# Sharp boundedness for vector-valued multilinear integral operators

## Abstract

In this paper, the sharp inequalities for some vector-valued multilinear integral operators are obtained. As applications, we get the weighted ${L}^{p}$ ($p>1$) norm inequalities and an $LlogL$-type estimate for the vector-valued multilinear operators.

MSC:42B20, 42B25.

## 1 Preliminaries and theorems

As the development of singular integral operators and their commutators, multilinear singular integral operators have been well studied (see ). In this paper, we study some vector-valued multilinear integral operators as follows.

Suppose that ${m}_{j}$ are positive integers ($j=1,\dots ,l$), ${m}_{1}+\cdots +{m}_{l}=m$ and ${A}_{j}$ are functions on ${R}^{n}$ ($j=1,\dots ,l$). Let ${F}_{t}\left(x,y\right)$ be defined on ${R}^{n}×{R}^{n}×\left[0,+\mathrm{\infty }\right)$. Set

${F}_{t}\left(f\right)\left(x\right)={\int }_{{R}^{n}}{F}_{t}\left(x,y\right)f\left(y\right)\phantom{\rule{0.2em}{0ex}}dy$

and

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

for every bounded and compactly supported function f, where

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

Let H be the Banach space $H=\left\{h:\parallel h\parallel <\mathrm{\infty }\right\}$ such that, for each fixed $x\in {R}^{n}$, ${F}_{t}\left(f\right)\left(x\right)$ and ${F}_{t}^{A}\left(f\right)\left(x\right)$ may be viewed as a mapping from $\left[0,+\mathrm{\infty }\right)$ to H. For $1, the vector-valued multilinear operator related to ${F}_{t}$ is defined by

${|{T}^{A}\left(f\right)\left(x\right)|}_{s}={\left(\sum _{i=1}^{\mathrm{\infty }}{\left({T}^{A}\left({f}_{i}\right)\left(x\right)\right)}^{s}\right)}^{1/s},$

where

${T}^{A}\left({f}_{i}\right)\left(x\right)=\parallel {F}_{t}^{A}\left({f}_{i}\right)\left(x\right)\parallel ,$

and ${F}_{t}$ satisfies: for fixed $\epsilon >0$,

$\parallel {F}_{t}\left(x,y\right)\parallel \le C{|x-y|}^{-n}$

and

$\parallel {F}_{t}\left(y,x\right)-{F}_{t}\left(z,x\right)\parallel \le C{|y-z|}^{\epsilon }{|x-z|}^{-n-\epsilon }$

if $2|y-z|\le |x-z|$. Set

${|T\left(f\right)\left(x\right)|}_{s}={\left(\sum _{i=1}^{\mathrm{\infty }}{|T\left({f}_{i}\right)\left(x\right)|}^{s}\right)}^{1/s}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{|f|}_{s}={\left(\sum _{i=1}^{\mathrm{\infty }}{|{f}_{i}\left(x\right)|}^{s}\right)}^{1/s}.$

Suppose that ${|T|}_{s}$ is bounded on ${L}^{p}\left({R}^{n}\right)$ for $1 and weak $\left({L}^{1},{L}^{1}\right)$-bounded.

Note that when $m=0$, ${T}^{A}$ is just a vector-valued multilinear commutator of T and A (see ). While when $m>0$, ${T}^{A}$ is a non-trivial generalization of the commutator. It is well known that multilinear operators are of great interest in harmonic analysis and have been studied by many authors (see ). In , Hu and Yang proved a variant sharp estimate for the multilinear singular integral operators. In , Pérez and Trujillo-Gonzalez proved a sharp estimate for some multilinear commutator. The main purpose of this paper is to prove a sharp inequality for the vector-valued multilinear integral operators. As applications, we obtain the weighted ${L}^{p}$ ($p>1$) norm inequalities and an $LlogL$-type estimate for the vector-valued 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

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

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 )

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

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 $0, we denote ${f}_{r}^{\mathrm{#}}$ by

${f}_{r}^{\mathrm{#}}\left(x\right)={\left[{\left({|f|}^{r}\right)}^{\mathrm{#}}\left(x\right)\right]}^{1/r}.$

Let M be the Hardy-Littlewood maximal operator, that is, $M\left(f\right)\left(x\right)={sup}_{x\in Q}{|Q|}^{-1}{\int }_{Q}|f\left(y\right)|\phantom{\rule{0.2em}{0ex}}dy$. For $k\in N$, we denote by ${M}^{k}$ the operator M iterated k times, i.e., ${M}^{1}\left(f\right)\left(x\right)=M\left(f\right)\left(x\right)$ and ${M}^{k}\left(f\right)\left(x\right)=M\left({M}^{k-1}\left(f\right)\right)\left(x\right)$ for $k\ge 2$.

Let Φ be a Young function and $\stackrel{˜}{\mathrm{\Phi }}$ be the complementary associated to Φ, we denote the Φ-average by, for a function f,

${\parallel f\parallel }_{\mathrm{\Phi },Q}=inf\left\{\lambda >0:\frac{1}{|Q|}{\int }_{Q}\mathrm{\Phi }\left(\frac{|f\left(y\right)|}{\lambda }\right)\phantom{\rule{0.2em}{0ex}}dy\le 1\right\}$

and the maximal function associated to Φ by

${M}_{\mathrm{\Phi }}\left(f\right)\left(x\right)=\underset{x\in Q}{sup}{\parallel f\parallel }_{\mathrm{\Phi },Q}.$

The Young functions to be used in this paper are $\mathrm{\Phi }\left(t\right)=exp\left({t}^{r}\right)-1$ and $\mathrm{\Psi }\left(t\right)=t{log}^{r}\left(t+e\right)$, the corresponding Φ-average and maximal functions are denoted by ${\parallel \cdot \parallel }_{exp{L}^{r},Q}$, ${M}_{exp{L}^{r}}$ and ${\parallel \cdot \parallel }_{L{\left(logL\right)}^{r},Q}$, ${M}_{L{\left(logL\right)}^{r}}$. We have the following inequality, for any $r>0$ and $m\in N$ (see )

$M\left(f\right)\le {M}_{L{\left(logL\right)}^{r}}\left(f\right),\phantom{\rule{2em}{0ex}}{M}_{L{\left(logL\right)}^{m}}\left(f\right)\approx {M}^{m+1}\left(f\right).$

