Linear distortion and rescaling for quasiregular values
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
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2024
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| _version_ | 1866909187399745536 |
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| author | Kangasniemi, Ilmari Onninen, Jani |
| author_facet | Kangasniemi, Ilmari Onninen, Jani |
| contents | 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 $f \colon Ω\to \mathbb{R}^n$ with a $(K, Σ)$-quasiregular value at $f(x_0)$ can be rescaled at $x_0$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_02073 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Linear distortion and rescaling for quasiregular values Kangasniemi, Ilmari Onninen, Jani Complex Variables Analysis of PDEs 30C65 (Primary) 35R45 (Secondary) 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 $f \colon Ω\to \mathbb{R}^n$ with a $(K, Σ)$-quasiregular value at $f(x_0)$ can be rescaled at $x_0$ 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. |
| title | Linear distortion and rescaling for quasiregular values |
| topic | Complex Variables Analysis of PDEs 30C65 (Primary) 35R45 (Secondary) |
| url | https://arxiv.org/abs/2404.02073 |