{
  "id": "1912.12784",
  "version": "v1",
  "published": "2019-12-30T02:26:20.000Z",
  "updated": "2019-12-30T02:26:20.000Z",
  "title": "Weighted Strichartz estimates with angular integrability and their applications",
  "authors": [
    "Jungkwon Kim",
    "Yoonjung Lee",
    "Ihyeok Seo"
  ],
  "comment": "17 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "The endpoint Strichartz estimate $\\|e^{it\\Delta} f\\|_{L_t^2 L_x^\\infty} \\lesssim \\|f\\|_{L^2}$ in dimension $2$ is known to be false. In this case, Tao showed an alternative of the form $\\|e^{it\\Delta} f\\|_{L_t^2L_\\rho^\\infty L_\\omega^2} \\lesssim \\|f\\|_{L^2}$ by introducing the mixed norms on the polar coordinates $x=\\rho\\omega$ with $\\rho>0$, $\\omega\\in\\mathbb{S}^1$. Motivated by this, we study the Strichartz estimates with angular integrability in higher dimensions. More generally, we consider a weighted mixed norm in the polar coordinates. As an application, the existence of solutions for the inhomogeneous nonlinear Schr\\\"odinger equation $i \\partial_{t} u + \\Delta u =\\lambda |x|^{-\\alpha} |u|^{\\beta} u$ for $L^2$ data is shown up to the $L^2$-critical case which has been left unsolved until quite recently. Our result here will provide more information on the solution such as the angular integrability.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-12-30T02:26:20.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "angular integrability",
      "weighted strichartz estimates",
      "application",
      "polar coordinates",
      "endpoint strichartz estimate"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}