Abstract. Let M be a real analytic manifold, Ω ⊂ M an open set, X a complexi- fication of M , P a pseudodifferential operator on X.

OPERATOR OVER ( C Ω|X ) T

X

Andrea D’Agnolo Giuseppe Zampieri

Abstract. Let M be a real analytic manifold, Ω ⊂ M an open set, X a complexi- fication of M , P a pseudodifferential operator on X.

Using the action of P over holomorphic functions on suitable domains of X, by [B-S], and the theory of representation of microfunctions at the boundary for (C

)

MX

, by [S-Z], [Z], we show that P defines in a natural manner a sheaf morphism of (C

)

MX

. Let us note that the hypotheses on ∂Ω are here weaker than in [K 2] where ∂Ω is supposed to be analytic.

We also easily prove that P is an isomorphism of (C

)

MX

out of T

X ∩char P (both by composition rule and by non-characteristic deformation).

We shall apply the method of this paper in our forthcoming works on regularity at the boundary for solutions of P (cf [D’A-Z]).

0. Preliminaries. Let M be a C ^ω -manifold, X a complexification of M . We denote by T ^∗ M , T ^∗ X the cotangent bundles to M , X, and by T _M ^∗ X the conormal bundle to M in X; in particular we denote by T _X ^∗ X the zero section of T ^∗ X. We set ˙ T ^∗ X = T ^∗ X \ T X ^∗ X.

For subsets S, V ⊂ X one denotes by C(S, V ) the normal cone to S along V and by N (S) the normal cone to S in X; these are objects of T X (cf. [K-S]).

We will denote by B ^M (resp C M|X ) the sheaf of hyperfunctions on M (resp of microfunctions).

Let Ω be an open subset of M and let C Ω|X be the complex of sheaves defined in [S] (cf also [K 1]). In this paper we shall assume that:

(1) Ω is C ^ω -convex,

(2) H ⁰ ( C Ω|X ) = ( C Ω|X ) T

X .

(One can prove that if Ω has a C ² −boundary then (1) and (2) are satisfied.) Let γ be an open subset of Ω × M T M X with convex conic fibers; a domain U ⊂ X is said to be an Ω-tuboid with profile γ iff C(X \ U, Ω) ∩ γ 1 = ∅ for some open set γ ₁ ⊂ T X with convex conic fiber such that γ 1 ⊃ σ(N(Ω)), ρ(γ 1 ) ⊃ γ. Here

T _M X ←− M × ^ρ X T X ←− T M ^σ are the canonical maps.

If one chooses coordinates x ∈ M, z = x + √

−1y ∈ X then U is an Ω-tuboid with profile γ iff for every γ ⁰ ⊂⊂ γ there exists ε = ε γ

such that

U ⊃ {(x, y) ∈ Ω × ^M γ ⁰ ; |y| < ε(dist(x, ∂Ω) ∧ 1)}.

Appeared in: Tsukuba J. Math. 15 (1991), no. 1, 175–184.

1

(Here we identify T M X ∼ = X in local coordinates.)

For example if Ω = {x 1 > 0 } and γ = Ω × {y n > 0 }, the set U = {(x, y) ∈ X; x ∈ Ω, y n x ₁ > y ⁰² } is a Ω-tuboid with profile γ.

We now recall how sections of ( C Ω|X ) _T

_X can be represented as boundary values of holomorphic functions (cf [S-Z],[Z]).

Take f ∈ π ∗ Γ γ

(( C Ω|X ) T

X )(S), S a ball in a local chart M ∼ = R ⁿ , π : T _M ^∗ X → M the projection, γ ⊂ Ω × M T M X open with convex conic fibers over S and with π(γ) = S ∩ Ω. Then one can write f as the boundary value b(F ), F ∈ O ^X (U ), U being both an Ω-tuboid with profile γ and a domain of holomorphy. At this subject we refer the reader to [Z]. Note here that the results of [S-Z], [Z] concerning the representation of the stalks of π _∗ Γ γ

(( C Ω|X ) T

X )

x , x ∈ π(γ), easily extend to global sections over vector spaces.

