{
  "id": "0907.4827",
  "version": "v6",
  "published": "2009-07-28T03:20:22.000Z",
  "updated": "2010-06-14T16:11:33.000Z",
  "title": "Kakeya-Nikodym averages and $L^p$-norms of eigenfunctions",
  "authors": [
    "Christopher D. Sogge"
  ],
  "comment": "18 pages, 2 figures, corrected a few typos. Final version to appear in Tohoku Mathematical Journal Centennial Issue",
  "categories": [
    "math.AP",
    "math.CA",
    "math.DG"
  ],
  "abstract": "We provide a necessary and sufficient condition that $L^p$-norms, $2<p<6$, of eigenfunctions of the square root of minus the Laplacian on 2-dimensional compact boundaryless Riemannian manifolds $M$ are small compared to a natural power of the eigenvalue $\\lambda$. The condition that ensures this is that their $L^2$ norms over $O(\\lambda^{-1/2})$ neighborhoods of arbitrary unit geodesics are small when $\\lambda$ is large (which is not the case for the highest weight spherical harmonics on $S^2$ for instance). The proof exploits Gauss' lemma and the fact that the bilinear oscillatory integrals in H\\\"ormander's proof of the Carleson-Sj\\\"olin theorem become better and better behaved away from the diagonal. Our results are related to a recent work of Bourgain who showed that $L^2$ averages over geodesics of eigenfunctions are small compared to a natural power of the eigenvalue $\\lambda$ provided that the $L^4(M)$ norms are similarly small. Our results imply that QUE cannot hold on a compact boundaryless Riemannian manifold $(M,g)$ of dimension two if $L^p$-norms are saturated for a given $2<p<6$. We also show that eigenfunctions cannot have a maximal rate of $L^2$-mass concentrating along unit portions of geodesics that are not smoothly closed.",
  "revisions": [
    {
      "version": "v6",
      "updated": "2010-06-14T16:11:33.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35P99",
      "35L20",
      "42C99"
    ],
    "keywords": [
      "kakeya-nikodym averages",
      "eigenfunctions",
      "compact boundaryless riemannian manifold",
      "natural power",
      "arbitrary unit geodesics"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0907.4827S"
    }
  }
}