Sard's theorem for mappings in Hölder and Sobolev spaces

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 Dubovitskii and Bates) as special cases. For Sobolev mappings between manifolds, we describe the structure of f ^?1(y).