Null sets for the capacity associated to Riesz kernels (Q1766863)

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)].