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 $L^{p,\kappa }(\omega )$, $1\le p<\mathrm{\infty }$, $0<\kappa <1$. A special attention is paid also to the multilinear commutator of this operator with BMO function.

MSC:42B20, 42B35.

1 Introduction

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

$$Tf(x)=p.v.{\int }_{{\mathbb{R}}^n}K(x,x-y)f(y)dy$$

(1.1)

and its multilinear commutator with the BMO function

$$[\overrightarrow{b},T]f(x)={\int }_{{\mathbb{R}}^n}\prod _{i=1}^N(b_i(x)-b_i(y))K(x,x-y)f(y)dy,$$

(1.2)

where $\overrightarrow{b}=(b_1,\dots ,b_n)$, $b_i\in \mathit{BMO}$, $1\le i\le N$. The variable kernel $K(x,\xi )$ 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 [1]. They generalized the classical Calderón and Zygmund kernel and the parabolic kernel studied by Jones in [2]. By introducing a new metric ρ, Fabes and Rieviève studied (1.1) in $L^p({\mathbb{R}}^n)$, where ${\mathbb{R}}^n$ was endowed with the topology induced by ρ and defined by ellipsoids.

By using this metric ρ, Softova in [3] obtained that the integral operator (1.1) and commutator were continuous in generalized Morrey spaces $L^{p,\omega }({\mathbb{R}}^n)$, $1<p<\mathrm{\infty }$, ω satisfying suitable conditions.

The multilinear commutator was introduced by Pérez and González [4] who proved the weighted Lebesgue estimates. Xu in [5] also showed that the multilinear commutators (1.2) were continuous in $L^{p,\omega }({\mathbb{R}}^n)$, $1<p<\mathrm{\infty }$.

The weighted Morrey spaces $L^{p,\kappa }(w)$ were introduced by Komori and Shirai [6]. Moreover, they showed some classical integral operators and corresponding commutators were bounded in weighted Morrey spaces. Recently, Wang [7–9] 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 [10] showed that multilinear operators were bounded on weighted Morrey spaces with the symbol of $b\in Lip(\beta )$. 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 $L^{p,\kappa }(\omega )$, $1<p<\mathrm{\infty }$, $0<\kappa <1$, 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 $L^{1,\kappa }(\omega )$, $0<\kappa <1$. Our main results are stated as follows.

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

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

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

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

If $K(x,\xi )$ is a constant kernel and a metric ρ is Euclidean one, this result is just Theorem 3.3 in [6].

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

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

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

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

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

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 [1], there exists a function ρ such that $\rho (x-y)$ 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 $\rho (x)$ centered at the origin and of radius r are the ellipsoids

$${\mathcal{E}}_r(0)=\{x\in {\mathbb{R}}^n:\frac{x_1^2}{r^{2{\alpha }_1}}+\cdots +\frac{x_n^2}{r^{2{\alpha }_n}}<1\}$$

with Lebesgue measure $|{\mathcal{E}}_r|=C(n)r^{|\alpha |}$. 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 $K(x,\xi ):{\mathbb{R}}^n\times {\mathbb{R}}^n\mathrm{\setminus }\{0\}\to \mathbb{R}$ is called a variable kernel with mixed homogeneity if

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

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

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

  $$K(x,{\mu }^{{\alpha }_1}{\xi }_1,\dots ,{\mu }^{{\alpha }_n}{\xi }_n)={\mu }^{-|\alpha |}K(x,\xi );$$

• (3) ${\int }_{{\mathrm{\Sigma }}_n}K(x,\xi )d{\sigma }_{\xi }=0$ and ${\int }_{{\mathrm{\Sigma }}_n}|K(x,\xi )|d{\sigma }_{\xi }<\mathrm{\infty }$;

• (ii) for every multiindex β, ${sup}_{\xi \in {\mathrm{\Sigma }}_n}|D_{\xi }^{\beta }K(x,\xi )|\le C(\beta )$ 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 [2] and discussed in [1].

