On the heterogeneous distortion inequality

We study Sobolev mappings f∈Wloc1,n(Rn,Rn), n≥ 2 , that satisfy the heterogeneous distortion inequality |Df(x)|n≤KJf(x)+σn(x)|f(x)|nfor almost every x∈ Rⁿ. Here K∈ [1 , ∞) is a constant and σ≥ 0 is a function in Llocn(Rn). Although we recover the class of K-quasiregular mappings when σ≡ 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 Hölder continuity estimate for all solutions, provided that σ∈ L^n-ε(Rⁿ) ∩ L^n+ε(Rⁿ) for some ε> 0. This gives an affirmative answer to a question of Astala, Iwaniec and Martin.