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Hardy inequalities


Author: Andreas Wannebo
Journal: Proc. Amer. Math. Soc. 109 (1990), 85-95
MSC: Primary 26D10; Secondary 46E35
DOI: https://doi.org/10.1090/S0002-9939-1990-1010807-1
MathSciNet review: 1010807
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Abstract: Here the following Hardy inequalities are studied

$\displaystyle \sum\limits_{k = 0}^{m - 1} {\int {\frac{{\vert{\nabla ^k}u{\vert... ...\Omega }\int {\vert{\nabla ^m}u{\vert^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 $.

References [Enhancements On Off] (What's this?)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1990-1010807-1
Article copyright: © Copyright 1990 American Mathematical Society

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