Bounded continuous weakly K-quasiregular mappings in W^{1,p} for p < nK/(K+1) exist that are not quasiregular, showing the almost-everywhere Jacobian sign condition is insufficient for orientation preservation below W^{1,n}.
Liu,Another proof of the Liouville theorem.Ann
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Bounded Continuous weak quasiregular mappings that fail to be quasiregular
Bounded continuous weakly K-quasiregular mappings in W^{1,p} for p < nK/(K+1) exist that are not quasiregular, showing the almost-everywhere Jacobian sign condition is insufficient for orientation preservation below W^{1,n}.