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) $\Vert u{\Vert }_i={\left(\frac{1}{2\pi }{\iint }_{{\Omega }_i}|\nabla U_i(z){|}^p{\lambda }_{{\Omega }_i}^{2-p}(z)dA(z)\right)}^{1/p}$, where $U_i$ is the harmonic extension of $u$ to ${\Omega }_i$ and ${\lambda }_{{\Omega }_i}$ is the hyperbolic density of ${\Omega }_i$; (ii) $\Vert u{\Vert }_e$, defined analogously on ${\Omega }_e$; and (iii) $\Vert u{\Vert }_{B_p(\Gamma )}={\left(\frac{1}{4{\pi }^2}{\iint }_{\Gamma \times \Gamma }\frac{|u(z)-u(\zeta ){|}^p}{|z-\zeta {|}^2}|dz||d\zeta |\right)}^{1/p}$. These three seminorms are known to be equivalent when $\Gamma$ is a chord-arc curve. We investigate the converse problem.