Carleson Measures and Reproducing Kernel Thesis in Dirichlet-type spaces

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.