MULTIPLE INTERPOLATION IN THE PRIVALOV CLASSES IN A DISK

Abstract

For all $0<q<+\infty$ the Privalov class $\Pi_q$ consists of all analytic functions $f$ in the unit disk such that
\begin{equation*}
\sup \limits_{0\leq r<1} \frac{1}{2\pi}\int_{-\pi}^{\pi}\left(\ln^{+}|f(re^{i\theta})| \right)^qd\theta<+\infty.
\end{equation*}
In this paper we solve the multiple interpolation problem in the Privalov class of analytic functions in the unit disk for all $0<q<1$. Namely, we find sufficient conditions for construction an explicit function that solves the interpolation problem in the Privalov class. In addition, we discuss the necessity of these conditions.