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Spherical maximal operator on symmetric spaces of constant curvature

Author(s): Amos Nevo; P. K. Ratnakumar
Journal: Trans. Amer. Math. Soc. 355 (2003), 1167-1182.
MSC (2000): Primary 43A85; Secondary 43A18
Posted: October 30, 2002
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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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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: 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
Posted: 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.
Copyright of article: Copyright 2002, American Mathematical Society

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