{
  "id": "0904.2880",
  "version": "v1",
  "published": "2009-04-19T04:14:40.000Z",
  "updated": "2009-04-19T04:14:40.000Z",
  "title": "An inverse theorem for the bilinear $L^2$ Strichartz estimate for the wave equation",
  "authors": [
    "Terence Tao"
  ],
  "comment": "20 pages, no figures. To be submitted in conjunction with other \"heatwave\" papers",
  "categories": [
    "math.AP"
  ],
  "abstract": "A standard bilinear $L^2$ Strichartz estimate for the wave equation, which underlies the theory of $X^{s,b}$ spaces of Bourgain and Klainerman-Machedon, asserts (roughly speaking) that if two finite-energy solutions to the wave equation are supported in transverse regions of the light cone in frequency space, then their product lies in spacetime $L^2$ with a quantitative bound. In this paper we consider the \\emph{inverse problem} for this estimate: if the product of two waves has large $L^2$ norm, what does this tell us about the waves themselves? The main result, roughly speaking, is that the lower-frequency wave is dispersed away from a bounded number of light rays. This result will be used in a forthcoming paper \\cite{tao:heatwave4} of the author on the global regularity problem for wave maps.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-04-19T04:14:40.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35L05"
    ],
    "keywords": [
      "wave equation",
      "strichartz estimate",
      "inverse theorem",
      "global regularity problem",
      "standard bilinear"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0904.2880T"
    }
  }
}