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

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

A classical result of Coifman, Rochberb and Weiss (see [1]) proved that the commutator $[b,T]$ is bounded on $L^p(R^n)$ ($1<p<\mathrm{\infty }$). In [2–4], 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 ${|S_{\psi ,\delta }^{\overrightarrow{b}}|}_r$ 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, 5–7]).

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

• (1) ${\int }_{R^n}\psi (x)dx=0$;

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

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

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

$$|S_{\psi ,\delta }^{\overrightarrow{b}}(f)(x){|}_r={\left(\sum _{i=1}^{\mathrm{\infty }}{(S_{\psi ,\delta }^{\overrightarrow{b}}(f_i)(x))}^r\right)}^{1/r},$$

where

$$S_{\psi ,\delta }^{\overrightarrow{b}}(f)(x)={\left({\int }_{\mathrm{\Gamma }(x)}\right|F_t^{\overrightarrow{b}}(f)(x,y){|}^2\frac{dydt}{t^{n+1}})}^{1/2}$$

and

$$F_t^{\tilde{b}}(f)(x)={\int }_{R^n}\left[\prod _{j=1}^m(b_j(x)-b_j(z))\right]{\psi }_t(y-z)f(z)dz.$$

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

$${\parallel b\parallel }_{\mathit{CMO}}=\underset{r>1}{sup}|Q(0,r){|}^{-1}{\int }_Q|b(y)-b_Q|dy<\mathrm{\infty }.$$

We have

$${\parallel b\parallel }_{\mathit{CMO}}\approx \underset{r>1}{sup}\underset{c\in C}{inf}|Q(0,r){|}^{-1}{\int }_Q|b(y)-c|dy.$$

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

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

Now we state our theorems as follows.

Theorem 1 Suppose $1<r<\mathrm{\infty }$, $0<\delta <n$, and $\overrightarrow{b}=(b_1,\dots ,b_m)$ for $b_j\in \mathit{BMO}$, $1\le j\le m$. Then $|S_{\psi ,\delta }^{\overrightarrow{b}}{|}_r$ is bounded from $L^{n/\delta }$ to $\mathit{BMO}(R^n)$.

Theorem 2 Let $1<r<\mathrm{\infty }$, $0<\delta <n$, $1<p<n/\delta$, and $\overrightarrow{b}=(b_1,\dots ,b_m)$, with $b_j\in \mathit{BMO}(R^n)$, for $1\le j\le m$. Then $|S_{\psi ,\delta }^{\overrightarrow{b}}{|}_r$ is bounded from $B_p^{\delta }(R^n)$ to $\mathit{CMO}(R^n)$.

3 Proofs of theorems

We begin with a preliminaries lemma.

Lemma 1 (see [3, 4])

Let $1<r<\mathrm{\infty }$, $0<\delta <n$, $1<p<n/\delta$, $1/q=1/p-\delta /n$. Then $|S_{\psi ,\delta }{|}_r$ is bounded from $L^p(R^n)$ to $L^q(R^n)$.

Proof of Theorem 1 It is only to prove that there exists a constant $C_Q$, the following inequality holds:

$$\frac{1}{|Q|}{\int }_Q\left|\right|S_{\psi ,\delta }^{\overrightarrow{b}}(f)(x){|}_r-C_Q|dx\le C{\parallel |f{|}_r\parallel }_{L^{n/\delta }}.$$

Fix a cube $Q=Q(x_0,r)$, let $f=g+h=\{g_i\}+\{h_i\}$ for $g_i=f_i{\chi }_Q$, $h_i=f_i{\chi }_{{(Q)}^c}$.

When $m=1$, set ${(b_1)}_Q={|Q|}^{-1}{\int }_Q{b}_1(y)dy$, then

$$F_t^{b_1}(f_i)(x,y)=(b_1(x)-{(b_1)}_Q)F_t(f_i)(y)-F_t\left((b_1-{(b_1)}_Q)g_i\right)(y)-F_t\left((b_1-{(b_1)}_Q)h_i\right)(y),$$

