\rule{2em}{0ex}}+\sum _{j=1}^{m-1}\sum _{\sigma \in {C}_{j}^{m}}{\left(-1\right)}^{m-j}{\left(\stackrel{\to }{b}\left(x\right)-{\stackrel{\to }{b}}_{Q}\right)}_{\sigma }{F}_{t}\left({\left(\stackrel{\to }{b}-{\stackrel{\to }{b}}_{Q}\right)}_{{\sigma }^{c}}{f}_{i}\right)\left(x,y\right),\end{array}$

by the Minkowski inequality, we have

$\begin{array}{r}||{S}_{\psi ,\delta }^{\stackrel{\to }{b}}\left(f\right)\left(x\right){|}_{r}-|{S}_{\psi ,\delta }\left(\left({\left({b}_{1}\right)}_{Q}-{b}_{1}\right)\cdots \left({\left({b}_{m}\right)}_{Q}-{b}_{m}\right)h\right)\left({x}_{0}\right){|}_{r}|\\ \phantom{\rule{1em}{0ex}}\le {\parallel {\chi }_{\mathrm{\Gamma }\left(x\right)}\left({b}_{1}\left(x\right)-{\left({b}_{1}\right)}_{Q}\right)\cdots \left({b}_{m}\left(x\right)-{\left({b}_{m}\right)}_{Q}\right){F}_{t}\left(f\right)\left(y\right)\parallel }_{r}\\ \phantom{\rule{2em}{0ex}}+\sum _{j=1}^{m-1}\sum _{\sigma \in {C}_{j}^{m}}{\parallel {\chi }_{\mathrm{\Gamma }\left(x\right)}{\left(\stackrel{\to }{b}\left(x\right)-{\stackrel{\to }{b}}_{Q}\right)}_{\sigma }{F}_{t}\left({\left(\stackrel{\to }{b}-{\stackrel{\to }{b}}_{Q}\right)}_{{\sigma }^{c}}f\right)\left(x,y\right)\parallel }_{r}\\ \phantom{\rule{2em}{0ex}}+{\parallel {\chi }_{\mathrm{\Gamma }\left(x\right)}{F}_{t}\left(\left({b}_{1}-{\left({b}_{1}\right)}_{Q}\right)\cdots \left({b}_{m}-{\left({b}_{m}\right)}_{Q}\right)g\right)\left(y\right)\parallel }_{r}\\ \phantom{\rule{2em}{0ex}}+{\parallel {\chi }_{\mathrm{\Gamma }\left(x\right)}{F}_{t}\left(\prod _{j=1}^{m}\left({b}_{j}-{\left({b}_{j}\right)}_{Q}\right)h\right)\left(y\right)-{\chi }_{\mathrm{\Gamma }\left({x}_{0}\right)}{F}_{t}\left(\prod _{j=1}^{m}\left({b}_{j}-{\left({b}_{j}\right)}_{Q}\right)h\right)\left(y\right)\parallel }_{r}\\ \phantom{\rule{1em}{0ex}}={M}_{1}\left(x\right)+{M}_{2}\left(x\right)+{M}_{3}\left(x\right)+{M}_{4}\left(x\right).\end{array}$

For ${M}_{1}\left(x\right)$, similar to the proof of $m=1$, we take $1, $1/q=1/p-\delta /n$, by the Hölder inequality and Lemma 1, we have

