{
  "id": "2005.04113",
  "version": "v1",
  "published": "2020-05-08T15:38:17.000Z",
  "updated": "2020-05-08T15:38:17.000Z",
  "title": "Surjectivity of Convolution Operators on Noncompact Symmetric Spaces",
  "authors": [
    "Fulton Gonzalez",
    "Tomoyuki Kakehi",
    "Jue Wang"
  ],
  "categories": [
    "math.FA",
    "math.RT"
  ],
  "abstract": "Let $\\mu$ be a $K$-invariant compactly supported distribution on a noncompact Riemannian symmetric space $X=G/K$. If the spherical Fourier transform $\\widetilde\\mu(\\lambda)$ is slowly decreasing, it is known that the right convolution operator $c_\\mu\\colon f\\mapsto f*\\mu$ maps $\\mathcal E(X)$ onto $\\mathcal E(X)$. In this paper, we prove the converse of this result. We also prove that $c_\\mu$ has a fundamental solution if and only if $\\widetilde\\mu(\\lambda)$ is slowly decreasing.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-05-08T15:38:17.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "noncompact symmetric spaces",
      "surjectivity",
      "noncompact riemannian symmetric space",
      "right convolution operator",
      "spherical fourier transform"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}