{
  "id": "2410.15043",
  "version": "v1",
  "published": "2024-10-19T09:02:08.000Z",
  "updated": "2024-10-19T09:02:08.000Z",
  "title": "Surjectivity of convolution operators on harmonic $NA$ groups",
  "authors": [
    "Effie Papageorgiou"
  ],
  "categories": [
    "math.FA"
  ],
  "abstract": "Let $\\mu$ be a radial compactly supported distribution on a harmonic $NA$ group. We prove that the right convolution operator $c_{\\mu}:f \\mapsto f* \\mu$ maps the space of smooth $\\mathfrak{v}$-radial functions onto itself if and only if the spherical Fourier transform $\\widetilde{\\mu}(\\lambda)$, $\\lambda \\in \\mathbb{C}$, is slowly decreasing. As an application, we prove that certain averages over spheres are surjective on the space of smooth $\\mathfrak{v}$-radial functions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-10-19T09:02:08.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "43A85",
      "43A90",
      "22E30"
    ],
    "keywords": [
      "radial functions",
      "surjectivity",
      "right convolution operator",
      "radial compactly supported distribution",
      "spherical fourier transform"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}