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 T, we study the set of functions such that lim r→1 − f (r w) fails to exist for every w ∈ E (such functions were first constructed by Lusin). We show that the set of Lusin-type functions, for a fixed zero-measure set E, contain algebras of algebraic dimension c (except for the zero function). When the set E is countable, we show also in the several-variable case that the Lusin-type functions contain infinite dimensional Banach spaces and, moreover, contain plenty of c-dimensional algebras. We also address the question for functions on infinitely many variables.