Spherical maximal operator on symmetric spaces of constant curvature

Nevo, Amos; Ratnakumar, P.

doi:10.1090/S0002-9947-02-03095-7

Spherical maximal operator on symmetric spaces of constant curvature
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by Amos Nevo and P. K. Ratnakumar

Trans. Amer. Math. Soc. 355 (2003), 1167-1182

DOI: https://doi.org/10.1090/S0002-9947-02-03095-7

Published electronically: 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 $f$, \[ \|{\mathcal M}f\|_{ n’,\infty }\leq C_n \|f \|_{n’,1}, n^\prime =\frac {n}{n-1}.\] The proof uses only geometric arguments and volume estimates, and applies uniformly in every dimension.

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Spherical maximal operator on symmetric spaces of constant curvature
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Abstract: