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.
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.
Accepted:
Published online:
@article{CRMATH_2009__347_11-12_627_0,
author = {Mart{\'\i}n, Joaquim and Milman, Mario},
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},
url = {http://www.numdam.org/articles/10.1016/j.crma.2009.04.011/}
}
TY - JOUR AU - Martín, Joaquim AU - Milman, Mario 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 UR - http://www.numdam.org/articles/10.1016/j.crma.2009.04.011/ DO - 10.1016/j.crma.2009.04.011 LA - en ID - CRMATH_2009__347_11-12_627_0 ER -
%0 Journal Article %A Martín, Joaquim %A Milman, Mario %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 %U http://www.numdam.org/articles/10.1016/j.crma.2009.04.011/ %R 10.1016/j.crma.2009.04.011 %G en %F CRMATH_2009__347_11-12_627_0
Martín, Joaquim; Milman, Mario. 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. http://www.numdam.org/articles/10.1016/j.crma.2009.04.011/
[1] Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry, Rev. Mat. Iberoamericana, Volume 22 (2006) no. 3, pp. 993-1067
[2] Interpolation of Operators, Academic Press, Boston, 1988
[3] Extremal properties of half-spaces for log-concave distributions, Ann. Probab., Volume 24 (1996) no. 1, pp. 35-48
[4] Some connections between isoperimetric and Sobolev-type inequalities, Mem. Amer. Math. Soc., Volume 129 (1997) no. 616
[5] Entropy bounds and isoperimetry, Mem. Amer. Math. Soc., Volume 176 (2005) no. 829
[6] Intrinsic bounds on some real-valued stationary random functions, Lecture Notes in Math., vol. 1153, 1985
[7] Spaces between and 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] The Concentration of Measure Phenomenon, Math. Surveys and Monographs, vol. 89, American Mathematical Society, 2001
[10] Self improving Sobolev–Poincaré inequalities, truncation and symmetrization, Potential Anal., Volume 29 (2008), pp. 391-408
[11] Isoperimetry and symmetrization for logarithmic Sobolev inequalities, J. Funct. Anal., Volume 256 (2009), pp. 149-178
[12] 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] On a rearrangement-invariant function set that appears in optimal Sobolev embeddings, J. Math. Anal. Appl., Volume 344 (2008), pp. 788-798
Cited by Sources:
