p-Dirichlet spaces over chord-arc domains

Abstract

Let $\Gamma$ be a rectifiable Jordan curve in the complex plane, let $\Omega_i$ and $\Omega_e$ be its interior and exterior domains, respectively, and let $1 < p < \infty$. Let $E$ be the vector space of restrictions to $\Gamma$ of functions in $C^1(\mathbb C)$. We consider the following three seminorms on $E$: (i) $\lVert u\rVert_i=\left(\frac{1}{2\pi}\iint_{\Omega_i}|\nabla U_i(z)|^p\lambda_{\Omega_i}^{2-p}(z)\,dA(z)\right)^{1/p}$, where $U_i$ is the harmonic extension of $u$ to $\Omega_i$ and $\lambda_{\Omega_i}$ is the hyperbolic density of $\Omega_i$; (ii) $\lVert u\rVert_e$, defined analogously on $\Omega_e$; and (iii) $\lVert u\rVert_{B_p(\Gamma)}=\left(\frac{1}{4\pi^2}\iint_{\Gamma\times\Gamma}\frac{|u(z)-u(\zeta)|^p}{|z-\zeta|^2}\,|dz|\,|d\zeta|\right)^{1/p}$. These three seminorms are known to be equivalent when $\Gamma$ is a chord-arc curve. We investigate the converse problem.