{
  "id": "1803.03040",
  "version": "v1",
  "published": "2018-03-08T11:08:33.000Z",
  "updated": "2018-03-08T11:08:33.000Z",
  "title": "Carleman estimates and boundedness of associated multiplier operators",
  "authors": [
    "Eunhee Jeong",
    "Yehyun Kwon",
    "Sanghyuk Lee"
  ],
  "comment": "19 pages, 1 figure",
  "categories": [
    "math.AP",
    "math.CA"
  ],
  "abstract": "Let $P(D)$ be the Laplacian $\\Delta,$ or the wave operator $\\square$. The following type of Carleman estimate is known to be true on a certain range of $p,q$: \\[ \\|e^{v\\cdot x}u\\|_{L^q(\\mathbb{R}^d)} \\le C\\|e^{v\\cdot x}P(D)u\\|_{L^p(\\mathbb{R}^d)} \\] with $C$ independent of $v\\in \\mathbb{R}^d$. The estimates are consequences of the uniform Sobolev type estimates for second order differential operators due to Kenig-Ruiz-Sogge \\cite{KRS} and Jeong-Kwon-Lee \\cite{JKL}. The range of $p,q$ for which the uniform Sobolev type estimates hold was completely characterized for the second order differential operators with nondegenerate principal part. But the optimal range of $p,q$ for which the Carleman estimate holds has not been clarified before. When $P(D)=\\Delta$, $\\square$, or the heat operator, we obtain a complete characterization of the admissible $p,q$ for the aforementioned type of Carleman estimate. For this purpose we investigate $L^p$-$L^q$ boundedness of related multiplier operators. As applications, we also obtain some unique continuation results.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-03-08T11:08:33.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42B15",
      "35B60"
    ],
    "keywords": [
      "carleman estimate",
      "associated multiplier operators",
      "second order differential operators",
      "boundedness",
      "uniform sobolev type estimates hold"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}