Sharp growth conditions for boundedness of maximal function in generalized Orlicz spaces

TL;DR

This paper establishes precise growth conditions on generalized Orlicz functions that ensure the boundedness of the Hardy-Littlewood maximal operator within these spaces, extending classical results to more general settings.

Contribution

It provides necessary and sufficient conditions on the generalized Orlicz functions for the maximal operator to be bounded in these spaces, including sharp growth criteria.

Findings

Boundedness characterized by almost increasing ratio $rac{(x,t)}{t^p}$

Equivalence to functions satisfying standard continuity properties

Necessary and sufficient conditions for boundedness in generalized Orlicz spaces

Abstract

We study sharp growth conditions for the boundedness of the Hardy-Littlewood maximal function in the generalized Orlicz spaces. We assume that the generalized Orlicz function $\phi(x, t)$ satisfies the standard continuity properties (A0), (A1) and (A2). We show that if the Hardy-Littlewood maximal function is bounded from the generalized Orlicz space to itself then $\phi(x,t)/ t^p$ is almost increasing for large $t$ for some $p>1$. Moreover we show that the Hardy-Littlewood maximal function is bounded from the generalized Orlicz space $L^\phi(\mathbb{R}^n)$ to itself if and only if $\phi$ is weakly equivalent to a generalized Orlicz function $\psi$ satisfying (A0), (A1) and (A2) for which $\psi(x,t)/ t^p$ is almost increasing for all $t>0$ and some $p>1$.