Morse–Sard theorem and Luzin N-property: a new synthesis result for Sobolev spaces

For a regular (in a sense) mapping v:ℝ^n →ℝ^d we study the following problem: let S be a subset of m-critical a set Z̃_v,m={ rank∇ v≤ m} and the equality ℋ^τ(S)=0 (or the inequality ℋ^τ(S)<∞) holds for some τ>0. Does it imply that ℋ^σ(v(S))=0 for some σ=σ(τ,m)? (Here ℋ^τ means the τ-dimensional Hausdorff measure.) For the classical classes C^k-smooth and C^k+α-Holder mappings this problem was solved in the papers by Bates and Moreira. We solve the problem for Sobolev W^k_p and fractional Sobolev W^k+α_p classes as well. Note that we study the Sobolev case under minimal integrability assumptions p=max(1,n/k), i.e., it guarantees in general only the continuity (not everywhere differentiability) of a mapping. In particular, there is an interesting and unexpected analytical phenomena here: if τ=n (i.e., in the case of Morse–Sard theorem), then the value σ(τ) is the same for the Sobolev W^k_p and for the classical C^k-smooth case. But if τ更多