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# The Weak Type (1,1) Estimates of Maximal Functions on the Laguerre Hypergroup

Published:2010-06-11
Printed: Sep 2010
• Huang Jizheng,
College of Sciences, North China University of Technology, Beijing 100144, China
• Heping Liu,
LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, China
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## Abstract

In this paper, we discuss various maximal functions on the Laguerre hypergroup $\mathbf{K}$ including the heat maximal function, the Poisson maximal function, and the Hardy--Littlewood maximal function which is consistent with the structure of hypergroup of $\mathbf{K}$. We shall establish the weak type $(1,1)$ estimates for these maximal functions. The $L^p$ estimates for $p>1$ follow from the interpolation. Some applications are included.
 Keywords: Laguerre hypergroup, maximal function, heat kernel, Poisson kernel
 MSC Classifications: 42B25 - Maximal functions, Littlewood-Paley theory 43A62 - Hypergroups

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