Partial Differential Equations
On logarithmic Sobolev inequalities for higher order fractional derivatives
Comptes Rendus. Mathématique, Volume 340 (2005) no. 3, pp. 205-208.

On $Rn$, we prove the existence of sharp logarithmic Sobolev inequalities with higher fractional derivatives. Let s be a positive real number. Any function f$Hs(Rn)$ satisfies

 $∫Rn|f(x)|2ln(|f(x)|2‖f‖22)dx+(n+nslnα+lnsΓ(n2)Γ(n2s))‖f‖22⩽α2πs‖(−Δ)s/2f‖22$
with $α>0$ be any number and where the operators $(−Δ)s/2$ in Fourier spaces are defined by $(−Δ)s/2fˆ(k):=(2π|k|)sfˆ(k)$.

Sur $Rn$, 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$Hs(Rn)$, on établit l'inégalité suivante :

 $∫Rn|f(x)|2ln(|f(x)|2‖f‖22)dx+(n+nslnα+lnsΓ(n2)Γ(n2s))‖f‖22⩽α2πs‖(−Δ)s/2f‖22.$
L'opérateur $(−Δ)s/2$ est defini dans les espaces de Fourier par $(−Δ)s/2fˆ(k):=(2π|k|)sfˆ(k)$.

Accepted:
Published online:
DOI: 10.1016/j.crma.2004.11.030

Athanase Cotsiolis 1; Nikolaos K. Tavoularis 1

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