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For certain semisimple Lie groups  G , it is known that convolution by an      L p ( G ) {L_p}(G)       function,      1 ≦  p &gt; 2 1 \leqq p &gt; 2      , is a bounded operator on      L 2 ( G ) {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 {L_p}       convolution theorem.

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