Hölder Continuity of Harmonic Quasiconformal Mappings

We prove that for harmonic quasiconformal mappings $α$-Hölder continuity on the boundary implies $α$-Hölder 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.

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