Trace operator on von Koch's snowflake
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arXiv
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| Format: | Preprint |
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2019
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| author | Kazaniecki, Krystian Wojciechowski, Michał |
| author_facet | Kazaniecki, Krystian Wojciechowski, Michał |
| contents | We study properties of the boundary trace operator on the Sobolev space $W^1_1(Ω)$. Using the density result by Koskela and Zhang, we define a surjective operator \mbox{$Tr: W^1_1(Ω_K)\rightarrow X(Ω_K)$}, where $Ω_K$ is von Koch's snowflake and $X(Ω_K)$ is a trace space with the quotient norm. Since $Ω_K$ is a uniform domain whose boundary is Ahlfors-regular with an exponent strictly bigger than one, it was shown by L. Malý that there exists a right inverse to $Tr$, i.e. a linear operator $S: X(Ω_K) \rightarrow W^1_1(Ω_K)$ such that $Tr \circ S= Id_{X(Ω_K)}$. In this paper we provide a different, purely combinatorial proof based on geometrical structure of von Koch's snowflake. Moreover we identify the isomorphism class of the trace space as $\ell_1$. As an additional consequence of our approach we obtain a simple proof of the Peetre's theorem about non-existence of the right inverse for domain $Ω$ with regular boundary, which explains Banach space geometry cause for this phenomenon. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1903_01100 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Trace operator on von Koch's snowflake Kazaniecki, Krystian Wojciechowski, Michał Functional Analysis 46E35 (Primary) 46B25, 46B03, 46B45 (Secondary) We study properties of the boundary trace operator on the Sobolev space $W^1_1(Ω)$. Using the density result by Koskela and Zhang, we define a surjective operator \mbox{$Tr: W^1_1(Ω_K)\rightarrow X(Ω_K)$}, where $Ω_K$ is von Koch's snowflake and $X(Ω_K)$ is a trace space with the quotient norm. Since $Ω_K$ is a uniform domain whose boundary is Ahlfors-regular with an exponent strictly bigger than one, it was shown by L. Malý that there exists a right inverse to $Tr$, i.e. a linear operator $S: X(Ω_K) \rightarrow W^1_1(Ω_K)$ such that $Tr \circ S= Id_{X(Ω_K)}$. In this paper we provide a different, purely combinatorial proof based on geometrical structure of von Koch's snowflake. Moreover we identify the isomorphism class of the trace space as $\ell_1$. As an additional consequence of our approach we obtain a simple proof of the Peetre's theorem about non-existence of the right inverse for domain $Ω$ with regular boundary, which explains Banach space geometry cause for this phenomenon. |
| title | Trace operator on von Koch's snowflake |
| topic | Functional Analysis 46E35 (Primary) 46B25, 46B03, 46B45 (Secondary) |
| url | https://arxiv.org/abs/1903.01100 |