Proof of the Kellogg's theorem Kellogg' theorem: Let $m\ge1,0<\alpha<1$. If $\Omega\in \mathbb{R}^n$ is of class $C^{m,\alpha}$, then $\log k\in C^{m-1,\alpha}(\partial\Omega)$, where $k(y)=k_{x_0}(y)=k(x_0,y)$ is the Poisson kernel for the domain $\Omega$ and $x_0\in\Omega$ is fixed.
I am looking for the proof of this theorem, since I can not find it in Kellogg's book:Foundations of Potential Theory. I will be grateful if someone can tell me where I can get a complete proof for it.

Kellog's original article on his theorem.
http://www.ams.org/journals/tran/1931-033-02/S0002-9947-1931-1501602-2/home.html