so

$$\begin{array}{r}|S_{\psi ,\delta }^{b_1}(f)(x){|}_r-|S_{\psi ,\delta }\left(({(b_1)}_{2Q}-b_1)h\right){(x_0)}_r|\\ \le {\left(\sum _{i=1}^{\mathrm{\infty }}{\parallel {\chi }_{\mathrm{\Gamma }(x)}(b_1(x)-{(b_1)}_Q)F_t(f_i)(y)\parallel }^r\right)}^{1/r}\\ +{\left(\sum _{i=1}^{\mathrm{\infty }}{\parallel {\chi }_{\mathrm{\Gamma }(x)}{F}_t\left(({(b_1)}_Q-b_1)g_i\right)(y)\parallel }^r\right)}^{1/r}\\ +{\parallel {\chi }_{\mathrm{\Gamma }(x)}{F}_t\left((b_1-{(b_1)}_Q)f_2\right)(y)-{\chi }_{\mathrm{\Gamma }(x_0)}{F}_t\left((b_1-{(b_1)}_Q)h\right)(y)\parallel }_r\\ =A(x)+B(x)+C(x).\end{array}$$

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

$$\begin{array}{rcl}\frac{1}{|Q|}{\int }_Q|A(x)|dx& =& \frac{1}{|Q|}{\int }_Q|b_1(x)-{(b_1)}_Q||S_{\psi ,\delta }(f)(x){|}_{r}dx\\ \le & {\left(\frac{1}{|Q|}{\int }_Q\right|b_1(x)-{(b_1)}_Q{|}^{q^{\prime }}dx)}^{1/q^{\prime }}\\ \times {\left(\frac{1}{|Q|}{\int }_{R^n}\right|S_{\psi ,\delta }(f)(x){|}_r^q{\chi }_Q(x)dx)}^{1/q}\\ \le & C{\parallel b_1\parallel }_{\mathit{BMO}}|Q{|}^{-1/q}{\left({\int }_{R^n}\right|f(x){|}_r^p{\chi }_Q(x)dx)}^{1/p}\\ \le & C{\parallel b_1\parallel }_{\mathit{BMO}}|Q{|}^{-1/q}\\ \times {\left[{\left({\int }_{R^n}\right|f(x){|}_r^{n/\delta }dx)}^{\delta p/n}{({\int }_Q{\chi }_Q(x)dx)}^{1-\delta p/n}\right]}^{1/p}\\ \le & C{\parallel b_1\parallel }_{\mathit{BMO}}|Q{|}^{-1/q}{\parallel |f{|}_r\parallel }_{L^{n/\delta }}|Q{|}^{(1-\delta p/n)/p}\\ \le & C{\parallel b_1\parallel }_{\mathit{BMO}}{\parallel |f{|}_r\parallel }_{L^{n/\delta }}.\end{array}$$

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

$$\begin{array}{r}\frac{1}{|Q|}{\int }_Q|B(x)|dx\\ =\frac{1}{|Q|}{\int }_Q|S_{\psi ,\delta }\left((b_1-{(b_1)}_Q)g\right)(x){|}_{r}dx\\ \le {\left(\frac{1}{|Q|}{\int }_{R^n}\right|S_{\psi ,\delta }{((b_1-{(b_1)}_Q)g)(x))}^{v}dx{|}_r)}^{1/v}\\ \le C|Q{|}^{-1/v}{\left({\int }_{R^n}\right|b_1(x)-{(b_1)}_Q{|}^u|f(x){|}_r^u{\chi }_Q(x)dx)}^{1/u}\\ \le C{\left(\frac{1}{|Q|}{\int }_Q\right|b_1(x)-{(b_1)}_Q{|}^{u}dx)}^{1/u}{\parallel |f{|}_r\parallel }_{L^{n/\delta }}\\ \le C{\parallel b_1\parallel }_{\mathit{BMO}}{\parallel |f{|}_r\parallel }_{L^{n/\delta }}.\end{array}$$

