Cyclicity and iterated logarithms in the Dirichlet space

TL;DR

This paper investigates conditions under which outer functions are cyclic in the Dirichlet space, linking cyclicity to properties of their logarithms and introducing iterated logarithm criteria for verification.

Contribution

It establishes new sufficient and necessary conditions for cyclicity of outer functions in the Dirichlet space using logarithmic and iterated logarithmic criteria.

Findings

Outer functions with log in the Pick-Smirnov class are cyclic.

Cyclicity can be checked via iterated logarithms for functions with bounded norm.

Conditions involving iterated logarithms are both sufficient and, under mild assumptions, necessary.

Abstract

Let $D(\mu)$ denote a harmonically weighted Dirichlet space on the unit disc $\mathbb D$. We show that outer functions $f\in D(\mu)$ are cyclic in $D(\mu)$, whenever $\log f$ belongs to the Pick-Smirnov class $N^+(D(\mu))$. 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 $\log(1+ \log(1/f))\in N^+(D(\mu))$. This condition can be checked by verifying that $\log(1+ \log(1/f))\in D(\mu)$. If $f$ satisfies a mild extra condition, then the conditions also become necessary for cyclicity.