Divergent series of Taylor coefficients on almost all slices (Q2418786)

Let \(\Omega\) be a bounded, balanced, strictly pseudoconvex domain with \(\mathcal C^2\) boundary. Given a function \(f\) holomorphic in \(\Omega\), the associated slice function is the function \(\mathbb B^1\ni \lambda \mapsto f(\lambda z)\) where \(\mathbb B^1\) is the \(1\)-dimensional unit ball and \(z\in\partial \Omega\). In this paper the authors prove the existence of a function holomorphic in \(\Omega\) and continuous up to \(\partial \Omega\) such that almost every slice function has a series of Taylor coefficients \(\{p_n(z)\}_{n\in\mathbb N}\) such that \(\sum_{n=0}^{+\infty}|p_n(z)|^s=+\infty\) for \(s\in (0,2)\) for \(\sigma\)-almost every \(z\in\partial \Omega\). Here \(\sigma\) denotes a standard circular invariant measure on \(\partial\Omega\) with \(\sigma(\partial \Omega)=1\). This paper improves the results in [\textit{P. Wojtaszczyk}, Proc. Am. Math. Soc. 85, 184--186 (1982; Zbl 0503.32005)] and exploits earlier results of the first author [\textit{P. Kot}, J. Convex Anal. 14, No. 4, 693--704 (2007; Zbl 1126.32006)]. Some typos are present and the definitions of some notions are not recalled. However, this does not cause excessives problem in reading the paper.