For f ∈ Γ Ω ( B ^M )

_x , x ^◦ ∈ ∂Ω, one denotes by SS Ω (f ) the support of f identified to a section of ( C Ω|X ) T

X in π ⁻¹ ( x). On account of the above characterization one ^◦ proves that given ( x, ^◦ √

−1 η) ^◦ ∈ T M ^∗ X one has ( x, ^◦ √

−1 η) / ^◦ ∈ SS Ω (f ) iff f = P b(F _j ) with F j holomorphic in U j , U j Ω-tuboid whose profile γ j verifies √

−1 η / ^◦ ∈ ((γ ^j )

x ) ^∗a . One also gets the following decomposition of micro-support.

Let f ∈ Γ Ω ( B M )(S) and decompose ˙π ⁻¹ (S) ∩ SS Ω (f ) ⊂ S

j γ _j ^∗a , ((γ _j ^∗a ) x closed proper convex cones ∀x ∈ π(γ j ) = S ∩ Ω). Then one can write f = P

b(F _j ), F j ∈ O X (U j ), U j Ω-tuboid (of holomorphy) with profile γ j .

Let P ∈ E X,t

be a pseudodifferential operator of order m defined in a neighbor- hood of a point t ^∗ ∈ ˙ T _M ^∗ X. Fix a system of coordinates near t ^∗ : x = (x ₁ , . . . , x n ) ∈ M ∼ = R ⁿ , z = x + √

−1y ∈ X ∼ = C ⁿ , (z; ζ) ∈ T ^∗ X ∼ = C ⁿ × C ⁿ , ζ = ξ + √

−1η, (x; √

−1η) ∈ T M ^∗ X ∼ = R ⁿ × √

−1R ⁿ , t ^∗ = ( x; ^◦ √

−1 η), ^◦ η = (0, . . . , 0, 1). ^◦ One can write the symbol of P as

X m

−∞

P l (z, ζ), P l ∈ O ^T

^X ( e U × f W ),

U e × f W open subset of T ^∗ X, f W ⊃ {ζ : |ζ ⁱ | ≤ k 0 |ζ ⁿ |, i = 1, . . . , n − 1}, e U 3 x, P ^◦ l

homogeneous in ζ of degree l, sup

{z∈ e U ,|ζ

|≤k

|ζ

|}

|ζ| ^l |P −l (z, ζ) | ≤ M 0 ^l+1 l!.

In [B-S] Bony and Schapira have shown that, under these conditions, if one fixes a complex hyperplane Σ = {z : hz, η ^◦ i = x ^◦ n + √

−1ε} it is possible to define an

“action” for P over holomorphic functions on suitable domains.

Definition 0.1. Let k > 0. An open convex subset U of C ⁿ is said to be k −Σ−plat if:

∀z ∈ U, ∀ez ∈ Σ such that |z ⁿ − ez ⁿ | ≥ k|z ⁱ − ez ⁱ |, i = 1, . . . , n − 1 we have e z ∈ U ∩ Σ.

We refer the reader to [B-S] for the definition of the operator P _Σ over holomorphic

functions defined in domains k ₀ − Σ−plat with diameter ≤ 1/M 0 .

1. Definition of the action. Let P ∈ E X (V ) be a pseudodifferential operator on an open set V ⊂ ˙T M ^∗ X.

Our aim is to define an action for P over sections of ( C Ω|X ) T

X . More precisely, if α is the map

α : π ⁻¹ (Γ _Ω ( B ^M )) −→ (C Ω|X ) T

X , we will show how P operates on α(π ⁻¹ (Γ _Ω ( B M )))  

V . To this end we will proceed in several steps.

