Douglas Weighted Dirichlet Spaces: Analytic and Probabilistic Aspects

Abstract

We introduce a class of weighted Dirichlet spaces on the unit disk, called Douglas weighted Dirichlet spaces, characterized by the existence of a Douglas type boundary representation of the weighted Dirichlet integral. This representation naturally induces a nonlocal Dirichlet form on the unit circle in the sense of Beurling Deny and Fukushima. Our main result shows that the reproducing kernel of the weighted Dirichlet space associated with a regular Douglas weighted Dirichlet form is the image of the Szego kernel under the corresponding 1 resolvent. This resolvent representation is new for superharmonic weights. We also develop the potential theory associated with Douglas weighted Dirichlet spaces. We characterize the capacity in terms of reproducing kernels. Finally, as an application of the fact that every Douglas weight induces a regular nonlocal Dirichlet form, together with the general theory of Dirichlet forms, we obtain an associated symmetric pure jump Hunt process. In the classical Dirichlet case, this process is the wrapped Cauchy process.