A single-point Reshetnyak's theorem

Abstract

We prove a single-value version of Reshetnyak's theorem. Namely, if a non-constant map $f\in W_{\text{loc}}^{1,n}(\Omega ,{\mathbb{R}}^n)$ from a domain $\Omega \subset {\mathbb{R}}^n$ satisfies the estimate $|Df(x){|}^n\le K{J}_f(x)+\Sigma (x)|f(x)-y_0{|}^n$ for some $K\ge 1$, $y_0\in {\mathbb{R}}^n$ and $\Sigma \in L_{\text{loc}}^{1+\epsilon }(\Omega )$, 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 Calderón problem.

19 pages, 1 figure

References in corpus (2)

• On the heterogeneous distortion inequality
• Corrigendum to "On the heterogeneous distortion inequality"