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Proceedings of the American Mathematical Society

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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
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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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