On the image size of singular maps. I

Author:
S. M. Bates

Journal:
Proc. Amer. Math. Soc. **114** (1992), 699-705

MSC:
Primary 58C25; Secondary 26B05

DOI:
https://doi.org/10.1090/S0002-9939-1992-1074748-8

MathSciNet review:
1074748

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Abstract: For fixed integers , we establish a sharp upper bound for the Hölder differentiability class of singular maps of subrank whose images have positive Lebesgue -measure.

**[Bates]***Towards a precise smoothness hypothesis in Sard's thoerem*, Proc. Amer. Math. Soc. (to appear).**[Falconer, 1985]***The geometry of fractal sets*, Cambridge Tracts in Math. 85, Cambridge Univ. Press, Cambridge and New York.**[Federer, 1969]***Geometric measure theory*, Grundlehren Math. Wiss., vol. 153, Springer-Verlag, New York and Berlin. MR**0257325 (41:1976)****[Hirsch, 1976]***Differential topology*, Graduate Texts in Math. vol. 33, Springer, New York and Berlin. MR**0448362 (56:6669)****[Kaufman, 1979]***A singular map of a cube onto a square*, J. Differential Geom.**14**, 593-594. MR**600614 (82a:26013)****[Norton, 1986]***A critical set with nonnull image has large Hausdorff dimension*, Trans. Amer. Math. Soc.**296**, 367-376. MR**837817 (87i:26011)****[Norton, 1987]***The fractal geometry of critical sets with nonnull image and the differentiablity of functions*, Ph.d. Thesis, University of California, Berkeley.**[Sternberg, 1964]***Lectures on differential geometry*, Prentice-Hall, Englewood Cliffs, NJ. MR**0193578 (33:1797)****[Whitney, 1935]***A function not constant on a connected set of critical points*, Duke Math J.**1**, 514-517. MR**1545896**

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DOI:
https://doi.org/10.1090/S0002-9939-1992-1074748-8

Article copyright:
© Copyright 1992
American Mathematical Society