Hardy inequalities
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- by Andreas Wannebo
- Proc. Amer. Math. Soc. 109 (1990), 85-95
- DOI: https://doi.org/10.1090/S0002-9939-1990-1010807-1
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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 $\Omega$.References
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Bibliographic Information
- © Copyright 1990 American Mathematical Society
- 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