Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-25T04:31:57.896Z Has data issue: false hasContentIssue false
  • English
  • Français

Regularity of spherical means and localization of spherical harmonic expansions

Part of: Nontrigonometric harmonic analysis Abstract 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
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

References

[1]Askey, R., Orthogonal polynomials and special functions (Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, 1975).CrossRefGoogle Scholar
[2]Bastis, A. I., ‘Almost everywhere convergence of expansions of the Laplace operator on the sphere’, Mat. Zametki 33 (1983), 857–862.Google Scholar
[3]Cazzaniga, F. and Giacalone, E., ‘Hardy's inequality for compact symmetric spaces’, Boll. Un. Mat. Ital. (6) 2-A (1983), 381–388.Google Scholar
[4]Coifman, R. R. and Weiss, G., Analyse harmonique non commutative sur certains espaces homogènes (Lecture Notes 242, Springer-Verlag, Berlin, Heidelberg and New York, 1971).CrossRefGoogle Scholar
[5]Coifman, R. R. and Weiss, G., ‘Extension of Hardy spaces and their use in analysis’, Bull. Amer. Math. Soc. 83 (1977), 569–645.CrossRefGoogle Scholar
[6]Colzani, L., ‘Hardy spaces on unit spheres’, Boll. Un. Mat. Ital., to appear.Google Scholar
[7]Cowling, M. and Mauceri, G., ‘On maximal functions’, Rend. Sem. Mat. Fis. Milano 49 (1979), 79–88.CrossRefGoogle Scholar
[8]Dreseler, B., ‘Norms of zonal spherical functions and Fourier series on compact symmetric spaces’, J. Funct. Anal. 44 (1981), 74–86.CrossRefGoogle Scholar
[9]Gangolli, R., ‘Positive definite kernels on homogeneous spaces and certain stochastic processes related to Levy's Brownian motion of several parameters’, Ann. Inst. H. Poincarè Sect. B 3 (1967), 121–225.Google Scholar
[10]Helgason, S., Differential geometry and symmetric spaces (Academic Press, New York and London, 1962).Google Scholar
[11]Meaney, C., ‘Divergent Jacobi polynomial series’, Proc. Amer. Math. Soc. 87 (1983), 459–462.CrossRefGoogle Scholar
[12]Meaney, C., ‘Localization of spherical harmonic expansions’, Mh. Math. 98 (1984), 65–74.CrossRefGoogle Scholar
[13]Oberlin, D. M. and Stein, E. M., ‘Mapping properties of the Radon transform’, Indiana Univ. Math. J. 31 (1982), 641–650.CrossRefGoogle Scholar
[14]Peyrière, J. and Sjölin, P., ‘Regularity of spherical means’, Ark. Mat. 16 (1978), 117–126.CrossRefGoogle Scholar
[15]Sjölin, P., ‘Regularity and integrability of spherical means’, Mh. Math. 96 (1983), 277–291.CrossRefGoogle Scholar
[16]Stein, E. M., ‘Maximal functions: spherical means’, Proc. Nat. Acad. Sci. U.S.A. 73 (1976), 2174–2175.CrossRefGoogle ScholarPubMed
[17]Szegö, G., Orthogonal polynomials (3rd edition, Amer. Math. Soc. Colloquium Publ. 23, 1967).Google Scholar