# Estimates of singular integrals and multilinear commutators in weighted Morrey spaces

## Abstract

Suppose T is a singular integral operator whose kernel is a variable kernel with mixed homogeneity; the purpose of this paper is to study the continuity of the operator in weighted Morrey spaces ${L}^{p,\kappa }\left(\omega \right)$, $1\le p<\mathrm{\infty }$, $0<\kappa <1$. A special attention is paid also to the multilinear commutator of this operator with BMO function.

MSC:42B20, 42B35.

## 1 Introduction

Let $K\left(x,\xi \right):{\mathbb{R}}^{n}×{\mathbb{R}}^{n}\mathrm{\setminus }\left\{0\right\}\to \mathbb{R}$ be a variable kernel. The singular integral operator is defined by

$Tf\left(x\right)=p.v.{\int }_{{\mathbb{R}}^{n}}K\left(x,x-y\right)f\left(y\right)\phantom{\rule{0.2em}{0ex}}dy$
(1.1)

and its multilinear commutator with the BMO function

$\left[\stackrel{\to }{b},T\right]f\left(x\right)={\int }_{{\mathbb{R}}^{n}}\prod _{i=1}^{N}\left({b}_{i}\left(x\right)-{b}_{i}\left(y\right)\right)K\left(x,x-y\right)f\left(y\right)\phantom{\rule{0.2em}{0ex}}dy,$
(1.2)

where $\stackrel{\to }{b}=\left({b}_{1},\dots ,{b}_{n}\right)$, ${b}_{i}\in \mathit{BMO}$, $1\le i\le N$. The variable kernel $K\left(x,\xi \right)$ depends on some parameter x and possesses ‘good’ properties with respect to the second variable ξ, which was firstly introduced by Fabes and Rieviève in . They generalized the classical Calderón and Zygmund kernel and the parabolic kernel studied by Jones in . By introducing a new metric ρ, Fabes and Rieviève studied (1.1) in ${L}^{p}\left({\mathbb{R}}^{n}\right)$, where ${\mathbb{R}}^{n}$ was endowed with the topology induced by ρ and defined by ellipsoids.

By using this metric ρ, Softova in  obtained that the integral operator (1.1) and commutator were continuous in generalized Morrey spaces ${L}^{p,\omega }\left({\mathbb{R}}^{n}\right)$, $1, ω satisfying suitable conditions.

The multilinear commutator was introduced by Pérez and González  who proved the weighted Lebesgue estimates. Xu in  also showed that the multilinear commutators (1.2) were continuous in ${L}^{p,\omega }\left({\mathbb{R}}^{n}\right)$, $1.

The weighted Morrey spaces ${L}^{p,\kappa }\left(w\right)$ were introduced by Komori and Shirai . Moreover, they showed some classical integral operators and corresponding commutators were bounded in weighted Morrey spaces. Recently, Wang  obtained that some other kind of integral operators (e.g., Bochner-Riesz operator, Marcinkiewicz operators etc.) and commutators were also bounded in weighted Morrey spaces. He Sha  showed that multilinear operators were bounded on weighted Morrey spaces with the symbol of $b\in Lip\left(\beta \right)$. The main purpose of this paper is to discuss the continuity of the singular integral operator whose kernel is a variable kernel with mixed homogeneity and its multilinear commutator in the weighted Morrey spaces ${L}^{p,\kappa }\left(\omega \right)$, $1, $0<\kappa <1$, where the weight function ω is ${A}_{p}$ weight. Furthermore, we shall give the weighted weak type estimate of theses operators in the weighted Morrey spaces ${L}^{1,\kappa }\left(\omega \right)$, $0<\kappa <1$. Our main results are stated as follows.

Theorem 1.1 Let $1, $0<\kappa <1$. If $w\in {A}_{p}$, then there exists a constant $C>0$ such that

${\parallel Tf\parallel }_{{L}^{p,\kappa }\left(w\right)}\le C{\parallel f\parallel }_{{L}^{p,\kappa }\left(w\right)}.$

When $p=1$, for any $\lambda >0$ and ellipsoid , there exists a constant $C>0$ such that

$\lambda w\left(\left\{x\in \mathcal{E}:|Tf\left(x\right)|>\lambda \right\}\right)\le C{\parallel f\parallel }_{{L}^{1,\kappa }\left(w\right)}.$

If $K\left(x,\xi \right)$ is a constant kernel and a metric ρ is Euclidean one, this result is just Theorem 3.3 in .

Theorem 1.2 Let $1, $0<\kappa <1$. If ${b}_{i}\in \mathit{BMO}\left({\mathbb{R}}^{n}\right)$, $1\le i\le N$, $w\in {A}_{p}$, then there exists a constant $C>0$ such that

${\parallel \left[\stackrel{\to }{b},T\right]f\parallel }_{{L}^{p,\kappa }\left(w\right)}\le C\parallel \stackrel{\to }{b}\parallel {\parallel f\parallel }_{{L}^{p,\kappa }\left(w\right)},$

where $\parallel \stackrel{\to }{b}\parallel ={\prod }_{i=1}^{N}{\parallel {b}_{i}\parallel }_{\ast }$. When $p=1$, for any $\lambda >0$ and ellipsoid , then there exists a constant $C>0$ such that

$\lambda w\left(\left\{x\in \mathcal{E}:|\left[\stackrel{\to }{b},T\right]f\left(x\right)|>\lambda \right\}\right)\le C\parallel \stackrel{\to }{b}\parallel {\parallel f\parallel }_{{L}^{\mathrm{\Phi },\kappa }\left(w\right)},$

where $\mathrm{\Phi }\left(t\right)=t{log}^{N}\left(e+t\right)$ and ${\parallel f\parallel }_{{L}^{\mathrm{\Phi },\kappa }\left(w\right)}={\parallel \mathrm{\Phi }\left(|f|\right)\parallel }_{{L}^{1,\kappa }\left(w\right)}$.

In what follows, we denote by C positive constants which are independent of the main parameters but may vary from line to line.

## 2 Some notations and lemmas

In this section, we introduce some basic definitions and lemmas needed for the proof of the main results.

Let ${\alpha }_{1},\dots ,{\alpha }_{n}$ be real numbers, ${\alpha }_{i}\ge 1$ and $|\alpha |={\sum }_{i=1}^{n}{\alpha }_{i}$. Following Fabes and Riviève , there exists a function ρ such that $\rho \left(x-y\right)$ defines a distance between any two points $x,y\in {\mathbb{R}}^{n}$. Thus ${\mathbb{R}}^{n}$ endowed with the metric ρ results in a homogeneous metric space [1, 3]. The balls with respect to $\rho \left(x\right)$ centered at the origin and of radius r are the ellipsoids

${\mathcal{E}}_{r}\left(0\right)=\left\{x\in {\mathbb{R}}^{n}:\frac{{x}_{1}^{2}}{{r}^{2{\alpha }_{1}}}+\cdots +\frac{{x}_{n}^{2}}{{r}^{2{\alpha }_{n}}}<1\right\}$

with Lebesgue measure $|{\mathcal{E}}_{r}|=C\left(n\right){r}^{|\alpha |}$. It is easy to see that the unit sphere with respect to this metric coincides with the unit sphere ${\mathrm{\Sigma }}_{n}$ with respect to the Euclidean one.

