Musielak-Orlicz-bumps and Bloom type estimates for commutators of Calderón-Zygmund and fractional integral operators on generalized Zygmund spaces via sparse operators

Abstract

We study continuity properties for commutators of Calderón-Zygmund and fractional integral operators between generalized Zygmund spaces of L log L type, in the variable exponent setting with different weights. In order to reach this goal we use two different approaches: the first one is related to generalized bump conditions on a pair of weights, allowing us to handle with a wide class of symbol involved with the commutator. The other approaches give Bloom type estimates restricting the class of symbols. The techniques involved in both type of results are related with the theory of sparse domination.