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

Pouriya Torkinejad Ziarati

TL;DR

This work addresses when a holomorphic function in the Dirichlet-type spaces $D_α(\\mathbb{B}_2)$ is cyclic, establishing explicit, geometry-driven thresholds $α_c$ tied to boundary zero sets. Using interpolation- and peak-set techniques from smooth ball algebras (Bruna, Ortega; Chaumat, Chollet), the authors construct critically cyclic functions in $D_2(\\mathbb{B}_2)$ whose cyclicity in $D_α$ switches at $α_c$. They prove sharp thresholds for three boundary geometries: transversal curves give $α_c=2-d$, complex-tangential curves give $α_c=2-d/2$, and totally real manifolds give $α_c=\\tfrac{3}{2}-d$, with $d$ the Hausdorff dimension of the zero set. The results illuminate the role of $α$-capacity in cyclicity and extend to higher dimensions, providing concrete constructions and a framework for critically cyclic functions across the Dirichlet scale.

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 $α_c \in [\frac12,2]$ 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.

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

TL;DR

This work addresses when a holomorphic function in the Dirichlet-type spaces is cyclic, establishing explicit, geometry-driven thresholds tied to boundary zero sets. Using interpolation- and peak-set techniques from smooth ball algebras (Bruna, Ortega; Chaumat, Chollet), the authors construct critically cyclic functions in whose cyclicity in switches at . They prove sharp thresholds for three boundary geometries: transversal curves give , complex-tangential curves give , and totally real manifolds give , with the Hausdorff dimension of the zero set. The results illuminate the role of -capacity in cyclicity and extend to higher dimensions, providing concrete constructions and a framework for critically cyclic functions across the Dirichlet scale.

Abstract

In this work, we construct examples of holomorphic functions in , the Dirichlet space on , for which there exists an index such that the function is cyclic in if and only if . To this end, we use the notion of \emph{interpolation sets} in smooth ball algebras, as studied by Bruna, Ortega, Chaumat, and Chollet.
Paper Structure (7 sections, 13 theorems, 76 equations)

This paper contains 7 sections, 13 theorems, 76 equations.

Table of Contents

  1. Introduction
  2. Preliminary Tools
  3. Basics and Notations
  4. Proof of Theorem \ref{['MainThm1']}
  5. Proof of Theorem \ref{['MainThm2']}
  6. Proof of Theorem \ref{['MainThm3']}
  7. Examples and Discussion

Key Result

Proposition 1

Bruna1986 Let $E$ be a $\mathcal{K}$-set contained in a simple transverse curve $\Gamma$. Then there exists $f\in A^1(\mathbb{B}_n)$ such that Moreover, $f$ has this property that $f^k \in A^k(\mathbb{B}_n)$ for $k$ being a positive integer.

Theorems & Definitions (35)

  • Proposition 1
  • Theorem 2
  • Proposition 3
  • Theorem 4
  • Theorem 5
  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Lemma 6
  • ...and 25 more