A Sard type theorem for Borel mappings

• Autorzy: Piotr Hajłasz
• Języki publikacji: EN

Abstrakty

EN

We find a condition for a Borel mapping $f:ℝ^m → ℝ^n$ which implies that the Hausdorff dimension of $f^{-1}(y)$ is less than or equal to m-n for almost all $y ∈ ℝ^n$.

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Colloquium Mathematicum
• Rocznik: 1994
• Tom: 67
• Numer: 2
• Strony: 217-221

Daty

wydano

1994

otrzymano

1993-09-16

Twórcy

autor

Piotr Hajłasz

• Institute of Mathematics, Warsaw University, Banacha 2, 02-097 Warszawa, Poland

Bibliografia

• [1] B. Bojarski and P. Hajłasz, in preparation.
• [2] Y. Burago and V. Zalgaller, Geometric Inequalities, Grundlehren Math. Wiss. 285, Springer, 1988.
• [3] S. Eilenberg, On $ϕ$-measures, Ann. Soc. Polon. Math. 17 (1938), 252-253.
• [4] S. Eilenberg and O. Harold, Continua of finite linear measure I, Amer. J. Math. 65 (1943), 137-146.
• [5] H. Federer, Geometric Measure Theory, Springer, 1969.
• [6] P. Hajłasz, A note on weak approximation of minors, Ann. Inst. H. Poincaré Anal. Non Linéaire, to appear.
• [7] P. Hajłasz, Sobolev mappings, co-area formula and related topics, preprint.
• [8] T. Jech, Set Theory, Acad. Press, 1978.
• [9] K. Kuratowski, Topology, Vol. 1, Acad. Press, 1966.
• [10] N. Lusin, Sur les ensembles analytiques, Fund. Math. 10 (1927), 1-95.
• [11] N. Lusin et W. Sierpiński, Sur quelques propriétés des ensembles \rm (A), Bull. Acad. Cracovie 4-5A (1918), 35-48.
• [12] P. Mattila, Hausdorff dimension, orthogonal projections and intersections with planes, Ann. Acad. Sci. Fenn. Ser. A I Math. 1 (1975), 227-244.
• [13] J. Milnor, Topology from the Differentiable Viewpoint, The Univ. Press of Virginia, 1965.
• [14] A. Sard, The measure of the critical values of differentiable maps, Bull. Amer. Math. Soc. 48 (1942), 883-890.
• [15] S. Sternberg, Lectures on Differential Geometry, Prentice-Hall, Englewood Cliffs, N.J., 1964.