Truncated microsupport and hyperbolic inequalities

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$\cal{M}$isacoherent$\cal{D}$-module, we obtain new estimates for the truncated microsupport of real analytic and hyperfunction solutions. When $\mathcal{M}$ 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 $\text{shm}$ when $Y$ is a hypersurface.