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On the semiadditivity of analytic capacity and planar Cantor sets. (English) Zbl 1053.30013
Beckner, William (ed.) et al., Harmonic analysis at Mount Holyoke. Proceedings of an AMS-IMS-SIAM joint summer research conference, Mount Holyoke College, South Hadley, MA, USA, June 25–July 5, 2001. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-2903-3/pbk). Contemp. Math. 320, 259-278 (2003).
This is an expository paper that describes the recent important progress in the theory of analytic capacity. The analytic capacity of a compact set \(E\) in the complex plane \(\mathbb C\) is defined by \[ \gamma (E)=\sup\{| f^\prime(\infty)| : f:\mathbb C\setminus E\to \mathbb C \text{ analytic with }| f| \leq 1 \}. \] X. Tolsa [Acta Math. 190, No.1, 105-149 (2003; )] proved that \(\gamma\) is semiadditive, that is, \[ \gamma(E\cup F)\leq C (\gamma(E)+\gamma(F)), \] for all compact sets \(E,F\), where \(C\) is an absolute constant. He thus confirmed an old, famous conjecture of A.G.Vitushkin. The paper under review contains an introduction to Tolsa’s proof. It discusses in detail the special case where \(E\) is the \(N\)th approximation of the planar Cantor set and presents the new ideas required for the general proof. It also describes the applications of semiadditivity to rational and harmonic approximation. Finally, it contains a list of interesting open problems.
For the entire collection see [Zbl 1013.00026].

MSC:
30C85 Capacity and harmonic measure in the complex plane
42B25 Maximal functions, Littlewood-Paley theory
31A15 Potentials and capacity, harmonic measure, extremal length and related notions in two dimensions
30E10 Approximation in the complex plane
30E20 Integration, integrals of Cauchy type, integral representations of analytic functions in the complex plane
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