On Laplace-Carleson embedding theorems

TL;DR

This paper establishes embedding theorems for weighted Bergman spaces, generalizing classical Carleson theorems, and explores applications to control theory via Laplace transform-induced embeddings in the right half-plane.

Contribution

It introduces new embedding theorems for a broad class of weighted Bergman spaces and characterizes bounded embeddings induced by Laplace transforms in control theory contexts.

Findings

Generalized classical Carleson embedding theorems

Provided necessary and sufficient conditions for bounded embeddings

Analyzed embeddings with measures supported on sectors or strips

Abstract

This paper gives embedding theorems for a very general class of weighted Bergman spaces: the results include a number of classical Carleson embedding theorems as special cases. We also consider little Hankel operators on these Bergman spaces. Next, a study is made of Carleson embeddings in the right half-plane induced by taking the Laplace transform of functions defined on the positive half-line (these embeddings have applications in control theory): particular attention is given to the case of a sectorial measure or a measure supported on a strip, and complete necessary and sufficient conditions for a bounded embedding are given in many cases.