I encountered this issue recently, but do not know of any general results to deal with it, so I would appreciate any pointers.

Let $\mathbb T=\{z\in\mathbb C\mid |z|=1\}$, and let $f:\mathbb T\to\mathbb C$ be continuous and injective, so its image $\mathbb T'$ is a Jordan loop. Under what (general) conditions can we ensure that there is a homeomorphism between the unit disc and the interior of $\mathbb T'$ whose extension to the boundary is $f$?

Moreover, if there are reasonable conditions that ensure this, and $f$ is $C^\infty$, can we further require some nice regularity (perhaps even $C^\infty$) of the extension as well?

Classical Topics in Complex Function Theory, page 187: books.google.com/books?id=BHc2b0iCoy8C&pg=PA187 and the book refers to Pommerenke,Boundary Behavior of Conformal Maps, Springer 1991 for details. $\endgroup$ – Theo Buehler Sep 23 '12 at 13:07