Abstract
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.
Citation
Fabio Berra. Marilina Carena. Gladis Pradolini. "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