Muckenhoupt weights and Lindelöf theorem for harmonic mappings

We extend the result of Lavrentiev which asserts that the harmonic measure and the arc-length measure are A∞A∞ equivalent in a chord-arc Jordan domain. By using this result we extend the classical result of Lindelöf to the class of quasiconformal (q.c.) harmonic mappings by proving the following assertion. Assume that f is a quasiconformal harmonic mapping of the unit disk U onto a Jordan domain. Then the function A(z)=arg⁡(∂φ(f(z))/z)A(z)=arg⁡(∂φ(f(z))/z) where z=reiφz=reiφ, is well-defined and smooth in U⁎={z:0<|z|<1}U⁎={z:0<|z|<1} and has a continuous extension to the boundary of the unit disk if and only if the image domain has C1C1 boundary.