Cyclicity and iterated logarithms in the Dirichlet space

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.