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# Asian Journal of Mathematics

## Volume 22 (2018)

### Number 2

### Special issue in honor of Ngaiming Mok (1 of 3)

Guest Editors: Jun-Muk Hwang, Korea Institute for Advanced Study; Yum-Tong Siu, Harvard University; Wing-Keung To, National University of Singapore; Stephen S.-T. Yau, Tsinghua University; Sai-Kee Yeung, Purdue University

### Smoothness of the Radon–Nikodym derivative of a convolution of orbital measures on compact symmetric spaces of rank one

Pages: 211 – 222

DOI: https://dx.doi.org/10.4310/AJM.2018.v22.n2.a1

#### Authors

#### Abstract

Let $G/K$ be a compact symmetric space of rank one. The aim of this paper is to give sufficient conditions for the $C^{\nu}$-smoothness of the Radon–Nikodym derivative $f_{a_1 , \dotsc , a_p} = d (\mu_{a_1} * \dotsc * \mu_{a_p} ) / d \mu_{\mathsf{G}}$ of the convolution $\mu_{a_1} * \dotsc * \mu_{a_p}$ of some orbital measures $\mu_{a_i}$, with respect to the Haar measure $\mu_{\mathsf{G}}$ of $G$. This generalizes some of the main results in [12], in the case of compact rank one symmetric spaces, where the absolute continuity of the measure $\mu_{a_1} * \dotsc * \mu_{a_p}$ with respect to $ d \mu_{\mathsf{G}}$ was considered. Our main result generalizes also the main results in [1] and [7], where the $L^2$-regularity was considered.

As a consequence of our main result, we give sufficient conditions for $f_{a_1 , \dotsc , a_p}$ to be in $L^q (G, d \mu_{\mathsf{G}})$ for all $q \geq 1$ and for the Fourier series of $f_{a_1 , \dotsc , a_p}$ to converge absolutely and uniformly to $f_{a_1 , \dotsc , a_p}$.

#### Keywords

orbital measures, Radon–Nikodym derivative, symmetric spaces

#### 2010 Mathematics Subject Classification

Primary 43A77, 43A90. Secondary 28C10, 53C35.

Received 28 October 2016

Accepted 2 June 2017

Published 15 June 2018