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 HI(C,). We study the boundary behavior of the derivatives of functions in H(b) and obtain weighted norm estimates of the form parallel to f((n))parallel to(L2(mu)) <= C parallel to f parallel to(H(b)), where f epsilon H(b) and mu is a Carleson-type measure on C+ boolean OR 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(lambda n)(b)} in H(b) under small perturbations of the points lambda(n).