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Lp-improving measures on compact non-abelian groups

Part of: Abstract harmonic analysis

Published online by Cambridge University Press:  09 April 2009

Kathryn E. Hare
Kathryn E. Hare
Affiliation:
Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
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Abstract

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A Borel measure μ on a compact group G is called Lp-improving if the operator Tμ: L2(G) → L2(G), defined by Tμ(f) = μ * f, maps into Lp(G) for some P > 2. We characterize Lp-improving measures on compact non-abelian groups by the eigenspaces of the operator Tμ if |Tμ|. This result is a generalization of our recent characterization of Lp-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 Lp-improving, while the other is a non-trivial example of an Lp improving measure.

MSC classification

Secondary: 43A05: Measures on groups and semigroups, etc. 43A46: Special sets (thin sets, Kronecker sets, Helson sets, Ditkin sets, Sidon sets, etc.)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 46 , Issue 3 , June 1989 , pp. 402 - 414
Copyright
Copyright © Australian Mathematical Society 1989

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