Values of finite distortion: Reshetnyak's theorem, the Liouville theorem, and the Lusin (N) -property
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arXiv
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| Format: | Preprint |
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2026
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| author | Kangasniemi, Ilmari Onninen, Jani Zhu, Yizhe |
| author_facet | Kangasniemi, Ilmari Onninen, Jani Zhu, Yizhe |
| contents | Let $Ω\subset \mathbb{R}^n$ be a domain and $f \in W^{1,n}_{\text{loc}} (Ω,\mathbb{R}^n)$. We say that $f$ has a value of finite distortion at $y_0 \in \mathbb{R}^n$ if there exist measurable functions $K \colon Ω\to [0,\infty)$ and $Σ\in L^1_{\text{loc}} (Ω)$ such that \[
\lvert Df(x) \rvert^n \le K(x) \det Df (x) + Σ(x) \lvert f(x)-y_0 \rvert^n
\quad \text{for a.e. } x \in Ω. \] This notion unifies the classical theory of mappings of finite distortion with the recently introduced theory of quasiregular values. Under sharp integrability assumptions on $K$ and $Σ$, we establish single-value analogues of Reshetnyak's theorem and the Liouville theorem. We also prove that mappings satisfying a more general distortion inequality with defect preserve sets of Lebesgue measure zero. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2602_03254 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Values of finite distortion: Reshetnyak's theorem, the Liouville theorem, and the Lusin (N) -property Kangasniemi, Ilmari Onninen, Jani Zhu, Yizhe Complex Variables 30C65 (Primary) 35R45, 46E35 (Secondary) Let $Ω\subset \mathbb{R}^n$ be a domain and $f \in W^{1,n}_{\text{loc}} (Ω,\mathbb{R}^n)$. We say that $f$ has a value of finite distortion at $y_0 \in \mathbb{R}^n$ if there exist measurable functions $K \colon Ω\to [0,\infty)$ and $Σ\in L^1_{\text{loc}} (Ω)$ such that \[ \lvert Df(x) \rvert^n \le K(x) \det Df (x) + Σ(x) \lvert f(x)-y_0 \rvert^n \quad \text{for a.e. } x \in Ω. \] This notion unifies the classical theory of mappings of finite distortion with the recently introduced theory of quasiregular values. Under sharp integrability assumptions on $K$ and $Σ$, we establish single-value analogues of Reshetnyak's theorem and the Liouville theorem. We also prove that mappings satisfying a more general distortion inequality with defect preserve sets of Lebesgue measure zero. |
| title | Values of finite distortion: Reshetnyak's theorem, the Liouville theorem, and the Lusin (N) -property |
| topic | Complex Variables 30C65 (Primary) 35R45, 46E35 (Secondary) |
| url | https://arxiv.org/abs/2602.03254 |