{
  "id": "1806.03912",
  "version": "v1",
  "published": "2018-06-11T11:21:57.000Z",
  "updated": "2018-06-11T11:21:57.000Z",
  "title": "On the unboundedness in $L^{q}$ of the Fourier transform of $L^{p}$ functions",
  "authors": [
    "Ahmed A. Abdelhakim"
  ],
  "categories": [
    "math.CA"
  ],
  "abstract": "The Hausdorff-Young inequality confirms the continuity of the Fourier transform $\\mathcal{F}$ from $L^{p}(\\mathbb{R}^{d})$ to $L^{q}(\\mathbb{R}^{d})$ exclusively when $1\\leq p\\leq 2$ and $q$ is its conjugate Lebesgue exponent. Let $\\Omega$ be a Lebesgue measurable bounded subset of $\\mathbb{R}^{d}$. We show that $\\mathcal{F}$ is bounded from $L^{p}(\\mathbb{R}^{d})$ to $L^{q}(\\Omega)$ precisely when $\\frac{1}{p}+\\frac{1}{q}\\geq 1$ and $1\\leq p\\leq 2$. We also prove that $\\mathcal{F}$ is unbounded from $L^{p}(\\mathbb{R}^{d})$ to $L^{q}(\\mathbb{R}^{d}\\setminus \\Omega)$ outside the admissible range of the Hausdorff-Young inequality. This is the case even if $L^{p}(\\mathbb{R}^{d})$ is replaced by the subspace of functions with compact support.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-06-11T11:21:57.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "fourier transform",
      "unboundedness",
      "hausdorff-young inequality confirms",
      "conjugate lebesgue exponent",
      "compact support"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}