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

Petteri Harjulehto¹ ; Arttu Karppinen¹

¹ University of Turku, Department of Mathematics and Statistics

Annales Fennici Mathematici, Tome 47 (2022) no. 1, pp. 489-505.

Résumé

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 $p>1$.  

Détail

DOI : 10.54330/afm.115122

Keywords: maximal functions, generalized Orlicz, Orlicz-Musielak, almost increasing