From $A_1$ to $A_\infty$: New mixed inequalities for certain maximal operators

TL;DR

This paper establishes new mixed inequalities for maximal operators associated with Young functions, extending previous conjectures and introducing an operator ratio that generalizes classical inequalities, with applications to fractional maximal operators.

Contribution

It introduces a novel mixed inequality involving a ratio of maximal operators for Young functions, extending known results to broader classes of weights and operators.

Findings

Proved mixed inequalities for maximal operators with Young functions.

Extended inequalities to cases where $v^r otin A_1$, but $v^r \\in A_\\infty$.

Applied results to generalized fractional maximal operators with $L\\log L$ type Young functions.

Abstract

In this article we prove mixed inequalities for maximal operators associated to Young functions, which are an improvement of a conjecture established in \cite{Berra}. Concretely, given $r\geq 1$, $u\in A_1$, $v^r\in A_\infty$ and a Young function $\Phi$ with certain properties, we have that inequality \[uv^r\left(\left\{x\in \mathbb{R}^n: \frac{M_\Phi(fv)(x)}{M_\Phi v(x)}>t\right\}\right)\leq C\int_{\mathbb{R}^n}\Phi\left(\frac{|f(x)|}{t}\right)u(x)v^r(x)\,dx\] holds for every positive $t$. The involved operator $\frac{M_\Phi(fv)(x)}{M_\Phi v(x)}$ seems to be an adequate extension when $v^r\in A_\infty$, since when we assume $v^r\in A_1$ we can replace $M_\Phi v$ by $v$, yielding a mixed inequality for $M_\Phi$ proved in \cite{Berra-Carena-Pradolini(MN)}. As an application, we furthermore exhibe and prove mixed inequalities for the generalized fractional maximal operator $M_{\gamma,\Phi}$, where $0<\gamma<n$ and $\Phi$ is a Young function of $L\log L$ type.