For $r\ge 1$, we denote that

${\parallel b\parallel }_{{osc}_{exp{L}^{r}}}=\underset{Q}{sup}{\parallel b-{b}_{Q}\parallel }_{exp{L}^{r},Q},$

the space ${Osc}_{exp{L}^{r}}$ is defined by

${Osc}_{exp{L}^{r}}=\left\{b\in {L}_{log}^{1}\left({R}^{n}\right):{\parallel b\parallel }_{{osc}_{exp{L}^{r}}}<\mathrm{\infty }\right\}.$

It has been known that (see )

${\parallel b-{b}_{2Q}\parallel }_{exp{L}^{r},{2}^{k}Q}\le Ck{\parallel b\parallel }_{{Osc}_{exp{L}^{r}}}.$

It is obvious that ${Osc}_{exp{L}^{r}}$ coincides with the $BMO$ space if $r=1$, and ${Osc}_{exp{L}^{r}}\subset BMO$ if $r>1$. We denote the Muckenhoupt weights by ${A}_{p}$ for $1\le p<\mathrm{\infty }$ (see ).

Now we state our main results as follows.

Theorem 1 Let $1, ${r}_{j}\ge 1$ and ${D}^{\alpha }{A}_{j}\in {Osc}_{exp{L}^{{r}_{j}}}$ for all α with $|\alpha |={m}_{j}$ and $j=1,\dots ,l$. Define $1/r=1/{r}_{1}+\cdots +1/{r}_{l}$. Then, for any $0, there exists a constant $C>0$ such that for any $f=\left\{{f}_{i}\right\}\in {C}_{0}^{\mathrm{\infty }}\left({R}^{n}\right)$ and $x\in {R}^{n}$,

${\left({|{T}^{A}\left(f\right)|}_{s}\right)}_{p}^{\mathrm{#}}\left(x\right)\le C\prod _{j=1}^{l}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{A}_{j}\parallel }_{{Osc}_{exp{L}^{{r}_{j}}}}\right){M}_{L{\left(logL\right)}^{1/r}}\left({|f|}_{s}\right)\left(x\right).$

Theorem 2 Let $1, ${r}_{j}\ge 1$ and ${D}^{\alpha }{A}_{j}\in {Osc}_{exp{L}^{{r}_{j}}}$ for all α with $|\alpha |={m}_{j}$ and $j=1,\dots ,l$.

1. (1)

If $1 and $w\in {A}_{p}$, then

${\parallel {|{T}^{A}\left(f\right)|}_{s}\parallel }_{{L}^{p}\left(w\right)}\le C\prod _{j=1}^{l}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{A}_{j}\parallel }_{{Osc}_{exp{L}^{{r}_{j}}}}\right){\parallel {|f|}_{s}\parallel }_{{L}^{p}\left(w\right)};$
2. (2)

If $w\in {A}_{1}$. Define $1/r=1/{r}_{1}+\cdots +1/{r}_{l}$ and $\mathrm{\Phi }\left(t\right)=t{log}^{1/r}\left(t+e\right)$. Then there exists a constant $C>0$ such that for all $\lambda >0$,

$\begin{array}{r}w\left(\left\{x\in {R}^{n}:{|{T}^{A}\left(f\right)\left(x\right)|}_{s}>\lambda \right\}\right)\\ \phantom{\rule{1em}{0ex}}\le C{\int }_{{R}^{n}}\mathrm{\Phi }\left[{\lambda }^{-1}\prod _{j=1}^{l}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{A}_{j}\parallel }_{{Osc}_{exp{L}^{{r}_{j}}}}\right){|f\left(x\right)|}_{s}\right]w\left(x\right)\phantom{\rule{0.2em}{0ex}}dx.\end{array}$

Remark The conditions in Theorems 1 and 2 are satisfied by many operators.

Now we give some examples.

Example 1 Littlewood-Paley operators.

Fix $\epsilon >0$ and $\mu >\left(3n+2\right)/n$. Let 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)

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

3. (3)

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

We denote that $\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 ${\chi }_{\mathrm{\Gamma }\left(x\right)}$. The Littlewood-Paley multilinear operators are defined by

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

and

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

where

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

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

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

and

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

which are the Littlewood-Paley operators (see ). Let H be the space

$H=\left\{h:\parallel h\parallel ={\left({\int }_{0}^{\mathrm{\infty }}{|h\left(t\right)|}^{2}\phantom{\rule{0.2em}{0ex}}dt/t\right)}^{1/2}<\mathrm{\infty }\right\}$

or

$H=\left\{h:\parallel h\parallel ={\left(\int {\int }_{{R}_{+}^{n+1}}{|h\left(y,t\right)|}^{2}\phantom{\rule{0.2em}{0ex}}dy\phantom{\rule{0.2em}{0ex}}dt/{t}^{n+1}\right)}^{1/2}<\mathrm{\infty }\right\}.$

Then, for each fixed $x\in {R}^{n}$, ${F}_{t}^{A}\left(f\right)\left(x\right)$ and ${F}_{t}^{A}\left(f\right)\left(x,y\right)$ may be viewed as the mapping from $\left[0,+\mathrm{\infty }\right)$ to H, and it is clear that

$\begin{array}{c}{g}_{\psi }^{A}\left(f\right)\left(x\right)=\parallel {F}_{t}^{A}\left(f\right)\left(x\right)\parallel ,\phantom{\rule{2em}{0ex}}{g}_{\psi }\left(f\right)\left(x\right)=\parallel {F}_{t}\left(f\right)\left(x\right)\parallel ,\hfill \\ {S}_{\psi }^{A}\left(f\right)\left(x\right)=\parallel {\chi }_{\mathrm{\Gamma }\left(x\right)}{F}_{t}^{A}\left(f\right)\left(x,y\right)\parallel ,\phantom{\rule{2em}{0ex}}{S}_{\psi }\left(f\right)\left(x\right)=\parallel {\chi }_{\mathrm{\Gamma }\left(x\right)}{F}_{t}\left(f\right)\left(y\right)\parallel \hfill \end{array}$

and

${g}_{\mu }^{A}\left(f\right)\left(x\right)=\parallel {\left(\frac{t}{t+|x-y|}\right)}^{n\mu /2}{F}_{t}^{A}\left(f\right)\left(x,y\right)\parallel ,\phantom{\rule{2em}{0ex}}{g}_{\mu }\left(f\right)\left(x\right)=\parallel {\left(\frac{t}{t+|x-y|}\right)}^{n\mu /2}{F}_{t}\left(f\right)\left(y\right)\parallel .$

It is easy to see that ${g}_{\psi }$, ${S}_{\psi }$ and ${g}_{\mu }$ satisfy the conditions of Theorems 1 and 2 (see ), thus Theorems 1 and 2 hold for ${g}_{\psi }^{A}$, ${S}_{\psi }^{A}$ and ${g}_{\mu }^{A}$.

