A sharp stability inequality of Liouville's theorem for quasiregular mappings on bounded domains

Abstract

Let $n\ge 3$ and fix $p>n$. We establish a sharp quantitative stability result in $W^{1,p}$ for Liouville's theorem for nonconstant, sense-preserving quasiregular mappings whose outer distortion $K$ is sufficiently close to one. We first reorganize Reshetnyak's classical local argument \cite{R1976}, drawing systematically on modern techniques such as those developed in \cite{FZ2022}. We then give two globalization schemes on bounded, connected domains. For bounded John domains, instead of Reshetnyak's original construction \cite{R19762}, which glues Möbius transformations on adjacent Whitney cubes, we use Whitney chains and Bojarski's enlarged-cube estimate to obtain a quantitatively sharp Euclidean stability result. Finally, we extend this idea to general bounded, connected open subsets of ${\mathbb{R}}^n$ via the Lorentzian representation of the Möbius group, yielding a compactified, weighted stability theorem whose underlying measure is absolutely continuous with respect to Lebesgue measure and has density bounded above by one.

We compress the original proof of Reshetnyak from 200+ pages to 84 pages. While preparing this manuscript, the author employed AI to improve the writing and to help draft technical and routine proofs, particularly those included in the Appendix. The core concepts, theoretical insights, and interpretations were developed and articulated solely by the author