0.0. In [Sat], M.Sato obtained a formula which describes the Fourier transform of a complex power of a relatively invariant polynomial of a prehomogeneous vector space over the real number field, up to an ambiguity of certain exponential factors. In [Gyo2] , I formulated conjectures which would give a finite field analogue of the theorem of M.Sato, without any ambiguity. Recently, J.Denef and I jointly have succeeded to prove these conjectures [DG] based on Laumon's product formula [Lau]. The

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... rpose of the present paper is to give an alternative approach based on the mixed Hodge theory. Our main result is Theorem 11, which includes as a special case Conjecture A of [Gyo2] up to an ambiguity of a constant factor of absolute value one. $\mathrm{T}\mathrm{l}\overline{\mathrm{l}}\mathrm{u}\mathrm{s}$ our result is less precise than [DG]. The result of the $1)\mathrm{r}\mathrm{e}\mathrm{S}\mathrm{e}\mathrm{n}\mathrm{t}$ paper was obtained around 1986 with help of M.Kashiwara, and thus seems more or less out of date, but I think it is still of some interest. The content was announced and outlined in [Gyo2]. 0.1. Our argument roughly goes as follows. Fix an isomorphism $(1-q)^{