Weighted Endpoint Estimates for Commutators of Calderón-Zygmund Operators

Abstract

Let $\delta \in (0,1]$ and $T$ be a $\delta$-Calderón-Zygmund operator. Let $w$ be in the Muckenhoupt class $A_{1+\delta /n}({\mathbb{R}}^n)$ satisfying ${\int }_{{\mathbb{R}}^n}\frac{w(x)}{1+|x{|}^n}dx<\infty$. When $b\in \mathrm{BMO}({\mathbb{R}}^n)$, it is well known that the commutator $[b,T]$ is not bounded from $H^1({\mathbb{R}}^n)$ to $L^1({\mathbb{R}}^n)$ if $b$ is not a constant function. In this article, the authors find out a proper subspace ${\mathrm{BMO}}_w({\mathbb{R}}^n)$ of $\mathrm{BMO}({\mathbb{R}}^n)$ such that, if $b\in {\mathrm{BMO}}_w({\mathbb{R}}^n)$, then $[b,T]$ is bounded from the weighted Hardy space $H_w^1({\mathbb{R}}^n)$ to the weighted Lebesgue space $L_w^1({\mathbb{R}}^n)$. Conversely, if $b\in \mathrm{BMO}({\mathbb{R}}^n)$ and the commutators of the classical Riesz transforms $\{[b,R_j]{\}}_{j=1}^n$ are bounded from $H_w^1({\mathbb{R}}^n)$ into $L_w^1({\mathbb{R}}^n)$, then $b\in {\mathrm{BMO}}_w({\mathbb{R}}^n)$.

11 pages; Submitted

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• Endpoint estimates for the commutators of multilinear Calderón-Zygmund operators with Dini type kernels