Bourgain–Brezis–Mironescu–Maz’ya–Shaposhnikova limit formulae for fractional Sobolev spaces via interpolation and extrapolation

• Published in: Calculus of variations and partial differential equations Vol. 62; no. 2
• Main Authors: Domínguez, Oscar, Milman, Mario
• 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; Extrapolation; Interpolation; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Smoothness; Sobolev space; Systems Theory; Theoretical; 46B20; Primary: 26A33; 42B10; 46B70; Secondary: 42B05
• ISSN: 0944-2669; 1432-0835
• DOI: 10.1007/s00526-022-02383-5

The real interpolation spaces between L p ( R n ) and H ˙ t , p ( R n ) (resp. H t , p ( R n ) ),   t > 0 , are characterized in terms of fractional moduli of smoothness, and the underlying seminorms are shown to be “the correct” fractional generalization of the classical Gagliardo seminorms. This is confirmed by the fact that, using the new spaces combined with interpolation and extrapolation methods, we are able to extend the Bourgain–Brezis–Mironescu–Maz’ya–Shaposhnikova limit formulae, as well as the Bourgain–Brezis–Mironescu convergence theorem, to fractional Sobolev spaces. On the other hand, we disprove a conjecture of Brazke et al. (Bourgain–Brezis–Mironescu convergence via Triebel–Lizorkin spaces. https://arxiv.org/abs/2109.04159 ) suggesting fractional convergence results given in terms of classical Gagliardo seminorms. We also solve a problem proposed in Brazke et al. (Bourgain–Brezis–Mironescu convergence via Triebel–Lizorkin spaces. https://arxiv.org/abs/2109.04159 ) concerning sharp forms of the fractional Sobolev embedding.