Superharmonically Weighted Dirichlet Spaces

In this paper, we consider weighted Dirichlet spaces $\cD_\omega$, where $\omega$ is a positive superharmonic weight on the unit disc $\DD$. These spaces include the standard weighted Dirichlet spaces $\cD_\alpha$ and appear in the description of their invariant subspaces. Our goal is to study the spaces $\cD_\omega$. 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 $\cD_\omega$. 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 $\cD_\omega$, and several properties on the capacity associated with $\cD_\omega$. Using these tools, we provide a description of invariant subspaces when the measure $\Delta \omega$ is finite measure or if the $\supp(\Delta \omega)\cap \TT$ is countable, where $\TT$ denotes the unit circle. Finally, we prove that a smooth outer function $f \in \cD_\alpha$ such that $\cZ (f)$ is "regular" is cyclic in $\cD_\alpha$ if and only if $c_{\alpha }(\cZ(f))= 0$.