Estimates for the optimal constants in multipolar Hardy inequalities for Schrödinger and Dirac operators

By expanding squares, we prove several Hardy inequalities with two critical singularities and constants which explicitly depend upon the distance between the two singularities. These inequalities involve the L norm. Such results are generalized to an arbitrary number of singularities and compared with standard results given by the IMS method. The generalized version of Hardy inequalities with several singularities is equivalent to some spectral information on a Schrödinger operator involving a potential with several inverse square singularities. We also give a generalized Hardy inequality for Dirac operators in the case of a potential having several singularities of Coulomb type, which are critical for Dirac operators. Introduction For N ≥ 3, the simplest form of Hardy’s inequality is easily obtained by the “expansion of the square” method as follows: for any function u ∈ H(R ), 0 ≤ ∫ RN ∣∣∣∣∇u+ α x |x|2 u ∣∣∣∣ 2 dx = ∫ RN |∇u| dx+ [ α − (N − 2)α ] ∫ RN |u|2 |x|2 dx , which shows for α = (N − 2)/2 that, for all u ∈ H(R ), ∫ RN |∇u| dx ≥ (N − 2) 2 4 ∫ RN |u|2 |x|2 dx , and it is well known that the constant (N − 2)/4 is optimal. With two singularities located at ±y ∈ R , from the above inequality we get without effort that ∫ RN |∇u|2 dx− (N−2) 2 8 ∫ RN ( 1 |x−y|2 + 1 |x+y|2 ) |u|2 dx = 1 2 ∑ ± ∫ RN |∇u±| − (N−2) 2 4 |u±| |x|2 dx ≥ 0 where u±(·) = u(· ± y). For a given function u with compact support and d := |y| large enough, it is however clear that the constant (N − 2)/8 can be replaced by (N − 2)/4. To improve upon (N − 2)/8 for general functions and in presence of two singularities, one has to break the scaling invariance by introducing a new scale. This can be done by adding a lower order term. One of the goals in this paper is to obtain estimates for the best constant λ = λ(μ, d), that is, the smallest positive constant, in the inequality μ ∫ RN ( 1 |x− y|2 + 1 |x+ y|2 ) |u| dx ≤ ∫