A bridge between Dubovitskii - Federer theorems and the coarea formula

Abstract

The Morse-Sard theorem requires that a mapping $v:R^n \to R^m$ is of class $C^k$, $k>n-m$. In 1957 Dubovitski\u{\i} generalized this result by proving that almost all level sets for a $C^k$ mapping have $H^s$-negligible intersection with its critical set, where $s=\max(n-m-k+1,0)$. Here the critical set, or $m$-critical set is defined as $Z_{v,m} = \{ x \in R^n : {\rm rank} \nabla v(x) < m \}$. Another generalization was obtained independently by Dubovitski\u{\i} and Federer in 1966, namely for $C^k$ mappings $v:R^n\to R^d$ and integers $m\le d$ they proved that the set of $m$-critical values $v(Z_{v,m})$ is $H^{b}$-negligible for $b= m-1+\frac{n-m+1}{k}$. They also established the sharpness of these results within the $C^k$ category. Here we prove that Dubovitski\u{\i}'s theorem can be generalized to the case of continuous mappings of the Sobolev-Lorentz class $W^{k}_{p,1}(R^n,R^d )$, $p=\frac{n}k$ (this is the minimal integrability assumption that guarantees the continuity of mappings). In this situation the mappings need not be everywhere differentiable and in order to handle the set of nondifferentiability points, we establish for such mappings an analog of the Luzin $N$-property with respect to lower dimensional Hausdorff content. Finally, we formulate and prove a~${\rm bridge\ theorem}$ that includes all the above results as particular cases. This result is new also for smooth mappings but is presented here in the general Sobolev context. The proofs of the results are based on our previous joint papers with J.~Bourgain (2013, 2015). Note, that in this paper some result concerning the Coarea formula was not formulated accurately. Now we put an Addendum consisting of three parts: first, we describe the accurate formulation of this result, then we give some historical remarks, and finally its relation to other results of the paper.