{
  "id": "1906.01556",
  "version": "v1",
  "published": "2019-06-04T16:17:35.000Z",
  "updated": "2019-06-04T16:17:35.000Z",
  "title": "A Note on Estimates for Elliptic Systems with $L^1$ Data",
  "authors": [
    "Bogdan Raiţă",
    "Daniel Spector"
  ],
  "comment": "8 pages",
  "categories": [
    "math.AP",
    "math.CA",
    "math.FA"
  ],
  "abstract": "In this paper we give necessary and sufficient conditions on the compatibility of a $k$th order homogeneous linear elliptic differential operator $\\mathbb{A}$ and differential constraint $\\mathcal{C}$ for solutions of \\begin{align*} \\mathbb{A} u=f\\quad\\text{subject to}\\quad \\mathcal{C} f=0\\quad\\text{ in }\\mathbb{R}^n \\end{align*} to satisfy the estimates \\begin{align*} \\|D^{k-j}u\\|_{L^{\\frac{n}{n-j}}(\\mathbb{R}^n)}\\leq c\\|f\\|_{L^1(\\mathbb{R}^n)} \\end{align*} for $j\\in \\{1,\\ldots,\\min\\{k,n-1\\}\\}$ and \\begin{align*} \\|D^{k-n}u\\|_{L^{\\infty}(\\mathbb{R}^n)}\\leq c\\|f\\|_{L^1(\\mathbb{R}^n)} \\end{align*} when $k\\geq n$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-06-04T16:17:35.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "elliptic systems",
      "order homogeneous linear elliptic differential",
      "th order homogeneous linear elliptic",
      "homogeneous linear elliptic differential operator",
      "sufficient conditions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}