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 ]
Annales de l'Institut Fourier, Tome 68 (2018) no. 4, p. 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( N ), W 1,p ( N ), B p, s ( N ) et C 0,1 ( 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( N ), W 1,p ( N ), B p, s ( N ) and C 0,1 ( 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.
Reçu le : 2017-03-01
Révisé le : 2017-09-25
Accepté le : 2017-11-07
Publié le : 2018-11-23
DOI : https://doi.org/10.5802/aif.3196
Classification:  46E35
Mots clés: Espaces fractionnaires, espaces de Besov d’ordres élevés, fonctionnelles non-locales, injections limites, non-compacité
@article{AIF_2018__68_4_1671_0,
     author = {BRASSEUR, Julien},
     title = {A Bourgain--Brezis--Mironescu characterization of higher order Besov--Nikol$^{\prime }$skii spaces},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {68},
     number = {4},
     year = {2018},
     pages = {1671-1714},
     doi = {10.5802/aif.3196},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_2018__68_4_1671_0}
}
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/item/AIF_2018__68_4_1671_0/

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