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 $b\in \textrm {BMO}(\mathbb R^n)$, 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 ${\mathcal {BMO}_w({\mathbb R}^n)}$ of $\mathrm {BMO}(\mathbb R^n)$ such that, if $b\in {\mathcal {BMO}_w({\mathbb R}^n)}$, 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 $b\in \textrm {BMO}({\mathbb R}^n)$ and the commutators of the classical Riesz transforms $\{[b,R_j]\}_{j=1}^n$ are bounded from $H^1_w({\mathbb R}^n)$ to $L^1_w({\mathbb R}^n)$, then $b\in {\mathcal {BMO}_w({\mathbb R}^n)}$.