p-Dirichlet spaces over chord-arc domains

Abstract

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

12 pages, 2 figures. arXiv admin note: text overlap with arXiv:2407.11577