Geodesic
The existence of inner functions in the ball
Sbornik. Mathematics, Tome 46 (1983) no. 2, pp. 143-159 Cet article a éte moissonné depuis la source Math-Net.Ru

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The following analogue of the well-known theorem of Rudin for the polydisk is proved for the ball. Given any positive lower semicontinuous integrable function $\varphi$ on the sphere $S\subset\mathbf C^d$, there is a positive singular measure $\mu$ on $S$ such that $\mu(S)=\|\varphi\|_{L^1(S)}$, and the difference between the Poisson integrals of the function $\varphi$ and the measure $\mu$ is a pluriharmonic function (in the unit ball $B$, with $S=\partial B$). This implies immediately the existence of an inner function in $B$. A certain weakened version of the Pick–Nevanlinna theorem on interpolation of inner functions is also obtained for $B$. The results obtained are applied to the Hardy classes $H^p$ ($0) in the ball and in the polydisk. Bibliography: 17 titles. 
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A. B. Aleksandrov. The existence of inner functions in the ball. Sbornik. Mathematics, Tome 46 (1983) no. 2, pp. 143-159. http://geodesic.mathdoc.fr/item/SM_1983_46_2_a0/

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