A single-point Reshetnyak's theorem

Abstract

We prove a single-value version of Reshetnyak's theorem. Namely, if a non-constant map $f \in W^{1,n}_{\text{loc}}(\Omega, \mathbb{R}^n)$ from a domain $\Omega \subset \mathbb{R}^n$ satisfies the estimate $\lvert Df(x) \rvert^n \leq K J_f(x) + \Sigma(x) \lvert f(x) - y_0 \rvert^n $ for some $K \geq 1$, $y_0\in \mathbb{R}^n$ and $\Sigma \in L^{1+\varepsilon}_{\text{loc}}(\Omega)$, 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 $\Sigma$ 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 Calder\'on problem.