The Dubovitskiı-Sard Theorem in Sobolev Spaces

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The Sard theorem from 1942 requires that a mapping f:Rⁿ → Rᵐ is of class Cᵏ, k > (n-m,0). In 1957 Duvovitskiı generalized Sard's theorem to the case of Cᵏ mappings for all k. Namely he proved that, for almost all y∈ Rᵐ, Hℓ(Cf ∩ f⁻¹(y))=0 where ℓ = (n-m-k+1,0), Hℓ denotes the Hausdorff measure, and Cf is the set of critical points of f. In 2001 De Pascale proved that the Sard theorem holds true for Sobolev mappings of the class W locᵏ,ᵖ(Rⁿ,Rᵐ), k>(n-m,0) and p>n. We will show that also Dubovitskiı's theorem can be generalized to the case of W locᵏ,ᵖ(Rⁿ,Rᵐ) mappings for all k and p>n.