{
  "id": "1602.04585",
  "version": "v1",
  "published": "2016-02-15T08:04:26.000Z",
  "updated": "2016-02-15T08:04:26.000Z",
  "title": "Extremal function for Moser-Trudinger type Inequality with Logarithmic weight",
  "authors": [
    "Prosenjit Roy"
  ],
  "comment": "To appear in \"Nonlinear Analysis- TMA\"",
  "doi": "10.1016/j.na.2016.01.024",
  "categories": [
    "math.AP"
  ],
  "abstract": "On the space of weighted radial Sobolev space, the following generalization of Moser-Trudinger type inequality was established by Calanchi and Ruf in dimension 2 : If $\\beta \\in [0,1)$ and $w_0(x) = |\\log |x||^\\beta $ then $$ \\sup_{\\int_B |\\grad u|^2w_0 \\leq 1 , u \\in H_{0,rad}^1(w_0,B)} \\int_B e^{\\alpha u^{\\frac{2}{1-\\beta}}} dx < \\infty,$$ if and only if $\\alpha \\leq \\alpha_\\beta = 2\\left[2\\pi (1-\\beta) \\right]^{\\frac{1}{1-\\beta}}.$ We prove the existence of an extremal function for the above inequality for the critical case when $\\alpha = \\alpha_\\beta$ thereby generalizing the result of Carleson-Chang who proved the case when $\\beta =0$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-02-15T08:04:26.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "moser-trudinger type inequality",
      "extremal function",
      "logarithmic weight",
      "weighted radial sobolev space",
      "carleson-chang"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Elsevier"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160204585R"
    }
  }
}