Sharp Sobolev type inequalities for higher fractional derivatives

Athanase Cotsiolis; Nikolaos Con. Tavoularis

doi:10.1016/S1631-073X(02)02576-1

Sharp Sobolev type inequalities for higher fractional derivatives
[Inégalités optimales de type Sobolev pour les dérivées fractionelles d'ordre supérieur]

Athanase Cotsiolis ¹ ; Nikolaos Con. Tavoularis ¹

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

Comptes Rendus. Mathématique, Volume 335 (2002) no. 10, pp. 801-804.

Résumé
Abstract

Sur $ℝ^{n}$ , n⩾1 et n≠2, on établit l'existence de meilleurs constantes dans les inégalités de Sobolev pour les dérivées fractionelles d'ordre supérieur. Soit s un reel positif. Pour n>2s et $q = \frac{2 n}{n - 2 s}$ toute fonction $f \in H^{s} (ℝ^{n})$ vérifie l'inégalité suivante

{∥ f ∥}_{q}^{2} ⩽ S_{n, s} {∥ {(- Δ)}^{s / 2} f ∥}_{2}^{2},

où S_n,s est la meilleure constante. L'opérateur (−Δ)^s est defini dans les espaces de Fourier par

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

.

On $ℝ^{n}$ , n⩾1 and n≠2, we prove the existence of a sharp constant for Sobolev inequalities with higher fractional derivatives. Let s be a positive real number. For n>2s and $q = \frac{2 n}{n - 2 s}$ any function $f \in H^{s} (ℝ^{n})$ satisfies

{∥ f ∥}_{q}^{2} ⩽ S_{n, s} {∥ {(- Δ)}^{s / 2} f ∥}_{2'}^{2}

where the operator (−Δ)^s in Fourier spaces is defined by

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

.

Accepté le : 2002-10-01
Publié le : 2002-10-20

DOI : 10.1016/S1631-073X(02)02576-1

Affiliations des auteurs :

Athanase Cotsiolis ¹ ; Nikolaos Con. Tavoularis ¹

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

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Athanase Cotsiolis; Nikolaos Con. Tavoularis. Sharp Sobolev type inequalities for higher fractional derivatives. Comptes Rendus. Mathématique, Volume 335 (2002) no. 10, pp. 801-804. doi : 10.1016/S1631-073X(02)02576-1. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(02)02576-1/

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Cité par 16 documents. Sources : zbMATH

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