Dimension of gradient measures

Dmitriy M. Stolyarov; Michal Wojciechowski

doi:10.1016/j.crma.2014.08.011

Mathematical analysis/Functional analysis

Dimension of gradient measures
[La dimension de mesures qui constituent le gradient d'une fonction]

Présenté par : Gilles Pisier

Dmitriy M. Stolyarov ^1,² ; Michal Wojciechowski ³

¹ St. Petersburg Department of Steklov Mathematical Institute RAS, Fontanka 27, St. Petersburg, Russia
² Chebyshev Laboratory (SPbU), 14th Line 29B, Vasilyevsky Island, St. Petersburg, Russia
³ Institute of Mathematics, Polish Academy of Sciences, 00-956 Warszawa, Poland

Comptes Rendus. Mathématique, Volume 352 (2014) no. 10, pp. 791-795.

Résumé
Abstract

Supposons que les dérivées pures (pas nécéssairement du même ordre) d'une fonction sur $R^{n}$ soient des mesures de Radon finies. On montre que leur dimension inférieure de Hausdorf est alors au moins $n - 1$ .

We prove that if pure derivatives of a function on $R^{n}$ are complex measures, then their lower Hausdorff dimension is at least $n - 1$ . The derivatives with respect to different coordinates may be of different order.

Reçu le : 2014-03-13
Accepté le : 2014-08-25
Publié le : 2014-09-18

DOI : 10.1016/j.crma.2014.08.011

Affiliations des auteurs :

Dmitriy M. Stolyarov ^{1, 2} ; Michal Wojciechowski ³

¹ St. Petersburg Department of Steklov Mathematical Institute RAS, Fontanka 27, St. Petersburg, Russia
² Chebyshev Laboratory (SPbU), 14th Line 29B, Vasilyevsky Island, St. Petersburg, Russia
³ Institute of Mathematics, Polish Academy of Sciences, 00-956 Warszawa, Poland

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     title = {Dimension of gradient measures},
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     pages = {791--795},
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     language = {en},
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Dmitriy M. Stolyarov; Michal Wojciechowski. Dimension of gradient measures. Comptes Rendus. Mathématique, Volume 352 (2014) no. 10, pp. 791-795. doi : 10.1016/j.crma.2014.08.011. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2014.08.011/

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Rami Ayoush; Michał Wojciechowski On dimension and regularity of vector-valued measures under Fourier analytic constraints, Illinois Journal of Mathematics, Volume 66 (2022) no. 3 | DOI:10.1215/00192082-10018154
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