Dichotomy of global density of Riesz capacity
Let Cα be the Riesz capacity of order α, 0 < α < n, in Rn. We consider the Riesz capacity density D(Cα,E, r) = inf x∈Rn Cα(E ∩ B(x, r)) Cα(B(x, r)) for a Borel setE ⊂ Rn, whereB(x, r) stands for the open ball with center at x and radiusr. In case 0< α ≤ 2, we show that limr→∞D(Cα,E, r) is equal to either 0 or 1; the first case occurs if and only ifD(Cα,E, r) is identically equal to 0 for all r > 0. Moreover, it is shown that the density with respect to more general open sets enjoy the same dichotomy. A decay estimate for α-capacitary potentials is also obtained.