The Zygmund Morse-sard Theorem

The classical Morse-Sard Theorem says that the set of critical values of f : R → R has Lebesgue measure zero if f ∈ Ck+1. We show the Ck+1 smoothness requirement can be weakened to Ck+Zygmund. This is corollary to the following theorem: For integers n > m > r > 0, let s = (n− r)/(m− r); if f : R → R belongs to the Lipschitz class Λs and E is a set of rank r for f , then f(E) has measure zero. 0. Introduction Let f : R → R be a differentiable function. Must the set of critical values of f have measure zero? The answer is “yes” provided that f is sufficiently smooth, and the classical theorem in this regard is the Morse-Sard theorem (often called simply “Sard’s theorem” — A.P. Morse [6] proved the theorem in 1939 for the real-valued case; A. Sard [10] then extended that result to the vector-valued case.) To state the theorem, we need some terminology. For f as above, a point x ∈ R is called a critical point if the linear mapping Df(x) is not surjective; a critical value is the image under f of a critical point. The set of all critical values is a subset of the target space R. The Morse-Sard Theorem. [6,10]. Let f : R → R be of class C. If k ≥ max{n−m+1, 1}, then the set of critical values of f has Lebesgue measure zero. We henceforth restrict our attention to the case n > m; smoothness is not an issue when n ≤ m. (In fact even measurability is not required if n ≤ m. See Varberg [12].) Prior to Morse’s work, H. Whitney had established in a famous paper [13] that some differentiability requirement is necessary by constructing an example of a C function f : R → R not constant on a connected set of critical points (and hence having a nontrivial interval of critical values). This example, and related ones in higher dimensions, shows that the smoothness hypothesis of the MorseSard theorem is sharp to the extent that one cannot replace the integer n−m+ 1 by any smaller integer. In fact, more is true: for every n,m with n > m > 0 there is a function f ∈ ⋂ α<1 C n−m,α(Rn,Rm) whose critical value set contains an open set [2,8]. (Notation: for α ∈ [0, 1), k a nonnegative integer, we say f ∈ Ck,α(R,R) if f : R → R is of class C and the kth derivative Df locally satisfies a Hölder condition with exponent α. We say f ∈ Ck,1(R,R) if the kth derivative is locally Lipschitz.) Typeset by AMS-TEX 1