Kernels of trace operators via fine continuity
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866909677207420928 |
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| author | Hinz, Michael Chandler-Wilde, Simon N. Hewett, David P. |
| author_facet | Hinz, Michael Chandler-Wilde, Simon N. Hewett, David P. |
| contents | We study traces of elements of fractional Sobolev spaces $H_p^α(\mathbb{R}^n)$ on closed subsets $Γ$ of $\mathbb{R}^n$, given as the supports of suitable measures $μ$. We prove that if these measures satisfy localized upper density conditions, then quasi continuous representatives vanish quasi everywhere on $Γ$ if and only if they vanish $μ$-almost everywhere on $Γ$. We use this result to characterize the kernel of the trace operator mapping from $H_p^α(\mathbb{R}^n)$ into the space of $μ$-equivalence classes of functions on $Γ$ as the closure of $C_c^\infty(\mathbb{R}^n\setminus Γ)$ in $H_p^α(\mathbb{R}^n)$. The measures do not have to satisfy a doubling condition. In particular, the set $Γ$ may be a finite union of closed sets having different Hausdorff dimensions. We provide corresponding results for fractional Sobolev spaces $H_p^α(Ω)$ on domains $Ω\subset \mathbb{R}^n$ satisfying the measure density condition. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_04536 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Kernels of trace operators via fine continuity Hinz, Michael Chandler-Wilde, Simon N. Hewett, David P. Functional Analysis Numerical Analysis Analysis of PDEs 28A12, 28A75, 28A80, 31B15, 31C40, 46E35, 46N40 We study traces of elements of fractional Sobolev spaces $H_p^α(\mathbb{R}^n)$ on closed subsets $Γ$ of $\mathbb{R}^n$, given as the supports of suitable measures $μ$. We prove that if these measures satisfy localized upper density conditions, then quasi continuous representatives vanish quasi everywhere on $Γ$ if and only if they vanish $μ$-almost everywhere on $Γ$. We use this result to characterize the kernel of the trace operator mapping from $H_p^α(\mathbb{R}^n)$ into the space of $μ$-equivalence classes of functions on $Γ$ as the closure of $C_c^\infty(\mathbb{R}^n\setminus Γ)$ in $H_p^α(\mathbb{R}^n)$. The measures do not have to satisfy a doubling condition. In particular, the set $Γ$ may be a finite union of closed sets having different Hausdorff dimensions. We provide corresponding results for fractional Sobolev spaces $H_p^α(Ω)$ on domains $Ω\subset \mathbb{R}^n$ satisfying the measure density condition. |
| title | Kernels of trace operators via fine continuity |
| topic | Functional Analysis Numerical Analysis Analysis of PDEs 28A12, 28A75, 28A80, 31B15, 31C40, 46E35, 46N40 |
| url | https://arxiv.org/abs/2507.04536 |