The Michigan Mathematical Journal
- Michigan Math. J.
- Volume 68, Issue 3 (2019), 527-564.
Mixed Weak Estimates of Sawyer Type for Commutators of Generalized Singular Integrals and Related Operators
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.
Michigan Math. J., Volume 68, Issue 3 (2019), 527-564.
Received: 28 July 2017
Revised: 23 February 2018
First available in Project Euclid: 7 June 2019
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Berra, Fabio; Carena, Marilina; Pradolini, Gladis. Mixed Weak Estimates of Sawyer Type for Commutators of Generalized Singular Integrals and Related Operators. Michigan Math. J. 68 (2019), no. 3, 527--564. doi:10.1307/mmj/1559894545. https://projecteuclid.org/euclid.mmj/1559894545