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$H^\infty (\mathbb{C}_ + )$. We study the boundary behavior of the derivatives of functions in H(b) and obtain weighted norm estimates of the form$||f^{(n)} ||_{L^2 (\mu )} \leqslant C||f||_{H(b)} ,$, where...

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Bibliographic Details
Published in:American journal of mathematics Vol. 132; no. 1; pp. 125 - 155
Main Authors: Baranov, Anton, Fricain, Emmanuel, Mashreghi, Javad
Format: Journal Article
Language:English
Published: Baltimore Johns Hopkins University Press 01.02.2010
Subjects:
Applied mathematics
Complex Variables
Embeddings
Entire functions
Functional Analysis
Mathematical functions
Mathematical inequalities
Mathematical integrals
Mathematical problems
Mathematical theorems
Mathematics
Operator theory
Property lines
Theorems
Unit ball
ISSN:0002-9327, 1080-6377, 1080-6377
Online Access:Get full text
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Summary:Let H(b) denote the de Branges-Rovnyak space associated with a function b in the unit ball of$H^\infty (\mathbb{C}_ + )$. We study the boundary behavior of the derivatives of functions in H(b) and obtain weighted norm estimates of the form$||f^{(n)} ||_{L^2 (\mu )} \leqslant C||f||_{H(b)} ,$, 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_{\lambda n}^b \} $in H(b) under small perturbations of the points$\lambda _n $.
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ISSN:0002-9327
1080-6377
1080-6377
DOI:10.1353/ajm.0.0094