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<p>Sobolev mappings exhibiting only pointwise quasiregularity-type bounds have arisen in various applications, leading to a recently developed theory of quasiregular values. In this article, we show that by using rescaling, one obtains a direct bridge between this theory and the classical theory of quasiregular maps. More precisely, we prove that a non-constant mapping <i>f</i>: Ω → ℝ<sup><i>n</i></sup> with a (<i>K</i>, Σ)-quasiregular value at <i>f</i>(<i>x</i><sub>0</sub>) can be rescaled at <i>x</i><sub>0</sub> to a non-constant <i>K</i>-quasiregular mapping. Our proof of this fact involves establishing a quasiregular-values version of the linear distortion bound of quasiregular mappings. A quasiregular values variant of the small <i>K</i>-theorem is obtained as an immediate corollary of our main result.</p>

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Linear distortion and rescaling for quasiregular values

  • 期刊论文

  • 发表日期：

    2026年4月19日

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  • Ilmari Kangasniemi,
  • Jani Onninen

摘要

Sobolev mappings exhibiting only pointwise quasiregularity-type bounds have arisen in various applications, leading to a recently developed theory of quasiregular values. In this article, we show that by using rescaling, one obtains a direct bridge between this theory and the classical theory of quasiregular maps. More precisely, we prove that a non-constant mapping f: Ω → ℝn with a (K, Σ)-quasiregular value at f(x0) can be rescaled at x0 to a non-constant K-quasiregular mapping. Our proof of this fact involves establishing a quasiregular-values version of the linear distortion bound of quasiregular mappings. A quasiregular values variant of the small K-theorem is obtained as an immediate corollary of our main result.