Havin-Mazya type uniqueness theorem for Dirichlet spaces

Abstract

Let μ be a positive finite Borel measure on the unit circle. The associated Dirichlet space D(μ) consists of holomorphic functions on the unit disc whose derivatives are square integrable when weighted against the Poisson integral of μ. We give a sufficient condition on a Borel subset E of the unit circle which ensures that E is a uniqueness set for D(μ). We also give some examples of positive Borel measures μ and uniqueness sets for D(μ).