Quasisymmetric graphs and Zygmund functions

TL;DR

This paper explores the relationship between quasisymmetric graphs, solutions to the reduced Beltrami equation, and a generalized Zygmund class, enabling the construction of new examples of quasiconformal maps using harmonic analysis tools.

Contribution

It establishes a novel connection between quasisymmetric graphs, Beltrami equations, and a generalized Zygmund class, facilitating new constructions in quasiconformal mapping theory.

Findings

Linked quasisymmetric graphs to solutions of the reduced Beltrami equation

Connected the class of quasisymmetric graphs to a generalized Zygmund class

Used harmonic analysis to construct nontrivial examples of quasiconformal maps

Abstract

A quasisymmetric graph is a curve whose projection onto a line is a quasisymmetric map. We show that this class of curves is related to solutions of the reduced Beltrami equation and to a generalization of the Zygmund class $\Lambda_*$. This relation makes it possible to use the tools of harmonic analysis to construct nontrivial examples of quasisymmetric graphs and of quasiconformal maps.