Bourgain-Brezis-Mironescu convergence via Triebel-Lizorkin spaces

• Verfasst von: Brazke, Denis [VerfasserIn]
• Verfasst von: Schikorra, Armin [VerfasserIn]
• Verfasst von: Yung, Po-Lam [VerfasserIn]
• Titel: Bourgain-Brezis-Mironescu convergence via Triebel-Lizorkin spaces
• Verf.angabe: Denis Brazke, Armin Schikorra, and Po-Lam Yung
• E-Jahr: 2021
• Jahr: 9 Sep 2021
• Umfang: 24 S.
• Fussnoten: Gesehen am 28.09.2022
• Titel Quelle: Enthalten in: De.arxiv.org
• Ort Quelle: [S.l.] : Arxiv.org, 1991
• Jahr Quelle: 2021
• Band/Heft Quelle: (2021), Artikel-ID 2109.04159, Seite 1-24
• DOI: doi:10.48550/arXiv.2109.04159
• Datenträger: Online-Ressource
• Sprache: eng
• Sach-SW: Mathematics - Analysis of PDEs
• Sach-SW: Mathematics - Classical Analysis and ODEs
• Sach-SW: Mathematics - Functional Analysis
• K10plus-PPN: 1817224743

Abstract

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.