Definition 2.1 The function $K\left(x,\xi \right):{\mathbb{R}}^{n}×{\mathbb{R}}^{n}\mathrm{\setminus }\left\{0\right\}\to \mathbb{R}$ is called a variable kernel with mixed homogeneity if

1. (i)

for every fixed x, the function $K\left(x,\cdot \right)$ is a constant kernel satisfying

2. (1)

$K\left(x,\cdot \right)\in {C}^{\mathrm{\infty }}\left({\mathbb{R}}^{n}\mathrm{\setminus }\left\{0\right\}\right)$;

3. (2)

for any $\mu >0$, ${\alpha }_{i}\ge 1$, $|\alpha |={\sum }_{i=1}^{n}{\alpha }_{i}$

$K\left(x,{\mu }^{{\alpha }_{1}}{\xi }_{1},\dots ,{\mu }^{{\alpha }_{n}}{\xi }_{n}\right)={\mu }^{-|\alpha |}K\left(x,\xi \right);$
4. (3)

${\int }_{{\mathrm{\Sigma }}_{n}}K\left(x,\xi \right)\phantom{\rule{0.2em}{0ex}}d{\sigma }_{\xi }=0$ and ${\int }_{{\mathrm{\Sigma }}_{n}}|K\left(x,\xi \right)|\phantom{\rule{0.2em}{0ex}}d{\sigma }_{\xi }<\mathrm{\infty }$;

5. (ii)

for every multiindex β, ${sup}_{\xi \in {\mathrm{\Sigma }}_{n}}|{D}_{\xi }^{\beta }K\left(x,\xi \right)|\le C\left(\beta \right)$ independent of x.

In the case ${\alpha }_{i}=1$, $1\le i\le n$, Definition 2.1 gives rise to the classical Calderón-Zygmund kernel. On the other hand, when ${\alpha }_{i}=1$, $1\le i\le n-1$ and ${\alpha }_{n}\ge 1$, we obtain the kernel studied by Jones in  and discussed in .

Definition 2.2 Let $1\le p<\mathrm{\infty }$, $0<\kappa <1$ and w be a weight function. Then a weighted Morrey space is defined by

${L}^{p,\kappa }\left(w\right):=\left\{f\in {L}_{\mathrm{loc}}^{1}\left(w\right):{\parallel f\parallel }_{{L}^{p,\kappa }\left(w\right)}<\mathrm{\infty }\right\},$

where

${\parallel f\parallel }_{{L}^{p,\kappa }\left(w\right)}=\underset{\mathcal{E}}{sup}{\left(\frac{1}{w{\left(\mathcal{E}\right)}^{\kappa }}{\int }_{\mathcal{E}}{|f\left(x\right)|}^{p}w\left(x\right)\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/p},$

the supremum is taken over all ellipsoid in ${\mathbb{R}}^{n}$.

Definition 2.3 For the function $b\in {L}_{\mathrm{loc}}^{1}\left({\mathbb{R}}^{n}\right)$ and any ellipsoid , b is called a BMO function if

${\parallel b\parallel }_{\ast }=\underset{\mathcal{E}}{sup}\frac{1}{|\mathcal{E}|}{\int }_{\mathcal{E}}|b\left(x\right)-{b}_{\mathcal{E}}|\phantom{\rule{0.2em}{0ex}}dx<\mathrm{\infty },$

where ${b}_{\mathcal{E}}=\frac{1}{|\mathcal{E}|}{\int }_{\mathcal{E}}b\left(y\right)\phantom{\rule{0.2em}{0ex}}dy$. The quantity ${\parallel b\parallel }_{\ast }$ is a norm in the BMO modulo constant function under which BMO results in a Banach space (see ).

Definition 2.4 Let $1. For any locally integrable function w and ellipsoid , if

$\left(\frac{1}{|\mathcal{E}|}{\int }_{\mathcal{E}}w\left(x\right)\phantom{\rule{0.2em}{0ex}}dx\right){\left(\frac{1}{|\mathcal{E}|}{\int }_{\mathcal{E}}w{\left(x\right)}^{\frac{1}{1-p}}\phantom{\rule{0.2em}{0ex}}dx\right)}^{p-1}<\mathrm{\infty }$

holds, then w belongs to the Muckenhoupt class ${A}_{p}$. We denote ${A}_{\mathrm{\infty }}={\bigcup }_{1.

When $p=1$, $w\in {A}_{1}$ if there exists $C>1$ such that

$Mw\left(x\right)\le Cw\left(x\right)$

for almost every $x\in {\mathbb{R}}^{n}$.

Remark 2.5 Given a weight function $w\in {A}_{p}$, $1\le p\le \mathrm{\infty }$, it also satisfies the doubling condition ${\mathrm{\Delta }}_{2}$: for any ellipsoid , there exists a constant $C>0$ such that $w\left(2\mathcal{E}\right)\le Cw\left(\mathcal{E}\right)$.

In fact, $w\in {\mathrm{\Delta }}_{2}$, we have the following inequality.

Lemma A [6, 12]

Suppose $w\in {\mathrm{\Delta }}_{2}$, there exists a constant $D>1$ such that

$w\left(2\mathcal{E}\right)\ge Dw\left(\mathcal{E}\right)$

for any ellipsoid .

Lemma B 

Suppose $w\in {A}_{\mathrm{\infty }}$, then the norm of $\mathit{BMO}\left(w\right)$ is equivalent to the norm of $\mathit{BMO}\left({\mathbb{R}}^{n}\right)$, where

$\mathit{BMO}\left(w\right)=\left\{b:{\parallel b\parallel }_{\ast ,w}=\underset{\mathcal{E}}{sup}\frac{1}{w\left(\mathcal{E}\right)}{\int }_{\mathcal{E}}|b\left(x\right)-{b}_{\mathcal{E},w}|w\left(x\right)\phantom{\rule{0.2em}{0ex}}dx<\mathrm{\infty }\right\},$

where ${b}_{\mathcal{E},w}=\frac{1}{w\left(\mathcal{E}\right)}{\int }_{\mathcal{E}}b\left(x\right)w\left(x\right)\phantom{\rule{0.2em}{0ex}}dx$.

Lemma C 

Let the ellipsoid $\mathcal{E}=\mathcal{E}\left({x}_{0},r\right)$ centered at ${x}_{0}$ with side length of r. For any positive integer i, ${2}^{i}\mathcal{E}$ denotes the ellipsoid centered at ${x}_{0}$ with side length of ${2}^{i}r$, we have the inequality

$|{b}_{{2}^{i}\mathcal{E}}-{b}_{\mathcal{E}}|\le Ci{\parallel b\parallel }_{\ast },$

where ${b}_{\mathcal{E},w}=\frac{1}{|\mathcal{E}|}{\int }_{\mathcal{E}}b\left(x\right)w\left(x\right)\phantom{\rule{0.2em}{0ex}}dx$.

Lemma D 

Suppose $1, $0<\kappa <1$ and $w\in {A}_{p}$, if $\overline{T}$ is the classical Calderón-Zygmund operator with a constant kernel, then the operator $\overline{T}$ is bounded on ${L}^{p,\kappa }\left(w\right)$.

If $p=1$, $0<\kappa <1$ and $w\in {A}_{1}$, then there exists a constant $C>0$ such that

$\lambda w\left(\left\{x\in \mathcal{E}:|\overline{T}f\left(x\right)|>\lambda \right\}\right)\le C{\parallel f\parallel }_{{L}^{1,\kappa }\left(w\right)}w{\left(\mathcal{E}\right)}^{\kappa }$

for all $\lambda >0$ and any ellipsoid .

