{
  "id": "2107.12177",
  "version": "v1",
  "published": "2021-07-22T19:56:41.000Z",
  "updated": "2021-07-22T19:56:41.000Z",
  "title": "Regularity of the Radon-Nikodym Derivative of a Convolution of Orbital Measures on Noncompact Symmetric Spaces",
  "authors": [
    "Boudjemaa Anchouche"
  ],
  "doi": "10.13140/RG.2.2.34657.97122",
  "categories": [
    "math.RT"
  ],
  "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}.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-07-22T19:56:41.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "43A85",
      "28C10",
      "43A77",
      "43A90",
      "53C35"
    ],
    "keywords": [
      "noncompact symmetric spaces",
      "orbital measures",
      "radon-nikodym derivative",
      "convolution",
      "regularity"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}