# Some sharp inequalities for multilinear integral operators

## Abstract

In this paper, some sharp inequalities for certain multilinear operators related to the Littlewood-Paley operator and the Marcinkiewicz operator are obtained. As an application, we obtain the $\left({L}^{p},{L}^{q}\right)$-norm inequalities and Morrey spaces boundedness for the multilinear operators.

MSC:42B20, 42B25.

## 1 Introduction and results

In this paper, we study some multilinear operators related to some integral operators, whose definitions are as follows.

Fix $n>\delta \ge 0$. We denote $\mathrm{\Gamma }\left(x\right)=\left\{\left(y,t\right)\in {R}_{+}^{n+1}:|x-y| and the characteristic function of $\mathrm{\Gamma }\left(x\right)$ by ${\chi }_{\mathrm{\Gamma }\left(x\right)}$. Suppose that ${m}_{j}$ are the positive integers ($j=1,\dots ,l$), ${m}_{1}+\cdots +{m}_{l}=m$ and ${A}_{j}$ are the functions on ${R}^{n}$ ($j=1,\dots ,l$). Let

${R}_{{m}_{j}+1}\left({A}_{j};x,y\right)={A}_{j}\left(x\right)-\sum _{|\alpha |\le {m}_{j}}\frac{1}{\alpha !}{D}^{\alpha }{A}_{j}\left(y\right){\left(x-y\right)}^{\alpha }.$

Definition 1 Let $\epsilon >0$ and ψ be a fixed function which satisfies the following properties:

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)

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

The multilinear Littlewood-Paley operator is defined by

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

where

${F}_{t}^{A}\left(f\right)\left(x,y\right)={\int }_{{R}^{n}}\frac{{\prod }_{j=1}^{l}{R}_{{m}_{j}+1}\left({A}_{j};x,z\right)}{{|x-z|}^{m}}{\psi }_{t}\left(y-z\right)f\left(z\right)\phantom{\rule{0.2em}{0ex}}dz$

and ${\psi }_{t}\left(x\right)={t}^{-n+\delta }\psi \left(x/t\right)$ for $t>0$. Set ${F}_{t}\left(f\right)\left(y\right)=f\ast {\psi }_{t}\left(y\right)$. We also define that

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

which is the Littlewood-Paley operator (see ).

Let H be the Hilbert space $H=\left\{h:\parallel h\parallel ={\left(\int {\int }_{{R}_{+}^{n+1}}{|h\left(y,t\right)|}^{2}\phantom{\rule{0.2em}{0ex}}dy\phantom{\rule{0.2em}{0ex}}dt/{t}^{n+1}\right)}^{1/2}<\mathrm{\infty }\right\}$. Then for each fixed $x\in {R}^{n}$, ${F}_{t}^{A}\left(f\right)\left(x,y\right)$ may be viewed as a mapping from $\left(0,+\mathrm{\infty }\right)$ to H, and it is clear that

${S}_{\psi }^{A}\left(f\right)\left(x\right)=\parallel {\chi }_{\mathrm{\Gamma }\left(x\right)}{F}_{t}^{A}\left(f\right)\left(x,y\right)\parallel ,\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}{S}_{\psi }\left(f\right)\left(x\right)=\parallel {\chi }_{\mathrm{\Gamma }\left(x\right)}{F}_{t}\left(f\right)\left(y\right)\parallel .$

Definition 2 Let $0<\gamma \le 1$ and Ω be homogeneous of degree zero on ${R}^{n}$ with ${\int }_{{S}^{n-1}}\mathrm{\Omega }\left({x}^{\prime }\right)\phantom{\rule{0.2em}{0ex}}d\sigma \left({x}^{\prime }\right)=0$. Assume that $\mathrm{\Omega }\in {Lip}_{\gamma }\left({S}^{n-1}\right)$, that is, there exists a constant $M>0$ such that for any $x,y\in {S}^{n-1}$, $|\mathrm{\Omega }\left(x\right)-\mathrm{\Omega }\left(y\right)|\le M{|x-y|}^{\gamma }$. The multilinear Marcinkiewicz operator is defined by

${\mu }_{S}^{A}\left(f\right)\left(x\right)={\left[\int {\int }_{\mathrm{\Gamma }\left(x\right)}{|{F}_{t}^{A}\left(f\right)\left(x,y\right)|}^{2}\frac{dy\phantom{\rule{0.2em}{0ex}}dt}{{t}^{n+3}}\right]}^{1/2},$

where

${F}_{t}^{A}\left(f\right)\left(x,y\right)={\int }_{|y-z|\le t}\frac{{\prod }_{j=1}^{l}{R}_{{m}_{j}+1}\left({A}_{j};x,z\right)}{{|x-z|}^{m}}\frac{\mathrm{\Omega }\left(y-z\right)}{{|y-z|}^{n-1-\delta }}f\left(z\right)\phantom{\rule{0.2em}{0ex}}dz.$

Set

${F}_{t}\left(f\right)\left(y\right)={\int }_{|y-z|\le t}\frac{\mathrm{\Omega }\left(y-z\right)}{{|y-z|}^{n-1-\delta }}f\left(z\right)\phantom{\rule{0.2em}{0ex}}dz.$

We also define that

${\mu }_{S}\left(f\right)\left(x\right)={\left(\int {\int }_{\mathrm{\Gamma }\left(x\right)}{|{F}_{t}\left(f\right)\left(y\right)|}^{2}\frac{dy\phantom{\rule{0.2em}{0ex}}dt}{{t}^{n+3}}\right)}^{1/2},$

which is the Marcinkiewicz operator (see ).

Let H be the Hilbert space $H=\left\{h:\parallel h\parallel ={\left(\int {\int }_{{R}_{+}^{n+1}}{|h\left(y,t\right)|}^{2}\phantom{\rule{0.2em}{0ex}}dy\phantom{\rule{0.2em}{0ex}}dt/{t}^{n+3}\right)}^{1/2}<\mathrm{\infty }\right\}$, then for each fixed $x\in {R}^{n}$, ${F}_{t}^{A}\left(f\right)\left(x,y\right)$ may be viewed as a mapping from $\left(0,+\mathrm{\infty }\right)$ to H, and it is clear that

${\mu }_{S}^{A}\left(f\right)\left(x\right)=\parallel {\chi }_{\mathrm{\Gamma }\left(x\right)}{F}_{t}^{A}\left(f\right)\left(x,y\right)\parallel ,\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}{\mu }_{S}\left(f\right)\left(x\right)=\parallel {\chi }_{\mathrm{\Gamma }\left(x\right)}{F}_{t}\left(f\right)\left(y\right)\parallel .$

Note that when $m=0$, ${S}_{\psi }^{A}$ and ${\mu }_{S}^{A}$ are just the multilinear commutators (see [3, 4]). While when $m>0$, ${S}_{\psi }^{A}$ and ${\mu }_{S}^{A}$ are non-trivial generalizations of the commutators. It is well known that multilinear operators are of great interest in harmonic analysis and have been widely studied by many authors (see ). In , Hu and Yang proved a variant sharp estimate for the multilinear singular integral operators. In , authors proved a sharp estimate for the multilinear commutator. The main purpose of this paper is to prove the sharp inequalities for the multilinear integral operators ${S}_{\psi }^{A}$ and ${\mu }_{S}^{A}$ when ${D}^{\alpha }{A}_{j}\in BMO\left({R}^{n}\right)$ for all α with $|\alpha |={m}_{j}$. As an application, we obtain the $\left({L}^{p},{L}^{q}\right)$-norm inequalities and Morrey spaces boundedness for the multilinear operators.

First, let us introduce some notations. Throughout this paper, Q will denote a cube of ${R}^{n}$ with sides parallel to the axes. For any locally integrable function f, the sharp function of f is defined by

${f}^{\mathrm{#}}\left(x\right)=\underset{Q\ni x}{sup}\frac{1}{|Q|}{\int }_{Q}|f\left(y\right)-{f}_{Q}|\phantom{\rule{0.2em}{0ex}}dy,$

where, and in what follows, ${f}_{Q}={|Q|}^{-1}{\int }_{Q}f\left(x\right)\phantom{\rule{0.2em}{0ex}}dx$. It is well-known that (see [14, 15])

${f}^{\mathrm{#}}\left(x\right)=\underset{Q\ni x}{sup}\underset{c\in C}{inf}\frac{1}{|Q|}{\int }_{Q}|f\left(y\right)-c|\phantom{\rule{0.2em}{0ex}}dy.$

We say that f belongs to $BMO\left({R}^{n}\right)$ if ${f}^{\mathrm{#}}$ belongs to ${L}^{\mathrm{\infty }}\left({R}^{n}\right)$ and ${\parallel f\parallel }_{BMO}={\parallel {f}^{\mathrm{#}}\parallel }_{{L}^{\mathrm{\infty }}}$. For $1\le p<\mathrm{\infty }$ and $0\le \delta , let

${M}_{\delta ,p}\left(f\right)\left(x\right)=\underset{Q\ni x}{sup}{\left(\frac{1}{{|Q|}^{1-p\delta /n}}{\int }_{Q}{|f\left(y\right)|}^{p}\phantom{\rule{0.2em}{0ex}}dy\right)}^{1/p};$

we write that ${M}_{\mu }\left(f\right)={M}_{n\mu ,1}\left(f\right)$, which is the fractional maximal operator.

Fixed $\lambda >0$. For $1\le p<\mathrm{\infty }$, let

${\parallel f\parallel }_{{L}^{p,\lambda }}=\underset{x\in {R}^{n},d>0}{sup}{\left(\frac{1}{{d}^{\lambda }}{\int }_{B\left(x,d\right)}{|f\left(y\right)|}^{p}\phantom{\rule{0.2em}{0ex}}dy\right)}^{1/p},$

where $B\left(x,d\right)=\left\{y\in {R}^{n}:|x-y|. The Morrey spaces are defined by (see )

${L}^{p,\lambda }\left({R}^{n}\right)=\left\{f\in {L}_{loc}^{1}\left({R}^{n}\right):{\parallel f\parallel }_{{L}^{p,\lambda }}<\mathrm{\infty }\right\}.$

As the Morrey space may be considered as an extension of the Lebesgue space, it is natural and important to study the boundedness of the multilinear integral operator on the Morrey space.

We shall prove the following theorems.

