A sharp stability inequality of Liouville's theorem for quasiregular mappings on bounded domains
Abstract
Let n ≥ 3 and fix p>n. We establish a sharp quantitative stability result in W1,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 ℝn 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.
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BibTeX
@misc{zhang2026sharp,
title = {{A sharp stability inequality of Liouville's theorem for quasiregular mappings on bounded domains}},
author = {Yi Ru-Ya Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.24041},
eprint = {2609.24041},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.24041},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A sharp stability inequality of Liouville's theorem for quasiregular mappings on bounded domains
AU - Yi Ru-Ya Zhang
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2609.24041
UR - https://arxiv.org/abs/2609.24041
AB - Let n ≥ 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 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.
ER -