  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.
