Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry

We introduce and study a notion of Orlicz hypercontractive semigroups. We analyze their relations with general F-Sobolev inequalities, thus extending Gross hypercontractivity theory. We provide criteria for these Sobolev type inequalities and for related properties. In particular, we implement in the context of probability measures the ideas of Maz'ja's capacity theory, and present equivalent forms relating the capacity of sets to their measure. Orlicz hypercontractivity efficiently describes the integrability improving properties of the Heat semigroup associated to the Boltzmann measures µa(dx) = (Za)-1 e-2|x|adx, when a I (1,2). As an application we derive accurate isoperimetric inequalities for their products. This completes earlier works by Bobkov-Houdre and Talagrand, and provides a scale of dimension free isoperimetric inequalities as well as comparison theorems

[1]  Entropy Bounds for Gibbs Measures with Non-Gaussian Tails☆ , 2001 .

[2]  M. Talagrand THE SUPREMUM OF SOME CANONICAL PROCESSES , 1994 .

[3]  M. Rao,et al.  Theory of Orlicz spaces , 1991 .

[4]  L. Gross Logarithmic Sobolev inequalities and contractivity properties of semigroups , 1993 .

[5]  B. Maurey Some deviation inequalities , 1990, math/9201216.

[6]  R. Latala,et al.  Between Sobolev and Poincaré , 2000, math/0003043.

[7]  Yih-Ho Pao,et al.  The Spectral Gap , 1969 .

[8]  M. Ledoux Concentration of measure and logarithmic Sobolev inequalities , 1999 .

[9]  Infinite Dimensional Isoperimetric Inequalities in Product Spaces with the Supremum Distance , 2004 .

[10]  D. Bakry L'hypercontractivité et son utilisation en théorie des semigroupes , 1994 .

[11]  M. Ledoux A simple analytic proof of an inequality by P. Buser , 1994 .

[12]  J. Rosen Sobolev Inequalities for Weight Spaces and Supercontractivity , 1976 .

[13]  M. Gromov,et al.  A topological application of the isoperimetric inequality , 1983 .

[14]  F. Barthe Levels of concentration between exponential and Gaussian , 2001 .

[15]  P. Cattiaux Hypercontractivity for perturbed diffusion semigroups , 2005, math/0510258.

[16]  E. Davies,et al.  Heat kernels and spectral theory , 1989 .

[17]  A. Grigor’yan,et al.  On isoperimetric profiles of product spaces , 2003 .

[18]  Feng-Yu Wang,et al.  Functional inequalities for uniformly integrable semigroups and application to essential spectrums , 2002 .

[19]  Patrick Cattiaux,et al.  On quadratic transportation cost inequalities , 2006 .

[20]  W. Beckner A generalized Poincaré inequality for Gaussian measures , 1989 .

[21]  M. Ledoux The concentration of measure phenomenon , 2001 .

[22]  M. Talagrand Concentration of measure and isoperimetric inequalities in product spaces , 1994, math/9406212.

[23]  G. Royer,et al.  Une initiation aux inégalités de Sobolev logarithmiques , 1999 .

[24]  Alice Guionnet,et al.  Lectures on Logarithmic Sobolev Inequalities , 2003 .

[25]  Christian Houdré,et al.  Some Connections Between Isoperimetric and Sobolev-Type Inequalities , 1997 .

[26]  R. Adams General logarithmic Sobolev inequalities and Orlicz imbeddings , 1979 .

[27]  S. Boucheron,et al.  Moment inequalities for functions of independent random variables , 2005, math/0503651.

[28]  M. Talagrand Transportation cost for Gaussian and other product measures , 1996 .

[29]  P. Cattiaux A Pathwise Approach of Some Classical Inequalities , 2004 .

[30]  Barry Simon,et al.  Ultracontractivity and the Heat Kernel for Schrijdinger Operators and Dirichlet Laplacians , 1987 .

[31]  Feng-Yu Wang Probability distance inequalities on Riemannian manifolds and path spaces , 2004 .

[32]  A. Guillin,et al.  Modified logarithmic Sobolev inequalities and transportation inequalities , 2004, math/0405520.

[33]  B. Muckenhoupt Hardy's inequality with weights , 1972 .

[34]  F. Barthe,et al.  Sobolev inequalities for probability measures on the real line , 2003 .

[35]  F. Barthe Extremal Properties of Central Half-Spaces for Product Measures , 2001 .

[36]  B. Zegarliński,et al.  Entropy Bounds and Isoperimetry , 2005 .

[37]  B. Simon,et al.  Hypercontractive semigroups and two dimensional self-coupled Bose fields , 1972 .

[38]  M. Ledoux,et al.  Sobolev inequalities in disguise , 1995 .

[39]  C. Houdré,et al.  Isoperimetric constants for product probability measures , 1997 .

[40]  Sergey G. Bobkov,et al.  Extremal properties of half-spaces for log-concave distributions , 1996 .

[41]  Djalil CHAFAÏ On Φ-entropies and Φ-Sobolev inequalities , 2002 .

[42]  S. Bobkov,et al.  Poincaré’s inequalities and Talagrand’s concentration phenomenon for the exponential distribution , 1997 .

[43]  Daniel W. Stroock,et al.  Moment estimates derived from Poincar'e and log-arithmic Sobolev inequalities , 1994 .

[44]  S. Bobkov,et al.  Exponential Integrability and Transportation Cost Related to Logarithmic Sobolev Inequalities , 1999 .

[45]  P. Cattiaux,et al.  Minimization of the Kullback information for some Markov processes , 1996 .

[46]  Feng-Yu Wang A Generalization of Poincaré and Log-Sobolev Inequalities , 2005 .

[47]  S. Aida,et al.  Logarithmic Sobolev Inequalities and Exponential Integrability , 1994 .

[48]  M. Röckner,et al.  Weak Poincaré Inequalities and L2-Convergence Rates of Markov Semigroups , 2001 .

[49]  M. Talagrand A new isoperimetric inequality and the concentration of measure phenomenon , 1991 .

[50]  Viktor Blåsjö,et al.  The Isoperimetric Problem , 2005, Am. Math. Mon..

[51]  Djalil CHAFAÏ Entropies, convexity, and functional inequalities : On Phi-entropies and Phi-Sobolev inequalities , 2004 .

[52]  Franck Barthe,et al.  Log-concave and spherical models in isoperimetry , 2002 .

[53]  Some Remarks on Ultracontractivity , 1993 .

[54]  M. Ledoux,et al.  Lévy–Gromov’s isoperimetric inequality for an infinite dimensional diffusion generator , 1996 .

[55]  Feng-Yu Wang,et al.  Functional Inequalities for Empty Essential Spectrum , 2000 .

[56]  L. Gross LOGARITHMIC SOBOLEV INEQUALITIES. , 1975 .

[57]  O. Rothaus Analytic inequalities, isoperimetric inequalities and logarithmic Sobolev inequalities , 1985 .