Boundary values, Fourier-Sato tranform and Laplace transform

Pierre Schapira

Abstract

In the first part of this paper we describe the well-known Fourier-Sato’s transform of conic sheaves and we show how this theory permits to define the boundary value of holomorphic cohomology classes associated to conic sheaves and in particular cohomology classes supported by non necessarily convex cones. As an example, we treat the case of quadratic cones.

Next, we recall the recent result of [7] on the Laplace transform of holomorphic cohomology classes associated to constructible sheaves, and give a link between all these constructions.

1 Introduction

Let M be a real analytic manifold of dimension n, X a complexification of M, γ a cone in T_{M}X, the normal bundle to M in X. When γ is open and convex (i.e. the fibers of γ over M are convex), Sato’s theory, as explained in [9], allows one to define the boundary value of holomorphic functions defined in tuboids with profile γ. But in fact, this theory also applies to the non open and non convex case, even if it has never been written down explicitely, and one of the aims of this paper is to fill up this gap of the literature. Hence, it should be clear that the results obtained here are more or less well-known from some specialists. Note that boundary values of holomorphic cohomology classes defined in non convex cones are constructed “à la main” in some situations in [2].

In order to describe our results, consider a (non necessarily convex) locally closed cone γ as above, and let ℂ_{γ}^{∧} denote the Fourier-Sato transform of the sheaf ℂ_{γ}. Assume that for some local system L on M, some integer d, and some locally closed cone λ in T_{M}^{∗}X, one has:

H^{n−d}(ℂ_{γ}^{∧}) ≃ ℂ_{λ} ⊗ L.

AMS classification: 32L25, 58G37

Boundary values, Fourier-Sato tranform and Laplace transform

(Here, we write L instead of π^{−1}L for short.) Then:

H_{γ}^{d}(T_{M}X; ν_{M}(O_{X})) ≃ Γ_{λ}(T_{M}^{∗}X; C_{M} ⊗ L^{∗} ⊗ or_{M})

where ν_{M}(O_{X}) is the specialization of the sheaf O_{X} along M, C_{M} is the sheaf of Sato’s microfunctions, L^{∗} is the dual of L and or_{M} is the orientation sheaf. As we shall see, this formula appears as an immediate corollary of Sato’s theory.

Then comes the problem of calculating the Fourier-Sato transform ℂ_{γ}^{∧} of ℂ_{γ}. We consider various examples and in particular the case of a quadratic cone. This last example has already been considered in a different language and with a different method by Faraut-Gindikin [3].

We end this paper by briefly recalling the main theorem of [7] on the Laplace transform of cohomology classes associated to conic ℝ-constructible sheaves (and in particular to tempered distributions supported by non convex cones) and show a link between all the above constructions.

2 Review on the Fourier-Sato transform

The Fourier transform for sheaves on sphere bundles has been introduced by Sato (see [9]), then extended to conic sheaves on vector bundles by [1]. We refer to [5] for a detailed construction and for historical comments.

On a topological space X, one denotes by D^{b}(ℂ_{X}) the derived category of the category of bounded complexes of sheaves of ℂ-vector spaces on X. If X is a real manifold, one denotes by or_{X} the orientation sheaf on X and by ω_{X} the dualizing complex, ω_{X} ≃ or_{X} [dimX]. If f : Y ⟶ X is a morphism of manifolds, one sets or_{Y/X} = or_{Y} ⊗ f^{−1}or_{X}.

Let τ : E ⟶ M be a finite dimensional real vector bundle over a real manifold M with fiber dimension n, π : E^{∗} ⟶ M the dual vector bundle. Denote by p_{1} and p_{2} the first and second projection defined on E ×_{M} E^{∗}, and define:

P = {(x, y) ∈ E ×_{M} E^{∗}; ⟨x, y⟩ ≥ 0}
P′ = {(x, y) ∈ E ×_{M} E^{∗}; ⟨x, y⟩ ≤ 0}

Consider the diagram:

[figure: E ×_{M} E^{∗}; p_{1}; p_{2}; E; τ; E^{∗}; π; M]

Denote by D_{ℝ^{+}}^{b}(ℂ_{E}) the full triangulated subcategory consisting of conic objects, that is, of objects F such that for all j, H^{j}(F) is locally constant on the orbits of the action of ℝ^{+} on E.

