Mixed weak estimates of Sawyer type for generalized maximal operators

TL;DR

This paper extends mixed weak estimates of Sawyer type to a broader class of maximal operators associated with Young functions, generalizing previous results and including operators related to Calderón-Zygmund commutators.

Contribution

It introduces new mixed weak estimates for maximal operators linked to Young functions, broadening the scope of prior Sawyer-type inequalities.

Findings

Extended mixed estimates to a wider class of maximal operators.

Recovered classical results as special cases when parameters are set.

Included estimates for operators related to Calderón-Zygmund commutators.

Abstract

We study mixed weak estimates of Sawyer type for maximal operators associated to the family of Young functions $\Phi(t)=t^r(1+\log^+t)^{\delta}$, where $r\geq 1$ and $\delta\geq 0$. More precisely, if $u$ and $v^r$ are $A_1$ weights, and $w$ is defined as $w=1/\Phi(v^{-1})$ then the following estimate \[uw\left(\left\{x\in \mathbb{R}^n: \frac{M_\Phi(fv)(x)}{v(x)} > t\right\}\right) \leq C\int_{\mathbb{R}^n} \Phi\left(\frac{|f(x)|v(x)}{t}\right)u(x) \,dx\] holds for every positive $t$. This extends mixed estimates to a wider class of maximal operators, since when we put $r=1$ and $\delta=0$ we recover a previous result for the Hardy-Littlewood maximal operator. This inequality generalizes some previous results proved by Cruz Uribe, Martell and P\'erez in (Int. Math. Res. Not. (30): 1849-1871, 2005). Moreover, it includes estimates for some maximal operators related with commutators of Calder\'on-Zygmund operators.