Quasiconformal maps, analytic capacity, and non linear potentials

Abstract

In this paper we prove that if ϕ:C→C is a K-quasiconformal map, with K>1, and E⊂C is a compact set contained in a ball B, then Ċ2K2K+1,2K+1K+1(E)diam(B)2K+1≥c−1(γ(ϕ(E))diam(ϕ(B)))2KK+1, where γ stands for the analytic capacity and Ċ2K2K+1,2K+1K+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 Ċ2K2K+1,2K+1K+1. This improves previous results that assert that E must have non-σ-finite Hausdorff measure of dimension 2K+1. We also show that the indices 2K2K+1, 2K+1K+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.