  Inspired by [6, 7], we study the boundary regularity of constant curvature
hypersurfaces in the hyperbolic space $\mathbb{H}^{n+1}$, which have prescribed
asymptotic boundary at infinity. Through constructing the boundary expansions
of the solutions, we derive the optimal regularity of the solutions. Moreover,
we obtain an equivalent condition that guarantees the smoothness of the
solutions.
