{
  "id": "2108.11840",
  "version": "v1",
  "published": "2021-08-26T15:10:03.000Z",
  "updated": "2021-08-26T15:10:03.000Z",
  "title": "Sobolev estimates for fractional parabolic equations with space-time non-local operators",
  "authors": [
    "Hongjie Dong",
    "Yanze Liu"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We obtain $L_p$ estimates for fractional parabolic equations with space-time non-local operators $$ \\partial_t^\\alpha u - Lu= f \\quad \\mathrm{in} \\quad (0,T) \\times \\mathbb{R}^d,$$ where $\\partial_t^\\alpha u$ is the Caputo fractional derivative of order $\\alpha \\in (0,1]$, $T\\in (0,\\infty)$, and $$Lu(t,x) := \\int_{ \\mathbb{R}^d} \\bigg( u(t,x+y)-u(t,x) - y\\cdot \\nabla_xu(t,x)\\chi^{(\\sigma)}(y)\\bigg)K(t,x,y)\\,dy $$ is an integro-differential operator in the spatial variables. Here we do not impose any regularity assumption on the kernel $K$ with respect to $t$ and $y$. We also derive a weighted mixed-norm estimate for the equations with operators that are local in time, i.e., $\\alpha = 1$, which extend the previous results by using a quite different method.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-08-26T15:10:03.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "fractional parabolic equations",
      "space-time non-local operators",
      "sobolev estimates",
      "integro-differential operator",
      "caputo fractional"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}