Comptes Rendus
Functional Analysis
Isoperimetry and symmetrization for Sobolev spaces on metric spaces
[Isopérimétrie et symetrisation dans des espaces de Sobolev sur les espaces métriques]
Comptes Rendus. Mathématique, Volume 347 (2009) no. 11-12, pp. 627-630.

En utilisant l'isopérimétrie nous obtenons des nouvelles inégalités de symetrisation qui nous permettent de fournir un cadre unifié pour étudier des inégalités de Sobolev dans des espaces métriques. Les applications incluent des inégalités de concentration, inégalités de Poincaré, et des versions métriques des principes de Pólya–Szegö et de Faber–Krahn.

Using isoperimetry we obtain new symmetrization inequalities that allow us to provide a unified framework to study Sobolev inequalities in metric spaces. The applications include concentration inequalities, Poincaré inequalities, as well as metric versions of the Pólya–Szegö and Faber–Krahn principles.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2009.04.011

Joaquim Martín 1 ; Mario Milman 2

1 Department of Mathematics, Universitat Autònoma de Barcelona, 08193 Bellaterra (Barcelona), Spain
2 Department of Mathematics, Florida Atlantic University, Boca Raton, FL 33431, USA
@article{CRMATH_2009__347_11-12_627_0,
     author = {Joaquim Mart{\'\i}n and Mario Milman},
     title = {Isoperimetry and symmetrization for {Sobolev} spaces on metric spaces},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {627--630},
     publisher = {Elsevier},
     volume = {347},
     number = {11-12},
     year = {2009},
     doi = {10.1016/j.crma.2009.04.011},
     language = {en},
}
TY  - JOUR
AU  - Joaquim Martín
AU  - Mario Milman
TI  - Isoperimetry and symmetrization for Sobolev spaces on metric spaces
JO  - Comptes Rendus. Mathématique
PY  - 2009
SP  - 627
EP  - 630
VL  - 347
IS  - 11-12
PB  - Elsevier
DO  - 10.1016/j.crma.2009.04.011
LA  - en
ID  - CRMATH_2009__347_11-12_627_0
ER  - 
%0 Journal Article
%A Joaquim Martín
%A Mario Milman
%T Isoperimetry and symmetrization for Sobolev spaces on metric spaces
%J Comptes Rendus. Mathématique
%D 2009
%P 627-630
%V 347
%N 11-12
%I Elsevier
%R 10.1016/j.crma.2009.04.011
%G en
%F CRMATH_2009__347_11-12_627_0
Joaquim Martín; Mario Milman. Isoperimetry and symmetrization for Sobolev spaces on metric spaces. Comptes Rendus. Mathématique, Volume 347 (2009) no. 11-12, pp. 627-630. doi : 10.1016/j.crma.2009.04.011. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.04.011/

[1] F. Barthe; P. Cattiaux; C. Roberto Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry, Rev. Mat. Iberoamericana, Volume 22 (2006) no. 3, pp. 993-1067

[2] C. Bennett; R. Sharpley Interpolation of Operators, Academic Press, Boston, 1988

[3] S.G. Bobkov Extremal properties of half-spaces for log-concave distributions, Ann. Probab., Volume 24 (1996) no. 1, pp. 35-48

[4] S.G. Bobkov; C. Houdré Some connections between isoperimetric and Sobolev-type inequalities, Mem. Amer. Math. Soc., Volume 129 (1997) no. 616

[5] S.G. Bobkov; B. Zegarlinski Entropy bounds and isoperimetry, Mem. Amer. Math. Soc., Volume 176 (2005) no. 829

[6] C. Borell Intrinsic bounds on some real-valued stationary random functions, Lecture Notes in Math., vol. 1153, 1985

[7] A.P. Calderón Spaces between L1 and L and the theorem of Marcinkiewicz, Studia Math., Volume 26 (1966), pp. 273-299

[8] J. Kalis, M. Milman, Symmetrization and sharp Sobolev inequalities in metric spaces, Rev. Mat. Complut., in press

[9] M. Ledoux The Concentration of Measure Phenomenon, Math. Surveys and Monographs, vol. 89, American Mathematical Society, 2001

[10] J. Martin; M. Milman Self improving Sobolev–Poincaré inequalities, truncation and symmetrization, Potential Anal., Volume 29 (2008), pp. 391-408

[11] J. Martin; M. Milman Isoperimetry and symmetrization for logarithmic Sobolev inequalities, J. Funct. Anal., Volume 256 (2009), pp. 149-178

[12] J. Martin; M. Milman; E. Pustylnik Sobolev inequalities: symmetrization and self-improvement via truncation, J. Funct. Anal., Volume 252 (2007), pp. 677-695

[13] E. Milman, On the role of convexity in isoperimetry, spectral-gap and concentration, preprint

[14] E. Pustylnik On a rearrangement-invariant function set that appears in optimal Sobolev embeddings, J. Math. Anal. Appl., Volume 344 (2008), pp. 788-798

Cité par Sources :

Commentaires - Politique