A new characterization of Sobolev spaces

Jean Bourgain; Hoai-Minh Nguyen

doi:10.1016/j.crma.2006.05.021

Mathematical Analysis

A new characterization of Sobolev spaces

Presented by: Jean Bourgain

Jean Bourgain ¹; Hoai-Minh Nguyen ²

¹ Institute for Advanced Study, Olden Lane, Princeton, NJ 08540, USA
² Laboratoire Jacques-Louis Lions, université Pierre et Marie Curie, 4, place Jussieu, 75252 Paris cedex 05, France

Comptes Rendus. Mathématique, Volume 343 (2006) no. 2, pp. 75-80.

Abstract
Résumé

Our main result is the following: Let $g \in L^{p} (R^{N})$ , $1 < p < + \infty$ , be such that

\sup_{n \in N} \underset{| g (x) - g (y) | > δ_{n}}{\int_{R^{N}} \int_{R^{N}}} \frac{δ_{n}^{p}}{{| x - y |}^{N + p}} d x d y < + \infty,

for some arbitrary sequence of positive numbers

{(δ_{n})}_{n \in N}

with

\lim_{n \to \infty} δ_{n} = 0

. Then

g \in W^{1, p} (R^{N})

.

This extends a result from H.-M. Nguyen (2006).

Notre résultat principal est le suivant : Soit une fonction $g \in L^{p} (R^{N})$ , $1 < p < + \infty$ , telle que

\sup_{n \in N} \underset{| g (x) - g (y) | > δ_{n}}{\int_{R^{N}} \int_{R^{N}}} \frac{δ_{n}^{p}}{{| x - y |}^{N + p}} d x d y < + \infty,

où

{(δ_{n})}_{n \in N}

est une suite arbitraire positive telle que

\lim_{n \to \infty} δ_{n} = 0

. Alors

g \in W^{1, p} (R^{N})

.

Cela étend un résultat de H.-M. Nguyen (2006).

Received: 2006-05-17
Published online: 2006-07-03

DOI: 10.1016/j.crma.2006.05.021

Author's affiliations:

Jean Bourgain ¹; Hoai-Minh Nguyen ²

¹ Institute for Advanced Study, Olden Lane, Princeton, NJ 08540, USA
² Laboratoire Jacques-Louis Lions, université Pierre et Marie Curie, 4, place Jussieu, 75252 Paris cedex 05, France

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     author = {Jean Bourgain and Hoai-Minh Nguyen},
     title = {A new characterization of {Sobolev} spaces},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {75--80},
     publisher = {Elsevier},
     volume = {343},
     number = {2},
     year = {2006},
     doi = {10.1016/j.crma.2006.05.021},
     language = {en},
}

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Jean Bourgain; Hoai-Minh Nguyen. A new characterization of Sobolev spaces. Comptes Rendus. Mathématique, Volume 343 (2006) no. 2, pp. 75-80. doi : 10.1016/j.crma.2006.05.021. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2006.05.021/

References
Cited by

[1] R.A. Adams Sobolev Spaces, Academic Press, New York, 1975

[2] H. Brezis Analyse Fonctionnelle. Théorie et applications, Mathématiques appliquées pour la maîtrise, Dunod, 2002

[3] J. Bourgain; H. Brezis; P. Mironescu Another look at Sobolev spaces (J.L. Menaldi; E. Rofman; A. Sulem, eds.), Optimal Control and Partial Differential Equations, A Volume in Honour of A. Bensoussan's 60th Birthday, IOS Press, 2001, pp. 439-455

[4] L. Evans; R.F. Gariepy Measure Theory and Fine Properties of Functions, Studies in Advanced Mathematics, CRC Press, 1992

[5] H.M. Nguyen Some new characterizations of Sobolev spaces, J. Funct. Anal., Volume 237 (2006), pp. 689-720

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