##
**Quasiconformal maps, analytic capacity, and non linear potentials.**
*(English)*
Zbl 1295.30052

A compact set \(E\subset\mathbb C\) is said to be \(K\)-removable (\(K\geq 1\)) if, for every open set \(\Omega\supset E\), every bounded \(K\)-quasiregular map \(f:\Omega\setminus E\mapsto\mathbb C\) admits a \(K\)-quasiregular extension to \(\Omega\). The case \(K=1\) corresponds to the compact sets that are removable for bounded analytic functions. It has been shown by Ahlfors that a compact set \(E\subset\mathbb C\) is removable for bounded analytic functions if and only if \(\gamma(E)=0\), where \(\gamma(E)\) denotes the analytic capacity of \(E\). The Painlevé problem for \(K\)-quasiregular maps consists in describing \(K\)-removable sets in metric and geometric terms. The first author [Acta Math. 190, No. 1, 105–149 (2003; Zbl 1060.30031); Ann. Math. (2) 162, No. 3, 1243–1304 (2005; Zbl 1097.30020)] obtained a solution of the classical Painlevé problem (\(K=1\)) in terms of the so-called curvature of measures. There is a well-known connection between \(K\)-removable sets and analytic capacity; \(E\) is \(K\)-removable if and only if \(\gamma(\phi(E))=0\) for every planar \(K\)-quasiconformal map. It has been considerably difficult to find sufficient conditions for removability and to construct “small” non-removable sets, in terms of geometric quantities.

In the paper under review, the authors give a sharp “metric” condition for \(K\)-removability in terms of Riesz capacities. They prove that if \(\phi:\mathbb C\mapsto\mathbb C\) is a \(K\)-quasiconformal map, with \(K>1\), and the compact set \(E\) is contained in a disk \(B\), then \[ \frac{\dot{C}_{\frac{2K}{2K+1},\frac{2K+1}{K+1}}(E)}{\text{{diam}}(B)^{\frac{2}{K+1}}}\geq c^{-1}\Big(\frac{\gamma(\phi(E))}{\text{{diam}}(\phi(B))}\Big)^{\frac{2K}{K+1}}, \] where \(\dot{C}_{\frac{2K}{2K+1},\frac{2K+1}{K+1}}\) is a Riesz capacity associated to a nonlinear potential and the constant \(c\) depends only on \(K\). The proof is long and technical, and includes distortion estimates (of capacities, measures and contents) under quasiconformal maps and a corona-type decomposition for measures with finite curvature and linear growth. Also, the authors introduce a new way of studying Riesz capacities via “Hausdorff-like” measures or contents. The strategies of proof of the main results are carefully described in separate parts of the paper.

In the paper under review, the authors give a sharp “metric” condition for \(K\)-removability in terms of Riesz capacities. They prove that if \(\phi:\mathbb C\mapsto\mathbb C\) is a \(K\)-quasiconformal map, with \(K>1\), and the compact set \(E\) is contained in a disk \(B\), then \[ \frac{\dot{C}_{\frac{2K}{2K+1},\frac{2K+1}{K+1}}(E)}{\text{{diam}}(B)^{\frac{2}{K+1}}}\geq c^{-1}\Big(\frac{\gamma(\phi(E))}{\text{{diam}}(\phi(B))}\Big)^{\frac{2K}{K+1}}, \] where \(\dot{C}_{\frac{2K}{2K+1},\frac{2K+1}{K+1}}\) is a Riesz capacity associated to a nonlinear potential and the constant \(c\) depends only on \(K\). The proof is long and technical, and includes distortion estimates (of capacities, measures and contents) under quasiconformal maps and a corona-type decomposition for measures with finite curvature and linear growth. Also, the authors introduce a new way of studying Riesz capacities via “Hausdorff-like” measures or contents. The strategies of proof of the main results are carefully described in separate parts of the paper.

Reviewer: Stamatis Pouliasis (Quebec)

### MSC:

30C62 | Quasiconformal mappings in the complex plane |

31A15 | Potentials and capacity, harmonic measure, extremal length and related notions in two dimensions |

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\textit{X. Tolsa} and \textit{I. Uriarte-Tuero}, Duke Math. J. 162, No. 8, 1503--1566 (2013; Zbl 1295.30052)

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