Carleson measures for Hilbert spaces of analytic functions on the complex half-plane

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• 作者：Andrzej S. Kucik ^{mmask@leeds.ac.uk" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Carleson measures ; Reproducing kernel Hilbert spaces ; Dirichlet space ; Control operators ; Admissibility ; Laplace transform
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2017
• 出版时间：1 January 2017
• 年：2017
• 卷：445
• 期：1
• 页码：476-497
• 全文大小：467 K

文摘

The notion of a Carleson measure   was introduced by Lennart Carleson in his proof of the Corona Theorem for H^∞(D)$H^{\infty }(\mathbb{D})$. In this paper we will define it for certain type of reproducing kernel Hilbert spaces of analytic functions of the complex half-plane, C₊${\mathbb{C}}_{+}$, which will include Hardy, Bergman and Dirichlet spaces. We will obtain several necessary or sufficient conditions for a positive Borel measure to be Carleson by preforming tests on reproducing kernels, weighted Bergman kernels, and studying the tree model obtained from a decomposition of the complex half-plane. The Dirichlet space will be investigated in detail as a special case. Finally, we will present a control theory application of Carleson measures in determining admissibility of controls in well-posed linear evolution equations.