Theorem 1 Let ${D}^{\alpha }{A}_{j}\in BMO\left({R}^{n}\right)$ for all α with $|\alpha |={m}_{j}$ and $j=1,\dots ,l$.

1. (1)

Then there exists a constant $C>0$ such that for any $f\in {C}_{0}^{\mathrm{\infty }}\left({R}^{n}\right)$, $1 and $x\in {R}^{n}$,

${\left({S}_{\psi }^{A}\left(f\right)\right)}^{\mathrm{#}}\left(x\right)\le C\prod _{j=1}^{l}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{A}_{j}\parallel }_{BMO}\right){M}_{\delta ,r}\left(f\right)\left(x\right);$
2. (2)

If $1 and $1/p-1/q=\delta /n$, then ${S}_{\psi }^{A}$ is bounded from ${L}^{p}\left({R}^{n}\right)$ to ${L}^{q}\left({R}^{n}\right)$, that is,

${\parallel {S}_{\psi }^{A}\left(f\right)\parallel }_{{L}^{q}}\le C\prod _{j=1}^{l}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{A}_{j}\parallel }_{BMO}\right){\parallel f\parallel }_{{L}^{p}};$
3. (3)

If $1, $0<\lambda , $1/q=1/p-\delta /\left(n-\lambda \right)$, then ${S}_{\psi }^{A}$ is bounded from ${L}^{p,\lambda }\left({R}^{n}\right)$ to ${L}^{q,\lambda }\left({R}^{n}\right)$, that is,

${\parallel {S}_{\psi }^{A}\left(f\right)\parallel }_{{L}^{q,\lambda }}\le C\prod _{j=1}^{l}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{A}_{j}\parallel }_{BMO}\right){\parallel f\parallel }_{{L}^{p,\lambda }}.$

Theorem 2 Let ${D}^{\alpha }{A}_{j}\in BMO\left({R}^{n}\right)$ for all α with $|\alpha |={m}_{j}$ and $j=1,\dots ,l$.

1. (1)

Then there exists a constant $C>0$ such that for any $f\in {C}_{0}^{\mathrm{\infty }}\left({R}^{n}\right)$, $1 and $x\in {R}^{n}$,

${\left({\mu }_{S}^{A}\left(f\right)\right)}^{\mathrm{#}}\left(x\right)\le C\prod _{j=1}^{l}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{A}_{j}\parallel }_{BMO}\right){M}_{\delta ,r}\left(f\right)\left(x\right);$
2. (2)

If $1 and $1/p-1/q=\delta /n$, then ${\mu }_{S}^{A}$ is bounded from ${L}^{p}\left({R}^{n}\right)$ to ${L}^{q}\left({R}^{n}\right)$, that is,

${\parallel {\mu }_{S}^{A}\left(f\right)\parallel }_{{L}^{q}}\le C\prod _{j=1}^{l}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{A}_{j}\parallel }_{BMO}\right){\parallel f\parallel }_{{L}^{p}};$
3. (3)

If $1, $0<\lambda , $1/q=1/p-\delta /\left(n-\lambda \right)$, then ${\mu }_{S}^{A}$ is bounded from ${L}^{p,\lambda }\left({R}^{n}\right)$ to ${L}^{q,\lambda }\left({R}^{n}\right)$, that is,

${\parallel {\mu }_{S}^{A}\left(f\right)\parallel }_{{L}^{q,\lambda }}\le C\prod _{j=1}^{l}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{A}_{j}\parallel }_{BMO}\right){\parallel f\parallel }_{{L}^{p,\lambda }}.$

Remark The conclusions of Theorems 1 and 2 are completely the same. Thus, they explain that the Littlewood-Paley and Marcinkiewicz operators have the many similar bondedness properties.

## 2 Proofs of theorems

To prove the theorems, we need the following lemmas.

Lemma 1 

Let A be a function on ${R}^{n}$ and ${D}^{\alpha }A\in {L}^{q}\left({R}^{n}\right)$ for all α with $|\alpha |=m$ and some $q>n$. Then

$|{R}_{m}\left(A;x,y\right)|\le C{|x-y|}^{m}\sum _{|\alpha |=m}{\left(\frac{1}{|\stackrel{˜}{Q}\left(x,y\right)|}{\int }_{\stackrel{˜}{Q}\left(x,y\right)}{|{D}^{\alpha }A\left(z\right)|}^{q}\phantom{\rule{0.2em}{0ex}}dz\right)}^{1/q},$

where $\stackrel{˜}{Q}$ is the cube centered at x and having side length $5\sqrt{n}|x-y|$.

Lemma 2 

Suppose that $1\le r and $1/q=1/p-\delta /n$. Then

${\parallel {M}_{\delta ,r}\left(f\right)\parallel }_{{L}^{q}}\le C{\parallel f\parallel }_{{L}^{p}}.$

Lemma 3 [16, 17]

Let $1 and $0<\lambda . Then the following estimates hold:

1. (a)

${\parallel M\left(f\right)\parallel }_{{L}^{p,\lambda }}\le C{\parallel {f}^{\mathrm{#}}\parallel }_{{L}^{p,\lambda }}$;

2. (b)

${\parallel {M}_{\mu }\left(f\right)\parallel }_{{L}^{q,\lambda }}\le C{\parallel f\parallel }_{{L}^{p,\lambda }}$ for $0<\mu <\left(n-\lambda \right)/np$ and $1/q=1/p-n\eta /\left(n-\lambda \right)$.

Lemma 4 Let $1 and $1/q=1/p-\delta /n$. Then ${S}_{\psi }$ and ${\mu }_{S}$ are all bounded from ${L}^{p}\left({R}^{n}\right)$ to ${L}^{q}\left({R}^{n}\right)$.

Proof For ${S}_{\psi }$, by Minkowski inequality and the condition of ψ, we have

$\begin{array}{rcl}{S}_{\psi }\left(f\right)\left(x\right)& \le & {\int }_{{R}^{n}}|f\left(z\right)|{\left({\int }_{\mathrm{\Gamma }\left(x\right)}{|{\psi }_{t}\left(y-z\right)|}^{2}\frac{dy\phantom{\rule{0.2em}{0ex}}dt}{{t}^{1+n}}\right)}^{1/2}\phantom{\rule{0.2em}{0ex}}dz\\ \le & C{\int }_{{R}^{n}}|f\left(z\right)|{\left({\int }_{0}^{\mathrm{\infty }}{\int }_{|x-y|\le t}\frac{{t}^{-2n+2\delta }}{{\left(1+|y-z|/t\right)}^{2n+2-2\delta }}\frac{dy\phantom{\rule{0.2em}{0ex}}dt}{{t}^{1+n}}\right)}^{1/2}\phantom{\rule{0.2em}{0ex}}dz\\ \le & C{\int }_{{R}^{n}}|f\left(z\right)|{\left({\int }_{0}^{\mathrm{\infty }}{\int }_{|x-y|\le t}\frac{{2}^{2n+2}{t}^{1-n}}{{\left(2t+|y-z|\right)}^{2n+2-2\delta }}\phantom{\rule{0.2em}{0ex}}dy\phantom{\rule{0.2em}{0ex}}dt\right)}^{1/2}\phantom{\rule{0.2em}{0ex}}dz,\end{array}$

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

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

we obtain

$\begin{array}{rcl}{S}_{\psi }\left(f\right)\left(x\right)& \le & C{\int }_{{R}^{n}}|f\left(z\right)|{\left({\int }_{0}^{\mathrm{\infty }}\frac{t\phantom{\rule{0.2em}{0ex}}dt}{{\left(t+|x-z|\right)}^{2n+2-2\delta }}\right)}^{1/2}\phantom{\rule{0.2em}{0ex}}dz\\ =& C{\int }_{{R}^{n}}\frac{|f\left(z\right)|}{{|x-z|}^{n-\delta }}\phantom{\rule{0.2em}{0ex}}dz.\end{array}$

For ${\mu }_{S}$, note that $|x-z|\le 2t$, $|y-z|\ge |x-z|-t\ge |x-z|-3t$ when $|x-y|\le t$, $|y-z|\le t$, we have

$\begin{array}{rcl}{\mu }_{S}\left(f\right)\left(x\right)& \le & {\int }_{{R}^{n}}{\left[\int {\int }_{|x-y|\le t}{\left(\frac{|\mathrm{\Omega }\left(y-z\right)\parallel f\left(z\right)|}{{|y-z|}^{n-1-\delta }}\right)}^{2}{\chi }_{\mathrm{\Gamma }\left(z\right)}\left(y,t\right)\frac{dy\phantom{\rule{0.2em}{0ex}}dt}{{t}^{n+3}}\right]}^{1/2}\phantom{\rule{0.2em}{0ex}}dz\\ \le & C{\int }_{{R}^{n}}|f\left(z\right)|{\left[\int {\int }_{|x-y|\le t}\frac{{\chi }_{\mathrm{\Gamma }\left(z\right)}\left(y,t\right){t}^{-n-3}}{{\left(|x-z|-3t\right)}^{2n-2-2\delta }}\phantom{\rule{0.2em}{0ex}}dy\phantom{\rule{0.2em}{0ex}}dt\right]}^{1/2}\phantom{\rule{0.2em}{0ex}}dz\\ \le & C{\int }_{{R}^{n}}\frac{|f\left(z\right)|}{{|x-z|}^{3/2}}{\left[{\int }_{|x-z|/2}^{\mathrm{\infty }}\frac{dt}{{\left(|x-z|-3t\right)}^{2n-2}}\right]}^{1/2}\phantom{\rule{0.2em}{0ex}}dz\\ \le & C{\int }_{{R}^{n}}\frac{|f\left(z\right)|}{{|x-z|}^{n-\delta }}\phantom{\rule{0.2em}{0ex}}dz.\end{array}$

Thus, the lemma follows from . □

Proof of Theorem 1 (1) It suffices to prove for $f\in {C}_{0}^{\mathrm{\infty }}\left({R}^{n}\right)$ and some constant ${C}_{0}$, the following inequality holds:

$\frac{1}{|Q|}{\int }_{Q}|{S}_{\psi }^{A}\left(f\right)\left(x\right)-{C}_{0}|\phantom{\rule{0.2em}{0ex}}dx\le C\prod _{j=1}^{l}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{A}_{j}\parallel }_{BMO}\right){M}_{\delta ,r}\left(f\right)\left(x\right).$

