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∞$A_{\infty }$ 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)$A(z)=\arg ?({\partial }_{\phi }(f(z))/z)$ where z=re^iφ$z=r{e}^{i\phi }$, is well-defined and smooth in U^?={z:0<|z|<1}${\mathbf{U}}^{?}=\{z:0<|z|<1\}$ and has a continuous extension to the boundary of the unit disk if and only if the image domain has C¹$C^1$ boundary.