$\begin{array}{r}\frac{1}{|Q|}{\int }_{Q}{M}_{1}\left(x\right)\phantom{\rule{0.2em}{0ex}}dx\\ \phantom{\rule{1em}{0ex}}\le {\left(\frac{1}{|Q|}{\int }_{Q}|\prod _{j=1}^{m}\left({b}_{j}\left(x\right)-{\left({b}_{j}\right)}_{Q}\right){|}^{{q}^{\prime }}\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/{q}^{\prime }}{\left(\frac{1}{|Q|}{\int }_{Q}|{S}_{\psi ,\delta }\left(f\right)\left(x\right){|}_{r}^{q}\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/q}\\ \phantom{\rule{1em}{0ex}}\le C{\parallel \stackrel{\to }{b}\parallel }_{\mathit{BMO}}|Q{|}^{-1/q}{\left({\int }_{Q}|f\left(x\right){|}_{r}^{p}\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/p}\\ \phantom{\rule{1em}{0ex}}\le C{\parallel \stackrel{\to }{b}\parallel }_{\mathit{BMO}}|Q{|}^{-1/q}{\left({\int }_{Q}|f\left(x\right){|}_{r}^{n/\delta }\phantom{\rule{0.2em}{0ex}}dx\right)}^{\delta /n}|Q{|}^{\left(1-\left(\delta p/n\right)\right)/p}\\ \phantom{\rule{1em}{0ex}}\le C{\parallel \stackrel{\to }{b}\parallel }_{\mathit{BMO}}{\parallel |f{|}_{r}\parallel }_{{L}^{n/\delta }}.\end{array}$

For ${M}_{2}\left(x\right)$, taking $1, $1/q=1/p-\delta /n$, we get

$\begin{array}{r}\frac{1}{|Q|}{\int }_{Q}{M}_{2}\left(x\right)\phantom{\rule{0.2em}{0ex}}dx\\ \phantom{\rule{1em}{0ex}}\le C\sum _{j=1}^{m-1}\sum _{\sigma \in {C}_{j}^{m}}{\parallel {\stackrel{\to }{b}}_{\sigma }\parallel }_{\mathit{BMO}}|Q{|}^{-1/q}{\left({\int }_{{R}^{n}}|{\left(b\left(x\right)-{b}_{Q}\right)}_{{\sigma }^{c}}f\left(x\right){|}_{r}^{p}{\chi }_{Q}\left(x\right)\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/p}\\ \phantom{\rule{1em}{0ex}}\le C\sum _{j=1}^{m-1}\sum _{\sigma \in {C}_{j}^{m}}{\parallel {\stackrel{\to }{b}}_{\sigma }\parallel }_{\mathit{BMO}}{\left(\frac{1}{|Q|}{\int }_{Q}|{\left(b\left(x\right)-{b}_{Q}\right)}_{{\sigma }^{c}}{|}^{q}\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/q}{\parallel |f{|}_{r}\parallel }_{{L}^{n/\delta }}\\ \phantom{\rule{1em}{0ex}}\le C\sum _{j=1}^{m-1}\sum _{\sigma \in {C}_{j}^{m}}{\parallel {\stackrel{\to }{b}}_{\sigma }\parallel }_{\mathit{BMO}}{\parallel {\stackrel{\to }{b}}_{{\sigma }^{c}}\parallel }_{\mathit{BMO}}{\parallel f\parallel }_{{L}^{n/\delta }}\\ \phantom{\rule{1em}{0ex}}\le C{\parallel \stackrel{\to }{b}\parallel }_{\mathit{BMO}}{\parallel |f{|}_{r}\parallel }_{{L}^{n/\delta }}.\end{array}$

For ${M}_{3}\left(x\right)$, taking $1, $1/q=1/p-\delta /n$, we obtain

$\begin{array}{rl}\frac{1}{|Q|}{\int }_{Q}{M}_{3}\left(x\right)\phantom{\rule{0.2em}{0ex}}dx& \le {\left(\frac{1}{|Q|}{\int }_{Q}|{S}_{\psi ,\delta }\left(\left({b}_{1}-{\left({b}_{1}\right)}_{Q}\right)\cdots \left({b}_{m}-{\left({b}_{m}\right)}_{Q}\right)g\right)\left(x\right){|}_{r}^{q}\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/q}\\ \le C|Q{|}^{-1/q}{\parallel \left({b}_{1}\left(x\right)-{\left({b}_{1}\right)}_{Q}\right)\cdots \left({b}_{m}\left(x\right)-{\left({b}_{m}\right)}_{Q}\right)|g\left(x\right){|}_{r}\parallel }_{{L}^{p}}\\ \le C{\left(\frac{1}{|Q|}{\int }_{Q}|\left({b}_{1}\left(x\right)-{\left({b}_{1}\right)}_{Q}\right)\cdots \left({b}_{m}\left(x\right)-{\left({b}_{m}\right)}_{Q}\right){|}^{q}\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/q}{\parallel |f{|}_{r}\parallel }_{{L}^{n/\delta }}\\ \le C{\parallel \stackrel{\to }{b}\parallel }_{\mathit{BMO}}{\parallel |f{|}_{r}\parallel }_{{L}^{n/\delta }}.\end{array}$

