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SMOOTHNESS OF CONVOLUTION PRODUCTS OF ORBITAL MEASURES ON RANK ONE COMPACT SYMMETRIC SPACES

Part of: Global differential geometry Lie groups Abstract harmonic analysis

Published online by Cambridge University Press:  12 February 2016

KATHRYN E. HARE  and
JIMMY HE
KATHRYN E. HARE*
Affiliation:
Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, Canada  N2L 3G1 email kehare@uwaterloo.ca
JIMMY HE
Affiliation:
Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, Canada  N2L 3G1 email jimmy.he@uwaterloo.ca
*
✉kehare@uwaterloo.ca
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Abstract

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We prove that all convolution products of pairs of continuous orbital measures in rank one, compact symmetric spaces are absolutely continuous and determine which convolution products are in $L^{2}$ (meaning that their density function is in $L^{2}$). We characterise the pairs whose convolution product is either absolutely continuous or in $L^{2}$ in terms of the dimensions of the corresponding double cosets. In particular, we prove that if $G/K$ is not $\text{SU}(2)/\text{SO}(2)$, then the convolution of any two regular orbital measures is in $L^{2}$, while in $\text{SU}(2)/\text{SO}(2)$ there are no pairs of orbital measures whose convolution product is in $L^{2}$.

Keywords

rank one symmetric spacespherical functionsorbital measureabsolute continuity

MSC classification

Primary: 43A80: Analysis on other specific Lie groups
Secondary: 22E30: Analysis on real and complex Lie groups 53C35: Symmetric spaces
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 94 , Issue 1 , August 2016 , pp. 131 - 143
Copyright
© 2016 Australian Mathematical Publishing Association Inc. 

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