An improved Hardy inequality for a nonlocal operator

Abstract

Let $0 Given $1 \le q < p$, there exists a positive constant $C\equivC(\Omega, q, N, s)$ such that $$\int\limits_{\mathbb{R}^N}\int\limits_{\mathbb{R}^N} \,\frac{|u(x)-u(y)|^{p}}{|x-y|^{N+ps}}\,dx\,dy - \Lambda_{N,p,s}\int\limits_{\mathbb{R}^N} \frac{|u(x)|^p}{|x|^{ps}}\,dx$$$$\geqC\int\limits_{\Omega}\int\limits_{\Omega}\frac{|u(x)-u(y)|^p}{|x-y|^{N+qs}}dxdy$$for all $u \in \mathcal{C}_0^\infty({\Omega})$.Here $\Lambda_{N,p,s}$ is the optimal constant in the Hardyinequality （1.1）.