Hardy's inequality and curvature

Abstract

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