Bourgain-Brezis-Mironescu Convergence via Triebel-Lizorkin Spaces

TL;DR

This paper explores the convergence of Bourgain--Brezis--Mironescu (BBM) in the context of Triebel-Lizorkin spaces, resolving apparent contradictions and establishing new embeddings and convergence results for fractional Sobolev spaces.

Contribution

It provides sharp embeddings of $W^{s,p}$ into Triebel-Lizorkin spaces with different parameters, extending BBM convergence results to $ ext{R}^N$ and introducing several new embedding theorems.

Findings

Established embeddings of $W^{s,p}$ into $F^{s}_{p,q}$ with sharp constants.

Extended BBM convergence results to higher dimensions ($ ext{R}^N$).

Discovered new BBM-type embedding and convergence theorems.

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.