Quasiconformal and harmonic mappings between Jordan domains

摘要

Let Ω and Ω₁ be Jordan domains, let μ ∈ (0, 1], and let f: W® W₁f: Omega mapsto Omega_1 be a harmonic homeomorphism. The object of the paper is to prove the following results: (a) If f is q.c. and ∂Ω, ∂Ω₁ ∈ C 1,μ , then f is Lipschitz; (b) if f is q.c., ∂Ω, ∂Ω₁ ∈ C 1,μ and Ω₁ is convex, then f is bi-Lipschitz; and (c) if Ω is the unit disk, Ω₁ is convex, and ∂Ω₁ ∈ C 1,μ , then f is quasiconformal if and only if its boundary function is bi-Lipschitz and the Hilbert transform of its derivative is in L ∞. These extend the results of Pavlović (Ann. Acad. Sci. Fenn. 27:365–372, 2002).