Without loss of generality, we may assume $l=2$. Fix a cube $Q=Q\left({x}_{0},d\right)$ and $\stackrel{˜}{x}\in Q$. Let $\stackrel{˜}{Q}=5\sqrt{n}Q$ and ${\stackrel{˜}{A}}_{j}\left(x\right)={A}_{j}\left(x\right)-{\sum }_{|\alpha |={m}_{j}}\frac{1}{\alpha !}{\left({D}^{\alpha }{A}_{j}\right)}_{\stackrel{˜}{Q}}{x}^{\alpha }$, then ${R}_{{m}_{j}}\left({A}_{j};x,y\right)={R}_{{m}_{j}}\left({\stackrel{˜}{A}}_{j};x,y\right)$ and ${D}^{\alpha }{\stackrel{˜}{A}}_{j}={D}^{\alpha }{A}_{j}-{\left({D}^{\alpha }{A}_{j}\right)}_{\stackrel{˜}{Q}}$ for $|\alpha |={m}_{j}$. We write, for ${f}_{1}=f{\chi }_{\stackrel{˜}{Q}}$ and ${f}_{2}=f{\chi }_{{R}^{n}\setminus \stackrel{˜}{Q}}$,

$\begin{array}{rcl}{F}_{t}^{A}\left(f\right)\left(x,y\right)& =& {\int }_{{R}^{n}}\frac{{\prod }_{j=1}^{2}{R}_{{m}_{j}+1}\left({\stackrel{˜}{A}}_{j};x,z\right)}{{|x-z|}^{m}}{\psi }_{t}\left(y-z\right)f\left(z\right)\phantom{\rule{0.2em}{0ex}}dz\\ =& {\int }_{{R}^{n}}\frac{{\prod }_{j=1}^{2}{R}_{{m}_{j}+1}\left({\stackrel{˜}{A}}_{j};x,z\right)}{{|x-z|}^{m}}{\psi }_{t}\left(y-z\right){f}_{2}\left(z\right)\phantom{\rule{0.2em}{0ex}}dz\\ +{\int }_{{R}^{n}}\frac{{\prod }_{j=1}^{2}{R}_{{m}_{j}}\left({\stackrel{˜}{A}}_{j};x,z\right)}{{|x-z|}^{m}}{\psi }_{t}\left(y-z\right){f}_{1}\left(z\right)\phantom{\rule{0.2em}{0ex}}dz\\ -\sum _{|{\alpha }_{1}|={m}_{1}}\frac{1}{{\alpha }_{1}!}{\int }_{{R}^{n}}\frac{{R}_{{m}_{2}}\left({\stackrel{˜}{A}}_{2};x,z\right){\left(x-z\right)}^{{\alpha }_{1}}}{{|x-z|}^{m}}{D}^{{\alpha }_{1}}{\stackrel{˜}{A}}_{1}\left(z\right){\psi }_{t}\left(y-z\right){f}_{1}\left(z\right)\phantom{\rule{0.2em}{0ex}}dz\\ -\sum _{|{\alpha }_{2}|={m}_{2}}\frac{1}{{\alpha }_{2}!}{\int }_{{R}^{n}}\frac{{R}_{{m}_{1}}\left({\stackrel{˜}{A}}_{1};x,z\right){\left(x-z\right)}^{{\alpha }_{2}}}{{|x-z|}^{m}}{D}^{{\alpha }_{2}}{\stackrel{˜}{A}}_{2}\left(z\right){\psi }_{t}\left(y-z\right){f}_{1}\left(z\right)\phantom{\rule{0.2em}{0ex}}dz\\ +\sum _{|{\alpha }_{1}|={m}_{1},|{\alpha }_{2}|={m}_{2}}\frac{1}{{\alpha }_{1}!{\alpha }_{2}!}{\int }_{{R}^{n}}\frac{{\left(x-z\right)}^{{\alpha }_{1}+{\alpha }_{2}}{D}^{{\alpha }_{1}}{\stackrel{˜}{A}}_{1}\left(z\right){D}^{{\alpha }_{2}}{\stackrel{˜}{A}}_{2}\left(z\right)}{{|x-z|}^{m}}{\psi }_{t}\left(y-z\right){f}_{1}\left(z\right)\phantom{\rule{0.2em}{0ex}}dz,\end{array}$

then

$\begin{array}{c}|{S}_{\psi }^{A}\left(f\right)\left(x\right)-{S}_{\psi }^{\stackrel{˜}{A}}\left({f}_{2}\right)\left({x}_{0}\right)|\hfill \\ \phantom{\rule{1em}{0ex}}=|\parallel {\chi }_{\mathrm{\Gamma }\left(x\right)}{F}_{t}^{A}\left(f\right)\left(x,y\right)\parallel -\parallel {\chi }_{\mathrm{\Gamma }\left({x}_{0}\right)}{F}_{t}^{\stackrel{˜}{A}}\left({f}_{2}\right)\left({x}_{0},y\right)\parallel |\hfill \\ \phantom{\rule{1em}{0ex}}\le \parallel {\chi }_{\mathrm{\Gamma }\left(x\right)}{F}_{t}^{A}\left(f\right)\left(x,y\right)-{\chi }_{\mathrm{\Gamma }\left({x}_{0}\right)}{F}_{t}^{\stackrel{˜}{A}}\left({f}_{2}\right)\left({x}_{0},y\right)\parallel \hfill \\ \phantom{\rule{1em}{0ex}}\le \parallel {\chi }_{\mathrm{\Gamma }\left(x\right)}{\int }_{{R}^{n}}\frac{{\prod }_{j=1}^{2}{R}_{{m}_{j}}\left({\stackrel{˜}{A}}_{j};x,z\right)}{{|x-z|}^{m}}{\psi }_{t}\left(y-z\right){f}_{1}\left(z\right)\phantom{\rule{0.2em}{0ex}}dz\parallel \hfill \\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}+\parallel {\chi }_{\mathrm{\Gamma }\left(x\right)}\sum _{|{\alpha }_{1}|={m}_{1}}\frac{1}{{\alpha }_{1}!}{\int }_{{R}^{n}}\frac{{R}_{{m}_{2}}\left({\stackrel{˜}{A}}_{2};x,z\right){\left(x-z\right)}^{{\alpha }_{1}}}{{|x-z|}^{m}}{D}^{{\alpha }_{1}}{\stackrel{˜}{A}}_{1}\left(z\right){\psi }_{t}\left(y-z\right){f}_{1}\left(z\right)\phantom{\rule{0.2em}{0ex}}dz\parallel \hfill \\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}+\parallel {\chi }_{\mathrm{\Gamma }\left(x\right)}\sum _{|{\alpha }_{2}|={m}_{2}}\frac{1}{{\alpha }_{2}!}{\int }_{{R}^{n}}\frac{{R}_{{m}_{1}}\left({\stackrel{˜}{A}}_{1};x,z\right){\left(x-z\right)}^{{\alpha }_{2}}}{{|x-z|}^{m}}{D}^{{\alpha }_{2}}{\stackrel{˜}{A}}_{2}\left(z\right){\psi }_{t}\left(y-z\right){f}_{1}\left(z\right)\phantom{\rule{0.2em}{0ex}}dz\parallel \hfill \\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}+\parallel {\chi }_{\mathrm{\Gamma }\left(x\right)}\sum _{|{\alpha }_{1}|={m}_{1},|{\alpha }_{2}|={m}_{2}}\frac{1}{{\alpha }_{1}!{\alpha }_{2}!}{\int }_{{R}^{n}}\frac{{\left(x-z\right)}^{{\alpha }_{1}+{\alpha }_{2}}{D}^{{\alpha }_{1}}{\stackrel{˜}{A}}_{1}\left(z\right){D}^{{\alpha }_{2}}{\stackrel{˜}{A}}_{2}\left(z\right)}{{|x-z|}^{m}}{\psi }_{t}\left(y-z\right){f}_{1}\left(z\right)\phantom{\rule{0.2em}{0ex}}dz\parallel \hfill \\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}+\parallel {\chi }_{\mathrm{\Gamma }\left(x\right)}{F}_{t}^{\stackrel{˜}{A}}\left({f}_{2}\right)\left(x,y\right)-{\chi }_{\mathrm{\Gamma }\left({x}_{0}\right)}{F}_{t}^{\stackrel{˜}{A}}\left({f}_{2}\right)\left({x}_{0},y\right)\parallel \hfill \\ \phantom{\rule{1em}{0ex}}:={I}_{1}\left(x\right)+{I}_{2}\left(x\right)+{I}_{3}\left(x\right)+{I}_{4}\left(x\right)+{I}_{5}\left(x\right),\hfill \end{array}$

thus,

$\begin{array}{c}\frac{1}{|Q|}{\int }_{Q}|{S}_{\psi }^{A}\left(f\right)\left(x\right)-{S}_{\psi }^{\stackrel{˜}{A}}\left({f}_{2}\right)\left({x}_{0}\right)|\phantom{\rule{0.2em}{0ex}}dx\hfill \\ \phantom{\rule{1em}{0ex}}\le \frac{1}{|Q|}{\int }_{Q}{I}_{1}\left(x\right)\phantom{\rule{0.2em}{0ex}}dx+\frac{C}{|Q|}{\int }_{Q}{I}_{2}\left(x\right)\phantom{\rule{0.2em}{0ex}}dx+\frac{C}{|Q|}{\int }_{Q}{I}_{3}\left(x\right)\phantom{\rule{0.2em}{0ex}}dx\hfill \\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}+\frac{C}{|Q|}{\int }_{Q}{I}_{4}\left(x\right)\phantom{\rule{0.2em}{0ex}}dx+\frac{1}{|Q|}{\int }_{Q}{I}_{5}\left(x\right)\phantom{\rule{0.2em}{0ex}}dx\hfill \\ \phantom{\rule{1em}{0ex}}:={I}_{1}+{I}_{2}+{I}_{3}+{I}_{4}+{I}_{5}.\hfill \end{array}$

Now, let us estimate ${I}_{1}$, ${I}_{2}$, ${I}_{3}$, ${I}_{4}$ and ${I}_{5}$, respectively. First, for $x\in Q$ and $z\in \stackrel{˜}{Q}$, by Lemma 1, we get

${R}_{{m}_{j}}\left({\stackrel{˜}{A}}_{j};x,z\right)\le C{|x-y|}^{{m}_{j}}\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{A}_{j}\parallel }_{BMO}.$

