The inverse Abel transform for SU(p,q)
type: Item
instance of: scholarly article
title: The inverse Abel transform for SU(p,q)
zbMATH Open document ID: 0608.43008
DOI: 10.1007/BF02384394
author: Christopher Meaney
published in: Arkiv för Matematik
publication date: 1986-01-01T00:00:00Z
review text: Let G/K be a Riemannian symmetric space of the noncompact type and \(G=KAN\) be an Iwasawa decomposition. The Abel transform is the mapping \(f\mapsto F_ f\) of \(C_ c^{\infty}(G//K)\) in \(C_ c^{\infty}(A)^ W\) (W is the Weyl group) which is defined by the formula \[ F_ f(a)=e^{\rho (\log a)}\int_{N}f(an) dn\quad. \] It is known that this mapping is an isomorphism (Gangolli). In the reviewed paper the inverse transform in the case \(G=SU(p,q)\) is described. It is a composition of some differential operator, a tensor product of fractional integrals (in number dim A) and multiplication with some fixed function. The proof is based on an explicit formula of Berezin and Karpelevich for the spherical functions. Moreover there is obtained an inversion formula for the Radon transform acting on K-invariant functions on the tangent space for G/K, \(G=SU(p,q)\). As an application of this latter formula a local regularity property for the corresponding Fourier transform is demonstrated.
Mathematics Subject Classification ID: 43A85
Mathematics Subject Classification ID: 44A15
Mathematics Subject Classification ID: 43A22
zbMATH DE Number: 3983837
zbMATH Keywords: semisimple Lie group
zbMATH Keywords: spherical transform
zbMATH Keywords: Riemannian symmetric space
zbMATH Keywords: Iwasawa decomposition
zbMATH Keywords: Abel transform
zbMATH Keywords: Radon transform
reviewed by: V. F. Molchanov
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3983837