Dirichlet spaces over chord-arc domains

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 [9] has shown that

$$\begin{aligned} \iint _{{\mathbb {D}}}|\nabla U_i|^2(z)dxdy&= \iint _{\bar{{\mathbb {C}}}\backslash \bar{{\mathbb {D}}}}|\nabla U_e|^2(z)dxdy\\&= \frac{1}{2\pi }\iint _{{\mathbb {S}}\times {\mathbb {S}}} \left| \frac{u(z_1)-u(z_2)}{z_1-z_2}\right| ^2|dz_1||dz_2|, \end{aligned}$$

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.