Bloch functions with wild boundary behaviour in $\C^N$
Stéphane Charpentier, Nicolas Espoullier, Rachid Zarouf · HAL (Le Centre pour la Communication Scientifique Directe) · 2024
We prove the existence of functions $f$ in the Bloch space of the unit ball $\B_N$ of $\C^N$ with the property that, given any measurable function $\vp$ on the unit sphere $\S_N$, there exists a sequence $(r_n)_n$, $r_n\in (0,1)$, converging to $1$, such that for every $w\in \B_N$,\[f(r_n(\zeta -w)+w) \to \vp(\zeta)\text{ as }n\to \infty\text{, for almost every }\zeta \in \S_N. \]The set of such functions is residual in the little Bloch space. A similar result is obtained for the Bloch space of the polydisc.