Thus, by the $\left({L}^{r},{L}^{q}\right)$-boundedness of ${S}_{\psi }$, for $1 and $1/q=1/r-\delta /n$, we obtain

$\begin{array}{rcl}{I}_{1}& \le & C\prod _{j=1}^{2}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{A}_{j}\parallel }_{BMO}\right)\frac{1}{|Q|}{\int }_{Q}|{S}_{\psi }\left({f}_{1}\right)\left(x\right)|\phantom{\rule{0.2em}{0ex}}dx\\ \le & C\prod _{j=1}^{2}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{A}_{j}\parallel }_{BMO}\right){\left(\frac{1}{|Q|}{\int }_{Q}{|{S}_{\psi }\left({f}_{1}\right)\left(x\right)|}^{q}\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/q}\\ \le & C\prod _{j=1}^{2}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{A}_{j}\parallel }_{BMO}\right){|Q|}^{-1/q}{\left({\int }_{Q}{|{f}_{1}\left(x\right)|}^{r}\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/r}\\ \le & C\prod _{j=1}^{2}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{A}_{j}\parallel }_{BMO}\right){M}_{\delta ,r}\left(f\right)\left(\stackrel{˜}{x}\right).\end{array}$

For ${I}_{2}$, denoting $r=pq$ for $1, $q>1$, $1/q+1/{q}^{\prime }=1$ and $1/s=1/p-\delta /n$, we have, by Hölder’s inequality,

$\begin{array}{rcl}{I}_{2}& \le & C\sum _{|{\alpha }_{2}|={m}_{2}}{\parallel {D}^{{\alpha }_{2}}{A}_{2}\parallel }_{BMO}\sum _{|{\alpha }_{1}|={m}_{1}}\frac{1}{|Q|}{\int }_{Q}|{S}_{\psi }\left({D}^{{\alpha }_{1}}{\stackrel{˜}{A}}_{1}{f}_{1}\right)\left(x\right)|\phantom{\rule{0.2em}{0ex}}dx\\ \le & C\sum _{|{\alpha }_{2}|={m}_{2}}{\parallel {D}^{{\alpha }_{2}}{A}_{2}\parallel }_{BMO}\sum _{|{\alpha }_{1}|={m}_{1}}{\left(\frac{1}{|Q|}{\int }_{{R}^{n}}{|{S}_{\psi }\left({D}^{{\alpha }_{1}}{\stackrel{˜}{A}}_{1}{f}_{1}\right)\left(x\right)|}^{s}\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/s}\\ \le & C\sum _{|{\alpha }_{2}|={m}_{2}}{\parallel {D}^{{\alpha }_{2}}{A}_{2}\parallel }_{BMO}\sum _{|{\alpha }_{1}|={m}_{1}}{|Q|}^{-1/s}{\left({\int }_{{R}^{n}}{|{D}^{{\alpha }_{1}}{\stackrel{˜}{A}}_{1}\left(x\right){f}_{1}\left(x\right)|}^{p}\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/p}\\ \le & C\sum _{|{\alpha }_{2}|={m}_{2}}{\parallel {D}^{{\alpha }_{2}}{A}_{2}\parallel }_{BMO}\\ ×\sum _{|{\alpha }_{1}|={m}_{1}}{\left(\frac{1}{|Q|}{\int }_{\stackrel{˜}{Q}}{|{D}^{{\alpha }_{1}}{\stackrel{˜}{A}}_{1}\left(x\right)|}^{p{q}^{\prime }}\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/p{q}^{\prime }}{\left(\frac{1}{{|Q|}^{1-r\delta /n}}{\int }_{\stackrel{˜}{Q}}{|f\left(x\right)|}^{pq}\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/pq}\\ \le & C\prod _{j=1}^{2}\left(\sum _{|\alpha |={m}_{j}}{\parallel {D}^{\alpha }{A}_{j}\parallel }_{BMO}\right){M}_{\delta ,r}\left(f\right)\left(\stackrel{˜}{x}\right).\end{array}$

For ${I}_{3}$, similar to the proof of ${I}_{2}$, we get

${I}_{3}\le C\prod _{j=1}^{2}\left(\sum _{|\alpha |={m}_{j}}{\parallel {D}^{\alpha }{A}_{j}\parallel }_{BMO}\right){M}_{\delta ,r}\left(f\right)\left(\stackrel{˜}{x}\right).$

Similarly, for ${I}_{4}$, denoting $r=p{q}_{3}$ for $1, ${q}_{1},{q}_{2},{q}_{3}>1$, $1/{q}_{1}+1/{q}_{2}+1/{q}_{3}=1$ and $1/s=1/p-\delta /n$, we obtain

$\begin{array}{rcl}{I}_{4}& \le & C\sum _{|{\alpha }_{1}|={m}_{1},|{\alpha }_{2}|={m}_{2}}\frac{1}{|Q|}{\int }_{Q}|{S}_{\psi }\left({D}^{{\alpha }_{1}}{\stackrel{˜}{A}}_{1}{D}^{{\alpha }_{2}}{\stackrel{˜}{A}}_{2}{f}_{1}\right)\left(x\right)|\phantom{\rule{0.2em}{0ex}}dx\\ \le & C\sum _{|{\alpha }_{1}|={m}_{1},|{\alpha }_{2}|={m}_{2}}{\left(\frac{1}{|Q|}{\int }_{{R}^{n}}{|{S}_{\psi }\left({D}^{{\alpha }_{1}}{\stackrel{˜}{A}}_{1}{D}^{{\alpha }_{2}}{\stackrel{˜}{A}}_{2}{f}_{1}\right)\left(x\right)|}^{s}\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/s}\\ \le & C\sum _{|{\alpha }_{1}|={m}_{1},|{\alpha }_{2}|={m}_{2}}{|Q|}^{-1/s}{\left({\int }_{{R}^{n}}{|{D}^{{\alpha }_{1}}{\stackrel{˜}{A}}_{1}\left(x\right){D}^{{\alpha }_{2}}{\stackrel{˜}{A}}_{2}\left(x\right){f}_{1}\left(x\right)|}^{p}\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/p}\\ \le & C\sum _{|{\alpha }_{1}|={m}_{1},|{\alpha }_{2}|={m}_{2}}{\left(\frac{1}{|Q|}{\int }_{\stackrel{˜}{Q}}{|{D}^{{\alpha }_{1}}{\stackrel{˜}{A}}_{1}\left(x\right)|}^{p{q}_{1}}\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/p{q}_{1}}{\left(\frac{1}{|Q|}{\int }_{\stackrel{˜}{Q}}{|{D}^{{\alpha }_{2}}{\stackrel{˜}{A}}_{2}\left(x\right)|}^{p{q}_{2}}\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/p{q}_{2}}\\ ×{\left(\frac{1}{{|Q|}^{1-r\delta /n}}{\int }_{\stackrel{˜}{Q}}{|f\left(x\right)|}^{p{q}_{3}}\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/p{q}_{3}}\\ \le & C\prod _{j=1}^{2}\left(\sum _{|\alpha |={m}_{j}}{\parallel {D}^{\alpha }{A}_{j}\parallel }_{BMO}\right){M}_{\delta ,r}\left(f\right)\left(\stackrel{˜}{x}\right).\end{array}$

For ${I}_{5}$, we write

$\begin{array}{c}{\chi }_{\mathrm{\Gamma }\left(x\right)}{F}_{t}^{\stackrel{˜}{A}}\left({f}_{2}\right)\left(x,y\right)-{\chi }_{\mathrm{\Gamma }\left({x}_{0}\right)}{F}_{t}^{\stackrel{˜}{A}}\left({f}_{2}\right)\left({x}_{0},y\right)\hfill \\ \phantom{\rule{1em}{0ex}}={\int }_{{R}^{n}}\left({\chi }_{\mathrm{\Gamma }\left(x\right)}-{\chi }_{\mathrm{\Gamma }\left({x}_{0}\right)}\right)\frac{{\prod }_{j=1}^{2}{R}_{{m}_{j}}\left({\stackrel{˜}{A}}_{j};x,z\right)}{{|x-z|}^{m}}{\psi }_{t}\left(y-z\right){f}_{2}\left(z\right)\phantom{\rule{0.2em}{0ex}}dz\hfill \\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}+{\chi }_{\mathrm{\Gamma }\left({x}_{0}\right)}{\int }_{{R}^{n}}\left(\frac{1}{{|x-z|}^{m}}-\frac{1}{{|{x}_{0}-z|}^{m}}\right)\prod _{j=1}^{2}{R}_{{m}_{j}}\left({\stackrel{˜}{A}}_{j};x,z\right){\psi }_{t}\left(y-z\right){f}_{2}\left(z\right)\phantom{\rule{0.2em}{0ex}}dz\hfill \\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}+{\chi }_{\mathrm{\Gamma }\left({x}_{0}\right)}{\int }_{{R}^{n}}\left({R}_{{m}_{1}}\left({\stackrel{˜}{A}}_{1};x,z\right)-{R}_{{m}_{1}}\left({\stackrel{˜}{A}}_{1};{x}_{0},z\right)\right)\frac{{R}_{{m}_{2}}\left({\stackrel{˜}{A}}_{2};x,z\right)}{{|{x}_{0}-z|}^{m}}{\psi }_{t}\left(y-z\right){f}_{2}\left(z\right)\phantom{\rule{0.2em}{0ex}}dz\hfill \\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}+{\chi }_{\mathrm{\Gamma }\left({x}_{0}\right)}{\int }_{{R}^{n}}\left({R}_{{m}_{2}}\left({\stackrel{˜}{A}}_{2};x,z\right)-{R}_{{m}_{2}}\left({\stackrel{˜}{A}}_{2};{x}_{0},z\right)\right)\frac{{R}_{{m}_{1}}\left({\stackrel{˜}{A}}_{1};{x}_{0},z\right)}{{|{x}_{0}-z|}^{m}}{\psi }_{t}\left(y-z\right){f}_{2}\left(z\right)\phantom{\rule{0.2em}{0ex}}dz\hfill \\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}-\sum _{|{\alpha }_{1}|={m}_{1}}\frac{1}{{\alpha }_{1}!}{\int }_{{R}^{n}}\left[\frac{{R}_{{m}_{2}}\left({\stackrel{˜}{A}}_{2};x,z\right){\left(x-z\right)}^{{\alpha }_{1}}{\chi }_{\mathrm{\Gamma }\left(x\right)}}{{|x-z|}^{m}}-\frac{{R}_{{m}_{2}}\left({\stackrel{˜}{A}}_{2};{x}_{0},z\right){\left({x}_{0}-z\right)}^{{\alpha }_{1}}{\chi }_{\mathrm{\Gamma }\left({x}_{0}\right)}}{{|{x}_{0}-z|}^{m}}\right]\hfill \\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}×{D}^{{\alpha }_{1}}{\stackrel{˜}{A}}_{1}\left(z\right){\psi }_{t}\left(y-z\right){f}_{2}\left(z\right)\phantom{\rule{0.2em}{0ex}}dz\hfill \\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}-\sum _{|{\alpha }_{2}|={m}_{2}}\frac{1}{{\alpha }_{2}!}{\int }_{{R}^{n}}\left[\frac{{R}_{{m}_{1}}\left({\stackrel{˜}{A}}_{1};x,z\right){\left(x-z\right)}^{{\alpha }_{2}}{\chi }_{\mathrm{\Gamma }\left(x\right)}}{{|x-z|}^{m}}-\frac{{R}_{{m}_{1}}\left({\stackrel{˜}{A}}_{1};{x}_{0},z\right){\left({x}_{0}-z\right)}^{{\alpha }_{2}}{\chi }_{\mathrm{\Gamma }\left({x}_{0}\right)}}{{|{x}_{0}-z|}^{m}}\right]\hfill \\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}×{D}^{{\alpha }_{2}}{\stackrel{˜}{A}}_{2}\left(z\right){\psi }_{t}\left(y-z\right){f}_{2}\left(z\right)\phantom{\rule{0.2em}{0ex}}dz\hfill \\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}+\sum _{|{\alpha }_{1}|={m}_{1},|{\alpha }_{2}|={m}_{2}}\frac{1}{{\alpha }_{1}!{\alpha }_{2}!}{\int }_{{R}^{n}}\left[\frac{{\left(x-z\right)}^{{\alpha }_{1}+{\alpha }_{2}}{\chi }_{\mathrm{\Gamma }\left(x\right)}}{{|x-z|}^{m}}-\frac{{\left({x}_{0}-z\right)}^{{\alpha }_{1}+{\alpha }_{2}}{\chi }_{\mathrm{\Gamma }\left({x}_{0}\right)}}{{|{x}_{0}-z|}^{m}}\right]\hfill \\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}×{D}^{{\alpha }_{1}}{\stackrel{˜}{A}}_{1}\left(z\right){D}^{{\alpha }_{2}}{\stackrel{˜}{A}}_{2}\left(z\right){\psi }_{t}\left(y-z\right){f}_{2}\left(z\right)\phantom{\rule{0.2em}{0ex}}dz\hfill \\ \phantom{\rule{1em}{0ex}}={I}_{5}^{\left(1\right)}+{I}_{5}^{\left(2\right)}+{I}_{5}^{\left(3\right)}+{I}_{5}^{\left(4\right)}+{I}_{5}^{\left(5\right)}+{I}_{5}^{\left(6\right)}+{I}_{5}^{\left(7\right)}.\hfill \end{array}$