For $C(x)$, we have

$$\begin{array}{rcl}C(x)& =& {\parallel {\chi }_{\mathrm{\Gamma }(x)}{F}_t\left((b_1-{(b_1)}_Q)f_2\right)(y)-{\chi }_{\mathrm{\Gamma }(x_0)}{F}_t\left((b_1-{(b_1)}_Q)h\right)(y)\parallel }_r\\ \le & {[\int {\int }_{R_{+}^{n+1}}{\left({\int }_{Q^c}\right|{\chi }_{\mathrm{\Gamma }(x)}-{\chi }_{\mathrm{\Gamma }(x_0)}\parallel b_1(z)-{(b_1)}_Q\parallel {\psi }_t(y-z)\left|\right|f(z){|}_{r}dz)}^2\frac{dydt}{t^{n+1}}]}^{1/2}\\ \le & C{\int }_{Q^c}|b_1(z)-{(b_1)}_Q||f(z){|}_r\\ \times |\int {\int }_{|x-y|\le t}\frac{t^{1-n}dydt}{{(t+|y-z|)}^{2n+2-2\delta }}-\int {\int }_{|x_0-y|\le t}\frac{t^{1-n}dydt}{{(t+|y-z|)}^{2n+2-2\delta }}{|}^{1/2}dz\\ \le & C{\int }_{Q^c}|b_1(z)-{(b_1)}_Q||f(z){|}_r\\ \times (\int {\int }_{|y|\le t,|x+y-z|\le t}|\frac{1}{{(t+|x+y-z|)}^{2n+2-2\delta }}\\ {-\frac{1}{{(t+|x_0+y-z|)}^{2n+2-2\delta }}\left|\frac{dydt}{t^{n-1}}\right)}^{1/2}dz\\ \le & {\int }_{Q^c}|b_1(z)-{(b_1)}_Q||f(z){|}_r{(\int {\int }_{|y|\le t,|x+y-z|\le t}\frac{|x-x_0|t^{1-n}}{{(t+|x+y-z|)}^{2n+3-2\delta }}dydt)}^{1/2}dz.\end{array}$$

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

$${\int }_0^{\mathrm{\infty }}\frac{tdt}{{(t+|x-z|)}^{2n+3-2\delta }}=C|x-z{|}^{-2n-1+2\delta },$$

then, for $x\in Q$,

$$\begin{array}{rcl}C(x)& \le & {\int }_{Q^c}|b_1(z)-{(b_1)}_Q||f(z){|}_r{(\int {\int }_{|y|\le t}\frac{2^{2n+3-2\delta }|x_0-x|t^{1-n}dydt}{{(2t+2|x+y-z|)}^{2n+3-2\delta }})}^{1/2}dz\\ \le & C{\int }_{Q^c}|b_1(z)-{(b_1)}_Q||f(z){|}_r|x-x_0{|}^{1/2}{(\int {\int }_{|y|\le t}\frac{t^{1-n}dydt}{{(t+|x-z|)}^{2n+3-2\delta }})}^{1/2}dz\\ \le & C{\int }_{Q^c}|b_1(z)-{(b_1)}_Q||f(z){|}_r|x-x_0{|}^{1/2}{\left({\int }_0^{\mathrm{\infty }}\frac{tdt}{{(t+|x-z|)}^{2n+3-2\delta }}\right)}^{1/2}dz\\ \le & C{\int }_{Q^c}|b_1(z)-{(b_1)}_Q||f(z){|}_r\frac{|x_0-x{|}^{1/2}}{|x_0-z{|}^{n+1/2-\delta }}dz\\ \le & C\sum _{k=1}^{\mathrm{\infty }}{\int }_{2^{k+1}Q\setminus 2^{k}Q}|b_1(z)-{(b_1)}_Q||f(z){|}_r\frac{|x_0-x{|}^{1/2}}{|x_0-z{|}^{n+1/2-\delta }}dz\\ \le & C\sum _{k=1}^{\mathrm{\infty }}{2}^{-k/2}\left|2^{k+1}Q{|}^{-1+\delta /n}{\int }_{2^{k+1}Q}\right|b_1(z)-{(b_1)}_Q\left|\right|f(z){|}_{r}dz\\ \le & C{\parallel b_1\parallel }_{\mathit{BMO}}\sum _{k=1}^{\mathrm{\infty }}k{2}^{-k/2}{\parallel |f{|}_r\parallel }_{L^{n/\delta }}\\ \le & C{\parallel b_1\parallel }_{\mathit{BMO}}{\parallel |f{|}_r\parallel }_{L^{n/\delta }},\end{array}$$

so that

$$\frac{1}{|Q|}{\int }_Q|C(x)|dx\le C{\parallel b_1\parallel }_{\mathit{BMO}}{\parallel |f{|}_r\parallel }_{L^{n/\delta }}.$$

