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

Let $\mathrm{\Omega }\subset {\mathbb{R}}^n$ be a domain and $f\in W_{\text{loc}}^{1,n}(\mathrm{\Omega },{\mathbb{R}}^n)$. We say that $f$ has a value of finite distortion at $y_0\in {\mathbb{R}}^n$ if there exist measurable functions $K\text{\hspace{0.17em}⁣}:\mathrm{\Omega }\to [0,\mathrm{\infty })$ and $\mathrm{\Sigma }\in L_{\text{loc}}^1(\mathrm{\Omega })$ such that $$\mid Df(x){\mid }^n\le K(x)\det Df(x)+\mathrm{\Sigma }(x)\mid f(x)-y_0{\mid }^n\text{for a.e. }x\in \mathrm{\Omega }\mathrm{.}$$ 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 $\mathrm{\Sigma }$, 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.