First we will make the operator act on cohomology classes of holomorphic func- tions defined on Ω-tuboids with prescribed profile. Since each germ of α(π ⁻¹ (Γ _Ω ( B ^M ))) is represented as boundary value of a holomorphic function defined on a domain as above we can interpret the previous action as an action over (α(π ⁻¹ (Γ _Ω ( B ^M )))) _(x, ^√ _−1η) , for every (x, √

−1η) ∈ V .

Finally we will show how P operates on α(π ⁻¹ (Γ Ω ( B M )))  

V gluing up the actions over each fiber.

Let ( x, ^◦ √

−1 η) ^◦ ∈ V with x ^◦ ∈ ∂Ω.

We fix a system of local coordinates so that η = (0, . . . , 0, 1), Ω = ^◦ {x; ρ(x) > 0}

(where ρ(x) = x ₁ − ϕ(x 2 , . . . , x _n ) for a convex function ϕ), and the symbol of P is defined on a set e U × e V as in the previous section.

Let

S = {x; |x − x ^◦ | < r},

with r < 1/M ₀ such that S ⊂ e U and let Γ be an open proper convex cone of ˙ R ⁿ such that

Γ ^∗a ⊂ {η; |η ⁱ | < kη ⁿ , i = 1, . . . , n − 1}, k < k ₀ 5 . For b > 0 let

(1.1) U = U Γ,b = ((S ∩ Ω) + √

−1Γ) ∩ {z; |y| < b ρ(x)}.

Let F ∈ O ^X (U ); we shall now define how P operates on f = α(b(F )).

For x ∈ ∂Ω ∩ S, η ∈ Γ ^∗a , |η| = 1, ν ∈ N + , let Σ = Σ x,η,ν = {z; hz − x, ηi = √

−1 1 ν }, Ω x = x + {x; x 1 > 1

k |x ⁰ |};

let ρ x be the function defined by Ω x = {x; ρ ^x > 0 }. Observe that Ω ∩ S = S

x

Ω x ∩ S for k small.

Consider

U x = U _x,Γ,b = ((S ∩ Ω ^x ) + √

−1Γ) ∩ {z; |y| < b ρ ^x (x) },

U x,ν = {z; |z n − e z n | ≥ |z i − e z i |, ez ∈ Σ _{x, η,ν}

= ⇒ ez ∈ U ∩ Σ _x,

_η,ν }, a k-Σ _{x, η,ν}

-plat set.

One can rewrite

U x,ν = {z = (z ⁰ , z n ) ∈ X; |z n −

√ −1

ν | ≤ k dist((z ⁰ ,

√ −1

ν ), Σ _x,

_η,ν \ U)},

and hence deduce at once the convexity of U _x,ν .

Remark 1.2. U x,ν is ( ¹ ₂ (k −k 0 ) −|η − ^◦ η |)-Σ _{x, η,ν}

-plat (due to |η − η ^◦ | < k, k < k 0 /5) thus U _x,ν contains the largest (k − k 0 )-Σ

x,

η,ν -plat set V such that V ∩ Σ ^x,η,ν = U x,ν ∩ Σ ^x,η,ν .

One has

Lemma 1.3. For every S ⁰ ⊂⊂ S there exists b ⁰ = b ⁰ _S

< b and for every Γ ⁰ ⊂⊂ Γ there exists b ⁰⁰ = b ⁰⁰ _Γ

so that for every x:

(1) S

ν

(U x,ν ∩ U x ) ⊃ (Ω x + √

−1 Γ) ∩ {z; y n < b ⁰ ρ x (x) } ∩ B ⁰ , (2) S

ν

(U _x,ν ∩{z : y n < _ν ¹ }) ⊃ {z; x ∈ Ω x , y ∈ −b ⁰⁰ ρ _x (x) η+Γ ^{◦ 0} , y _n < b ⁰ ρ _x (x) }∩B ⁰ , where B ⁰ = S ⁰ + √

−1 R ⁿ .

