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Equivalent conditions for the $ L\sb{p}$ convolution theorem on semisimple groups


Author: Ronald L. Lipsman
Journal: Proc. Amer. Math. Soc. 34 (1972), 497-503
MSC: Primary 43A80; Secondary 22E20, 47G05
DOI: https://doi.org/10.1090/S0002-9939-1972-0310557-X
MathSciNet review: 0310557
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Abstract: For certain semisimple Lie groups G, it is known that convolution by an $ {L_p}(G)$ function, $ 1 \leqq p < 2$, is a bounded operator on $ {L_2}(G)$. This result is a consequence of the so-called ``analytic continuation of the principal series'' which has been carried out on these groups. However, this continuation procedure does not generalize readily to arbitrary semisimple groups. In an attempt to bypass the continuation and obtain the convolution theorem in an alternate manner, we derive in this paper several equivalent conditions for this $ {L_p}$ convolution theorem.


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  • [1] J. M. G. Fell, The dual spaces of $ {C^ \ast }$-algebras, Trans. Amer. Math. Soc. 94 (1960), 365-403. MR 26 #4201. MR 0146681 (26:4201)
  • [2] Harish-Chandra, Harmonic analysis on semisimple Lie groups, Bull. Amer. Math. Soc. 76 (1970), 529-551. MR 41 #1933. MR 0257282 (41:1933)
  • [3] R. A. Kunze and E. M. Stein a) Uniformly bounded representations and harmonic analysis of the $ 2 \times 2$ real unimodular group, Amer. J. Math. 82 (1960), 1-62. MR 29 #1287; b) Uniformly bounded representations. II. Analytic continuation of the principal series of representations of the $ n \times n$ complex unimodular group, Amer. J. Math. 83 (1961), 723-786. MR 29 #1288.
  • [4] R. L. Lipsman a) Uniformly bounded representations of $ {\text{SL}}(2,C)$, Amer. J. Math. 91 (1969), 47-66. MR 39 #355; b) Harmonic analysis on $ {\text{SL}}(2,{\mathbf{C}})$, J. Functional Anal. 3 (1969), 126-155. MR 38 #5997; c) An indicator diagram for locally compact unimodular groups, Duke Math. J. 36 (1969), 765-780. MR 41 #8592; d) An indicator diagram. II. The $ {L_p}$ convolution theorem for connected unimodular groups, Duke Math. J. 37 (1970), 459-466. MR 42 #1947; e) The dual topology for the principal and discrete series on semisimple groups, Trans. Amer. Math. Soc. 152 (1970), 399-417. MR 42 #4673; f) On the characters and equivalence of continuous series representations, J. Math. Soc. Japan 23 (1971), 452-480. MR 0238995 (39:355)
  • [5] N. W. Rickert, Convolutions of $ {L_2}$ functions, Colloq. Math. 19 (1968), 301-303. MR 37 #4509. MR 0228930 (37:4509)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1972-0310557-X
Keywords: Semisimple Lie groups, Cartan subgroups, parabolic groups, operator-valued Fourier transform, Plancherel measure, unitary representations, $ {L_p}$ convolution theorem, direct integrals
Article copyright: © Copyright 1972 American Mathematical Society

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