On the best possible remaining term in the Hardy Inequality

Abstract

We give a necessary and sufficient condition on a radially symmetric potential $V$ on $\Omega$ that makes it an admissible candidate for an improved Hardy inequality of the following form: \begin{equation}\label{gen-hardy.0} \hbox{$\int_{\Omega}|\nabla u |^{2}dx - (\frac{n-2}{2})^{2} \int_{\Omega}\frac{|u|^{2}}{|x|^{2}}dx\geq c\int_{\Omega} V(|x|)|u|^{2}dx$ \quad for all $u \in H^{1}_{0}(\Omega)$.} \end{equation}