Definition 2.1. Let F ∈ D_{ℝ^{+}}^{b}(ℂ_{E}), G ∈ D_{ℝ^{+}}^{b}(ℂ_{E^{∗}}). One sets:

F^{∧} = Rp_{2}_{!}(p_{1}^{−1}F)_{P′}
G^{∨} = Rp_{1}_{!}(p_{2}^{−1}G)_{P} ⊗ or_{E/M} [n]

The main result of the theory is the following.

Theorem 2.2. (i) There is a natural isomorphism

F^{∧} ≃ Rp_{2}_{∗}RΓ_{P} (p_{1}^{−1}F),

(ii) the two functors (·)^{∧} and (·)^{∨} are inverse to each other, hence define equivalences of categories

D_{ℝ^{+}}^{b}(ℂ_{E}) ≃ D_{ℝ^{+}}^{b}(ℂ_{E^{∗}}).

(iii) In particular, let F_{1} and F_{2} belong to D_{ℝ^{+}}^{b}(ℂ_{E}). Then there is a natural isomorphism:

RHom (F_{1}, F_{2}) ≃ RHom (F_{1}^{∧}, F_{2}^{∧}).

Moreover, the following formulas are of constant use.

Rτ_{!}F ≃ RΓ_{M}(F)|_{M} ≃ Rπ_{∗}(F^{∧}) ≃ (F^{∧})|_{M}  (2.1)
Rτ_{∗}F ≃ F |_{M} ≃ Rπ_{!}(F^{∧}) ⊗ or_{E^{∗}/M}[n] ≃ RΓ_{M}(F^{∧}) ⊗ or_{E^{∗}/M}  (2.2)

Example 2.3. (i) Let γ be a closed proper convex cone in E with M ⊂ γ. Then:

(ℂ_{γ})^{∧} ≃ ℂ_{Intγ^{∘}}.

Here γ^{∘} is the polar cone to γ, a closed convex cone in E^{∗} and Intγ^{∘} denotes its interior.

(ii) Let γ be an open convex cone in E. Then:

(ℂ_{γ})^{∧} ≃ ℂ_{γ^{∘a}} ⊗ or_{E^{∗}/M} [−n].

Here λ^{a} = −λ, the image of λ by the antipodal map.

(iii) In particular, applying Theorem 2.2 (iii), we get for F a conic sheaf on E, and γ an open convex cone:

RΓ(γ; F) ≃ RΓ_{γ^{∘a}}(E^{∗}; F^{∧} ⊗ or_{E^{∗}/M}) [n].

A useful property of the Fourier-Sato transform is that it commutes to base change. More precely, consider a morphism of manifolds: f : N ⟶ M. It defines morphisms of vector bundles:

[figure: f^{∗}E; f_{τ}; τ; E; τ; N; f; M; f^{∗}E^{∗}; f_{π}; π; E^{∗}; π; N; f; M]

Theorem 2.4. Let F ∈ D_{ℝ^{+}}^{b}(ℂ_{E}) and let G ∈ D_{ℝ^{+}}^{b}(ℂ_{f^{∗}E}). Then there are natural isomorphisms:

(f_{τ}^{−1}F)^{∧} ≃ f_{π}^{−1}(F^{∧})
(Rf_{τ}_{!}G)^{∧} ≃ Rf_{π}_{!}(G^{∧})

3 Review on specialization and microlocalization

We shall describe here the functor of specialization and its Fourier transform, Sato’s functor of microlocalization. For a detailed exposition, see [5].

Let X be a real manifold (let’s say of class C^{∞}), M a closed submanifold. Denote by τ : T_{M}X ⟶ M and π : T_{M}^{∗}X ⟶ M the normal bundle and the conormal bundle to M in X, respectively. Let F ∈ D^{b}(ℂ_{X}). The specialization of F along M, denoted ν_{M}(F), is an object of D_{ℝ^{+}}^{b}(ℂ_{T_{M} X}). Its cohomology objects are described as follows. If V is an open cone in T_{M}X, then

H^{j}(V; ν_{M}(F)) ≃ \underrightarrow{lim}_{U} H^{j}(U; F)

where U ranges over the family of open subsets of X which are “tangent” to V, that is, open tuboids in X with wedge M whose “profiles” is V. (For a precise definition, refer to [5].)

