Translator Disclaimer
August 2019 Mixed Weak Estimates of Sawyer Type for Commutators of Generalized Singular Integrals and Related Operators
Fabio Berra, Marilina Carena, Gladis Pradolini
Michigan Math. J. 68(3): 527-564 (August 2019). DOI: 10.1307/mmj/1559894545

Abstract

We study mixed weak-type inequalities for the commutator [b,T], where b is a BMO function, and T is a Calderón–Zygmund operator. More precisely, we prove that, for every t>0,

uv({xRn:|[b,T](fv)(x)v(x)|>t})CRnΦ(|f(x)|t)u(x)v(x)dx, where Φ(t)=t(1+log+t), uA1, and vA(u). 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 ϕ we also consider singular integral operators T whose kernels satisfy a Lϕ-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 Tbm.

We also obtain a mixed estimation for a wide class of maximal operators associated to certain Young functions of LlogL 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.

Citation

Download 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

Information

Received: 28 July 2017; Revised: 23 February 2018; Published: August 2019
First available in Project Euclid: 7 June 2019

zbMATH: 07130698
MathSciNet: MR3990170
Digital Object Identifier: 10.1307/mmj/1559894545

Subjects:
Primary: 42B20, 42B25

Rights: Copyright © 2019 The University of Michigan

JOURNAL ARTICLE
38 PAGES

This article is only available to subscribers.
It is not available for individual sale.
+ SAVE TO MY LIBRARY

SHARE
Vol.68 • No. 3 • August 2019
Back to Top