On sets of extreme functions for Fatou’s theorem

Abstract:

Bounded holomorphic functions on the disk have radial limits in almost every direction, as follows from Fatou’s theorem. Given a zero-measure set $E$ in the torus $\mathbb T$, we study the set of functions such that $\lim _{r \to 1^{-}} f(r \, w)$ fails to exist for every $w\in E$ (such functions were first constructed by Lusin). We show that the set of Lusin-type functions, for a fixed zero-measure set $E$, contains algebras of algebraic dimension $\mathfrak {c}$ (except for the zero function). When the set $E$ is countable, we show also in the several-variable case that the set of Lusin-type functions contains infinite dimensional Banach spaces and, moreover, contains plenty of $\mathfrak {c}$-dimensional algebras. We also address the question for functions of infinitely many variables.