Hardy’s Inequality and Curvature

A Hardy inequality of the form ∫ Ω |∇f(x)|dx ≥ ( p− 1 p )p ∫ Ω {1 + a(δ, ∂Ω)(x)} |f(x)| p δ(x)p dx, for all f ∈ C∞ 0 (Ω \ R(Ω)), is considered for p ∈ (1,∞), where Ω is a domain in R, n ≥ 2, R(Ω) is the ridge of Ω, and δ(x) is the distance from x ∈ Ω to the boundary ∂Ω. The main emphasis is on determining the dependance of a(δ, ∂Ω) on the geometric properties of ∂Ω. A Hardy inequality is also established for any doubly connected domain Ω in R in terms of a uniformization of Ω, that is, any conformal univalent map of Ω onto an annulus.