{
  "id": "1503.07238",
  "version": "v1",
  "published": "2015-03-25T00:05:52.000Z",
  "updated": "2015-03-25T00:05:52.000Z",
  "title": "Localized $L^p$-estimates of eigenfunctions: A note on an article of Hezari and Rivière",
  "authors": [
    "Christopher D. Sogge"
  ],
  "comment": "11 pages",
  "categories": [
    "math.AP",
    "math.CA",
    "math.DG"
  ],
  "abstract": "We use a straightforward variation on a recent argument of Hezari and Rivi\\`ere~\\cite{HR} to obtain localized $L^p$-estimates for all exponents larger than or equal to the the critical exponent $p_c=\\tfrac{2(n+1)}{n-1}$. We are able to this directly by just using the $L^{p}$-bounds for spectral projection operators from our much earlier work \\cite{Seig}. The localized bounds we obtain here imply, for instance, that, for a density one sequence of eigenvalues on a manifold whose geodesic flow is ergodic, all of the $L^p$, $2<p\\le \\infty$, bounds of the corresponding eigenfunctions are relatively small compared to the general ones in \\cite{Seig}, which are saturated on round spheres. The connection with quantum ergodicity was established for exponents $2<p<p_c$ in the recent results of the author \\cite{SK} and Blair and the author \\cite{BS2}; however, the article of Hezari and Rivi\\`ere~\\cite{HR} was the first one to make this connection (in the case of negatively curved manifolds) for the critical exponent, $p_c$. As is well known, and we indicate here, bounds for the critical exponent, $p_c$, imply ones for all of the other exponents $2<p\\le \\infty$. The localized estimates involve $L^2$-norms over small geodesic balls $B_r$ of radius $r$, and we shall go over what happens for these in certain model cases on the sphere and on manifolds of nonpositive curvature. We shall also state a problem as to when one can improve on the trivial $O(r^{\\frac12})$ estimates for these $L^2(B_r)$ bounds. If $r=\\lambda^{-1}$, one can improve on the trivial estimates if one has improved $L^{p_c}(M)$ bounds just by using H\\\"older's inequality; however, obtaining improved bounds for $r\\gg \\lambda^{-1}$ seems to be subtle.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-03-25T00:05:52.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "58J51"
    ],
    "keywords": [
      "critical exponent",
      "eigenfunctions",
      "spectral projection operators",
      "small geodesic balls",
      "earlier work"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}