On the image size of singular maps. I
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- by S. M. Bates PDF
- Proc. Amer. Math. Soc. 114 (1992), 699-705 Request permission
Abstract:
For fixed integers $n > m > {r_l} + 1 > 0$, we establish a sharp upper bound for the Hölder differentiability class of singular maps $f:{{\mathbf {R}}^n} \to {{\mathbf {R}}^m}$ of subrank ${r_l}$ whose images have positive Lebesgue $m$-measure.References
-
Towards a precise smoothness hypothesis in Sard’s thoerem, Proc. Amer. Math. Soc. (to appear).
The geometry of fractal sets, Cambridge Tracts in Math. 85, Cambridge Univ. Press, Cambridge and New York.
- Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York, Inc., New York, 1969. MR 0257325
- Morris W. Hirsch, Differential topology, Graduate Texts in Mathematics, No. 33, Springer-Verlag, New York-Heidelberg, 1976. MR 0448362
- R. Kaufman, A singular map of a cube onto a square, J. Differential Geometry 14 (1979), no. 4, 593–594 (1981). MR 600614
- 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 differentiablity of functions, Ph.d. Thesis, University of California, Berkeley.
- 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
Additional Information
- © Copyright 1992 American Mathematical Society
- 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