  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.
