Dirichlet spaces over chord-arc domains
Huaying Wei, Michel Zinsmeister

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 is a function with compact support in the plane, we let be its restriction to the unit circle , and denote by the harmonic extensions of respectively in the interior and the exterior of 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 . On a rectifiable Jordan curve 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 (semi-)norms are equivalent if and only if is a chord-arc…
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Taxonomy
TopicsAnalytic and geometric function theory · Holomorphic and Operator Theory · Advanced Mathematical Modeling in Engineering
