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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


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.


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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.


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

Primary: 42B20, 42B25

Rights: Copyright © 2019 The University of Michigan


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Vol.68 • No. 3 • August 2019
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