Dichotomy of global density of Riesz capacity (Q2809350)

Let \(B(x,r)\) denote the open ball in \(\mathbb{R}^{n}\) of centre \(x\) and radius \(r\), and let \(C_{\alpha }\) denote Riesz capacity of order \(\alpha \in (0,n)\). For any Borel set \(E\subset \mathbb{R}^{n}\), let \(\underline{D}{(C}_{\alpha }(E,r))\) denote the infimum over all \(x\in \mathbb{R}^{n}\) of the quotient \(C_{\alpha }(E\cap B(x,r))/C_{\alpha }(B(x,r))\). It is shown that \(\lim_{r\rightarrow \infty }\underline{D}{(C}_{\alpha }(E,r))\in \{0,1\}\) provided \(0<\alpha \leq 2\). Further, the value \(0\) arises here if and only if \(\underline{D}{(C}_{\alpha }(E,\cdot ))\equiv 0\). A generalization is also established in which balls are replaced by other families of sets. Results of this nature had previously been established for \(L^{p}\)-capacity by the author and \textit{T. Itoh} [Proc. Am. Math. Soc. 143, No. 12, 5381--5393 (2015; Zbl 1333.31011)].