Dichotomy of global density of Riesz capacity

Let C-alpha be the Riesz capacity of order alpha, 0 < alpha < n, in R-n. We consider the Riesz capacity density (D) under bar (C-alpha, E, r) = inf x is an element of R-n C-alpha (E boolean AND B(x,r))/C-alpha(B(x,r)) for a Borel set E subset of R-n, where B(x, r) stands for the open ball with center at x and radius r. In case 0 < alpha <= 2, we show that lim(r ->infinity)<(D)under bar> (C alpha, E, r) is either 0 or 1; the first case occurs if and only if (D) under bar (C alpha, E, r) is identically zero for all r > 0. Moreover, it is shown that the densities with respect to more general open sets enjoy the same dichotomy. A decay estimate for a-capacitary potentials is also obtained.