Embeddings into de Branges-Rovnyak spaces

Abstract

We study conditions for containment of a given space $X$ of analytic functions on the unit disk $\mathbb{D}$ in the de Branges-Rovnyak space $\mathcal{H}(b)$. We deal with the non-extreme case in which $b$ admits a Pythagorean mate $a$, and derive a multiplier boundedness criterion on the function $\varphi =b/a$ which implies the containment $X\subset \mathcal{H}(b)$. With our criterion, we are able to characterize the containment of the Hardy space ${\mathcal{H}}^p$ inside $\mathcal{H}(b)$, for $p\in [2,\infty ]$. The end-point cases have previously been considered by Sarason, and we show that in his result, stating that $\varphi \in {\mathcal{H}}^2$ is equivalent to ${\mathcal{H}}^{\infty }\subset \mathcal{H}(b)$, one can in fact replace ${\mathcal{H}}^{\infty }$ by BMOA. We establish various other containment results, and study in particular the case of the Dirichlet space $\mathcal{D}$, containment of which is characterized by a Carleson measure condition. In this context, we show that matters are not as simple as in the case of the Hardy spaces, and we carefully work out an example.