Fourier transform and Radon transform for mixed Hodge modules

We give a generalization to bi-filtered $\mathcal{D}$-modules underlying mixed Hodge modules of the relation between microlocalization along $f_1,\mathrm{.}\mathrm{.}\mathrm{.},f_r\in {\mathcal{O}}_X(X)$ and vanishing cycles along $g={\sum }_{i=1}^r{y}_i{f}_i$. This leads to an interesting isomorphism between localization triangles. As an application, we use these results to compare the $k$-plane Radon transform and the Fourier-Laplace transform for mixed Hodge modules. This is then applied to the Hodge module structure of certain GKZ systems.