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
[Sur les inégalités de Sobolev logarithmiques pour les dérivées fractionnelles d'ordre supérieur]

Présenté par : 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 (VO)
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)

.

Reçu le : 2004-10-18
Accepté le : 2004-11-16
Publié le : 2005-01-11

DOI : 10.1016/j.crma.2004.11.030

Affiliations des auteurs :

Athanase Cotsiolis ¹ ; Nikolaos K. Tavoularis ¹

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

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     author = {Athanase Cotsiolis and Nikolaos K. Tavoularis},
     title = {On logarithmic {Sobolev} inequalities for higher order fractional derivatives},
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     pages = {205--208},
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     number = {3},
     year = {2005},
     doi = {10.1016/j.crma.2004.11.030},
     language = {en},
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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/

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