{
  "id": "2401.04129",
  "version": "v1",
  "published": "2024-01-05T14:52:22.000Z",
  "updated": "2024-01-05T14:52:22.000Z",
  "title": "Gradient stability of Caffarelli-Kohn-Nirenberg inequality involving weighted p-Laplace",
  "authors": [
    "Shengbing Deng",
    "Xingliang Tian"
  ],
  "comment": "38 pages. Any suggestions and comments are welcome!",
  "categories": [
    "math.AP"
  ],
  "abstract": "The best constant and extremal functions are well known of the following Caffarelli-Kohn-Nirenberg inequality \\[ \\int_{\\mathbb{R}^N}|\\nabla u|^p\\frac{\\mathrm{d}x}{|x|^{\\mu}}\\geq \\mathcal{S} \\left(\\int_{\\mathbb{R}^N}|u|^r\\frac{\\mathrm{d}x}{|x|^s} \\right)^{\\frac{p}{r}}, \\quad \\mbox{for all}\\quad u\\in C^\\infty_c(\\mathbb{R}^N), \\] where $1<p<p+\\mu<N$, $\\frac{\\mu}{p}\\leq \\frac{s}{r}<\\frac{\\mu}{p}+1$, $r=\\frac{p(N-s)}{N-p-\\mu}$. An important task is investigating the stability of extremals for this inequality. Firstly, we give the classification to the linearized problem related to the extremals which shows the extremals are non-degenerate. Then we investigate the gradient type remainder term of previous inequality by using spectral estimate combined with a compactness argument which partially extends the work of Wei and Wu [Math. Ann., 2022] to a general $p$-Laplace case, and also the work of Figalli and Zhang [Duke Math. J., 2022] to a weighted case.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-01-05T14:52:22.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "caffarelli-kohn-nirenberg inequality",
      "gradient stability",
      "weighted p-laplace",
      "gradient type remainder term",
      "spectral estimate"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 38,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}