Embeddings into de Branges–Rovnyak spaces
Tom 278 / 2024
Streszczenie
We study conditions for the containment of a given space 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.