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

Abstract

For a regular (in a sense) mapping $v:\mathbb{R}^n \to \mathbb{R}^d$ we study the following problem: {\sl let $S$ be a subset of $m$-critical a set $\tilde Z_{v,m}=\{{\rm rank} \nabla v\le m\}$ and the equality $\mathcal{H}^\tau(S)=0$ (or the inequality $\mathcal{H}^\tau(S)<\infty$) holds for some $\tau>0$. Does it imply that $\mathcal{H}^{\sigma}(v(S))=0$ for some $\sigma=\sigma(\tau,m)$?} (Here $\mathcal{H}^\tau$ means the $\tau$-dimensional Hausdorff measure.) For the classical classes $C^k$-smooth and $C^{k+\alpha}$-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 $W^{k+\alpha}_p$ 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 {\it the continuity} (not everywhere differentiability) of a mapping. In particular, there is an interesting and unexpected analytical phenomena here: if $\tau=n$ (i.e., in the case of Morse--Sard theorem), then the value $\sigma(\tau)$ is the same for the Sobolev $W^k_p$ and for the classical $C^k$-smooth case. But if $\tau<n$, then the value $\sigma$ depends on $p$ also; the value $\sigma$ for $C^k$ case could be obtained as the limit when $p\to\infty$. The similar phenomena holds for Holder continuous $C^{k+\alpha}$ and for the fractional Sobolev $W^{k+\alpha}_p$ classes. 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).