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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: 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

P. K. Ratnakumar
Affiliation: Institute of advanced studies in mathematics, Technion–Israel Institute of Technology, Haifa 32900, Israel

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

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