{
  "id": "1302.4070",
  "version": "v5",
  "published": "2013-02-17T14:19:11.000Z",
  "updated": "2014-09-11T13:25:34.000Z",
  "title": "Estimates for Fourier transforms of surface measures in R^3 and PDE applications",
  "authors": [
    "Michael Greenblatt"
  ],
  "comment": "28 pages. v5: Corrected 2 typos in the statement of Theorem 2.1",
  "categories": [
    "math.CA",
    "math.AP"
  ],
  "abstract": "A local two-dimensional resolution of singularities theorem and arguments based on the Van der Corput lemma are used to give new estimates for the decay rate of the Fourier transform of a locally defined smooth hypersurface measure in R^3, as well as to provide new proofs of some known estimates. These are then used to give L^q bounds on solutions to certain PDE problems in terms of the L^p norms of their initial data for various values of p and q. Unlike much of the earlier work in this subject, no use is made of the adapted coordinate systems that have been often been used to study two-dimensional oscillatory integrals; all of the needed information is furnished by the resolution of singularities theorem.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2014-02-01T02:13:42.000Z",
      "comment": "28 pages. v4: Improved title, several improvements to the exposition",
      "journal": null,
      "doi": null
    },
    {
      "version": "v5",
      "updated": "2014-09-11T13:25:34.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42B10",
      "42B20",
      "42B37"
    ],
    "keywords": [
      "fourier transform",
      "surface measures",
      "pde applications",
      "van der corput lemma",
      "study two-dimensional oscillatory integrals"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1302.4070G"
    }
  }
}