A single-point Reshetnyak's theorem
Ilmari Kangasniemi, Jani Onninen
TL;DR
This work develops a pointwise, single-value version of Reshetnyak's theorem for maps $f\in W^{1,n}_{\text{loc}}(\Omega,\mathbb{R}^n)$ that satisfy a heterogeneous distortion inequality $|Df|^n \le KJ_f + \Sigma|f-y_0|^n$ with $\Sigma\in L^{1+\varepsilon}_{\text{loc}}$. The main result shows that, for a fixed $y_0$, either $f\equiv y_0$ or the preimage $f^{-1}\{y_0\}$ is discrete with positive local indices and local openness near preimages; this yields a higher-dimensional analogue of Reshetnyak's theorem and connects to a Liouville-type theorem and a Calderón-problem–related argument principle. A key part of the argument uses a degree-theoretic framework for maps with totally disconnected fibers, establishing positivity of the local degree for small preimage components and ruling out negative or zero degrees via a local regularity analysis. The paper also proves an equivalence: if the pointwise inequality holds for every $y_0$ with some $\Sigma_{y_0}\in L^{1+\varepsilon}_{\text{loc}}$, then $f$ is $K$-quasiregular, and provides sharp examples illustrating the necessity of the integrability condition and the nuanced behavior of quasiregular values. Overall, the results extend quasiregular theory to a pointwise setting and yield tools applicable to inverse problems like Calderón’s problem in higher dimensions.
Abstract
We prove a single-value version of Reshetnyak's theorem. Namely, if a non-constant map $f \in W^{1,n}_{\text{loc}}(Ω, \mathbb{R}^n)$ from a domain $Ω\subset \mathbb{R}^n$ satisfies the estimate $\lvert Df(x) \rvert^n \leq K J_f(x) + Σ(x) \lvert f(x) - y_0 \rvert^n $ for some $K \geq 1$, $y_0\in \mathbb{R}^n$ and $Σ\in L^{1+\varepsilon}_{\text{loc}}(Ω)$, then $f^{-1}\{y_0\}$ is discrete, the local index $i(x, f)$ is positive in $f^{-1}\{y_0\}$, and every neighborhood of a point of $f^{-1}\{y_0\}$ is mapped to a neighborhood of $y_0$. Assuming this estimate for a fixed $K$ at every $y_0 \in \mathbb{R}^n$ is equivalent to assuming that the map $f$ is $K$-quasiregular, even if the choice of $Σ$ is different for each $y_0$. Since the estimate also yields a single-value Liouville theorem, it hence appears to be a good pointwise definition of $K$-quasiregularity. As a corollary of our single-value Reshetnyak's theorem, we obtain a higher-dimensional version of the argument principle that played a key part in the solution to the Calderón problem.