Weighted bound for commutators

Let $K$ be the Calderón-Zygmund convolution kernel on $\mathbb{R}^d (d\geq2)$. 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 $|x|^α\,(-2<α<0)$ for $d=2$. Moreover, we prove that $T_a$ is bounded on weighted $L^p(\mathbb{R}^d)\,(1<p<\infty)$ for all $d\ge2$.

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