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Uniformly fat sets. (English) Zbl 0668.31002

Given \(\alpha >0\) and \(p>1\) such that \(0<\alpha p<n\), a set E in \(R^ n\) is called (\(\alpha\),p) locally uniformly fat if there are \(r_ 0>0\) and \(\lambda >0\) such that \(R_{\alpha,p}(E(x,r))\geq \lambda\) for every \(x\in E\) and \(0<r<r_ 0,\) where \(R_{\alpha,p}\) is the (\(\alpha\),p) Riesz capacity and \(E(x,r)=\{x+r^{-1}(y-x):y\in E,\quad | y-x| <r\}.\) E is called (\(\alpha\),p) uniformly fat if we can take \(r_ 0=\infty\) in the above. If \(\beta q<\alpha p\) \((\beta >0\), \(q>1)\), then \(R_{\beta,q}\) is weaker than \(R_{\alpha,p}\), so that a (\(\beta\),q) locally uniformly fat set is (\(\alpha\),p) locally uniformly fat. The main theorem of the present paper asserts that given an (\(\alpha\),p) locally uniformly fat set E, there is \(\epsilon >0\) such that E is (\(\beta\),q) locally uniformly fat whenever \(\alpha p-\epsilon <\beta q<\alpha p.\) As an application of this theorem, a Sobolev type inequality is proved for a domain whose complement is (1,p) uniformly fat. Another application is concerned with the mutual distance of Fukete points of an (\(\alpha\),2) locally uniformly fat compact set with respect to the Riesz kernel of order \(2\alpha\) \((0<\alpha <n/2)\).
Reviewer: F.Y.Maeda

MSC:

31C15 Potentials and capacities on other spaces
31B15 Potentials and capacities, extremal length and related notions in higher dimensions
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