Given a complex harmonic function of finite Dirichlet energy on a quasidisk, it has boundary values in a certain conformally invariant sense by a construction of H. Osborn. We call the set of such boundary values the Douglas-Osborn space. One may then solve the Dirichlet problem on the complement for these boundary values. This defines a reflection of harmonic functions. We show that quasidisks are precisely those Jordan curves for which this reflection is well-defined and bounded. Furthermore, the reflection is an isometry with respect to the Dirichlet norm.
We then use a limiting Cauchy integral along level curves of Green's function to show that the Plemelj-Sokhotski jump formula holds on quasicircles with boundary data in the Douglas-Osborn space, which also enable us to prove the well-posedness of a Riemann-Hilbert problem with boundary data in Douglas-Osborn space on quasicircles.