A bridge between Dubovitskiĭ–Federer theorems and the coarea formula

Abstract

The Morse–Sard theorem requires that a mapping v:Rn→Rm is of class Ck, k>max⁡(n−m,0). In 1957 Dubovitskiĭ generalized this result by proving that almost all level sets for a Ck mapping have Hs-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 Zv,m={x∈Rn:rank∇v(x)<m}. Another generalization was obtained independently by Dubovitskiĭ and Federer in 1966, namely for Ck mappings v:Rn→Rd and integers m≤d they proved that the set of m-critical values v(Zv,m) is Hq∘-negligible for q∘=m−1+n−m+1k. They also established the sharpness of these results within the Ck category.Here we prove that Dubovitskiĭ's theorem can be generalized to the case of continuous mappings of the Sobolev–Lorentz class Wp,1k(Rn,Rd), p=nk (this is the minimal integrability assumption that guarantees the continuity of mappings). In this situation the mappings need not to 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 bridge theorem that includes all the above results as particular cases. As a limiting case in this bridge theorem we also establish a new coarea type formula: if E⊂{x∈Rn:rank∇v(x)≤m}, then∫EJmv(x)dx=∫RdHn−m(E∩v−1(y))dHm(y). The mapping v is Rd-valued, with arbitrary d, and the formula is obtained without any restrictions on the image v(Rn) (such as m-rectifiability or σ-finiteness with respect to the m-Hausdorff measure). These last results are new also for smooth mappings, but are presented here in the general Sobolev context.The proofs of the results are based on our previous joint papers with J. Bourgain (2013, 2015).