Embedding model and de Branges-Rovnyak spaces in Dirichlet spaces

TL;DR

This paper characterizes the embeddings between de Branges-Rovnyak spaces and Dirichlet spaces using boundary spectrum and measure support, providing complete criteria and discussing applications.

Contribution

It offers a complete characterization of embeddings between de Branges-Rovnyak spaces and Dirichlet spaces based on boundary spectrum and measure support.

Findings

Complete characterization of embeddings between $H(b)$ and $ ext{Dirichlet}( u)$.

Criteria for embedding between model spaces $K_u$ and local Dirichlet spaces $ ext{Dirichlet}_ au$.

Discussion of applications of these embedding results.

Abstract

In this paper we study embeddings between de Branges-Rovnyak spaces $H(b)$ and harmonically weighted Dirichlet spaces $\mathcal{D}(\mu)$ in terms of the boundary spectrum of $b$ and the support of the measure $\mu$, by using elementary reproducing kernel estimates. We completely characterize the embedding between the model spaces $K_u$ and the local Dirichlet spaces $\mathcal{D}_\zeta$, and we discuss some applications.