A remark on the fractional Hardy inequality with a remainder term

Abdellaoui, Boumediene; Peral, Ireneo; Primo, Ana

doi:10.1016/j.crma.2014.02.003

Partial differential equations

A remark on the fractional Hardy inequality with a remainder term
[Une remarque sur l'inégalité de Hardy fractionnaire avec reste]

Présenté par : Yves Meyer

Abdellaoui, Boumediene ¹ ; Peral, Ireneo ² ; Primo, Ana ²

¹ Laboratoire d'analyse nonlinéaire et mathématiques appliquées, faculté des sciences, université Abou-Bakr-Belkaïd, Tlemcen 13000, Algeria
² Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain

Comptes Rendus. Mathématique, Tome 352 (2014) no. 4, pp. 299-303.

Résumé
Abstract

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

Soient $N ⩾ 1$ , $0 < s < 1$ , $N > 2 s$ , et $Ω \subset R^{N}$ un domaine borné. Alors, pour tout $1 < q < 2$ , il existe une constante positive $C \equiv C (Ω, q, N, s)$ telle que, pour tout $u \in C_{0}^{\infty} (Ω)$ ,

a_{N, s} \int_{R^{N}} \int_{R^{N}} \frac{{(u (x) - u (y))}^{2}}{{| x - y |}^{N + 2 s}} d x d y - Λ_{N, s} \int_{R^{N}} \frac{u^{2} (x)}{{| x |}^{2 s}} d x ⩾ C (Ω, q, N, s) \int_{Ω} \int_{Ω} \frac{{(u (x) - u (y))}^{2}}{{| x - y |}^{N + q s}} d x d y,

avec

a_{N, s} = 2^{2 s - 1} π^{- \frac{N}{2}} \frac{Γ (\frac{N + 2 s}{2})}{| Γ (- s) |} et Λ_{N, s} = 2^{2 s} \frac{Γ^{2} (\frac{N + 2 s}{4})}{Γ^{2} (\frac{N - 2 s}{4})} .

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

Let $N ⩾ 1$ , $0 < s < 1$ , $N > 2 s$ , and $Ω \subset R^{N}$ a bounded domain. Then for all $1 < q < 2$ , there exists a positive constant $C = C (Ω, q, N, s)$ such that for all $u \in C_{0}^{\infty} (Ω)$ ,

a_{N, s} \int_{R^{N}} \int_{R^{N}} \frac{{(u (x) - u (y))}^{2}}{{| x - y |}^{N + 2 s}} d x d y - Λ_{N, s} \int_{R^{N}} \frac{u^{2} (x)}{{| x |}^{2 s}} d x ⩾ C (Ω, q, N, s) \int_{Ω} \int_{Ω} \frac{{(u (x) - u (y))}^{2}}{{| x - y |}^{N + q s}} d x d y,

(1)

where

a_{N, s} = 2^{2 s - 1} π^{- \frac{N}{2}} \frac{Γ (\frac{N + 2 s}{2})}{| Γ (- s) |} and Λ_{N, s} = 2^{2 s} \frac{Γ^{2} (\frac{N + 2 s}{4})}{Γ^{2} (\frac{N - 2 s}{4})} .

Reçu le : 2013-12-20
Accepté le : 2014-02-04
Publié le : 2014-02-20

DOI : 10.1016/j.crma.2014.02.003

Affiliations des auteurs :

Abdellaoui, Boumediene ¹ ; Peral, Ireneo ² ; Primo, Ana ²

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     title = {A remark on the fractional {Hardy} inequality with a remainder term},
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Abdellaoui, Boumediene; Peral, Ireneo; Primo, Ana. A remark on the fractional Hardy inequality with a remainder term. Comptes Rendus. Mathématique, Tome 352 (2014) no. 4, pp. 299-303. doi : 10.1016/j.crma.2014.02.003. http://www.numdam.org/articles/10.1016/j.crma.2014.02.003/

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