| Summary: | Given a mixed Hodge module $\mathcal{N}$ and a meromorphic function $f$ on a complex manifold, we associate to these data a filtration (the irregular Hodge filtration) on the exponentially twisted holonomic module $\mathcal{N}\otimes \mathcal{E}^{f}$, which extends the construction of Esnault et al. ($E_{1}$-degeneration of the irregular Hodge filtration (with an appendix by Saito), J. reine angew. Math. (2015), doi:10.1515/crelle-2014-0118). We show the strictness of the push-forward filtered ${\mathcal{D}}$-module through any projective morphism ${\it\pi}:X\rightarrow Y$, by using the theory of mixed twistor ${\mathcal{D}}$-modules of Mochizuki. We consider the example of the rescaling of a regular function $f$, which leads to an expression of the irregular Hodge filtration of the Laplace transform of the Gauss–Manin systems of $f$ in terms of the Harder–Narasimhan filtration of the Kontsevich bundles associated with $f$.
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