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 ∞ (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+, 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. Mathematics Subject Classification (2010). Primary 30H25, 93B28; Secondary 28E99, 30H10, 30H20, 46C15, 93B05.

Laplace–Carleson embeddings and weighted infinite-time admissibility

“…And our problem of determining admissibility of control or observation operators is reduced to the characterisation of Carleson measures for A 2 (m) , allowing us to consider L 2 w -admissibility for nondecreasing weights, which were not included in the Zen space context. This has been partially done in [13] and we aim to extend the results obtained there to the non-Hilbertian case of A p (C + , (ν n ) m n=0 ) in the next section.…”

“…As a special case, the definitions of so-called Zen spaces and their generalisation are provided, presenting their connection to the admissibility concept. In Section 4 boundedness of Carleson embeddings for these generalised spaces is studied, following a similar analysis from [13]. And finally, in Section 5 boundedness of Laplace-Carleson embeddings for sectorial measures is characterised there, and we believe that Theorem 6 is the most important result of this paper.…”

“…Although in most common physical applications the norms used are usually the unweighted L 1 or L 2 norms, it may sometimes be useful to consider weighted L p norms. The main purpose of this paper is to generalise admissibility criteria, obtained in [10] for weighted L 2 -and unweighted L p -admissibility (given in terms of Carleson measures and Carleson-Laplace embeddings described in [9]) to weighted L p (0, ∞) case, applying and generalising recent results from [13], concerning the spaces defined and studied in [12]. A powerful boundedness criterion for the Laplace-Carleson embeddings for weighted L p spaces, containing the earlier version from [9] as a special case, is also proved here.…”

Weighted Composition Operators on Spaces of Analytic Functions on the Complex Half-Plane

“…The classical Hardy and weighted Bergman spaces defined on the open unit disk of the complex plane and the open right complex half-plane may be viewed as discrete and continuous counterparts (this is discussed for example in [13] and [14]). The continuous case is sometimes more appropriate when we consider applications of these spaces (see [10], [11] and [15]).…”

“…then we say that µ is a Carleson measure for A p ν . The notion of a Carleson measure has been introduced by Lennart Carleson in [2] to solve the corona problem, and since then has found many other applications (for example, in the context of spaces of analytic functions on the half-plane, it is used to describe the admissibility criterion for control and observation operators, see [10], [11], [14], [15]). Lemma 1.…”