Hardy’s inequality and curvature

A Hardy inequality of the form Ω ∇f (x) p 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 Rn, 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 dependence of a(δ, ∂Ω) on the geometric properties of ∂Ω. A Hardy inequality is also established for any doubly connected domain Ω in R2 in terms of a uniformization of Ω, that is, any conformal univalent map of Ω onto an annulus. © 2011 Elsevier Inc. All rights reserved