Metric Properties of Capacities

Abstract

Many problems have definitive solutions in terms of capacities, but the latter have the drawback that their geometrical meaning is not transparent. For this reason we devote most of this chapter to comparing the (α, p)-capacities C _{α, p} for 1 < p < ∞ and 0 < αp ≤ N to the more geometric quantities known as Hausdorff measures. As we now know, C ^{α, p} is associated not only to the Sobolev spaces B _α ^{p, p} and Bessel potential spaces L^{α, p}, but also to the Besov spaces and the Lizorkin—Triebel spaces F _α ^{p, q}, 1 < q < ∞.