Carleson Measures and Reproducing Kernel Thesis in Dirichlet-type spaces

TL;DR

This paper characterizes Carleson measures for Dirichlet-type spaces with finite point mass measures and establishes a reproducing kernel thesis, advancing understanding of these function spaces.

Contribution

It provides a new characterization of Carleson measures and a reproducing kernel thesis for Dirichlet-type spaces with finite point mass measures, generalizing previous results.

Findings

New characterization of Carleson measures for $ ext{Dirichlet}$-type spaces

Reproducing kernel thesis established for these spaces

Extension of Richter and Sundberg's representation theorem

Abstract

In this paper, using a generalization of a Richter and Sundberg representation theorem, we give a new characterization of Carleson measures for the Dirichlet-type space $\mathcal D(\mu)$ when $\mu$ is a finite sum of point masses. A reproducing kernel thesis result is also established in this case.