Sharp growth conditions for boundedness of maximal function 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 $\varphi (x,t)$ 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 $\varphi (x,t)/t^p$ 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 $L^{\varphi }({\mathbb{R}}^n)$ to itself if and only if $\varphi$ 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 $p>1$.