On the best possible remaining term in the Hardy inequality

• Published in: Proceedings of the National Academy of Sciences - PNAS 2008-09, Vol.105 (37), p.13746-13751
• Main Authors: Ghoussoub, Nassif, Moradifam, Amir
• Format: Article
• Language: English
• Subjects: Decreasing functions; Differentiable functions; Differential equations; Linear algebra; Mathematical constants; Mathematical functions; Mathematical inequalities; Numerical analysis; Ordinary differential equations; Physical Sciences; Sufficient conditions; Theorems
• ISSN: 0027-8424; 1091-6490
• DOI: 10.1073/pnas.0803703105

We give a necessary and sufficient condition on a radially symmetric potential V on a bounded domain Ω of {Ropf}n that makes it an admissible candidate for an improved Hardy inequality of the following type. For every [set membership] H[Formula: see text](Ω) Formula 1 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 infr[rightward arrow]oln(r)[integral operator][Formula: see text],sV(s)ds>-[infinity],then there exists ρ:=ρ(Ω) > 0 such that the above inequality holds for the scaled potential vρ(x)=v([Formula: see text]).On the other hand, if lim r[rightward arrow]₀ ln(r)[integral operator][Formula: see text],sV(s)ds=-[infinity], then there is no ρ > 0 for which the inequality holds for Vρ.