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. More precisely, we prove that, for every t>0, uv({x∈Rn:|[b,T](fv)(x)v(x)|>t})≤C∫RnΦ(|f(x)|t)u(x)v(x)dx, where Φ(t)=t(1+log+t), u∈A1, and v∈A∞(u). 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 Tbm. 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.