ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

8. Analysis

1991 MS Classification: 30, 34 Dyn’kin, Evsey, Technion, Haifa, Israel Smoothness of quasiconformal mappings at a point A plane quasiconformal mapping is a homeomorphic solution of the Beltrami equation fz̄ = μf_{z}. The classical TeichmullerBelinski theorem asserts that f has a finite complex derivative at z = 0 if μ vanishes at 0 in a sense. The only known proof of the theorem admits no modification or refinement. We present a new proof, providing sharp estimates for the remainder term as well as generalizations of the theorem to higher orders of smoothness. As an application, one can obtain a sharp criterion for the existence of the second derivative or a version of the Warschawski asymptotic formula with remainder term. As another application, we present a new proof of the recent Tkachenko result on spectra of Hill operators.