Proof. (1) can be easily proven by taking b ⁰ such that S

ν

(U x,ν ∩U ^x ) ∩{z; x ∈ S ⁰ , y n <

b ρ _x (x) } is a convex set. As for (2) one sees that for some b ⁰⁰ one has that for every z in the right hand side of (2) there exists ν such that:

|w n − z n | > k|w ⁰ − z ⁰ |, w ∈ Σ _x,

_η,ν = ⇒ w ∈ U x ∩ Σ _x,

_η,ν = U x,ν . To this end b ⁰⁰ has to be chosen small with respect to ε where ε is such that

Γ ^0∗a ⊂ {η; |η ⁰ | < (k − ε)η n }



In particular S

ν

(U x,ν ∩ U x ) is a Ω x -tuboid with profile Ω x + √

−1 Γ and S

ν

(U x,ν ∩ {z : y ⁿ < ¹ _ν }) is a Ω ^x -tuboid with profile Ω x + √

−1 R ⁿ .

Remark 1.4. We also observe that (U x,ν+1 ∩ U x,ν ) is k-Σ _x,

_η,ν -plat and (U x,ν +1 ∩ U x,ν ) ∩ Σ _{x, η,ν}

₊₁ = (U x ∩ U x,ν ) ∩ Σ _{x, η,ν}

₊₁ . Set U e _x,ν = U _x,ν ∩ {z; y ∈ −b ⁰⁰ ρ _x (x) η + Γ ^{◦ 0} , y _n < b ⁰ ρ _x (x) },

and note that, by a suitable choice of b ⁰ , b ⁰⁰ , e U x,ν ∩ B ⁰ and S

ν

U e x,ν ∩ B ⁰ are Stein do- mains (cf the statement before Remark 1.2). Let us consider P _Σ

x,◦

η,ν

F ∈ O ^X (U x,ν ∩ U x ) as defined in [B-S]. By [B-S,Th´eor`eme 2.5.1], and by Remark 1.4, we get

(1.5) P Σ

x,◦

η,ν

F − P Σ

x,◦

η,ν0

F ∈ O X (U x,ν ∩ U x,ν

).

Put

H ν,ν

= P _Σ

x,◦

η,ν

F − P Σ

x,◦ η,ν0

F ; these functions satisfy

H ν,ν

+ H ν

,ν = 0

H _ν,ν

+ H _ν

_,ν

+ H _ν

_,ν = 0.

Since H ¹ ( S

ν

U e x,ν ∩ B ⁰ , O X ) = 0 then there exists G ν ∈ O X ( e U x,ν ∩ B ⁰ ) such that G ν − G ^ν

= H ν,ν

in e U x,ν ∩ e U x,ν

∩ B ⁰ .

Summarizing up, to every F ∈ O X (U ), we can associate a function

(1.6) P _Σ

x,◦

η

F ∈ O ^X ( ∪ ^ν U e x,ν ∩ U ^x ∩ B ⁰ ) setting

P _Σ

x,◦ η

F    _e

U

∩U

∩B

= P _Σ

x,◦

η,ν

F − G ^ν | U e

∩U

∩B

. This function satisfies

P Σ

x,◦

η

F = P Σ

x,◦

η,ν

F in H _X ¹ _\U

x

( e U x,ν ∩ B ⁰ , O X ) ← ^∼ Γ ( e U x,ν ∩ U x ∩ B ⁰ , O X ) Γ ( e U x,ν ∩ B ⁰ , O ^X ) . Remark 1.7.

(i) Let {µ ν }, µ ν & 0, be another sequence and let P _Σ ⁰

x,◦ η

F be a function defined as in (1.6) (with 1/ν replaced by µ ν ). Similarly as before we get

P _Σ ⁰

x,◦ η

F − P Σ

x,◦

η

F ∈ O ^X ( ∪ ^ν (U x,ν ∩ U ^x,µ

)) . On the other hand, since

N 

∪ ^ν ( e U x,ν ∩ e U x,µ

) 

∩ σ((S ⁰ ∩ Ω ^x ) × ^M T M ) ˙ 6= ∅, then α(b(P Σ

x,◦

η

F )) = α(b(P _Σ ⁰

x,◦ η

F )) in π ⁻¹ (Ω x ∩ S ⁰ ).

