# Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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by Andreas Wannebo
Proc. Amer. Math. Soc. 109 (1990), 85-95 Request permission

## 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 $\Omega$.
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