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

TL;DR

This paper extends the concept of Carleson measures to various Hilbert spaces of analytic functions on the complex half-plane, providing conditions for measures and exploring applications in control theory.

Contribution

It introduces Carleson measures for reproducing kernel Hilbert spaces on the half-plane and applies these results to control theory problems.

Findings

Characterization of Carleson measures via kernel tests

Conditions for measures in Hardy, Bergman, and Dirichlet spaces

Application to control admissibility in linear evolution equations

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.