Regularity of the Radon-Nikodym Derivative of a Convolution of Orbital Measures on Noncompact Symmetric Spaces

Abstract

Let $G/K$ be a Riemannian symmetric space of noncompact type, and let ${\nu }_{a_j}$, $j=1,...,r$ be some orbital measures on $G$ (see the definition below). The aim of this paper is to study the $L^2$-regularity (resp. $C^k$-smoothness) of the Radon-Nikodym derivative of the convolution ${\nu }_{a_1}\ast ...\ast {\nu }_{a_r}$ with respect to a fixed left Haar measure ${\mu }_G$ on $G$. As a consequence of a result of Ragozin, \cite{ragozin}, we prove that if $r\ge {\max }_{1\le i\le s}\dim G_i/K_i$, then ${\nu }_{a_1}\ast ...\ast {\nu }_{a_r}$ is absolutely continuous with respect to ${\mu }_G$, i.e., $d({\nu }_{a_1}\ast ...\ast {\nu }_{a_r})/d{\mu }_G$ is in $L^1(G)$, where $G_i/K_i$, $i=1,...,s$, are the irreducible components in the de Rham decomposition of $G/K$. The aim of this paper is to prove that $d({\nu }_{a_1}\ast ...\ast {\nu }_{a_r})/d{\mu }_G$ is in $L^2(G)$ (resp. $C^k\left(G\right)$) for $r\ge {\max }_{1\le i\le s}\dim \left(G_i/K_i\right)+1$\, (resp. $r\ge {\max }_{1\le i\le s}\dim \left(G_i/K_i\right)+k+1$). The case of a compact symmetric space of rank one was considered in \cite{AGP} and \cite{AG}, and the case of a complex Grassmannian was considered in \cite{AA}.