For ${M}_{4}\left(x\right)$, we have

$\begin{array}{rcl}{M}_{4}\left(x\right)& \le & C{\int }_{{Q}^{c}}|{x}_{0}-x{|}^{1/2}|{x}_{0}-z{|}^{-\left(n+1/2-\delta \right)}|\prod _{j=1}^{m}\left({b}_{j}\left(z\right)-{\left({b}_{j}\right)}_{Q}\right)||f\left(z\right){|}_{r}\phantom{\rule{0.2em}{0ex}}dz\\ \le & C\sum _{k=1}^{\mathrm{\infty }}{\int }_{{2}^{k}Q\mathrm{\setminus }{2}^{k-1}Q}|{x}_{0}-x{|}^{1/2}|{x}_{0}-z{|}^{-\left(n+1/2-\delta \right)}|\prod _{j=1}^{m}\left({b}_{j}\left(z\right)-{\left({b}_{j}\right)}_{Q}\right)||f\left(z\right){|}_{r}\phantom{\rule{0.2em}{0ex}}dz\\ \le & C\sum _{k=1}^{\mathrm{\infty }}{\int }_{{2}^{k}Q\mathrm{\setminus }{2}^{k-1}Q}\frac{|{x}_{0}-x{|}^{1/2}}{|{x}_{0}-z{|}^{n+1/2-\delta }}|\prod _{j=1}^{m}\left({b}_{j}\left(z\right)-{\left({b}_{j}\right)}_{Q}\right)||f\left(z\right){|}_{r}\phantom{\rule{0.2em}{0ex}}dz\\ \le & C\sum _{k=1}^{\mathrm{\infty }}{2}^{-k/2}\frac{1}{|{2}^{k}Q{|}^{1-\delta /n}}{\int }_{{2}^{k}Q}|\prod _{j=1}^{m}\left({b}_{j}\left(z\right)-{\left({b}_{j}\right)}_{Q}\right)||f\left(z\right){|}_{r}\phantom{\rule{0.2em}{0ex}}dz\\ \le & C\sum _{k=1}^{\mathrm{\infty }}{2}^{-k/2}{\left({\int }_{{2}^{k}Q}|f\left(z\right){|}_{r}^{n/\delta }\phantom{\rule{0.2em}{0ex}}dz\right)}^{\delta /n}\\ ×{\left(\frac{1}{|{2}^{k}Q|}{\int }_{{2}^{k}Q}|\prod _{j=1}^{m}\left({b}_{j}\left(z\right)-{\left({b}_{j}\right)}_{Q}\right){|}^{n/\left(n-\delta \right)}\phantom{\rule{0.2em}{0ex}}dz\right)}^{\left(n-\delta \right)/n}\\ \le & C\sum _{k=1}^{\mathrm{\infty }}{k}^{m}{2}^{-k/2}\prod _{j=1}^{m}{\parallel {b}_{j}\parallel }_{\mathit{BMO}}{\parallel |f{|}_{r}\parallel }_{{L}^{n/\delta }}\\ \le & C{\parallel \stackrel{\to }{b}\parallel }_{\mathit{BMO}}{\parallel |f{|}_{r}\parallel }_{{L}^{n/\delta }},\end{array}$

so

$\frac{1}{|Q|}{\int }_{Q}|{M}_{4}\left(x\right)|\phantom{\rule{0.2em}{0ex}}dx\le C{\parallel \stackrel{\to }{b}\parallel }_{\mathit{BMO}}{\parallel |f{|}_{r}\parallel }_{{L}^{n/\delta }}.$

