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

Abstract

Let T be a bounded operator on Lp(ℝn). Under the assumption that the kernel of T satisfies some Hörmander-type estimates, we obtain a boundedness criterion for the multilinear commutators Tb→ on the weighted Lebesgue spaces Lp(ω) with b→∈BMO(ℝn) and ω belonging to the Muckenhoupt weight class Ap∕m′. Further, for b→∈CMO(ℝn), the vanishing mean oscillation space, a criterion of Lp-weighted compactness of Tb→ is established. As applications, the weighted Lp-boundedness and Lp-compactness criteria can be applied to the 𝜃-type Calderón–Zygmund operator and its commutators.