Relations of the spaces $A^p(\mathrm{\Omega })$ and $C^p(\mathrm{\partial }\mathrm{\Omega })$

In this paper we prove that for functions $f\in A(D)$ there is an equivalence between the continuous extension of their derivatives over the boundary and the differentiability of the map $t\mapsto f(e^{it})$. More specifically, we are able to prove that $A^p(D)=A(D)\cap C^p(\mathbb{T})$ by making use of the Poisson representation. Moreover, we extend our results over Jordan domains bounded by an analytic Jordan curve by using what we initially prove on the disk in combination with the Osgood- Caratheodory theorem.