By Lemma 1 and the following inequality (see )

we know that, for $x\in Q$ and $z\in {2}^{k+1}\stackrel{˜}{Q}\setminus {2}^{k}\stackrel{˜}{Q}$,

$\begin{array}{rcl}|{R}_{m}\left(\stackrel{˜}{A};x,z\right)|& \le & C{|x-z|}^{m}\sum _{|\alpha |=m}\left({\parallel {D}^{\alpha }A\parallel }_{BMO}+|{\left({D}^{\alpha }A\right)}_{\stackrel{˜}{Q}\left(x,z\right)}-{\left({D}^{\alpha }A\right)}_{\stackrel{˜}{Q}}|\right)\\ \le & Ck{|x-z|}^{m}\sum _{|\alpha |=m}{\parallel {D}^{\alpha }A\parallel }_{BMO}.\end{array}$

Note that $|x-z|\sim |{x}_{0}-z|$ for $x\in Q$ and $z\in {R}^{n}\setminus \stackrel{˜}{Q}$, we obtain, similar to the proof of Lemma 4,

$\begin{array}{c}\parallel {I}_{5}^{\left(1\right)}\parallel \le {\int }_{{R}^{n}}\left(\int {\int }_{{R}_{+}^{n+1}}\left[\frac{{\prod }_{j=1}^{2}|{R}_{{m}_{j}}\left({\stackrel{˜}{A}}_{j};x,z\right)||{\psi }_{t}\left(y-z\right)||{f}_{2}\left(z\right)|}{{|x-z|}^{m}}\hfill \\ \phantom{\parallel {I}_{5}^{\left(1\right)}\parallel \le }×{{|{\chi }_{\mathrm{\Gamma }\left(x\right)}\left(y,t\right)-{\chi }_{\mathrm{\Gamma }\left({x}_{0}\right)}\left(y,t\right)|\right]}^{2}\frac{dy\phantom{\rule{0.2em}{0ex}}dt}{{t}^{n+1}}\right)}^{1/2}\phantom{\rule{0.2em}{0ex}}dz\hfill \\ \phantom{\parallel {I}_{5}^{\left(1\right)}\parallel }\le C{\int }_{{R}^{n}}\frac{{\prod }_{j=1}^{2}|{R}_{{m}_{j}}\left({\stackrel{˜}{A}}_{j};x,z\right)||{f}_{2}\left(z\right)|}{{|{x}_{0}-z|}^{m}}\hfill \\ \phantom{\parallel {I}_{5}^{\left(1\right)}\parallel \le }×|{\int }_{\mathrm{\Gamma }\left(x\right)}\frac{{t}^{1-n}dy\phantom{\rule{0.2em}{0ex}}dt}{{\left(t+|y-z|\right)}^{2n+2-2\delta }}-{\int }_{\mathrm{\Gamma }\left({x}_{0}\right)}\frac{{t}^{1-n}dy\phantom{\rule{0.2em}{0ex}}dt}{{\left(t+|y-z|\right)}^{2n+2-2\delta }}{|}^{1/2}\phantom{\rule{0.2em}{0ex}}dz\hfill \\ \phantom{\parallel {I}_{5}^{\left(1\right)}\parallel }\le C{\int }_{{R}^{n}}\frac{{\prod }_{j=1}^{2}|{R}_{{m}_{j}}\left({\stackrel{˜}{A}}_{j};x,z\right)||{f}_{2}\left(z\right)|}{{|{x}_{0}-z|}^{m}}\hfill \\ \phantom{\parallel {I}_{5}^{\left(1\right)}\parallel \le }×{\left(\int {\int }_{|y|\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\hfill \\ \phantom{\parallel {I}_{5}^{\left(1\right)}\parallel }\le C{\int }_{{R}^{n}}\frac{{\prod }_{j=1}^{2}|{R}_{{m}_{j}}\left({\stackrel{˜}{A}}_{j};x,z\right)||{f}_{2}\left(z\right)|}{{|{x}_{0}-z|}^{m}}{\left(\int {\int }_{|y|\le t}\frac{|x-{x}_{0}|{t}^{1-n}dy\phantom{\rule{0.2em}{0ex}}dt}{{\left(t+|x+y-z|\right)}^{2n+3-2\delta }}\right)}^{1/2}\phantom{\rule{0.2em}{0ex}}dz\hfill \\ \phantom{\parallel {I}_{5}^{\left(1\right)}\parallel }\le C{\int }_{{R}^{n}}\frac{{\prod }_{j=1}^{2}{R}_{{m}_{j}}\left({\stackrel{˜}{A}}_{j};x,z\right)|{f}_{2}\left(z\right)|{|x-{x}_{0}|}^{1/2}}{{|{x}_{0}-z|}^{m+n+1/2-\delta }}\phantom{\rule{0.2em}{0ex}}dz\hfill \\ \phantom{\parallel {I}_{5}^{\left(1\right)}\parallel }\le C\prod _{j=1}^{2}\left(\sum _{|\alpha |={m}_{j}}{\parallel {D}^{\alpha }{A}_{j}\parallel }_{BMO}\right)\sum _{k=0}^{\mathrm{\infty }}{\int }_{{2}^{k+1}\stackrel{˜}{Q}\setminus {2}^{k}\stackrel{˜}{Q}}{k}^{2}\frac{{|x-{x}_{0}|}^{1/2}}{{|{x}_{0}-z|}^{n+1/2-\delta }}|f\left(z\right)|\phantom{\rule{0.2em}{0ex}}dz\hfill \\ \phantom{\parallel {I}_{5}^{\left(1\right)}\parallel }\le C\prod _{j=1}^{2}\left(\sum _{|\alpha |={m}_{j}}{\parallel {D}^{\alpha }{A}_{j}\parallel }_{BMO}\right)\sum _{k=1}^{\mathrm{\infty }}{k}^{2}{2}^{-k/2}\frac{1}{{|{2}^{k}\stackrel{˜}{Q}|}^{1-\delta /n}}{\int }_{{2}^{k}\stackrel{˜}{Q}}|f\left(z\right)|\phantom{\rule{0.2em}{0ex}}dz\hfill \\ \phantom{\parallel {I}_{5}^{\left(1\right)}\parallel }\le C\prod _{j=1}^{2}\left(\sum _{|\alpha |={m}_{j}}{\parallel {D}^{\alpha }{A}_{j}\parallel }_{BMO}\right){M}_{\delta ,r}\left(f\right)\left(\stackrel{˜}{x}\right);\hfill \\ \parallel {I}_{5}^{\left(2\right)}\parallel \le C{\int }_{{R}^{n}}\frac{|x-{x}_{0}|}{{|{x}_{0}-z|}^{m+n+1-\delta }}\prod _{j=1}^{2}|{R}_{{m}_{j}}\left({\stackrel{˜}{A}}_{j};x,z\right)||{f}_{2}\left(z\right)|\phantom{\rule{0.2em}{0ex}}dz\hfill \\ \phantom{\parallel {I}_{5}^{\left(2\right)}\parallel }\le C\prod _{j=1}^{2}\left(\sum _{|\alpha |={m}_{j}}{\parallel {D}^{\alpha }{A}_{j}\parallel }_{BMO}\right)\sum _{k=0}^{\mathrm{\infty }}{\int }_{{2}^{k+1}\stackrel{˜}{Q}\setminus {2}^{k}\stackrel{˜}{Q}}{k}^{2}\frac{|x-{x}_{0}|}{{|{x}_{0}-z|}^{n+1-\delta }}|f\left(z\right)|\phantom{\rule{0.2em}{0ex}}dz\hfill \\ \phantom{\parallel {I}_{5}^{\left(2\right)}\parallel }\le C\prod _{j=1}^{2}\left(\sum _{|\alpha |={m}_{j}}{\parallel {D}^{\alpha }{A}_{j}\parallel }_{BMO}\right)\sum _{k=1}^{\mathrm{\infty }}{k}^{2}{2}^{-k}\frac{1}{{|{2}^{k}\stackrel{˜}{Q}|}^{1-\delta /n}}{\int }_{{2}^{k}\stackrel{˜}{Q}}|f\left(z\right)|\phantom{\rule{0.2em}{0ex}}dz\hfill \\ \phantom{\parallel {I}_{5}^{\left(2\right)}\parallel }\le C\prod _{j=1}^{2}\left(\sum _{|\alpha |={m}_{j}}{\parallel {D}^{\alpha }{A}_{j}\parallel }_{BMO}\right){M}_{\delta ,r}\left(f\right)\left(\stackrel{˜}{x}\right).\hfill \end{array}$

