{
  "id": "2012.13964",
  "version": "v1",
  "published": "2020-12-27T15:35:52.000Z",
  "updated": "2020-12-27T15:35:52.000Z",
  "title": "Fractional-order operators with real kernel, partial integration over a halfspace",
  "authors": [
    "Gerd Grubb"
  ],
  "comment": "18 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "For a strongly elliptic pseudodifferential operator $L$ of order $2a$ ($0<a<1$) with real kernel, we show an integration-by-parts formula in the model case where the operator is $x$-independent with homogeneous symbol, considered on the halfspace $R^n_+$. The new aspect, compared to $(-\\Delta )^a$, is that $L$ is allowed to have both an even and an odd part. Hence it satisfies a $\\mu $-transmission condition where generally $\\mu \\ne a$. We present a complex method, relying on a factorization in factors holomorphic in $\\xi_n$ in the lower or upper complex halfplane, using order-reducing operators combined with a decomposition principle originating from Wiener and Hopf. This is in contrast to a real, computational method presented very recently by Dipierro, Ros-Oton, Serra and Valdinoci. Our method allows $\\mu $ in a larger range than they consider. Moreover, we deduce a \"halfways Green's formula\" for $L$: $$ \\int_{R^n_+} Lu\\,\\bar v\\,dx-\\int_{R^n_+}u\\,\\overline{ L^*v}\\,dx=c\\int_{R^{n-1}}\\gamma_0(u/x_n^{\\mu -1 })\\,{\\gamma_0(\\bar v/x_n^{\\mu ^*})}\\, dx', $$ when $u$ solves the nonhomogeneous Dirichlet problem for $L$, and $v$ solves the homogeneous Dirichlet problem for $L^*$; here $\\mu ^*=2a-\\mu $.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-12-27T15:35:52.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35S15",
      "47G30",
      "35J25",
      "60G52"
    ],
    "keywords": [
      "real kernel",
      "partial integration",
      "fractional-order operators",
      "upper complex halfplane",
      "strongly elliptic pseudodifferential operator"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}