On the heterogeneous distortion inequality

We study Sobolev mappings $f \in W_{\mathrm{loc}}^{1,n} (\mathbb{R}^n, \mathbb{R}^n)$, $n \ge 2$, that satisfy the heterogeneous distortion inequality $$\left|Df(x)\right|^n \leq K J_f(x) + \sigma^n(x) \left|f(x)\right|^n$$ for almost every $x \in \mathbb{R}^n$. Here $K \in [1, \infty)$ is a constant and $\sigma \geq 0$ is a function in $L^n_{\mathrm{loc}}(\mathbb{R}^n)$. Although we recover the class of $K$-quasiregular mappings when $\sigma \equiv 0$, the theory of arbitrary solutions is significantly more complicated, partly due to the unavailability of a robust degree theory for non-quasiregular solutions. Nonetheless, we obtain a Liouville-type theorem and the sharp H\"older continuity estimate for all solutions, provided that $\sigma \in L^{n-\varepsilon}(\mathbb{R}^n) \cap L^{n+\varepsilon}(\mathbb{R}^n)$ for some $\varepsilon >0$. This gives an affirmative answer to a question of Astala, Iwaniec and Martin.