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

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.

Keywords: Carleson measures, reproducing kernel Hilbert spaces, Dirichlet space, control operators, admissibility, Laplace transform