A note on Trudinger-Moser Functions and Reproducing Kernel Hilbert Spaces
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866915619243294720 |
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| author | Costa, David G. Tehrani, Hossein |
| author_facet | Costa, David G. Tehrani, Hossein |
| contents | After a brief review of the definition of the Trudinger-Moser functions in dimension $N=2$ and some basic notions in the theory of ``Reproducing Kernel Hilbert Spaces (RKHS)'', we will show that there is a close connection between those two topics. More precisely, among other things, we start by considering a properly chosen multiple of the classical Trudinger-Moser family of functions in dimension $N=2$, which we denote by
$ γ_t (r) := \frac{1}{2π}\min\,\{ log \frac{1}{r}, log \frac{1}{t} \}\,, $ where $0 < t , r < 1$, and using the theory of RKHS we will show that $γ_t$ can be seen as a ``bounded'' (linear) evaluation functional $u \longrightarrow u(t)$ for functions $u$ in a suitable Hilbert Space ${\cal H}$. A slightly different definition for a ''Trudinger-Moser'' type function will also be considered for $N\geq 3$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_11904 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A note on Trudinger-Moser Functions and Reproducing Kernel Hilbert Spaces Costa, David G. Tehrani, Hossein Functional Analysis After a brief review of the definition of the Trudinger-Moser functions in dimension $N=2$ and some basic notions in the theory of ``Reproducing Kernel Hilbert Spaces (RKHS)'', we will show that there is a close connection between those two topics. More precisely, among other things, we start by considering a properly chosen multiple of the classical Trudinger-Moser family of functions in dimension $N=2$, which we denote by $ γ_t (r) := \frac{1}{2π}\min\,\{ log \frac{1}{r}, log \frac{1}{t} \}\,, $ where $0 < t , r < 1$, and using the theory of RKHS we will show that $γ_t$ can be seen as a ``bounded'' (linear) evaluation functional $u \longrightarrow u(t)$ for functions $u$ in a suitable Hilbert Space ${\cal H}$. A slightly different definition for a ''Trudinger-Moser'' type function will also be considered for $N\geq 3$. |
| title | A note on Trudinger-Moser Functions and Reproducing Kernel Hilbert Spaces |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2511.11904 |