{
  "id": "2008.05639",
  "version": "v1",
  "published": "2020-08-13T01:49:51.000Z",
  "updated": "2020-08-13T01:49:51.000Z",
  "title": "Fractional Integration and Optimal Estimates for Elliptic Systems",
  "authors": [
    "Felipe Hernandez",
    "Daniel Spector"
  ],
  "comment": "22 pages",
  "categories": [
    "math.AP",
    "math.FA"
  ],
  "abstract": "In this paper we prove the following optimal Lorentz embedding for the Riesz potentials: Let $\\alpha \\in (0,d)$. There exists a constant $C=C(\\alpha,d)>0$ such that \\[ \\|I_\\alpha F \\|_{L^{d/(d-\\alpha),1}(\\mathbb{R}^d;\\mathbb{R}^d)} \\leq C \\|F\\|_{L^1(\\mathbb{R}^d;\\mathbb{R}^d)} \\] for all fields $F \\in L^1(\\mathbb{R}^d;\\mathbb{R}^d)$ such that $\\operatorname*{div} F=0$ in the sense of distributions. We then show how this result implies optimal Lorentz regularity for a Div-Curl system (which can be applied, for example to obtain new estimates for the magnetic field in Maxwell's equations), as well as for a vector-valued Poisson equation in the divergence free case.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-08-13T01:49:51.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "optimal estimates",
      "fractional integration",
      "elliptic systems",
      "result implies optimal lorentz regularity",
      "divergence free case"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}