Some sets of first category in product Calderón-Lozanovski\uı spaces on hypergroups

Abstract

Let $K$ be a locally compact hypergroup with a left Haar measure $\mu$ and $\Omega$ be a Banach ideal of $\mu$-measurable complex-valued functions on $K$. For Young functions $\{{\phi }_i{\}}_{i=1,2,3}$, let ${\Omega }_{{\phi }_i}(K)$ be the corresponding Calderón--Lozanovski\uı space associated with ${\phi }_i$ on $K$. Motivated by the remarkable work of Akbarbaglu et al. in [Adv. Math. 312 (2017), 737-763], in this article, the authors give several sufficient conditions for the sets $$\left\{(f,g)\in {\Omega }_{{\phi }_1}(K)\times {\Omega }_{{\phi }_2}(K):|f|\ast |g|\in {\Omega }_{{\phi }_3}(K)\right\}$$ and $$\left\{(f,g)\in {\Omega }_{{\phi }_1}(K)\times {\Omega }_{{\phi }_2}(K):\exists x\in U,(|f|\ast |g|)(x)<\infty \right\}$$ to be of first category in the sense of Baire, where $U\subset K$ denotes a compact set. All these results are new even for Orlicz(-Lorentz) spaces on hypergroups.

25 pages; Summitted