Bourgain–Brezis–Mironescu convergence via Triebel-Lizorkin spaces

• Published in: Calculus of variations and partial differential equations Vol. 62; no. 2
• Main Authors: Brazke, Denis, Schikorra, Armin, Yung, Po-Lam
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.03.2023; Springer Nature B.V
• Subjects: Analysis; Calculus of Variations and Optimal Control; Optimization; Control; Convergence; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Systems Theory; Theoretical
• ISSN: 0944-2669; 1432-0835
• DOI: 10.1007/s00526-022-02382-6

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 → 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 ≠ 2 this seems like a contradiction, but we resolve this by providing embeddings of W s , p into F p , q s for q ∈ { p , 2 } with sharp constants with respect to s ∈ ( 0 , 1 ) . As a consequence we obtain an 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.