## Tokyo Journal of Mathematics

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

Takeshi KAWAZOE

#### 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.

#### Article information

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

Dates
First available in Project Euclid: 20 November 2017

https://projecteuclid.org/euclid.tjm/1511221566

Mathematical Reviews number (MathSciNet)
MR3830810

Zentralblatt MATH identifier
06966860

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

#### Citation

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. https://projecteuclid.org/euclid.tjm/1511221566

#### References

• J.-Ph. Anker, Sharp estimates for some functions of the Laplacian on noncompact symmetric spaces, Duke Math. J. 65 (1992), 257–297.
• J.-Ph. Anker and L. Ji, Heat kernel and green function estimates for noncompact symmetric spaces, Geom. Funct. Anal. 9 (1999), 1035–1091.
• W. R. Bloom and Z. Xu, The Hardy-Littlewood maximal function for Chebli-Trimeche hypergroups, Contemp. Math. 183 (1995), 45–70.
• M. Cowling, The Kunze-Stein phenomenon, Ann. of Math. 107 (1978), 209–234.
• M. Cowling, Hertz's “principle de majoration” and Kunze-Stein phenomenon, Harmonic Analysis and Number Theory, CMS Conf. Proc. 21, A. M. S., Providence, RI, 1997, 73–88.
• J. C. Clerc and E. M. Stein, $L^p$-multipliers for non-compact symmetric spaces, Proc. Nat. Acad. Sci. USA 71 (1974), 3911–3912.
• M. Flensted-Jensen and T. Koornwinder, The convolution structure for Jacobi function expansions, Ark. Mat. 11 (1973), 245–262.
• R. A. Hunt, On $L(p,q)$ spaces, L'Enseignement Math. 12 (1966), 249–276.
• A. D. Ionescu, An endpoint estimate for the Kunze-Stein phenomenon and the related maximal operators, Ann. of Math. 152 (2000), 259–275.
• T. Kawazoe, $H^1$-estimates of the Littlewood-Paley and Lusin functions for Jacobi analysis, Anal. Theory Appl. 25 (2009), 201–229.
• T. Kawazoe and J. Liu, On a weak $L^1$ property of maximal operators on non-compact semisimple Lie groups, Tokyo J. Math. 25 (2002), 165–180.
• T. Koornwinder, A new proof of a Paley-Wiener type theorem for the Jacobi transform, Ark. Mat. 13 (1975), 145–159.
• J. Liu, Maximal functions associated with the Jacobi transform, Bull. London Math. 32 (2000), 1–7.
• J. Liu, The Kunze-Stein phenomenon associated with Jacobi transforms, Proc. Amer. Math. Soc. 133 (2005), 1817–1821.
• J-O. Strömberg, Weak type $L^1$ estimates for maximal functions on non-compact symmetric spaces, Ann. Math. 114 (1981), 115–126.