Convex Sobolev inequalities and spectral gap
Link to item filehttp://hal.archives-ouvertes.fr/hal-00004418/en/
Sujetperturbation; interpolation; entropy production method; logarithmic Sobolev inequalities; spectral gap inequalities; Poincaré inequalities; convex Sobolev inequalities; generalized Poincaré inequalities
Journal issueComptes rendus mathématique
PublisherAcadémie des Sciences / Elsevier Masson SAS
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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.
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