Analysis on quasidisks; a unified approach through transmission and jump problems

EMS surveys in mathematical sciences（2020）

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Abstract

We give an exposition of results from a crossroad between geometric function theory, harmonic analysis, boundary value problems and approximation theory, which characterize quasicircles. We will specifically expose the interplay between the jump decomposition, singular integral operators and approximation by Faber series. Our unified point of view is made possible by the the concept of transmission.

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Key words

Conformally non-tangential boundary value, Bergman space, Dirichlet space, Faber operator, Grunsky operator, jump problem, Schiffer operator, transmission, quasisymmetry, quasicircles

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