Weighted Hardy Space Estimates for Commutators of Calderón–Zygmund Operators

Abstract

Let δ ∈ (0,1] and T be a δ-Calderón–Zygmund operator. Let p ∈ (0,1] be such that p(1 + δ/n) > 1, and assume that w belongs to the Muckenhoupt weight class $A_{p(1+\delta /n)}(\mathbb {R}^{n})$ with the property ${\int \limits }_{\mathbb {R}^{n}}\frac {w(x)}{(1+|x|)^{np}}dx<\infty$. When $b\in \text {BMO}(\mathbb {R}^{n})$, it is well-know that the commutator [b,T] is not bounded from $H^{p}(\mathbb {R}^{n})$ into $L^{p}(\mathbb {R}^{n})$ if b is not a constant function. In this paper, we find a proper subspace $\mathcal {BMO}_{w,p}(\mathbb {R}^{n})$ of $\text {BMO}(\mathbb {R}^{n})$ such that, if $b\in \mathcal {BMO}_{w,p}(\mathbb {R}^{n})$, then [b,T] is bounded from the weighted Hardy space ${H_{w}^{p}}(\mathbb {R}^{n})$ into the weighted Lebesgue space ${L_{w}^{p}}(\mathbb {R}^{n})$. Conversely, if $b\in \text {BMO}(\mathbb {R}^{n})$ and the commutators $\{[b,R_{j}]\}_{j=1}^{n}$ of the classical Riesz transforms are bounded from ${H^{p}_{w}}(\mathbb {R}^{n})$ into ${L^{p}_{w}}(\mathbb {R}^{n})$, then $b\in \mathcal {BMO}_{w,p}(\mathbb {R}^{n})$.