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    This article has been cited by the following publications. This list is generated based on data provided by CrossRef.
    Mejjaoli, Hatem and Trimèche, Khalifa 2016. Qualitative uncertainty principles for the generalized Fourier transform associated to a Dunkl type operator on the real line. Analysis and Mathematical Physics, Vol. 6, Issue. 2, p. 141.
    Mejjaoli, Hatem 2015. Cowling–Price’s and Hardy’s uncertainty Principles for the generalized Fourier transform associated to a Cherednik type operator on the real line. Archiv der Mathematik, Vol. 104, Issue. 4, p. 377.
    Chabeh, W. Mourou, M.A. and Rebhi, S. 2014. Hardy and Cowling–Price theorems associated with the Jacobi–Cherednik operator. Bulletin des Sciences Mathématiques, Vol. 138, Issue. 5, p. 602.
    Mejjaoli, Hatem 2014. Qualitative uncertainty principles for the Opdam–Cherednik transform. Integral Transforms and Special Functions, Vol. 25, Issue. 7, p. 528.
    Mejjaoli, Hatem and Trimèche, Khalifa 2008. A variant of Cowling–Price's theorem for the Dunkl transform on <mml:math altimg="si1.gif" overflow="scroll" xmlns:xocs="http://www.elsevier.com/xml/xocs/dtd" xmlns:xs="http://www.w3.org/2001/XMLSchema" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns="http://www.elsevier.com/xml/ja/dtd" xmlns:ja="http://www.elsevier.com/xml/ja/dtd" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:tb="http://www.elsevier.com/xml/common/table/dtd" xmlns:sb="http://www.elsevier.com/xml/common/struct-bib/dtd" xmlns:ce="http://www.elsevier.com/xml/common/dtd" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:cals="http://www.elsevier.com/xml/common/cals/dtd"><mml:mi mathvariant="double-struck">R</mml:mi></mml:math>. Journal of Mathematical Analysis and Applications, Vol. 345, Issue. 2, p. 593.
    Fitouhi, Ahmed Bettaibi, Néji and Bettaieb, Rym H. 2008. On Hardy’s inequality for symmetric integral transforms and analogous. Applied Mathematics and Computation, Vol. 198, Issue. 1, p. 346.
    Attour, Latifa Bou and Trimèche, Khalifa 2005. Uncertainty principle and (L p , L q )sufficient pairs on Chébli–Trimèche hypergroups. Integral Transforms and Special Functions, Vol. 16, Issue. 8, p. 625.
    CHETTAOUI, C. OTHMANI, Y. and TRIMÈCHE, K. 2004. AN ANALOGUE OF COWLING–PRICE'S THEOREM AND HARDY'S THEOREM FOR THE GENERALIZED FOURIER TRANSFORM ASSOCIATED WITH THE SPHERICAL MEAN OPERATOR. Analysis and Applications, Vol. 02, Issue. 03, p. 177.
    Gallardo, Léonard and Trimèche, Khalifa 2004. An LP version of Hardy's theorem for the Dunkl transform. Journal of the Australian Mathematical Society, Vol. 77, Issue. 03, p. 371.
    Chouchane*, F. Mili†, M. and Trimèche‡, K. 2004. AnLpversion of Hardy's theorem for the Jacobi–Dunkl transform. Integral Transforms and Special Functions, Vol. 15, Issue. 3, p. 225.
    Ben Farah, Slaim and Mokni, Kamel 2003. Uncertainty principle and Lp–Lq-sufficient pairs on noncompact real symmetric spaces. Comptes Rendus Mathematique, Vol. 336, Issue. 11, p. 889.
    Kawazoe, T. and Liu, Jianming 2003. Heat Kernel and Hardy's Theorem for Jacobi Transform. Chinese Annals of Mathematics, Vol. 24, Issue. 03, p. 359.
    Gallardo, Léonard and Trimèche, Khalifa 2002. Un analogue d'un théorème de Hardy pour la transformation de Dunkl. Comptes Rendus Mathematique, Vol. 334, Issue. 10, p. 849.
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An Lp Version of the Hardy Theorem for Motion Groups

  • Masaaki Eguchi (a1), Shin Koizumi (a2) and Keisaku Kimahara (a3)
Abstract

We describe a generalization of the Hardy theorem on the motion group. We prove that for some weight functions νω growing very rapidly and a measurable function f, the finiteness of the Lp-norm of vf and the Lq-norm of ωf implies f=0 (almost everywhere).

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COPYRIGHT: © Australian Mathematical Society 2000
References
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[1]Amrein, W. O. and Berthier, A., ‘On support properties of Lp -functions and their Fourier transforms’, J. Funct. Anal. 24 (1977), 258–267.
[2]Cowling, M. G. and Price, J. F., Generalizations of Heisenberg's inequalify, Lecture Notes in Math. 992 (Springer, Berlin, 1983) pp. 443–449.
[3]Dym, H. and McKean, H. P., Fourier series and integral (Academic Press, New York, 1972).
[4]Eguchi, M., Koizumi, S. and Kumahara, K., ‘An analogue of the Hardy theorem for the Cartan motion group’, Proc. Japan Acad. 74 (1998), 149–151.
[5]Eguchi, M., Kumahara, K. and Muta, Y., ‘A subspace of Schwartz space on motion group’, Hiroshima Math. J. 10 (1980), 691–698.
[6]Kumahara, K., ‘Fourier transforms on the motion groups’, J. Math. Soc. Japan. 26 (1976), 18–32.
[7]Sitaram, A. and Sundari, M., ‘An analogue of Hardy's theorem for very rapidly decreasing functions on semi-simple Lie groups’, Pacific J. Math. 177 (1997), 187–200.
[8]Sundari, M., ‘Hardy's theorem for the n-dimensional Euclidean motion group’, Proc. Amer Math. Soc. 126 (1998), 1199–1204.
[9]Warner, G., Harmonic analysis on semi-simple Lie groups, vol. I (Springer, New York, 1972).
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Journal of the Australian Mathematical Society
  • ISSN: 1446-7887
  • EISSN: 1446-8107
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