This completes the proof of Theorem 1. □

Proof of Theorem 2 It is only to prove that there exists a constant ${C}_{Q}$, for any of the cubes $Q=Q\left(0,d\right)$ ($d>1$), the following inequality holds:

$\frac{1}{|Q|}{\int }_{Q}||{S}_{\psi ,\delta }^{\stackrel{\to }{b}}\left(f\right)\left(x\right){|}_{r}-{C}_{Q}|\phantom{\rule{0.2em}{0ex}}dx\le C{\parallel f\parallel }_{{B}_{p}^{\delta }}.$

Fix a cube $Q=Q\left(0,d\right)$ ($d>1$). Let $f=g+h=\left\{{g}_{i}\right\}+\left\{{h}_{i}\right\}$, where ${g}_{i}={f}_{i}{\chi }_{Q}$, ${h}_{i}={f}_{i}{\chi }_{{\left(Q\right)}^{c}}$ and ${\stackrel{\to }{b}}_{Q}=\left({\left({b}_{1}\right)}_{Q},\dots ,{\left({b}_{m}\right)}_{Q}\right)$. For ${\left({b}_{j}\right)}_{Q}=|Q{|}^{-1}{\int }_{Q}|{b}_{j}\left(y\right)|\phantom{\rule{0.2em}{0ex}}dy$, $1\le j\le m$, we have

$\begin{array}{rcl}{F}_{t}^{\stackrel{\to }{b}}\left({f}_{i}\right)\left(x,y\right)& =& {\int }_{{R}^{n}}\left[\prod _{j=1}^{m}\left({b}_{1}\left(x\right)-{b}_{1}\left(z\right)\right)\right]{\psi }_{t}\left(y-z\right){f}_{i}\left(z\right)\phantom{\rule{0.2em}{0ex}}dz\\ =& \left({b}_{1}\left(x\right)-{\left({b}_{1}\right)}_{Q}\right)\cdots \left({b}_{m}\left(x\right)-{\left({b}_{m}\right)}_{Q}\right){F}_{t}\left({f}_{i}\right)\left(y\right)\\ +{\left(-1\right)}^{m}{F}_{t}\left(\left({b}_{1}-{\left({b}_{1}\right)}_{Q}\right)\cdots \left({b}_{m}-{\left({b}_{m}\right)}_{Q}\right){g}_{i}\right)\left(y\right)\\ +{\left(-1\right)}^{m}{F}_{t}\left(\left({b}_{1}-{\left({b}_{1}\right)}_{Q}\right)\cdots \left({b}_{m}-{\left({b}_{m}\right)}_{Q}\right){h}_{i}\right)\left(y\right)\\ +\sum _{j=1}^{m-1}\sum _{\sigma \in {C}_{j}^{m}}{\left(-1\right)}^{m-j}{\left(\stackrel{\to }{b}\left(x\right)-{\stackrel{\to }{b}}_{Q}\right)}_{\sigma }{F}_{t}\left({\left(\stackrel{\to }{b}-{\stackrel{\to }{b}}_{Q}\right)}_{{\sigma }^{c}}{f}_{i}\right)\left(x,y\right).\end{array}$

