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

On the heterogeneous distortion inequality