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

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

where

$${\parallel f\parallel }_{L^{p,\kappa }(w)}=\underset{\mathcal{E}}{sup}{(\frac{1}{w{(\mathcal{E})}^{\kappa }}{\int }_{\mathcal{E}}{|f(x)|}^{p}w(x)dx)}^{1/p},$$

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

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

$${\parallel b\parallel }_{\ast }=\underset{\mathcal{E}}{sup}\frac{1}{|\mathcal{E}|}{\int }_{\mathcal{E}}|b(x)-b_{\mathcal{E}}|dx<\mathrm{\infty },$$

where $b_{\mathcal{E}}=\frac{1}{|\mathcal{E}|}{\int }_{\mathcal{E}}b(y)dy$. The quantity ${\parallel b\parallel }_{\ast }$ is a norm in the BMO modulo constant function under which BMO results in a Banach space (see [11]).

Definition 2.4 Let $1<p<\mathrm{\infty }$. For any locally integrable function w and ellipsoid ℰ, if

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

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

When $p=1$, $w\in A_1$ if there exists $C>1$ such that

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

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 ℰ, there exists a constant $C>0$ such that $w(2\mathcal{E})\le Cw(\mathcal{E})$.

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

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

for any ellipsoid ℰ.

Lemma B [13]

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

$$\mathit{BMO}(w)=\{b:{\parallel b\parallel }_{\ast ,w}=\underset{\mathcal{E}}{sup}\frac{1}{w(\mathcal{E})}{\int }_{\mathcal{E}}|b(x)-b_{\mathcal{E},w}|w(x)dx<\mathrm{\infty }\},$$

where $b_{\mathcal{E},w}=\frac{1}{w(\mathcal{E})}{\int }_{\mathcal{E}}b(x)w(x)dx$.

Lemma C [14]

Let the ellipsoid $\mathcal{E}=\mathcal{E}(x_0,r)$ 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 $b_{\mathcal{E},w}=\frac{1}{|\mathcal{E}|}{\int }_{\mathcal{E}}b(x)w(x)dx$.

Lemma D [6]

Suppose $1<p<\mathrm{\infty }$, $0<\kappa <1$ 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 $L^{p,\kappa }(w)$.

If $p=1$, $0<\kappa <1$ and $w\in A_1$, then there exists a constant $C>0$ such that

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

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

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

$$M_{\mathrm{\Phi }}f(x)=\underset{x\in \mathcal{E}}{sup}{\parallel f\parallel }_{\mathrm{\Phi },\mathcal{E}}=\underset{x\in \mathcal{E}}{sup}\frac{1}{|\mathcal{E}|}{\int }_{\mathcal{E}}\mathrm{\Phi }(|f|)(x)dx.$$

From the above definition, observe that $Mf(x)\le M_{\mathrm{\Phi }}f(x)\le M(\mathrm{\Phi }(|f|))(x)$. 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 [4].

Lemma E Let $1<p<\mathrm{\infty }$ and $w\in A_p$. Suppose $b_j\in \mathit{BMO}({\mathbb{R}}^n)$, $1\le j\le N$, then there exists a constant $C>0$ such that

$${\int }_{{\mathbb{R}}^n}{|[\overrightarrow{b},\overline{T}](f)(x)|}^{p}w(x)dx\le C{\parallel \overrightarrow{b}\parallel }^p{\int }_{{\mathbb{R}}^n}{|f(x)|}^{p}w(x)dx.$$

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

Lemma F Let $w\in A_{\mathrm{\infty }}$. There exists a constant $C>0$ such that

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

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 $\lambda >0$,

$$w(x\in {\mathbb{R}}^n:|[\overrightarrow{b},\overline{T}](f)(x)|>\lambda )\le C{\int }_{{\mathbb{R}}^n}\mathrm{\Phi }(|f|)(x)w(x)dx,$$

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

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

$$g_m=C_{m+n-1}^{n-1}-C_{m+n-3}^{n-1}\le C(n)m^{n-2},m\ge 2.$$

(2.1)

