Hardy's inequality and curvature

A Hardy inequality of the form \[\int_{\tildeΩ} |\nabla f({\bf{x}})|^p d {\bf{x}} \ge (\frac{p-1}{p})^p \int_{\tildeΩ} \{1 + a(δ, \partial \tildeΩ)(\x)\}\frac{|f({\bf{x}})|^p}{δ({\bf{x}})^p} d{\bf{x}}, \] for all $f \in C_0^{\infty}({\tildeΩ})$, is considered for $p\in (1,\infty)$, where ${\tildeΩ}$ can be either $Ω$ or $\mathbb{R}^n \setminus Ω$ with $Ω$ a domain in $\mathbb{R}^n$, $n \ge 2$, and $δ({\bf{x}})$ is the distance from ${\bf{x}} \in {\tildeΩ}$ to the boundary $\partial {\tildeΩ}.$ The main emphasis is on determining the dependance of $a(δ, \partial {\tildeΩ})$ on the geometric properties of $\partial {\tildeΩ}.$ A Hardy inequality is also established for any doubly connected domain $Ω$ in $\mathbb{R}^2$ in terms of a uniformisation of $Ω,$ that is, any conformal univalent map of $Ω$ onto an annulus.

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