When $m>1$, let ${\overrightarrow{b}}_Q=({(b_1)}_Q,\dots ,{(b_m)}_Q)\in R^n$, where

$${(b_j)}_Q=|Q{|}^{-1}{\int }_Q{b}_j(y)dy,1\le j\le m,$$

let $f=g+h=\{g_i\}+\{h_i\}$ for $g_i=f_i{\chi }_Q$, $h_i=f_i{\chi }_{{(Q)}^c}$. We have

$$\begin{array}{r}{F}_t^{\overrightarrow{b}}(f_i)(x,y)\\ ={\int }_{R^n}\left[\prod _{j=1}^m(b_1(x)-b_1(z))\right]{\psi }_t(y-z)f_i(z)dz\\ =(b_1(x)-{(b_1)}_Q)\cdots (b_m(x)-{(b_m)}_Q)F_t(f_i)(y)\\ +{(-1)}^m{F}_t((b_1-{(b_1)}_Q)\cdots (b_m-{(b_m)}_Q)f_i)(y)\\ +\sum _{j=1}^{m-1}\sum _{\sigma \in C_j^m}{(-1)}^{m-j}{(\overrightarrow{b}(x)-{\overrightarrow{b}}_Q)}_{\sigma }\\ \times {\int }_{R^n}{(\overrightarrow{b}(z)-{\overrightarrow{b}}_Q)}_{{\sigma }^c}{\psi }_t(y-z)f_i(z)dz\\ =(b_1(x)-{(b_1)}_Q)\cdots (b_m(x)-{(b_m)}_Q)F_t(f_i)(y)\\ +{(-1)}^m{F}_t((b_1-{(b_1)}_Q)\cdots (b_m-{(b_m)}_Q)g_i)(y)\\ +{(-1)}^m{F}_t((b_1-{(b_1)}_Q)\cdots (b_m-{(b_m)}_Q)h_i)(y)\\ +\sum _{j=1}^{m-1}\sum _{\sigma \in C_j^m}{(-1)}^{m-j}{(\overrightarrow{b}(x)-{\overrightarrow{b}}_Q)}_{\sigma }{F}_t\left({(\overrightarrow{b}-{\overrightarrow{b}}_Q)}_{{\sigma }^c}{f}_i\right)(x,y),\end{array}$$

by the Minkowski inequality, we have

$$\begin{array}{r}\left|\right|S_{\psi ,\delta }^{\overrightarrow{b}}(f)(x){|}_r-|S_{\psi ,\delta }(({(b_1)}_Q-b_1)\cdots ({(b_m)}_Q-b_m)h)(x_0){|}_r|\\ \le {\parallel {\chi }_{\mathrm{\Gamma }(x)}(b_1(x)-{(b_1)}_Q)\cdots (b_m(x)-{(b_m)}_Q)F_t(f)(y)\parallel }_r\\ +\sum _{j=1}^{m-1}\sum _{\sigma \in C_j^m}{\parallel {\chi }_{\mathrm{\Gamma }(x)}{(\overrightarrow{b}(x)-{\overrightarrow{b}}_Q)}_{\sigma }{F}_t\left({(\overrightarrow{b}-{\overrightarrow{b}}_Q)}_{{\sigma }^c}f\right)(x,y)\parallel }_r\\ +{\parallel {\chi }_{\mathrm{\Gamma }(x)}{F}_t((b_1-{(b_1)}_Q)\cdots (b_m-{(b_m)}_Q)g)(y)\parallel }_r\\ +{\parallel {\chi }_{\mathrm{\Gamma }(x)}{F}_t\left(\prod _{j=1}^m(b_j-{(b_j)}_Q)h\right)(y)-{\chi }_{\mathrm{\Gamma }(x_0)}{F}_t\left(\prod _{j=1}^m(b_j-{(b_j)}_Q)h\right)(y)\parallel }_r\\ =M_1(x)+M_2(x)+M_3(x)+M_4(x).\end{array}$$

