Hardyʼs inequality and curvature

A Hardy inequality of the form∫Ω|∇f(x)|pdx⩾(p−1p)p∫Ω{1+a(δ,∂Ω)(x)}|f(x)|pδ(x)pdx, for all f∈C0∞(Ω∖R(Ω)), is considered for p∈(1,∞)p∈(1,∞), where Ω   is a domain in RnRn, n⩾2n⩾2, R(Ω)R(Ω) is the ridge of Ω  , and δ(x)δ(x) is the distance from x∈Ωx∈Ω to the boundary ∂Ω  . The main emphasis is on determining the dependence of a(δ,∂Ω)a(δ,∂Ω) on the geometric properties of ∂Ω. A Hardy inequality is also established for any doubly connected domain Ω   in R2R2 in terms of a uniformization of Ω, that is, any conformal univalent map of Ω onto an annulus.