On certain porous sets in the Orlicz space of a locally compact group

Ibrahim Akbarbaglu ¹   ; Saeid Maghsoudi ¹

¹ Department of Mathematics University of Zanjan Zanjan 45195-313, Iran

Colloquium Mathematicum, Tome 129 (2012) no. 1, pp. 99-111

Résumé

Let $G$ be a locally compact group with a fixed left Haar measure. Given Young functions $\varphi $ and $\psi $, we consider the Orlicz spaces $L^\varphi (G)$ and $L^\psi (G)$ on a non-unimodular group $G$, and, among other things, we prove that under mild conditions on $\varphi $ and $\psi $, the set $\{(f,g)\in L^\varphi (G)\times L^\psi (G): f\ast g$ is well defined on $G\} $ is $\sigma $-$c$-lower porous in $L^\varphi (G)\times L^\psi (G)$. This answers a question raised by Głąb and Strobin in 2010 in a more general setting of Orlicz spaces. We also prove a similar result for non-compact locally compact groups.

Détail

DOI : 10.4064/cm129-1-7

Keywords: locally compact group fixed haar measure given young functions varphi psi consider orlicz spaces varphi psi non unimodular group among other things prove under mild conditions varphi psi set varphi times psi ast defined sigma c lower porous varphi times psi answers question raised strobin general setting orlicz spaces prove similar result non compact locally compact groups