For $M_1(x)$, similar to the proof of $m=1$, we take $1<p<n/\delta$, $1/q=1/p-\delta /n$, by the Hölder inequality and Lemma 1, we have

$$\begin{array}{r}\frac{1}{|Q|}{\int }_Q{M}_1(x)dx\\ \le {\left(\frac{1}{|Q|}{\int }_Q\right|\prod _{j=1}^m(b_j(x)-{(b_j)}_Q){|}^{q^{\prime }}dx)}^{1/q^{\prime }}{\left(\frac{1}{|Q|}{\int }_Q\right|S_{\psi ,\delta }(f)(x){|}_r^{q}dx)}^{1/q}\\ \le C{\parallel \overrightarrow{b}\parallel }_{\mathit{BMO}}|Q{|}^{-1/q}{\left({\int }_Q\right|f(x){|}_r^{p}dx)}^{1/p}\\ \le C{\parallel \overrightarrow{b}\parallel }_{\mathit{BMO}}|Q{|}^{-1/q}{\left({\int }_Q\right|f(x){|}_r^{n/\delta }dx)}^{\delta /n}|Q{|}^{(1-(\delta p/n))/p}\\ \le C{\parallel \overrightarrow{b}\parallel }_{\mathit{BMO}}{\parallel |f{|}_r\parallel }_{L^{n/\delta }}.\end{array}$$

For $M_2(x)$, taking $1<p<n/\delta$, $1/q=1/p-\delta /n$, we get

$$\begin{array}{r}\frac{1}{|Q|}{\int }_Q{M}_2(x)dx\\ \le C\sum _{j=1}^{m-1}\sum _{\sigma \in C_j^m}{\parallel {\overrightarrow{b}}_{\sigma }\parallel }_{\mathit{BMO}}|Q{|}^{-1/q}{\left({\int }_{R^n}\right|{(b(x)-b_Q)}_{{\sigma }^c}f(x){|}_r^p{\chi }_Q(x)dx)}^{1/p}\\ \le C\sum _{j=1}^{m-1}\sum _{\sigma \in C_j^m}{\parallel {\overrightarrow{b}}_{\sigma }\parallel }_{\mathit{BMO}}{\left(\frac{1}{|Q|}{\int }_Q\right|{(b(x)-b_Q)}_{{\sigma }^c}{|}^{q}dx)}^{1/q}{\parallel |f{|}_r\parallel }_{L^{n/\delta }}\\ \le C\sum _{j=1}^{m-1}\sum _{\sigma \in C_j^m}{\parallel {\overrightarrow{b}}_{\sigma }\parallel }_{\mathit{BMO}}{\parallel {\overrightarrow{b}}_{{\sigma }^c}\parallel }_{\mathit{BMO}}{\parallel f\parallel }_{L^{n/\delta }}\\ \le C{\parallel \overrightarrow{b}\parallel }_{\mathit{BMO}}{\parallel |f{|}_r\parallel }_{L^{n/\delta }}.\end{array}$$

For $M_3(x)$, taking $1<p<n/\delta$, $1/q=1/p-\delta /n$, we obtain

$$\begin{array}{rl}\frac{1}{|Q|}{\int }_Q{M}_3(x)dx& \le {\left(\frac{1}{|Q|}{\int }_Q\right|S_{\psi ,\delta }((b_1-{(b_1)}_Q)\cdots (b_m-{(b_m)}_Q)g)(x){|}_r^{q}dx)}^{1/q}\\ \le C|Q{|}^{-1/q}{\parallel (b_1(x)-{(b_1)}_Q)\cdots (b_m(x)-{(b_m)}_Q)|g(x){|}_r\parallel }_{L^p}\\ \le C{\left(\frac{1}{|Q|}{\int }_Q\right|(b_1(x)-{(b_1)}_Q)\cdots (b_m(x)-{(b_m)}_Q){|}^{q}dx)}^{1/q}{\parallel |f{|}_r\parallel }_{L^{n/\delta }}\\ \le C{\parallel \overrightarrow{b}\parallel }_{\mathit{BMO}}{\parallel |f{|}_r\parallel }_{L^{n/\delta }}.\end{array}$$

