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

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

💡 Research Summary

The paper investigates the phenomenon of “critical cyclicity” for holomorphic functions in Dirichlet‑type spaces on the unit ball in ℂ². A function f in a Banach space A of holomorphic functions is called cyclic if the linear span of all products pf, where p ranges over polynomials, is dense in A. In Dirichlet‑type spaces Dα(ℬ₂) the norm becomes stronger as α increases, making cyclicity harder to achieve. The authors focus on functions that are bounded away from zero in the interior, so the only obstruction to cyclicity comes from the boundary behavior of the function, specifically the zero set of its non‑tangential boundary values f*.

The main goal is to construct explicit examples of functions f∈D₂(ℬ₂) for which there exists a critical index αc∈