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 $A_\infty$ 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 $\mathbf{U}$ onto a Jordan domain. Then the function $A(z)=\arg(\partial_φ(f(z))/z)$ where $z=re^{iφ}$, is well-defined and smooth in $\mathbf{U}^*=\{z: 0<|z|<1\}$ and has a continuous extension to the boundary of the unit disk if and only if the image domain has $C^1$ boundary.

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