{
  "id": "math/0505021",
  "version": "v1",
  "published": "2005-05-02T11:04:21.000Z",
  "updated": "2005-05-02T11:04:21.000Z",
  "title": "Multipliers spaces and pseudo-differential operators",
  "authors": [
    "Sadek Gala"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "Let $\\sigma(x,\\xi) $ be a sufficiently regular function defined on $R^d \\times R^d.$ The pseudo-differential operator with symbol $\\sigma$ is defined on the Schwartz class by the formula: \\[f\\to\\sigma f(x)=\\int_{R^d} \\sigma(x,\\xi) \\hat{f}(\\xi)e^{2\\pi ix\\xi}d\\xi, \\] where $\\hat{f}(\\xi)=\\int_{R^d} f(x)e^{-2\\pi ix\\xi}dx$ is the Fourier transform of $f.$ In this paper, we shall consider the regularity of the following type : \\begin{description} \\item[(a)] $| \\partial_{\\xi}^{\\alpha}\\sigma(x,\\xi) | \\leq A_{\\alpha}(1+| \\xi|) ^{-| \\alpha|},$ \\item[(b)] $| \\partial_{\\xi}^{\\alpha}\\sigma(x+y,\\xi) -\\partial_{\\xi}^{\\alpha}\\sigma(x,\\xi) | \\leq A_{\\alpha}\\omega(| y|) (1+| \\xi|) ^{-| \\alpha|},$ \\end{description} %",
  "revisions": [
    {
      "version": "v1",
      "updated": "2005-05-02T11:04:21.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42Bxx",
      "31Bxx"
    ],
    "keywords": [
      "pseudo-differential operator",
      "multipliers spaces",
      "schwartz class",
      "fourier transform",
      "sufficiently regular function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}