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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Spherical maximal operator on symmetric spaces of constant curvature


Authors: Amos Nevo and P. K. Ratnakumar
Journal: Trans. Amer. Math. Soc. 355 (2003), 1167-1182
MSC (2000): Primary 43A85; Secondary 43A18
Published electronically: October 30, 2002
MathSciNet review: 1938751
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Abstract | References | Similar Articles | Additional Information

Abstract: We prove an endpoint weak-type maximal inequality for the spherical maximal operator applied to radial funcions on symmetric spaces of constant curvature and dimension $n\ge 2$. More explicitly, in the Lorentz space associated with the natural isometry-invariant measure, we show that, for every radial function $f$,

\begin{displaymath}\Vert{\mathcal M}f\Vert _{\,n^{\prime},\infty}\leq C_n \Vert f \Vert _{n^{\prime},1},\,\,\,\, n^\prime=\frac{n}{n-1}.\end{displaymath}

The proof uses only geometric arguments and volume estimates, and applies uniformly in every dimension.


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  • [B1] J. BOURGAIN, Averages in the plane over convex curves and maximal operators, J. d'Analyse Math. 47 (1986), 69 - 85. MR 88f:42036
  • [B2] J. BOURGAIN, Estimations de certaines fonctions maximales. C. R. Acad. Sci. Paris, Sér. I Math. 301, 499-502, 1985. MR 87b:42023
  • [BS] C. BENNETT and R. SHARPLEY, Interpolation of Operators, Pure and Applied Math., vol. 129, Academic Press, New York, 1988. MR 89e:46001
  • [C] M. COWLING, On Littlewood-Paley-Stein theory, Proceedings of the seminar in Harmonic Analysis (Pisa, 1980) Rend. Circ. Math. Palermo (2) (1981) suppl. 1, pp. 21-55. MR 83h:42024
  • [CN] M. COWLING and A. NEVO, Uniform estimates for spherical functions on complex semisimple Lie groups. Geometric and Functional Analysis 11 (2001), 900-932.
  • [Ch] I. CHAVEL, Eigenvalues in Riemannian Geometry, Academic Press, 1984. MR 86g:58140
  • [FJ-K] M. FLENSTED-JENSEN AND T. H. KOORNWINDER, The convolution structure for Jacobi function expansion, Arkiv für Matematik, vol. 11 (1973) pp. 245-262. MR 49:5688
  • [H1] S. HELGASON, Differential geometry, Lie groups and symmetric spaces, Academic Press, New York, 1978. MR 80k:53081
  • [H2] S. HELGASON, Geometric analysis on symmetric spaces, Amer. Math. Soc. Mathematical Surveys and Monographs Series, Vol. 39, 1994. MR 96h:43009
  • [Hi] N. J. HICKS, Notes on Differential Geometry, Van Nostrand, 1965. MR 31:3936
  • [I] A. D. IONESCU, Fourier integral operators on non-compact symmetric spaces of real rank one. J. Funct. Anal., vol. 174, pp. 274-300 (2000). MR 2001h:43009
  • [Iv] B. IVERSEN, Hyperbolic geometry, London Math. Soc. Student Texts, vol. 25, Cambridge Univ. Press, 1992. MR 94b:51023
  • [KR] R. KERMAN and P. K. RATNAKUMAR, Spherical means of radial functions, Preprint.
  • [K] T. H. KOORNWINDER, Jacobi functions and analysis on non-compact semisimple Lie groups. In : R. A. Askey et al. (eds.), Special Functions : Group Theoretical Aspects and Applications, pp. 1-85 , 1984, D. Reidel Publishing Company. MR 86m:33018
  • [L] M. LECKBAND, A note on the spherical maximal operator for radial functions, Proc. Amer. Math. Soc. vol. 100 (1987) pp. 635-640. MR 88i:42032
  • [N1] A. NEVO, Pointwise ergodic theorem for radial averages on simple Lie groups I, Duke Math J. 76 (1994), 113 - 140. MR 96c:28027
  • [N2] A. NEVO, Pointwise ergodic theorem for radial averages on simple Lie groups II, Duke Math J. 86 (1997), 239-259. MR 98m:28041
  • [NS] A. NEVO and E. M. STEIN, Analogs of Wiener's ergodic theorems for semisimple groups I, Ann. of Math. 145 (1997) 565-595. MR 98m:22007
  • [NT] A. NEVO and S. THANGAVELU, Pointwise ergodic theorems for radial averages on the Heisenberg group, Adv. in Math. vol. 127 (1997) pp. 307-334. MR 98f:22005
  • [S1] E. M. STEIN, Maximal functions: spherical means, Proc. Nat. Acad. Sci. U.S.A, 73 (1976), 2174 - 2175. MR 54:8133a
  • [S2] E. M. STEIN, Harmonic Analysis : Real Variable Methods, Orthogonality and Oscillatory Integrals, Princeton University Press, Princeton, New Jersey, 1993. MR 95c:42002
  • [SWa] E. M. STEIN and S. WAINGER, Problems in harmonic analysis related to curvature, Bull. Amer. Math. Soc. vol. 84, (1978) pp. 1239-1295. MR 80k:42023
  • [SWe] E. M. STEIN and G. WEISS, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Univ. Press, Princeton, N.J. (1971). MR 46:4102
  • [S] J. O. STROMBERG, Weak type $L^1$ estimate for maximal functions on non compact symmetric spaces, Ann. of Math., vol. 114 (1981) 115 - 126. MR 82k:43010
  • [T] A. TORCHINSKY, Real Variable Methods in Harmonic Analysis, Academic Press, San Diego (1986). MR 88e:42001
  • [W] N. J. WILDBERGER, Hypergroups, symmetric spaces, and wrapping maps. In : Probability measures on groups and related structures, vol. XI, pp. 406-425, World Sci. Publishing, River Edge, NJ, 1995. MR 98b:43014

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Additional Information

Amos Nevo
Affiliation: Institute of advanced studies in mathematics, Technion–Israel Institute of Technology, Haifa 32900, Israel
Email: anevo@tx.technion.ac.il

P. K. Ratnakumar
Affiliation: Institute of advanced studies in mathematics, Technion–Israel Institute of Technology, Haifa 32900, Israel
Email: pkrsm@uohyd.ernet.in

DOI: http://dx.doi.org/10.1090/S0002-9947-02-03095-7
PII: S 0002-9947(02)03095-7
Keywords: Symmetric spaces, constant curvature, spherical means, maximal function
Received by editor(s): June 5, 2000
Published electronically: October 30, 2002
Additional Notes: The first author was supported by Technion V.P.R. fund—E. and J. Bishop research fund, and the second author was supported by the fund for the promotion of research at the Technion.
Article copyright: © Copyright 2002 American Mathematical Society