Definition 2.6 Let $\mathrm{\Phi }\left(t\right)=t{log}^{N}\left(t+e\right)$. The Orlicz maximal operator ${M}_{\mathrm{\Phi }}$ is given by

${M}_{\mathrm{\Phi }}f\left(x\right)=\underset{x\in \mathcal{E}}{sup}{\parallel f\parallel }_{\mathrm{\Phi },\mathcal{E}}=\underset{x\in \mathcal{E}}{sup}\frac{1}{|\mathcal{E}|}{\int }_{\mathcal{E}}\mathrm{\Phi }\left(|f|\right)\left(x\right)\phantom{\rule{0.2em}{0ex}}dx.$

From the above definition, observe that $Mf\left(x\right)\le {M}_{\mathrm{\Phi }}f\left(x\right)\le M\left(\mathrm{\Phi }\left(|f|\right)\right)\left(x\right)$. This inequality will be relevant in our work.

Aside from the properties of an ${A}_{p}$ weight function and a BMO function, we need some estimates of multilinear commutators. The following results were proved by Pérez and González .

Lemma E Let $1 and $w\in {A}_{p}$. Suppose ${b}_{j}\in \mathit{BMO}\left({\mathbb{R}}^{n}\right)$, $1\le j\le N$, then there exists a constant $C>0$ such that

${\int }_{{\mathbb{R}}^{n}}{|\left[\stackrel{\to }{b},\overline{T}\right]\left(f\right)\left(x\right)|}^{p}w\left(x\right)\phantom{\rule{0.2em}{0ex}}dx\le C{\parallel \stackrel{\to }{b}\parallel }^{p}{\int }_{{\mathbb{R}}^{n}}{|f\left(x\right)|}^{p}w\left(x\right)\phantom{\rule{0.2em}{0ex}}dx.$

Although the commutators with a BMO function are not of weak type $\left(1,1\right)$, we have the following inequality.

Lemma F Let $w\in {A}_{\mathrm{\infty }}$. There exists a constant $C>0$ such that where $\mathrm{\Phi }\left(t\right)=t{log}^{N}\left(e+t\right)$.

By the above inequality, we have the following result.

Lemma G Let $w\in {A}_{1}$. There exists a constant $C>0$ such that, for all $\lambda >0$,

$w\left(x\in {\mathbb{R}}^{n}:|\left[\stackrel{\to }{b},\overline{T}\right]\left(f\right)\left(x\right)|>\lambda \right)\le C{\int }_{{\mathbb{R}}^{n}}\mathrm{\Phi }\left(|f|\right)\left(x\right)w\left(x\right)\phantom{\rule{0.2em}{0ex}}dx,$

where $\mathrm{\Phi }\left(t\right)=t{log}^{N}\left(e+t\right)$.

Finally, we need the spherical harmonics and their properties (see more detail in [1, 15, 16]). Recall that any homogeneous polynomial $P:{\mathbb{R}}^{n}\to \mathbb{R}$ of degree m that satisfies $\mathrm{\Delta }P=0$ is called an n-dimensional solid harmonic of degree m. Its restriction to the unit sphere ${\mathrm{\Sigma }}_{n}$ will be called an n-dimensional spherical harmonic of degree m. Denote by ${H}_{m}$ the space of all n-dimensional spherical harmonics of degree m. In general, it results in a finite-dimensional linear space with ${g}_{m}=dim{H}_{m}$ such that ${g}_{0}=1$, ${g}_{1}=n$ and

${g}_{m}={C}_{m+n-1}^{n-1}-{C}_{m+n-3}^{n-1}\le C\left(n\right){m}^{n-2},\phantom{\rule{1em}{0ex}}m\ge 2.$
(2.1)

Furthermore, let ${\left\{{Y}_{sm}\right\}}_{s=1}^{{g}_{m}}$ be an orthonormal basis of ${H}_{m}$. Then ${\left\{{Y}_{sm}\right\}}_{s=1m=0}^{{g}_{m}\mathrm{\infty }}$ is a complete orthonormal system in ${L}^{2}\left({\mathrm{\Sigma }}_{n}\right)$ and

$\underset{x\in {\mathrm{\Sigma }}_{n}}{sup}|{D}_{x}^{\beta }{Y}_{sm}\left(x\right)|\le C\left(n\right){m}^{|\beta |+\left(n-2\right)/2},\phantom{\rule{1em}{0ex}}m=1,2,\dots .$
(2.2)

If, for instance, $\varphi \in {C}^{\mathrm{\infty }}\left({\mathrm{\Sigma }}_{n}\right)$, then ${\mathrm{\Sigma }}_{s,m}{b}_{sm}{Y}_{sm}\left(x\right)$ is the Fourier series expansion of $\varphi \left(x\right)$ with respect to ${\left\{{Y}_{sm}\right\}}_{s,m}$ (${\mathrm{\Sigma }}_{s,m}$ substitutes ${\mathrm{\Sigma }}_{m=0}^{\mathrm{\infty }}{\mathrm{\Sigma }}_{s=1}^{{g}_{m}}$) and

${b}_{sm}={\int }_{{\mathrm{\Sigma }}_{n}}\varphi \left(x\right){Y}_{sm}\left(x\right)\phantom{\rule{0.2em}{0ex}}d\sigma ,\phantom{\rule{1em}{0ex}}|{b}_{sm}|\le C\left(n,l\right){m}^{-2l}\underset{y\in {\mathrm{\Sigma }}_{n}}{\underset{|\beta |=2l}{sup}}|{D}_{y}^{\beta }\varphi \left(y\right)|,$
(2.3)

for any integer l. In particular, the expansion of ϕ into spherical harmonics converges uniformly to ϕ. For more detail, we can see .

## 3 Proof of the theorems

In this section, we shall use the complete orthonormal system in ${L}^{2}\left({\mathrm{\Sigma }}_{n}\right)$ and some lemmas as above to finish the theorems.

Proof of Theorem 1.1 In order to ensure the existence of the operator (1.1) in ${L}^{p,\kappa }\left(w\right)$, $1\le p<\mathrm{\infty }$, we restrict our consideration to the function $f\in {L}^{p,\kappa }\left(w\right)$, for which the norm of ${L}^{p}\left(w\right)$ is finite. For the sake of convenience, we still denote these spaces by ${L}^{p,\kappa }\left(w\right)$. Let $x,y\in {\mathbb{R}}^{n}$ and $\overline{y}=y/\rho \left(y\right)\in {\mathrm{\Sigma }}_{n}$. In view of the properties of the kernel K with respect to the second variable and the complete of $\left\{{Y}_{sm}\left(x\right)\right\}$ in ${L}^{2}\left({\mathrm{\Sigma }}_{n}\right)$, we get

$\begin{array}{rcl}K\left(x,x-y\right)& =& \rho {\left(x-y\right)}^{-|\alpha |}K\left(x,\overline{x-y}\right)\\ =& \rho {\left(x-y\right)}^{-|\alpha |}\sum _{s,m}{b}_{sm}\left(x\right){Y}_{sm}\left(\overline{x-y}\right).\end{array}$

Replacing the kernel with its series expansion, (1.1) can be written as

$\begin{array}{rcl}Tf\left(x\right)& =& \underset{ϵ\to 0}{lim}{T}_{ϵ}f\left(x\right)\\ =& \underset{ϵ\to 0}{lim}{\int }_{\rho \left(x-y\right)>ϵ}\sum _{s,m}{b}_{sm}\left(x\right)\rho {\left(x-y\right)}^{-|\alpha |}{Y}_{sm}\left(\overline{x-y}\right)f\left(y\right)\phantom{\rule{0.2em}{0ex}}dy.\end{array}$