Example 2 Marcinkiewicz operators.

Fix $\lambda >max\left(1,2n/\left(n+2\right)\right)$ and $0<\gamma \le 1$. Let 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 $\mathrm{\Omega }\in {Lip}_{\gamma }\left({S}^{n-1}\right)$. The Marcinkiewicz multilinear operators are defined by

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

and

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

where

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

and

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

Set

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

We also define that

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

and

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

which are the Marcinkiewicz operators (see ). Let H be the space

$H=\left\{h:\parallel h\parallel ={\left({\int }_{0}^{\mathrm{\infty }}{|h\left(t\right)|}^{2}\phantom{\rule{0.2em}{0ex}}dt/{t}^{3}\right)}^{1/2}<\mathrm{\infty }\right\}$

or

$H=\left\{h:\parallel h\parallel ={\left(\int {\int }_{{R}_{+}^{n+1}}{|h\left(y,t\right)|}^{2}\phantom{\rule{0.2em}{0ex}}dy\phantom{\rule{0.2em}{0ex}}dt/{t}^{n+3}\right)}^{1/2}<\mathrm{\infty }\right\}.$

Then it is clear that

$\begin{array}{c}{\mu }_{\mathrm{\Omega }}^{A}\left(f\right)\left(x\right)=\parallel {F}_{t}^{A}\left(f\right)\left(x\right)\parallel ,\phantom{\rule{2em}{0ex}}{\mu }_{\mathrm{\Omega }}\left(f\right)\left(x\right)=\parallel {F}_{t}\left(f\right)\left(x\right)\parallel ,\hfill \\ {\mu }_{S}^{A}\left(f\right)\left(x\right)=\parallel {\chi }_{\mathrm{\Gamma }\left(x\right)}{F}_{t}^{A}\left(f\right)\left(x,y\right)\parallel ,\phantom{\rule{2em}{0ex}}{\mu }_{S}\left(f\right)\left(x\right)=\parallel {\chi }_{\mathrm{\Gamma }\left(x\right)}{F}_{t}\left(f\right)\left(y\right)\parallel \hfill \end{array}$

and

${\mu }_{\lambda }^{A}\left(f\right)\left(x\right)=\parallel {\left(\frac{t}{t+|x-y|}\right)}^{n\lambda /2}{F}_{t}^{A}\left(f\right)\left(x,y\right)\parallel ,\phantom{\rule{2em}{0ex}}{\mu }_{\lambda }\left(f\right)\left(x\right)=\parallel {\left(\frac{t}{t+|x-y|}\right)}^{n\lambda /2}{F}_{t}\left(f\right)\left(y\right)\parallel .$

It is easy to see that ${\mu }_{\mathrm{\Omega }}$, ${\mu }_{S}$ and ${\mu }_{\lambda }$ satisfy the conditions of Theorems 1 and 2 (see [13, 14]), thus Theorems 1 and 2 hold for ${\mu }_{\mathrm{\Omega }}^{A}$, ${\mu }_{S}^{A}$ and ${\mu }_{\lambda }^{A}$.

Example 3 Bochner-Riesz operators.

Let $\delta >\left(n-1\right)/2$, ${B}_{t}^{\delta }\left(\stackrel{ˆ}{f}\right)\left(\xi \right)={\left(1-{t}^{2}{|\xi |}^{2}\right)}_{+}^{\delta }\stackrel{ˆ}{f}\left(\xi \right)$ and ${B}_{t}^{\delta }\left(z\right)={t}^{-n}{B}^{\delta }\left(z/t\right)$ for $t>0$. Set

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

The maximal Bochner-Riesz multilinear operators are defined by

${B}_{\delta ,\ast }^{A}\left(f\right)\left(x\right)=\underset{t>0}{sup}|{B}_{\delta ,t}^{A}\left(f\right)\left(x\right)|.$

We also define that

${B}_{\delta ,\ast }\left(f\right)\left(x\right)=\underset{t>0}{sup}|{B}_{t}^{\delta }\left(f\right)\left(x\right)|,$

which is the maximal Bochner-Riesz operator (see ). Let H be the space $H=\left\{h:\parallel h\parallel ={sup}_{t>0}|h\left(t\right)|<\mathrm{\infty }\right\}$, then

${B}_{\delta ,\ast }^{A}\left(f\right)\left(x\right)=\parallel {B}_{\delta ,t}^{A}\left(f\right)\left(x\right)\parallel ,\phantom{\rule{2em}{0ex}}{B}_{\ast }^{\delta }\left(f\right)\left(x\right)=\parallel {B}_{t}^{\delta }\left(f\right)\left(x\right)\parallel .$

It is easy to see that ${B}_{\delta ,\ast }^{A}$ satisfies the conditions of Theorems 1 and 2 (see ), thus Theorems 1 and 2 hold for ${B}_{\delta ,\ast }^{A}$.

## 2 Some lemmas

We give some preliminary 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

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

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

Lemma 2 ([, p.485])

Let $0 and for any function $f\ge 0$, we define that, for $1/r=1/p-1/q$,

${\parallel f\parallel }_{W{L}^{q}}=\underset{\lambda >0}{sup}\lambda {|\left\{x\in {R}^{n}:f\left(x\right)>\lambda \right\}|}^{1/q},\phantom{\rule{2em}{0ex}}{N}_{p,q}\left(f\right)=\underset{E}{sup}{\parallel f{\chi }_{E}\parallel }_{{L}^{p}}/{\parallel {\chi }_{E}\parallel }_{{L}^{r}},$

where the sup is taken for all measurable sets E with $0<|E|<\mathrm{\infty }$. Then

${\parallel f\parallel }_{W{L}^{q}}\le {N}_{p,q}\left(f\right)\le {\left(q/\left(q-p\right)\right)}^{1/p}{\parallel f\parallel }_{W{L}^{q}}.$

Lemma 3 ()

Let ${r}_{j}\ge 1$ for $j=1,\dots ,m$, we denote that $1/r=1/{r}_{1}+\cdots +1/{r}_{m}$. Then

$\frac{1}{|Q|}{\int }_{Q}|{f}_{1}\left(x\right)\cdots {f}_{m}\left(x\right)g\left(x\right)|\phantom{\rule{0.2em}{0ex}}dx\le {\parallel f\parallel }_{exp{L}^{{r}_{1}},Q}\cdots {\parallel f\parallel }_{exp{L}^{{r}_{m}},Q}{\parallel g\parallel }_{L{\left(logL\right)}^{1/r},Q}.$

## 3 Proof of the theorem

It is only to prove Theorem 1.

