Weighted bound for commutators

Abstract

Let $K$ be the Calderón-Zygmund convolution kernel on ${\mathbb{R}}^d(d\ge 2)$. Define the commutator associated with $K$ and $a\in L^{\infty }({\mathbb{R}}^d)$ by \[ T_af(x)=p.v. \int K(x-y)m_{x,y}a\cdot f(y)dy. \] Recently, Grafakos and Honz\'ık [5] proved that $T_a$ is of weak type (1,1) for $d=2$. In this paper, we show that $T_a$ is also weighted weak type (1,1) with the weight $|x{|}^{\alpha }(-2<\alpha <0)$ for $d=2$. Moreover, we prove that $T_a$ is bounded on weighted $L^p({\mathbb{R}}^d)(1<p<\infty )$ for all $d\ge 2$.

Some misprints and the reference [6] of published version are corrected. Published in J.Geom.Anal(2015)

References in corpus (1)

• A weak type bound for a singular integral