Comptes Rendus
no. 11
Complex analysis/Partial differential equations
On the higher dimensional harmonic analog of the Levinson log⁡log theorem
[Sur l'analogue harmonique du théorème log⁡log de Levinson pour plusieurs dimensions]

Présenté par : Jean-Pierre Kahane

Alexander Logunov1
1 Saint Petersburg State University, Chebyshev Laboratory, 14th Line 29B, Vasilyevsky Island, St. Petersburg, Russia
Comptes Rendus. Mathématique, Volume 352 (2014) no. 11, pp. 889-893.

Soit M:(0,1)→[e,+∞) une fonction décroissante telle que ∫01log⁡log⁡M(y)dy<+∞. Considérons l'ensemble HM de toutes les fonctions u qui sont harmoniques dans P:={(x,y)∈Rn:x∈Rn−1,y∈R,|x|<1,|y|<1} et satisfont |u(x,y)|≤M(|y|). On montre que HM est une famille normale dans P.

Let M: (0,1)→[e,+∞) be a decreasing function such that ∫01log⁡log⁡M(y)dy<+∞. Consider the set HM of all functions u harmonic in P:={(x,y):x∈Rn−1,y∈R,|x|<1,|y|<1} and satisfying |u(x,y)|≤M(|y|). We prove that HM is a normal family in P.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2014.09.019
Alexander Logunov 1

1 Saint Petersburg State University, Chebyshev Laboratory, 14th Line 29B, Vasilyevsky Island, St. Petersburg, Russia 
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Alexander Logunov. On the higher dimensional harmonic analog of the Levinson $ \mathrm{log}\mathrm{log}$ theorem. Comptes Rendus. Mathématique, Volume 352 (2014) no. 11, pp. 889-893. doi : 10.1016/j.crma.2014.09.019. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2014.09.019/

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