A WEIGHTED COMPACTNESS CRITERION FOR COMMUTATORS ASSOCIATED WITH GENERALIZED CALDERÓN–ZYGMUND OPERATORS

Abstract

Let $T$ be a bounded operator on $L^p({ℝ}^n)$. Under the assumption that the kernel of $T$ satisfies some Hörmander-type estimates, we obtain a boundedness criterion for the multilinear commutators $T_{\overrightarrow{b}}$ on the weighted Lebesgue spaces $L^p(\omega )$ with $\overrightarrow{b}\in \text{BMO}({ℝ}^n)$ and $\omega$ belonging to the Muckenhoupt weight class $A_{p∕m^{\prime }}$. Further, for $\overrightarrow{b}\in \text{CMO}({ℝ}^n)$, the vanishing mean oscillation space, a criterion of $L^p$-weighted compactness of $T_{\overrightarrow{b}}$ is established. As applications, the weighted $L^p$-boundedness and $L^p$-compactness criteria can be applied to the $𝜃$-type Calderón–Zygmund operator and its commutators.