Furthermore, let ${\{Y_{sm}\}}_{s=1}^{g_m}$ be an orthonormal basis of $H_m$. Then ${\{Y_{sm}\}}_{s=1m=0}^{g_m\mathrm{\infty }}$ is a complete orthonormal system in $L^2({\mathrm{\Sigma }}_n)$ and

$$\underset{x\in {\mathrm{\Sigma }}_n}{sup}|D_x^{\beta }{Y}_{sm}(x)|\le C(n)m^{|\beta |+(n-2)/2},m=1,2,\dots .$$

(2.2)

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

$$b_{sm}={\int }_{{\mathrm{\Sigma }}_n}\varphi (x)Y_{sm}(x)d\sigma ,|b_{sm}|\le C(n,l)m^{-2l}\underset{y\in {\mathrm{\Sigma }}_n}{\underset{|\beta |=2l}{sup}}|D_y^{\beta }\varphi (y)|,$$

(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 [15].

3 Proof of the theorems

In this section, we shall use the complete orthonormal system in $L^2({\mathrm{\Sigma }}_n)$ 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 $L^{p,\kappa }(w)$, $1\le p<\mathrm{\infty }$, we restrict our consideration to the function $f\in L^{p,\kappa }(w)$, for which the norm of $L^p(w)$ is finite. For the sake of convenience, we still denote these spaces by $L^{p,\kappa }(w)$. Let $x,y\in {\mathbb{R}}^n$ and $\overline{y}=y/\rho (y)\in {\mathrm{\Sigma }}_n$. In view of the properties of the kernel K with respect to the second variable and the complete of $\{Y_{sm}(x)\}$ in $L^2({\mathrm{\Sigma }}_n)$, we get

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

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

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

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

$$Tf(x)=\sum _{s,m}{b}_{sm}(x)\underset{ϵ\to 0}{lim}{\int }_{\rho (x-y)>ϵ}{H}_{sm}(x-y)f(y)dy=\sum _{s,m}{b}_{sm}(x)T_{sm}f(x),$$

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

$$|H_{sm}(x)|\le C(n,\alpha )m^{\frac{n-2}{2}}{\rho }^{-|\alpha |};|\mathrm{\nabla }{H}_{sm}(x)|\le C(n,\alpha )m^{\frac{n}{2}}{\rho }^{-|\alpha |-1}.$$

From Lemma D, it follows

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

for $1<p<\mathrm{\infty }$. Consequently, by the above inequality and (2.1)-(2.3), we show

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

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

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

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

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

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 $K(x,y)$, the operator $[\overrightarrow{b},T]f(x)$ is divided into

$$[\overrightarrow{b},T]f(x)=\sum _{s,m}{b}_{sm}(x)[\overrightarrow{b},T_{sm}]f(x).$$

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

Let $1<p<\mathrm{\infty }$. For any ellipsoid ℰ, we only need to obtain the inequality

$${\int }_{\mathcal{E}}{|[\overrightarrow{b},T_{sm}]f(x)|}^{p}w(x)dx\le C{m}^{\frac{mp}{2}}{\parallel b\parallel }^{p}w{(\mathcal{E})}^k{\parallel f\parallel }_{L^{p,\kappa }(w)}^p.$$

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

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

where the integer l is chosen greater than $l>\frac{3n}{4}$. Next, fix the above ellipsoid $\mathcal{E}=\mathcal{E}(x_0,r)$ 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

$$\begin{array}{rl}{\int }_{\mathcal{E}}{|[\overrightarrow{b},T_{sm}](f)(x)|}^{p}w(x)dx& \le C{\int }_{\mathcal{E}}\{{|[\overrightarrow{b},T_{sm}](f_1)(x)|}^p+{|[\overrightarrow{b},T_{sm}](f_2)(x)|}^p\}w(x)dx\\ =C\{I+\mathit{II}\}.\end{array}$$

(3.1)

