Suppose we have a map $ h : S^1\to \mathbb{C} $ that we know is a $\mathcal{C^r} $ map ( in the sense of a map between 1-manifold ( or in the sense of a $2\pi$ periodic map from $\mathbb{R}\to \mathbb{R} $ ). Consider the complex harmonic extension $H$ of that map onto the whole closed unit disk $\mathbb{D} $. Is $H, $ as a map from the closed unit disk, a $\mathcal{C^k}(\mathbb{\bar{D}}) $map ? From PDE or complex analysis books, $H$ is definitely a $ \mathcal{C^\infty}(\mathbb{D})\cap \mathcal{C}(\mathbb{\bar{D}}) $ map , but my question is : is there an open set $U$ containing the closed unit disk $\bar{D}$ such that $H \in \mathcal{C^\infty}(U)$ ?

Just suggest a reference if the answer is pretty long . Thanks.

method of translations(here, method of rotations). Therefore $H\in{\mathcal C}^r(\overline D)$ as well. – Denis Serre Apr 8 '11 at 15:11