# Improvements on Sawyer type estimates for generalized maximal functions

## TL;DR

This paper extends Sawyer's estimates for generalized maximal functions involving Young functions, establishing mixed inequalities under certain weight conditions, which aids in understanding their boundedness in harmonic analysis.

## Contribution

It generalizes Sawyer's maximal operator estimates to a broader class of Young functions with mixed weight inequalities, enhancing the theoretical framework.

## Key findings

- Proved mixed inequalities for $M_$ with Young functions.
- Extended Sawyer's estimates to more general weights and functions.
- Provided tools for analyzing boundedness of generalized maximal operators.

## Abstract

In this paper we prove mixed inequalities for the maximal operator $M_\Phi$, for general Young functions $\Phi$ with certain additional properties, improving and generalizing some previous estimates for the Hardy-Littlewood maximal operator proved by E. Sawyer. We show that given $r\geq 1$, if $u,v^r$ are weights belonging to the $A_1$-Muckenhoupt class and $\Phi$ is a Young function as above, then the inequality \[uv^r\left(\left\{x\in \mathbb{R}^n: \frac{M_\Phi(fv)(x)}{v(x)}>t\right\}\right)\leq C\int_{\mathbb{R}^n}\Phi\left(\frac{|f(x)|}{t}\right)u(x)v^r(x)\,dx\] holds for every positive $t$. A motivation for studying these type of estimates is to find an alternative way to prove the boundedness properties of $M_\Phi$. Moreover, it is well-known that for the particular case $\Phi(t)=t(1+\log^+t)^m$ with $m\in\mathbb{N}$ these maximal functions control, in some sense, certain operatos in Harmonic Analysis.