Bourgain-Brezis-Mironescu Convergence via Triebel-Lizorkin Spaces

Abstract

We study a convergence result of Bourgain--Brezis--Mironescu (BBM) using Triebel-Lizorkin spaces. It is well known that as spaces $W^{s,p}=F_{p,p}^s$, and $H^{1,p}=F_{p,2}^1$. When $s\to 1$, the $F_{p,p}^s$ norm becomes the $F_{p,p}^1$ norm but BBM showed that the $W^{s,p}$ norm becomes the $H^{1,p}=F_{p,2}^1$ norm. Naively, for $p\ne 2$ this seems like a contradiction, but we resolve this by providing embeddings of $W^{s,p}$ into $F_{p,q}^s$ 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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Cited by in corpus (2)

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