Optimizing Improved Hardy Inequalities

Let Ω be a bounded domain in RN, N⩾3, containing the origin. Motivated by a question of Brezis and Vázquez, we consider an Improved Hardy Inequality with best constant b, that we formally write as: −Δ⩾(N−22)21∣x∣2+bV(x). We first give necessary conditions on the potential V, under which the previous inequality can or cannot be further improved. We show that the best constant b is never achieved in H01(Ω), and in particular that the existence or not of further correction terms is not connected to the nonachievement of b in H01(Ω). Our analysis reveals that the original inequality can be repeatedly improved by adding on the right-hand side specific potentials. This leads to an infinite series expansion of Hardyʹs inequality. The series obtained is in some sense optimal. In establishing these results we derive various sharp improved Hardy–Sobolev Inequalities.