{
  "id": "1703.08128",
  "version": "v1",
  "published": "2017-03-23T16:17:27.000Z",
  "updated": "2017-03-23T16:17:27.000Z",
  "title": "Schur multipliers on $\\mathcal{B}(L^p,L^q)$",
  "authors": [
    "Clément Coine"
  ],
  "comment": "25 pages",
  "categories": [
    "math.FA"
  ],
  "abstract": "Let $(\\Omega_1, \\mathcal{F}_1, \\mu_1)$ and $(\\Omega_2, \\mathcal{F}_2, \\mu_2)$ be two measure spaces and let $1 \\leq p,q \\leq +\\infty$. We give a definition of Schur multipliers on $\\mathcal{B}(L^p(\\Omega_1), L^q(\\Omega_2))$ which extends the definition of classical Schur multipliers on $\\mathcal{B}(\\ell_p,\\ell_q)$. Our main result is a characterization of Schur multipliers in the case $1\\leq q \\leq p \\leq +\\infty$. When $1 < q \\leq p < +\\infty$, $\\phi \\in L^{\\infty}(\\Omega_1 \\times \\Omega_2)$ is a Schur multiplier on $\\mathcal{B}(L^p(\\Omega_1), L^q(\\Omega_2))$ if and only if there are a measure space (a probability space when $p\\neq q$) $(\\Omega,\\mu)$, $a\\in L^{\\infty}(\\mu_1, L^{p}(\\mu))$ and $b\\in L^{\\infty}(\\mu_2, L^{q'}(\\mu))$ such that, for almost every $(s,t) \\in \\Omega_1 \\times \\Omega_2$, $$\\phi(s,t)=\\left\\langle a(s), b(t) \\right\\rangle.$$ Here, $L^{\\infty}(\\mu_1, L^{r}(\\mu))$ denotes the Bochner space on $\\Omega_1$ valued in $L^r(\\mu)$. This result is new, even in the classical case. As a consequence, we give new inclusion relationships between the spaces of Schur multipliers on $\\mathcal{B}(\\ell_p,\\ell_q)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-03-23T16:17:27.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "measure space",
      "main result",
      "classical schur multipliers",
      "inclusion relationships",
      "probability space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}