Blaschke Products and Zero Sets in Weighted Dirichlet Spaces

In this paper, we deal with superharmonically weighted Dirichlet spaces \(\mathcal {D}_{\omega }\). First, we prove that the classical Dirichlet space is the largest, among all these spaces, which contains no infinite Blaschke product. Next, we give new sufficient conditions on a Blaschke sequence to be a zero set for \(\mathcal {D}_{\omega }\). Our conditions improve Shapiro-Shields condition for \(\mathcal {D}_{\alpha }\), when α ∈ (0,1).