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

Andrzej S. Kucik

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arXiv · 1604.06015

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

Andrzej S. Kucik

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Abstract

The notion of a Carleson measure was introduced by Lennart Carleson in his proof of the Corona Theorem for $H^\infty(\mathbb{D})$. In this paper we will define it for certain type of reproducing kernel Hilbert spaces of analytic functions of the complex half-plane, $\mathbb{C}_+$, 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.

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BibTeX RIS

Andrzej S. Kucik. 2016-04-20. Carleson measures for Hilbert spaces of analytic functions on the complex half-plane. https://arxiv.org/abs/1604.06015

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