On the best possible remaining term in the Hardy inequality.

We give a necessary and sufficient condition on a radially symmetric potential V on a bounded domain Omega of (n) that makes it an admissible candidate for an improved Hardy inequality of the following type. For every element in H(1)(0)(Omega) integral(Omega) |vector differential u|2 dx - ((n - 2)/2)2 integral(Omega) |u|2/|x|2 dx > or = c integral(Omega) V(x)|u|2 dx. A characterization of the best possible constant c(V) is also given. This result yields easily the improved Hardy's inequalities of Brezis-Vázquez [Brezis H, Vázquez JL (1997) Blow up solutions of some nonlinear elliptic problems. Revista Mat Univ Complutense Madrid 10:443-469], Adimurthi et al. [Adimurthi, Chaudhuri N, Ramaswamy N (2002) An improved Hardy Sobolev inequality and its applications. Proc Am Math Soc 130:489-505], and Filippas-Tertikas [Filippas S, Tertikas A (2002) Optimizing improved Hardy inequalities. J Funct Anal 192:186-233] as well as the corresponding best constants. Our approach clarifies the issue behind the lack of an optimal improvement while yielding the following sharpening of known integrability criteria: If a positive radial function V satisfies lim inf(r-->o) ln(r) integral(r)(o),sV(s) ds > -infinity, then there exists rho: = rho(Omega) > 0 such that the above inequality holds for the scaled potential v(rho)(x) = v((|x|)(rho)). On the other hand, if lim (r-->0) ln(r) integral(r)(o),sV(s) ds = -infinity, then there is no rho > 0 for which the inequality holds for V(rho).