{
  "id": "1206.5110",
  "version": "v2",
  "published": "2012-06-22T11:00:36.000Z",
  "updated": "2012-11-11T14:14:08.000Z",
  "title": "Optimal constants and extremisers for some smoothing estimates",
  "authors": [
    "Neal Bez",
    "Mitsuru Sugimoto"
  ],
  "comment": "21 pages, statement of Theorem 1.6 fixed, further minor modifications, references added",
  "categories": [
    "math.AP",
    "math.CA"
  ],
  "abstract": "We establish new results concerning the existence of extremisers for a broad class of smoothing estimates of the form $\\|\\psi(|\\nabla|) \\exp(it\\phi(|\\nabla|)f \\|_{L^2(w)} \\leq C\\|f\\|_{L^2}$, where the weight $w$ is radial and depends only on the spatial variable; such a smoothing estimate is of course equivalent to the $L^2$-boundedness of a certain oscillatory integral operator $S$ depending on $(w,\\psi,\\phi)$. Furthermore, when $w$ is homogeneous, and for certain $(\\psi,\\phi)$, we provide an explicit spectral decomposition of $S^*S$ and consequently recover an explicit formula for the optimal constant $C$ and a characterisation of extremisers. In certain well-studied cases when $w$ is inhomogeneous, we obtain new expressions for the optimal constant.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2012-11-11T14:14:08.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35B45",
      "35P10",
      "35B65"
    ],
    "keywords": [
      "optimal constant",
      "smoothing estimate",
      "extremisers",
      "oscillatory integral operator",
      "explicit spectral decomposition"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1206.5110B"
    }
  }
}