Ahlfors function and holomorphic maps on an annulus shappe region

The Ahlfors function, which in the case of the annulus is a 2-to-1 mapping of the annulus onto the unit disk, which extends to be holomorphic on a neighborhood of the annulus and which maps each boundary circle to the unit circle. See this post.

Now suppose we have a smooth function $u$ defined on an annulus $K$. I wonder, since Ahlfors function is 2-to-1, whether we can define two functions $u_1$ and $u_2$ on unit disk such that

1. $u_1$ and $u_2$ are smooth functions on the unit disk
2. $\int_{K}u dA= \int_{D} u_1dA+\int_{D}u_2dA?$ ($D$ is the unit disk)

Indeed we can probably define $\{u_1(x),u_2(x)\}=u(f^{-1}(x))$, $x\in D$ where $f$ is the Ahlfors function, and make this definition well-defined.

If not, can we construct such functions $u_1$ and $u_2$ under other stronger assumptions?