H lder continuity of harmonic quasiconformal mappings

We prove that for harmonic quasiconformal mappings α-H lder continuity on the boundary implies α-H lder continuity of the map itself. Our result holds for the class of uniformly perfect bounded domains, in fact we can allow that a portion of the boundary is thin in the sense of capacity. The problem for general bounded domains remains open. MSC 2010: 30C65.