The smoothness of convolutions of zonal measures on compact symmetric spaces

Sanjiv K. Gupta; Kathryn E. Hare

doi:10.1016/j.jmaa.2013.01.070

The smoothness of convolutions of zonal measures on compact symmetric spaces

Sanjiv K. Gupta^*, Kathryn E. Hare

^*Corresponding author for this work

Mathematics

Research output: Contribution to journal › Article › peer-review

3 Citations (Scopus)

Abstract

We prove that for every irreducible, compact symmetric space, G_c/K, of rank r, the convolution of any (2r+1) continuous, K-bi-invariant measures is absolutely continuous with respect to the Haar measure on G_c. We also prove that the convolution of (r+1) continuous, K-invariant measures on the -1 eigenspace in the Cartan decomposition of the Lie algebra of G_c is absolutely continuous with respect to Lebesgue measure. These results are nearly sharp.

Original language	English
Pages (from-to)	668-678
Number of pages	11
Journal	Journal of Mathematical Analysis and Applications
Volume	402
Issue number	2
DOIs	https://doi.org/10.1016/j.jmaa.2013.01.070
Publication status	Published - Jun 15 2013

Keywords

Absolutely continuous
Double coset
Symmetric space
Zonal measure

ASJC Scopus subject areas

Analysis
Applied Mathematics

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10.1016/j.jmaa.2013.01.070

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title = "The smoothness of convolutions of zonal measures on compact symmetric spaces",

abstract = "We prove that for every irreducible, compact symmetric space, Gc/K, of rank r, the convolution of any (2r+1) continuous, K-bi-invariant measures is absolutely continuous with respect to the Haar measure on Gc. We also prove that the convolution of (r+1) continuous, K-invariant measures on the -1 eigenspace in the Cartan decomposition of the Lie algebra of Gc is absolutely continuous with respect to Lebesgue measure. These results are nearly sharp.",

keywords = "Absolutely continuous, Double coset, Symmetric space, Zonal measure",

author = "Gupta, {Sanjiv K.} and Hare, {Kathryn E.}",

note = "Funding Information: The first author would like to thank the Dept. of Pure Mathematics for their hospitality while some of this research was done. This research was supported in part by NSERC and the Sultan Qaboos University . ",

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T1 - The smoothness of convolutions of zonal measures on compact symmetric spaces

AU - Gupta, Sanjiv K.

AU - Hare, Kathryn E.

N1 - Funding Information: The first author would like to thank the Dept. of Pure Mathematics for their hospitality while some of this research was done. This research was supported in part by NSERC and the Sultan Qaboos University .

PY - 2013/6/15

Y1 - 2013/6/15

N2 - We prove that for every irreducible, compact symmetric space, Gc/K, of rank r, the convolution of any (2r+1) continuous, K-bi-invariant measures is absolutely continuous with respect to the Haar measure on Gc. We also prove that the convolution of (r+1) continuous, K-invariant measures on the -1 eigenspace in the Cartan decomposition of the Lie algebra of Gc is absolutely continuous with respect to Lebesgue measure. These results are nearly sharp.

AB - We prove that for every irreducible, compact symmetric space, Gc/K, of rank r, the convolution of any (2r+1) continuous, K-bi-invariant measures is absolutely continuous with respect to the Haar measure on Gc. We also prove that the convolution of (r+1) continuous, K-invariant measures on the -1 eigenspace in the Cartan decomposition of the Lie algebra of Gc is absolutely continuous with respect to Lebesgue measure. These results are nearly sharp.

KW - Absolutely continuous

KW - Double coset

KW - Symmetric space

KW - Zonal measure

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M3 - Article

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VL - 402

SP - 668

EP - 678

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

IS - 2

ER -

The smoothness of convolutions of zonal measures on compact symmetric spaces

Abstract

Keywords

ASJC Scopus subject areas

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