{
  "id": "1510.07724",
  "version": "v1",
  "published": "2015-10-26T23:51:18.000Z",
  "updated": "2015-10-26T23:51:18.000Z",
  "title": "Refined and Microlocal Kakeya-Nikodym Bounds of Eigenfunctions in Higher Dimensions",
  "authors": [
    "Matthew D. Blair",
    "Christopher D. Sogge"
  ],
  "comment": "26 pages",
  "categories": [
    "math.AP",
    "math.CA"
  ],
  "abstract": "We prove a Kakeya-Nikodym bound on eigenfunctions and quasimodes, which sharpens a result of the authors and extends it to higher dimensions. As in the prior work, the key intermediate step is to prove a microlocal version of these estimates, which involves a phase space decomposition of these modes which is essentially invariant under the bicharacteristic/geodesic flow. In a companion paper, it will be seen that these sharpened estimates yield improved $L^q(M)$ bounds on eigenfunctions in the presence of nonpositive curvature when $2 < q < \\frac{2(d+1)}{d-1}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-10-26T23:51:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35P99",
      "42B37"
    ],
    "keywords": [
      "microlocal kakeya-nikodym bounds",
      "higher dimensions",
      "eigenfunctions",
      "phase space decomposition",
      "sharpened estimates yield"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv151007724B"
    }
  }
}