On some universal Morse-Sard type Theorem

Abstract

The classical Morse--Sard theorem claims that for a mapping $v:\mathbb R^n\to\mathbb R^{m+1}$ of class $C^k$ the measure of critical values $v(Z_{v,m})$ is zero under condition $k\ge n-m$. Here the critical set, or $m$-critical set is defined as $Z_{v,m} = \{ x \in \mathbb R^n : \, {\rm rank}\,\nabla v(x)\le m \}$. Further Dubovitski\u{\i} in 1957 and independently Federer and Dubovitski\u{\i} 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 $C^k$ category. Here we formulate and prove a \textit{bridge theorem} that includes all the above results as particular cases: namely, if a function $v:\mathbb R^n\to\mathbb R^d$ belongs to the Holder class $C^{k,\alpha}$, $0\le\alpha\le1$, then for every $q>m$ the identity $$\mathcal H^{\mu}(Z_{v,m}\cap v^{-1}(y))=0$$ holds for $\mathcal H^q$-almost all $y\in\mathbb R^d$, where $\mu=n-m-(k+\alpha)(q-m)$. The result is new even for the classical $C^k$-case (when $\alpha=0$); a similar result is established for the Sobolev classes of mappings $W^k_p(\mathbb R^n,\mathbb R^d)$ 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).