(ii) In the same way one proves that α(b(P _Σ

x,◦ η

F ))   

π

(Ω

∩S

) does not depend neither on the choice of the constant b nor on the sets Γ, S (as long as (1.2) is satisfied).

And now we consider Ω instead of Ω x . Due to the convexity of Ω we observe that

U = b



 ∪

x ∈∂Ω∩S ν

U e x,ν



 ∩ B ⁰

is still a Stein domain.

Reasoning as before we get

Proposition 1.8. To any F ∈ O ^X (U ) we can associate a holomorphic function P Σ

η

F ∈ O X ( b U ∩ U), such that

P _Σ

η

F = P _Σ

x,◦

η

F in H _X ¹

\( b U ∩U) ( ∪ ^ν U e x,ν ∩ B ⁰ , O ^X ).

Note that, on the same line as Lemma 1.3, one proves that b U ∩ U is a Ω-tuboid with profile Ω + √

−1Γ.

Remark 1.9. By its very definition it is clear that α(b(P _Σ

η

F ))   

π

(W ) (for W ⊂ Ω) equals P _Σ

η

α(b(F ))   

π

(W ) defined in [B-S].

The compatibility with the action of P on C M|X is thus assured.

Let f ∈ α(π ⁻¹ Γ _Ω ( B ^M ))(W ), W = S + √

−1 intΓ ^∗a ; on account of the first section we can write f = b(F ), F ∈ O X (U ), U being an Ω −tuboid with profile (S ∩ Ω) + √

−1 Γ, i.e. ∀S ⁰ ⊂⊂ S, ∀Γ ⁰ ⊂⊂ Γ, ∃b so that U ⊃ U ⁰ where (1.10) U ⁰ = ((S ⁰ ∩ Ω) + √

−1 Γ ⁰ ) ∩ {z : |y| < b ρ(x)}.

According to Proposition 1.8 we can define an holomorphic function P _Σ

η

(F   

U

) and by Remark 1.7 P _Σ

η

(F   

U

)    

S+ √

−1 intΓ

does not depend on U ⁰ .

It does not depend on the choice of the representative F neither. In fact, if one has b(F − F ⁰ ) | W = 0 then b(F − F ⁰ ) = P

j b(F j ), F j ∈ O ^X (U _j ⁰ ), U _j ⁰ = ((S ⁰ ∩ Ω) +

√ −1 Γ ⁰ j ) ∩ {z : |y| < b ^j ρ(x) } where Γ ⁰ j ⊂⊂ Γ ^j with Γ ^∗ _j ∩ intΓ ^∗ = ∅.

Reasoning as in the proof of Proposition 1.8 we get P Σ

η

F j ∈ O X (U j ∩ {z : |y| < b ⁰ j ρ(x) } ∩ (S ⁰⁰ + √

−1 R ⁿ )) and thus

b(P _Σ

η

F − P Σ

F ⁰ )   

S

+ √

−1(R

\(∪

Γ

)) = 0;

for S ⁰⁰ + √

−1(R ⁿ \ (∪ ^j Γ ⁰ _j ^∗a )) % W we have thus given an action of P over f.

Remark 1.11. By similar arguments as before one gets:

SS Ω α(b(P Σ

η

F )) ⊂ SS Ω α(b(F )).

And now we have to define an action over generic sections.

