Bloch functions with wild boundary behaviour in $\mathbb{C}^N$

TL;DR

This paper constructs Bloch functions in several complex variables with highly irregular boundary behavior, showing such functions are abundant in the space and can approximate any boundary function along certain sequences.

Contribution

It demonstrates the existence and abundance of Bloch functions with wild boundary behavior in $\,\mathbb{C}^N$, extending known results to higher dimensions and the polydisc.

Findings

Existence of Bloch functions with prescribed boundary limits along sequences.

Residual set of such functions in the little Bloch space.

Extension of results to the polydisc setting.

Abstract

We prove the existence of functions $f$ in the Bloch space of the unit ball $\mathbb{B}_N$ of $\mathbb{C}^N$ with the property that, given any measurable function $\varphi$ on the unit sphere $\mathbb{S}_N$, there exists a sequence $(r_n)_n$, $r_n\in (0,1)$, converging to $1$, such that for every $w\in \mathbb{B}_N$, $$f(r_n(\zeta -w)+w) \to \varphi(\zeta)\text{ as }n\to \infty\text{, for almost every }\zeta \in \mathbb{S}_N.$$ The set of such functions is residual in the little Bloch space. A similar result is obtained for the Bloch space of the polydisc.