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Sard’s theorem for mappings in Hölder and Sobolev spaces. (English) Zbl 1098.46024

Let \(M^m\) and \(N^n\) be smooth Riemannian manifolds of dimension \(m\), respectively, \(n\). The classical theorem of Sard is the following: Let \(f:M^m\longrightarrow N^n\) be of class \(C^k\), and let \(S=\text{ Crit}f\). If \(k>\text{ max}(m-n,0)\), then \({\mathcal{H}}^n(f(S))=0\), where \({\mathcal{H}}^s\) denotes the \(s\)-dimensional Hausdorff measure and \(\text{ Crit}f:=\{x\in M^m\mid \text{ rank}~ Df(x)<n\}\) denotes the set of critical points of \(f\).
In this paper the authors prove various generalizations of Sard’s classical theorem to mappings \(f:M^m\longrightarrow N^n\) between manifolds in Hölder and Sobolev classes. It turns out that if \(f\in C^{k,\lambda}(M^m,N^n)\), then, for arbitrary \(k\) and \(\lambda\), one can obtain estimates of the Hausdorff measure of the set of critical points in a typical level set \(f^{-1}(y)\). The estimates from this paper contain Sard’s theorem (and improvements due to Dubovitskii and Bates) as special cases. The structure of \(f^{-1}(y)\) is described for Sobolev mappings between manifolds.

MSC:

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
58C25 Differentiable maps on manifolds
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