Some sets of first category in product Calderón-Lozanovskiĭ spaces on hypergroups

Let $K$ be a locally compact hypergroup with a left Haar measure $\mu$ and $\mathrm{\Omega }$ be a Banach ideal of $\mu$-measurable complex-valued functions on $K$. For Young functions $\{{\phi }_i{\}}_{i=1,2,3}$, let ${\mathrm{\Omega }}_{{\phi }_i}(K)$ be the corresponding Calderón–Lozanovskiĭ 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 $$\{(f,g)\in {\mathrm{\Omega }}_{{\phi }_1}(K)\times {\mathrm{\Omega }}_{{\phi }_2}(K):\mathrm{\mid }f\mathrm{\mid }\ast \mathrm{\mid }g\mathrm{\mid }\in {\mathrm{\Omega }}_{{\phi }_3}(K)\}$$ and $$\{(f,g)\in {\mathrm{\Omega }}_{{\phi }_1}(K)\times {\mathrm{\Omega }}_{{\phi }_2}(K):\mathrm{\exists }x\in U,(\mathrm{\mid }f\mathrm{\mid }\ast \mathrm{\mid }g\mathrm{\mid })(x)<\mathrm{\infty }\}$$ 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.