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June 2018 Applications of an Inverse Abel Transform for Jacobi Analysis: Weak-$L^1$ Estimates and the Kunze-Stein Phenomenon
Takeshi KAWAZOE
Tokyo J. Math. 41(1): 77-112 (June 2018). DOI: 10.3836/tjm/1502179242

Abstract

For the Jacobi hypergroup $({\bf R}_+,\Delta,*)$, the weak-$L^1$ estimate of the Hardy-Littlewood maximal operator was obtained by W. Bloom and Z. Xu, later by J. Liu, and the endpoint estimate for the Kunze-Stein phenomenon was obtained by J. Liu. In this paper we shall give alternative proofs based on the inverse Abel transform for the Jacobi hypergroup. The point is that the Abel transform reduces the convolution $*$ to the Euclidean convolution. More generally, let $T$ be the Hardy-Littlewood maximal operator, the Poisson maximal operator or the Littlewood-Paley $g$-function for the Jacobi hypergroup, which are defined by using $*$. Then we shall give a standard shape of $Tf$ for $f\in L^1(\Delta)$, from which its weak-$L^1$ estimate follows. Concerning the endpoint estimate of the Kunze-Stein phenomenon, though Liu used the explicit form of the kernel of the convolution, we shall give a proof without using the kernel form.

Citation

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Takeshi KAWAZOE. "Applications of an Inverse Abel Transform for Jacobi Analysis: Weak-$L^1$ Estimates and the Kunze-Stein Phenomenon." Tokyo J. Math. 41 (1) 77 - 112, June 2018. https://doi.org/10.3836/tjm/1502179242

Information

Published: June 2018
First available in Project Euclid: 20 November 2017

zbMATH: 06966860
MathSciNet: MR3830810
Digital Object Identifier: 10.3836/tjm/1502179242

Subjects:
Primary: 22E30
Secondary: 43A62

Rights: Copyright © 2018 Publication Committee for the Tokyo Journal of Mathematics

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Vol.41 • No. 1 • June 2018
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