Null sets for the capacity associated to Riesz kernels
type: Item
instance of: scholarly article
title: Null sets for the capacity associated to Riesz kernels
zbMATH Open document ID: 1070.42009
author: L. Prat
published in: Illinois Journal of Mathematics
publication date: 2005-03-01T00:00:00Z
review text: Let \(0<\alpha <n\) and let \(E \subset {\mathbb R}^{n}\) be a compact set. The capacity \(\gamma_{\alpha}(E)\) of \(E\) associated to the Riesz kernels in \({\mathbb R}^n\) is defined by \[ \gamma_{\alpha}(E)=\sup |\langle T, 1\rangle|, \] where the supremum is taken over all real distributions \(T\) supported on \(E\) such that the function \(T\ast \frac{x_{k}}{|x|^{1+ \alpha}}\) is bounded by \(1\) for all \(k=1,\dots,n\). The case \(\alpha=n-1\) corresponds to the Lipschitz harmonic capacity, and if \(n=2\), \(\gamma_{1}\) is comparable to the analytic capacity. In a previous work [Int. Math. Res. Not. 2004, No. 19, 937--981 (2004)], the author proved that (a) if \(0<\alpha<1\), and \(H^{\alpha}(E)<\infty\) (where \(H^{\alpha}\) denotes the \(\alpha\)-dimensional Hausdorff measure), then \(\gamma_{\alpha}(E)=0\), and (b) if \(\alpha\in(0,n)\) is not an integer, \(\gamma_\alpha(E)=0\) whenever \(E\) is the Ahlfors-David regular set with dimension \(\alpha\). The paper under review is to extend (b) to compact sets (with finite \(\alpha\)-dimensional Hausdorff measure) that satisfy a very weak density condition. The proof is based on a modification of a covering theorem of \textit{H. Pajot} [Bull. Soc. Math. Fr. 125, No. 1, 15--53 (1997; Zbl 0890.28004)].
reviewed by: Lixin Yan
Mathematics Subject Classification ID: 42B20
Mathematics Subject Classification ID: 28A12
Mathematics Subject Classification ID: 31B15
zbMATH DE Number: 2140259
zbMATH Keywords: analytic capacity
zbMATH Keywords: Lipschitz harmonic capacity
zbMATH Keywords: Riesz kernel
zbMATH Keywords: Ahlfors-David regular set
zbMATH Keywords: Hausdorff measure
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 2140259