From the properties of (2.1)-(2.3), the series expansion ${\sum }_{s,m}|{b}_{sm}\left(x\right){Y}_{sm}\left(\overline{x-y}\right)|\le C\left(n,\alpha \right){m}^{3\left(n-2\right)/2-2l}$, where the integer l is preliminarily chosen greater than $\left(3n-4\right)/4$. Along with the $\rho {\left(x-y\right)}^{-|\alpha |}f\left(y\right)\in {L}^{1}\left({\mathbb{R}}^{n}\right)$ for a.a. $x\in {\mathbb{R}}^{n}$, by the Fubini dominated convergence theorem, we have

$Tf\left(x\right)=\sum _{s,m}{b}_{sm}\left(x\right)\underset{ϵ\to 0}{lim}{\int }_{\rho \left(x-y\right)>ϵ}{H}_{sm}\left(x-y\right)f\left(y\right)\phantom{\rule{0.2em}{0ex}}dy=\sum _{s,m}{b}_{sm}\left(x\right){T}_{sm}f\left(x\right),$

where ${H}_{sm}\left(x-y\right)=\rho {\left(x-y\right)}^{-|\alpha |}{Y}_{sm}\left(\overline{x-y}\right)$. Instead of the operators $Tf\left(x\right)$, we shall study the existence and boundedness in ${L}^{p,\kappa }\left(\omega \right)$ of the operators ${T}_{sm}f\left(x\right)$ with a kernel ${H}_{sm}\left(\cdot \right)$. Observe that ${H}_{sm}\left(\cdot \right)$ is a constant kernel and satisfies

$|{H}_{sm}\left(x\right)|\le C\left(n,\alpha \right){m}^{\frac{n-2}{2}}{\rho }^{-|\alpha |};\phantom{\rule{2em}{0ex}}|\mathrm{\nabla }{H}_{sm}\left(x\right)|\le C\left(n,\alpha \right){m}^{\frac{n}{2}}{\rho }^{-|\alpha |-1}.$

From Lemma D, it follows

${\parallel {T}_{sm}f\left(x\right)\parallel }_{{L}^{p,\kappa }\left(\omega \right)}\le C\left(n,\alpha \right){m}^{\frac{n}{2}}{\parallel f\left(x\right)\parallel }_{{L}^{p,\kappa }\left(\omega \right)}$

for $1. Consequently, by the above inequality and (2.1)-(2.3), we show

$\begin{array}{rl}{\parallel Tf\left(x\right)\parallel }_{{L}^{p,\kappa }\left(\omega \right)}& \le C\sum _{s,m}{\parallel {b}_{sm}\left(x\right)\parallel }_{{L}^{\mathrm{\infty }}}{\parallel {T}_{sm}f\left(x\right)\parallel }_{{L}^{p,\kappa }\left(\omega \right)}\\ \le C\sum _{s,m}{m}^{-2l+\frac{n}{2}}{\parallel f\left(x\right)\parallel }_{{L}^{p,\kappa }\left(\omega \right)}\\ \le C{\parallel f\parallel }_{{L}^{p,\kappa }\left(\omega \right)},\end{array}$

where the integer l is preliminary chosen greater that $l>\frac{3n}{4}$. For $p=1$, by Lemma D, we have

$\lambda w\left(\left\{x\in \mathcal{E}:|{T}_{sm}f\left(x\right)|>\lambda \right\}\right)\le C\left(n,\alpha \right){m}^{\frac{n}{2}}{\parallel f\parallel }_{{L}^{1,\kappa }\left(w\right)}$

for any $\lambda >0$ and ellipsoid . Therefore, one gets

$\begin{array}{rl}\lambda w\left(\left\{x\in \mathcal{E}:|Tf\left(x\right)|>\lambda \right\}\right)& \le C\sum _{s,m}{\parallel {b}_{sm}\left(x\right)\parallel }_{{L}^{\mathrm{\infty }}}\lambda w\left(\left\{x\in \mathcal{E}:|{T}_{sm}f\left(x\right)|>\lambda \right\}\right)\\ \le C\sum _{s,m}{m}^{-2l+\frac{n}{2}}{\parallel f\parallel }_{{L}^{1,\kappa }\left(w\right)}\\ \le C{\parallel f\left(x\right)\parallel }_{{L}^{1,\kappa }\left(w\right)},\end{array}$

thus we complete the proof of Theorem 1.1. □

Next we begin with the second theorem, for which further discussion is needed.

Proof of Theorem 1.2 As above, we use the series expansion of a kernel $K\left(x,y\right)$, the operator $\left[\stackrel{\to }{b},T\right]f\left(x\right)$ is divided into

$\left[\stackrel{\to }{b},T\right]f\left(x\right)=\sum _{s,m}{b}_{sm}\left(x\right)\left[\stackrel{\to }{b},{T}_{sm}\right]f\left(x\right).$

Instead of the operator $\left[\stackrel{\to }{b},T\right]f\left(x\right)$, we only consider the existence and boundedness in ${L}^{p,\kappa }\left(w\right)$ of the operators $\left[\stackrel{\to }{b},{T}_{sm}\right]f\left(x\right)$.

Let $1. For any ellipsoid , we only need to obtain the inequality

${\int }_{\mathcal{E}}{|\left[\stackrel{\to }{b},{T}_{sm}\right]f\left(x\right)|}^{p}w\left(x\right)\phantom{\rule{0.2em}{0ex}}dx\le C{m}^{\frac{mp}{2}}{\parallel b\parallel }^{p}w{\left(\mathcal{E}\right)}^{k}{\parallel f\parallel }_{{L}^{p,\kappa }\left(w\right)}^{p}.$

In fact, by the series expansion of a kernel $K\left(x,y\right)$, we have

$\begin{array}{rl}{\parallel \left[\stackrel{\to }{b},T\right]f\left(x\right)\parallel }_{{L}^{p,\kappa }\left(\omega \right)}& \le C\sum _{s,m}{\parallel {b}_{sm}\left(x\right)\parallel }_{{L}^{\mathrm{\infty }}}{\parallel \left[\stackrel{\to }{b},{T}_{sm}\right]f\left(x\right)\parallel }_{{L}^{p,\kappa }\left(\omega \right)}\\ \le C\sum _{s,m}{m}^{-2l+\frac{n}{2}}{\parallel f\parallel }_{{L}^{p,\kappa }\left(\omega \right)}\le C{\parallel f\parallel }_{{L}^{p,\kappa }\left(\omega \right)},\end{array}$

where the integer l is chosen greater than $l>\frac{3n}{4}$. Next, fix the above ellipsoid $\mathcal{E}=\mathcal{E}\left({x}_{0},r\right)$ and decompose $f={f}_{1}+{f}_{2}$, where ${f}_{1}=f{\chi }_{2\mathcal{E}}$, ${\chi }_{2\mathcal{E}}$ denotes the characteristic function of $2\mathcal{E}$, then we have

$\begin{array}{rl}{\int }_{\mathcal{E}}{|\left[\stackrel{\to }{b},{T}_{sm}\right]\left(f\right)\left(x\right)|}^{p}w\left(x\right)\phantom{\rule{0.2em}{0ex}}dx& \le C{\int }_{\mathcal{E}}\left\{{|\left[\stackrel{\to }{b},{T}_{sm}\right]\left({f}_{1}\right)\left(x\right)|}^{p}+{|\left[\stackrel{\to }{b},{T}_{sm}\right]\left({f}_{2}\right)\left(x\right)|}^{p}\right\}w\left(x\right)\phantom{\rule{0.2em}{0ex}}dx\\ =C\left\{I+\mathit{II}\right\}.\end{array}$
(3.1)

