  The value function of an optimal stopping problem for jump diffusions is
known to be a generalized solution of a variational inequality. Assuming that
the diffusion component of the process is nondegenerate and a mild assumption
on the singularity of the L\'{e}vy measure, this paper shows that the value
function of this optimal stopping problem on an unbounded domain with
finite/infinite variation jumps is in $W^{2,1}_{p, loc}$ with $p\in(1,
\infty)$. As a consequence, the smooth-fit property holds.
