The Michigan Mathematical Journal

Mixed Weak Estimates of Sawyer Type for Commutators of Generalized Singular Integrals and Related Operators

Fabio Berra, Marilina Carena, and Gladis Pradolini

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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.

Article information

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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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Primary: 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.) 42B25: Maximal functions, Littlewood-Paley theory


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.

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