Moduli of continuity of harmonic quasiregular mappings on bounded domains

Abstract

We prove that Ωu(δ) ≤ CΩf (δ), where u: Ω → Rn is the harmonic extension of a continuous map f: ω → Rn, if u is a K-quasiregular map and ω is bounded in Rn with C2 boundary. Here C is a constant depending only on n, Ωf and K and Ωh denotes the modulus of continuity of h. We also prove a version of this result for ΛΩ-extension domains with c-uniformly perfect boundary and quasiconformal mappings.