The microlocalization of F along M, denoted μ_{M}(F), is the Fourier-Sato transform of ν_{M}(F), hence is an object of D_{ℝ^{+}}^{b}(ℂ_{T_{M}^{∗} X}). It satisfies:

Rπ_{∗}μ_{M}(F) ≃ RΓ_{M}(F),

H^{j}(μ_{M}(F))_{(x_{0};ξ_{0})} ≃ \underrightarrow{lim}_{U,Z} H_{U∩Z}^{j}(U; F).

In the last formula, (x_{0}; ξ_{0}) ∈ T_{M}^{∗}X, U ranges over the family of open neighborghoods of x_{0} in X and Z ranges over the family of closed tuboids in X with wedge M whose profiles λ in T_{M}X satisfy (x_{0}; ξ_{0}) ∈ Intλ^{∘a}. (For a precise definition, refer to [5].)

Now assume M is a real analytic manifold of dimension n and X is a complexification of M. (This situation may be immediately generalized by considering “totally real submanifolds” of X.) In such a case, one of the main results of Sato’s theory asserts that the object μ_{M}(O_{X}) is concentrated in degree n (see [9] and also [8]). This leads to the following definition.

Definition 3.1. (i) The sheaf of Sato’s hyperfunctions on M is defined by:

B_{M} = H_{M}^{n}(O_{X}) ⊗ or_{M},

(ii) The sheaf of Sato’s microfunctions on T_{M}^{∗}X is defined by:

C_{M} = H^{n}(μ_{M}(O_{X})) ⊗ or_{M}.

Hence, a hyperfunction is nothing but a microfunction globally defined in the fibers of π. Denote by spec the natural isomorphism:

spec : B_{M} ≃ π_{∗}C_{M}.

If u is an hyperfunction, Sato defines its analytic wave front set as:

W F (u) = supp(spec(u)),

a closed conic subset of T_{M}^{∗}X.

Since Γ_{M}(T_{M}^{∗}X; C_{M}) ≃ A_{M} (the sheaf of real analytic functions on M), one sees that a hyperfunction is a real analytic function if and only if its wave front set is contained in the zero-section.

An important property (proved by Kashiwara) is that the sheaf C_{M} it is conically flabby (see [9]).

Applying Theorem 2.2, one gets:

Corollary 3.2. Let F ∈ D_{ℝ^{+}}^{b}(ℂ_{T_{M} X}). Then:

Rτ_{∗}RHom (F, ν_{M}(O_{X})) ≃ Rπ_{∗}RHom (F^{∧} ⊗ or_{M} [n], C_{M}).

This corollary applies in particular when F = ℂ_{γ}, γ being a cone in T_{M}X.

4 Boundary values

In this section, we shall apply Corollary 3.2 to various geometrical situations.

As above, let M be a real analytic manifold of dimension n and let X be a complexification of M. First, assume to be given a morphism (a “boundary values morphism”):

b : or_{M} [−n] ⟶ F.  (4.1)

It defines ℂ_{M} ⟶ F ⊗ or_{M} [n] and also, applying the Fourier-Sato transform on the vector bundle T_{M}X:

b : ℂ_{M} ⟶ F^{∧} ⊗ or_{M} [n].

Using Corollary 3.2, we get the commutative diagram:

[figure: Rτ_{∗}RHom (F, ν_{M}(O_{X})); b; ∼; Rπ_{∗}RHom (F^{∧} ⊗ or_{M} [n], C_{M}); B_{M}]

Next, we specialize our study to the case where F is associated to a cone. Let γ be a locally closed cone of T_{M}X, d an integer, and assume that there exists a locally closed cone λ in T_{M}^{∗}X and a local system L on M such that

H^{n−d}(ℂ_{γ}^{∧}) ≃ ℂ_{λ} ⊗ L.  (4.2)

Denote by L^{∗} the dual local system to L.

Proposition 4.1. Assume (4.2). Then there is a natural isomorphism:

H_{γ}^{d}(T_{M}X; ν_{M}(O_{X})) ≃ Γ_{λ}(T_{M}^{∗}X; C_{M} ⊗ or_{M} ⊗ L^{∗}).  (4.3)

Proof. Applying Corollary 3.2, we get the isomorphisms:

RΓ_{γ}(T_{M}X; ν_{M}(O_{X})) ≃ RHom (ℂ_{γ}, ν_{M}(O_{X}))
≃ RHom (ℂ_{γ}^{∧} ⊗ or_{M} [n], C_{M}).

