Weighted estimates for commutators of anisotropic Calderón-Zygmund operators

Abstract

Let T be an anisotropic Calderón-Zygmund operator and $b\in Li{p}_{\alpha ,w}({\mathbb{R}}^n,A)$ with $0<\alpha <1$ and $Li{p}_{\alpha ,w}({\mathbb{R}}^n,A)$ being an anisotropic weighted Lipschitz space. The goal of the paper is to give five boundedness theorems of the commutator $[b,T]$. Precisely, $[b,T]$ is bounded from $L_w^q\left({\mathbb{R}}^n\right)$ to $L_{w^{1-qr}}^{qr}\left({\mathbb{R}}^n\right)$, where $w\in {\mathbb{A}}_{qr}\left(A\right)$, $1/qr=1/q-\alpha /n$, $1<q<n/\alpha$ and $1<r<\mathrm{\infty }$; $[b,T]$ is bounded from $L_w^q\left({\mathbb{R}}^n\right)$ to $L_{w^{1-r}}^r\left({\mathbb{R}}^n\right)$, when $w\in {\mathbb{A}}_1\left(A\right)$ or $w\in {\mathbb{A}}_r\left(A\right)$, $1/r=1/q-\alpha /n$, $1<q<n/\alpha$ and $1<q<r<\mathrm{\infty }$; $[b,T]$ is bounded from anisotropic weighted Hardy space $H_w^p({\mathbb{R}}^n,A)$ to $L_{w^{1-r}}^r\left({\mathbb{R}}^n\right)$, if $w\in {\mathbb{A}}_1\left(A\right)$ or $w\in {\mathbb{A}}_r\left(A\right)$, $1/r=1/p-\alpha /n$ and $n/(n+\alpha )<p\le 1<r<\mathrm{\infty }$; $[b,T]$ is bounded from $H_w^{n/(n+\alpha )}({\mathbb{R}}^n,A)$ to weak Lebesgue space $L^{1,\mathrm{\infty }}\left({\mathbb{R}}^n\right)$ with $w\in {\mathbb{A}}_1\left(A\right)$, $p=n/(n+\alpha )$, which are extensions of isotropic settings and new even for the isotropic weighted and anisotropic unweighted settings.

COMMUNICATED BY:

• J. Shi

Keywords:

• Expansive dilation
• Muckenhoupt weights
• weighted Hardy spaces
• Calderón-Zygmund operators
• commutators

AMS Subject Classifications:

• 42B35
• 46E30
• 42B25
• 42B30

Acknowledgments

The authors of the paper would like to express grateful thanks to the anonymous referees for their professional comments and suggestions, which are greatly helpful to improve the quality of the paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This project is supported by the National Natural Science Foundation of China [grant number 11671414]. This project is also supported by the “Basic Innovation” Program of Graduate Students of Guangzhou University [grant number 2018GDJC-D01].