Hyperfunctions and harmonic analysis on symmetric spaces
type: Item
instance of: scholarly article
title: Hyperfunctions and harmonic analysis on symmetric spaces
zbMATH Open document ID: 0555.43002
author: Henrik Schlichtkrull
published in: Progress in Mathematics
publication date: 1984-01-01T00:00:00Z
review text: Ever since \textit{M. Kashiwara}, \textit{A. Kowata}, \textit{K. Minomura}, \textit{K. Okamoto}, \textit{T. Oshima} and \textit{M. Tanaka} in [Ann. Math., II. Ser. 107, 1-39 (1978; Zbl 0377.43012)] have demonstrated the strength of an approach based on the use of the theory of hyperfunctions to problems in harmonic analysis by proving Helgason's conjecture, and appreciation for that technique has been growing among harmonic analysis. The book reviewed is apparently the first monograph devoted specifically to applications of the theory of hyperfunctions in harmonic analysis. The applications considered are based on results concerning systems of PDEs with regular singularity and used for solving two problems in the analysis on symmetric spaces. The first is that of representing eigenfunctions of invariant differential operators on symmetric space as Poisson transforms of their hyperfunction boundary values. This was originally posed by \textit{S. Helgason} [Adv. Math. 5, 1-154 (1970; Zbl 0209.254)] in the context of Riemannian symmetric spaces and in its full generality remained open untill 1978, when it was positively answered in the remarkable work of Kashiwara et al. quoted above. The solution has been preceded by some partial results by Helgason and several teams composed of certain of the future coauthors of the final and conclusive version. The book of Schlichtkrull brings (with some modifications) an original proof of Kashiwara's et al. solution to the Helgason conjecture in an almost complete form - almost referring to the omission of a certain set of singular eigenvalues, what would have required yet more refined machinery, along with some extensions due to the author. The book is split into 8 chapters each concluded with a section of historical and bibliographical notes - these are: 1) Hyperfunctions and microlocal analysis - an introduction, 2) Differential equations with regular singularities, 3) Riemannian symmetric spaces and invariant differential opera…
Mathematics Subject Classification ID: 43-02
Mathematics Subject Classification ID: 43A85
Mathematics Subject Classification ID: 46F15
Mathematics Subject Classification ID: 22E46
Mathematics Subject Classification ID: 22E45
Mathematics Subject Classification ID: 58J15
zbMATH DE Number: 3884738
zbMATH Keywords: hyperfunctions
zbMATH Keywords: PDEs with regular singularity
zbMATH Keywords: symmetric spaces
zbMATH Keywords: eigenfunctions
zbMATH Keywords: invariant differential operators
zbMATH Keywords: Poisson transforms
zbMATH Keywords: Helgason conjecture
zbMATH Keywords: compactification
zbMATH Keywords: boundary values
zbMATH Keywords: Flensted-Jensen discrete series
reviewed by: A. Strasburger
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 3884738