By using Lemma E, we get

$$\begin{array}{rl}I& \le {\int }_{{\mathbb{R}}^n}{|[\overrightarrow{b},T_{sm}](f_1)(x)|}^{p}w(x)dx\\ \le C{m}^{\frac{np}{2}}{\parallel \overrightarrow{b}\parallel }^{p}w{(\mathcal{E})}^{\kappa }{\parallel f\parallel }_{L^{p,\kappa }(w)}^p.\end{array}$$

(3.2)

For the term II, without loss of generality, we can assume $N=2$. Thus, the operator $[\overrightarrow{b},T_{sm}]$ can be divided into four parts,

$$\begin{array}{rcl}[\overrightarrow{b},T_{sm}]f_2(x)& =& (b_1(x)-{\lambda }_1)(b_2(x)-{\lambda }_2){\int }_{{\mathbb{R}}^n}{H}_{sm}(x-y)f_2(y)dy\\ +{\int }_{{\mathbb{R}}^n}{H}_{sm}(x-y)(b_1(y)-{\lambda }_1)(b_2(y)-{\lambda }_2)f_2(y)dy\\ -(b_1(x)-{\lambda }_1){\int }_{{\mathbb{R}}^n}{H}_{sm}(x-y)(b_2(y)-{\lambda }_2)f_2(y)dy\\ -(b_2(x)-{\lambda }_2){\int }_{{\mathbb{R}}^n}{H}_{sm}(x-y)(b_1(y)-{\lambda }_1)f_2(y)dy\\ =& {\mathit{II}}_1(x)+{\mathit{II}}_2(x)+{\mathit{II}}_3(x)+{\mathit{II}}_4(x),\end{array}$$

(3.3)

where ${\lambda }_i={(b_i)}_{\mathcal{E},w}=\frac{1}{w(\mathcal{E})}{\int }_{\mathcal{E}}{b}_i(x)w(x)dx$, $i=1,2$. For the term ${\mathit{II}}_1(x)$, observing that $x\in \mathcal{E}$ and $y\in {\mathbb{R}}^n\mathrm{\setminus }2\mathcal{E}$, we have $\rho (x_0-y)\le C\rho (x-y)$. Thus, it yields

$$\begin{array}{rcl}{\int }_{\mathcal{E}}{|{\mathit{II}}_1(x)|}^{p}w(x)dx& \le & C{m}^{\frac{np}{2}}{\int }_{\mathcal{E}}{\left|(b_1(x)-{(b_1)}_{\mathcal{E},w})(b_2(x)-{(b_2)}_{\mathcal{E},w})\right|}^{p}w(x)dx\\ \times {\left({\int }_{{\mathbb{R}}^n\mathrm{\setminus }2\mathcal{E}}\frac{|f(y)|}{\rho {(x_0-y)}^{|\alpha |}}dy\right)}^p\\ \le & C{m}^{\frac{np}{2}}w(\mathcal{E}){(\frac{1}{w(\mathcal{E})}{\int }_{\mathcal{E}}{|b_1(x)-{(b_1)}_{\mathcal{E},w}|}^{2p}w(x)dx)}^{\frac{1}{2}}\\ \times {(\frac{1}{w(\mathcal{E})}{\int }_{\mathcal{E}}{|b_2(x)-{(b_2)}_{\mathcal{E},w}|}^{2p}w(x)dx)}^{\frac{1}{2}}\\ \times {\left(\sum _{j=1}^{\mathrm{\infty }}{\int }_{2^{j+1}\mathcal{E}\mathrm{\setminus }{2}^j\mathcal{E}}\frac{|f(y)|}{\rho {(x_0-y)}^{|\alpha |}}dy\right)}^p\\ \le & C{m}^{\frac{np}{2}}{\parallel b_1\parallel }_{\ast }^p{\parallel b_2\parallel }_{\ast }^{p}w(\mathcal{E})(\sum _{j=1}^{\mathrm{\infty }}\frac{1}{|2^j\mathcal{E}|}{({\int }_{2^{j+1}\mathcal{E}}{|f(y)|}^{p}w(y)dy)}^{\frac{1}{p}}\\ \times {{\left({\int }_{2^{j+1}\mathcal{E}}w{(y)}^{-\frac{p^{\prime }}{p}}dx\right)}^{\frac{1}{p^{\prime }}})}^p,\end{array}$$