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

${\left(\frac{1}{|Q|}{\int }_{Q}{|{|{T}^{A}\left(f\right)\left(x\right)|}_{s}-{C}_{0}|}^{p}\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/p}\le C\prod _{j=1}^{l}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{A}_{j}\parallel }_{{Osc}_{exp{L}^{{r}_{j}}}}\right){M}_{L{\left(logL\right)}^{1/r}}\left({|f|}_{s}\right)\left(x\right).$

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 ${\stackrel{˜}{A}}_{j}\left(x\right)={A}_{j}\left(x\right)-{\sum }_{|\alpha |=m}\frac{1}{\alpha !}{\left({D}^{\alpha }{A}_{j}\right)}_{\stackrel{˜}{Q}}{x}^{\alpha }$, then ${R}_{m}\left({A}_{j};x,y\right)={R}_{m}\left({\stackrel{˜}{A}}_{j};x,y\right)$ and ${D}^{\alpha }{\stackrel{˜}{A}}_{j}={D}^{\alpha }{A}_{j}-{\left({D}^{\alpha }{A}_{j}\right)}_{\stackrel{˜}{Q}}$ for $|\alpha |={m}_{j}$. We split $f=g+h=\left\{{g}_{i}\right\}+\left\{{h}_{i}\right\}$ for ${g}_{i}={f}_{i}{\chi }_{\stackrel{˜}{Q}}$ and ${h}_{i}={f}_{i}{\chi }_{{R}^{n}\setminus \stackrel{˜}{Q}}$. Write

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

Then, by Minkowski’s inequality, we have

$\begin{array}{r}{\left[\frac{1}{|Q|}{\int }_{Q}|{|{T}^{A}\left(f\right)\left(x\right)|}_{s}-{|{T}^{\stackrel{˜}{A}}\left(h\right)\left({x}_{0}\right)|}_{s}{|}^{p}\phantom{\rule{0.2em}{0ex}}dx\right]}^{1/p}\\ \phantom{\rule{1em}{0ex}}\le {\left[\frac{1}{|Q|}{\int }_{Q}|{\parallel {F}_{t}^{A}\left(f\right)\left(x\right)\parallel }_{s}-{\parallel {F}_{t}^{\stackrel{˜}{A}}\left(h\right)\left({x}_{0}\right)\parallel }_{s}{|}^{p}\phantom{\rule{0.2em}{0ex}}dx\right]}^{1/p}\\ \phantom{\rule{1em}{0ex}}\le {\left[\frac{1}{|Q|}{\int }_{Q}{\left(\sum _{i=1}^{\mathrm{\infty }}{\parallel {F}_{t}^{A}\left({f}_{i}\right)\left(x\right)-{F}_{t}^{\stackrel{˜}{A}}\left({h}_{i}\right)\left({x}_{0}\right)\parallel }^{s}\right)}^{p/s}\phantom{\rule{0.2em}{0ex}}dx\right]}^{1/p}\\ \phantom{\rule{1em}{0ex}}\le {\left[\frac{C}{|Q|}{\int }_{Q}{\left(\sum _{i=1}^{\mathrm{\infty }}{\parallel {\int }_{{R}^{n}}\frac{{\prod }_{j=1}^{2}{R}_{{m}_{j}}\left({\stackrel{˜}{A}}_{j};x,y\right)}{{|x-y|}^{m}}{F}_{t}\left(x,y\right){g}_{i}\left(y\right)\phantom{\rule{0.2em}{0ex}}dy\parallel }^{s}\right)}^{p/s}\phantom{\rule{0.2em}{0ex}}dx\right]}^{1/p}\\ \phantom{\rule{2em}{0ex}}+\left[\frac{C}{|Q|}{\int }_{Q}\left(\sum _{i=1}^{\mathrm{\infty }}\parallel \sum _{|{\alpha }_{1}|={m}_{1}}\frac{1}{{\alpha }_{1}!}\\ {{\phantom{\rule{2em}{0ex}}×{\int }_{{R}^{n}}\frac{{R}_{{m}_{2}}\left({\stackrel{˜}{A}}_{2};x,y\right){\left(x-y\right)}^{{\alpha }_{1}}}{{|x-y|}^{m}}{D}^{{\alpha }_{1}}{\stackrel{˜}{A}}_{1}\left(y\right){F}_{t}\left(x,y\right){g}_{i}\left(y\right)\phantom{\rule{0.2em}{0ex}}dy{\parallel }^{s}\right)}^{p/s}\phantom{\rule{0.2em}{0ex}}dx\right]}^{1/p}\\ \phantom{\rule{2em}{0ex}}+\left[\frac{C}{|Q|}{\int }_{Q}\left(\sum _{i=1}^{\mathrm{\infty }}\parallel \sum _{|{\alpha }_{2}|={m}_{2}}\frac{1}{{\alpha }_{2}!}\\ {{\phantom{\rule{2em}{0ex}}×{\int }_{{R}^{n}}\frac{{R}_{{m}_{1}}\left({\stackrel{˜}{A}}_{1};x,y\right){\left(x-y\right)}^{{\alpha }_{2}}}{{|x-y|}^{m}}{D}^{{\alpha }_{2}}{\stackrel{˜}{A}}_{2}\left(y\right){F}_{t}\left(x,y\right){g}_{i}\left(y\right)\phantom{\rule{0.2em}{0ex}}dy{\parallel }^{s}\right)}^{p/s}\phantom{\rule{0.2em}{0ex}}dx\right]}^{1/p}\\ \phantom{\rule{2em}{0ex}}+\left[\frac{C}{|Q|}{\int }_{Q}\left(\sum _{i=1}^{\mathrm{\infty }}\parallel \sum _{|{\alpha }_{1}|={m}_{1},|{\alpha }_{2}|={m}_{2}}\frac{1}{{\alpha }_{1}!{\alpha }_{2}!}\\ {{\phantom{\rule{2em}{0ex}}×{\int }_{{R}^{n}}\frac{{\left(x-y\right)}^{{\alpha }_{1}+{\alpha }_{2}}{D}^{{\alpha }_{1}}{\stackrel{˜}{A}}_{1}\left(y\right){D}^{{\alpha }_{2}}{\stackrel{˜}{A}}_{2}\left(y\right)}{{|x-y|}^{m}}{F}_{t}\left(x,y\right){g}_{i}\left(y\right)\phantom{\rule{0.2em}{0ex}}dy{\parallel }^{s}\right)}^{p/s}\phantom{\rule{0.2em}{0ex}}dx\right]}^{1/p}\\ \phantom{\rule{2em}{0ex}}+{\left[\frac{C}{|Q|}{\int }_{Q}{\left(\sum _{i=1}^{\mathrm{\infty }}{\parallel {F}_{t}^{\stackrel{˜}{A}}\left({h}_{i}\right)\left(x\right)-{F}_{t}^{\stackrel{˜}{A}}\left({h}_{i}\right)\left({x}_{0}\right)\parallel }^{s}\right)}^{p/s}\phantom{\rule{0.2em}{0ex}}dx\right]}^{1/p}\\ \phantom{\rule{1em}{0ex}}:={I}_{1}+{I}_{2}+{I}_{3}+{I}_{4}+{I}_{5}.\end{array}$

