Hardy inequalities

Abstract:

Here the following Hardy inequalities are studied \[ \sum \limits _{k = 0}^{m - 1} {\int {\frac {{|{\nabla ^k}u{|^p}{d_{\partial \Omega }}{{(x)}^t}}}{{{d_{\partial \Omega }}{{(x)}^{(m - k)p}}}}dx \leq {A_\Omega }\int {|{\nabla ^m}u{|^p}{d_{\partial \Omega }}{{(x)}^t}dx} } } \] for $u \in C_0^\infty (\Omega )$, an open proper subset of ${{\mathbf {R}}^N}$ and $t < {t_0}$, some small positive ${t_0}$. This inequality has previously been shown to hold for bounded Lipschitz domains. The question discussed is, How general can $\Omega$ be and allow the same inequality? A sufficient condition is given in the form of a local Maz’ja capacity condition. In ${{\mathbf {R}}^2}$, or generally if $p > N - 1$, this is satisfied for any $\Omega$ which is deformable to a point. Furthermore, if $p > N$ the condition is satisfied for all $\Omega$.