A. Balinsky, W.D. Evans, and R.T. Lewis
Hardy's Inequality and Curvature
(505K, AMS=TeX)

ABSTRACT.  A Hardy inequality of the form 
 \[ 
 \int_{\Omega} |
abla f({f{x}})|^p d {f{x}} \ge \left(rac{p-1}{p}
ight)^p \int_{\Omega} 
 \{1 + a(\delta, \partial \Omega)(\x)\}rac{|f({f{x}})|^p}{\delta({f{x}})^p} d{f{x}}, 
 \] 
 for all $f \in C_0^{\infty}({\Omega\setminus{\mathcal{R}(\Omega)}}),$ is considered for $p\in (1,\infty)$, 
 where ${\Omega}$ is a domain in $\mathbb{R}^n$, $n \ge 2$, $\mathcal{R}(\Omega)$ is the 	extit{ridge} of $\Omega$, 
 and $\delta({f{x}})$ is the distance from ${f{x}} \in {\Omega} $ 
 to the boundary $ \partial {\Omega}.$ The main emphasis is on 
 determining the dependance of $a(\delta, \partial {\Omega})$ on the geometric properties 
 of $\partial {\Omega}.$ A Hardy inequality is also 
 established for any doubly connected domain $\Omega$ in 
 $\mathbb{R}^2$ in terms of a uniformization of $\Omega,$ that is, 
 any conformal univalent map of $\Omega$ onto an annulus. }