Weighted Bound for Commutators

Abstract

Let \(K\) be the Calderon–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 $$\begin{aligned} T_af(x)=p.v. \int K(x-y)m_{x,y}a\cdot f(y)dy. \end{aligned}$$ Recently, Grafakos and Honzik [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 \(d\ge 2\).