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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| Veröffentlicht in: | American journal of mathematics Jg. 132; H. 1; S. 125 - 155 |
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| Format: | Journal Article |
| Sprache: | Englisch |
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Johns Hopkins University Press
01.02.2010
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| ISSN: | 0002-9327, 1080-6377, 1080-6377 |
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| Abstract | 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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| AbstractList | Let $\mathcal{H}(b)$ 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 $\mathcal{H}(b)$ and obtain weighted norm estimates of the form $\|f^{(n)}\|_{L^2(\mu)} \le C\|f\|_{\mathcal{H}(b)}$, where $f \in \mathcal{H}(b)$ and $\mu$ 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 $\mathcal{H}(b)$ 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 $\{k^b_{\lambda_n}\}$ in $\mathcal{H}(b)$ under small perturbations of the points $\lambda_n$. Let ${\cal H}(b)$ denote the de Branges--Rovnyak space associated with a function $b$ in the unit ball of $H^\infty({\Bbb C}_+)$. We study the boundary behavior of the derivatives of functions in ${\cal H}(b)$ and obtain weighted norm estimates of the form $\|f^{(n)}\|_{L^2(\mu)} \le C\|f\|_{{\cal H}(b)}$, where $f \in {\cal H}(b)$ and $\mu$ is a Carleson-type measure on ${\Bbb C}_+\cup{\Bbb R}$. We provide several applications of these inequalities. We apply them to obtain embedding theorems for ${\cal H}(b)$ 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 $\{k^b_{\lambda_n}\}$ in ${\cal H}(b)$ under small perturbations of the points $\lambda_n$. Let H(b) denote the de Branges-Rovnyak space associated with a function b in the unit ball of H^sup ∞^(C^sub +^). We study the boundary behavior of the derivatives of functions in H(b) and obtain weighted norm estimates of the form ..., where f ∈ H(b) and μ is a Carleson-type measure on C^sub +^ ∪ 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 {...} in H(b) under small perturbations of the points λ^sub n^. [PUBLICATION ABSTRACT] 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 $. |
| Author | Fricain, Emmanuel Baranov, Anton Mashreghi, Javad |
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| Copyright | Copyright 2010 The Johns Hopkins University Press Copyright © 2010 The Johns Hopkins University Press. Copyright Johns Hopkins University Press Feb 2010 Distributed under a Creative Commons Attribution 4.0 International License |
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| DOI | 10.1353/ajm.0.0094 |
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| Snippet | Let H(b) denote the de Branges-Rovnyak space associated with a function b in the unit ball of$H^\infty (\mathbb{C}_ + )$. We study the boundary behavior of the... Let ${\cal H}(b)$ denote the de Branges--Rovnyak space associated with a function $b$ in the unit ball of $H^\infty({\Bbb C}_+)$. We study the boundary... Let H(b) denote the de Branges-Rovnyak space associated with a function b in the unit ball of H^sup ∞^(C^sub +^). We study the boundary behavior of the... Let $\mathcal{H}(b)$ denote the de Branges--Rovnyak space associated with a function $b$ in the unit ball of $H^\infty(\mathbb{C}_+)$. We study the boundary... |
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| SubjectTerms | 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 |
| Title | Weighted Norm Inequalities for de Branges-Rovnyak Spaces and Their Applications |
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