Exact Hardy constant for the Riemannian distance in Schwarzschild exterior geometry

Determine the exact value of the optimal constant κ(n) in the Hardy inequality ∫_ℰ |∇_g ψ|^2 dv_g ≥ κ(n) ∫_ℰ |ψ|^2/d^2 dv_g on the Riemannian manifold (ℰ, g), where ℰ = {x ∈ ℝ^n : |x| > 1} is the exterior of the Schwarzschild black hole, g is the reduced Schwarzschild metric g = (r^{n−2}/(r^{n−2}−1)) dr ⊗ dr + r^2 g_{S^{n−1}}, d is the Riemannian distance to the event horizon given by d(r) = ∫_1^r √(ξ^{n−2}/(ξ^{n−2}−1)) dξ, and ψ ranges over locally absolutely continuous functions satisfying ψ(x)(|x|−1)^{−1/4} → 0 as |x| → 1 and ψ(x)|x|^{(n−2)/2} → 0 as |x| → ∞.

Background

The paper establishes a sharp Hardy inequality on the exterior of a Schwarzschild black hole using a tailor-made weight δ(r), with optimal constant ((n−2)/2)^2. By comparing δ to the Riemannian distance d(r) from the event horizon, the authors deduce a Hardy inequality involving d(r) with some positive constant κ(n).

However, the method only provides lower bounds via infimum comparisons of d/δ, and does not identify the exact optimal constant. In dimension n=3, the authors suggest (based on plotting) that the minimum of d/δ occurs at r = R = (4/3)^{1/(n−2)}, yielding a numerical lower bound κ(3) ≳ 0.117, but they emphasize that this does not determine the optimal constant.