  We prove that the k-truncated microsupport of the specialization of a complex
of sheaves $F$ along a submanifold is contained in the normal cone to the
conormal bundle along the k-truncated microsupport of $F$. In the complex case,
applying our estimates to $F=\text{R}\cal{H}\text{om}_{\cal{D}}(\cal{M},
\sho)$, where $\cal{M}$ is a coherent $\cal{D}$-module, we obtain new estimates
for the truncated microsupport of real analytic and hyperfunction solutions.
When $\cal{M}$ is regular along $Y$ we also obtain estimates for the truncated
microsupport of the holomorphic solutions of the induced system along $Y$ as
well as for the nearby-cycle sheaf of $\shm$ when $Y$ is a hypersurface.