since $w\in A_p$, and by the definition of a weighted Morrey space, we get

$$\begin{array}{rcl}{\int }_{\mathcal{E}}{|{\mathit{II}}_1(x)|}^{p}w(x)dx& \le & C{m}^{\frac{np}{2}}{\parallel \overrightarrow{b}\parallel }^{p}w(\mathcal{E}){\left(\sum _{j=1}^{\mathrm{\infty }}w{\left(2^{j+1}\mathcal{E}\right)}^{\frac{-1}{p}}{({\int }_{2^{j+1}\mathcal{E}}{|f(y)|}^{p}w(y)dy)}^{\frac{1}{p}}\right)}^p\\ \le & C{m}^{\frac{np}{2}}{\parallel \overrightarrow{b}\parallel }^{p}w(\mathcal{E}){\left(\sum _{j=1}^{\mathrm{\infty }}w{\left(2^{j+1}\mathcal{E}\right)}^{\frac{\kappa -1}{p}}{\parallel f\parallel }_{L^{p,\kappa }(w)}\right)}^p\\ \le & C{m}^{\frac{np}{2}}{\parallel \overrightarrow{b}\parallel }^{p}w(\mathcal{E}){\left(\sum _{j=1}^{\mathrm{\infty }}{D}^{j\frac{\kappa -1}{p}}w{(\mathcal{E})}^{\frac{\kappa -1}{p}}{\parallel f\parallel }_{L^{p,\kappa }(w)}\right)}^p\\ \le & C{m}^{\frac{np}{2}}{\parallel \overrightarrow{b}\parallel }^{p}w{(\mathcal{E})}^{\kappa }{\parallel f\parallel }_{L^{p,\kappa }(w)}^p.\end{array}$$

(3.4)

The third inequality is obtained by Lemma A.

For ${\mathit{II}}_2(x)$, note that ${\lambda }_i={(b_i)}_{\mathcal{E},w}=\frac{1}{w(\mathcal{E})}{\int }_{\mathcal{E}}{b}_i(x)w(x)dx$, $i=1,2$. By Hölder’s inequality and $\rho (x_0-y)\le C\rho (x-y)$, we get

$$\begin{array}{rcl}{\int }_{\mathcal{E}}{|{\mathit{II}}_2(x)|}^{p}w(x)dx& \le & C{m}^{\frac{np}{2}}w(\mathcal{E}){\left({\int }_{{\mathbb{R}}^n\mathrm{\setminus }2\mathcal{E}}\frac{|(b_1(y)-{(b_1)}_{\mathcal{E},w})(b_2(y)-{(b_2)}_{\mathcal{E},w})|}{\rho {(x_0-y)}^{|\alpha |}}|f(y)|dy\right)}^p\\ \le & C{m}^{\frac{np}{2}}w(\mathcal{E})\left(\sum _{j=1}^{\mathrm{\infty }}\frac{1}{|2^j\mathcal{E}|}{\int }_{2^{j+1}\mathcal{E}\mathrm{\setminus }{2}^j\mathcal{E}}\right|(b_1(y)-{(b_1)}_{\mathcal{E},w})\\ \times {(b_2(y)-{(b_2)}_{\mathcal{E},w})\left||f(y)|dy\right)}^p\\ \le & C{m}^{\frac{np}{2}}w(\mathcal{E})(\sum _{j=1}^{\mathrm{\infty }}\frac{1}{|2^j\mathcal{E}|}{({\int }_{2^{j+1}\mathcal{E}}{|f(y)|}^{p}w(y)dy)}^{\frac{1}{p}}\\ \times {\left({\int }_{2^{j+1}\mathcal{E}}{\left|(b_1(y)-{(b_1)}_{\mathcal{E},w})\right|}^{2p^{\prime }}w{(y)}^{-\frac{p^{\prime }}{p}}dy\right)}^{\frac{1}{2p^{\prime }}}\\ \times {{\left({\int }_{2^{j+1}\mathcal{E}}{\left|(b_2(y)-{(b_2)}_{\mathcal{E},w})\right|}^{2p^{\prime }}w{(y)}^{-\frac{p^{\prime }}{p}}dy\right)}^{\frac{1}{2p^{\prime }}})}^p.\end{array}$$

