An improved Hardy inequality for a nonlocal operator

Boumediene  Abdellaoui; Fethi  Mahmoudi

doi:10.3934/dcds.2016.36.1143

Article Contents

2016, Volume 36, Issue 3: 1143-1157. Doi: 10.3934/dcds.2016.36.1143

This issue Previous Article Next Article Pure discrete spectrum for a class of one-dimensional substitution tiling systems

An improved Hardy inequality for a nonlocal operator

Boumediene Abdellaoui^1, and
Fethi Mahmoudi^2,

1.
Laboratoire D'Analyse Nonlinéaire et Mathématiques Appliquées, Département de Mathématiques, Faculté des sciences, Université About Baker Belkad, Tlemcen 13000
2.
Centro de Modelamiento Matemático (CMM), Universidad de Chile, Beauchef 851, Santiago

Received: January 2015

Revised: June 2015

Published: May 2017

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Abstract

Let $0 < s < 1$ and $1< p < 2$ be such that $ps < N$ and let $\Omega$ be a bounded domain containing the origin. In this paper we prove the following improved Hardy inequality:
Given $1 \le q < p$, there exists a positive constant $C\equiv C(\Omega, q, N, s)$ such that $$ \int\limits_{\mathbb{R}^N}\int\limits_{\mathbb{R}^N} \, \frac{|u(x)-u(y)|^{p}}{|x-y|^{N+ps}}\,dx\,dy - \Lambda_{N,p,s} \int\limits_{\mathbb{R}^N} \frac{|u(x)|^p}{|x|^{ps}}\,dx$$$$\geq C \int\limits_{\Omega}\int\limits_{\Omega}\frac{|u(x)-u(y)|^p}{|x-y|^{N+qs}}dxdy $$ for all $u \in \mathcal{C}_0^\infty({\Omega})$. Here $\Lambda_{N,p,s}$ is the optimal constant in the Hardy inequality （1.1）.

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Mathematics Subject Classification: 49J35, 35A15, 35S15.

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