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

Abstract

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 (n)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 {k b λn } in H(b) under small perturbations of the points λ n . 2000 Mathematics Subject Classification. Primary: 46E15, 46E22, Secondary: 30D55, 47A15. Key words and phrases. Bernstein's inequality, de Branges-Rovnyak space, model subspace, reproducing kernel, embedding theorem, Riesz basis. This work was supported by funds from NSERC (Canada), Jacques Cartier Center (France) and RFBR (Russia).

On certain Riesz families in vector-valued de Branges–Rovnyak spaces

“…Let us also mention that in the scalar de Branges-Rovnyak spaces, using a different approach based on Bernstein type inequalities, a stability result was found in [3] without the assumption (4.1). However, the techniques used therein do not seem adaptable to the vector case.…”

“…(1) span{k b λ,e : λ ∈ D, e ∈ E} = H(b); (2) H ∩ H * = {0}; (3) H ∨ H * = H; (4) ker(P R b * |R * ) = {0} (R, R * defined by (3.6)).…”

A short introduction to de Branges–Rovnyak spaces

“…The last paper is used in [5] to obtain weighted norm inequalities for functions in de Branges-Rovnyak spaces.…”

Two-Isometries and de Branges–Rovnyak Spaces

“…Sarason has studied this question ( [9]) only for an example, namely b(z) = (1 + z)/2. By Theorem 1 and Richter wandering subspace Theorem [6, Theorem 1] we have the following Corollary 2 Let b has the form given by (1), the closed invariant subspaces of S|H(b) are wandering that is…”

“…on T, then H(b) = H 2 bH 2 , see [10]. We refer to [1][2][3][4] for some recent results on these spaces. We denote by S the shift operator on H 2 .…”