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

$${(\frac{1}{W(2^{j+1}\mathcal{E})}{\int }_{2^{j+1}\mathcal{E}}{|b_i(y)-{(b_i)}_{\mathcal{E},w}|}^{2p^{\prime }}W(y)dy)}^{\frac{1}{2p^{\prime }}}\le Cj{\parallel b_i\parallel }_{\ast }.$$

Thus, since $w\in A_p$, it yields

$$\begin{array}{rcl}{\int }_{\mathcal{E}}{|{\mathit{II}}_2(x)|}^{p}w(x)dx& \le & C{m}^{\frac{np}{2}}w(\mathcal{E})(\sum _{j=1}^{\mathrm{\infty }}\frac{j^2}{|2^j\mathcal{E}|}{\parallel b_1\parallel }_{\ast }{\parallel b_2\parallel }_{\ast }W{\left(2^{j+1}\mathcal{E}\right)}^{\frac{1}{p^{\prime }}}\\ \times {{({\int }_{2^{j+1}\mathcal{E}}{|f(y)|}^{p}w(y)dy)}^{\frac{1}{p}})}^p\\ \le & C{m}^{\frac{np}{2}}w(\mathcal{E}){\parallel \overrightarrow{b}\parallel }^p{\parallel f\parallel }_{L^{p,\kappa }(w)}^p{\left(\sum _{j=1}^{\mathrm{\infty }}\frac{j^2}{w{(2^{j+1}\mathcal{E})}^{\frac{1-\kappa }{p}}}\right)}^p\\ \le & C{m}^{\frac{np}{2}}w{(\mathcal{E})}^{\kappa }{\parallel \overrightarrow{b}\parallel }^p{\parallel f\parallel }_{L^{p,\kappa }(w)}^p.\end{array}$$

(3.5)

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

For ${\mathit{II}}_3(x)$, by the inequality $\rho (x_0-y)\le C\rho (x-y)$ 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

$$\begin{array}{rcl}{\int }_{{\mathbb{R}}^n\mathrm{\setminus }2\mathcal{E}}\frac{|b_2(y)-{\lambda }_2|}{\rho {(x_0-y)}^{|\alpha |}}|f(y)|dy& \le & C\sum _{j=1}^{\mathrm{\infty }}\frac{1}{|2^j\mathcal{E}|}{\int }_{2^{j+1}\mathcal{E}}|b_2(y)-{\lambda }_2||f(y)|dy\\ \le & C\sum _{j=1}^{\mathrm{\infty }}\frac{1}{|2^j\mathcal{E}|}{({\int }_{2^{j+1}\mathcal{E}}{|f(y)|}^{p}w(y)dy)}^{\frac{1}{p}}\\ \times {\left({\int }_{2^{j+1}\mathcal{E}}{|b_2(y)-{\lambda }_2|}^{p^{\prime }}w{(y)}^{-\frac{p^{\prime }}{p}}dy\right)}^{\frac{1}{p^{\prime }}}\\ \le & C{\parallel b_2\parallel }_{\ast }{\parallel f\parallel }_{L^{p,\kappa }(w)}\sum _{j=1}^{\mathrm{\infty }}j\frac{w{(2^{j+1}\mathcal{E})}^{\frac{\kappa }{p}}}{|2^j\mathcal{E}|}{\left({\int }_{2^{j+1}\mathcal{E}}w{(y)}^{-\frac{p^{\prime }}{p}}dy\right)}^{\frac{1}{p^{\prime }}}\\ \le & C{\parallel b_2\parallel }_{\ast }{\parallel f\parallel }_{L^{p,\kappa }(w)}\sum _{j=1}^{\mathrm{\infty }}\frac{j}{w{(2^{j+1}\mathcal{E})}^{\frac{1-\kappa }{p}}},\end{array}$$

