p-Dirichlet spaces over chord-arc domains

Abstract

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