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

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

Thus, by Lemma 2 and the weak type $\left(1,1\right)$ of ${|T|}_{s}$, we obtain

$\begin{array}{rcl}{I}_{1}& \le & C\prod _{j=1}^{2}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{A}_{j}\parallel }_{{Osc}_{exp{L}^{{r}_{j}}}}\right){\left(\frac{1}{|Q|}{\int }_{Q}{|T\left(g\right)\left(x\right)|}_{s}^{p}\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/p}\\ =& C\prod _{j=1}^{2}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{A}_{j}\parallel }_{{Osc}_{exp{L}^{{r}_{j}}}}\right){|Q|}^{-1}\frac{{\parallel {|T\left(g\right)|}_{s}{\chi }_{Q}\parallel }_{{L}^{p}}}{{|Q|}^{1/p-1}}\\ \le & C\prod _{j=1}^{2}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{A}_{j}\parallel }_{{Osc}_{exp{L}^{{r}_{j}}}}\right){|Q|}^{-1}{\parallel {|T\left(g\right)|}_{s}\parallel }_{W{L}^{1}}\\ \le & C\prod _{j=1}^{2}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{A}_{j}\parallel }_{{Osc}_{exp{L}^{{r}_{j}}}}\right){|Q|}^{-1}{\parallel {|g|}_{s}\parallel }_{{L}^{1}}\\ \le & C\prod _{j=1}^{2}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{A}_{j}\parallel }_{{Osc}_{exp{L}^{{r}_{j}}}}\right)M\left({|f|}_{s}\right)\left(\stackrel{˜}{x}\right)\\ \le & C\prod _{j=1}^{2}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{A}_{j}\parallel }_{{Osc}_{exp{L}^{{r}_{j}}}}\right){M}_{L{\left(logL\right)}^{1/r}}\left({|f|}_{s}\right)\left(\stackrel{˜}{x}\right).\end{array}$

For ${I}_{2}$, note that ${\parallel {\chi }_{Q}\parallel }_{exp{L}^{{r}_{2}},Q}\le C$, similar to the proof of ${I}_{1}$ and by using Lemma 3, we get

$\begin{array}{rcl}{I}_{2}& \le & C\sum _{|{\alpha }_{2}|={m}_{2}}{\parallel {D}^{{\alpha }_{2}}{A}_{2}\parallel }_{{Osc}_{exp{L}^{{r}_{2}}}}\sum _{|{\alpha }_{1}|={m}_{1}}{\left(\frac{1}{|Q|}{\int }_{{R}^{n}}{|T\left({D}^{{\alpha }_{1}}{\stackrel{˜}{A}}_{1}g\right)\left(x\right)|}_{s}^{p}\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/p}\\ \le & C\sum _{|{\alpha }_{2}|={m}_{2}}{\parallel {D}^{{\alpha }_{2}}{A}_{2}\parallel }_{{Osc}_{exp{L}^{{r}_{2}}}}\sum _{|{\alpha }_{1}|={m}_{1}}{|Q|}^{-1}{\parallel {|T\left({D}^{{\alpha }_{1}}{\stackrel{˜}{A}}_{1}g\right)\left(x\right)|}_{s}{\chi }_{Q}\parallel }_{W{L}^{1}}\\ \le & C\sum _{|{\alpha }_{2}|={m}_{2}}{\parallel {D}^{{\alpha }_{2}}{A}_{2}\parallel }_{{Osc}_{exp{L}^{{r}_{2}}}}\sum _{|{\alpha }_{1}|={m}_{1}}\frac{1}{|Q|}{\int }_{{R}^{n}}|{D}^{{\alpha }_{1}}{\stackrel{˜}{A}}_{1}\left(x\right)|{|g\left(x\right)|}_{s}\phantom{\rule{0.2em}{0ex}}dx\\ \le & C\sum _{|{\alpha }_{2}|={m}_{2}}{\parallel {D}^{{\alpha }_{2}}{A}_{2}\parallel }_{{Osc}_{exp{L}^{{r}_{2}}}}{\parallel {\chi }_{Q}\parallel }_{exp{L}^{{r}_{2}},Q}\\ ×\sum _{|{\alpha }_{1}|={m}_{1}}{\parallel {D}^{{\alpha }_{1}}{A}_{1}-{\left({D}^{{\alpha }_{1}}{A}_{1}\right)}_{\stackrel{˜}{Q}}\parallel }_{exp{L}^{{r}_{1}},\stackrel{˜}{Q}}{\parallel {|f|}_{s}\parallel }_{L{\left(logL\right)}^{1/r},\stackrel{˜}{Q}}\\ \le & C\prod _{j=1}^{2}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{A}_{j}\parallel }_{{Osc}_{exp{L}^{{r}_{j}}}}\right){M}_{L{\left(logL\right)}^{1/r}}\left({|f|}_{s}\right)\left(\stackrel{˜}{x}\right).\end{array}$

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

${I}_{3}\le C\prod _{j=1}^{2}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{A}_{j}\parallel }_{{Osc}_{exp{L}^{{r}_{j}}}}\right){M}_{L{\left(logL\right)}^{1/r}}\left({|f|}_{s}\right)\left(\stackrel{˜}{x}\right).$