Now apply H^{d} to both sides. Since the sheaf C_{M} is conically flabby, it is injective in the category of conic sheaves of ℂ-vector spaces on T_{M}^{∗}X. Hence we get the isomorphism:

H_{γ}^{d}(T_{M}X; ν_{M}(O_{X})) ≃ Hom (H^{n−d}(ℂ_{γ}^{∧}) ⊗ or_{M}, C_{M}),  (4.4)

and the result follows. q.e.d.

Remark 4.2. (i) In practice, L will be of rank one, and locally, it can be forgotten, as well as or_{M}.

(ii) The result is of particular interest when λ is a closed cone and L is of rank one. In such a case, the right hand-side of (4.3) is nothing but the space of hyperfunctions on M with wave front set contained in λ (locally on M, see the remark above).

(iii) If for some j, H^{n−j}(ℂ_{γ}^{∧}) = 0, then H^{j}(γ; ν_{M}(O_{X})) = 0.

(iv) Using formula (2.1), we get that if γ is an open cone in T_{M}X, then H^{n−d}(M; Rτ_{!}ℂ_{γ}⊗ or_{M})≃H^{n−d}(T_{M}^{∗}X; ℂ_{γ}^{∧} ⊗ or_{M}). Using equation (4.4), we get the pairing:

H^{n−d}(M; Rτ_{!}ℂ_{γ} ⊗ or_{M}) ⊗ H^{d}(γ; ν_{M}(O_{X})) ⟶ Γ(M; B_{M}).

Now assume M is a vector space, identify X to M × \sqrt{−1}M and let γ = M × γ̃, where γ̃ is an open cone of \sqrt{−1}M. Then H^{n−d}(M; Rτ_{!}ℂ_{γ} ⊗ or_{M}) ≃ H_{c}^{n−d}(\sqrt{−1}M; ℂγ̃) ≃ H_{d}(γ̃) (homology of γ̃ in degree d). Hence we get a pairing:

H_{d}(γ̃) ⊗ H^{d}(γ; ν_{M}(O_{X})) ⟶ Γ(M; B_{M}).

Given a cycle c ∈ H_{d}(γ̃), we get a map H^{d}(γ; ν_{M}(O_{X})) ⟶ Γ(M; B_{M}). This map is already considered (in some special situations) in [2] where its injectivity (which, of course, does not hold in general) is discussed.

(v) Let γ be an open tube in X, γ̄ its closure, and assume that γ is topologically convex and γ̄ contains M. The natural morphism ℂγ̄ ⟶ ℂ_{M} defines by duality the boundary values morphism (of 4.1) or_{M} [−n] ⟶ ℂ_{γ}, from which one deduces the morphism Γ(γ; O_{X}) ⟶ B(M). This purely topological construction of the boundary value morphism first appeared in [10] (see also [5] pp. 467).

Example 4.3. (i) Let γ be an open convex cone in T_{M}X. Then:

Γ(γ; ν_{M}(O_{X})) ≃ Γ_{γ^{∘a}}(T_{M}^{∗}X; C_{M}),

and H^{j}((γ; ν_{M}(O_{X})) = 0 for j ≠ 0.

(ii) Assume n > 1 and choose γ = T_{M}X \ M. The Fourier-Sato’s functor applied to the exact sequence of sheaves ℂ_{γ} ⟶ ℂ_{T_{M} X} ⟶ ℂ_{M} ⟶_{+1} gives rise to the

distinguished triangle: ℂ_{γ}^{∧} ⟶ or_{M}[−n] ⟶ ℂ_{T_{M}^{∗} X} ⟶_{+1} . Hence, H^{1}(ℂ_{γ}^{∧}) ≃ ℂ_{T_{M}^{∗} X},

H^{n−1}(ℂ_{γ}^{∧}) ≃ or_{M}, and the other groups are zero. We get:

H^{0}(γ; ν_{M}(O_{X})) ≃ Γ_{M}(T_{M}^{∗}X; C_{M}) ≃ Γ(M; A_{M})
H^{n−1}(γ; ν_{M}(O_{X})) ≃ Γ(T_{M}^{∗}X; C_{M} ⊗ or_{M}) ≃ Γ(M; B_{M} ⊗ or_{M})
H^{j}(γ; ν_{M}(O_{X})) ≃ 0 for j ≠ 1, n − 1.

(Of course, this result is classical, and can easily be obtained without the explicit use of the Fourier-Sato transform.)

