I am reading Gilbarg-Trudinger's book "Elliptic Partial Differential Equations of Second Order". I do not understand the proof of Theorem 4.13.
Theorem 4.13 is a special case of Kellogg's theorem in a ball. Its proof is to use the Kelvin transform to transfer a ball to a halfspace. I do not know how to get the regularity of $v$ which is defined as $$v(x)=\frac{1}{|x|^{n-2}}u(\frac{x}{|x|^2}),$$ and satisfies $$\Delta v=\frac{1}{|x|^{n+2}}f(\frac{x}{|x|^2}),$$ since I am not clear the regularity of $\Delta v$.
After Corollary 4.14, the book says that
"As a byproduct of the proof of Theorem 4.13, we see that Lemma 4.4 may be improved in the sense that if $f\in C^\alpha(\overline{B})$, its Newtonian potential in $B$ will belong to $C^{2,\alpha}(\overline{B})$."
I do not know how to use the proof of Theorem 4.13.
Could anyone give some hints?
