Mixed Weak Estimates of Sawyer Type for Commutators of Generalized Singular Integrals and Related Operators

摘要

We study mixed weak-type inequalities for the commutator [b, T], where b is a BMO function, and T is a Calderon-Zygmund operator. More precisely, we prove that, for every t > 0, uv({x∈R~n:|([b,T](f v)(x))/(v(x))|＞t}) ≤C∫_(R~n)Φ((|f(x)|)/t)u(x)v(x)dx, where Φ(t) =t(1 + log~+ t), u ∈ A_1, and v ∈ A_∞(u)- Our technique involves the classical Calderon-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 L~Φ-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_b~m. We also obtain a mixed estimation for a wide class of maximal operators associated to certain Young functions of L log L 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.