{
  "id": "1905.09057",
  "version": "v1",
  "published": "2019-05-22T10:40:08.000Z",
  "updated": "2019-05-22T10:40:08.000Z",
  "title": "Harmonic Measure and the Analyst's Traveling Salesman Theorem",
  "authors": [
    "Jonas Azzam"
  ],
  "categories": [
    "math.CA",
    "math.AP",
    "math.MG"
  ],
  "abstract": "We study how generalized Jones $\\beta$-numbers relate to harmonic measure. Firstly, we generalize a result of Garnett, Mourgoglou and Tolsa by showing that domains in $\\mathbb{R}^{d+1}$ whose boundaries are lower $d$-content regular admit Corona decompositions for harmonic measure if and only if the square sum $\\beta_{\\partial\\Omega}$ of the generalized Jones $\\beta$-numbers is finite. Using this, we give estimates on the fluctuation of Green's function in a uniform domain in terms of the $\\beta$-numbers. As a corollary, for bounded NTA domains , if $B_{\\Omega}=B(x_{\\Omega},c\\mathrm{diam} \\Omega)$ is so that $2B_{\\Omega}\\subseteq \\Omega$, we obtain that \\[ (\\mathrm{diam} \\partial\\Omega)^{d} + \\int_{\\Omega\\backslash B_{\\Omega}} \\left|\\frac{\\nabla^2 G_{\\Omega}(x_{\\Omega},x)}{G_{\\Omega}(x_{\\Omega},x)}\\right|^{2} \\mathrm{dist}(x,\\Omega^c)^{3} dx \\sim \\mathscr{H}^{d}(\\partial\\Omega). \\] Secondly, we also use $\\beta$-numbers to estimate how much harmonic measure fails to be $A_{\\infty}$-weight for semi-uniform domains with Ahlfors regular boundaries.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-05-22T10:40:08.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "31A15",
      "28A75",
      "28A78",
      "31B05",
      "35J25"
    ],
    "keywords": [
      "analysts traveling salesman theorem",
      "harmonic measure",
      "content regular admit corona decompositions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}