Weighted Norm Inequalities for De Branges-rovnyak Spaces and Their Applications

Let H(b) denote the de Branges–Rovnyak space associated with a function b in the unit ball of H∞(C+). We study the boundary behavior of the derivatives of functions in H(b) and obtain weighted norm estimates of the form ‖f ‖L2(μ) ≤ C‖f‖H(b), where f ∈ H(b) and μ is a Carleson-type measure on C+ ∪ 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 {kb λn} in H(b) under small perturbations of the points λn.