Diffeomorphic approximation of planar Sobolev homeomorphisms in rearrangement invariant spaces

Let Ω ⊆ ℝ2 be a domain, let X be a rearrangement invariant space and let f ∈ W 1 X (Ω, ℝ2 ) be a homeomorphism between Ω and f (Ω). Then there exists a sequence of diffeomorphisms f k converging to f in the space W 1 X (Ω, ℝ2 ).