manuscripta math. 118, 383–397 (2005) © Springer-Verlag 2005 Bogdan Bojarski · Piotr Hajlasz · Pawel Strzelecki Sard’s theorem for mappings in Hölder and Sobolev spaces Received: 8 February 2005 / Revised version: 8 July 2005 Published online: 5 October 2005 Abstract. We prove various generalizations of classical Sard’s theorem to mappings f : M m → N n between manifolds in Hölder and Sobolev classes. It turns out that if f ∈ C k,λ (M m ,N n ), then—for arbitrary k and λ—one can obtain estimates of the Hausdorff measure of the set of critical points in a typical level set f -1 (y). The classical theorem of Sard holds true for f ∈ C k with sufficiently large k, i.e., k> max(m - n, 0); our estimates contain Sard’s theorem (and improvements due to Dubovitski˘ ı and Bates) as special cases. For Sobolev mappings between manifolds, we describe the structure of f -1 (y). 1. Introduction Throughout the paper we assume that M m and N n are smooth Riemannian mani- folds of dimension m and n respectively. In 1942 A. Sard [22] (see also Sternberg’s book [23]) proved the following theorem. Theorem 1.1. Let f : M m → N n be of class C k , and let S = Crit f . If k> max(m - n, 0), then H n (f (S)) = 0. Here and in the sequel H s denotes the s -dimensional Hausdorff measure (we shall follow the convention that H s ≡ the counting measure for all s ≤ 0) and, for a C 1 mapping f : M m → N n , Crit f : ={x ∈ M m | rank Df (x) < n} denotes the set of critical points of f . All the authors were partially supported by KBN grant no 2 P03A 028 22. Piotr Hajlasz was also supported by NSF grant DMS-0500966. B. Bojarski: Institute of Mathematics, Polish Academy of Sciences, ul. ´ Sniadeckich 8, 00–950 Warszawa, Poland. e-mail: bojarski@impan.gov.pl P. Hajlasz: Department of Mathematics, University of Pittsburgh, 301 Thackeray Hall, Pittsburgh, PA 15260, USA. e-mail: hajlasz@pitt.edu P. Strzelecki: Institute of Mathematics, Warsaw University, ul. Banacha 2, 02–097 Warsz- awa, Poland. e-mail: pawelst@mimuw.edu.pl Mathematics Subject Classification (2000): Primary: 46E35 Secondary: 41A63, 41A80, 41A99, 31B15 DOI: 10.1007/s00229-005-0590-1