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

Zelený, Miroslav

Zelený, Miroslav

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

Abstract
Résumé

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.

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.

MR 2410378 | Zbl 1154.26016

DOI : https://doi.org/10.5802/aif.2356
Classification: 26B05, 28A75
Keywords: Denjoy–Clarkson property, gradient, Hausdorff measure, infinite game

@article{AIF_2008__58_2_405_0,
     author = {Zelen\'y, Miroslav},
     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},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {58},
     number = {2},
     year = {2008},
     pages = {405-428},
     doi = {10.5802/aif.2356},
     mrnumber = {2410378},
     zbl = {1154.26016},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2008__58_2_405_0}
}

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, Volume 58 (2008) no. 2, pp. 405-428. doi : 10.5802/aif.2356. http://www.numdam.org/item/AIF_2008__58_2_405_0/

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