{
  "id": "2106.09253",
  "version": "v1",
  "published": "2021-06-17T05:26:31.000Z",
  "updated": "2021-06-17T05:26:31.000Z",
  "title": "Stability of Caffarelli-Kohn-Nirenberg inequality",
  "authors": [
    "Juncheng Wei",
    "Yuanze Wu"
  ],
  "comment": "29 pages; comments welcome",
  "categories": [
    "math.AP"
  ],
  "abstract": "In this paper, we consider the Caffarelli-Kohn-Nirenberg (CKN) inequality: \\begin{eqnarray*} \\bigg(\\int_{{\\mathbb R}^N}|x|^{-b(p+1)}|u|^{p+1}dx\\bigg)^{\\frac{2}{p+1}}\\leq C_{a,b,N}\\int_{{\\mathbb R}^N}|x|^{-2a}|\\nabla u|^2dx \\end{eqnarray*} where $N\\geq3$, $a<\\frac{N-2}{2}$, $a\\leq b\\leq a+1$ and $p=\\frac{N+2(1+a-b)}{N-2(1+a-b)}$. It is well-known that up to dilations $\\tau^{\\frac{N-2}{2}-a}u(\\tau x)$ and scalar multiplications $Cu(x)$, the CKN inequality has a unique extremal function $W(x)$ which is positive and radially symmetric in the parameter region $b_{FS}(a)\\leq b<a+1$ with $a<0$ and $a\\leq b<a+1$ with $a\\geq0$ and $a+b>0$, where $b_{FS}(a)$ is the Felli-Schneider curve. We prove that in the above parameter region the following stabilities hold: \\begin{enumerate} \\item[$(1)$] \\quad stability of CKN inequality in the functional inequality setting $$dist_{D^{1,2}_{a}}^2(u, \\mathcal{Z})\\lesssim\\|u\\|^2_{D^{1,2}_a({\\mathbb R}^N)}-C_{a,b,N}^{-1}\\|u\\|^2_{L^{p+1}(|x|^{-b(p+1)},{\\mathbb R}^N)}$$ where $\\mathcal{Z}= \\{ c W_\\tau\\mid c\\in\\bbr\\backslash\\{0\\}, \\tau>0\\}$; \\item[$(2)$]\\quad stability of CKN inequality in the critical point setting (in the class of nonnegative functions) \\begin{eqnarray*} dist_{D_a^{1,2}}(u, \\mathcal{Z}_0^\\nu)\\lesssim\\left\\{\\aligned &\\Gamma(u),\\quad p>2\\text{ or }\\nu=1,\\\\ &\\Gamma(u)|\\log\\Gamma(u)|^{\\frac12},\\quad p=2\\text{ and }\\nu\\geq2,\\\\ &\\Gamma(u)^{\\frac{p}{2}},\\quad 1<p<2\\text{ and }\\nu\\geq2, \\endaligned\\right. \\end{eqnarray*} where $\\Gamma (u)=\\|div(|x|^{-a}\\nabla u)+|x|^{-b(p+1)}|u|^{p-1}u\\|_{(D^{1,2}_a)^{'}}$ and $$\\mathcal{Z}_0^\\nu=\\{(W_{\\tau_1},W_{\\tau_2},\\cdots,W_{\\tau_\\nu})\\mid \\tau_i>0\\}.$$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-06-17T05:26:31.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "caffarelli-kohn-nirenberg inequality",
      "ckn inequality",
      "parameter region",
      "unique extremal function",
      "scalar multiplications"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 29,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}