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We use filtered log-$\mathscr{D}$-modules to represent the (dual)localization of Saito's Mixed Hodge Modules along a smooth hypersurface, andshow that they also behave well under the direct image functor and the dualfunctor in the derived category of filtered log-$\mathscr{D}$-modules. Theresults of this paper can be used to generalize the result of M. Popa and C.Schnell about Kodaira dimension and zeros of holomorphic one-forms into the logsetting.

Logarithmic Comparison with smooth boundary divisor in Mixed Hodge Modules