Hilbertian Hardy-Sobolev spaces on a half-plane

In this paper we deal with a scale of reproducing kernel Hilbert spaces \(H^{(n)}_2\), \(n\ge 0\), which are linear subspaces of the classical Hilbertian Hardy space on the right-hand half-plane \(\mathbb{C}^+\). They are obtained as ranges of the Laplace transform in extended versions of the Paley-...

Full description

Saved in:
Bibliographic Details
Published in:arXiv.org 2024-01
Main Authors: Galé, José E, Matache, Valentin, Miana, Pedro J, Sánchez--Lajusticia, Luis
Format: Article
Language:English
Subjects:
Composition
Continuity (mathematics)
Half planes
Hilbert space
Operators (mathematics)
Sobolev space
Subspaces
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:In this paper we deal with a scale of reproducing kernel Hilbert spaces \(H^{(n)}_2\), \(n\ge 0\), which are linear subspaces of the classical Hilbertian Hardy space on the right-hand half-plane \(\mathbb{C}^+\). They are obtained as ranges of the Laplace transform in extended versions of the Paley-Wiener theorem which involve absolutely continuous functions of higher degree. An explicit integral formula is given for the reproducing kernel \(K_{z,n}\) of \(H^{(n)}_2\), from which we can find the estimate \(\Vert K_{z,n}\Vert\sim\vert z\vert^{-1/2}\) for \(z\in\mathbb{C}^+\). Then composition operators \(C_\varphi :H_2^{(n)} \to H_2^{(n)}\), \(C_\varphi f=f\circ \varphi \), on these spaces are discussed, giving some necessary and some sufficient conditions for analytic maps \(\varphi: \mathbb{C}^+\to \mathbb{C}^+\) to induce bounded composition operators.
ISSN:2331-8422