Weighted Boundedness of Commutators of Generalized Calderόn-Zygmund Operators

Abstract

$[b,T]$ denotes the commutator of generalized Calder\u00f3n-Zygmund operators $T$ with Lipschitz function $b$, where $b \u2208 \rm{Lip}_\u03b2(R^n)$, $(0<\u03b2\u22641)$ and $T$ is a $\u03b8(t)$\u2212type Calder\u00f3n-Zygmund operator. The commutator $[b,T]$ generated by $b$ and $T$ is defined by\u00a0

$$[b,T] f(x)=b(x)Tf(x)\u2212T(bf)(x)=\u222bk(x,y)(b(x)\u2212b(y))f(y)dy.$$

In this paper, the authors discuss the boundedness of the commutator $[b,T]$ on weighted Hardy spaces and weighted Herz type Hardy spaces and prove that $[b,T]$ is bounded from $H^p(\u03c9^p)$ to $L^q(\u03c9^q)$, and from $H\dot{K}^{\u03b1,p}_{q_1}(\u03c9_1,\u03c9^{q_1}_2)$ to $\dot{K}^{\u03b1,p}_{q_2}(\u03c9_1,\u03c9^{q_2}_2)$. The results extend and generalize the well-known ones in [7].