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Regularity of spherical means and localization of spherical harmonic expansions

Part of: Abstract harmonic analysis Nontrigonometric harmonic analysis

Published online by Cambridge University Press:  09 April 2009

Leonardo Colzani
Leonardo Colzani
Affiliation:
Dipartimento di Matematica “F. Enriques”, Università degli Studi di Milano, via C. Saldini, 50 20133 Milano, Italia
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Abstract

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Let G/K be a compact symmetric space, and let G = KAK be a Cartan decomposition of G. For f in L1(G) we define the spherical means f(g, t) = ∫k∫k ∫(gktk′) dk dk′, g ∈ G, t ∈ A. We prove that if f is in Lp(G), 1 ≤ p ≤ 2, then for almost every g ∈ G the functions t → f(g, t) belong to certain Soblev spaces on A. From these regularity results for the spherical means we deduce, if G/K is a compact rank one symmetric space, a theorem on the almost everywhere localization of spherical harmonic expansions of functions in L2 (G/K).

Keywords

symmetric spacesspherical meanslocalization

MSC classification

Secondary: 42C10: Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.) 43A15: $L^p$-spaces and other function spaces on groups, semigroups, etc.
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 41 , Issue 3 , December 1986 , pp. 287 - 297
Copyright
Copyright © Australian Mathematical Society 1986

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