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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
##### Keywords:
harmonic function; normal family; universal series
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##### References:
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