We study a convergence result of Bourgain–Brezis–Mironescu (BBM) using Triebel-Lizorkin spaces. It is well known that as spaces Ws,p=Fp,ps\documentclass[12pt]{minimal}
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\begin{document}$$W^{s,p} = F^{s}_{p,p}$$\end{document}, and H1,p=Fp,21\documentclass[12pt]{minimal}
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\begin{document}$$H^{1,p} = F^{1}_{p,2}$$\end{document}. When s→1\documentclass[12pt]{minimal}
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\begin{document}$$s\rightarrow 1$$\end{document}, the Fp,ps\documentclass[12pt]{minimal}
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\begin{document}$$F^{s}_{p,p}$$\end{document} norm becomes the Fp,p1\documentclass[12pt]{minimal}
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\begin{document}$$F^{1}_{p,p}$$\end{document} norm but BBM showed that the Ws,p\documentclass[12pt]{minimal}
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\begin{document}$$W^{s,p}$$\end{document} norm becomes the H1,p=Fp,21\documentclass[12pt]{minimal}
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\begin{document}$$H^{1,p} = F^{1}_{p,2}$$\end{document} norm. Naively, for p≠2\documentclass[12pt]{minimal}
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\begin{document}$$p \ne 2$$\end{document} this seems like a contradiction, but we resolve this by providing embeddings of Ws,p\documentclass[12pt]{minimal}
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\begin{document}$$W^{s,p}$$\end{document} into Fp,qs\documentclass[12pt]{minimal}
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\begin{document}$$F^{s}_{p,q}$$\end{document} for q∈{p,2}\documentclass[12pt]{minimal}
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\begin{document}$$q \in \{p,2\}$$\end{document} with sharp constants with respect to s∈(0,1)\documentclass[12pt]{minimal}
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\begin{document}$$s \in (0,1)$$\end{document}. As a consequence we obtain an RN\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {R}}^N$$\end{document}-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.
