Some sets of first category in product Calder\'{o}n-Lozanovski\u{\i} spaces on hypergroups

TL;DR

This paper investigates conditions under which certain sets of function pairs in product Calderón-Lozanovski spaces on hypergroups are of first category, extending known results to new classes of spaces.

Contribution

It provides new sufficient conditions for sets of function pairs to be of first category in product Calderón-Lozanovski spaces on hypergroups, including Orlicz and Lorentz spaces.

Findings

Identifies conditions for sets to be of first category in these spaces

Extends results to Orlicz and Lorentz spaces on hypergroups

Provides new insights into the structure of function 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 $\{\varphi_i\}_{i=1,2,3}$, let $\Omega_{\varphi_i}(K)$ be the corresponding Calder\'{o}n--Lozanovski\u{\i} space associated with $\varphi_i$ 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 $$\left\{(f,g)\in\Omega_{\varphi_1}(K)\times\Omega_{\varphi_2}(K):\ |f|\ast |g|\in\Omega_{\varphi_3}(K)\right\}$$ and $$\left\{(f,g)\in\Omega_{\varphi_1}(K)\times\Omega_{\varphi_2}(K):\ \exists\,x\in U,\ (|f|\ast |g|)(x)<\infty\right\}$$ 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.