{
  "id": "1301.7468",
  "version": "v1",
  "published": "2013-01-30T23:16:01.000Z",
  "updated": "2013-01-30T23:16:01.000Z",
  "title": "On Kakeya-Nikodym averages, $L^p$-norms and lower bounds for nodal sets of eigenfunctions in higher dimensions",
  "authors": [
    "Matthew D. Blair",
    "Christopher D. Sogge"
  ],
  "comment": "29 pages, 1 figure",
  "categories": [
    "math.AP",
    "math.CA",
    "math.DG"
  ],
  "abstract": "We extend a result of the second author \\cite[Theorem 1.1]{soggekaknik} to dimensions $d \\geq 3$ which relates the size of $L^p$-norms of eigenfunctions for $2<p<\\frac{2(d+1)}{d-1}$ to the amount of $L^2$-mass in shrinking tubes about unit-length geodesics. The proof uses bilinear oscillatory integral estimates of Lee \\cite{leebilinear} and a variable coefficient variant of an \"$\\veps$ removal lemma\" of Tao and Vargas \\cite{tv1}. We also use H\\\"ormander's \\cite{HorOsc} $L^2$ oscillatory integral theorem and the Cartan-Hadamard theorem to show that, under the assumption of nonpositive curvature, the $L^2$-norm of eigenfunctions $e_\\la$ over unit-length tubes of width $\\la^{-\\frac12}$ goes to zero. Using our main estimate, we deduce that, in this case, the $L^p$-norms of eigenfunctions for the above range of exponents is relatively small. As a result, we can slightly improve the known lower bounds for nodal sets in dimensions $d\\ge3$ of Colding and Minicozzi \\cite{CM} in the special case of (variable) nonpositive curvature.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-01-30T23:16:01.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35P99"
    ],
    "keywords": [
      "nodal sets",
      "lower bounds",
      "higher dimensions",
      "kakeya-nikodym averages",
      "eigenfunctions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 29,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1301.7468B"
    }
  }
}