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

Abstract

Abstract. We construct a Sobolev homeomorphisms F ∈ W 1,2 ((0, 1) 4 , R 4 ) which fails the 2-dimensional Lusin's condition on H 2 -positively many hyperplanes, i.e. there exists C 1 ⊂ [0, 1] 2 with H 2 (C 1 ) > 0, such that for each (z, w) ∈ C 1 there is a set A (z,w) ⊂ [0, 1] 2 with H 2 (A (z,w) ) = 0 and H 2 F (A (z,w) × {(z, w)}) > 0.