{
  "id": "1711.00879",
  "version": "v1",
  "published": "2017-11-02T18:38:58.000Z",
  "updated": "2017-11-02T18:38:58.000Z",
  "title": "Weighted a priori estimates for elliptic equations",
  "authors": [
    "Maria Eugenia Cejas",
    "Ricardo Duran"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We give a simpler proof of the a priori estimates obtained in the paper by Duran, Sanmartino and Toschi for solutions of elliptic problems in weighted Sobolev norms when the weights belong to the Muckenhoupt class $A_p$. The argument is a generalization to bounded domains of the one used in $\\mathbb{R}^n$ to prove the continuity of singular integral operators in weighted norms. In the case of singular integral operators it is known that the $A_p$ condition is also necessary for the continuity. We do not know whether this is also true for the a priori estimates in bounded domains but we are able to prove a weaker result when the operator is the Laplacian or a power of it. We prove that a necessary condition is that the weight belongs to the local $A_p$ class.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-11-02T18:38:58.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "priori estimates",
      "elliptic equations",
      "singular integral operators",
      "bounded domains",
      "weaker result"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}