On the modulus of continuity of harmonic quasiregular mappings on the unit ball in Rn

Abstract

We show that, for a class of moduli functions ω(δ), 0 ≤ δ ≤ 2, the property ׀φ(ξ) - φ(η) ׀≤ ω (׀ξ - η׀), ξ, ηЄ Sn-1 implies the corresponding property ׀u(x)-u(y) ׀≤ Cω(׀x-y׀) x, y Є Bn; for u = P[φ], provided u is a quasiregular mapping. Our class of moduli functions includes φ(δ) = δα (0 < α ≤ 1), so our result generalizes earlier results on Hölder continuity (see [1]) and Lipschitz continuity (see [2]).