By the Minkowski inequality, we have

$\begin{array}{r}||{S}_{\psi ,\delta }^{\stackrel{\to }{b}}\left(f\right)\left(x\right){|}_{r}-|{S}_{\psi ,\delta }\left(\left({\left({b}_{1}\right)}_{Q}-{b}_{1}\right)\cdots \left({\left({b}_{m}\right)}_{Q}-{b}_{m}\right)h\right)\left({x}_{0}\right){|}_{r}|\\ \phantom{\rule{1em}{0ex}}\le {\parallel {\chi }_{\mathrm{\Gamma }\left(x\right)}\left({b}_{1}\left(x\right)-{\left({b}_{1}\right)}_{Q}\right)\cdots \left({b}_{m}\left(x\right)-{\left({b}_{m}\right)}_{Q}\right){F}_{t}\left(f\right)\left(y\right)\parallel }_{r}\\ \phantom{\rule{2em}{0ex}}+\sum _{j=1}^{m-1}\sum _{\sigma \in {C}_{j}^{m}}{\parallel {\chi }_{\mathrm{\Gamma }\left(x\right)}{\left(\stackrel{\to }{b}\left(x\right)-{\stackrel{\to }{b}}_{Q}\right)}_{\sigma }{F}_{t}\left({\left(\stackrel{\to }{b}-{\stackrel{\to }{b}}_{Q}\right)}_{{\sigma }^{c}}f\right)\left(x,y\right)\parallel }_{r}\\ \phantom{\rule{2em}{0ex}}+{\parallel {\chi }_{\mathrm{\Gamma }\left(x\right)}{F}_{t}\left(\left({b}_{1}-{\left({b}_{1}\right)}_{Q}\right)\cdots \left({b}_{m}-{\left({b}_{m}\right)}_{Q}\right)g\right)\left(y\right)\parallel }_{r}\\ \phantom{\rule{2em}{0ex}}+{\parallel {\chi }_{\mathrm{\Gamma }\left(x\right)}{F}_{t}\left(\prod _{j=1}^{m}\left({b}_{j}-{\left({b}_{j}\right)}_{Q}\right)h\right)\left(y\right)-{\chi }_{\mathrm{\Gamma }\left({x}_{0}\right)}{F}_{t}\left(\prod _{j=1}^{m}\left({b}_{j}-{\left({b}_{j}\right)}_{Q}\right)h\right)\left(y\right)\parallel }_{r}\\ \phantom{\rule{1em}{0ex}}={H}_{1}\left(x\right)+{H}_{2}\left(x\right)+{H}_{3}\left(x\right)+{H}_{4}\left(x\right).\end{array}$

For ${H}_{1}\left(x\right)$, take $1/q=1/p-\delta /n$, by the Hölder inequality and Lemma 1, we have

$\begin{array}{r}\frac{1}{|Q|}{\int }_{Q}{H}_{1}\left(x\right)\phantom{\rule{0.2em}{0ex}}dx\\ \phantom{\rule{1em}{0ex}}\le {\left(\frac{1}{|Q|}{\int }_{Q}|\prod _{j=1}^{m}\left({b}_{j}\left(x\right)-{\left({b}_{j}\right)}_{Q}\right){|}^{{q}^{\prime }}\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/{q}^{\prime }}{\left(\frac{1}{|Q|}{\int }_{Q}|{S}_{\psi ,\delta }\left(f\right)\left(x\right){|}_{r}^{q}\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/q}\\ \phantom{\rule{1em}{0ex}}\le C{\parallel \stackrel{\to }{b}\parallel }_{\mathit{BMO}}|Q{|}^{-1/q}{\left({\int }_{{R}^{n}}|f\left(x\right){|}_{r}^{p}{\chi }_{Q}\left(x\right)\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/p}\\ \phantom{\rule{1em}{0ex}}\le C{\parallel \stackrel{\to }{b}\parallel }_{\mathit{BMO}}{d}^{-n\left(1/p-\delta /n\right)}{\parallel |f{|}_{r}{\chi }_{Q}\parallel }_{{L}^{p}}\\ \phantom{\rule{1em}{0ex}}\le C{\parallel \stackrel{\to }{b}\parallel }_{\mathit{BMO}}{\parallel |f{|}_{r}\parallel }_{{B}_{p}^{\delta }}.\end{array}$

