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.

令T为各向异性 Calderón-Zygmund 算子，并且$b\in \mathrm{大号}\mathrm{一世}{p}_{\alpha ,w}({\mathbb{R}}^n,\mathrm{一种})$和$0<\alpha <1$和$\mathrm{大号}\mathrm{一世}{p}_{\alpha ,w}({\mathbb{R}}^n,\mathrm{一种})$是一个各向异性加权 Lipschitz 空间。本文的目标是给出交换子的五个有界定理$[b,吨]$. 恰恰，$[b,吨]$是有界的${\mathrm{大号}}_w^q\left({\mathbb{R}}^n\right)$到${\mathrm{大号}}_{w^{1-qr}}^{qr}\left({\mathbb{R}}^n\right)$， 在哪里$w\in {\mathbb{一种}}_{qr}\left(\mathrm{一种}\right)$,$1/qr=1/q-\alpha /n$,$1<q<n/\alpha$和$1<r<\mathrm{\infty }$;$[b,吨]$是有界的${\mathrm{大号}}_w^q\left({\mathbb{R}}^n\right)$到${\mathrm{大号}}_{w^{1-r}}^r\left({\mathbb{R}}^n\right)$， 什么时候$w\in {\mathbb{一种}}_1\left(\mathrm{一种}\right)$或者$w\in {\mathbb{一种}}_r\left(\mathrm{一种}\right)$,$1/r=1/q-\alpha /n$,$1<q<n/\alpha$和$1<q<r<\mathrm{\infty }$;$[b,吨]$以各向异性加权哈代空间为界$H_w^p({\mathbb{R}}^n,\mathrm{一种})$到${\mathrm{大号}}_{w^{1-r}}^r\left({\mathbb{R}}^n\right)$， 如果$w\in {\mathbb{一种}}_1\left(\mathrm{一种}\right)$或者$w\in {\mathbb{一种}}_r\left(\mathrm{一种}\right)$,$1/r=1/p-\alpha /n$和$n/(n+\alpha )<p\le 1<r<\mathrm{\infty }$;$[b,吨]$是有界的$H_w^{n/(n+\alpha )}({\mathbb{R}}^n,\mathrm{一种})$到弱勒贝格空间${\mathrm{大号}}^{1,\mathrm{\infty }}\left({\mathbb{R}}^n\right)$和$w\in {\mathbb{一种}}_1\left(\mathrm{一种}\right)$,$p=n/(n+\alpha )$，它们是各向同性设置的扩展，甚至对于各向同性加权和各向异性未加权设置也是新的。