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 $$W^{s,p} = F^{s}_{p,p}$$ , and $$H^{1,p} = F^{1}_{p,2}$$ . When $$s\rightarrow 1$$ , 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 $$p \ne 2$$ 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.