Embeddings into de Branges–Rovnyak spaces

We study conditions for the 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 $\phi = b/a$ 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 $\phi \in \mathcal H^2$ is equivalent to $\mathcal H^\infty \subset \mathcal H(b)$, one can in fact replace $\mathcal H^\infty $ by $\mathbf{BMOA}$. We establish various other containment results, and study in particular the case of the Dirichlet space $\mathcal D$, whose containment 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.