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SMOOTHNESS OF CONVOLUTION PRODUCTS OF ORBITAL MEASURES ON RANK ONE COMPACT SYMMETRIC SPACES
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 94 / Issue 1 / August 2016
- Published online by Cambridge University Press:
- 12 February 2016, pp. 131-143
- Print publication:
- August 2016
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- Article
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- You have access
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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 
$L^{2}$.
