Convex Sobolev inequalities and spectral gap

Bartier, Jean-Philippe; Dolbeault, Jean

Bartier, Jean-Philippe; Dolbeault, Jean (2006), Convex Sobolev inequalities and spectral gap, Comptes rendus mathématique, 342, 5, p. 307-312. http://dx.doi.org/10.1016/j.crma.2005.12.004

Type

Article accepté pour publication ou publié

External document link

http://hal.archives-ouvertes.fr/hal-00004418/en/

Date

2006

Journal name

Comptes rendus mathématique

Volume

342

Number

Publisher

Académie des Sciences / Elsevier Masson SAS

Pages

307-312

Abstract (EN)

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 with constants which are uniformly bounded in the limit approaching the logarithmic Sobolev inequalities. We recover the case of the logarithmic Sobolev inequalities as a special case.

Subjects / Keywords

perturbation; interpolation; entropy production method; logarithmic Sobolev inequalities; spectral gap inequalities; Poincaré inequalities; convex Sobolev inequalities; generalized Poincaré inequalities