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.


30C62 Quasiconformal mappings in the complex plane
31A15 Potentials and capacity, harmonic measure, extremal length and related notions in two dimensions
Full Text: DOI arXiv Euclid


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