New Estimates of Rychkov's Universal Extension Operator for Lipschitz Domains and Some Applications
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2021
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| _version_ | 1866913305867583488 |
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| author | Shi, Ziming Yao, Liding |
| author_facet | Shi, Ziming Yao, Liding |
| contents | Given a bounded Lipschitz domain $Ω\subset\mathbb R^n$, Rychkov showed that there is a linear extension operator $\mathcal E$ for $Ω$ which is bounded in Besov and Triebel-Lizorkin spaces. In this paper we introduce some new estimates for the extension operator $\mathcal E$ and give some applications. We prove the equivalent norms $\|f\|_{\mathscr A_{pq}^s(Ω)}\approx\sum_{|α|\le m}\|\partial^αf\|_{\mathscr A_{pq}^{s-m}(Ω)}$ for general Besov and Triebel-Lizorkin spaces. We also derive some quantitative smoothing estimates of the extended function and all its derivatives on $\overlineΩ^c$ up to the boundary. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2110_14477 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | New Estimates of Rychkov's Universal Extension Operator for Lipschitz Domains and Some Applications Shi, Ziming Yao, Liding Classical Analysis and ODEs Functional Analysis 46E35 (primary), 42B25, 46F10 (secondary) Given a bounded Lipschitz domain $Ω\subset\mathbb R^n$, Rychkov showed that there is a linear extension operator $\mathcal E$ for $Ω$ which is bounded in Besov and Triebel-Lizorkin spaces. In this paper we introduce some new estimates for the extension operator $\mathcal E$ and give some applications. We prove the equivalent norms $\|f\|_{\mathscr A_{pq}^s(Ω)}\approx\sum_{|α|\le m}\|\partial^αf\|_{\mathscr A_{pq}^{s-m}(Ω)}$ for general Besov and Triebel-Lizorkin spaces. We also derive some quantitative smoothing estimates of the extended function and all its derivatives on $\overlineΩ^c$ up to the boundary. |
| title | New Estimates of Rychkov's Universal Extension Operator for Lipschitz Domains and Some Applications |
| topic | Classical Analysis and ODEs Functional Analysis 46E35 (primary), 42B25, 46F10 (secondary) |
| url | https://arxiv.org/abs/2110.14477 |