QUASICONFORMAL MAPS, ANALYTIC CAPACITY, AND NON LINEAR POTENTIALS

In this paper we prove that if phi : C -> C is a K-quasiconformal map, with K > 1, and E subset of C is a compact set contained in a ball B, then (C) over dot(2K/2K+1,) (2K+1/K+1)(E)/diam(B)2/K+1 >= c(-1) (gamma(phi(E))/diam(phi(B)))(2/K+1), where gamma stands for the analytic capacity and (C) over dot(2K/2K+1,) (2K+1/K+1) is a capacity associated to a nonlinear Riesz potential. As a consequence, if E is not K-removable (i.e., removable for bounded K-quasiregular maps), it has positive capacity (C) over dot(2K/2K+1,) (2K+1/K+1). This improves previous results that assert that E must have non-a-finite Hausdorff measure of dimension 2/K+1. We also show that the indices 2K/2K+1, 2K+1/K+1 are sharp, and that Hausdorff gauge functions do not appropriately discriminate which sets are K-removable. So essentially we solve the problem of finding sharp "metric" conditions for K-removability.