On the boundedness and unboundedness of certain convolution operators on nilpotent Lie groups
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Abstract:
One method of proving irreducibility of the “principal series” representations of semisimple Lie groups involves showing that a certain nonintegrable function on a nilpotent subgroup $X$ cannot be regularized to give a bounded convolution operator on ${L_2}(X)$. This note gives an elementary proof of this unboundedness property for the groups $X$ which occur in real-rank one semisimple groups.References
- G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, Cambridge, at the University Press, 1952. 2d ed. MR 0046395
- A. W. Knapp and E. M. Stein, Intertwining operators for semisimple groups, Ann. of Math. (2) 93 (1971), 489–578. MR 460543, DOI 10.2307/1970887
Additional Information
- © Copyright 1973 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 39 (1973), 409-413
- MSC: Primary 22E30
- DOI: https://doi.org/10.1090/S0002-9939-1973-0320227-0
- MathSciNet review: 0320227