By using Lemma E, we get

$\begin{array}{rl}I& \le {\int }_{{\mathbb{R}}^{n}}{|\left[\stackrel{\to }{b},{T}_{sm}\right]\left({f}_{1}\right)\left(x\right)|}^{p}w\left(x\right)\phantom{\rule{0.2em}{0ex}}dx\\ \le C{m}^{\frac{np}{2}}{\parallel \stackrel{\to }{b}\parallel }^{p}w{\left(\mathcal{E}\right)}^{\kappa }{\parallel f\parallel }_{{L}^{p,\kappa }\left(w\right)}^{p}.\end{array}$
(3.2)

For the term II, without loss of generality, we can assume $N=2$. Thus, the operator $\left[\stackrel{\to }{b},{T}_{sm}\right]$ can be divided into four parts,

$\begin{array}{rcl}\left[\stackrel{\to }{b},{T}_{sm}\right]{f}_{2}\left(x\right)& =& \left({b}_{1}\left(x\right)-{\lambda }_{1}\right)\left({b}_{2}\left(x\right)-{\lambda }_{2}\right){\int }_{{\mathbb{R}}^{n}}{H}_{sm}\left(x-y\right){f}_{2}\left(y\right)\phantom{\rule{0.2em}{0ex}}dy\\ +{\int }_{{\mathbb{R}}^{n}}{H}_{sm}\left(x-y\right)\left({b}_{1}\left(y\right)-{\lambda }_{1}\right)\left({b}_{2}\left(y\right)-{\lambda }_{2}\right){f}_{2}\left(y\right)\phantom{\rule{0.2em}{0ex}}dy\\ -\left({b}_{1}\left(x\right)-{\lambda }_{1}\right){\int }_{{\mathbb{R}}^{n}}{H}_{sm}\left(x-y\right)\left({b}_{2}\left(y\right)-{\lambda }_{2}\right){f}_{2}\left(y\right)\phantom{\rule{0.2em}{0ex}}dy\\ -\left({b}_{2}\left(x\right)-{\lambda }_{2}\right){\int }_{{\mathbb{R}}^{n}}{H}_{sm}\left(x-y\right)\left({b}_{1}\left(y\right)-{\lambda }_{1}\right){f}_{2}\left(y\right)\phantom{\rule{0.2em}{0ex}}dy\\ =& {\mathit{II}}_{1}\left(x\right)+{\mathit{II}}_{2}\left(x\right)+{\mathit{II}}_{3}\left(x\right)+{\mathit{II}}_{4}\left(x\right),\end{array}$
(3.3)

where ${\lambda }_{i}={\left({b}_{i}\right)}_{\mathcal{E},w}=\frac{1}{w\left(\mathcal{E}\right)}{\int }_{\mathcal{E}}{b}_{i}\left(x\right)w\left(x\right)\phantom{\rule{0.2em}{0ex}}dx$, $i=1,2$. For the term ${\mathit{II}}_{1}\left(x\right)$, observing that $x\in \mathcal{E}$ and $y\in {\mathbb{R}}^{n}\mathrm{\setminus }2\mathcal{E}$, we have $\rho \left({x}_{0}-y\right)\le C\rho \left(x-y\right)$. Thus, it yields

$\begin{array}{rcl}{\int }_{\mathcal{E}}{|{\mathit{II}}_{1}\left(x\right)|}^{p}w\left(x\right)\phantom{\rule{0.2em}{0ex}}dx& \le & C{m}^{\frac{np}{2}}{\int }_{\mathcal{E}}{|\left({b}_{1}\left(x\right)-{\left({b}_{1}\right)}_{\mathcal{E},w}\right)\left({b}_{2}\left(x\right)-{\left({b}_{2}\right)}_{\mathcal{E},w}\right)|}^{p}w\left(x\right)\phantom{\rule{0.2em}{0ex}}dx\\ ×{\left({\int }_{{\mathbb{R}}^{n}\mathrm{\setminus }2\mathcal{E}}\frac{|f\left(y\right)|}{\rho {\left({x}_{0}-y\right)}^{|\alpha |}}\phantom{\rule{0.2em}{0ex}}dy\right)}^{p}\\ \le & C{m}^{\frac{np}{2}}w\left(\mathcal{E}\right){\left(\frac{1}{w\left(\mathcal{E}\right)}{\int }_{\mathcal{E}}{|{b}_{1}\left(x\right)-{\left({b}_{1}\right)}_{\mathcal{E},w}|}^{2p}w\left(x\right)\phantom{\rule{0.2em}{0ex}}dx\right)}^{\frac{1}{2}}\\ ×{\left(\frac{1}{w\left(\mathcal{E}\right)}{\int }_{\mathcal{E}}{|{b}_{2}\left(x\right)-{\left({b}_{2}\right)}_{\mathcal{E},w}|}^{2p}w\left(x\right)\phantom{\rule{0.2em}{0ex}}dx\right)}^{\frac{1}{2}}\\ ×{\left(\sum _{j=1}^{\mathrm{\infty }}{\int }_{{2}^{j+1}\mathcal{E}\mathrm{\setminus }{2}^{j}\mathcal{E}}\frac{|f\left(y\right)|}{\rho {\left({x}_{0}-y\right)}^{|\alpha |}}\phantom{\rule{0.2em}{0ex}}dy\right)}^{p}\\ \le & C{m}^{\frac{np}{2}}{\parallel {b}_{1}\parallel }_{\ast }^{p}{\parallel {b}_{2}\parallel }_{\ast }^{p}w\left(\mathcal{E}\right)\left(\sum _{j=1}^{\mathrm{\infty }}\frac{1}{|{2}^{j}\mathcal{E}|}{\left({\int }_{{2}^{j+1}\mathcal{E}}{|f\left(y\right)|}^{p}w\left(y\right)\phantom{\rule{0.2em}{0ex}}dy\right)}^{\frac{1}{p}}\\ ×{{\left({\int }_{{2}^{j+1}\mathcal{E}}w{\left(y\right)}^{-\frac{{p}^{\prime }}{p}}\phantom{\rule{0.2em}{0ex}}dx\right)}^{\frac{1}{{p}^{\prime }}}\right)}^{p},\end{array}$

since $w\in {A}_{p}$, and by the definition of a weighted Morrey space, we get

$\begin{array}{rcl}{\int }_{\mathcal{E}}{|{\mathit{II}}_{1}\left(x\right)|}^{p}w\left(x\right)\phantom{\rule{0.2em}{0ex}}dx& \le & C{m}^{\frac{np}{2}}{\parallel \stackrel{\to }{b}\parallel }^{p}w\left(\mathcal{E}\right){\left(\sum _{j=1}^{\mathrm{\infty }}w{\left({2}^{j+1}\mathcal{E}\right)}^{\frac{-1}{p}}{\left({\int }_{{2}^{j+1}\mathcal{E}}{|f\left(y\right)|}^{p}w\left(y\right)\phantom{\rule{0.2em}{0ex}}dy\right)}^{\frac{1}{p}}\right)}^{p}\\ \le & C{m}^{\frac{np}{2}}{\parallel \stackrel{\to }{b}\parallel }^{p}w\left(\mathcal{E}\right){\left(\sum _{j=1}^{\mathrm{\infty }}w{\left({2}^{j+1}\mathcal{E}\right)}^{\frac{\kappa -1}{p}}{\parallel f\parallel }_{{L}^{p,\kappa }\left(w\right)}\right)}^{p}\\ \le & C{m}^{\frac{np}{2}}{\parallel \stackrel{\to }{b}\parallel }^{p}w\left(\mathcal{E}\right){\left(\sum _{j=1}^{\mathrm{\infty }}{D}^{j\frac{\kappa -1}{p}}w{\left(\mathcal{E}\right)}^{\frac{\kappa -1}{p}}{\parallel f\parallel }_{{L}^{p,\kappa }\left(w\right)}\right)}^{p}\\ \le & C{m}^{\frac{np}{2}}{\parallel \stackrel{\to }{b}\parallel }^{p}w{\left(\mathcal{E}\right)}^{\kappa }{\parallel f\parallel }_{{L}^{p,\kappa }\left(w\right)}^{p}.\end{array}$
(3.4)

The third inequality is obtained by Lemma A.

