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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
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Abstract: For a $\mathcal{C}^1$-smooth bump function $b \colon {\mathbb R}^{2} \to \mathbb{R}$ we show that the gradient range $\nabla b( {\mathbb R}^{2} )$ is the closure of its interior, provided that $\nabla b$ admits a modulus of continuity $\omega = \omega (t)$ satisfying $\omega (t)/\sqrt{t} \to 0$ as $t \searrow 0$. The result is a consequence of a more general result about gradient ranges of bump functions $b \colon {\mathbb R}^{n} \to \mathbb{R}$of the same degree of smoothness. For such bump functions we show that for open sets $G \subset {\mathbb R}^{n}$, either the intersection $\nabla b( {\mathbb R}^{n}) \cap G$ 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 $n$ of the space.

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Additional Information

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

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

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