J Austral Math Soc Ser A 46 pp402--414, 1989.

L^p-Improving Measures on Compact Non-Abelian Groups

Kathryn E. Hare

Abstract

A Borel measure m on a compact groupG is called L^p-improving of the operator T_m : L²(G) ® L²(G), defined by T_m(f) = m*f, maps into L^p(G) for some p > 2. We characterize L^p-improving measures on compact non-abelian groups by the eigenspaces of the operator T_m if T_m is normal, and otherwise by the eigenspaces of |T_m|. This result is a generalization of our recent characterization of L^p-improving measures on compact abelian groups. Two examples of Riesz product-like measures are constructed. In contrast with the abelian case one of these is not L^p-improving, while the other is a non-trivial example of an L^p-improving measure.

1980 AMS Subject Classification (1985 Revision): 43A05, 43A46

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Authors

Kathryn E. Hare

Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada.