Regularization of Relative Holonomic $\mathcal{D}$-Modules

Let X and S be complex analytic manifolds where S plays the role of a parameter space. Using the sheaf \mathcal{D}_{X\times S/S}^{\infty} of relative differential operators of infinite order, we construct functorially the regular holonomic \mathcal{D}_{X\times S/S} -module \mathcal{M}_{{\mathrm{reg}}} associated to a relative holonomic \mathcal{D}_{X\times S/S} -module \mathcal{M} , extending to the relative case classical theorems by Kashiwara–Kawai: denoting by \mathcal{M}^{\infty} the tensor product of \mathcal{M} by \mathcal{D}_{X\times S/S}^{\infty} we make \mathcal{M}^{\infty} explicit in terms of the sheaf of holomorphic solutions of \mathcal{M} . As a consequence of the relative Riemann–Hilbert correspondence we conclude that \mathcal{M}^{\infty} and \mathcal{M}_{{\mathrm{reg}}}^{\infty} are isomorphic.