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

TL;DR

This paper constructs specific holomorphic functions in the Dirichlet space on the ball that exhibit a critical cyclicity index, advancing understanding of cyclic functions in complex analysis.

Contribution

It introduces examples of functions with a critical cyclicity index in Dirichlet spaces on the ball, utilizing interpolation sets in smooth ball algebras.

Findings

Existence of functions with a critical cyclicity index in Dirichlet spaces.

Cyclicity depends on the parameter alpha relative to a critical value.

Application of interpolation sets in smooth ball algebras to construct examples.

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 $\alpha_c \in [\frac12,2]$ such that the function is cyclic in $D_\alpha(\B_2)$ if and only if $\alpha \leq \alpha_c$. To this end, we use the notion of \emph{interpolation sets} in smooth ball algebras, as studied by Bruna, Ortega, Chaumat, and Chollet.