[Borne inférieure de densité en lien avec une conjecture de A. Zygmund sur des bases de dérivation à variation lipschitzienne]
On désigne par la grassmannienne constituée des sous-espaces vectoriels de dimension dans , par la mesure de Lebesgue dans , par la mesure de Hausdorff -dimensionnelle dans et par la mesure de Lebesgue de la boule euclidienne unité de . Nous montrons que si est borélien et est lipschitzien, alors
pour -presque tout . Il en résulte en particulier que est -négligeable si et seulement si , pour -presque tout .
Let be the Grassmannian consisting of -dimensional vector subspaces of , let be the Lebesgue measure in , let be the -dimensional Hausdorff measure in , and let be the Lebesgue measure of the Euclidean unit ball of . We establish that, if is Borel measurable and is Lipschitzian, then
for -almost every . In particular, it follows that is -negligible if and only if , for -almost every .
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Keywords: Lebesgue measure, Nikodým set, negligible set, derivation basis, Zygmund conjecture, Lipschitz differentiation
Mot clés : Mesure de Lebesgue, ensemble de Nikodým, ensemble négligeable, base de derivation, conjecture de Zygmund, differentiation lipschitzienne
@article{JEP_2022__9__1473_0, author = {De Pauw, Thierry}, title = {Density estimate from below in relation to a~conjecture of {A.} {Zygmund} on {Lipschitz~differentiation}}, journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques}, pages = {1473--1512}, publisher = {Ecole polytechnique}, volume = {9}, year = {2022}, doi = {10.5802/jep.211}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/jep.211/} }
TY - JOUR AU - De Pauw, Thierry TI - Density estimate from below in relation to a conjecture of A. Zygmund on Lipschitz differentiation JO - Journal de l’École polytechnique — Mathématiques PY - 2022 SP - 1473 EP - 1512 VL - 9 PB - Ecole polytechnique UR - http://archive.numdam.org/articles/10.5802/jep.211/ DO - 10.5802/jep.211 LA - en ID - JEP_2022__9__1473_0 ER -
%0 Journal Article %A De Pauw, Thierry %T Density estimate from below in relation to a conjecture of A. Zygmund on Lipschitz differentiation %J Journal de l’École polytechnique — Mathématiques %D 2022 %P 1473-1512 %V 9 %I Ecole polytechnique %U http://archive.numdam.org/articles/10.5802/jep.211/ %R 10.5802/jep.211 %G en %F JEP_2022__9__1473_0
De Pauw, Thierry. Density estimate from below in relation to a conjecture of A. Zygmund on Lipschitz differentiation. Journal de l’École polytechnique — Mathématiques, Tome 9 (2022), pp. 1473-1512. doi : 10.5802/jep.211. http://archive.numdam.org/articles/10.5802/jep.211/
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