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

BERRA, FABIO; CARENA, MARILINA; PRADOLINI, GLADIS

MICHIGAN MATHEMATICAL JOURNAL

MICHIGAN MATHEMATICAL JOURNAL

Lugar: Ann Arbor; Año: 2019 vol. 68 p. 527 - 564

0026-2285

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 allow 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 $arphi$ we also consider singular integral operators T whose kernels satisfy a $L^{arphi}$-Hörmander property, and we find sufficient conditions on $arphi$ such that a mixed weak estimateholds 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 $Llog L$ 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.