p-Dirichlet spaces over chord-arc domains

Abstract

Let $\Gamma$ be a rectifiable Jordan curve in the complex plane, $\Omega_i$ and $\Omega_e$ respectively the interior and exterior domains of $\Gamma$, and $p\geq 2$. Let $E$ be the vector space of functions defined on $\Gamma$ consisting of restrictions to $\Gamma$ of functions in $C^1(\mathbb C)$. We define three semi-norms on $E$: \begin{enumerate} \item $\Vert u\|_i=\left(\frac{1}{2\pi}\iint_{\Omega_i}|\nabla U_i(z)|^p\lambda_{\Omega_i}^{2-p}(z) dxdy\right)^{1/p},$ where $U_i$ is the harmonic extension of $u\in E$ to $\Omega_i$ and $\lambda_{\Omega_i}$ is the density of hyperbolic metric of domain $\Omega_i$, \item $\|u\|_e$ defined similarly for the exterior domain $\Omega_e$, \item $\|u\|_{B_p(\Gamma)} =\left(\frac{1}{4\pi^2}\iint_{\Gamma\times\Gamma}\frac{|u(z)-u(\zeta)|^p}{|z-\zeta|^2}|dz| |d\zeta|\right)^{1/p}$. \end{enumerate} The equivalences of these three semi-norms are well-known when $\Gamma$ is the unit circle. We prove that they are equivalent if and only if $\Gamma$ is a chord-arc curve.