Some sets of first category in product Calderón-Lozanovski\uı spaces on hypergroups
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866910841119440896 |
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| author | Liu, Jun Lu, Yaqian Zhang, Chi |
| author_facet | Liu, Jun Lu, Yaqian Zhang, Chi |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_16516 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Some sets of first category in product Calderón-Lozanovski\uı spaces on hypergroups Liu, Jun Lu, Yaqian Zhang, Chi Classical Analysis and ODEs Functional Analysis 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. |
| title | Some sets of first category in product Calderón-Lozanovski\uı spaces on hypergroups |
| topic | Classical Analysis and ODEs Functional Analysis |
| url | https://arxiv.org/abs/2502.16516 |