5 Fourier-Sato transform of quadratic cones

Let E denote the the Euclidian space ℝ^{n}, with coordinates x = (x_{1}, …, x_{n}), and set x = (x′, x′′) with x′ = (x_{1}, …, x_{p}) and x′′ = (x_{p+1}, …, x_{n}). Let q = n − p and assume q > 1. We denote by u = (u′, u′′) the dual coordinates on the dual space E^{∗}.

Let γ denote the closed solid cone in E,

γ = {x; x′^{2} − x′′^{2} ≥ 0}

and let λ denote the closed solid cone in E^{∗}:

λ = {u; u′^{2} − u′′^{2} ≤ 0}.

Lemma 5.1. We have:

ℂ_{γ}^{∧} ≃ ℂ_{λ} [−p].

Proof. For u ∈ E^{∗}, set:

γ_{u} = {x ∈ γ; ⟨x, u⟩ ≤ 0}.

Then (ℂ_{γ}^{∧})_{u} ≃ RΓ_{c}(E; ℂ_{γ_{u}}). We have (see [5], Ex III 5):

RΓ_{c}(E; ℂ_{γ}) ≃ ℂ[−p].

Hence it is enough to check that:

(i) if u ∉ λ, RΓ_{c}(E; ℂ_{γ_{u}}) = 0, (ii) if u ∈ λ \ {0}, the morphism

RΓ_{c}(E; ℂ_{γ}) ⟶ RΓ_{c}(E; ℂ_{γ_{u}})

is an isomorphism.

Let us prove (i). Let u = (u′, u′′). We may assume u′′ = 0. Let f be the projection ℝ^{p} × ℝ^{q} ⟶ ℝ^{p}, (x′, x′′) ↦ x′, and set γ̃_{u} = f(γ_{u}). The fibers of f above γ̃_{u} are closed balls and γ̃_{u} is a closed half plane. Hence RΓ_{c}(γ̃_{u}; ℂγ̃_{u}) = 0, and (i) follows. Let us prove (ii). We may assume u = (0, …, 0, 1). Let f be the projection ℝ^{n} ⟶ ℝ^{n−1}, x ↦ (x_{1}, ⋯, x_{n−1}). Set γ̃_{u}^{−} = f(γ \ γ_{u}). Then the fibers of f above γ̃_{u}^{−} are intervals (0, a] for some a ∈ ℝ. Hence Rf_{!}ℂ_{γ\γ_{u}} = 0, and we obtain RΓ_{c}(M; ℂ_{γ\γ_{u}}) = 0, which implies (ii). q.e.d.

Now assume M is an open subset of E, X a complexification of M (of course this situation could be generalized to manifolds). We change our notations and denote by γ the cone M + \sqrt{−1}\{y; y′^{2} − y′′^{2} ≥ 0\} in T_{M}X and similarly for λ in T_{M}^{∗}X.

Applying Lemma 5.1 and Proposition 4.1, we get:

Proposition 5.2. One has the isomorphism:

H_{γ}^{q}(T_{M}X; ν_{M}O_{X}) ≃ Γ_{λ}(T_{M}^{∗}X; C_{M}).

Remark 5.3. The case of quadratic cones has already been considered by Faraut-Gindikin [3], with a different formulation and a different method. Their method consists in representing cohomology classes in the non convex tube γ as an integral of holomorphic functions in convex tubes.

6 Formal and temperate cohomology

In order to state the next result on the Laplace transform, we need to briefly recall the functors of formal (see [6]) and temperate cohomology (see [4]).

Let M be a real manifold, and let ℝ-cons(ℂ_{M}) denote the category of ℝ-cons-tructible sheaves on M, D_{ℝ−c}^{b}(ℂ_{M}) its derived category. The functors T hom(·, Db_{M})

of [4] and the dual functor · ⊗^{w} C_{M}^{∞} of [6], are defined on the category ℝ-cons(ℂ_{M}), with values in the category Mod(D_{M}) of D_{M}-modules on M. (The first functor is contravariant).

They are characterized as follows. Denote by Db_{M} the sheaf of Schwartz’s distributions on M and by C_{M}^{∞} the sheaf of C^{∞} functions on M. Let Z (resp. U) be a closed (resp. open) subanalytic subset of M. Then these two functors are exact and moreover:

T hom(ℂ_{Z}, Db_{M}) = Γ_{Z}(Db_{M}),
ℂ_{U} ⊗^{w} C_{M}^{∞} = I_{M\U}^{∞},

where Γ_{Z}(Db_{M}) denotes as usual the subsheaf of Db_{M} of sections supported by Z and I_{M\U}^{∞} denotes the ideal of C_{M}^{∞} of sections vanishing up to order infinity on M \U.

