Superharmonically Weighted Dirichlet Spaces

In this paper, we consider weighted Dirichlet spaces ${\text{cD}}_{\omega }$, where $\omega$ is a positive superharmonic weight on the unit disc $\text{DD}$. These spaces include the standard weighted Dirichlet spaces ${\text{cD}}_{\alpha }$ and appear in the description of their invariant subspaces. Our goal is to study the spaces ${\text{cD}}_{\omega }$. We show that an explicit description of invariant subspaces reduces to the description of those generated by a bounded outer function, and then to the problem of describing cyclic functions, known as the Brown–Shields conjecture. We develop tools, analogous to those used in the harmonic case, that are needed to treat this problem for superharmonically weighted Dirichlet spaces ${\text{cD}}_{\omega }$. In particular, we obtain a formula for the Dirichlet integral of outer functions of Carleson–Richter–Sundberg type, estimates for the norm of the reproducing kernel of ${\text{cD}}_{\omega }$, and several properties on the capacity associated with ${\text{cD}}_{\omega }$. Using these tools, we provide a description of invariant subspaces when the measure $\mathrm{\Delta }\omega$ is finite measure or if the $\text{supp}(\mathrm{\Delta }\omega )\cap \text{TT}$ is countable, where $\text{TT}$ denotes the unit circle. Finally, we prove that a smooth outer function $f\in {\text{cD}}_{\alpha }$ such that $\text{cZ}(f)$ is “regular” is cyclic in ${\text{cD}}_{\alpha }$ if and only if $c_{\alpha }(\text{cZ}(f))=0$.