Carathéodory and Smirnov type theorems for harmonic mappings of the unit disk onto surfaces

Abstract

Abstract. We prove representation theorems, the versions of Smirnov's theorem and Carathéo-dory type theorem for harmonic homeomorphisms of the unit disk onto Jordan surfaces with rectifiable boundaries. Further we establish the classical isoperimetric inequality and the Riesz-Zygmund inequality for Jordan harmonic surfaces without any smoothness assumptions on the boundary.

Quasiconformal harmonic mappings between Dini-smooth Jordan domains

“…(see e.g. [32,18]). Therefore, by having in mind the quasinconformality, we get that g ′ , h ′ ∈ H 1 (D).…”

Quasiconformal mappings with controlled Laplacian and Hölder continuity

“…Thus we may assume that ∇w ∈ L q (B n ) with q > 2n. At this stage (9) shows that ∆h ∈ L p∧(q/2) (B n ). As p ∧ (q/2) = p, Lemma 1(a) verifies that ∇h ∈ L np/(n−p) (B n ).…”

Schwarz Lemma, and Distortion for Harmonic Functions Via Length and Area

“…Proof. Since the Gaussian curvature of S is negative, by a version of isoperimetric inequality (see for example Theorem 3.4 [9]), we get (i1). Set J u = √ EG − F 2 .…”