For ${H}_{2}\left(x\right)$, taking $1, $1/v=1/u-\delta /n$, we get

$\begin{array}{rl}\frac{1}{|Q|}{\int }_{Q}{H}_{2}\left(x\right)\phantom{\rule{0.2em}{0ex}}dx\le & C\sum _{j=1}^{m-1}\sum _{\sigma \in {C}_{j}^{m}}{\left(\frac{1}{|Q|}{\int }_{Q}|{\left(b\left(x\right)-{b}_{Q}\right)}_{\sigma }{|}^{{v}^{\prime }}\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/{v}^{\prime }}\\ ×{\left(\frac{1}{|Q|}{\int }_{Q}|{S}_{\psi ,\delta }\left({\left(b-{b}_{Q}\right)}_{{\sigma }^{c}}f\right)\left(x\right){|}_{r}^{u}\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/u}\\ \le & C\sum _{j=1}^{m-1}\sum _{\sigma \in {C}_{j}^{m}}{\parallel {\stackrel{\to }{b}}_{\sigma }\parallel }_{\mathit{BMO}}|Q{|}^{-1/v}{\left({\int }_{{R}^{n}}|{\left(b\left(x\right)-{b}_{Q}\right)}_{{\sigma }^{c}}f\left(x\right){|}_{r}^{u}{\chi }_{Q}\left(x\right)\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/u}\\ \le & C\sum _{j=1}^{m-1}\sum _{\sigma \in {C}_{j}^{m}}{\parallel {\stackrel{\to }{b}}_{\sigma }\parallel }_{\mathit{BMO}}|Q{|}^{\left(\delta /n-1/p\right)}{\parallel |f{|}_{r}{\chi }_{Q}\parallel }_{{L}^{p}}\\ ×{\left(\frac{1}{|Q|}{\int }_{Q}|{\left(b\left(x\right)-{b}_{Q}\right)}_{{\sigma }^{c}}{|}^{pr/\left(p-r\right)}\phantom{\rule{0.2em}{0ex}}dx\right)}^{\left(p-u\right)/pu}\\ \le & C\sum _{j=1}^{m-1}\sum _{\sigma \in {C}_{j}^{m}}{\parallel {\stackrel{\to }{b}}_{\sigma }\parallel }_{\mathit{BMO}}{\parallel {\stackrel{\to }{b}}_{{\sigma }^{c}}\parallel }_{\mathit{BMO}}{d}^{-n\left(1/p-\delta /n\right)}{\parallel |f{|}_{r}{\chi }_{Q}\parallel }_{{L}^{p}}\\ \le & C{\parallel \stackrel{\to }{b}\parallel }_{\mathit{BMO}}{\parallel |f{|}_{r}\parallel }_{{B}_{p}^{\delta }}.\end{array}$

For ${H}_{3}\left(x\right)$, taking $1, $1/v=1/u-\delta /n$, we obtain

$\begin{array}{r}\frac{1}{|Q|}{\int }_{Q}{H}_{3}\left(x\right)\phantom{\rule{0.2em}{0ex}}dx\\ \phantom{\rule{1em}{0ex}}\le {\left(\frac{1}{|Q|}{\int }_{Q}|{S}_{\psi ,\delta }\left(\left({b}_{1}-{\left({b}_{1}\right)}_{Q}\right)\cdots \left({b}_{m}-{\left({b}_{m}\right)}_{Q}\right)g\right)\left(x\right){|}_{r}^{v}\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/v}\\ \phantom{\rule{1em}{0ex}}\le C|Q{|}^{-1/v}{\left({\int }_{Q}|\left({b}_{1}\left(x\right)-{\left({b}_{1}\right)}_{Q}\right)\cdots \left({b}_{m}\left(x\right)-{\left({b}_{m}\right)}_{Q}\right)g\left(x\right){|}_{r}^{u}\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/u}\\ \phantom{\rule{1em}{0ex}}\le C|Q{|}^{-1/v}{\parallel \left({b}_{1}\left(x\right)-{\left({b}_{1}\right)}_{Q}\right)\cdots \left({b}_{m}\left(x\right)-{\left({b}_{m}\right)}_{Q}\right)|f{|}_{r}{\chi }_{Q}\parallel }_{{L}^{v}}\\ \phantom{\rule{1em}{0ex}}\le C{\parallel \stackrel{\to }{b}\parallel }_{\mathit{BMO}}{\parallel |f{|}_{r}\parallel }_{{B}_{p}^{\delta }}.\end{array}$

