Analytic capacity, bilipschitz maps and Cantor sets. (English) Zbl 1063.30025

Garnett, John; Verdera, Joan

Math. Res. Lett. 10, No. 4, 515-522 (2003).

Let \(E\) be a compact set in the plane and let \(\gamma(E)\) denote its analytic capacity. The authors state the conjecture that \[ \gamma(T(E))\leq C(K)\,\gamma(E) \] for every bilipschitz homeomorphism \(T:E\to T(E)\); the constant \(C(K)\) depends only on the bilipschitz constant \(K\). The authors remark that the conjecture is true when \(E\) has finite one-dimensional Hausdorff measure. Their main result is that the conjecture is also true for a well-known, wide class of Cantor-type sets described in detail in the paper. Although this result involves only a rather specific kind of sets, its proof depends on the recent very deep work of X.Tolsa [“Painlevé’s problem and the semiadditivity of analytic capacity”, Acta Math. 190, No.1, 105–149 (2003; Zbl 1060.30031)] and other deep results, and involves technical computations.

Reviewer: Dimitrios Betsakos (Thessaloniki)

MSC:

30C85

Capacity and harmonic measure in the complex plane

Keywords:

analytic capacity; Cantor set; bilipschitz map; Menger curvature