The Denjoy-Clarkson property with respect to Hausdorff measures for the gradient mapping of functions of several variables

Zelený, Miroslav

doi:10.5802/aif.2356

The Denjoy-Clarkson property with respect to Hausdorff measures for the gradient mapping of functions of several variables
[La propriété de Denjoy-Clarkson par rapport aux mesures de Hausdorff et le gradient des fonctions de plusieurs variables]

Zelený, Miroslav ¹

¹ Charles University Faculty of Mathematics and Physics Sokolovská 83 186 75 Praha 8 (Czech Republic)

Annales de l'Institut Fourier, Tome 58 (2008) no. 2, pp. 405-428.

Résumé
Abstract

On construit une fonction différentiable $f : R^{n} \to R$ ( $n \geq 2$ ) telle que l’ensemble ${(\nabla f)}^{- 1} (B (0, 1))$ est non vide et sa dimension de Hausdorff est $1$ . C’est une réponse à une question posée par Z. Buczolich.

We construct a differentiable function $f : R^{n} \to R$ ( $n \geq 2$ ) 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.

MR Zbl

DOI : 10.5802/aif.2356

Classification : 26B05, 28A75
Keywords: Denjoy–Clarkson property, gradient, Hausdorff measure, infinite game
Mot clés : propriété de Denjoy-Clarkson, gradient, mesure de Hausdorff, jeu infini

Affiliations des auteurs :

Zelený, Miroslav ¹

¹ Charles University Faculty of Mathematics and Physics Sokolovská 83 186 75 Praha 8 (Czech Republic)

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     title = {The {Denjoy-Clarkson} property with respect to {Hausdorff} measures for the gradient mapping of functions of several variables},
     journal = {Annales de l'Institut Fourier},
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     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {58},
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Zelený, Miroslav. The Denjoy-Clarkson property with respect to Hausdorff measures for the gradient mapping of functions of several variables. Annales de l'Institut Fourier, Tome 58 (2008) no. 2, pp. 405-428. doi : 10.5802/aif.2356. http://www.numdam.org/articles/10.5802/aif.2356/

Bibliographie
Cité par

[1] Bruckner, A. Differentiation of real functions, CRM Monograph Series, 5, American Mathematical Society, Providence, RI, 1994 | MR | Zbl

[2] Buczolich, Z. The $n$ -dimensional gradient has the 1-dimensional Denjoy-Clarkson property, Real Anal. Exchange, Volume 18 (1992-93), pp. 221-224 | MR | Zbl

[3] Buczolich, Z. Another note on the gradient problem of C. E. Weil, Real Anal. Exchange, Volume 22 (1996-97), pp. 775-784 | MR | Zbl

[4] Buczolich, Z. Solution to the gradient problem of C. E. Weil, Rev. Mat. Iberoamericana, Volume 21 (2005), pp. 889-910 | EuDML | MR | Zbl

[5] Clarkson, J. A. A property of derivatives, Bull. Amer. Math. Soc., Volume 53 (1947), pp. 124-125 | DOI | MR | Zbl

[6] Denjoy, A. Sur une proprieté des fonctions dérivées, Enseignement Math., Volume 18 (1916), pp. 320-328 | JFM

[7] Deville, R.; Matheron, É. Infinite games, Banach space geometry and the eikonal equation, Proc. Lond. Math. Soc. (3), Volume 95 (2007) no. 1, pp. 49-68 | DOI | MR | Zbl

[8] Engelking, R. General Topology, Heldermann Verlag, Berlin, 1989 | MR | Zbl

[9] Holický, P.; Malý, J.; Weil, C. E.; Zajíček, L. A note on the gradient problem, Real Anal. Exchange, Volume 22 (1996-97), pp. 225-235 | MR | Zbl

[10] Malý, J. The Darboux property for gradients, Real Anal. Exchange, Volume 22 (1996/97) no. 1, pp. 167-173 | MR | Zbl

[11] Malý, J.; Zelený, M. A note on Buczolich’s solution of the Weil gradient problem: a construction based on an infinite game, Acta Math. Hungar., Volume 113 (2006), pp. 145-158 | DOI | Zbl

[12] Mattila, P. Geometry of sets and measures in Euclidean spaces, Cambridge Studies in Advanced Mathematics, Volume 44, Cambridge University Press, Cambridge, 1995 | MR | Zbl

[13] Weil, C. E. Query 1, Real Anal. Exchange, Volume 16 (1990-91), pp. 373

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