{"title":"Dirichlet spaces over chord-arc domains","authors":"Huaying Wei, Michel Zinsmeister","doi":"arxiv-2407.11577","DOIUrl":null,"url":null,"abstract":"If $U$ is a $C^{\\infty}$ function with compact support in the plane, we let\n$u$ be its restriction to the unit circle $\\mathbb{S}$, and denote by\n$U_i,\\,U_e$ the harmonic extensions of $u$ respectively in the interior and the\nexterior of $\\mathbb S$ on the Riemann sphere. About a hundred years ago,\nDouglas has shown that \\begin{align*} \\iint_{\\mathbb{D}}|\\nabla U_i|^2(z)dxdy&=\n\\iint_{\\bar{\\mathbb{C}}\\backslash\\bar{\\mathbb{D}}}|\\nabla U_e|^2(z)dxdy &= \\frac{1}{2\\pi}\\iint_{\\mathbb S\\times\\mathbb\nS}\\left|\\frac{u(z_1)-u(z_2)}{z_1-z_2}\\right|^2|dz_1||dz_2|, \\end{align*} thus\ngiving three ways to express the Dirichlet norm of $u$. On a rectifiable Jordan\ncurve $\\Gamma$ we have obvious analogues of these three expressions, which will\nof course not be equal in general. The main goal of this paper is to show that\nthese $3$ (semi-)norms are equivalent if and only if $\\Gamma$ is a chord-arc\ncurve.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"39 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Complex Variables","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2407.11577","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
If $U$ is a $C^{\infty}$ function with compact support in the plane, we let
$u$ be its restriction to the unit circle $\mathbb{S}$, and denote by
$U_i,\,U_e$ the harmonic extensions of $u$ respectively in the interior and the
exterior of $\mathbb S$ on the Riemann sphere. About a hundred years ago,
Douglas has shown that \begin{align*} \iint_{\mathbb{D}}|\nabla U_i|^2(z)dxdy&=
\iint_{\bar{\mathbb{C}}\backslash\bar{\mathbb{D}}}|\nabla U_e|^2(z)dxdy &= \frac{1}{2\pi}\iint_{\mathbb S\times\mathbb
S}\left|\frac{u(z_1)-u(z_2)}{z_1-z_2}\right|^2|dz_1||dz_2|, \end{align*} thus
giving three ways to express the Dirichlet norm of $u$. On a rectifiable Jordan
curve $\Gamma$ we have obvious analogues of these three expressions, which will
of course not be equal in general. The main goal of this paper is to show that
these $3$ (semi-)norms are equivalent if and only if $\Gamma$ is a chord-arc
curve.