On the analytic and Cauchy capacities
- 15 Downloads
We give new sufficient conditions for a compact set E ⊆ C to satisfy γ(E) = γc(E), where γ is the analytic capacity and γc is the Cauchy capacity. As a consequence, we provide examples of compact plane sets such that the above equality holds but the Ahlfors function is not the Cauchy transform of any complex Borel measure supported on the set.
Unable to display preview. Download preview PDF.
- S. Ya. Havinson, Analytic capacity of sets, joint nontriviality of various classes of analytic functions and the Schwarz lemma in arbitrary domains, Mat. Sb. (N.S.) 54 (96) (1961), 3–50; translation in Amer. Math. Soc. Transl. (2) 43 (1964), 215–266.Google Scholar
- S. Ya. Havinson, Representation and approximation of functions on thin sets, 1966 Contemporary Problems in Theory Anal. Functions, Izdat. “Nauka”, Moscow, pp. 314–318.Google Scholar
- T. Murai, Analyic capacity for arcs, Proceedings of the International Congress ofMathematicians, Math. Soc. Japan, Tokyo, 1991, pp. 901–911.Google Scholar
- X. Tolsa, Analytic capacity, rectifiability and the Cauchy integral, International Congress of Mathematicians,Vol. 1, Eur. Math. Soc. Zúrich, 2006, pp. 1505–1527.Google Scholar