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 $r \geq \, \max_{1\leq i \leq s}\dim {G_i}/K_i$, then $\nu_{a_{1}}\ast...\ast\nu_{a_{r}}$ is absolutely continuous with respect to $\mu_G$, i.e., $d\big(\nu_{a_{1}}\ast...\ast\nu_{a_{r}}\big)/d\mu_G$ 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 $d\big(\nu_{a_{1}}\ast...\ast\nu_{a_{r}}\big)/d\mu_G$ is in $L^2(G)$ (resp. $C^k\left(G \right) $) for $r \geq \max_{1\leq i \leq s}\dim \left( {G_i}/{K_i}\right) + 1$\, (resp. $r \geq \max_{1\leq i \leq s} \dim\left( {G_i}/{K_i}\right) +k+1$). 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}.