Global smoothness of quasiconformal mappings in the Triebel-Lizorkin scale

TL;DR

This paper investigates the regularity of quasiconformal mappings in planar domains within the Triebel-Lizorkin scale, establishing optimal geometric and smoothness conditions for global smoothness of these mappings.

Contribution

It provides new optimal conditions linking boundary geometry and Beltrami coefficient smoothness to global regularity in Triebel-Lizorkin spaces, extending previous results.

Findings

Identifies optimal boundary and Beltrami coefficient conditions for regularity

Establishes global regularity results in Triebel-Lizorkin spaces below smoothness 1

Applies to principal solutions with Beltrami coefficients supported in the domain

Abstract

We study quasiconformal mappings in planar domains $\Omega$ and their regularity properties described in terms of Sobolev, Bessel potential or Triebel-Lizorkin scales. This leads to optimal conditions, in terms of the geometry of the boundary $\partial \Omega$ and of the smoothness of the Beltrami coefficient, that guarantee the global regularity of the mappings in these classes. In the Triebel-Lizorkin class with smoothness below $1$, the same conditions give global regularity in $\Omega$ for the principal solutions with Beltrami coefficient supported in $\Omega$.