Similarly, for ${I}_{4}$, by using Lemma 3, we get

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

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

$\begin{array}{r}{F}_{t}^{\stackrel{˜}{A}}\left({h}_{i}\right)\left(x\right)-{F}_{t}^{\stackrel{˜}{A}}\left({h}_{i}\right)\left({x}_{0}\right)\\ \phantom{\rule{1em}{0ex}}={\int }_{{R}^{n}}\left(\frac{{F}_{t}\left(x,y\right)}{{|x-y|}^{m}}-\frac{{F}_{t}\left({x}_{0},y\right)}{{|{x}_{0}-y|}^{m}}\right)\prod _{j=1}^{2}{R}_{{m}_{j}}\left({\stackrel{˜}{A}}_{j};x,y\right){h}_{i}\left(y\right)\phantom{\rule{0.2em}{0ex}}dy\\ \phantom{\rule{2em}{0ex}}+{\int }_{{R}^{n}}\left({R}_{{m}_{1}}\left({\stackrel{˜}{A}}_{1};x,y\right)-{R}_{{m}_{1}}\left({\stackrel{˜}{A}}_{1};{x}_{0},y\right)\right)\frac{{R}_{{m}_{2}}\left({\stackrel{˜}{A}}_{2};x,y\right)}{{|{x}_{0}-y|}^{m}}{F}_{t}\left({x}_{0},y\right){h}_{i}\left(y\right)\phantom{\rule{0.2em}{0ex}}dy\\ \phantom{\rule{2em}{0ex}}+{\int }_{{R}^{n}}\left({R}_{{m}_{2}}\left({\stackrel{˜}{A}}_{2};x,y\right)-{R}_{{m}_{2}}\left({\stackrel{˜}{A}}_{2};{x}_{0},y\right)\right)\frac{{R}_{{m}_{1}}\left({\stackrel{˜}{A}}_{1};{x}_{0},y\right)}{{|{x}_{0}-y|}^{m}}{F}_{t}\left({x}_{0},y\right){h}_{i}\left(y\right)\phantom{\rule{0.2em}{0ex}}dy\\ \phantom{\rule{2em}{0ex}}-\sum _{|{\alpha }_{1}|={m}_{1}}\frac{1}{{\alpha }_{1}!}{\int }_{{R}^{n}}\left[\frac{{R}_{{m}_{2}}\left({\stackrel{˜}{A}}_{2};x,y\right){\left(x-y\right)}^{{\alpha }_{1}}}{{|x-y|}^{m}}{F}_{t}\left(x,y\right)-\frac{{R}_{{m}_{2}}\left({\stackrel{˜}{A}}_{2};{x}_{0},y\right){\left({x}_{0}-y\right)}^{{\alpha }_{1}}}{{|{x}_{0}-y|}^{m}}{F}_{t}\left({x}_{0},y\right)\right]\\ \phantom{\rule{2em}{0ex}}×{D}^{{\alpha }_{1}}{\stackrel{˜}{A}}_{1}\left(y\right){h}_{i}\left(y\right)\phantom{\rule{0.2em}{0ex}}dy\\ \phantom{\rule{2em}{0ex}}-\sum _{|{\alpha }_{2}|={m}_{2}}\frac{1}{{\alpha }_{2}!}{\int }_{{R}^{n}}\left[\frac{{R}_{{m}_{1}}\left({\stackrel{˜}{A}}_{1};x,y\right){\left(x-y\right)}^{{\alpha }_{2}}}{{|x-y|}^{m}}{F}_{t}\left(x,y\right)-\frac{{R}_{{m}_{1}}\left({\stackrel{˜}{A}}_{1};{x}_{0},y\right){\left({x}_{0}-y\right)}^{{\alpha }_{2}}}{{|{x}_{0}-y|}^{m}}{F}_{t}\left({x}_{0},y\right)\right]\\ \phantom{\rule{2em}{0ex}}×{D}^{{\alpha }_{2}}{\stackrel{˜}{A}}_{2}\left(y\right){h}_{i}\left(y\right)\phantom{\rule{0.2em}{0ex}}dy\\ \phantom{\rule{2em}{0ex}}+\sum _{|{\alpha }_{1}|={m}_{1},|{\alpha }_{2}|={m}_{2}}\frac{1}{{\alpha }_{1}!{\alpha }_{2}!}{\int }_{{R}^{n}}\left[\frac{{\left(x-y\right)}^{{\alpha }_{1}+{\alpha }_{2}}}{{|x-y|}^{m}}{F}_{t}\left(x,y\right)-\frac{{\left({x}_{0}-y\right)}^{{\alpha }_{1}+{\alpha }_{2}}}{{|{x}_{0}-y|}^{m}}{F}_{t}\left({x}_{0},y\right)\right]\\ \phantom{\rule{2em}{0ex}}×{D}^{{\alpha }_{1}}{\stackrel{˜}{A}}_{1}\left(y\right){D}^{{\alpha }_{2}}{\stackrel{˜}{A}}_{2}\left(y\right){h}_{i}\left(y\right)\phantom{\rule{0.2em}{0ex}}dy\\ \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)}.\end{array}$

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

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

Note that $|x-y|\sim |{x}_{0}-y|$ for $x\in Q$ and $y\in {R}^{n}\setminus \stackrel{˜}{Q}$, we obtain, by the condition of ${F}_{t}$,

$\begin{array}{rcl}\parallel {I}_{5}^{\left(1\right)}\parallel & \le & C{\int }_{{R}^{n}}\left(\frac{|x-{x}_{0}|}{{|{x}_{0}-y|}^{m+n+1}}+\frac{{|x-{x}_{0}|}^{\epsilon }}{{|{x}_{0}-y|}^{m+n+\epsilon }}\right)\prod _{j=1}^{2}{R}_{{m}_{j}}\left({\stackrel{˜}{A}}_{j};x,y\right)|{h}_{i}\left(y\right)|\phantom{\rule{0.2em}{0ex}}dy\\ \le & C\prod _{j=1}^{2}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{A}_{j}\parallel }_{{Osc}_{exp{L}^{{r}_{j}}}}\right)\\ ×\sum _{k=0}^{\mathrm{\infty }}{\int }_{{2}^{k+1}\stackrel{˜}{Q}\setminus {2}^{k}\stackrel{˜}{Q}}{k}^{2}\left(\frac{|x-{x}_{0}|}{{|{x}_{0}-y|}^{n+1}}+\frac{{|x-{x}_{0}|}^{\epsilon }}{{|{x}_{0}-y|}^{n+\epsilon }}\right)|{f}_{i}\left(y\right)|\phantom{\rule{0.2em}{0ex}}dy\\ \le & C\prod _{j=1}^{2}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{A}_{j}\parallel }_{{Osc}_{exp{L}^{{r}_{j}}}}\right)\sum _{k=1}^{\mathrm{\infty }}{k}^{2}\left({2}^{-k}+{2}^{-\epsilon k}\right)\frac{1}{|{2}^{k}\stackrel{˜}{Q}|}{\int }_{{2}^{k}\stackrel{˜}{Q}}|{f}_{i}\left(y\right)|\phantom{\rule{0.2em}{0ex}}dy.\end{array}$