For ${I}_{5}^{\left(3\right)}$ and ${I}_{5}^{\left(4\right)}$, by the formula (see )

${R}_{m}\left(\stackrel{˜}{A};x,z\right)-{R}_{m}\left(\stackrel{˜}{A};{x}_{0},z\right)=\sum _{|\beta |

and Lemma 1, we have

$|{R}_{m}\left(\stackrel{˜}{A};x,z\right)-{R}_{m}\left(\stackrel{˜}{A};{x}_{0},z\right)|\le C\sum _{|\beta |

Thus, similar to the proof of Lemma 4,

$\begin{array}{c}\parallel {I}_{5}^{\left(3\right)}\parallel \le C\prod _{j=1}^{2}\left(\sum _{|\alpha |={m}_{j}}{\parallel {D}^{\alpha }{A}_{j}\parallel }_{BMO}\right)\sum _{k=0}^{\mathrm{\infty }}{\int }_{{2}^{k+1}\stackrel{˜}{Q}\setminus {2}^{k}\stackrel{˜}{Q}}k\frac{|x-{x}_{0}|}{{|{x}_{0}-z|}^{n+1-\delta }}|f\left(y\right)|\phantom{\rule{0.2em}{0ex}}dy\hfill \\ \phantom{\parallel {I}_{5}^{\left(3\right)}\parallel }\le C\prod _{j=1}^{2}\left(\sum _{|\alpha |={m}_{j}}{\parallel {D}^{\alpha }{A}_{j}\parallel }_{BMO}\right){M}_{\delta ,r}\left(f\right)\left(\stackrel{˜}{x}\right);\hfill \\ \parallel {I}_{5}^{\left(4\right)}\parallel \le C\prod _{j=1}^{2}\left(\sum _{|\alpha |={m}_{j}}{\parallel {D}^{\alpha }{A}_{j}\parallel }_{BMO}\right){M}_{\delta ,r}\left(f\right)\left(\stackrel{˜}{x}\right).\hfill \end{array}$

Similarly, we get

$\begin{array}{c}\parallel {I}_{5}^{\left(5\right)}\parallel \le C\sum _{|{\alpha }_{1}|={m}_{1}}{\int }_{{R}^{n}}\parallel \left[\frac{{R}_{{m}_{2}}\left({\stackrel{˜}{A}}_{2};x,z\right){\left(x-z\right)}^{{\alpha }_{1}}{\chi }_{\mathrm{\Gamma }\left(x\right)}}{{|x-z|}^{m}}-\frac{{R}_{{m}_{2}}\left({\stackrel{˜}{A}}_{2};{x}_{0},z\right){\left({x}_{0}-z\right)}^{{\alpha }_{1}}{\chi }_{\mathrm{\Gamma }\left({x}_{0}\right)}}{{|{x}_{0}-z|}^{m}}\right]\hfill \\ \phantom{\parallel {I}_{5}^{\left(5\right)}\parallel \le }×{\psi }_{t}\left(y-z\right)\parallel |{D}^{{\alpha }_{1}}{\stackrel{˜}{A}}_{1}\left(z\right)||{f}_{2}\left(z\right)|\phantom{\rule{0.2em}{0ex}}dz\hfill \\ \phantom{\parallel {I}_{5}^{\left(5\right)}\parallel }\le C\sum _{|\alpha |={m}_{2}}{\parallel {D}^{\alpha }{A}_{2}\parallel }_{BMO}\sum _{|{\alpha }_{1}|={m}_{1}}\sum _{k=1}^{\mathrm{\infty }}k\left({2}^{-k/2}+{2}^{-k}\right)\hfill \\ \phantom{\parallel {I}_{5}^{\left(5\right)}\parallel \le }×{\left(\frac{1}{|{2}^{k}\stackrel{˜}{Q}|}{\int }_{{2}^{k}\stackrel{˜}{Q}}{|{D}^{{\alpha }_{1}}{\stackrel{˜}{A}}_{1}\left(y\right)|}^{{r}^{\prime }}\phantom{\rule{0.2em}{0ex}}dy\right)}^{1/{r}^{\prime }}{\left(\frac{1}{{|{2}^{k}\stackrel{˜}{Q}|}^{1-r\delta /n}}{\int }_{{2}^{k}\stackrel{˜}{Q}}{|f\left(y\right)|}^{r}\phantom{\rule{0.2em}{0ex}}dy\right)}^{1/r}\hfill \\ \phantom{\parallel {I}_{5}^{\left(5\right)}\parallel }\le C\prod _{j=1}^{2}\left(\sum _{|\alpha |={m}_{j}}{\parallel {D}^{\alpha }{A}_{j}\parallel }_{BMO}\right){M}_{\delta ,r}\left(f\right)\left(\stackrel{˜}{x}\right);\hfill \\ \parallel {I}_{5}^{\left(6\right)}\parallel \le C\prod _{j=1}^{2}\left(\sum _{|\alpha |={m}_{j}}{\parallel {D}^{\alpha }{A}_{j}\parallel }_{BMO}\right){M}_{\delta ,r}\left(f\right)\left(\stackrel{˜}{x}\right).\hfill \end{array}$

For ${I}_{5}^{\left(7\right)}$, taking ${q}_{1},{q}_{2}>1$ such that $1/r+1/{q}_{1}+1/{q}_{2}=1$, then

$\begin{array}{rcl}\parallel {I}_{5}^{\left(7\right)}\parallel & \le & C\sum _{|{\alpha }_{1}|={m}_{1},|{\alpha }_{2}|={m}_{2}}{\int }_{{R}^{n}}\parallel \left[\frac{{\left(x-z\right)}^{{\alpha }_{1}+{\alpha }_{2}}{\chi }_{\mathrm{\Gamma }\left(x\right)}}{{|x-z|}^{m}}-\frac{{\left({x}_{0}-z\right)}^{{\alpha }_{1}+{\alpha }_{2}}{\chi }_{\mathrm{\Gamma }\left({x}_{0}\right)}}{{|{x}_{0}-z|}^{m}}\right]{\psi }_{t}\left(y-z\right)\parallel \\ ×|{D}^{{\alpha }_{1}}{\stackrel{˜}{A}}_{1}\left(z\right)||{D}^{{\alpha }_{2}}{\stackrel{˜}{A}}_{2}\left(z\right)||{f}_{2}\left(z\right)|\phantom{\rule{0.2em}{0ex}}dz\\ \le & C\sum _{|{\alpha }_{1}|={m}_{1},|{\alpha }_{2}|={m}_{2}}\sum _{k=1}^{\mathrm{\infty }}k\left({2}^{-k/2}+{2}^{-k}\right){\left(\frac{1}{{|{2}^{k}\stackrel{˜}{Q}|}^{1-p\delta /n}}{\int }_{{2}^{k}\stackrel{˜}{Q}}{|f\left(y\right)|}^{r}\phantom{\rule{0.2em}{0ex}}dy\right)}^{1/r}\\ ×{\left(\frac{1}{|{2}^{k}\stackrel{˜}{Q}|}{\int }_{{2}^{k}\stackrel{˜}{Q}}{|{D}^{{\alpha }_{1}}{\stackrel{˜}{A}}_{1}\left(y\right)|}^{{q}_{1}}\phantom{\rule{0.2em}{0ex}}dy\right)}^{1/{q}_{1}}{\left(\frac{1}{|{2}^{k}\stackrel{˜}{Q}|}{\int }_{{2}^{k}\stackrel{˜}{Q}}{|{D}^{{\alpha }_{2}}{\stackrel{˜}{A}}_{2}\left(y\right)|}^{{q}_{2}}\phantom{\rule{0.2em}{0ex}}dy\right)}^{1/{q}_{2}}\\ \le & C\prod _{j=1}^{2}\left(\sum _{|\alpha |={m}_{j}}{\parallel {D}^{\alpha }{A}_{j}\parallel }_{BMO}\right){M}_{\delta ,r}\left(f\right)\left(\stackrel{˜}{x}\right).\end{array}$

Thus

$\parallel {I}_{5}\parallel \le C\prod _{j=1}^{2}\left(\sum _{|\alpha |={m}_{j}}{\parallel {D}^{\alpha }{A}_{j}\parallel }_{BMO}\right){M}_{\delta ,r}\left(f\right)\left(\stackrel{˜}{x}\right).$

We choose $1 in (1), then (2) follows from Lemma 2. For (3), taking $1 in (1) and by Lemma 3, we obtain

