On the radial behavior of functions holomorphic in a ball (Q1921651)

Let \(B\) be the unit ball in \(\mathbb{C}^n\), \(S = \partial B\). Let \(N(B)\) and \(H^p (B)\), \(0<p \leq \infty\), be Nevanlinna and Hardy classes, respectively. For an arbitrary positive singular measure \(\mu\) on \(S\) the author constructs a function \(f \in N(B)\) such that \(\mu\)-a.e. \(\lim \sup_{r\to 1-} |f(r\zeta) |= \infty\) and \(\lim \inf_{r\to 1-} |f(r\zeta) |=0\), \(\zeta \in S\) (Theorem 1). He considers the sets such that the above-mentioned equalities hold everywhere if \(f\in \cap_{p<\infty} H^p(B)\). At last the following theorem is proved. Theorem 11. If \(\mu\) is an arbitrary singular measure on \(S\) then there exist holomorphic functions \(f\) and \(g\) such that \(\mu\)-a.e. the radial values of \(f\) are equal to \(\infty\) and the radial values of \(g\) are equal to 0. Note that Theorems 1 and 11 are the answers on Rudin's questions formulated in 1986.