t-structures for relative D-modules and t-exactness of the de Rham functor

This paper is a contribution to the study of relative holonomic D-modules. Contrary to the absolute case, the standard t-structure on holonomic D-modules is not preserved by duality and hence the solution functor is no longer t-exact with respect to the canonical, resp. middle-perverse, t-structure. We provide an explicit description of these dual t-structures. We use this description to prove that the solution functor as well as the relative Riemann-Hilbert functor are t-exact with respect to the dual t-structure and to the middle-perverse one while the de Rham functor is t-exact for the canonical, resp. middle-perverse, t-structure and their duals.