Some sets of first category in product Calderón-Lozanovskiĭ spaces on hypergroups
Jun Liu, Yaqian Lu, Chi Zhang
TL;DR
This work addresses a quantitative and Baire-category perspective on the convolution of Calderón–Lozanovskiĭ and related spaces over hypergroups. By establishing sufficient growth and structural conditions on the Young functions and on the hypergroup itself (including divideontimes and adapted Leptin-type conditions), the authors show that the sets of pairs of functions whose convolution remains in a Calderón–Lozanovskiĭ space, as well as those for which the convolution is well defined at a point in a compact set, are of first category in the corresponding product spaces. The results generalize and unify known conclusions from Agbarbaglu–Maghsoudi and related work to the broader hypergroup setting, and they recover classical cases for locally compact groups including Orlicz and Orlicz–Lorentz spaces. The findings contribute to the understanding of the Lp-conjecture and its quantitative versions in a broad interpolation framework, with potential implications for harmonic analysis on hypergroups and related interpolation theory.
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 $\{\varphi_i\}_{i=1,2,3}$, let $Ω_{\varphi_i}(K)$ be the corresponding Calderón--Lozanovskiĭ 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 $$\left\{(f,g)\inΩ_{\varphi_1}(K)\timesΩ_{\varphi_2}(K):\ |f|\ast |g|\inΩ_{\varphi_3}(K)\right\}$$ and $$\left\{(f,g)\inΩ_{\varphi_1}(K)\timesΩ_{\varphi_2}(K):\ \exists\,x\in U,\ (|f|\ast |g|)(x)<\infty\right\}$$ 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.