The Denjoy-Clarkson property with respect to Hausdorff measures for the gradient mapping of functions of several variables
- [1] Charles University Faculty of Mathematics and Physics Sokolovská 83 186 75 Praha 8 (Czech Republic)
Annales de l’institut Fourier (2008)
- Volume: 58, Issue: 2, page 405-428
- ISSN: 0373-0956
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topZelený, Miroslav. "The Denjoy-Clarkson property with respect to Hausdorff measures for the gradient mapping of functions of several variables." Annales de l’institut Fourier 58.2 (2008): 405-428. <http://eudml.org/doc/10320>.
@article{Zelený2008,
abstract = {We construct a differentiable function $f:\{\mathbf\{R\}\}^n \rightarrow \{\mathbf\{R\}\}$ ($n\ge 2$) such that the set $(\nabla f)^\{-1\}(B(0,1))$ is a nonempty set of Hausdorff dimension $1$. This answers a question posed by Z. Buczolich.},
affiliation = {Charles University Faculty of Mathematics and Physics Sokolovská 83 186 75 Praha 8 (Czech Republic)},
author = {Zelený, Miroslav},
journal = {Annales de l’institut Fourier},
keywords = {Denjoy–Clarkson property; gradient; Hausdorff measure; infinite game; Denjoy-Clarkson property},
language = {eng},
number = {2},
pages = {405-428},
publisher = {Association des Annales de l’institut Fourier},
title = {The Denjoy-Clarkson property with respect to Hausdorff measures for the gradient mapping of functions of several variables},
url = {http://eudml.org/doc/10320},
volume = {58},
year = {2008},
}
TY - JOUR
AU - Zelený, Miroslav
TI - The Denjoy-Clarkson property with respect to Hausdorff measures for the gradient mapping of functions of several variables
JO - Annales de l’institut Fourier
PY - 2008
PB - Association des Annales de l’institut Fourier
VL - 58
IS - 2
SP - 405
EP - 428
AB - We construct a differentiable function $f:{\mathbf{R}}^n \rightarrow {\mathbf{R}}$ ($n\ge 2$) such that the set $(\nabla f)^{-1}(B(0,1))$ is a nonempty set of Hausdorff dimension $1$. This answers a question posed by Z. Buczolich.
LA - eng
KW - Denjoy–Clarkson property; gradient; Hausdorff measure; infinite game; Denjoy-Clarkson property
UR - http://eudml.org/doc/10320
ER -
References
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- J. Malý, The Darboux property for gradients, Real Anal. Exchange 22 (1996/97), 167-173 Zbl0879.26042MR1433604
- J. Malý, M. Zelený, A note on Buczolich’s solution of the Weil gradient problem: a construction based on an infinite game, Acta Math. Hungar. 113 (2006), 145-158 Zbl1127.26006
- P. Mattila, Geometry of sets and measures in Euclidean spaces, Cambridge Studies in Advanced Mathematics 44 (1995), Cambridge University Press, Cambridge Zbl0819.28004MR1333890
- C. E. Weil, Query 1, Real Anal. Exchange 16 (1990-91)
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