Weighted bound for commutators

Let $K$ be the Calder'on-Zygmund convolution kernel on $$\mathbb{R}$^d (d\geq2)$\mathrm{.}Definethecommutatorassociatedwith$K$and$a\in L^\infty($\mathbb{R}$^d)$ by $$T_{a}f(x)=p\mathrm{.}v\mathrm{.}\int K(x-y)m_{x,y}a\cdot f(y)dy\mathrm{.}$$ Recently, Grafakos and Honz'{\i}k [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 $\mathrm{\mid }x{\mathrm{\mid }}^{\alpha }(-2<\alpha <0)$ for $d=2$. Moreover, we prove that $T_a$ is bounded on weighted $L^p({\mathbb{R}}^d)(1<p<\mathrm{\infty })$ for all $d\ge 2$.