Mixed weak estimates of Sawyer type for commutators of generalized singular integrals and related operators

BERRA, FABIO; CARENA, MARILINA; PRADOLINI, GLADIS

MICHIGAN MATHEMATICAL JOURNAL

MICHIGAN MATHEMATICAL JOURNAL

Lugar: Ann Arbor; Año: 2019

0026-2285

We study mixed weak type inequalities for the commutator [b, T], where b is aBMO function and T is a Calderón-Zygmund operator. Our technique involves the classical Calderón-Zygmund decomposition, which allow us to give a direct proof without taking into account theassociated maximal operator. We use this result to prove an analogous inequality for higher order commutators.For a given Young function arphi we also consider singular integral operators T whose kernels satisfy a L^{arphi}-Hörmander property, and we find sufficient conditions on arphi such that a mixed weak estimate holds for T and also for its higher order commutators T_b^m.We also obtain a mixed estimation for a wide class of maximal operators associated to certain Young functions of LlogL type which are in intimate relation with the commutators.This last estimate involves an arbitrary weight u and a radial function v which is not even locally integrable.