### 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. As a consequence, if E is not K-removable (i.e., removable for bounded K-quasiregular maps), it has positive capacity C C 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, 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. © 2013.

Original language | English |
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Pages (from-to) | 1503-1566 |

Journal | Duke Mathematical Journal |

Volume | 162 |

Issue number | 8 |

DOIs | |

Publication status | Published - 1 Jun 2013 |

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## Cite this

Tolsa, X., & Uriarte-Tuero, I. (2013). Quasiconformal maps, analytic capacity, and non linear potentials.

*Duke Mathematical Journal*,*162*(8), 1503-1566. https://doi.org/10.1215/00127094-2208869