A simple proof of the Morse-Sard theorem in Sobolev spaces

Alessio Figalli
2008 Proceedings of the American Mathematical Society  
In this paper we give a new simple proof of a result of Luigi De Pascale, which states that the Morse-Sard Theorem holds under the hypothesis of Sobolev regularity. Moreover, as our proof is independent of the Morse-Sard Theorem with C k regularity, our result implies the classical Morse-Sard Theorem. The Morse-Sard Theorem is concerned with the size of the image of the critical values of a differentiable function. To recall it and to state our result, we need some definitions. Definition 1 Let
more » ... s. Definition 1 Let Ω ⊂ R n be open and let for a critical point x. The set of all the critical points is called the critical set. Let us denote by L m the m-dimensional Lebesgue measure. We can now recall the classical Morse-Sard Theorem (for a proof, see [1, Paragraph 15]): *
doi:10.1090/s0002-9939-08-09321-0 fatcat:m7qgqivcxneq7jqzu5ssvbl5r4