Applications of an inverse Abel transform for Jacobi analysis: weak-\(L^1\) estimates and the Kunze-Stein phenomenon
type: Item
instance of: scholarly article
title: Applications of an inverse Abel transform for Jacobi analysis: weak-\(L^1\) estimates and the Kunze-Stein phenomenon
zbMATH Open document ID: 1404.43012
author: Takeshi Kawazoe
published in: Tokyo Journal of Mathematics
publication date: 2018-10-26T00:00:00Z
review text: It is known and truly justified that the Abel transform reduces the convolution * to the Euclidean convolution, which in general can be spelled in the sense that the Hardy-Littlewood maximal operator \(T\), the Poisson maximal operator or the Littlewood-Paley g-function for the Jacobi hypergroup are defined by using the convolution *. Theorems 1, 2, and 3 are generalizations, based on the concept of the harmonic analysis on \(G\) (a non-compact semisimple Lie group) of real rank one corresponds to the Jacobi analysis, as a result of which these weak-\(L^1(\Delta)\) estimates and the Kunze-Stein phenomena hold for the Jacobi hypergroup with special \(\alpha\), \(\beta\), in the sense that the proofs are given for more general \(\alpha\), \(\beta\). Notations used in these theorems are explained. One may refer to \textit{J. Liu} [Bull. Lond. Math. Soc. 32, No. 5, 582--588 (2000; Zbl 1029.42012); Proc. Am. Math. Soc. 133, No. 6, 1817--1821 (2005; Zbl 1062.43022)], cited in the references on page 111 of the present paper, for simple proofs of Theorems 1 and 3, given on page 78 of the paper. Derivations of some other simple proofs of the theorems are the specific objective of the paper, which is based on an inversion formula for the Abel transform [\textit{T. Kawazoe}, Anal. Theory Appl. 25, No. 3, 201--229 (2009; Zbl 1212.22007)]. A standard shape of \(Tf\) for \(f\in L^1(\Delta)\) is given, from which its weak-\(L^1\) is estimated. Instead of Liu [loc. cit.] using the explicit form of the kernel of the convolution, the author of the present paper gives a proof without using the kernel form.
reviewed by: Deshna Loonker
Mathematics Subject Classification ID: 43A62
Mathematics Subject Classification ID: 22E30
Mathematics Subject Classification ID: 44A99
Mathematics Subject Classification ID: 44A35
zbMATH DE Number: 6966860
zbMATH Keywords: Abel transform
zbMATH Keywords: Kunze-Stein phenomenon
zbMATH Keywords: Jacobi analysis
zbMATH Keywords: weak \(L^1\) estimates
zbMATH Keywords: Poisson maximal operator
zbMATH Keywords: convolution
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.3836/tjm/1502179242
OpenAlex ID: W2768625076
cites work: Sharp estimates for some functions of the Laplacian on noncompact symmetric spaces
cites work: Heat kernel and Green function estimates on noncompact symmetric spaces
cites work: Q4833863
cites work: The Kunze-Stein phenomenon
cites work: Q4213399
cites work: L p -multipliers for Noncompact Symmetric Spaces
cites work: The convolution structure for Jacobi function expansions
cites work: On L(p,q) spaces
cites work: An endpoint estimate for the Kunze-Stein phenomenon and related maximal operators
cites work: H 1-estimates of Littlewood-Paley and Lusin functions for Jacobi analysis
cites work: On a weak \(L^1\) property of maximal operators on non-compact semisimple Lie groups
cites work: A new proof of a Paley-Wiener type theorem for the Jacobi transform
cites work: Maximal Functions Associated with The Jacobi Transform
cites work: The Kunze-Stein phenomenon associated with Jacobi transforms
cites work: Weak type \(L^ 1\) estimates for maximal functions on non-compact symmetric spaces
DOI: 10.3836/TJM/1502179242
Recommended article: The Kunze-Stein phenomenon associated with Jacobi transforms
Recommended article: On estimates for the Jacobi transform in the space $L^{p}(\mathbb{R}^{+}, J^{\alpha,\beta}(x)dx)$
Recommended article: Q4892686
Recommended article: H 1-estimates of Littlewood-Paley and Lusin functions for Jacobi analysis
Recommended article: An analogue of Beurling's theorem for the Jacobi transform
Recommended article: Approximation of the multidimensional Jacobi transform in \(L_2\) by partial integrals
Recommended article: AnLpversion of Hardy's theorem for the Jacobi–Dunkl transform
Recommended article: On some theorems of the Jacobi-Lipschitz class for the Jacobi transform
Recommended article: Q4011673
Recommended article: The finite continuous Jacobi transform and its inverse
description: scientific article; zbMATH DE number 6966860