Hardy's inequality and curvature

A Hardy inequality of the form \[∫Ω |∇ f({x})|^p d {x} ≥ (p-1/p)^p ∫Ω \{1 + a(δ, ∂ Ω)()\}{|f({x})|^p}{δ({x})^p} d{x}, \] for all f ∈ C₀∞(Ω), is considered for p∈ (1,∞), where Ω can be either $Ω$ or Rⁿ Ω with $Ω$ a domain in Rⁿ, n ≥ 2, 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 R² in terms of a uniformisation of $Ω,$ that is, any conformal univalent map of $Ω$ onto an annulus.