  Let $\mathbb{X}$ be a Jordan domain satisfying hyperbolic growth conditions.
Assume that $\varphi$ is a homeomorphism from the boundary $\partial
\mathbb{X}$ of $\mathbb{X}$ onto the unit circle. Denote by $h$ the harmonic
diffeomorphic extension of $\varphi $ from $\mathbb{X}$ onto the unit disk. We
establish the optimal Orlicz-Sobolev regularity and weighted Sobolev estimate
of $h.$ These generalize the Sobolev regularity of $h$ by Koski-Onninen [21,
Theorem 3.1].
