Quasiconformal maps, analytic capacity, and non linear potentials

Abstract

In this paper we prove that if $\varphi :\mathbb{C}\to \mathbb{C}$ is a $K$-quasiconformal map, with $K>1$, and $E\subset \mathbb{C}$ is a compact set contained in a ball $B$, then

$$\frac{{Ċ}_{\frac{2K}{2K+1},\frac{2K+1}{K+1}}\left(E\right)}{diam(B{)}^{\frac{2}{K+1}}}\ge c^{-1}(\frac{\gamma \left(\varphi \right(E\left)\right)}{diam\left(\varphi \right(B\left)\right)}{)}^{\frac{2K}{K+1}},$$

where $\gamma$ stands for the analytic capacity and ${Ċ}_{\frac{2K}{2K+1},\frac{2K+1}{K+1}}$ 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 ${Ċ}_{\frac{2K}{2K+1},\frac{2K+1}{K+1}}$. This improves previous results that assert that $E$ must have non-$\sigma$-finite Hausdorff measure of dimension $\frac{2}{K+1}$. We also show that the indices $\frac{2K}{2K+1}$, $\frac{2K+1}{K+1}$ 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.