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{${\int }_{\Omega }|\nabla u{|}^{2}dx-(\frac{n-2}{2}{)}^2{\int }_{\Omega }\frac{|u{|}^2}{|x{|}^2}dx\ge c{\int }_{\Omega }V(|x|)|u{|}^{2}dx$ \quad for all $u\in H_0^1(\Omega )$.} \end{equation}

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