Thus, by Minkowski’s inequality, we get

$\begin{array}{rcl}{\left(\sum _{i=1}^{\mathrm{\infty }}{\parallel {I}_{5}^{\left(1\right)}\parallel }^{s}\right)}^{1/s}& \le & C\prod _{j=1}^{2}\left(\sum _{|\alpha |={m}_{j}}{\parallel {D}^{\alpha }{A}_{j}\parallel }_{{Osc}_{exp{L}^{{r}_{j}}}}\right)\\ ×\sum _{k=1}^{\mathrm{\infty }}{k}^{2}\left({2}^{-k}+{2}^{-\epsilon k}\right)\frac{1}{|{2}^{k}\stackrel{˜}{Q}|}{\int }_{{2}^{k}\stackrel{˜}{Q}}{|f\left(y\right)|}_{s}\phantom{\rule{0.2em}{0ex}}dy\\ \le & C\prod _{j=1}^{2}\left(\sum _{|\alpha |={m}_{j}}{\parallel {D}^{\alpha }{A}_{j}\parallel }_{{Osc}_{exp{L}^{{r}_{j}}}}\right)M\left({|f|}_{s}\right)\left(\stackrel{˜}{x}\right)\\ \le & C\prod _{j=1}^{2}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{A}_{j}\parallel }_{{Osc}_{exp{L}^{{r}_{j}}}}\right){M}_{L{\left(logL\right)}^{1/r}}\left({|f|}_{s}\right)\left(\stackrel{˜}{x}\right).\end{array}$

For ${I}_{5}^{\left(2\right)}$, by the formula (see )

${R}_{{m}_{j}}\left(\stackrel{˜}{A};x,y\right)-{R}_{{m}_{j}}\left(\stackrel{˜}{A};{x}_{0},y\right)=\sum _{|\beta |<{m}_{j}}\frac{1}{\beta !}{R}_{{m}_{j}-|\beta |}\left({D}^{\beta }\stackrel{˜}{A};x,{x}_{0}\right){\left(x-y\right)}^{\beta }$

and Lemma 1, we have

$|{R}_{{m}_{j}}\left(\stackrel{˜}{A};x,y\right)-{R}_{{m}_{j}}\left(\stackrel{˜}{A};{x}_{0},y\right)|\le C\sum _{|\beta |<{m}_{j}}\sum _{|\alpha |={m}_{j}}{|x-{x}_{0}|}^{{m}_{j}-|\beta |}{|x-y|}^{|\beta |}{\parallel {D}^{\alpha }A\parallel }_{{Osc}_{exp{L}^{{r}_{j}}}},$

thus

$\begin{array}{rl}{\left(\sum _{i=1}^{\mathrm{\infty }}{\parallel {I}_{5}^{\left(2\right)}\parallel }^{s}\right)}^{1/s}& \le C\prod _{j=1}^{2}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{A}_{j}\parallel }_{{Osc}_{exp{L}^{{r}_{j}}}}\right)\sum _{k=0}^{\mathrm{\infty }}{\int }_{{2}^{k+1}\stackrel{˜}{Q}\setminus {2}^{k}\stackrel{˜}{Q}}k\frac{|x-{x}_{0}|}{{|{x}_{0}-y|}^{n+1}}{|f\left(y\right)|}_{s}\phantom{\rule{0.2em}{0ex}}dy\\ \le C\prod _{j=1}^{2}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{A}_{j}\parallel }_{{Osc}_{exp{L}^{{r}_{j}}}}\right){M}_{L{\left(logL\right)}^{1/r}}\left({|f|}_{s}\right)\left(\stackrel{˜}{x}\right).\end{array}$

Similarly,

${\left(\sum _{i=1}^{\mathrm{\infty }}{\parallel {I}_{5}^{\left(3\right)}\parallel }^{s}\right)}^{1/s}\le C\prod _{j=1}^{2}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{A}_{j}\parallel }_{{Osc}_{exp{L}^{{r}_{j}}}}\right){M}_{L{\left(logL\right)}^{1/r}}\left({|f|}_{s}\right)\left(\stackrel{˜}{x}\right).$

For ${I}_{5}^{\left(4\right)}$, similar to the proof of ${I}_{5}^{\left(1\right)}$, ${I}_{5}^{\left(2\right)}$ and ${I}_{2}$, we get

