Equivalent conditions for the 𝐿_{𝑝} convolution theorem on semisimple groups

Lipsman, Ronald L.

doi:10.1090/S0002-9939-1972-0310557-X

Equivalent conditions for the $L_{p}$ convolution theorem on semisimple groups
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Proc. Amer. Math. Soc. 34 (1972), 497-503 Request permission

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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Uniformly bounded representations and harmonic analysis of the

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29

Uniformly bounded representations

Analytic continuation of the principal series of representations of the

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