New Sharp Gagliardo–Nirenberg–Sobolev Inequalities and an Improved Borell–Brascamp–Lieb Inequality

We propose a new Borell-Brascamp-Lieb inequality which leads to novel sharp Euclidean inequalities such as Gagliardo-Nirenberg-Sobolev inequalities in R^n and in the half-space R^n_+. This gives a new bridge between the geometric pont of view of the Brunn-Minkowski inequality and the functional point of view of the Sobolev type inequalities. In this way we unify, simplify and generalize results by S. Bobkov-M. Ledoux, M. del Pino-J. Dolbeault and B. Nazaret.

[1]  S. Zugmeyer.  Sharp trace Gagliardo–Nirenberg–Sobolev inequalities for convex cones, and convex domains , 2017, Annales de l'Institut Henri Poincaré C, Analyse non linéaire.

[2]  K. Allegaert,et al.  (Preprint) , 2018 .

[3]  A. Figalli.  The Monge-ampere Equation and Its Applications , 2017 .

[4]  Arnaud Guillin,et al.  Dimensional improvements of the logarithmic Sobolev, Talagrand and Brascamp-Lieb inequalities , 2015, 1507.01086.

[5]  M. Ledoux,et al.  Analysis and Geometry of Markov Diffusion Operators , 2013 .

[6]  V. H. Nguyen.  Sharp weighted Sobolev and Gagliardo–Nirenberg inequalities on half‐spaces via mass transport and consequences , 2013, 1307.1363.

[7]  C. Villani.  Optimal Transport: Old and New , 2008 .

[8]  S. Bobkov,et al.  From Brunn–Minkowski to sharp Sobolev inequalities , 2008 .

[9]  I. Gentil.  From the Prékopa-Leindler inequality to modified logarithmic Sobolev inequality , 2007, 0710.5025.

[10]  B. Nazaret.  Best constant in Sobolev trace inequalities on the half-space , 2006 .

[11]  Stephen P. Boyd,et al.  Convex Optimization , 2004, Algorithms and Theory of Computation Handbook.

[12]  C. Villani,et al.  A MASS-TRANSPORTATION APPROACH TO SHARP SOBOLEV AND GAGLIARDO-NIRENBERG INEQUALITIES , 2004 .

[13]  I. Gentil.  The general optimal Lp-Euclidean logarithmic Sobolev inequality by Hamilton–Jacobi equations , 2003 .

[14]  C. Villani.  Topics in Optimal Transportation , 2003 .

[15]  Manuel del Pino,et al.  The optimal Euclidean Lp-Sobolev logarithmic inequality , 2003 .

[16]  Manuel del Pino,et al.  Best constants for Gagliardo–Nirenberg inequalities and applications to nonlinear diffusions☆ , 2002 .

[17]  V. V. Buldygin,et al.  Brunn-Minkowski inequality , 2000 .

[18]  Dario Cordero-Erausquin,et al.  Some Applications of Mass Transport to Gaussian-Type Inequalities , 2002 .

[19]  S. Bobkov,et al.  From Brunn-Minkowski to Brascamp-Lieb and to logarithmic Sobolev inequalities , 2000 .

[20]  R. McCann.  A Convexity Principle for Interacting Gases , 1997 .

[21]  F. Barthe.  Inégalités fonctionnelles et géométriques obtenues par transport des mesures , 1997 .

[22]  R. McCann.  Existence and uniqueness of monotone measure-preserving maps , 1995 .

[23]  R. McCann.  A convexity theory for interacting gases and equilibrium crystals , 1994 .

[24]  Y. Brenier.  Polar Factorization and Monotone Rearrangement of Vector-Valued Functions , 1991 .

[25]  G. Talenti,et al.  Best constant in Sobolev inequality , 1976 .

[26]  E. Lieb,et al.  On extensions of the Brunn-Minkowski and Prékopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation , 1976 .

[27]  T. Aubin,et al.  Problèmes isopérimétriques et espaces de Sobolev , 1976 .

[28]  C. Borell.  Convex set functions ind-space , 1975 .

[29]  G. M.,et al.  Partial Differential Equations I , 2023, Applied Mathematical Sciences.