{
  "id": "math/0609024",
  "version": "v1",
  "published": "2006-09-01T10:18:18.000Z",
  "updated": "2006-09-01T10:18:18.000Z",
  "title": "$L\\sp p$-$L\\sp q$ regularity of Fourier integral operators with caustics",
  "authors": [
    "Andrew Comech"
  ],
  "comment": "24 pages, 1 figure",
  "journal": "Trans. Amer. Math. Soc. 356 (2004), no. 9, 3429--3454",
  "categories": [
    "math.AP"
  ],
  "abstract": "The caustics of Fourier integral operators are defined as caustics of the corresponding Schwartz kernels (Lagrangian distributions on $X\\times Y$). The caustic set $\\Sigma(C)$ of the canonical relation $C$ is characterized as the set of points where the rank of the projection $\\pi:C\\to X\\times Y$ is smaller than its maximal value, $dim(X\\times Y)-1$. We derive the $L\\sp p(Y)\\to L\\sp q(X)$ estimates on Fourier integral operators with caustics of corank 1 (such as caustics of type $A\\sb{m+1}$, $m\\in\\N$). For the values of $p$ and $q$ outside of certain neighborhood of the line of duality, $q=p'$, the $L\\sp p\\to L\\sp q$ estimates are proved to be caustics-insensitive. We apply our results to the analysis of the blow-up of the estimates on the half-wave operator just before the geodesic flow forms caustics.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2006-09-01T10:18:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35S30"
    ],
    "keywords": [
      "fourier integral operators",
      "regularity",
      "geodesic flow forms caustics",
      "lagrangian distributions",
      "caustic set"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "AMS",
      "journal": "Trans. Amer. Math. Soc."
    },
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math......9024C"
    }
  }
}