Sard’s theorem for mappings in Hölder and Sobolev spaces
نویسندگان
چکیده
We prove various generalizations of classical Sard’s theorem to mappings f : M → N between manifolds in Hölder and Sobolev classes. It turns out that if f ∈ C(M,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 with sufficiently large k, i.e., k > max(m−n, 0); our estimates contain Sard’s theorem (and improvements due to Dubovitskiı̆ and Bates) as special cases. For Sobolev mappings between manifolds, we describe the structure of f−1(y).
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تاریخ انتشار 2005