The existence of a bounded linear extension operator for $L^{s,p}(\mathbb{R}^n)$ when $\frac{n}{p}<\{s\}$
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866913539992584192 |
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| author | Li, Han |
| author_facet | Li, Han |
| contents | Let $L^{s,p}(\mathbb{R}^n)$ denote the homogeneous Sobolev-Slobodeckij space. In this paper, we demonstrate the existence of a bounded linear extension operator from the jet space $J^{\lfloor s \rfloor}_E L^{s,p}(\mathbb{R}^n)$ to $L^{s,p}(\mathbb{R}^n)$ for any $E \subseteq \mathbb{R}^n$, $p \in [1, \infty)$, and $s \in (0, \infty)$ satisfying $\frac{n}{p} < \{s\}$, where $\{s\}$ represents the fractional part of $s$. Our approach builds upon the classical Whitney extension operator and uses the method of exponentially decreasing paths. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_07367 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The existence of a bounded linear extension operator for $L^{s,p}(\mathbb{R}^n)$ when $\frac{n}{p}<\{s\}$ Li, Han Classical Analysis and ODEs 46E35 Let $L^{s,p}(\mathbb{R}^n)$ denote the homogeneous Sobolev-Slobodeckij space. In this paper, we demonstrate the existence of a bounded linear extension operator from the jet space $J^{\lfloor s \rfloor}_E L^{s,p}(\mathbb{R}^n)$ to $L^{s,p}(\mathbb{R}^n)$ for any $E \subseteq \mathbb{R}^n$, $p \in [1, \infty)$, and $s \in (0, \infty)$ satisfying $\frac{n}{p} < \{s\}$, where $\{s\}$ represents the fractional part of $s$. Our approach builds upon the classical Whitney extension operator and uses the method of exponentially decreasing paths. |
| title | The existence of a bounded linear extension operator for $L^{s,p}(\mathbb{R}^n)$ when $\frac{n}{p}<\{s\}$ |
| topic | Classical Analysis and ODEs 46E35 |
| url | https://arxiv.org/abs/2410.07367 |