||DICHOTOMY OF GLOBAL CAPACITY DENSITY
Aikawa, HiroakiItoh, Tsubasa
Proceedings of the American mathematical society
5393 , 2015-12 , American Mathematical Society
Let 1 < p < infinity and let d mu(x) = w(x) dx be a p-admissible weight in R-n, n >= 2. By Cap(p, mu)(E, D) we denote the variational (p, mu)-capacity of condenser (E, D). We show a dichotomy of the global density with respect to Cap(p, mu). One of our results is as follows: Let lambda > 1 and let B(x, r) stand for the open ball with center at x and radius r. Then lim(r ->infinity) (inf(x is an element of Rn) Cap(p, mu)(E boolean AND B(x, r), B(x, lambda r))/Cap(p, mu)(B(x, r), B(x, lambda r))) is equal to either 0 or 1; the first case occurs if and only if inf(x is an element of Rn) Cap(p, mu)(E boolean AND B(x, r(0)), B(x, lambda r))/Cap(p, mu)(B(x, r), B(x, lambda r)) is identically equal to 0. This provides a sharp contrast between capacity and Lebesgue measure.