Quasiconformal Maps, Analytic Capacity, and Non Linear Potentials

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 Ċ 2K 2K+1 , 2K+1 K+1 (E) diam(B) 2 K+1 ≥ c−1 ( γ(φ(E)) diam(φ(B)) ) 2K K+1 , where γ stands for the analytic capacity and Ċ 2K 2K+1 , 2K+1 K+1 is a capacity associated to a non linear Riesz potential. As a consequence, if E not K-removable, it has positive capacity Ċ 2K 2K+1 , 2K+1 K+1 . This improves previous results that assert that E must have non σ-finite Hausdorff measure of dimension 2/(K +1). We also show that the indices 2K 2K+1 , 2K+1 K+1 are sharp.