Examples of critically cyclic functions in the Dirichlet spaces of the ball

Abstract

In this work, we construct examples of holomorphic functions in $D_2(\B_2)$, the Dirichlet space on $\B_2$, for which there exists an index ${\alpha }_c\in [\frac{1}{2},2]$ such that the function is cyclic in $D_α(\B_2)$ if and only if $\alpha \le {\alpha }_c$. To this end, we use the notion of \emph{interpolation sets} in smooth ball algebras, as studied by Bruna, Ortega, Chaumat, and Chollet.