Calderón–Zygmund Operators and Commutators on Weighted Lorentz Spaces

Abstract

We find necessary conditions (which are also sufficient, for some particular cases) for a pair of weights u and w such that a Calder_on-Zygmund operator T, or its commutator [b; T], with b 2 BMO, is bounded on the weighted Lorentz spaces _p u(w), for 1 < p < 1. This result completes the study already known for the Hardy-Littlewood maximal operator and the Hilbert transform, and hence unifies the weighted theories for the Ap and Bp classes.