For $M_4(x)$, we have

$$\begin{array}{rcl}{M}_4(x)& \le & C{\int }_{Q^c}|x_0-x{|}^{1/2}|x_0-z{|}^{-(n+1/2-\delta )}\left|\prod _{j=1}^m(b_j(z)-{(b_j)}_Q)\right||f(z){|}_{r}dz\\ \le & C\sum _{k=1}^{\mathrm{\infty }}{\int }_{2^{k}Q\mathrm{\setminus }{2}^{k-1}Q}|x_0-x{|}^{1/2}|x_0-z{|}^{-(n+1/2-\delta )}\left|\prod _{j=1}^m(b_j(z)-{(b_j)}_Q)\right||f(z){|}_{r}dz\\ \le & C\sum _{k=1}^{\mathrm{\infty }}{\int }_{2^{k}Q\mathrm{\setminus }{2}^{k-1}Q}\frac{|x_0-x{|}^{1/2}}{|x_0-z{|}^{n+1/2-\delta }}\left|\prod _{j=1}^m(b_j(z)-{(b_j)}_Q)\right||f(z){|}_{r}dz\\ \le & C\sum _{k=1}^{\mathrm{\infty }}{2}^{-k/2}\frac{1}{|2^{k}Q{|}^{1-\delta /n}}{\int }_{2^{k}Q}\left|\prod _{j=1}^m(b_j(z)-{(b_j)}_Q)\right||f(z){|}_{r}dz\\ \le & C\sum _{k=1}^{\mathrm{\infty }}{2}^{-k/2}{\left({\int }_{2^{k}Q}\right|f(z){|}_r^{n/\delta }dz)}^{\delta /n}\\ \times {\left(\frac{1}{|2^{k}Q|}{\int }_{2^{k}Q}\right|\prod _{j=1}^m(b_j(z)-{(b_j)}_Q){|}^{n/(n-\delta )}dz)}^{(n-\delta )/n}\\ \le & C\sum _{k=1}^{\mathrm{\infty }}{k}^m{2}^{-k/2}\prod _{j=1}^m{\parallel b_j\parallel }_{\mathit{BMO}}{\parallel |f{|}_r\parallel }_{L^{n/\delta }}\\ \le & C{\parallel \overrightarrow{b}\parallel }_{\mathit{BMO}}{\parallel |f{|}_r\parallel }_{L^{n/\delta }},\end{array}$$

so

$$\frac{1}{|Q|}{\int }_Q|M_4(x)|dx\le C{\parallel \overrightarrow{b}\parallel }_{\mathit{BMO}}{\parallel |f{|}_r\parallel }_{L^{n/\delta }}.$$

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 $Q=Q(0,d)$ ($d>1$), the following inequality holds:

$$\frac{1}{|Q|}{\int }_Q\left|\right|S_{\psi ,\delta }^{\overrightarrow{b}}(f)(x){|}_r-C_Q|dx\le C{\parallel f\parallel }_{B_p^{\delta }}.$$

Fix a cube $Q=Q(0,d)$ ($d>1$). Let $f=g+h=\{g_i\}+\{h_i\}$, where $g_i=f_i{\chi }_Q$, $h_i=f_i{\chi }_{{(Q)}^c}$ and ${\overrightarrow{b}}_Q=({(b_1)}_Q,\dots ,{(b_m)}_Q)$. For ${(b_j)}_Q=|Q{|}^{-1}{\int }_Q|b_j(y)|dy$, $1\le j\le m$, we have

$$\begin{array}{rcl}{F}_t^{\overrightarrow{b}}(f_i)(x,y)& =& {\int }_{R^n}\left[\prod _{j=1}^m(b_1(x)-b_1(z))\right]{\psi }_t(y-z)f_i(z)dz\\ =& (b_1(x)-{(b_1)}_Q)\cdots (b_m(x)-{(b_m)}_Q)F_t(f_i)(y)\\ +{(-1)}^m{F}_t((b_1-{(b_1)}_Q)\cdots (b_m-{(b_m)}_Q)g_i)(y)\\ +{(-1)}^m{F}_t((b_1-{(b_1)}_Q)\cdots (b_m-{(b_m)}_Q)h_i)(y)\\ +\sum _{j=1}^{m-1}\sum _{\sigma \in C_j^m}{(-1)}^{m-j}{(\overrightarrow{b}(x)-{\overrightarrow{b}}_Q)}_{\sigma }{F}_t\left({(\overrightarrow{b}-{\overrightarrow{b}}_Q)}_{{\sigma }^c}{f}_i\right)(x,y).\end{array}$$

