Anton Ottosson: Characterization of quasisymmetric circle homeomorphisms using the welding Grunsky matrix

Abstract: In a recent preprint, Fan, Viklund, and Wang introduced the welding Grunsky matrix \(\Lambda_\varphi\) of an orientation-preserving circle homeomorphism \(\varphi\). Among other results, they showed that \(\Lambda_\varphi\) defines a bounded linear operator on the homogeneous Sobolev space \(\dot H^{1/2}(\mathbb S^1)\) whenever \(\varphi\) is quasisymmetric, and in this case they were able to relate it to the composition operator \(C_\varphi\) introduced by Nag and Sullivan in 1995. In this talk I will discuss my proof of the converse statement, that quasisymmetry is a necessary condition for \(\Lambda_\varphi\) to define such an operator, after providing the necessary background material on quasiconformal mappings, conformal welding, and the composition operator.