Weighted norm inequalities for de Branges--Rovnyak spaces and their applications

Abstract

Let $\mathcal{H}(b)$ denote the de Branges--Rovnyak space associated with a function $b$ in the unit ball of $H^{\infty }({\mathbb{C}}_{+})$. We study the boundary behavior of the derivatives of functions in $\mathcal{H}(b)$ and obtain weighted norm estimates of the form $\Vert f^{(n)}{\Vert }_{L^2(\mu )}\le C\Vert f{\Vert }_{\mathcal{H}(b)}$, where $f\in \mathcal{H}(b)$ and $\mu$ is a Carleson-type measure on ${\mathbb{C}}_{+}\cup \mathbb{R}$. We provide several applications of these inequalities. We apply them to obtain embedding theorems for $\mathcal{H}(b)$ spaces. These results extend Cohn and Volberg--Treil embedding theorems for the model (star-invariant) subspaces which are special classes of de Branges--Rovnyak spaces. We also exploit the inequalities for the derivatives to study stability of Riesz bases of reproducing kernels $\{k_{{\lambda }_n}^b\}$ in $\mathcal{H}(b)$ under small perturbations of the points ${\lambda }_n$.