{
  "id": "2011.10160",
  "version": "v1",
  "published": "2020-11-20T00:42:39.000Z",
  "updated": "2020-11-20T00:42:39.000Z",
  "title": "Sharp convergence for sequences of nonelliptic Schrödinger means",
  "authors": [
    "Wenjuan Li",
    "Huiju Wang",
    "Dunyan Yan"
  ],
  "categories": [
    "math.CA",
    "math.AP"
  ],
  "abstract": "We consider pointwise convergence of nonelliptic Schr\\\"{o}dinger means $e^{it_{n}\\square}f(x)$ for $f \\in H^{s}(\\mathbb{R}^{2})$ and decreasing sequences $\\{t_{n}\\}_{n=1}^{\\infty}$ converging to zero, where \\[{e^{it_{n}\\square }}f\\left( x \\right): = \\int_{{\\mathbb{R}^2}} {{e^{i\\left( {x \\cdot \\xi + t_{n}{{ \\xi_{1}\\xi_{2} }}} \\right)}}\\widehat{f}} \\left( \\xi \\right)d\\xi .\\] We prove that when $0<s < \\frac{1}{2}$, \\[\\mathop {\\lim }\\limits_{n \\to \\infty} {e^{it_{n}\\square }}f\\left( x \\right) = f(x) \\hspace{0.2cm} a.e.\\hspace{0.2cm} x\\in \\mathbb{R}^2\\] holds for all $f \\in {H^s}\\left( {{\\mathbb{R}^2}} \\right)$ if and only if $\\{t_{n}\\}_{n=1}^{\\infty} \\in \\ell^{r(s), \\infty}(\\mathbb{N})$, $r(s)=\\frac{s}{1-s}$. Moreover, our result remains valid in general dimensions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-11-20T00:42:39.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42B20",
      "42B25",
      "35S10"
    ],
    "keywords": [
      "nonelliptic schrödinger means",
      "sharp convergence",
      "result remains valid",
      "general dimensions",
      "pointwise convergence"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}