Quasiadditivity of Riesz capacity.

Abstract

{ inf{∥f ∥ p : kα ∗ f (x) ≥ 1 on E, f ≥ 0} if 1 < p < ∞, inf{∥µ∥ : kα ∗ µ(x) ≥ 1 on E, µ ≥ 0} if p = 1. In view of [7] we see that Rα,1(E) is equal to the usual (outer) α-capacity Cα(E). It is obvious that Rα,p is countably subadditive, i.e. Rα,p(E) ≤ ∑ k Rα,p(Ek) with E = ∪ k Ek. The main purpose of this paper is to investigate for what decompositions the inequality Rα,p(E) ≥ N ∑ k Rα,p(Ek)