Cyclicity and iterated logarithms in the Dirichlet space

TL;DR

The paper develops cyclicity criteria for outer functions in the superharmonically weighted Dirichlet spaces $D(\mu)$. It shows that if $f\in D(\mu)$ is outer with $\log f\in N^+(D(\mu))$, then $f$ is cyclic, and for $\|f\|_\infty\le1$ this cyclicity can be verified via iterated logarithms, e.g. $\log\bigl(1+\log\frac{1}{f}\bigr)\in N^+(D(\mu))$, which is testable by membership in $D(\mu)$. The work builds a network of equivalences among log-type conditions and their iterates, and provides a sufficient criterion for the converse under mild hypotheses, extending cyclicity criteria from the classical Dirichlet space to the full class of $D(\mu)$ spaces and linking to Pick/Nevanlinna theory. It also analyzes the Nevanlinna-class membership of iterated logarithms and discusses implications for Brown–Shields-type questions. Overall, the results deepen understanding of how logarithmic growth and iterates control cyclicity in these reproducing-kernel, Pick-spaces settings, with potential applications to function theory on the disk and related spaces.

Abstract

Let $D(μ)$ denote a harmonically weighted Dirichlet space on the unit disc $\mathbb D$. We show that outer functions $f\in D(μ)$ are cyclic in $D(μ)$, whenever $\log f$ belongs to the Pick-Smirnov class $N^+(D(μ))$. If $f$ has $H^\infty$-norm less than or equal to 1, then cyclicity can also be checked via iterated logarithms. For example, we show that such outer functions $f$ are cyclic, whenever $\log(1+ \log(1/f))\in N^+(D(μ))$. This condition can be checked by verifying that $\log(1+ \log(1/f))\in D(μ)$. If $f$ satisfies a mild extra condition, then the conditions also become necessary for cyclicity.

Theorems & Definitions (23)

• Theorem 1.1
• Corollary 1.2
• Corollary 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• proof