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

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

Thus, we conclude

$${\int }_{\mathcal{E}}{|{\mathit{II}}_3(x)|}^{p}w(x)dx\le C{m}^{\frac{np}{2}}w{(\mathcal{E})}^{\kappa }{\parallel \overrightarrow{b}\parallel }^p{\parallel f\parallel }_{L^{p,\kappa }(w)}^p.$$

(3.6)

In the same way, we shall get the result of ${\mathit{II}}_4(x)$

$${\int }_{\mathcal{E}}{|{\mathit{II}}_4(x)|}^{p}w(x)dx\le C{m}^{\frac{np}{2}}w{(\mathcal{E})}^{\kappa }{\parallel \overrightarrow{b}\parallel }^p{\parallel f\parallel }_{L^{p,\kappa }(w)}^p.$$

(3.7)

Which together with (3.1)-(3.7), for $1<p<\mathrm{\infty }$, 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

$$\begin{array}{rcl}\lambda w\left(\{x\in \mathcal{E}:|[\overrightarrow{b},T]f(x)|>\lambda \}\right)& \le & C\sum _{s,m}{\parallel b_{sm}\parallel }_{L^{\mathrm{\infty }}}\lambda w\left(\{x\in \mathcal{E}:|[\overrightarrow{b},T_{sm}]f(x)|>\lambda \}\right)\\ \le & C\sum _{s,m}{m}^{-2l}[\lambda w\left(\{x\in \mathcal{E}:|[\overrightarrow{b},T_{sm}]f_1(x)|>\lambda /2\}\right)\\ +\lambda w\left(\{x\in \mathcal{E}:|[\overrightarrow{b},T_{sm}]f_2(x)|>\lambda /2\}\right)]\\ =& C\sum _{s,m}{m}^{-2l}[\mathit{III}+\mathit{IV}]\end{array}$$

(3.8)

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

$$\begin{array}{rl}\mathit{III}& \le C{\int }_{{\mathbb{R}}^n}\mathrm{\Phi }(|f_1|)(x)w(x)dx\\ \le Cw{(\mathcal{E})}^{\kappa }{\parallel f\parallel }_{L^{\mathrm{\Phi },\kappa }(w)}.\end{array}$$

(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

$$w\left(\{x\in \mathcal{E}:|[\overrightarrow{b},T_{sm}]f_2(x)|>1\}\right)\le Cw{(\mathcal{E})}^{\kappa }{\parallel f\parallel }_{L^{\mathrm{\Phi },\kappa }(w)}.$$

In fact, by Lemma F, we get

$$\begin{array}{rl}w\left(\{x\in \mathcal{E}:|[\overrightarrow{b},T_{sm}]f_2(x)|>1\}\right)& \le \underset{t>0}{sup}\frac{1}{\mathrm{\Phi }(\frac{1}{t})}w\left(\{x\in \mathcal{E}:|[\overrightarrow{b},T_{sm}]f_2(x)|>t\}\right)\\ \le C\underset{t>0}{sup}\frac{1}{\mathrm{\Phi }(\frac{1}{t})}w\left(\{x\in \mathcal{E}:M_{\mathrm{\Phi }}{f}_2(x)>t\}\right)\\ =C\underset{t>0}{sup}\frac{1}{\mathrm{\Phi }(\frac{1}{t})}w\left(\{x\in \mathcal{E}:M(\mathrm{\Phi }|f_2|)(x)>t\}\right),\end{array}$$

(3.10)

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

$${\int }_{x:Mf(x)>t}\varphi (t)dx\le \frac{C}{t}{\int }_{{\mathbb{R}}^n}|f(x)|M\varphi (x)dx,$$