For ${H}_{4}\left(x\right)$, we have

$\begin{array}{rcl}{I}_{4}\left(x\right)& \le & {\left[\int {\int }_{{R}_{+}^{n+1}}{\left({\int }_{{Q}^{c}}|{\chi }_{\mathrm{\Gamma }\left(x\right)}-{\chi }_{\mathrm{\Gamma }\left({x}_{0}\right)}|\prod _{j=1}^{m}|{b}_{j}\left(z\right)-{\left({b}_{j}\right)}_{Q}||{\psi }_{t}\left(y-z\right)||f\left(z\right){|}_{r}\phantom{\rule{0.2em}{0ex}}dz\right)}^{2}\frac{dy\phantom{\rule{0.2em}{0ex}}dt}{{t}^{n+1}}\right]}^{1/2}\\ \le & C\sum _{k=0}^{\mathrm{\infty }}{\int }_{{2}^{k+1}Q\setminus {2}^{k}Q}|{x}_{0}-x{|}^{1/2}|{x}_{0}-z{|}^{-\left(n+1/2-2\delta \right)}|\prod _{j=1}^{m}\left({b}_{j}\left(z\right)-{\left({b}_{j}\right)}_{Q}\right)||f\left(z\right){|}_{r}\phantom{\rule{0.2em}{0ex}}dz\\ \le & C\sum _{k=1}^{\mathrm{\infty }}{2}^{-k/2}|{2}^{k+1}Q{|}^{-1+\delta /n}{\int }_{{2}^{k+1}Q}|\prod _{j=1}^{m}\left({b}_{j}\left(z\right)-{\left({b}_{j}\right)}_{Q}\right)||f\left(z\right){|}_{r}\phantom{\rule{0.2em}{0ex}}dz\\ \le & C\sum _{k=1}^{\mathrm{\infty }}{k}^{m}{2}^{-k/2}|{2}^{k}Q{|}^{-\left(1/p-\delta /n\right)}{\parallel \stackrel{\to }{b}\parallel }_{\mathit{BMO}}{\parallel |f{|}_{r}{\chi }_{{2}^{k}Q}\parallel }_{{L}^{p}}\\ \le & C{\parallel \stackrel{\to }{b}\parallel }_{\mathit{BMO}}{\parallel |f{|}_{r}\parallel }_{{B}_{p}^{\delta }},\end{array}$

so

$\frac{1}{|Q|}{\int }_{Q}|{H}_{4}\left(x\right)|\phantom{\rule{0.2em}{0ex}}dx\le C{\parallel \stackrel{\to }{b}\parallel }_{\mathit{BMO}}{\parallel |f{|}_{r}\parallel }_{{B}_{p}^{\delta }}.$

This completes the proof of Theorem 2. □

## References

1. Coifman R, Rochberg R, Weiss G: Factorization theorems for Hardy spaces in several variables. Ann. Math. 1976, 103: 611–635. 10.2307/1970954

2. Garcia-Cuerva J, Rubio de Francia JL North-Holland Mathematics Studies 116. In Weighted Norm Inequalities and Related Topics. North-Holland, Amsterdam; 1985.

3. Liu LZ: The continuity of commutators on Triebel-Lizorkin spaces. Integral Equ. Oper. Theory 2004, 49: 65–76. 10.1007/s00020-002-1186-8

4. Liu LZ, Wu BS: Weighted boundedness for commutator of Littewood-Paley integral on some Hardy spaces. Southeast Asian Bull. Math. 2004, 28: 643–650.

5. Liu LZ:Weighted weak type $\left({H}^{1},{L}^{1}\right)$ estimates for commutators of Littlewood-Paley operator. Indian J. Math. 2003, 45: 71–78.

6. Pérez C, Trujillo-Gonzalez R: Sharp Weighted estimates for multilinear commutators. J. Lond. Math. Soc. 2002, 65: 672–692. 10.1112/S0024610702003174

7. Stein EM: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton University Press, Princeton; 1993.

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Correspondence to Weiping Kuang.

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Kuang, W. Endpoint estimates for vector-valued multilinear commutator of fractional area integral operator. J Inequal Appl 2013, 513 (2013). https://doi.org/10.1186/1029-242X-2013-513

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• DOI: https://doi.org/10.1186/1029-242X-2013-513

### Keywords

• vector-valued multilinear commutator
• Triebel-Lizorkin space
• Lipschitz space
• Lebesgue space
• $\mathit{BMO}\left({R}^{n}\right)$ 