A Simple Proof of the Morse-sard Theorem in Sobolev Spaces

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 MorseSard Theorem with Ck 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 Ω ⊂ R be open and let f : Ω → R be a C function. A point x ∈ Ω is said to be a critical point if Df(x) is not of maximum rank. A point y ∈ f(Ω) is said to be a critical value if y = f(x) for a critical point x. The set of all the critical points is called the critical set. Let us denote by L the m-dimensional Lebesgue measure. We can now recall the classical Morse-Sard Theorem (for a proof, see [1, Paragraph 15]): Theorem 2 (Morse-Sard). Let Ω ⊂ R be open and let f : Ω → R be a Cn−m+1 function, with n ≥ m (C if m > n). Then the set of critical values of f has L-measure zero. After that theorem, many generalizations have been proved and, at the same time, many counterexamples have been found in the case of not sufficient regularity. In particular, in [2] the same conclusion of the Morse-Sard Theorem has been proved under the only assumption of a Cn−m,1 regularity, while in [3] only a Wn−m+1,p regularity, with p > n, is assumed (see [3] for more historical notes). Here we give a simple proof of the result in [3]. We remark that, as our proof is independent of Theorem 2, our result implies the classical Morse-Sard Theorem. In the proof of our theorem, we will need a refined version of the classical Morrey inequality (for a proof, see [4, paragraph 4.5.3]): Lemma 3. Let Ω ⊂ R be an open subset and let B(x, r) be a ball contained in Ω. Then for any y ∈ B(x, r) we have (1) |u(x)− u(y)| ≤ Cr1− n p (∫ B(x,r) |Du(z)| dz ) 1 p ∀u ∈ W . Received by the editors April 6, 2006, and, in revised form, June 20, 2006. 2000 Mathematics Subject Classification. Primary 58C25; Secondary 46T20.