## The Michigan Mathematical Journal

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

#### 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 $t\gt 0$,

$$uv(\{x\in\mathbb{R}^{n}:\vert \frac{[b,T](fv)(x)}{v(x)}\vert \gt t\})\leq C\int_{\mathbb{R}^{n}}\Phi (\frac{|f(x)|}{t})u(x)v(x)\,dx,$$ where $\Phi(t)=t(1+\log^{+}{t})$, $u\in A_{1}$, and $v\in A_{\infty}(u)$. 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 $\phi$ we also consider singular integral operators $T$ whose kernels satisfy a $L^{\phi }$-Hörmander property, and we find sufficient conditions on $\phi$ such that a mixed weak estimate holds 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 $L\log L$ 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.

#### Article information

Source
Michigan Math. J., Volume 68, Issue 3 (2019), 527-564.

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

Permanent link to this document
https://projecteuclid.org/euclid.mmj/1559894545

Digital Object Identifier
doi:10.1307/mmj/1559894545

Mathematical Reviews number (MathSciNet)
MR3990170

#### Citation

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

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