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

TL;DR

This paper establishes a connection between boundary differentiability of functions in the disk algebra and the smoothness of their boundary maps, extending results to Jordan domains using Poisson representation and classical theorems.

Contribution

It proves the equivalence of boundary differentiability and boundary map smoothness for functions in the disk algebra and extends these results to Jordan domains.

Findings

Equivalence between boundary derivative extension and boundary map differentiability.

Characterization of $A^p(D)$ as intersection of $A(D)$ and $C^p( ext{boundary})$.

Extension of results to Jordan domains using classical theorems.

Abstract

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.