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We construct a Sobolev homeomorphisms F ∈ W ((0, 1),R) which fails the 2-dimensional Lusin’s condition on H-positively many hyperplanes, i.e. there exists C1 ⊂ [0, 1] with H(C1) > 0, such that for each (z, w) ∈ C1 there is a set A(z,w) ⊂ [0, 1] with H(A(z,w)) = 0 and H ( F (A(z,w) × {(z, w)}) ) > 0.

Sobolev homeomorphisms in W^1,k and the Lusin's condition (N) on k-dimensional subspaces