Comptes Rendus
no. 12
Functional analysis
Positivity improvement and Gaussian kernels
[Positivité améliorée et noyaux gaussiens]

Présenté par : Gilles Pisier

Franck Barthe1 ; Paweł Wolff2, 3
1 Institut de mathématiques de Toulouse, CNRS UMR 5219, Université Paul-Sabatier, 31062 Toulouse cedex 9, France
2 Institute of Mathematics, University of Warsaw, ul. Banacha 2, 02-097 Warszawa, Poland
3 Institute of Mathematics, Polish Academy of Sciences, ul. Śniadeckich 8, 00-956 Warszawa, Poland
Comptes Rendus. Mathématique, Volume 352 (2014) no. 12, pp. 1017-1021.

Nous montrons qu'une propriété d'amélioration de la positivité par les opérateurs multilinéaires à noyaux gaussiens peut être déterminée, avec des constantes exactes, en testant l'opérateur uniquement sur les fonctions gaussiennes. Ce résultat peut être considéré comme une forme inverse du théorème de Lieb sur les maximiseurs des noyaux gaussiens.

We show that a positivity improving property of multilinear operators with Gaussian kernels can be determined, with sharp constants, by testing Gaussian functions only. This result can be considered as a reversed form of Lieb's theorem on maximizers of Gaussian kernels.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2014.09.016
Franck Barthe 1 ; Paweł Wolff 2, 3

1 Institut de mathématiques de Toulouse, CNRS UMR 5219, Université Paul-Sabatier, 31062 Toulouse cedex 9, France
2 Institute of Mathematics, University of Warsaw, ul. Banacha 2, 02-097 Warszawa, Poland
3 Institute of Mathematics, Polish Academy of Sciences, ul. Śniadeckich 8, 00-956 Warszawa, Poland 
@article{CRMATH_2014__352_12_1017_0,
     author = {Franck Barthe and Pawe{\l} Wolff},
     title = {Positivity improvement and {Gaussian} kernels},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1017--1021},
     publisher = {Elsevier},
     volume = {352},
     number = {12},
     year = {2014},
     doi = {10.1016/j.crma.2014.09.016},
     language = {en},
}
TY  - JOUR
AU  - Franck Barthe
AU  - Paweł Wolff
TI  - Positivity improvement and Gaussian kernels
JO  - Comptes Rendus. Mathématique
PY  - 2014
SP  - 1017
EP  - 1021
VL  - 352
IS  - 12
PB  - Elsevier
DO  - 10.1016/j.crma.2014.09.016
LA  - en
ID  - CRMATH_2014__352_12_1017_0
ER  - 
%0 Journal Article
%A Franck Barthe
%A Paweł Wolff
%T Positivity improvement and Gaussian kernels
%J Comptes Rendus. Mathématique
%D 2014
%P 1017-1021
%V 352
%N 12
%I Elsevier
%R 10.1016/j.crma.2014.09.016
%G en
%F CRMATH_2014__352_12_1017_0
Franck Barthe; Paweł Wolff. Positivity improvement and Gaussian kernels. Comptes Rendus. Mathématique, Volume 352 (2014) no. 12, pp. 1017-1021. doi : 10.1016/j.crma.2014.09.016. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2014.09.016/

[1] S. Alesker; S. Dar; V. Milman A remarkable measure preserving diffeomorphism between two convex bodies in Rn, Geom. Dedic., Volume 74 (1999) no. 2, pp. 201-212

[2] K. Ball Volumes of sections of cubes and related problems, Geometric Aspects of Functional Analysis (1987-1988), Lecture Notes in Math., vol. 1376, Springer, Berlin, 1989, pp. 251-260

[3] K. Ball Volume ratios and a reverse isoperimetric inequality, J. Lond. Math. Soc. (2), Volume 44 (1991) no. 2, pp. 351-359

[4] F. Barthe An extremal property of the mean width of the simplex, Math. Ann., Volume 310 (1998) no. 4, pp. 685-693

[5] F. Barthe On a reverse form of the Brascamp–Lieb inequality, Invent. Math., Volume 134 (1998) no. 2, pp. 335-361

[6] F. Barthe Optimal Young's inequality and its converse: a simple proof, Geom. Funct. Anal., Volume 8 (1998) no. 2, pp. 234-242

[7] F. Barthe; N. Huet On Gaussian Brunn–Minkowski inequalities, Stud. Math., Volume 191 (2009) no. 3, pp. 283-304

[8] W. Beckner Inequalities in Fourier analysis, Ann. Math. (2), Volume 102 (1975) no. 1, pp. 159-182

[9] J. Bennett; A. Carbery; M. Christ; T. Tao The Brascamp–Lieb inequalities: finiteness, structure and extremals, Geom. Funct. Anal., Volume 17 (2008) no. 5, pp. 1343-1415

[10] C. Borell Positivity improving operators and hypercontractivity, Math. Z., Volume 180 (1982) no. 2, pp. 225-234

[11] H.J. Brascamp; E.H. Lieb Best constants in Young's inequality, its converse, and its generalization to more than three functions, Adv. Math., Volume 20 (1976) no. 2, pp. 151-173

[12] E.A. Carlen; D. Cordero-Erausquin Subadditivity of the entropy and its relation to Brascamp–Lieb type inequalities, Geom. Funct. Anal., Volume 19 (2009) no. 2, pp. 373-405

[13] E.A. Carlen; E.H. Lieb; M. Loss A sharp analog of Young's inequality on SN and related entropy inequalities, J. Geom. Anal., Volume 14 (2004) no. 3, pp. 487-520

[14] W.-K. Chen; N. Dafnis; G. Paouris Improved Hölder and reverse Hölder inequalities for correlated Gaussian random vectors, 2013 (preprint) | arXiv

[15] J. Lehec Short probabilistic proof of the Brascamp–Lieb and Barthe theorems, Can. Math. Bull., Volume 57 (2014) no. 3, pp. 585-597

[16] E.H. Lieb Gaussian kernels have only Gaussian maximizers, Invent. Math., Volume 102 (1990) no. 1, pp. 179-208

[17] L.H. Loomis; H. Whitney An inequality related to the isoperimetric inequality, Bull. Amer. Math. Soc., Volume 55 (1949), pp. 961-962

[18] E. Nelson The free Markoff field, J. Funct. Anal., Volume 12 (1973), pp. 211-227

Cité par Sources :

Commentaires - Politique