These functors being exact, they extend naturally to the derived category D_{ℝ−c}^{b}(ℂ_{X}). We keep the same notations to denote the derived functors.

Now let X be a complex manifold and denote by \overline{X} the complex conjugate manifold and by X_{ℝ} the real underlying manifold. Let O_{X} be the sheaf of holomorphic functions on X, let D_{X} be the sheaf of finite order holomorphic differential operators on X. The functors of moderate and formal cohomology are defined for F ∈ D_{ℝ−c}^{b}(ℂ_{X_{ℝ}}) by:

T hom(F, O_{X}) = RHom_{D\overline{X}}(O\overline{X}, T hom(F, Db_{X_{ℝ}}))
F ⊗^{w} O_{X} = RHom_{D\overline{X}}(O\overline{X}, F ⊗^{w} C_{X_{ℝ}}^{∞}).

7 Laplace transform

Consider now a complex vector space 𝔼 of complex dimension n, and denote by j : 𝔼 ↪ ℙ its projective compactification. Let D_{ℝ−c,ℝ^{+}}^{b}(ℂ_{𝔼}) denote the full triangulated subcategory of D_{ℝ−c}^{b}(ℂ_{𝔼}) consisting of ℝ^{+}-conic objects (i.e. objects whose cohomology is ℝ-constructible and locally constant on the orbits of the action of ℝ^{+} on 𝔼).

Let F ∈ D_{ℝ−c,ℝ^{+}}^{b}(ℂ_{𝔼}) and set for short

THom (F, O_{𝔼}) = RΓ(ℙ; T hom(j_{!}F, O_{ℙ}))

F ⊗^{W} O_{𝔼} = RΓ(ℙ; j_{!}F ⊗^{w} O_{ℙ})

Remark that if E is a real n-dimensional vector space and 𝔼 its complexification, then THom (or_{E}[−n], O_{𝔼}) ≃ S′(E), the Schwartz’s space of tempered distributions

on E and ℂ_{E} ⊗^{W} O_{𝔼} ≃ S(E), the space of rapidly decreasing C^{∞}-functions on E.

Theorem 7.1. ([7]) The Laplace transform extends naturally as isomorphisms:

T L : THom (F, O_{𝔼}) ≃ THom (F^{∧}[n], O_{𝔼^{∗}})  (7.1)
W L : F ⊗^{W} O_{𝔼} ≃ F^{∧}[n] ⊗^{W} O_{𝔼^{∗}}.  (7.2)

We may summarize the previous isomorphisms by the following diagrams.

[figure: E + \sqrt{−1}E^{∗} ≃ T_{E}^{∗}𝔼; \sqrt{−1}E^{∗} + E ≃ T_{\sqrt{−1}E^{∗}}^{∗}𝔼^{∗}; E + \sqrt{−1}E ≃ T_{E}𝔼; \sqrt{−1}E^{∗} + E^{∗} ≃ T_{\sqrt{−1}E^{∗}}𝔼^{∗}; E; \sqrt{−1}E^{∗}]

Let F ∈ D_{ℝ−c,ℝ^{+}}^{b}(ℂ_{𝔼}) and assume to be given a “boundary value morphism” b : ℂ_{E} [−n] ⟶ F. We get the diagram, in which we neglect or_{E} for short:

[figure: RHom (F^{∧} [n], C_{E}); RHom (F, C_{\sqrt{−1}E^{∗}}); RHom (F, ν_{E}(O_{𝔼})); Sato; ∼; RHom (F^{∧} [n], ν_{\sqrt{−1}E^{∗}}(O_{𝔼}^{∗})); Sato; ∼; THom (F, O_{𝔼}); b; Laplace; ∼; THom (F^{∧}[n], O_{𝔼^{∗}}); b; S′(E); F ourier; ∼; S′(\sqrt{−1}E^{∗})]

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Institut de Mathématiques; Analyse Algébrique; Université Pierre et Marie Curie; Case 247; 4, place Jussieu; F-75252 Paris Cedex 05

email: schapira@math.jussieu.fr