Havin-Mazya type uniqueness theorem for Dirichlet spaces

TL;DR

This paper establishes a sufficient condition for certain subsets of the unit circle to be uniqueness sets for Dirichlet spaces associated with positive Borel measures, expanding understanding of boundary behavior of holomorphic functions.

Contribution

It provides a new criterion for identifying uniqueness sets in Dirichlet spaces linked to positive Borel measures, with illustrative examples.

Findings

A sufficient condition for a Borel subset to be a uniqueness set.

Examples of measures and sets satisfying the uniqueness criteria.

Enhanced understanding of boundary uniqueness in Dirichlet spaces.

Abstract

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