We study a class of nonlocal functionals in the spirit of the recent characterization of the Sobolev spaces derived by Bourgain, Brezis and Mironescu. We show that it provides a common roof to the description of the , , and 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.
On étudie une classe de fonctionnelles non-locales dans l’esprit de la récente caractérisation des espaces de Sobolev obtenue par Bourgain, Brezis et Mironescu. On montre que celle-ci fournit un cadre unifié qui permet de décrire simultanément les espaces , , et , 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.
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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
@article{AIF_2018__68_4_1671_0, author = {BRASSEUR, Julien}, 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}, volume = {68}, number = {4}, year = {2018}, doi = {10.5802/aif.3196}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3196/} }
TY - JOUR AU - BRASSEUR, Julien TI - A Bourgain–Brezis–Mironescu characterization of higher order Besov–Nikol$^{\prime }$skii spaces JO - Annales de l'Institut Fourier PY - 2018 SP - 1671 EP - 1714 VL - 68 IS - 4 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3196/ DO - 10.5802/aif.3196 LA - en ID - AIF_2018__68_4_1671_0 ER -
%0 Journal Article %A BRASSEUR, Julien %T A Bourgain–Brezis–Mironescu characterization of higher order Besov–Nikol$^{\prime }$skii spaces %J Annales de l'Institut Fourier %D 2018 %P 1671-1714 %V 68 %N 4 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.3196/ %R 10.5802/aif.3196 %G en %F 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, Volume 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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