For ${\mathit{II}}_{2}\left(x\right)$, note that ${\lambda }_{i}={\left({b}_{i}\right)}_{\mathcal{E},w}=\frac{1}{w\left(\mathcal{E}\right)}{\int }_{\mathcal{E}}{b}_{i}\left(x\right)w\left(x\right)\phantom{\rule{0.2em}{0ex}}dx$, $i=1,2$. By Hölder’s inequality and $\rho \left({x}_{0}-y\right)\le C\rho \left(x-y\right)$, we get

$\begin{array}{rcl}{\int }_{\mathcal{E}}{|{\mathit{II}}_{2}\left(x\right)|}^{p}w\left(x\right)\phantom{\rule{0.2em}{0ex}}dx& \le & C{m}^{\frac{np}{2}}w\left(\mathcal{E}\right){\left({\int }_{{\mathbb{R}}^{n}\mathrm{\setminus }2\mathcal{E}}\frac{|\left({b}_{1}\left(y\right)-{\left({b}_{1}\right)}_{\mathcal{E},w}\right)\left({b}_{2}\left(y\right)-{\left({b}_{2}\right)}_{\mathcal{E},w}\right)|}{\rho {\left({x}_{0}-y\right)}^{|\alpha |}}|f\left(y\right)|\phantom{\rule{0.2em}{0ex}}dy\right)}^{p}\\ \le & C{m}^{\frac{np}{2}}w\left(\mathcal{E}\right)\left(\sum _{j=1}^{\mathrm{\infty }}\frac{1}{|{2}^{j}\mathcal{E}|}{\int }_{{2}^{j+1}\mathcal{E}\mathrm{\setminus }{2}^{j}\mathcal{E}}|\left({b}_{1}\left(y\right)-{\left({b}_{1}\right)}_{\mathcal{E},w}\right)\\ ×{\left({b}_{2}\left(y\right)-{\left({b}_{2}\right)}_{\mathcal{E},w}\right)||f\left(y\right)|\phantom{\rule{0.2em}{0ex}}dy\right)}^{p}\\ \le & C{m}^{\frac{np}{2}}w\left(\mathcal{E}\right)\left(\sum _{j=1}^{\mathrm{\infty }}\frac{1}{|{2}^{j}\mathcal{E}|}{\left({\int }_{{2}^{j+1}\mathcal{E}}{|f\left(y\right)|}^{p}w\left(y\right)\phantom{\rule{0.2em}{0ex}}dy\right)}^{\frac{1}{p}}\\ ×{\left({\int }_{{2}^{j+1}\mathcal{E}}{|\left({b}_{1}\left(y\right)-{\left({b}_{1}\right)}_{\mathcal{E},w}\right)|}^{2{p}^{\prime }}w{\left(y\right)}^{-\frac{{p}^{\prime }}{p}}\phantom{\rule{0.2em}{0ex}}dy\right)}^{\frac{1}{2{p}^{\prime }}}\\ ×{{\left({\int }_{{2}^{j+1}\mathcal{E}}{|\left({b}_{2}\left(y\right)-{\left({b}_{2}\right)}_{\mathcal{E},w}\right)|}^{2{p}^{\prime }}w{\left(y\right)}^{-\frac{{p}^{\prime }}{p}}\phantom{\rule{0.2em}{0ex}}dy\right)}^{\frac{1}{2{p}^{\prime }}}\right)}^{p}.\end{array}$

Indeed, by Lemma B we know $\mathit{BMO}\left({\mathbb{R}}^{n}\right)$ is equivalent to $\mathit{BMO}\left(w\right)$, $w\in {A}_{\mathrm{\infty }}$. Let $W={w}^{-\frac{{p}^{\prime }}{p}}\in {A}_{{p}^{\prime }}\subset {A}_{\mathrm{\infty }}$, ${b}_{i}\in \mathit{BMO}\left({\mathbb{R}}^{n}\right)$, $i=1,2$. For any ellipsoid , by using Lemma B and Lemma C, we show

${\left(\frac{1}{W\left({2}^{j+1}\mathcal{E}\right)}{\int }_{{2}^{j+1}\mathcal{E}}{|{b}_{i}\left(y\right)-{\left({b}_{i}\right)}_{\mathcal{E},w}|}^{2{p}^{\prime }}W\left(y\right)\phantom{\rule{0.2em}{0ex}}dy\right)}^{\frac{1}{2{p}^{\prime }}}\le Cj{\parallel {b}_{i}\parallel }_{\ast }.$

Thus, since $w\in {A}_{p}$, it yields

$\begin{array}{rcl}{\int }_{\mathcal{E}}{|{\mathit{II}}_{2}\left(x\right)|}^{p}w\left(x\right)\phantom{\rule{0.2em}{0ex}}dx& \le & C{m}^{\frac{np}{2}}w\left(\mathcal{E}\right)\left(\sum _{j=1}^{\mathrm{\infty }}\frac{{j}^{2}}{|{2}^{j}\mathcal{E}|}{\parallel {b}_{1}\parallel }_{\ast }{\parallel {b}_{2}\parallel }_{\ast }W{\left({2}^{j+1}\mathcal{E}\right)}^{\frac{1}{{p}^{\prime }}}\\ ×{{\left({\int }_{{2}^{j+1}\mathcal{E}}{|f\left(y\right)|}^{p}w\left(y\right)\phantom{\rule{0.2em}{0ex}}dy\right)}^{\frac{1}{p}}\right)}^{p}\\ \le & C{m}^{\frac{np}{2}}w\left(\mathcal{E}\right){\parallel \stackrel{\to }{b}\parallel }^{p}{\parallel f\parallel }_{{L}^{p,\kappa }\left(w\right)}^{p}{\left(\sum _{j=1}^{\mathrm{\infty }}\frac{{j}^{2}}{w{\left({2}^{j+1}\mathcal{E}\right)}^{\frac{1-\kappa }{p}}}\right)}^{p}\\ \le & C{m}^{\frac{np}{2}}w{\left(\mathcal{E}\right)}^{\kappa }{\parallel \stackrel{\to }{b}\parallel }^{p}{\parallel f\parallel }_{{L}^{p,\kappa }\left(w\right)}^{p}.\end{array}$
(3.5)

The last inequality is obtained by Lemma A and the D’Alembert judge method of positive series.

