Let $f:S^n\to C$ be a continuous function, $n\geq 1$. When $n=1$, this is a well-known theorem, called Kellog's theorem (or sometimes Kellog-Warschawski's theorem) which states the following

Theorem: Fix $k \geq 0, 0<\alpha<1$. Let $f\in C^{k,\alpha}(S^1)$. Then its harmonic extension $H(f)$, which is the solution to the Dirichlet problem on the unit disk $D$ with boundary value $f$, is in $C^{k+1, \alpha}(D)$.

My main question is: is the above true for $n\geq 2$ as well? Any refernces/ suggestions?

While I don't know exactly a complete reference for the proof, but I have read the following theorem mentioned in the book "Boundary Behaviour of Conformal maps" by Christian Pommerenke which states:

Let $F:D\to\Omega $ be a conformal homeomorphism of $D$ onto a Jordan domain $\Omega$ whose boundary curve $\partial\Omega$ has a $C^{k,\alpha}$ -parametrization. Then $f\in C^{k+1, \alpha}(D)$. Note that any conformal homeomorphism $F$ of $D$ onto a Jordan domain extends to the boundary of $D$, by Caratheodory's extension theorem.