  In this paper, we consider the obstacle problem for the fractional Laplace
operator $(-\Delta)^s$ in the Euclidian space $\mathbb{R}^n$ in the case where
$1<s<2$. As first observed in \cite{Y}, the problem can be extended to the
upper half-space $\mathbb{R}_+^{n+1}$ to obtain a thin obstacle problem for the
weighted biLaplace operator $\Delta^2_b U$, where $\Delta_b U=y^{-b}\nabla
\cdot (y^b \nabla U)$. Such a problem arises in connection with unilateral
phenomena for elastic, homogenous, and isotropic flat plates. We establish the
well-posedness and $C_{loc}^{1,1}(\R^n) \cap H^{1+s}(\R^n)$-regularity of the
solution. By writing the solutions in terms of Riesz potentials of suitable
local measures, we can base our proofs on tools from potential theory, such as
a continuity principle and a maximum principle. Finally, we deduce the
regularity of the extension problem to the higher dimensional upper half space.
This gives an extension of Schild's work in \cite{Sc1} and \cite{Sc2} from the
case $b=0$ to the general case $-1<b<1$.
