Toward a precise smoothness hypothesis in Sard’s theorem

Toward a precise smoothness hypothesis in Sard’s theorem
HTML articles powered by AMS MathViewer

by S. M. Bates

Proc. Amer. Math. Soc. 117 (1993), 279-283

DOI: https://doi.org/10.1090/S0002-9939-1993-1112486-4

PDF | Request permission

The familiar ${{\mathbf {C}}^k}$ smoothness hypothesis in the Morse-Sard Theorem can be weakened to ${{\mathbf {C}}^{k - 1,1}}$.

References

S. M. Bates, On the image size of singular maps. I, Proc. Amer. Math. Soc. 114 (1992), no. 3, 699–705. MR 1074748, DOI 10.1090/S0002-9939-1992-1074748-8

On the image size of singular maps

Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York, Inc., New York, 1969. MR 0257325
Shaul Grinberg, On invariant linear functionals, Nederl. Akad. Wetensch. Indag. Math. 47 (1985), no. 3, 299–304. MR 814882, DOI 10.1016/1385-7258(85)90041-1
R. Kaufman, A singular map of a cube onto a square, J. Differential Geometry 14 (1979), no. 4, 593–594 (1981). MR 600614, DOI 10.4310/jdg/1214435238
Alec Norton, A critical set with nonnull image has large Hausdorff dimension, Trans. Amer. Math. Soc. 296 (1986), no. 1, 367–376. MR 837817, DOI 10.1090/S0002-9947-1986-0837817-2

The fractal geometry of critical sets with nonnull image and the differentiability of functions

Arthur Sard, The measure of the critical values of differentiable maps, Bull. Amer. Math. Soc. 48 (1942), 883–890. MR 7523, DOI 10.1090/S0002-9904-1942-07811-6
Shlomo Sternberg, Lectures on differential geometry, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. MR 0193578
Hassler Whitney, A function not constant on a connected set of critical points, Duke Math. J. 1 (1935), no. 4, 514–517. MR 1545896, DOI 10.1215/S0012-7094-35-00138-7
Y. Yomdin, The geometry of critical and near-critical values of differentiable mappings, Math. Ann. 264 (1983), no. 4, 495–515. MR 716263, DOI 10.1007/BF01456957