Hardy's inequality and curvature

Abstract

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 $f\in C_0^{\infty }(\tilde{\Omega })$, 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 $\delta (\mathbf{x})$ is the distance from $\mathbf{x}\in \tilde{\Omega }$ 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.