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arXiv · 2301.09701

On sets of extreme functions for Fatou's theorem

Thiago R. Alves·Leonardo Brito·Daniel Carando
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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.

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Thiago R. Alves, Leonardo Brito, Daniel Carando. 2023-01-23. On sets of extreme functions for Fatou's theorem. https://arxiv.org/abs/2301.09701

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