A Bourgain–Brezis–Mironescu characterization of higher order Besov–Nikol ' skii spaces
Annales de l'Institut Fourier, Volume 68 (2018) no. 4, pp. 1671-1714.

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.

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.

Published online:
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)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
     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/}
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
%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/

[1] Andreu-Vaillo, Fuensanta; Mazón, José M.; Rossi, Julio D.; Toledo-Melero, J. Julián Nonlocal Diffusion Problems, Mathematical Surveys and Monographs, 165, American Mathematical Society, 2010 | Zbl

[2] Aubert, Gilles; Kornprobst, Pierre Can the nonlocal characterization of Sobolev spaces by Bourgain et al. be useful to solve variational problems?, SIAM J. Numer. Anal., Volume 47 (2009) no. 2, pp. 844-860 | Zbl

[3] Berestycki, Henri; Coville, Jérôme; Vo, Hoang-Hung Persistence criteria for populations with non-local dispersion, J. Math. Biol., Volume 72 (2016) no. 7, pp. 1693-1745 | Zbl

[4] Bojarski, Bogdan; Ihnatsyeva, Lizaveta; Kinnunen, Juha How to recognize polynomials in higher order Sobolev spaces, Math. Scand., Volume 112 (2013) no. 2, pp. 161-181 | Zbl

[5] Borghol, Rouba Some properties of Sobolev spaces, Asymptotic Anal., Volume 51 (2007) no. 3, pp. 303-318 | Zbl

[6] Bourgain, Jean; Brezis, Haim; Mironescu, Petru Another look at Sobolev spaces, Optimal control and partial differential equations (Paris, 2000) (2001), pp. 439-455 | Zbl

[7] Brezis, Haïm How to recognize constant functions. Connections with Sobolev spaces, Uspekhi Mat. Nauk, Volume 57 (2002), pp. 59-74 | Zbl

[8] Bruckner, Andrew M.; Ostrow, E. Some function classes related to the class of convex functions, Pac. J. Math., Volume 12 (1962) no. 4, pp. 1203-1215 | Zbl

[9] Chiron, David On the definitions of Sobolev and BV spaces into singular spaces and the trace problem, Commun. Contemp. Math., Volume 7 (2007) no. 4, pp. 473-513 | Zbl

[10] Dávila, Juan On an open question about functions of bounded variations, Calc. Var. Partial Differ. Equ., Volume 15 (2002), pp. 519-527 | Zbl

[11] Ferreira, Rita; Kreisbeck, Carolin; Ribeiro, Ana Margarida Characterization of polynomials and higher-order Sobolev spaces in terms of functionals involving difference quotients, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods, Volume 112 (2015), pp. 199-214 | Zbl

[12] Fiscella, Alessio; Servadei, Raffaella; Valdinoci, Enrico Density properties for fractional Sobolev spaces, Ann. Acad. Sci. Fenn., Math., Volume 40 (2015), pp. 235-253 | Zbl

[13] Gilboa, Guy; Osher, Stanley Nonlocal linear image regularization and supervised segmentation, Multiscale Model. Simul., Volume 6 (2007) no. 2, pp. 595-630 | Zbl

[14] Gilboa, Guy; Osher, Stanley Nonlocal operators with applications to image processing, Multiscale Model. Simul., Volume 7 (2008) no. 3, pp. 1005-1028 | Zbl

[15] Karadzhov, Georgi E.; Milman, Mario; Xiao, Jie Limits of higher-order Besov spaces and sharp reiteration theorems, J. Funct. Anal., Volume 221 (2005) no. 2, pp. 323-339 | Zbl

[16] Kolyada, Viktor I.; Lerner, Andrei K. On limiting embeddings of Besov spaces, Stud. Math., Volume 171 (2005) no. 1, pp. 1-13 | Zbl

[17] Kowalski, Jan Krzysztof A method of approximation of Besov spaces, Stud. Math., Volume 96 (1990) no. 2, pp. 183-193 | Zbl

[18] Kuczma, Marek An Introduction to the Theory of Functional Equations and Inequalities. Cauchy’s Equation and Jensen’s Inequality, Birkhäuser, 2009, xiv+595 pages (edited by A. Gilányi) | Zbl

[19] Lamy, Xavier; Mironescu, Petru Characterization of function spaces via low regularity mollifiers, Discrete Contin. Dyn. Syst., Volume 35 (2015) no. 12, pp. 6015-6030 | Zbl

[20] Maz’ya, Vladimir; Shaposhnikova, Tatyana On the Bourgain, Brezis and Mironescu theorem concerning limiting embeddings of fractional Sobolev spaces, J. Funct. Anal., Volume 195 (2002) no. 2, pp. 230-238 | Zbl

[21] Nikolʼskii, Sergei M. Approximation of Functions of Several Variables and Embedding Theorems, Die Grundlehren der mathematischen Wissenschaften, Springer, 1975, viii+420 pages | Zbl

[22] Ponce, Augusto C. An estimate in the spirit of Poincaré’s inequality, J. Eur. Math. Soc., Volume 6 (2004), pp. 1-15 | Zbl

[23] Ponce, Augusto C. A new approach to Sobolev spaces and connections to Gamma-convergence, Calc. Var. Partial Differ. Equ., Volume 19 (2004) no. 3, pp. 229-255 | Zbl

[24] Simon, J. Sobolev, Besov and Nikolskii fractional spaces: Imbeddings and Comparisons for Vector Valued Spaces on an Interval, Ann. Mat. Pura Appl., Volume 157 (1990) no. 1, pp. 117-148 | Zbl

[25] Spector, Daniel Characterization of Sobolev and BV Spaces, Dissertation, Volume 78 (2011)

[26] Stasyuk, Sergei A.; Yanchenko, S. Y. Approximation of functions from Nikolskii-Besov type classes of generalized mixed smoothness, Anal. Math., Volume 41 (2015), pp. 311-334 | Zbl

[27] Trageser, Jeremy Local and nonlocal models in Thin-Plate and Bridge Dynamics, Dissertation, Theses and Student Research Paper in Mathematics, Volume 64 (2015)

[28] Triebel, Hans Theory of Function Spaces, Monographs in Mathematics, Birkhäuser, 1983 no. 78 | Zbl

[29] Triebel, Hans Fractals and Spectra Related to Fourier Analysis and Function Spaces, Monographs in Mathematics, Birkhäuser, 1997 no. 91, viii+271 pages | Zbl

[30] Triebel, Hans Interpolation theory, Function Spaces, Differential Operators, Monographs in Mathematics, Wiley-VCH, 1998

[31] Triebel, Hans The Structure of Functions, Monographs in Mathematics, Birkhäuser, 2001 no. 97, xii+425 pages | Zbl

[32] Triebel, Hans Limits of Besov norms, Arch. Math., Volume 96 (2011) no. 2, pp. 169-175 | Zbl

Cited by Sources: