A note on Dirichlet-type spaces and cyclic vectors in the unit ball of $\mathbb{C}^2$

TL;DR

This paper extends the study of cyclic vectors in Dirichlet-type spaces from the unit disk to the unit ball in complex two-dimensional space, identifying classes of cyclic and non-cyclic functions and emphasizing capacity conditions.

Contribution

It generalizes previous results to higher dimensions, providing new insights into cyclic vectors in Dirichlet-type spaces on the unit ball.

Findings

Identifies classes of cyclic functions in the unit ball setting

Highlights the importance of capacity conditions for cyclicity

Extends prior results from the unit disk to the unit ball

Abstract

We extend results of B\'en\'eteau, Condori, Liaw, Seco, and the author concerning cyclic vectors in Dirichlet-type spaces to the setting of the unit ball, identifying some classes of cyclic and non-cyclic functions, and noting the necessity of certain capacity conditions.