$\begin{array}{rcl}{\parallel {S}_{\psi }^{A}\left(f\right)\parallel }_{{L}^{q,\lambda }}& \le & C{\parallel M\left({S}_{\psi }^{A}\left(f\right)\right)\parallel }_{{L}^{q,\lambda }}\le C{\parallel {\left({S}_{\psi }^{A}\left(f\right)\right)}^{\mathrm{#}}\parallel }_{{L}^{q,\lambda }}\\ \le & C\prod _{j=1}^{2}\left(\sum _{|\alpha |={m}_{j}}{\parallel {D}^{\alpha }{A}_{j}\parallel }_{BMO}\right){\parallel {M}_{\delta ,r}\left(f\right)\parallel }_{{L}^{q,\lambda }}\\ \le & C\prod _{j=1}^{2}\left(\sum _{|\alpha |={m}_{j}}{\parallel {D}^{\alpha }{A}_{j}\parallel }_{BMO}\right){\parallel {\left({M}_{r\delta /n}\left({|f|}^{r}\right)\right)}^{1/r}\parallel }_{{L}^{q,\lambda }}\\ \le & C\prod _{j=1}^{2}\left(\sum _{|\alpha |={m}_{j}}{\parallel {D}^{\alpha }{A}_{j}\parallel }_{BMO}\right){\parallel {M}_{r\delta /n}\left({|f|}^{r}\right)\parallel }_{{L}^{q/r,\lambda }}^{1/r}\\ \le & C\prod _{j=1}^{2}\left(\sum _{|\alpha |={m}_{j}}{\parallel {D}^{\alpha }{A}_{j}\parallel }_{BMO}\right){\parallel {|f|}^{r}\parallel }_{{L}^{p/r,\lambda }}^{1/r}\\ \le & C\prod _{j=1}^{2}\left(\sum _{|\alpha |={m}_{j}}{\parallel {D}^{\alpha }{A}_{j}\parallel }_{BMO}\right){\parallel f\parallel }_{{L}^{p,\lambda }}.\end{array}$

This completes the proof of Theorem 1. □

Proof of Theorem 2 It is only to prove (1). Let Q, $\stackrel{˜}{Q}$, ${\stackrel{˜}{A}}_{j}\left(x\right)$, ${f}_{1}$ and ${f}_{2}$ be the same as the proof of Theorem 1. We write

$\begin{array}{c}{F}_{t}^{A}\left(f\right)\left(x,y\right)\hfill \\ \phantom{\rule{1em}{0ex}}={\int }_{{R}^{n}}\frac{{\prod }_{j=1}^{2}{R}_{{m}_{j}+1}\left({\stackrel{˜}{A}}_{j};x,z\right)}{{|x-z|}^{m}}\frac{\mathrm{\Omega }\left(y-z\right)}{{|y-z|}^{n-1-\delta }}{f}_{2}\left(z\right)\phantom{\rule{0.2em}{0ex}}dz\hfill \\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}+{\int }_{{R}^{n}}\frac{{\prod }_{j=1}^{2}{R}_{{m}_{j}}\left({\stackrel{˜}{A}}_{j};x,z\right)}{{|x-z|}^{m}}\frac{\mathrm{\Omega }\left(y-z\right)}{{|y-z|}^{n-1-\delta }}{f}_{1}\left(z\right)\phantom{\rule{0.2em}{0ex}}dz\hfill \\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}-\sum _{|{\alpha }_{1}|={m}_{1}}\frac{1}{{\alpha }_{1}!}{\int }_{{R}^{n}}\frac{{R}_{{m}_{2}}\left({\stackrel{˜}{A}}_{2};x,z\right){\left(x-z\right)}^{{\alpha }_{1}}{D}^{{\alpha }_{1}}{\stackrel{˜}{A}}_{1}\left(z\right)}{{|x-z|}^{m}}\frac{\mathrm{\Omega }\left(y-z\right)}{{|y-z|}^{n-1-\delta }}{f}_{1}\left(z\right)\phantom{\rule{0.2em}{0ex}}dz\hfill \\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}-\sum _{|{\alpha }_{2}|={m}_{2}}\frac{1}{{\alpha }_{2}!}{\int }_{{R}^{n}}\frac{{R}_{{m}_{1}}\left({\stackrel{˜}{A}}_{1};x,z\right){\left(x-z\right)}^{{\alpha }_{2}}{D}^{{\alpha }_{2}}{\stackrel{˜}{A}}_{2}\left(z\right)}{{|x-z|}^{m}}\frac{\mathrm{\Omega }\left(y-z\right)}{{|y-z|}^{n-1-\delta }}{f}_{1}\left(z\right)\phantom{\rule{0.2em}{0ex}}dz\hfill \\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}+\sum _{|{\alpha }_{1}|={m}_{1},|{\alpha }_{2}|={m}_{2}}\frac{1}{{\alpha }_{1}!{\alpha }_{2}!}{\int }_{{R}^{n}}\frac{{\left(x-z\right)}^{{\alpha }_{1}+{\alpha }_{2}}{D}^{{\alpha }_{1}}{\stackrel{˜}{A}}_{1}\left(z\right){D}^{{\alpha }_{2}}{\stackrel{˜}{A}}_{2}\left(z\right)}{{|x-z|}^{m}}\frac{\mathrm{\Omega }\left(y-z\right)}{{|y-z|}^{n-1-\delta }}{f}_{1}\left(z\right)\phantom{\rule{0.2em}{0ex}}dz,\hfill \end{array}$

then

$\begin{array}{c}\frac{1}{|Q|}{\int }_{Q}|{\mu }_{S}^{A}\left(f\right)\left(x\right)-{\mu }_{S}^{\stackrel{˜}{A}}\left({f}_{2}\right)\left({x}_{0}\right)|\phantom{\rule{0.2em}{0ex}}dx\hfill \\ \phantom{\rule{1em}{0ex}}\le \frac{1}{|Q|}{\int }_{Q}\parallel {\chi }_{\mathrm{\Gamma }\left(x\right)}{F}_{t}^{A}\left(f\right)\left(x,y\right)-{\chi }_{\mathrm{\Gamma }\left({x}_{0}\right)}{F}_{t}^{\stackrel{˜}{A}}\left({f}_{2}\right)\left({x}_{0},y\right)\parallel \phantom{\rule{0.2em}{0ex}}dx\hfill \\ \phantom{\rule{1em}{0ex}}\le \frac{1}{|Q|}{\int }_{Q}\parallel {\chi }_{\mathrm{\Gamma }\left(x\right)}{\int }_{{R}^{n}}\frac{{\prod }_{j=1}^{2}{R}_{{m}_{j}}\left({\stackrel{˜}{A}}_{j};x,z\right)}{{|x-z|}^{m}}\frac{\mathrm{\Omega }\left(y-z\right)}{{|y-z|}^{n-1-\delta }}{f}_{1}\left(z\right)\phantom{\rule{0.2em}{0ex}}dz\parallel \phantom{\rule{0.2em}{0ex}}dx\hfill \\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}+\frac{1}{|Q|}{\int }_{Q}\parallel {\chi }_{\mathrm{\Gamma }\left(x\right)}\sum _{|{\alpha }_{1}|={m}_{1}}\frac{1}{{\alpha }_{1}!}\hfill \\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}×{\int }_{{R}^{n}}\frac{{R}_{{m}_{2}}\left({\stackrel{˜}{A}}_{2};x,z\right){\left(x-z\right)}^{{\alpha }_{1}}{D}^{{\alpha }_{1}}{\stackrel{˜}{A}}_{1}\left(z\right)}{{|x-z|}^{m}}\frac{\mathrm{\Omega }\left(y-z\right)}{{|y-z|}^{n-1-\delta }}{f}_{1}\left(z\right)\phantom{\rule{0.2em}{0ex}}dz\parallel \phantom{\rule{0.2em}{0ex}}dx\hfill \\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}+\frac{1}{|Q|}{\int }_{Q}\parallel {\chi }_{\mathrm{\Gamma }\left(x\right)}\sum _{|{\alpha }_{2}|={m}_{2}}\frac{1}{{\alpha }_{2}!}\hfill \\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}×{\int }_{{R}^{n}}\frac{{R}_{{m}_{1}}\left({\stackrel{˜}{A}}_{1};x,z\right){\left(x-z\right)}^{{\alpha }_{2}}{D}^{{\alpha }_{2}}{\stackrel{˜}{A}}_{2}\left(z\right)}{{|x-z|}^{m}}\frac{\mathrm{\Omega }\left(y-z\right)}{{|y-z|}^{n-1-\delta }}{f}_{1}\left(z\right)\phantom{\rule{0.2em}{0ex}}dz\parallel \phantom{\rule{0.2em}{0ex}}dx\hfill \\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}+\frac{1}{|Q|}{\int }_{Q}\parallel {\chi }_{\mathrm{\Gamma }\left(x\right)}\sum _{|{\alpha }_{1}|={m}_{1},|{\alpha }_{2}|={m}_{2}}\frac{1}{{\alpha }_{1}!{\alpha }_{2}!}\hfill \\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}×{\int }_{{R}^{n}}\frac{{\left(x-z\right)}^{{\alpha }_{1}+{\alpha }_{2}}{D}^{{\alpha }_{1}}{\stackrel{˜}{A}}_{1}\left(z\right){D}^{{\alpha }_{2}}{\stackrel{˜}{A}}_{2}\left(z\right)}{{|x-z|}^{m}}\frac{\mathrm{\Omega }\left(y-z\right)}{{|y-z|}^{n-1-\delta }}{f}_{1}\left(z\right)\phantom{\rule{0.2em}{0ex}}dz\parallel \phantom{\rule{0.2em}{0ex}}dx\hfill \\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}+\frac{1}{|Q|}{\int }_{Q}\parallel {\chi }_{\mathrm{\Gamma }\left(x\right)}{F}_{t}^{\stackrel{˜}{A}}\left({f}_{2}\right)\left(x,y\right)-{\chi }_{\mathrm{\Gamma }\left({x}_{0}\right)}{F}_{t}^{\stackrel{˜}{A}}\left({f}_{2}\right)\left({x}_{0},y\right)\parallel \phantom{\rule{0.2em}{0ex}}dx\hfill \\ \phantom{\rule{1em}{0ex}}:={J}_{1}+{J}_{2}+{J}_{3}+{J}_{4}+{J}_{5}.\hfill \end{array}$

