On some universal Morse–Sard type theorems

The classical Morse–Sard theorem claims that for a mapping v:from Rn to Rm+1 of class C^k the measure of critical values v(Zv,m) is zero under condition k≥n−m. Here the critical set, or m-critical set is defined as Zv,m={x∈Rn:rank(Nabla v)(x)≤m}. Further Dubovitskiĭ in 1957 and independently Federer and Dubovitskiĭ in 1967 found some elegant extensions of this theorem to the case of other (e.g., lower) smoothness assumptions. They also established the sharpness of their results within the Ck category. Here we formulate and prove a bridge theorem that includes all the above results as particular cases: namely, if a function v:Rn→Rd belongs to the Hölder class Ck,alpha, 0≤alpha≤1, then for every q>m the identity H^mu(Zv,m \cap v−1(y))=0 holds for Hq-almost all y in Rd, where mu=n−m−(k+alpha)(q−m). Intuitively, the sense of this bridge theorem is very close to Heisenberg's uncertainty principle in theoretical physics: the more precise is the information we receive on measure of the image of the critical set, the less precisely the preimages are described, and vice versa. The result is new even for the classical Ck-case (when alpha=0); similar result is established for the Sobolev classes of mappings W_p^k(Rn,Rd) with minimal integrability assumptions p=max⁡(1,n/k), i.e., it guarantees in general only the continuity (not everywhere differentiability) of a mapping. However, using some N-properties for Sobolev mappings, established in our previous paper, we obtained that the sets of nondifferentiability points of Sobolev mappings are fortunately negligible in the above bridge theorem. We cover also the case of fractional Sobolev spaces. The proofs of the most results are based on our previous joint papers with J. Bourgain and J. Kristensen (2013, 2015). We also crucially use very deep Y. Yomdin's entropy estimates of near critical values for polynomials (based on algebraic geometry tools).