For ${\mathit{II}}_{3}\left(x\right)$, by the inequality $\rho \left({x}_{0}-y\right)\le C\rho \left(x-y\right)$ since $w\in {A}_{p}\subset {A}_{\mathrm{\infty }}$, by Lemma B, we have By Hölder’s inequality, Lemma B and Lemma C, we get

$\begin{array}{rcl}{\int }_{{\mathbb{R}}^{n}\mathrm{\setminus }2\mathcal{E}}\frac{|{b}_{2}\left(y\right)-{\lambda }_{2}|}{\rho {\left({x}_{0}-y\right)}^{|\alpha |}}|f\left(y\right)|\phantom{\rule{0.2em}{0ex}}dy& \le & C\sum _{j=1}^{\mathrm{\infty }}\frac{1}{|{2}^{j}\mathcal{E}|}{\int }_{{2}^{j+1}\mathcal{E}}|{b}_{2}\left(y\right)-{\lambda }_{2}||f\left(y\right)|\phantom{\rule{0.2em}{0ex}}dy\\ \le & C\sum _{j=1}^{\mathrm{\infty }}\frac{1}{|{2}^{j}\mathcal{E}|}{\left({\int }_{{2}^{j+1}\mathcal{E}}{|f\left(y\right)|}^{p}w\left(y\right)\phantom{\rule{0.2em}{0ex}}dy\right)}^{\frac{1}{p}}\\ ×{\left({\int }_{{2}^{j+1}\mathcal{E}}{|{b}_{2}\left(y\right)-{\lambda }_{2}|}^{{p}^{\prime }}w{\left(y\right)}^{-\frac{{p}^{\prime }}{p}}\phantom{\rule{0.2em}{0ex}}dy\right)}^{\frac{1}{{p}^{\prime }}}\\ \le & C{\parallel {b}_{2}\parallel }_{\ast }{\parallel f\parallel }_{{L}^{p,\kappa }\left(w\right)}\sum _{j=1}^{\mathrm{\infty }}j\frac{w{\left({2}^{j+1}\mathcal{E}\right)}^{\frac{\kappa }{p}}}{|{2}^{j}\mathcal{E}|}{\left({\int }_{{2}^{j+1}\mathcal{E}}w{\left(y\right)}^{-\frac{{p}^{\prime }}{p}}\phantom{\rule{0.2em}{0ex}}dy\right)}^{\frac{1}{{p}^{\prime }}}\\ \le & C{\parallel {b}_{2}\parallel }_{\ast }{\parallel f\parallel }_{{L}^{p,\kappa }\left(w\right)}\sum _{j=1}^{\mathrm{\infty }}\frac{j}{w{\left({2}^{j+1}\mathcal{E}\right)}^{\frac{1-\kappa }{p}}},\end{array}$

indeed for $0<\kappa <1$, by using Lemma A, we have that

$\sum _{j=1}^{\mathrm{\infty }}\frac{j}{w{\left({2}^{j+1}\mathcal{E}\right)}^{\frac{1-\kappa }{p}}}\le \sum _{j=1}^{\mathrm{\infty }}\frac{j}{{D}^{\left(j+1\right)\frac{1-\kappa }{p}}}w{\left(\mathcal{E}\right)}^{\frac{\kappa -1}{p}}\le Cw{\left(\mathcal{E}\right)}^{\frac{\kappa -1}{p}}.$

Thus, we conclude

${\int }_{\mathcal{E}}{|{\mathit{II}}_{3}\left(x\right)|}^{p}w\left(x\right)\phantom{\rule{0.2em}{0ex}}dx\le C{m}^{\frac{np}{2}}w{\left(\mathcal{E}\right)}^{\kappa }{\parallel \stackrel{\to }{b}\parallel }^{p}{\parallel f\parallel }_{{L}^{p,\kappa }\left(w\right)}^{p}.$
(3.6)

In the same way, we shall get the result of ${\mathit{II}}_{4}\left(x\right)$

${\int }_{\mathcal{E}}{|{\mathit{II}}_{4}\left(x\right)|}^{p}w\left(x\right)\phantom{\rule{0.2em}{0ex}}dx\le C{m}^{\frac{np}{2}}w{\left(\mathcal{E}\right)}^{\kappa }{\parallel \stackrel{\to }{b}\parallel }^{p}{\parallel f\parallel }_{{L}^{p,\kappa }\left(w\right)}^{p}.$
(3.7)

Which together with (3.1)-(3.7), for $1, the proof of Theorem 1.2 is finished.

Now, we are in a position to consider the case $p=1$. In general, the singularity of the commutator is stronger than the singular integral, and the endpoint case $p=1$ of the commutator is not even obtained. Thus, the result for the case $p=1$ of the multilinear commutator is interesting. We split f as above by $f={f}_{1}+{f}_{2}$, which yields

$\begin{array}{rcl}\lambda w\left(\left\{x\in \mathcal{E}:|\left[\stackrel{\to }{b},T\right]f\left(x\right)|>\lambda \right\}\right)& \le & C\sum _{s,m}{\parallel {b}_{sm}\parallel }_{{L}^{\mathrm{\infty }}}\lambda w\left(\left\{x\in \mathcal{E}:|\left[\stackrel{\to }{b},{T}_{sm}\right]f\left(x\right)|>\lambda \right\}\right)\\ \le & C\sum _{s,m}{m}^{-2l}\left[\lambda w\left(\left\{x\in \mathcal{E}:|\left[\stackrel{\to }{b},{T}_{sm}\right]{f}_{1}\left(x\right)|>\lambda /2\right\}\right)\\ +\lambda w\left(\left\{x\in \mathcal{E}:|\left[\stackrel{\to }{b},{T}_{sm}\right]{f}_{2}\left(x\right)|>\lambda /2\right\}\right)\right]\\ =& C\sum _{s,m}{m}^{-2l}\left[\mathit{III}+\mathit{IV}\right]\end{array}$
(3.8)

for any ellipsoid , $\lambda >0$ and integer $l>0$. For the term III, we use Lemma G. It follows that

$\begin{array}{rl}\mathit{III}& \le C{\int }_{{\mathbb{R}}^{n}}\mathrm{\Phi }\left(|{f}_{1}|\right)\left(x\right)w\left(x\right)\phantom{\rule{0.2em}{0ex}}dx\\ \le Cw{\left(\mathcal{E}\right)}^{\kappa }{\parallel f\parallel }_{{L}^{\mathrm{\Phi },\kappa }\left(w\right)}.\end{array}$
(3.9)

For the last term IV, without loss of generality, we still suppose $N=2$. By homogeneity, it is enough to assume $\lambda /2={\parallel {b}_{1}\parallel }_{\ast }={\parallel {b}_{2}\parallel }_{\ast }=1$, and hence we only need to prove

$w\left(\left\{x\in \mathcal{E}:|\left[\stackrel{\to }{b},{T}_{sm}\right]{f}_{2}\left(x\right)|>1\right\}\right)\le Cw{\left(\mathcal{E}\right)}^{\kappa }{\parallel f\parallel }_{{L}^{\mathrm{\Phi },\kappa }\left(w\right)}.$

