On some Fréchet spaces associated to the functions satisfying Mulholland inequality
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912898300772352 |
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| author | Singh, Lav Kumar Peperko, Aljosa |
| author_facet | Singh, Lav Kumar Peperko, Aljosa |
| contents | In this article we explore a new growth condition on Young functions, which we call Mulholland condition, pertaining to the mathematician H.P Mulholland, who studied these functions for the first time, albeit in a different context. We construct a non-trivial Young function $Ω$ which satisfies Mulholland condition and $Δ_2$-condition. We then associate exotic $F$-norms to the vector space $X_1\oplus X_2$, where $X_1$ and $X_2$ are Banach spaces, using the function $Ω$. This $F$-spaces contains the Banach space $X_1$ and $X_2$ as a maximal Banach subspace. Further, the Banach envelope $(X_1\oplus X_2,||.||_{Ω_o})$ of this $F$-space corresponds to the Young function $Ω_o$ who characteristic function is an asymptotic line to the characteristic function of the Young function $Ω$. Thus these $F$-spaces serves as "interpolation space" for Banach spaces $X_1$ and $(X_1\oplus X_2, ||.||_{Ω_o})$ in some sense. These $F$-space are well behaved in regards to Hahn-Banach extension property, which is lacking in classical $F$-spaces like $L^p$ and $H^p$ for $0<p<1$. Towards the end, some direct sums for Orlicz spaces are discussed. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_01661 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On some Fréchet spaces associated to the functions satisfying Mulholland inequality Singh, Lav Kumar Peperko, Aljosa Functional Analysis Metric Geometry 46A16, 46E30, 46B70 In this article we explore a new growth condition on Young functions, which we call Mulholland condition, pertaining to the mathematician H.P Mulholland, who studied these functions for the first time, albeit in a different context. We construct a non-trivial Young function $Ω$ which satisfies Mulholland condition and $Δ_2$-condition. We then associate exotic $F$-norms to the vector space $X_1\oplus X_2$, where $X_1$ and $X_2$ are Banach spaces, using the function $Ω$. This $F$-spaces contains the Banach space $X_1$ and $X_2$ as a maximal Banach subspace. Further, the Banach envelope $(X_1\oplus X_2,||.||_{Ω_o})$ of this $F$-space corresponds to the Young function $Ω_o$ who characteristic function is an asymptotic line to the characteristic function of the Young function $Ω$. Thus these $F$-spaces serves as "interpolation space" for Banach spaces $X_1$ and $(X_1\oplus X_2, ||.||_{Ω_o})$ in some sense. These $F$-space are well behaved in regards to Hahn-Banach extension property, which is lacking in classical $F$-spaces like $L^p$ and $H^p$ for $0<p<1$. Towards the end, some direct sums for Orlicz spaces are discussed. |
| title | On some Fréchet spaces associated to the functions satisfying Mulholland inequality |
| topic | Functional Analysis Metric Geometry 46A16, 46E30, 46B70 |
| url | https://arxiv.org/abs/2507.01661 |