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

$$\begin{array}{rl}w\left(\{x\in \mathcal{E}:M(\mathrm{\Phi }|f_2|)(x)>t\}\right)& \le \frac{1}{t}{\int }_{\{x\in {\mathbb{R}}^n:M(\mathrm{\Phi }|f_2|)(x)>t\}}{\chi }_{\mathcal{E}}(x)w(x)dx\\ \le \frac{C}{t}{\int }_{{\mathbb{R}}^n}\mathrm{\Phi }(|f_2|)(x)M(w{\chi }_{\mathcal{E}})(x)dx\\ =\frac{C}{t}({\int }_{3\mathcal{E}}+{\int }_{{\mathbb{R}}^n\mathrm{\setminus }3\mathcal{E}})\mathrm{\Phi }(|f_2|)(x)M(w{\chi }_{\mathcal{E}})(x)dx\\ =\frac{C}{t}({\mathit{IV}}_1+{\mathit{IV}}_2).\end{array}$$

(3.11)

For ${\mathit{IV}}_1$, since $w\in A_1$, it follows that

$$\begin{array}{rl}{\mathit{IV}}_1& \le C{\int }_{3\mathcal{E}}\mathrm{\Phi }(|f|)(x)w(x)dx\\ \le Cw{(3\mathcal{E})}^{\kappa }{\parallel \mathrm{\Phi }(|f|)\parallel }_{L^{1,\kappa }(w)}\\ \le Cw{(\mathcal{E})}^{\kappa }{\parallel f\parallel }_{L^{\mathrm{\Phi },\kappa }}.\end{array}$$

(3.12)

To estimate the term ${\mathit{IV}}_2$, we first consider the form

$$\frac{1}{|\mathcal{F}|}{\int }_{\mathcal{E}\cap \mathcal{F}}w(y)dy$$

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

$$\begin{array}{rl}\frac{1}{|\mathcal{F}|}{\int }_{\mathcal{E}\cap \mathcal{F}}w(y)dy& \le C\frac{1}{\rho {(x-x_0)}^{|\alpha |}}{\int }_{\mathcal{E}}w(y)dy\\ =\frac{C}{\rho {(x-x_0)}^{|\alpha |}}w(\mathcal{E}).\end{array}$$

Therefore, we obtain

$$M(w{\chi }_{\mathcal{E}})(x)\le \frac{C}{\rho {(x-x_0)}^{|\alpha |}}w(\mathcal{E}).$$

Since $w\in A_1$ satisfies the doubling condition and Lemma A, we estimate the term ${\mathit{IV}}_2$ as follows:

$$\begin{array}{rl}{\mathit{IV}}_2& \le C{\int }_{{\mathbb{R}}^n\mathrm{\setminus }3\mathcal{E}}\frac{\mathrm{\Phi }(|f|)(x)}{\rho {(x-x_0)}^{|\alpha |}}w(\mathcal{E})dx\\ \le Cw(\mathcal{E})\sum _{j=1}^{\mathrm{\infty }}\frac{1}{|3^j\mathcal{E}|}{\int }_{3^{j+1}\mathcal{E}}\mathrm{\Phi }(|f|)(x)dx\\ \le Cw(\mathcal{E}){\parallel \mathrm{\Phi }(|f|)\parallel }_{L^{1,\kappa }(w)}\sum _{j=1}^{\mathrm{\infty }}\frac{1}{w{(3^j\mathcal{E})}^{1-\kappa }}\\ \le Cw{(\mathcal{E})}^{\kappa }{\parallel f\parallel }_{L^{\mathrm{\Phi },\kappa }}.\end{array}$$

(3.13)

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

$$w\left(\{x\in \mathcal{E}:|[\overrightarrow{b},T_{sm}]f_2(x)|>1\}\right)\le Cw{(\mathcal{E})}^{\kappa }{\parallel f\parallel }_{L^{\mathrm{\Phi },\kappa }(w)}.$$

Thus, the proof of Theorem 1.2 is completed. □