On logarithmic Sobolev inequalities for higher order fractional derivatives

Athanase Cotsiolis; Nikolaos K. Tavoularis

doi:10.1016/j.crma.2004.11.030

Partial Differential Equations

On logarithmic Sobolev inequalities for higher order fractional derivatives

Presented by: Thierry Aubin

Athanase Cotsiolis ¹; Nikolaos K. Tavoularis ¹

¹ Department of Mathematics, University of Patras, Patras 26110, Greece

Comptes Rendus. Mathématique, Volume 340 (2005) no. 3, pp. 205-208.

Abstract (Original language)
Résumé

On $R^{n}$ , we prove the existence of sharp logarithmic Sobolev inequalities with higher fractional derivatives. Let s be a positive real number. Any function f ∈ $H^{s} (R^{n})$ satisfies

\int_{R^{n}} | f (x) |^{2} \ln (\frac{{| f (x) |}^{2}}{‖ f ‖_{2}^{2}}) d x + (n + \frac{n}{s} \ln α + \ln \frac{s Γ (\frac{n}{2})}{Γ (\frac{n}{2 s})}) ‖ f ‖_{2}^{2} ⩽ \frac{α^{2}}{π^{s}} ‖ {(- Δ)}^{s / 2} f ‖_{2}^{2}

with

α > 0

be any number and where the operators

{(- Δ)}^{s / 2}

in Fourier spaces are defined by

\hat{{(- Δ)}^{s / 2} f} (k) : = {(2 π | k |)}^{s} \hat{f} (k)

.

Sur $R^{n}$ , on établi l'existence d'inégalités de Sobolev logarithmiques optimales pour les dérivées fractionnelles d'ordre supérieur. Soit s et α deux réels positifs. Pour toute fonction f ∈ $H^{s} (R^{n})$ , on établit l'inégalité suivante :

\int_{R^{n}} | f (x) |^{2} \ln (\frac{{| f (x) |}^{2}}{‖ f ‖_{2}^{2}}) d x + (n + \frac{n}{s} \ln α + \ln \frac{s Γ (\frac{n}{2})}{Γ (\frac{n}{2 s})}) ‖ f ‖_{2}^{2} ⩽ \frac{α^{2}}{π^{s}} ‖ {(- Δ)}^{s / 2} f ‖_{2}^{2} .

L'opérateur

{(- Δ)}^{s / 2}

est defini dans les espaces de Fourier par

\hat{{(- Δ)}^{s / 2} f} (k) : = {(2 π | k |)}^{s} \hat{f} (k)

.

Received: 2004-10-18
Accepted: 2004-11-16
Published online: 2005-01-11

DOI: 10.1016/j.crma.2004.11.030

Author's affiliations:

Athanase Cotsiolis ¹; Nikolaos K. Tavoularis ¹

¹ Department of Mathematics, University of Patras, Patras 26110, Greece

@article{CRMATH_2005__340_3_205_0,
     author = {Athanase Cotsiolis and Nikolaos K. Tavoularis},
     title = {On logarithmic {Sobolev} inequalities for higher order fractional derivatives},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {205--208},
     publisher = {Elsevier},
     volume = {340},
     number = {3},
     year = {2005},
     doi = {10.1016/j.crma.2004.11.030},
     language = {en},
}

TY  - JOUR
AU  - Athanase Cotsiolis
AU  - Nikolaos K. Tavoularis
TI  - On logarithmic Sobolev inequalities for higher order fractional derivatives
JO  - Comptes Rendus. Mathématique
PY  - 2005
SP  - 205
EP  - 208
VL  - 340
IS  - 3
PB  - Elsevier
DO  - 10.1016/j.crma.2004.11.030
LA  - en
ID  - CRMATH_2005__340_3_205_0
ER  -

%0 Journal Article
%A Athanase Cotsiolis
%A Nikolaos K. Tavoularis
%T On logarithmic Sobolev inequalities for higher order fractional derivatives
%J Comptes Rendus. Mathématique
%D 2005
%P 205-208
%V 340
%N 3
%I Elsevier
%R 10.1016/j.crma.2004.11.030
%G en
%F CRMATH_2005__340_3_205_0

Athanase Cotsiolis; Nikolaos K. Tavoularis. On logarithmic Sobolev inequalities for higher order fractional derivatives. Comptes Rendus. Mathématique, Volume 340 (2005) no. 3, pp. 205-208. doi : 10.1016/j.crma.2004.11.030. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.11.030/

References
Cited by

[1] C. Ané; S. Blachère; D. Chafaï; P. Fougères; I. Gentil; F. Malrieu; C. Roberto; G. Scheffer Sur les inégalités de Sobolev logarithmiques, 10, Société Mathématique de France, Paris, 2000

[2] Th. Aubin Some Nonlinear Problems in Riemannian Geometry, Springer, 1998

[3] W. Beckner Inequalities in Fourier analysis, Ann. Math., Volume 102 (1975), pp. 159-182

[4] W. Beckner Sharp Sobolev inequalities on the sphere and the Moser–Trudinger inequality, Ann. Math., Volume 138 (1993) no. 2, pp. 213-243

[5] W. Beckner; M. Pearson On sharp Sobolev embedding and the logarithmic Sobolev inequalities, Bull. London Math. Soc., Volume 30 (1998), pp. 80-84

[6] A. Cotsiolis; N.K. Tavoularis Sharp Sobolev type inequalities for higher fractional derivatives, C. R. Acad. Sci. Paris, Ser. I, Volume 335 (2002), pp. 801-804

[7] A. Cotsiolis; N.K. Tavoularis Best constants for Sobolev inequalities for higher order fractional derivatives, J. Math. Anal. Appl., Volume 295 (2004), pp. 225-236

[8] R. Gilles Une initiation aux inégalités de Sobolev logarithmiques, Cours Spécialisés, 5, Société Mathématique de France, Paris, 1999

[9] L. Gross Logarithmic Sobolev inequalities, Am. J. Math., Volume 97 (1975) no. 761, pp. 1061-1083

[10] E. Lieb; M. Loss Analysis, Amer. Math. Soc., 2001

[11] M. Del Pino; J. Dolbeault; I. Gentil Nonlinear diffusions, hypercontractivity and the optimal $L^{p}$ -Euclidean logarithmic Sobolev inequality, J. Math. Anal. Appl., Volume 293 (2004) no. 2, pp. 375-388

[12] A.J. Stam Some inequalities satisfied by the quantities of information of Fisher and Shannon, Inform. Control, Volume 2 (1959), pp. 255-269

[13] N.K. Tavoularis, Thèse de Doctorat de l'Université de Patras, Spécialité Mathématiques, 2004

Cited by Sources:

Comments - Policy