A representation theorem for standard weighted harmonic mappings with an integer exponent and its applications

In this paper, we study α¯-harmonic mappings with an integer exponent and obtain a representation theorem which determines the relation between α¯-harmonic mappings and Euclidean harmonic mappings. As two applications of this representation theorem, we obtain a counterexample of the Radó–Kneser–Choquet theorem for α¯-harmonic mappings and show that the Lipschitz continuity of an α¯-harmonic mapping with an integer exponent is determined by the Euclidean harmonic mapping with same boundary.