Applications of an inverse Abel transform for Jacobi analysis: weak-\(L^1\) estimates and the Kunze-Stein phenomenon
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.
scientific article; zbMATH DE number 6966860
scientific article