A Bourgain–Brezis–Mironescu characterization of higher order Besov–Nikol${}^{\text{'}}$skii spaces

BRASSEUR, Julien

Annales de l'Institut Fourier, Tome 68 (2018) no. 4, pp. 1671-1714.

On étudie une classe de fonctionnelles non-locales dans l’esprit de la récente caractérisation des espaces de Sobolev $W^{1,p}$ obtenue par Bourgain, Brezis et Mironescu. On montre que celle-ci fournit un cadre unifié qui permet de décrire simultanément les espaces $BV\left({ℝ}^N\right)$, $W^{1,p}\left({ℝ}^N\right)$, $B_{p,\infty }^s\left({ℝ}^N\right)$ et $C^{0,1}\left({ℝ}^N\right)$, et on obtient de nouvelles caractérisations de ces espaces. On établit également un résultat de non-compacité ainsi que de nouvelles (non-)injections limites entre espaces de Lipschitz et de Besov qui étendent les résultats connus.

We study a class of nonlocal functionals in the spirit of the recent characterization of the Sobolev spaces $W^{1,p}$ derived by Bourgain, Brezis and Mironescu. We show that it provides a common roof to the description of the $BV\left({ℝ}^N\right)$, $W^{1,p}\left({ℝ}^N\right)$, $B_{p,\infty }^s\left({ℝ}^N\right)$ and $C^{0,1}\left({ℝ}^N\right)$ scales and we obtain new equivalent characterizations for these spaces. We also establish a non-compactness result for sequences and new (non-)limiting embeddings between Lipschitz and Besov spaces which extend the previous known results.

Classification : 46E35
Keywords: Fractional spaces, higher order Besov spaces, Nikol$^{\prime }$skii spaces, nonlocal functionals, limiting embeddings, non-compactness
Mot clés : Espaces fractionnaires, espaces de Besov d’ordres élevés, fonctionnelles non-locales, injections limites, non-compacité

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