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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.) (English. French summary) Zbl 1303.31002
The classical log log theorem asserts the normality of families of holomorphic functions satisfying a certain majorization condition. The present paper presents an analogue for harmonic functions. More precisely, let \(H_{M}\) denote the collection of harmonic functions \(u\) on \([-1,1]^{n}\) satisfying \(|u(x)|\leq M(|x_{n}|)\), where \(M:(0,1)\rightarrow [ e,\infty )\) is decreasing and \(\int_{0}^{1}\log \log M(y)dy<\infty \). Then \(H_{M}\) is shown to be a normal family. This is established using axial symmetrization arguments. As an application the author improves a recent result of the reviewer and D. Khavinson [C. R., Math., Acad. Sci. Paris 352, No. 2, 99–103 (2014; Zbl 1297.30083)] concerning universal polynomial expansions of harmonic functions.

MSC:
31B05 Harmonic, subharmonic, superharmonic functions in higher dimensions
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