By the Minkowski inequality, we have

$$\begin{array}{r}\left|\right|S_{\psi ,\delta }^{\overrightarrow{b}}(f)(x){|}_r-|S_{\psi ,\delta }(({(b_1)}_Q-b_1)\cdots ({(b_m)}_Q-b_m)h)(x_0){|}_r|\\ \le {\parallel {\chi }_{\mathrm{\Gamma }(x)}(b_1(x)-{(b_1)}_Q)\cdots (b_m(x)-{(b_m)}_Q)F_t(f)(y)\parallel }_r\\ +\sum _{j=1}^{m-1}\sum _{\sigma \in C_j^m}{\parallel {\chi }_{\mathrm{\Gamma }(x)}{(\overrightarrow{b}(x)-{\overrightarrow{b}}_Q)}_{\sigma }{F}_t\left({(\overrightarrow{b}-{\overrightarrow{b}}_Q)}_{{\sigma }^c}f\right)(x,y)\parallel }_r\\ +{\parallel {\chi }_{\mathrm{\Gamma }(x)}{F}_t((b_1-{(b_1)}_Q)\cdots (b_m-{(b_m)}_Q)g)(y)\parallel }_r\\ +{\parallel {\chi }_{\mathrm{\Gamma }(x)}{F}_t\left(\prod _{j=1}^m(b_j-{(b_j)}_Q)h\right)(y)-{\chi }_{\mathrm{\Gamma }(x_0)}{F}_t\left(\prod _{j=1}^m(b_j-{(b_j)}_Q)h\right)(y)\parallel }_r\\ =H_1(x)+H_2(x)+H_3(x)+H_4(x).\end{array}$$

For $H_1(x)$, take $1/q=1/p-\delta /n$, by the Hölder inequality and Lemma 1, we have

$$\begin{array}{r}\frac{1}{|Q|}{\int }_Q{H}_1(x)dx\\ \le {\left(\frac{1}{|Q|}{\int }_Q\right|\prod _{j=1}^m(b_j(x)-{(b_j)}_Q){|}^{q^{\prime }}dx)}^{1/q^{\prime }}{\left(\frac{1}{|Q|}{\int }_Q\right|S_{\psi ,\delta }(f)(x){|}_r^{q}dx)}^{1/q}\\ \le C{\parallel \overrightarrow{b}\parallel }_{\mathit{BMO}}|Q{|}^{-1/q}{\left({\int }_{R^n}\right|f(x){|}_r^p{\chi }_Q(x)dx)}^{1/p}\\ \le C{\parallel \overrightarrow{b}\parallel }_{\mathit{BMO}}{d}^{-n(1/p-\delta /n)}{\parallel |f{|}_r{\chi }_Q\parallel }_{L^p}\\ \le C{\parallel \overrightarrow{b}\parallel }_{\mathit{BMO}}{\parallel |f{|}_r\parallel }_{B_p^{\delta }}.\end{array}$$

For $H_2(x)$, taking $1<u<p<n/\delta$, $1/v=1/u-\delta /n$, we get

$$\begin{array}{rl}\frac{1}{|Q|}{\int }_Q{H}_2(x)dx\le & C\sum _{j=1}^{m-1}\sum _{\sigma \in C_j^m}{\left(\frac{1}{|Q|}{\int }_Q\right|{(b(x)-b_Q)}_{\sigma }{|}^{v^{\prime }}dx)}^{1/v^{\prime }}\\ \times {\left(\frac{1}{|Q|}{\int }_Q\right|S_{\psi ,\delta }\left({(b-b_Q)}_{{\sigma }^c}f\right)(x){|}_r^{u}dx)}^{1/u}\\ \le & C\sum _{j=1}^{m-1}\sum _{\sigma \in C_j^m}{\parallel {\overrightarrow{b}}_{\sigma }\parallel }_{\mathit{BMO}}|Q{|}^{-1/v}{\left({\int }_{R^n}\right|{(b(x)-b_Q)}_{{\sigma }^c}f(x){|}_r^u{\chi }_Q(x)dx)}^{1/u}\\ \le & C\sum _{j=1}^{m-1}\sum _{\sigma \in C_j^m}{\parallel {\overrightarrow{b}}_{\sigma }\parallel }_{\mathit{BMO}}|Q{|}^{(\delta /n-1/p)}{\parallel |f{|}_r{\chi }_Q\parallel }_{L^p}\\ \times {\left(\frac{1}{|Q|}{\int }_Q\right|{(b(x)-b_Q)}_{{\sigma }^c}{|}^{pr/(p-r)}dx)}^{(p-u)/pu}\\ \le & C\sum _{j=1}^{m-1}\sum _{\sigma \in C_j^m}{\parallel {\overrightarrow{b}}_{\sigma }\parallel }_{\mathit{BMO}}{\parallel {\overrightarrow{b}}_{{\sigma }^c}\parallel }_{\mathit{BMO}}{d}^{-n(1/p-\delta /n)}{\parallel |f{|}_r{\chi }_Q\parallel }_{L^p}\\ \le & C{\parallel \overrightarrow{b}\parallel }_{\mathit{BMO}}{\parallel |f{|}_r\parallel }_{B_p^{\delta }}.\end{array}$$

