Improvements on Sawyer type estimates for generalized maximal functions

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 $r\ge 1$, if $u,v^r$ are weights belonging to the $A_1$-Muckenhoupt class and $\Phi$ is a Young function as above, then the inequality \[uv^r\left(\left\{x\in \mathbb{R}^n: \frac{M_Φ(fv)(x)}{v(x)}>t\right\}\right)\leq C\int_{\mathbb{R}^n}Φ\left(\frac{|f(x)|}{t}\right)u(x)v^r(x)\,dx\] 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.

18 pages. arXiv admin note: text overlap with arXiv:1808.04333