no. 4
A Bourgain–Brezis–Mironescu characterization of higher order Besov–Nikol ' skii spaces
[Caractérisation de type Bourgain–Brezis–Mironescu des espaces de Besov–Nikolskii d’ordres élevés]
BRASSEUR, Julien1
1 INRA Avignon, unité BioSP and Aix-Marseille Univ, CNRS Centrale Marseille, I2M Marseille (France)
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(ℝ N ), W 1,p (ℝ N ), B p,∞ s (ℝ N ) et C 0,1 (ℝ N ), 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(ℝ N ), W 1,p (ℝ N ), B p,∞ s (ℝ N ) and C 0,1 (ℝ N ) 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.

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DOI : 10.5802/aif.3196
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
BRASSEUR, Julien 1

1 INRA Avignon, unité BioSP and Aix-Marseille Univ, CNRS Centrale Marseille, I2M Marseille (France)
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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     title = {A {Bourgain{\textendash}Brezis{\textendash}Mironescu} characterization of higher order {Besov{\textendash}Nikol}$^{\prime }$skii spaces},
     journal = {Annales de l'Institut Fourier},
     pages = {1671--1714},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
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BRASSEUR, Julien. A Bourgain–Brezis–Mironescu characterization of higher order Besov–Nikol$^{\prime }$skii spaces. Annales de l'Institut Fourier, Tome 68 (2018) no. 4, pp. 1671-1714. doi : 10.5802/aif.3196. https://aif.centre-mersenne.org/articles/10.5802/aif.3196/

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