{
  "id": "2002.02260",
  "version": "v1",
  "published": "2020-02-06T13:54:53.000Z",
  "updated": "2020-02-06T13:54:53.000Z",
  "title": "$L^2$ estimates and existence theorems for the $\\bar{\\partial}$ operators in infinite dimensions, I",
  "authors": [
    "Jiayang Yu",
    "Xu Zhang"
  ],
  "categories": [
    "math.FA"
  ],
  "abstract": "In this paper we introduce a probability measure on $\\ell^p$ ($p\\in [1,\\infty)$) by which we define the $(s,t)$-forms on $\\ell^p$ and the $\\bar{\\partial}$ operator from $(s,t)$-forms to $(s,t+1)$-forms. Then we can employ the $L^2$-method to establish the exactness of the $\\bar{\\partial}$ operator. As a consequence, the counterexamples given by Coeur\\'{e} and Lempert which are not solvable on any open set are solvable for the $\\bar{\\partial}$ operator defined in this paper.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-02-06T13:54:53.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "infinite dimensions",
      "existence theorems",
      "probability measure",
      "open set",
      "consequence"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}