Bourgain-Brezis-Mironescu convergence via Triebel-Lizorkin spaces
We study a convergence result of Bourgain--Brezis--Mironescu (BBM) using Triebel-Lizorkin spaces. It is well known that as spaces $W^{s,p} = F^{s}_{p,p}$, and $H^{1,p} = F^{1}_{p,2}$. When $s\to 1$, the $F^{s}_{p,p}$ norm becomes the $F^{1}_{p,p}$ norm but BBM showed that the $W^{s,p}$ norm becomes...
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| Hauptverfasser: | , , |
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| Dokumenttyp: | Article (Journal) Kapitel/Artikel |
| Sprache: | Englisch |
| Veröffentlicht: |
9 Sep 2021
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| In: |
Arxiv
Year: 2021, Pages: 1-24 |
| DOI: | 10.48550/arXiv.2109.04159 |
| Online-Zugang: | Verlag, lizenzpflichtig, Volltext: https://doi.org/10.48550/arXiv.2109.04159 Verlag, lizenzpflichtig, Volltext: http://arxiv.org/abs/2109.04159 
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| Verfasserangaben: | Denis Brazke, Armin Schikorra, and Po-Lam Yung |
MARC
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| 520 | |a We study a convergence result of Bourgain--Brezis--Mironescu (BBM) using Triebel-Lizorkin spaces. It is well known that as spaces $W^{s,p} = F^{s}_{p,p}$, and $H^{1,p} = F^{1}_{p,2}$. When $s\to 1$, the $F^{s}_{p,p}$ norm becomes the $F^{1}_{p,p}$ norm but BBM showed that the $W^{s,p}$ norm becomes the $H^{1,p} = F^{1}_{p,2}$ norm. Naively, for |p \neq 2$ this seems like a contradiction, but we resolve this by providing embeddings of $W^{s,p}$ into $F^{s}_{p,q}$ for |q \in \{p,2\}$ with sharp constants with respect to |s \in (0,1)$. As a consequence we obtain an $\mathbb{R}^N$-version of the BBM-result, and obtain several more embedding and convergence theorems of BBM-type that to the best of our knowledge are unknown. | ||
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