11-81 A. Balinsky, W.D. Evans, and R.T. Lewis

Hardy's Inequality and Curvature (505K, AMS=TeX) May 23, 11

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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. }

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