Values of finite distortion: Reshetnyak's theorem and the Lusin (N) -property

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. We establish a single-value analogue of Reshetnyak's theorem in this setting. Specifically, if $f$ is nonconstant and has a value of finite distortion at $y_0$, with $K \in L^p_{\text{loc}}(Ω) $, $Σ/K \in L^q_{\text{loc}}(Ω)$, $p>n-1$, and $p^{-1}+q^{-1}<1$, then the preimage $f^{-1}\{y_0\}$ is discrete, the local topological index is positive at every point of $f^{-1}\{y_0\}$, and every neighborhood of a point in $f^{-1}\{y_0\}$ is mapped onto a neighborhood of $y_0$. We also prove that mappings satisfying a more general distortion inequality with defect preserve sets of Lebesgue measure zero.