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

Let ${\cal H}(b)$ denote the de Branges--Rovnyak space associated with a function $b$ in the unit ball of $H^\infty({\Bbb C}_+)$. We study the boundary behavior of the derivatives of functions in ${\cal H}(b)$ and obtain weighted norm estimates of the form $\|f^{(n)}\|_{L^2(\mu)} \le C\|f\|_{{\cal H}(b)}$, where $f \in {\cal H}(b)$ and $\mu$ is a Carleson-type measure on ${\Bbb C}_+\cup{\Bbb R}$. We provide several applications of these inequalities. We apply them to obtain embedding theorems for ${\cal H}(b)$ 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 $\{k^b_{\lambda_n}\}$ in ${\cal H}(b)$ under small perturbations of the points $\lambda_n$.