$\begin{array}{r}{\left(\sum _{i=1}^{\mathrm{\infty }}{\parallel {I}_{5}^{\left(4\right)}\parallel }^{s}\right)}^{1/s}\\ \phantom{\rule{1em}{0ex}}\le C\sum _{|{\alpha }_{1}|={m}_{1}}{\int }_{{R}^{n}\setminus \stackrel{˜}{Q}}\parallel \frac{{\left(x-y\right)}^{{\alpha }_{1}}{F}_{t}\left(x,y\right)}{{|x-y|}^{m}}-\frac{{\left({x}_{0}-y\right)}^{{\alpha }_{1}}{F}_{t}\left({x}_{0},y\right)}{{|{x}_{0}-y|}^{m}}\parallel \\ \phantom{\rule{2em}{0ex}}×|{R}_{{m}_{2}}\left({\stackrel{˜}{A}}_{2};x,y\right)||{D}^{{\alpha }_{1}}{\stackrel{˜}{A}}_{1}\left(y\right)|{|f\left(y\right)|}_{s}\phantom{\rule{0.2em}{0ex}}dy\\ \phantom{\rule{2em}{0ex}}+C\sum _{|{\alpha }_{1}|={m}_{1}}{\int }_{{R}^{n}\setminus \stackrel{˜}{Q}}|{R}_{{m}_{2}}\left({\stackrel{˜}{A}}_{2};x,y\right)-{R}_{{m}_{2}}\left({\stackrel{˜}{A}}_{2};{x}_{0},y\right)|\\ \phantom{\rule{2em}{0ex}}×\frac{\parallel {\left({x}_{0}-y\right)}^{{\alpha }_{1}}{F}_{t}\left({x}_{0},y\right)\parallel }{{|{x}_{0}-y|}^{m}}|{D}^{{\alpha }_{1}}{\stackrel{˜}{A}}_{1}\left(y\right)|{|f\left(y\right)|}_{s}\phantom{\rule{0.2em}{0ex}}dy\\ \phantom{\rule{1em}{0ex}}\le C\sum _{|{\alpha }_{2}|={m}_{2}}{\parallel {D}^{{\alpha }_{2}}{A}_{2}\parallel }_{{Osc}_{exp{L}^{{r}_{2}}}}\\ \phantom{\rule{2em}{0ex}}×\sum _{|{\alpha }_{1}|={m}_{1}}\sum _{k=1}^{\mathrm{\infty }}k\left({2}^{-k}+{2}^{-\epsilon k}\right)\frac{1}{|{2}^{k}\stackrel{˜}{Q}|}{\int }_{{2}^{k}\stackrel{˜}{Q}}|{D}^{{\alpha }_{1}}{\stackrel{˜}{A}}_{1}\left(y\right)|{|f\left(y\right)|}_{s}\phantom{\rule{0.2em}{0ex}}dy\\ \phantom{\rule{1em}{0ex}}\le C\sum _{|{\alpha }_{2}|={m}_{2}}{\parallel {D}^{{\alpha }_{2}}{A}_{2}\parallel }_{{Osc}_{exp{L}^{{r}_{2}}}}\sum _{|{\alpha }_{1}|={m}_{1}}\sum _{k=1}^{\mathrm{\infty }}k\left({2}^{-k}+{2}^{-\epsilon k}\right)\\ \phantom{\rule{2em}{0ex}}×{\parallel {D}^{{\alpha }_{1}}{A}_{1}-{\left({D}^{{\alpha }_{1}}{A}_{1}\right)}_{\stackrel{˜}{Q}}\parallel }_{exp{L}^{{r}_{1}},{2}^{k}\stackrel{˜}{Q}}{\parallel {|f|}_{s}\parallel }_{L{\left(logL\right)}^{1/r},{2}^{k}\stackrel{˜}{Q}}\\ \phantom{\rule{1em}{0ex}}\le C\prod _{j=1}^{2}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{A}_{j}\parallel }_{{Osc}_{exp{L}^{{r}_{j}}}}\right){M}_{L{\left(logL\right)}^{1/r}}\left({|f|}_{s}\right)\left(\stackrel{˜}{x}\right).\end{array}$

Similarly,

${\left(\sum _{i=1}^{\mathrm{\infty }}{\parallel {I}_{5}^{\left(5\right)}\parallel }^{s}\right)}^{1/s}\le C\prod _{j=1}^{2}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{A}_{j}\parallel }_{{Osc}_{exp{L}^{{r}_{j}}}}\right){M}_{L{\left(logL\right)}^{1/r}}\left({|f|}_{s}\right)\left(\stackrel{˜}{x}\right).$

For ${I}_{5}^{\left(6\right)}$, by using Lemma 3, we obtain

$\begin{array}{r}{\left(\sum _{i=1}^{\mathrm{\infty }}{\parallel {I}_{5}^{\left(6\right)}\parallel }^{s}\right)}^{1/s}\\ \phantom{\rule{1em}{0ex}}\le C\sum _{|{\alpha }_{1}|={m}_{1},|{\alpha }_{2}|={m}_{2}}{\int }_{{R}^{n}\setminus \stackrel{˜}{Q}}\parallel \frac{{\left(x-y\right)}^{{\alpha }_{1}+{\alpha }_{2}}{F}_{t}\left(x,y\right)}{{|x-y|}^{m}}-\frac{{\left({x}_{0}-y\right)}^{{\alpha }_{1}+{\alpha }_{2}}{F}_{t}\left({x}_{0},y\right)}{{|{x}_{0}-y|}^{m}}\parallel \\ \phantom{\rule{2em}{0ex}}×|{D}^{{\alpha }_{1}}{\stackrel{˜}{A}}_{1}\left(y\right)||{D}^{{\alpha }_{2}}{\stackrel{˜}{A}}_{2}\left(y\right)|{|f\left(y\right)|}_{s}\phantom{\rule{0.2em}{0ex}}dy\\ \phantom{\rule{1em}{0ex}}\le C\sum _{|{\alpha }_{1}|={m}_{1},|{\alpha }_{2}|={m}_{2}}\sum _{k=1}^{\mathrm{\infty }}\left({2}^{-k}+{2}^{-\epsilon k}\right)\frac{1}{|{2}^{k}\stackrel{˜}{Q}|}{\int }_{{2}^{k}\stackrel{˜}{Q}}|{D}^{{\alpha }_{1}}{\stackrel{˜}{A}}_{1}\left(y\right)||{D}^{{\alpha }_{2}}{\stackrel{˜}{A}}_{2}\left(y\right)|{|f\left(y\right)|}_{s}\phantom{\rule{0.2em}{0ex}}dy\\ \phantom{\rule{1em}{0ex}}\le C\prod _{j=1}^{2}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{A}_{j}\parallel }_{{Osc}_{exp{L}^{{r}_{j}}}}\right){M}_{L{\left(logL\right)}^{1/r}}\left({|f|}_{s}\right)\left(\stackrel{˜}{x}\right).\end{array}$

Thus

${I}_{5}\le C\prod _{j=1}^{2}\left(\sum _{|{\alpha }_{j}|={m}_{j}}{\parallel {D}^{{\alpha }_{j}}{A}_{j}\parallel }_{{Osc}_{exp{L}^{{r}_{j}}}}\right){M}_{L{\left(logL\right)}^{1/r}}\left({|f|}_{s}\right)\left(\stackrel{˜}{x}\right).$

This completes the proof of Theorem 1. □

By Theorem 1 and the ${L}^{p}$-boundedness of ${M}_{L{\left(logL\right)}^{1/r}}$, we may obtain the conclusions (1), (2) of Theorem 2.

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