In fact, by Lemma F, we get

$\begin{array}{rl}w\left(\left\{x\in \mathcal{E}:|\left[\stackrel{\to }{b},{T}_{sm}\right]{f}_{2}\left(x\right)|>1\right\}\right)& \le \underset{t>0}{sup}\frac{1}{\mathrm{\Phi }\left(\frac{1}{t}\right)}w\left(\left\{x\in \mathcal{E}:|\left[\stackrel{\to }{b},{T}_{sm}\right]{f}_{2}\left(x\right)|>t\right\}\right)\\ \le C\underset{t>0}{sup}\frac{1}{\mathrm{\Phi }\left(\frac{1}{t}\right)}w\left(\left\{x\in \mathcal{E}:{M}_{\mathrm{\Phi }}{f}_{2}\left(x\right)>t\right\}\right)\\ =C\underset{t>0}{sup}\frac{1}{\mathrm{\Phi }\left(\frac{1}{t}\right)}w\left(\left\{x\in \mathcal{E}:M\left(\mathrm{\Phi }|{f}_{2}|\right)\left(x\right)>t\right\}\right),\end{array}$
(3.10)

where $\mathrm{\Phi }\left(t\right)=t{log}^{N}\left(e+t\right)$. We use the Fefferman-Stein maximal inequality

${\int }_{x:Mf\left(x\right)>t}\varphi \left(t\right)\phantom{\rule{0.2em}{0ex}}dx\le \frac{C}{t}{\int }_{{\mathbb{R}}^{n}}|f\left(x\right)|M\varphi \left(x\right)\phantom{\rule{0.2em}{0ex}}dx,$

for any functions f and $\varphi \ge 0$. This yields

$\begin{array}{rl}w\left(\left\{x\in \mathcal{E}:M\left(\mathrm{\Phi }|{f}_{2}|\right)\left(x\right)>t\right\}\right)& \le \frac{1}{t}{\int }_{\left\{x\in {\mathbb{R}}^{n}:M\left(\mathrm{\Phi }|{f}_{2}|\right)\left(x\right)>t\right\}}{\chi }_{\mathcal{E}}\left(x\right)w\left(x\right)\phantom{\rule{0.2em}{0ex}}dx\\ \le \frac{C}{t}{\int }_{{\mathbb{R}}^{n}}\mathrm{\Phi }\left(|{f}_{2}|\right)\left(x\right)M\left(w{\chi }_{\mathcal{E}}\right)\left(x\right)\phantom{\rule{0.2em}{0ex}}dx\\ =\frac{C}{t}\left({\int }_{3\mathcal{E}}+{\int }_{{\mathbb{R}}^{n}\mathrm{\setminus }3\mathcal{E}}\right)\mathrm{\Phi }\left(|{f}_{2}|\right)\left(x\right)M\left(w{\chi }_{\mathcal{E}}\right)\left(x\right)\phantom{\rule{0.2em}{0ex}}dx\\ =\frac{C}{t}\left({\mathit{IV}}_{1}+{\mathit{IV}}_{2}\right).\end{array}$
(3.11)

For ${\mathit{IV}}_{1}$, since $w\in {A}_{1}$, it follows that

$\begin{array}{rl}{\mathit{IV}}_{1}& \le C{\int }_{3\mathcal{E}}\mathrm{\Phi }\left(|f|\right)\left(x\right)w\left(x\right)\phantom{\rule{0.2em}{0ex}}dx\\ \le Cw{\left(3\mathcal{E}\right)}^{\kappa }{\parallel \mathrm{\Phi }\left(|f|\right)\parallel }_{{L}^{1,\kappa }\left(w\right)}\\ \le Cw{\left(\mathcal{E}\right)}^{\kappa }{\parallel f\parallel }_{{L}^{\mathrm{\Phi },\kappa }}.\end{array}$
(3.12)

To estimate the term ${\mathit{IV}}_{2}$, we first consider the form

$\frac{1}{|\mathcal{F}|}{\int }_{\mathcal{E}\cap \mathcal{F}}w\left(y\right)\phantom{\rule{0.2em}{0ex}}dy$

for any $x\in {\mathbb{R}}^{n}\mathrm{\setminus }3\mathcal{E}$, $x\in \mathcal{F}$ and $\mathcal{F}\cap \mathcal{E}\ne \mathrm{\varnothing }$. By simple geometric observation, we have

$\begin{array}{rl}\frac{1}{|\mathcal{F}|}{\int }_{\mathcal{E}\cap \mathcal{F}}w\left(y\right)\phantom{\rule{0.2em}{0ex}}dy& \le C\frac{1}{\rho {\left(x-{x}_{0}\right)}^{|\alpha |}}{\int }_{\mathcal{E}}w\left(y\right)\phantom{\rule{0.2em}{0ex}}dy\\ =\frac{C}{\rho {\left(x-{x}_{0}\right)}^{|\alpha |}}w\left(\mathcal{E}\right).\end{array}$

Therefore, we obtain

$M\left(w{\chi }_{\mathcal{E}}\right)\left(x\right)\le \frac{C}{\rho {\left(x-{x}_{0}\right)}^{|\alpha |}}w\left(\mathcal{E}\right).$

Since $w\in {A}_{1}$ satisfies the doubling condition and Lemma A, we estimate the term ${\mathit{IV}}_{2}$ as follows:

$\begin{array}{rl}{\mathit{IV}}_{2}& \le C{\int }_{{\mathbb{R}}^{n}\mathrm{\setminus }3\mathcal{E}}\frac{\mathrm{\Phi }\left(|f|\right)\left(x\right)}{\rho {\left(x-{x}_{0}\right)}^{|\alpha |}}w\left(\mathcal{E}\right)\phantom{\rule{0.2em}{0ex}}dx\\ \le Cw\left(\mathcal{E}\right)\sum _{j=1}^{\mathrm{\infty }}\frac{1}{|{3}^{j}\mathcal{E}|}{\int }_{{3}^{j+1}\mathcal{E}}\mathrm{\Phi }\left(|f|\right)\left(x\right)\phantom{\rule{0.2em}{0ex}}dx\\ \le Cw\left(\mathcal{E}\right){\parallel \mathrm{\Phi }\left(|f|\right)\parallel }_{{L}^{1,\kappa }\left(w\right)}\sum _{j=1}^{\mathrm{\infty }}\frac{1}{w{\left({3}^{j}\mathcal{E}\right)}^{1-\kappa }}\\ \le Cw{\left(\mathcal{E}\right)}^{\kappa }{\parallel f\parallel }_{{L}^{\mathrm{\Phi },\kappa }}.\end{array}$
(3.13)

The last inequality is similar to (3.4). Noting that $t\mathrm{\Phi }\left(\frac{1}{t}\right)>1$, from (3.8)-(3.11), we conclude

$w\left(\left\{x\in \mathcal{E}:|\left[\stackrel{\to }{b},{T}_{sm}\right]{f}_{2}\left(x\right)|>1\right\}\right)\le Cw{\left(\mathcal{E}\right)}^{\kappa }{\parallel f\parallel }_{{L}^{\mathrm{\Phi },\kappa }\left(w\right)}.$

Thus, the proof of Theorem 1.2 is completed. □

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## Acknowledgements

XFY research was partially supported by the National Natural Sciences Foundation of China (10961015; 11161021). XSZ research was supported by the National Natural Sciences Foundation of China (11161021).

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Correspondence to Xiao Feng Ye.

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

XFY conceived of the study and drafted the manuscript. XSZ participated in the discussion. All authors read and approved the final manuscript.

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Ye, X.F., Zhu, X.S. Estimates of singular integrals and multilinear commutators in weighted Morrey spaces. J Inequal Appl 2012, 302 (2012). https://doi.org/10.1186/1029-242X-2012-302

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• DOI: https://doi.org/10.1186/1029-242X-2012-302

### Keywords

• mixed homogeneity
• multilinear commutators
• weighted Morrey spaces
• BMO
• Orlicz maximal operator 