  A complex harmonic function of finite Dirichlet energy on a Jordan domain 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 attempt to solve the Dirichlet problem on the complement for these
boundary values. This defines a reflection of harmonic functions. We show that
quasicircles are precisely those Jordan curves for which this reflection is
defined and bounded. 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. This enables us to
prove the well-posedness of a Riemann-Hilbert problem with boundary data in the
Douglas-Osborn space on quasicircles.

Complex Variables