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

TL;DR

This paper investigates the smoothness and regularity of the Radon-Nikodym derivative of convoluted orbital measures on noncompact symmetric spaces, establishing conditions for absolute continuity, $L^2$-regularity, and $C^k$-smoothness.

Contribution

It extends previous results by providing new conditions under which the Radon-Nikodym derivative is in $L^2$ and $C^k$, generalizing the understanding of orbital measure convolutions on noncompact symmetric spaces.

Findings

Convolution of orbital measures is absolutely continuous if the number of measures exceeds the maximum dimension of irreducible components.

The Radon-Nikodym derivative belongs to $L^2(G)$ when the number of convolutions exceeds the maximum dimension plus one.

Smoothness up to $C^k$ is achieved when the number of convolutions exceeds the maximum dimension plus $k+1$.

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}.