Gradient ranges of bumps on the plane

Authors:
Jan Kolár and Jan Kristensen

Journal:
Proc. Amer. Math. Soc. **133** (2005), 1699-1706

MSC (2000):
Primary 26B05; Secondary 46G05

Published electronically:
December 20, 2004

MathSciNet review:
2120251

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: For a -smooth bump function we show that the gradient range is the closure of its interior, provided that admits a modulus of continuity satisfying as . The result is a consequence of a more general result about gradient ranges of bump functions of the same degree of smoothness. For such bump functions we show that for open sets , either the intersection is empty or its topological dimension is at least two. The proof relies on a new Morse-Sard type result where the smoothness hypothesis is independent of the dimension of the space.

**1.**Daniel Azagra and Mar Jiménez-Sevilla,*On the size of the sets of gradients of bump functions and starlike bodies on the Hilbert space*, Bull. Soc. Math. France**130**(2002), no. 3, 337–347. MR**1943881****2.**M. FABIAN, O. KALENDA & J. KOLÁSR. Filling analytic sets by the derivatives of -smooth bumps. *Proc. Amer. Math. Soc.*, to appear.**3.**T. Gaspari,*On the range of the derivative of a real-valued function with bounded support*, Studia Math.**153**(2002), no. 1, 81–99. MR**1948929**, 10.4064/sm153-1-6**4.**Witold Hurewicz and Henry Wallman,*Dimension Theory*, Princeton Mathematical Series, v. 4, Princeton University Press, Princeton, N. J., 1941. MR**0006493****5.**Philip Hartman and Louis Nirenberg,*On spherical image maps whose Jacobians do not change sign*, Amer. J. Math.**81**(1959), 901–920. MR**0126812****6.**J. KOL´ASR & J. KRISTENSEN. The set of gradients of a bump. Max-Planck-Institute MIS, Leipzig, Preprint Nr. 64/2002. **7.**K. Kuratowski,*Topology. Vol. II*, New edition, revised and augmented. Translated from the French by A. Kirkor, Academic Press, New York-London; Państwowe Wydawnictwo Naukowe Polish Scientific Publishers, Warsaw, 1968. MR**0259835****8.**Elias M. Stein,*Singular integrals and differentiability properties of functions*, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR**0290095**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
26B05,
46G05

Retrieve articles in all journals with MSC (2000): 26B05, 46G05

Additional Information

**Jan Kolár**

Affiliation:
Department of Mathematical Analysis, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75 Praha 8, Czech Republic

Email:
kolar@karlin.mff.cuni.cz

**Jan Kristensen**

Affiliation:
Mathematical Institute, 24-29 St Giles’, University of Oxford, Oxford OX1 3LB, United Kingdom

Email:
kristens@maths.ox.ac.uk

DOI:
http://dx.doi.org/10.1090/S0002-9939-04-07747-0

Keywords:
Gradient range,
derivative,
bump,
Morse-Sard theorem

Received by editor(s):
November 5, 2002

Received by editor(s) in revised form:
February 2, 2004

Published electronically:
December 20, 2004

Communicated by:
David Preiss

Article copyright:
© Copyright 2004
American Mathematical Society