Weighted Norm Inequalities for de Branges-Rovnyak Spaces and Their Applications - Baranov, Anton; Fricain, Emmanuel; Mashreghi, Javad (American journal of mathematics, 2010)
Let H(b) denote the de Branges-Rovnyak space associated with a function b in the unit ball of$H^∞ (\mathbb{C}_ + )$. We study the boundary behavior of the derivatives of functions in H(b) and obtain weighted norm estimates of the form$||f^{(n)} ||_{L^2 (μ )} \leqslant C||f||_{H(b)} ,$, where f ∈ H(b) and µ 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 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_{λ n}^b \} $in H(b) under small perturbations of the points$λ _n $.