For $H_3(x)$, taking $1<u<p<n/\delta$, $1/v=1/u-\delta /n$, we obtain

$$\begin{array}{r}\frac{1}{|Q|}{\int }_Q{H}_3(x)dx\\ \le {\left(\frac{1}{|Q|}{\int }_Q\right|S_{\psi ,\delta }((b_1-{(b_1)}_Q)\cdots (b_m-{(b_m)}_Q)g)(x){|}_r^{v}dx)}^{1/v}\\ \le C|Q{|}^{-1/v}{\left({\int }_Q\right|(b_1(x)-{(b_1)}_Q)\cdots (b_m(x)-{(b_m)}_Q)g(x){|}_r^{u}dx)}^{1/u}\\ \le C|Q{|}^{-1/v}{\parallel (b_1(x)-{(b_1)}_Q)\cdots (b_m(x)-{(b_m)}_Q)|f{|}_r{\chi }_Q\parallel }_{L^v}\\ \le C{\parallel \overrightarrow{b}\parallel }_{\mathit{BMO}}{\parallel |f{|}_r\parallel }_{B_p^{\delta }}.\end{array}$$

For $H_4(x)$, we have

$$\begin{array}{rcl}{I}_4(x)& \le & {[\int {\int }_{R_{+}^{n+1}}{({\int }_{Q^c}|{\chi }_{\mathrm{\Gamma }(x)}-{\chi }_{\mathrm{\Gamma }(x_0)}|\prod _{j=1}^m|b_j(z)-{(b_j)}_Q\left|\right|{\psi }_t(y-z)\left|\right|f(z){|}_{r}dz)}^2\frac{dydt}{t^{n+1}}]}^{1/2}\\ \le & C\sum _{k=0}^{\mathrm{\infty }}{\int }_{2^{k+1}Q\setminus 2^{k}Q}|x_0-x{|}^{1/2}|x_0-z{|}^{-(n+1/2-2\delta )}\left|\prod _{j=1}^m(b_j(z)-{(b_j)}_Q)\right||f(z){|}_{r}dz\\ \le & C\sum _{k=1}^{\mathrm{\infty }}{2}^{-k/2}\left|2^{k+1}Q{|}^{-1+\delta /n}{\int }_{2^{k+1}Q}\right|\prod _{j=1}^m(b_j(z)-{(b_j)}_Q)\left|\right|f(z){|}_{r}dz\\ \le & C\sum _{k=1}^{\mathrm{\infty }}{k}^m{2}^{-k/2}|2^{k}Q{|}^{-(1/p-\delta /n)}{\parallel \overrightarrow{b}\parallel }_{\mathit{BMO}}{\parallel |f{|}_r{\chi }_{2^{k}Q}\parallel }_{L^p}\\ \le & C{\parallel \overrightarrow{b}\parallel }_{\mathit{BMO}}{\parallel |f{|}_r\parallel }_{B_p^{\delta }},\end{array}$$

so

$$\frac{1}{|Q|}{\int }_Q|H_4(x)|dx\le C{\parallel \overrightarrow{b}\parallel }_{\mathit{BMO}}{\parallel |f{|}_r\parallel }_{B_p^{\delta }}.$$

This completes the proof of Theorem 2. □