Given a $C^1$ function $g(z)$ on the unit circle, I would like to find $f^+$ holomorphic on $\mathbb D$ and continuous on $\overline{\mathbb D}$, $f^-$ holomorphic on $\mathbb C\setminus \overline{\mathbb D}$ and continuous on $\mathbb C\setminus \mathbb D$ such that $g=f^+-f^-$ on $\mathbb S^1$.
I guess both $f^+$ and $f^-$ is given by the Cauchy integral formula $\frac{1}{2\pi i}\oint \frac{g(w)}{w-z}dw$. It is clear that $f^+$ and $f^-$ are holomorphic on the interior and exterior of the disk, but it isn't clear to me how to show $f^+$ and $f^-$ are continuous at the boundary and their difference is $g$. The problem is $g$ is only a $C^1$ function but not a holomorphic function on a neighborhood of the unit circle. Can anyone help? Thanks.