Partial differential equations
A remark on the fractional Hardy inequality with a remainder term
Comptes Rendus. Mathématique, Volume 352 (2014) no. 4, pp. 299-303.

We prove in this note the following sharpened fractional Hardy inequality:

Let $N⩾1$, $0, $N>2s$, and $Ω⊂RN$ a bounded domain. Then for all $1, there exists a positive constant $C=C(Ω,q,N,s)$ such that for all $u∈C0∞(Ω)$,

 $aN,s∫RN∫RN(u(x)−u(y))2|x−y|N+2sdxdy−ΛN,s∫RNu2(x)|x|2sdx⩾C(Ω,q,N,s)∫Ω∫Ω(u(x)−u(y))2|x−y|N+qsdxdy,$ (1)
where
 $aN,s=22s−1π−N2Γ(N+2s2)|Γ(−s)|andΛN,s=22sΓ2(N+2s4)Γ2(N−2s4).$

Dans cette note, nous proposons l'amélioration suivante de l'inégalité de Hardy fractionnaire :

Soient $N⩾1$, $0, $N>2s$, et $Ω⊂RN$ un domaine borné. Alors, pour tout $1, il existe une constante positive $C≡C(Ω,q,N,s)$ telle que, pour tout $u∈C0∞(Ω)$,

 $aN,s∫RN∫RN(u(x)−u(y))2|x−y|N+2sdxdy−ΛN,s∫RNu2(x)|x|2sdx⩾C(Ω,q,N,s)∫Ω∫Ω(u(x)−u(y))2|x−y|N+qsdxdy,$
avec
 $aN,s=22s−1π−N2Γ(N+2s2)|Γ(−s)|etΛN,s=22sΓ2(N+2s4)Γ2(N−2s4).$

Accepted:
Published online:
DOI: 10.1016/j.crma.2014.02.003

Boumediene Abdellaoui 1; Ireneo Peral 2; Ana Primo 2

1 Laboratoire d'analyse nonlinéaire et mathématiques appliquées, faculté des sciences, université Abou-Bakr-Belkaïd, Tlemcen 13000, Algeria
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Boumediene Abdellaoui; Ireneo Peral; Ana Primo. A remark on the fractional Hardy inequality with a remainder term. Comptes Rendus. Mathématique, Volume 352 (2014) no. 4, pp. 299-303. doi : 10.1016/j.crma.2014.02.003. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2014.02.003/

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