Harmonic reflection in quasicircles and well-posedness of a Riemann–Hilbert problem on quasidisks

Abstract

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.2010 Mathematics Subject Classification. Primary 35Q15, 30C62, 30E25, 31C25 ; Secondary 31A20.

A Model of the Teichmüller space of genus-zero bordered surfaces by period maps

“…The operators P (Ω ± ) were shown to be well-defined maps which are bounded with respect to the Dirichlet semi-norm (1.3). It was also shown in [19] that the Faber operator is an isomorphism precisely for quasicircles. This remarkable result is originally due to Y. Shen [21], with a somewhat different formulation of the operator; closely related results for convergence of Faber series on quasidisks were obtained by A. Ç avuş [3].…”

Transmission of harmonic functions through quasicircles on compact Riemann surfaces

“…In the case when Γ is a Jordan curve separating the Riemann sphere into two components, the authors showed that the transmission exists and is bounded if and only if Γ is a quasicircle [28]. Our proof here of the general case uses sewing techniques for Riemann surfaces.…”

Dirichlet spaces of domains bounded by quasicircles

“…They also disproved a conjecture of Anderson that the Faber operator is a bounded isomorphism of Besov spaces B p for general Jordan curves. The formulation of Schippers and Staubach as a map between Dirichlet spaces for general quasicircles (in terms of the composition operator and Cauchy-type projection) required the jump formula on quasicircles as well as the existence of a bounded reflection of Dirichlet-bounded harmonic functions, obtained in [17,18].…”

“…It is not clear whether H + (Γ) and H − (Γ) are the same in general. However, on quasidisks we have by a result in[17, Theorem 2.14] thatH + (Γ) = H − (Γ)in the following sense. Fix conformal maps f : D + → Ω + and g : D − → Ω − .…”

“…For h ∈ H(Γ) let E ± h denote the unique elements of D(Ω ± ) whose boundary values in the sense of Osborn agree with those of h.Remark 2.2. In fact, the condition that H + (Γ) = H − (Γ) characterizes quasidisks in a certain sense, see[17, Theorem 2.14].…”