Similar to the proof of Theorem 1, we get

$\begin{array}{c}{J}_{1}\le C\prod _{j=1}^{2}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{A}_{j}\parallel }_{BMO}\right)\frac{1}{|Q|}{\int }_{Q}|{\mu }_{S}\left({f}_{1}\right)\left(x\right)|\phantom{\rule{0.2em}{0ex}}dx\hfill \\ \phantom{{J}_{1}}\le C\prod _{j=1}^{2}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{A}_{j}\parallel }_{BMO}\right){\left(\frac{1}{|Q|}{\int }_{Q}{|{\mu }_{S}\left({f}_{1}\right)\left(x\right)|}^{q}\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/q}\hfill \\ \phantom{{J}_{1}}\le C\prod _{j=1}^{2}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{A}_{j}\parallel }_{BMO}\right){M}_{\delta ,r}\left(f\right)\left(\stackrel{˜}{x}\right);\hfill \\ {J}_{2}\le C\sum _{|{\alpha }_{2}|={m}_{2}}{\parallel {D}^{{\alpha }_{2}}{A}_{2}\parallel }_{BMO}\sum _{|{\alpha }_{1}|={m}_{1}}\frac{1}{|Q|}{\int }_{Q}|{\mu }_{S}\left({D}^{{\alpha }_{1}}{\stackrel{˜}{A}}_{1}{f}_{1}\right)\left(x\right)|\phantom{\rule{0.2em}{0ex}}dx\hfill \\ \phantom{{J}_{2}}\le C\sum _{|{\alpha }_{2}|={m}_{2}}{\parallel {D}^{{\alpha }_{2}}{A}_{2}\parallel }_{BMO}\sum _{|{\alpha }_{1}|={m}_{1}}{\left(\frac{1}{|Q|}{\int }_{{R}^{n}}{|{\mu }_{S}\left({D}^{{\alpha }_{1}}{\stackrel{˜}{A}}_{1}{f}_{1}\right)\left(x\right)|}^{s}\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/s}\hfill \\ \phantom{{J}_{2}}\le C\prod _{j=1}^{2}\left(\sum _{|\alpha |={m}_{j}}{\parallel {D}^{\alpha }{A}_{j}\parallel }_{BMO}\right){M}_{\delta ,r}\left(f\right)\left(\stackrel{˜}{x}\right);\hfill \\ {J}_{3}\le C\prod _{j=1}^{2}\left(\sum _{|\alpha |={m}_{j}}{\parallel {D}^{\alpha }{A}_{j}\parallel }_{BMO}\right){M}_{\delta ,r}\left(f\right)\left(\stackrel{˜}{x}\right);\hfill \\ {J}_{4}\le C\sum _{|{\alpha }_{1}|={m}_{1},|{\alpha }_{2}|={m}_{2}}\frac{1}{|Q|}{\int }_{Q}|{\mu }_{S}\left({D}^{{\alpha }_{1}}{\stackrel{˜}{A}}_{1}{D}^{{\alpha }_{2}}{\stackrel{˜}{A}}_{2}{f}_{1}\right)\left(x\right)|\phantom{\rule{0.2em}{0ex}}dx\hfill \\ \phantom{{J}_{4}}\le C\sum _{|{\alpha }_{1}|={m}_{1},|{\alpha }_{2}|={m}_{2}}{\left(\frac{1}{|Q|}{\int }_{{R}^{n}}{|{\mu }_{S}\left({D}^{{\alpha }_{1}}{\stackrel{˜}{A}}_{1}{D}^{{\alpha }_{2}}{\stackrel{˜}{A}}_{2}{f}_{1}\right)\left(x\right)|}^{s}\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/s}\hfill \\ \phantom{{J}_{4}}\le C\prod _{j=1}^{2}\left(\sum _{|\alpha |={m}_{j}}{\parallel {D}^{\alpha }{A}_{j}\parallel }_{BMO}\right){M}_{\delta ,p}\left(f\right)\left(\stackrel{˜}{x}\right).\hfill \end{array}$

For ${J}_{5}$, we write

$\begin{array}{c}{\chi }_{\mathrm{\Gamma }\left(x\right)}{F}_{t}^{\stackrel{˜}{A}}\left({f}_{2}\right)\left(x,y\right)-{\chi }_{\mathrm{\Gamma }\left({x}_{0}\right)}{F}_{t}^{\stackrel{˜}{A}}\left({f}_{2}\right)\left({x}_{0},y\right)\hfill \\ \phantom{\rule{1em}{0ex}}={\int }_{{R}^{n}}\left({\chi }_{\mathrm{\Gamma }\left(x\right)}-{\chi }_{\mathrm{\Gamma }\left({x}_{0}\right)}\right)\frac{{\prod }_{j=1}^{2}{R}_{{m}_{j}}\left({\stackrel{˜}{A}}_{j};x,z\right)}{{|x-z|}^{m}}\frac{\mathrm{\Omega }\left(y-z\right)}{{|y-z|}^{n-1-\delta }}{f}_{2}\left(z\right)\phantom{\rule{0.2em}{0ex}}dz\hfill \\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}+{\chi }_{\mathrm{\Gamma }\left({x}_{0}\right)}{\int }_{{R}^{n}}\left(\frac{1}{{|x-z|}^{m}}-\frac{1}{{|{x}_{0}-z|}^{m}}\right)\prod _{j=1}^{2}{R}_{{m}_{j}}\left({\stackrel{˜}{A}}_{j};x,z\right)\frac{\mathrm{\Omega }\left(y-z\right)}{{|y-z|}^{n-1-\delta }}{f}_{2}\left(z\right)\phantom{\rule{0.2em}{0ex}}dz\hfill \\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}+{\chi }_{\mathrm{\Gamma }\left({x}_{0}\right)}{\int }_{{R}^{n}}\left({R}_{{m}_{1}}\left({\stackrel{˜}{A}}_{1};x,z\right)-{R}_{{m}_{1}}\left({\stackrel{˜}{A}}_{1};{x}_{0},z\right)\right)\frac{{R}_{{m}_{2}}\left({\stackrel{˜}{A}}_{2};x,z\right)}{{|{x}_{0}-z|}^{m}}\frac{\mathrm{\Omega }\left(y-z\right)}{{|y-z|}^{n-1-\delta }}{f}_{2}\left(z\right)\phantom{\rule{0.2em}{0ex}}dz\hfill \\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}+{\chi }_{\mathrm{\Gamma }\left({x}_{0}\right)}{\int }_{{R}^{n}}\left({R}_{{m}_{2}}\left({\stackrel{˜}{A}}_{2};x,z\right)-{R}_{{m}_{2}}\left({\stackrel{˜}{A}}_{2};{x}_{0},z\right)\right)\frac{{R}_{{m}_{1}}\left({\stackrel{˜}{A}}_{1};{x}_{0},z\right)}{{|{x}_{0}-z|}^{m}}\frac{\mathrm{\Omega }\left(y-z\right)}{{|y-z|}^{n-1-\delta }}{f}_{2}\left(z\right)\phantom{\rule{0.2em}{0ex}}dz\hfill \\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}-\sum _{|{\alpha }_{1}|={m}_{1}}\frac{1}{{\alpha }_{1}!}{\int }_{{R}^{n}}\left[\frac{{R}_{{m}_{2}}\left({\stackrel{˜}{A}}_{2};x,z\right){\left(x-z\right)}^{{\alpha }_{1}}{\chi }_{\mathrm{\Gamma }\left(x\right)}}{{|x-z|}^{m}}-\frac{{R}_{{m}_{2}}\left({\stackrel{˜}{A}}_{2};{x}_{0},z\right){\left({x}_{0}-z\right)}^{{\alpha }_{1}}{\chi }_{\mathrm{\Gamma }\left({x}_{0}\right)}}{{|{x}_{0}-z|}^{m}}\right]\hfill \\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}×\frac{\mathrm{\Omega }\left(y-z\right)}{{|y-z|}^{n-1-\delta }}{D}^{{\alpha }_{1}}{\stackrel{˜}{A}}_{1}\left(z\right){f}_{2}\left(z\right)\phantom{\rule{0.2em}{0ex}}dz\hfill \\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}-\sum _{|{\alpha }_{2}|={m}_{2}}\frac{1}{{\alpha }_{2}!}{\int }_{{R}^{n}}\left[\frac{{R}_{{m}_{1}}\left({\stackrel{˜}{A}}_{1};x,z\right){\left(x-z\right)}^{{\alpha }_{2}}{\chi }_{\mathrm{\Gamma }\left(x\right)}}{{|x-z|}^{m}}-\frac{{R}_{{m}_{1}}\left({\stackrel{˜}{A}}_{1};{x}_{0},z\right){\left({x}_{0}-z\right)}^{{\alpha }_{2}}{\chi }_{\mathrm{\Gamma }\left({x}_{0}\right)}}{{|{x}_{0}-z|}^{m}}\right]\hfill \\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}×\frac{\mathrm{\Omega }\left(y-z\right)}{{|y-z|}^{n-1-\delta }}{D}^{{\alpha }_{2}}{\stackrel{˜}{A}}_{2}\left(z\right){f}_{2}\left(z\right)\phantom{\rule{0.2em}{0ex}}dz\hfill \\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}+\sum _{|{\alpha }_{1}|={m}_{1},|{\alpha }_{2}|={m}_{2}}\frac{1}{{\alpha }_{1}!{\alpha }_{2}!}{\int }_{{R}^{n}}\left[\frac{{\left(x-z\right)}^{{\alpha }_{1}+{\alpha }_{2}}{\chi }_{\mathrm{\Gamma }\left(x\right)}}{{|x-z|}^{m}}-\frac{{\left({x}_{0}-z\right)}^{{\alpha }_{1}+{\alpha }_{2}}{\chi }_{\mathrm{\Gamma }\left({x}_{0}\right)}}{{|{x}_{0}-z|}^{m}}\right]\hfill \\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}×\frac{\mathrm{\Omega }\left(y-z\right)}{{|y-z|}^{n-1-\delta }}{D}^{{\alpha }_{1}}{\stackrel{˜}{A}}_{1}\left(z\right){D}^{{\alpha }_{2}}{\stackrel{˜}{A}}_{2}\left(z\right){f}_{2}\left(z\right)\phantom{\rule{0.2em}{0ex}}dz.\hfill \end{array}$

Then, similar to the proof of Lemma 4 and Theorem 1, we get

$\parallel {J}_{5}\parallel \le C\prod _{j=1}^{2}\left(\sum _{|\alpha |={m}_{j}}{\parallel {D}^{\alpha }{A}_{j}\parallel }_{BMO}\right){M}_{\delta ,r}\left(f\right)\left(\stackrel{˜}{x}\right).$

The same argument as the proof of Theorem 1 will give the proof of (2) and (3), we omit the details and finish the proof. □

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Lu, D. Some sharp inequalities for multilinear integral operators. J Inequal Appl 2013, 445 (2013). https://doi.org/10.1186/1029-242X-2013-445

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• DOI: https://doi.org/10.1186/1029-242X-2013-445

### Keywords

• multilinear operator
• Littlewood-Paley operator
• Marcinkiewicz operator
• Morrey space
• BMO 