We study mixed weak-type inequalities for the commutator , where is a BMO function, and is a Calderón–Zygmund operator. More precisely, we prove that, for every ,
where , , and . 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 we also consider singular integral operators whose kernels satisfy a -Hörmander property, and we find sufficient conditions on such that a mixed weak estimate holds for and also for its higher order commutators .
We also obtain a mixed estimation for a wide class of maximal operators associated to certain Young functions of type which are in intimate relation with the commutators. This last estimate involves an arbitrary weight and a radial function which is not even locally integrable.
"Mixed Weak Estimates of Sawyer Type for Commutators of Generalized Singular Integrals and Related Operators." Michigan Math. J. 68 (3) 527 - 564, August 2019. https://doi.org/10.1307/mmj/1559894545