Tokyo Journal of Mathematics

Applications of an Inverse Abel Transform for Jacobi Analysis: Weak-$L^1$ Estimates and the Kunze-Stein Phenomenon


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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.

Article information

Tokyo J. Math., Volume 41, Number 1 (2018), 77-112.

First available in Project Euclid: 20 November 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 22E30: Analysis on real and complex Lie groups [See also 33C80, 43-XX]
Secondary: 43A62: Hypergroups


KAWAZOE, Takeshi. Applications of an Inverse Abel Transform for Jacobi Analysis: Weak-$L^1$ Estimates and the Kunze-Stein Phenomenon. Tokyo J. Math. 41 (2018), no. 1, 77--112. doi:10.3836/tjm/1502179242.

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