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

Abstract

We study mixed weak type inequalities for the commutator [b,T], where b is a BMO function and T is a Calderón-Zygmund operator. Our technique involves the classical Calderón–Zygmund decomposition, which allows us to give a direct proof without taking into account the associated maximal operator. We use this result to prove an analogous inequality for higher-order commutators. For a given Young function φ we also consider singular integral operators T whose kernels satisfy a Lφ-Hörmander property, and we find sufficient conditions on φ such that a mixed weak estimate holds for T and also for its higher order commutators T m b . 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.