Décroissance rapide de la distribution $f^&lgr;$

Abstract

Let X be an open subset of C$^n$ and (f$_1$, …,f$_p$): X → C$^p$ be a holomorphic mapping. We prove that if (x$^0$,0, λ$^0$) ∈ T$^*$ × C$^p$ does not belong to the characteristic variety of the D$_X$ [λ]-module D$_X$[λ]f$^&lgr;$, then there exists a conic neighborhood V × Γ of (x$^0$, λ$^0$) such the function (λ$_1$, …, λ$_p$) [map ] ∫ | f$_1$ |$^lgr_1$ … | f$_p$ | $^lgr_p$ ω is rapidely decreasing in | Im λ | for λ ∈ Γ with Re λ bounded, for any (n,n)-form ω of class C$^∞$ with compact support in V. The following partial converse of this result is also established: if s [map ] ∫$_f=s$ φ is of class C$^∞$ in C$^p$ for all (n,n)-forms φ of class C$^∞$ with compact support in X, then d f$_1$ ∧ … ∧ d f$_p$ (x) ≠ 0, ∀ x ∈ X.