Density estimate from below in relation to a conjecture of A. Zygmund on Lipschitz differentiation
Journal de l’École polytechnique — Mathématiques, Volume 9 (2022), pp. 1473-1512.

Let 𝔾(n,m) be the Grassmannian consisting of m-dimensional vector subspaces of  n , let n be the Lebesgue measure in n , let m be the m-dimensional Hausdorff measure in n , and let α(m)= m (B(0,1)) be the Lebesgue measure of the Euclidean unit ball of  m . We establish that, if A n is Borel measurable and W 0 :A𝔾(n,m) is Lipschitzian, then

lim sup r0 + m AB(x,r)(x+W 0 (x)) α(m)r m 1 2 n ,

for n -almost every xA. In particular, it follows that A is n -negligible if and only if m (A(x+W 0 (x))=0, for n -almost every xA.

On désigne par 𝔾(n,m) la grassmannienne constituée des sous-espaces vectoriels de dimension m dans n , par n la mesure de Lebesgue dans n , par m la mesure de Hausdorff m-dimensionnelle dans n et par α(m)= m (B(0,1)) la mesure de Lebesgue de la boule euclidienne unité de m . Nous montrons que si A n est borélien et W 0 :A𝔾(n,m) est lipschitzien, alors

lim sup r0 + m AB(x,r)(x+W 0 (x)) α(m)r m 1 2 n ,

pour n -presque tout xA. Il en résulte en particulier que A est n -négligeable si et seulement si m (A(x+W 0 (x))=0, pour n -presque tout xA.

Received:
Accepted:
Published online:
DOI: 10.5802/jep.211
Classification: 28A75, 26B15
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

Thierry De Pauw 1

1 Université Paris Cité and Sorbonne Université, CNRS, IMJ-PRG F-75013 Paris, France
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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     title = {Density estimate from below in relation to a~conjecture of {A.} {Zygmund} on {Lipschitz~differentiation}},
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Thierry De Pauw. Density estimate from below in relation to a conjecture of A. Zygmund on Lipschitz differentiation. Journal de l’École polytechnique — Mathématiques, Volume 9 (2022), pp. 1473-1512. doi : 10.5802/jep.211. https://jep.centre-mersenne.org/articles/10.5802/jep.211/

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