{
  "id": "1610.06639",
  "version": "v1",
  "published": "2016-10-21T01:17:21.000Z",
  "updated": "2016-10-21T01:17:21.000Z",
  "title": "Localized $L^p$-estimates for eigenfunctions: II",
  "authors": [
    "Christopher D. Sogge"
  ],
  "comment": "12 pages",
  "categories": [
    "math.AP",
    "math.CA",
    "math.DG",
    "math.SP"
  ],
  "abstract": "If $(M,g)$ is a compact Riemannian manifold of dimension $n\\ge 2$ we give necessary and sufficient conditions for improved $L^p(M)$-norms of eigenfunctions for all $2<p\\ne p_c=\\tfrac{2(n+1)}{n-1}$, the critical exponent. Since improved $L^{p_c}(M)$ bounds imply improvement all other exponents, these conditions are necessary for improved bounds for the critical space. We also show that improved $L^{p_c}(M)$ bounds are valid if these conditions are met and if the half-wave operators, $U(t)$, have no caustics when $t\\ne 0$. The problem of finding a necessary and sufficient condition for $L^{p_c}(M)$ improvement remains an interesting open problem.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-10-21T01:17:21.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "eigenfunctions",
      "sufficient condition",
      "compact riemannian manifold",
      "interesting open problem",
      "half-wave operators"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}