Sobolev Versus Homogeneous Sobolev Extension
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| Format: | Preprint |
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2024
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| _version_ | 1866915024572776448 |
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| author | Koskela, Pekka Mishra, Riddhi Zhu, Zheng |
| author_facet | Koskela, Pekka Mishra, Riddhi Zhu, Zheng |
| contents | In this paper, we study the relationship between Sobolev extension domains and homogeneous Sobolev extension domains. Precisely, we obtain the following results.
1- Let $1\leq q\leq p\leq \infty$. Then a bounded $(L^{1, p}, L^{1, q})$-extension domain is also a $(W^{1, p}, W^{1, q})$-extension domain.
2- Let $1\leq q\leq p<q^\star\leq \infty$ or $n< q \leq p\leq \infty$. Then a bounded domain is a $(W^{1, p}, W^{1, q})$-extension domain if and only if it is an $(L^{1, p}, L^{1, q})$-extension domain.
3- For $1\leq q<n$ and $q^\star<p\leq \infty$, there exists a bounded domain $Ω\subset\mathbb{R}^n$ which is a $(W^{1, p}, W^{1, q})$-extension domain but not an $(L^{1, p}, L^{1, q})$-extension domain for $1 \leq q <p\leq n$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_11470 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Sobolev Versus Homogeneous Sobolev Extension Koskela, Pekka Mishra, Riddhi Zhu, Zheng Functional Analysis 46E35, 30L99 In this paper, we study the relationship between Sobolev extension domains and homogeneous Sobolev extension domains. Precisely, we obtain the following results. 1- Let $1\leq q\leq p\leq \infty$. Then a bounded $(L^{1, p}, L^{1, q})$-extension domain is also a $(W^{1, p}, W^{1, q})$-extension domain. 2- Let $1\leq q\leq p<q^\star\leq \infty$ or $n< q \leq p\leq \infty$. Then a bounded domain is a $(W^{1, p}, W^{1, q})$-extension domain if and only if it is an $(L^{1, p}, L^{1, q})$-extension domain. 3- For $1\leq q<n$ and $q^\star<p\leq \infty$, there exists a bounded domain $Ω\subset\mathbb{R}^n$ which is a $(W^{1, p}, W^{1, q})$-extension domain but not an $(L^{1, p}, L^{1, q})$-extension domain for $1 \leq q <p\leq n$. |
| title | Sobolev Versus Homogeneous Sobolev Extension |
| topic | Functional Analysis 46E35, 30L99 |
| url | https://arxiv.org/abs/2411.11470 |