On sets of extreme functions for Fatou's theorem

TL;DR

This paper investigates the structure and size of sets of bounded holomorphic functions that lack radial limits on specific zero-measure sets, revealing rich algebraic and linear structures within these sets.

Contribution

It demonstrates that for fixed zero-measure sets, the functions failing to have radial limits form large algebraic and linear structures, including algebras of continuum dimension and infinite-dimensional Banach spaces.

Findings

Sets of Lusin-type functions contain algebras of algebraic dimension continuum.

For countable sets, these functions include infinite-dimensional Banach spaces.

The results extend to functions in infinitely many variables.

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 $E$, contain 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 Lusin-type functions contain infinite dimensional Banach spaces and, moreover, contain plenty of $\mathfrak{c}$-dimensional algebras. We also address the question for functions on infinitely many variables.