Quasiconformal maps, analytic capacity, and non linear potentials

TL;DR

This paper establishes a precise relationship between quasiconformal maps, analytic capacity, and nonlinear potentials, providing sharp criteria for the removability of sets under quasiregular maps.

Contribution

It introduces a new capacity linked to nonlinear Riesz potentials that characterizes K-removability, improving previous Hausdorff measure-based results.

Findings

Proves a lower bound relating nonlinear capacity and analytic capacity under quasiconformal maps.

Shows that non K-removable sets have positive nonlinear capacity, refining earlier measure-based criteria.

Demonstrates the sharpness of the capacity indices and the limitations of Hausdorff gauge functions.

Abstract

In this paper we prove that if $\phi:\C\to\C$ is a $K$-quasiconformal map, with $K>1$, and $E\subset \C$ is a compact set contained in a ball $B$, then $$\frac{\dot C_{\frac{2K}{2K+1},\frac{2K+1}{K+1}}(E)}{\diam(B)^{\frac2{K+1}}} \geq c^{-1} (\frac{\gamma(\phi(E))}{\diam(\phi(B))})^{\frac{2K}{K+1}},$$ where $\gamma$ stands for the analytic capacity and $\dot C_{\frac{2K}{2K+1},\frac{2K+1}{K+1}}$ is a capacity associated to a non linear Riesz potential. As a consequence, if $E$ is not $K$-removable (i.e. removable for bounded $K$-quasiregular maps), it has positive capacity $\dot C_{frac{2K}{2K+1},\frac{2K+1}{K+1}}$. This improves previous results that assert that $E$ must have non $\sigma$-finite Hausdorff measure of dimension $2/(K+1)$. 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.