Dirichlet spaces over chord-arc domains

TL;DR

This paper investigates the equivalence of three Dirichlet norms associated with harmonic extensions over chord-arc domains, extending classical results from the unit circle to more general rectifiable Jordan curves.

Contribution

It establishes that the three Dirichlet norms are equivalent precisely when the boundary curve is a chord-arc curve, generalizing Douglas's classical results.

Findings

Three Dirichlet norms are equivalent iff the boundary is a chord-arc curve.

Extends classical harmonic extension results from circles to chord-arc domains.

Provides a characterization of chord-arc curves via Dirichlet space norms.

Abstract

If $U$ is a $C^{\infty}$ function with compact support in the plane, we let $u$ be its restriction to the unit circle $\mathbb{S}$, and denote by $U_i,\,U_e$ the harmonic extensions of $u$ respectively in the interior and the exterior of $\mathbb S$ on the Riemann sphere. About a hundred years ago, Douglas has shown that \begin{align*} \iint_{\mathbb{D}}|\nabla U_i|^2(z)dxdy&= \iint_{\bar{\mathbb{C}}\backslash\bar{\mathbb{D}}}|\nabla U_e|^2(z)dxdy &= \frac{1}{2\pi}\iint_{\mathbb S\times\mathbb S}\left|\frac{u(z_1)-u(z_2)}{z_1-z_2}\right|^2|dz_1||dz_2|, \end{align*} thus giving three ways to express the Dirichlet norm of $u$. On a rectifiable Jordan curve $\Gamma$ we have obvious analogues of these three expressions, which will of course not be equal in general. The main goal of this paper is to show that these $3$ (semi-)norms are equivalent if and only if $\Gamma$ is a chord-arc curve.