The existence of inner functions in the ball

A. B. Aleksandrov

Sbornik. Mathematics, Tome 46 (1983) no. 2, pp. 143-159

Résumé

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.