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 $μ$ and $Ω$ be a Banach ideal of $μ$-measurable complex-valued functions on $K$. For Young functions $\{φ_i\}_{i=1,2,3}$, let $Ω_{φ_i}(K)$ be the corresponding Calderón--Lozanovski\uı space associated with $φ_i$ 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 $$\left\{(f,g)\inΩ_{φ_1}(K)\timesΩ_{φ_2}(K):\ |f|\ast |g|\inΩ_{φ_3}(K)\right\}$$ and $$\left\{(f,g)\inΩ_{φ_1}(K)\timesΩ_{φ_2}(K):\ \exists\,x\in U,\ (|f|\ast |g|)(x)<\infty\right\}$$ 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.