{
  "id": "2001.11944",
  "version": "v1",
  "published": "2020-01-31T16:43:15.000Z",
  "updated": "2020-01-31T16:43:15.000Z",
  "title": "Calderon-Zygmund type estimates for nonlocal PDE with Hölder continuous kernel",
  "authors": [
    "Tadele Mengesha",
    "Armin Schikorra",
    "Sasikarn Yeepo"
  ],
  "categories": [
    "math.AP",
    "math.FA"
  ],
  "abstract": "We study interior $L^p$-regularity theory, also known as Calderon-Zygmund theory, of the equation \\[ \\int_{\\mathbb{R}^n} \\int_{\\mathbb{R}^n} \\frac{K(x,y)\\ (u(x)-u(y))\\, (\\varphi(x)-\\varphi(y))}{|x-y|^{n+2s}}\\, dx\\, dy = \\langle f, \\varphi \\rangle \\quad \\varphi \\in C_c^\\infty(\\mathbb{R}^n). \\] For $s \\in (0,1)$, $t \\in [s,2s]$, $p \\in [2,\\infty)$, $K$ an elliptic, symmetric, H\\\"older continuous kernel, if $f \\in \\left (H^{t,p'}_{00}(\\Omega)\\right )^\\ast$, then the solution $u$ belongs to $H^{2s-t,p}_{loc}(\\Omega)$ as long as $2s-t < 1$. The increase in differentiability is independent of the H\\\"older coefficient of $K$. For example, our result shows that if $f\\in L^{p}_{loc}$ then $u\\in H^{2s-\\delta,p}_{loc}$ for any $\\delta\\in (0, s]$ as long as $2s-\\delta < 1$. This is different than the classical analogue of divergence-form equations ${\\rm div}(\\bar{K} \\nabla u) = f$ (i.e. $s=1$) where a $C^\\gamma$-H\\\"older continuous coefficient $\\bar{K}$ only allows for estimates of order $H^{1+\\gamma}$. In fact, it is another appearance of the differential stability effect observed in many forms by many authors for this kind of nonlocal equations -- only that in our case we do not get a \"small\" differentiability improvement, but all the way up to $\\min\\{2s-t,1\\}$. The proof argues by comparison with the (much simpler) equation \\[ \\int_{\\mathbb{R}^n} K(z,z) (-\\Delta)^{\\frac{t}{2}} u(z) \\, (-\\Delta)^{\\frac{2s-t}{2}} \\varphi(z)\\, dz = \\langle g,\\varphi\\rangle \\quad \\varphi \\in C_c^\\infty(\\mathbb{R}^n). \\] and showing that as long as $K$ is H\\\"older continuous and $s,t, 2s-t \\in (0,1)$ then the \"commutator\" \\[ \\int_{\\mathbb{R}^n} K(z,z) (-\\Delta)^{\\frac{t}{2}} u(z) \\, (-\\Delta)^{\\frac{2s-t}{2}} \\varphi(z)\\, dz - c\\int_{\\mathbb{R}^n} \\int_{\\mathbb{R}^n} \\frac{K(x,y)\\ (u(x)-u(y))\\, (\\varphi(x)-\\varphi(y))}{|x-y|^{n+2s}}\\, dx\\, dy \\] behaves like a lower order operator.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-01-31T16:43:15.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "calderon-zygmund type estimates",
      "hölder continuous kernel",
      "nonlocal pde",
      "differential stability effect",
      "lower order operator"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}