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Global smoothness of quasiconformal mappings in the Triebel-Lizorkin scale

Kari Astala, Martí Prats, Eero Saksman

TL;DR

This work resolves global regularity questions for planar quasiconformal mappings in Triebel–Lizorkin, Besov, and Sobolev scales by linking boundary geometry to Beltrami-coefficient smoothness. The authors deploy a Fredholm framework built around the Beurling transform restricted to a domain and use Stoilow factorization to relate domain-regularity to the regularity of principal and conformal components, proving sharp results: if the domain boundary is a $B^{s+1-1/p}_{p,p}$-domain and the Beltrami coefficient satisfies $\mu\in F^s_{p,q}(\Omega)$ (with $sp>2$ and $\|\mu\|_\infty<1$), then the associated quasiconformal map $f$ belongs to $F^{s+1}_{p,q}(\Omega)$; analogues hold for Sobolev and (subcritical) Triebel–Lizorkin scales for principal mappings. The analysis introduces Dorronsoro's beta coefficients to measure boundary flatness, establishes polynomial control of Beurling iterates, and proves compactness of the commutator and Beurling reflection, enabling a robust Fredholm invertibility argument. The results extend the understanding of how boundary regularity and coefficient smoothness govern global regularity, including principal mappings in the Triebel–Lizorkin setting, with broad implications for conformal and quasiconformal mapping theory on Besov domains.

Abstract

We study quasiconformal mappings in planar domains $Ω$ 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 $\partial Ω$ 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 $Ω$ for the principal solutions with Beltrami coefficient supported in $Ω$.

Global smoothness of quasiconformal mappings in the Triebel-Lizorkin scale

TL;DR

This work resolves global regularity questions for planar quasiconformal mappings in Triebel–Lizorkin, Besov, and Sobolev scales by linking boundary geometry to Beltrami-coefficient smoothness. The authors deploy a Fredholm framework built around the Beurling transform restricted to a domain and use Stoilow factorization to relate domain-regularity to the regularity of principal and conformal components, proving sharp results: if the domain boundary is a -domain and the Beltrami coefficient satisfies (with and ), then the associated quasiconformal map belongs to ; analogues hold for Sobolev and (subcritical) Triebel–Lizorkin scales for principal mappings. The analysis introduces Dorronsoro's beta coefficients to measure boundary flatness, establishes polynomial control of Beurling iterates, and proves compactness of the commutator and Beurling reflection, enabling a robust Fredholm invertibility argument. The results extend the understanding of how boundary regularity and coefficient smoothness govern global regularity, including principal mappings in the Triebel–Lizorkin setting, with broad implications for conformal and quasiconformal mapping theory on Besov domains.

Abstract

We study quasiconformal mappings in planar domains 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 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 , the same conditions give global regularity in for the principal solutions with Beltrami coefficient supported in .

Paper Structure

This paper contains 17 sections, 45 theorems, 192 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Structure of the paper
  3. Preliminaries
  4. Notation
  5. Definitions of function spaces
  6. Besov domains and auxiliary results on Besov spaces
  7. Properties of Triebel-Lizorkin spaces
  8. Regularity of the Riemann mapping
  9. Quasiconformal mappings on domains
  10. Proof of Theorem
  11. Proof of Theorem modulo key auxiliary results
  12. Dorronsoro's Betas
  13. Boundedness of the truncated iterates
  14. Compactness of the commutator
  15. Beurling Reflection
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $s > 0$ and $1 < p < \infty$ with $sp>2$. Suppose $\Omega, \Omega'$ are simply connected, bounded $B^{s+1-\frac{1}{p}}_{p,p}$-domains and $f:\Omega \to\Omega'$ is a quasiconformal mapping, with $\mu_f\in W^{s,p}(\Omega)$. Then $f\in W^{s+1,p}(\Omega)$.

Figures (4)

  • Figure 1.1: Stoilow decomposition and associated Riemann mappings when $\Omega' = \Omega$.
  • Figure 1.2: Light green represents the results conjectured in , dark green stands for the known results, red is for the range of indices where the result is known to be false, and light red where it is conjectured to fail: In case $s\in{\mathbb N}$ (see PratsQuasiconformal) or $s<1$ (see Theorem above), Conjecture holds. In case $\frac{2}{p}-1<s<\frac{2}{p}$ there are radial stretchings which are counterexamples, see ClopFaracoMateuOrobitgZhong.
  • Figure 5.1: The minimal normalized area comprised between a degree 2 polynomial and the function $f$ in the interval $3I$ is $\beta_{(2)}(f,I)$.
  • Figure 5.2: The candidate intervals in $\mathcal{J}_Q$ can be identified with the bold red segments, see Definition , which correspond to only a bounded number of different local parameterizations of the boundary.

Theorems & Definitions (89)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Conjecture 1.6: see PratsTesi
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • ...and 79 more