Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



The Kunze-Stein phenomenon associated with Jacobi transforms

Author: Jianming Liu
Journal: Proc. Amer. Math. Soc. 133 (2005), 1817-1821
MSC (2000): Primary 33C45; Secondary 43A90, 42B25, 22E30
Published electronically: January 14, 2005
MathSciNet review: 2120282
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Recently A. D. Ionescu (2000) established the endpoint estimate for the Kunze-Stein phenomenon, which states that if $G$ is a noncompact connected semisimple Lie group of real rank one with finite center, then

\begin{displaymath}L^{2,1}(G)\ast L^{2,1}(G)\subseteq L^{2,\infty }(G). \end{displaymath}

In this paper, we will prove the corresponding result for the Jacobi transform. Our method is analytical, in which we do not use the structure of Lie groups.

References [Enhancements On Off] (What's this?)

  • [1] M. Cowling, The Kunze-Stein phenomenon, Ann. of Math. 107 (1978), 209-234. MR 0507240 (58:22398)
  • [2] M. Cowling, Herz's ``principe de majoration" and the Kunze-Stein phenomenon, Harmonic analysis and Number Theory, CMS Conf. Proc. 21, A. M. S., Providence, RI, 1997, pp. 73-88. MR 1472779 (98k:22040)
  • [3] A. D. Ionescu, An endpoint estimate for the Kunze-Stein phenomenon and the related maximal operators, Ann. of Math. 152 (2000), 259-275.MR 1792296 (2001m:22017)
  • [4] T. H. Koornwinder, Jacobi functions and analysis on noncompact semisimple Lie Groups, Special functions, R. Askey et al. (eds.), D. Reidel Publishing Company, Dordrecht, 1984, pp. 1-84. MR 0774055 (86m:33018)
  • [5] J. Liu, Maximal functions associated with the Jacobi transform, Bull. of London Math. Soc. 32(5) (2000), 582-588. MR 1767711 (2001e:42028)
  • [6] N. Lohoué and T. Rychener, Some function spaces on symmetric spaces related to convolution operators, J. Function. Anal. 55 (1984), 200-219.MR 0733916 (85d:22024)
  • [7] R. A. Hunt, On $L(p,q)$ Spaces, L'Enseignement Math. 12 (1966), 249-276.MR 0223874 (36:6921)
  • [8] E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Math. Series, No 32, Princeton Univ. Press, Princeton, NJ, 1971.MR 0304972 (46:4102)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 33C45, 43A90, 42B25, 22E30

Retrieve articles in all journals with MSC (2000): 33C45, 43A90, 42B25, 22E30

Additional Information

Jianming Liu
Affiliation: LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, People’s Republic of China

Keywords: Kunze-Stein phenomenon, Jacobi transform, Lorentz space
Received by editor(s): July 19, 2003
Received by editor(s) in revised form: February 24, 2004
Published electronically: January 14, 2005
Additional Notes: This research was supported by the National Natural Science Foundation of China, Projects 10001002 and 10371004
Communicated by: Andreas Seeger
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society