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

2008/02/06 by Anton Baranov, Baranov, Anton, Emmanuel Fricain +3
Mathematics · #Complex Variables (math.CV) #FOS: Mathematics #Functional Analysis (math.FA) #math.CV #math.FA

paper · pdf · doi:10.48550/arxiv.0802.0789

arxiv created 2008/02/06 · arxiv updated 2009/12/01

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⁾‖_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 \k^b_λn\ in H(b) under small perturbations of the points λ_n.

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