[Inégalités de Sobolev convexes et trou spectral]
Cette Note est consacrée à la preuve d'inégalités de Sobolev convexes (ou inégalités de Poincaré généralisées) qui interpolent entre des inégalités de trou spectral (ou de Poincaré) et des inégalités de Sobolev logarithmiques. Nous étendons à la famille des inégalités de Sobolev convexes toute entière des résultats qui ont été obtenus récemment par Cattiaux, et Carlen et Loss pour des inégalités de Sobolev logarithmiques. Sous des conditions locales sur la densité de la mesure par rapport à une mesure de référence, nous démontrons que les inégalités de trou spectral entraînent toutes les inégalités de Sobolev convexes, y compris dans le cas limite des inégalités de Sobolev logarithmiques.
This Note is devoted to the proof of convex Sobolev (or generalized Poincaré) inequalities which interpolate between spectral gap (or Poincaré) inequalities and logarithmic Sobolev inequalities. We extend to the whole family of convex Sobolev inequalities results which have recently been obtained by Cattiaux, and Carlen and Loss for logarithmic Sobolev inequalities. Under local conditions on the density of the measure with respect to a reference measure, we prove that spectral gap inequalities imply all convex Sobolev inequalities including in the limit case corresponding to the logarithmic Sobolev inequalities.
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@article{CRMATH_2006__342_5_307_0,
author = {Bartier, Jean-Philippe and Dolbeault, Jean},
title = {Convex {Sobolev} inequalities and spectral gap},
journal = {Comptes Rendus. Math\'ematique},
pages = {307--312},
publisher = {Elsevier},
volume = {342},
number = {5},
year = {2006},
doi = {10.1016/j.crma.2005.12.004},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2005.12.004/}
}
TY - JOUR AU - Bartier, Jean-Philippe AU - Dolbeault, Jean TI - Convex Sobolev inequalities and spectral gap JO - Comptes Rendus. Mathématique PY - 2006 SP - 307 EP - 312 VL - 342 IS - 5 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2005.12.004/ DO - 10.1016/j.crma.2005.12.004 LA - en ID - CRMATH_2006__342_5_307_0 ER -
%0 Journal Article %A Bartier, Jean-Philippe %A Dolbeault, Jean %T Convex Sobolev inequalities and spectral gap %J Comptes Rendus. Mathématique %D 2006 %P 307-312 %V 342 %N 5 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2005.12.004/ %R 10.1016/j.crma.2005.12.004 %G en %F CRMATH_2006__342_5_307_0
Bartier, Jean-Philippe; Dolbeault, Jean. Convex Sobolev inequalities and spectral gap. Comptes Rendus. Mathématique, Tome 342 (2006) no. 5, pp. 307-312. doi : 10.1016/j.crma.2005.12.004. http://www.numdam.org/articles/10.1016/j.crma.2005.12.004/
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