On reduced Beltrami equations and linear families of quasiregular mappings

TL;DR

This paper investigates linear classes of planar quasiregular mappings, confirming a conjecture on reduced Beltrami equations and establishing the uniqueness of associated Beltrami equations for linear families.

Contribution

It provides a positive resolution to a conjecture on reduced Beltrami equations and proves a Wronsky-type theorem for linear Beltrami systems, demonstrating the uniqueness of related equations.

Findings

Confirmed a conjecture on reduced Beltrami equations

Proved a Wronsky-type theorem for linear Beltrami systems

Established the uniqueness of the Beltrami equation for linear quasiregular families

Abstract

This paper studies linear classes of planar quasiregular mappings. We give a positive answer to a conjecture of K. Astala, T. Iwaniec, and G. Martin (2009) on reduced Beltrami equations. Moreover, we use it to prove a Wronsky-type theorem for general linear Beltrami systems. This is a key to show that the associated Beltrami equation of a linear quasiregular family is unique.