Given an open subset V ⁰ ⊂ V ∩ ˙π ⁻¹ (Ω) and f ∈ α(π ⁻¹ Γ _Ω ( B M ))(V ⁰ ), we can find an open covering {V t

} t

∈V

, V t

= S t

+ √

−1 int(Γ t

) ^∗a so that f | _V

= b(F t

) with F t

∈ O X (U _t ⁰

) (U _t ⁰

as in (1.1) with S, Γ replaced by S _t ⁰

, Γ ⁰ _t

respectively).

Let t ^∗i = (x ⁱ , √

−1 η ⁱ ) i = 1, 2; then

Proposition 1.12. α(b(P Σ

F t

) − b(P Σ

F t

)) = 0 in V t

∩ V t

. Proof. Let t ^∗3 ∈ V t

∩ V t

then it is enough to show that

(1.13) α 

b(P _Σ

F _t

) 

t

= α 

b(P _Σ

F _t

) 

t

.

First notice that it is possible to take F _t

= F _t

according to the discussion following Remark 1.9.

If η ¹ = η ³ (1.13) is then obvious.

If x ¹ = x ³ (1.13) follows from Remark 1.2, [B-S,Remarque 2.5.3] and from an

analogous of Lemma 1.3. 

We have then shown that the sections {b(P Σ

F t

) } t

define a section of ( C Ω|X ) T

X (V ⁰ ) which, of course, will be denoted by P f .

Let V be an open set of ˙ T _M ^∗ X with proper convex hull, and let f be an element of Γ (V ⁰ , α(π ⁻¹ (Γ Ω B M ))), for V ⁰ ⊂ V .

Then we can represent f = α(b(F )) | V

, F ∈ O ^X (U ⁰ ) where U ⁰ is a Ω-tuboid with profile int(V ^0∗a ). (In fact one can find e f ∈ B ^M such that SS e f ⊂ V ⁰ and

α( e f    _Ω )   

V

− f = 0.)

Summarizing up the above results one gets:

Theorem 1.14. Let P ∈ E(V ) then P is a sheaf endomorphism of α(π ⁻¹ Γ Ω B M )  

V . Corollary 1.15. Let ∂Ω be analytic; then P is a sheaf endomorphism of ( C Ω|X ) T

X

V . Proof. Since ( C Ω|X ) T

X is conically flabby (cf [S-Z]), then

α : π ⁻¹ Γ Ω B M −→ (C Ω|X ) T

X

is surjective. 

Remark 1.16. In describing the action of P over C M|X one can replace the lan- guage of P Σ by the language of the γ-topology (cf [K-S]).

Thus let X γ be the space X endowed with the γ-topology and φ γ : X −→ X ^γ the canonical map. Let Ω 1 ⊃ Ω 0 be two γ-open sets and set V = int(Ω 1 \ Ω 0 ) × intγ ^∗a . We have:

(1) O X ∼ = φ ⁻¹ _γ RΓ _Ω

\Ω

Rφ _{γ ∗} O X in D ^b (X; V );

(2) φ ⁻¹ _γ RΓ _Ω

\Ω

Rφ _{γ ∗} O ^X is a Γ (D × intγ ^∗a , E ^X )-module (D ⊂ X a γ-round set).

By applying µhom(Z M , •) ⊗ ω M|X [n] to both sides of (1) one gets the conclusion.

According to a private communication by P. Schapira the same procedure could be applied for Z M replaced by Z _Ω .

2. Elliptic regularity at the boundary. Let P, Q be pseudodifferential opera- tors in an open set V ⊂ ˙ T _M ^∗ X and let P · Q denote their formal composition (cf [B-S]).

If Q has negative order we have by [B-S,Proposition 2.1.2] that P _Σ

x,◦

η,ν

◦ Q Σ

x,◦

η,ν

= (P · Q) Σ

x,◦

η,ν

, in O ^X and hence

(2.1) P · Q = P ◦ Q, in ( C Ω|X ) T

X .

In particular if P is a pseudodifferential operator of positive order whose principal symbol p never vanishes on V , we get:

(2.2) P is an isomorphism of α(π ⁻¹ Γ _Ω ( B ^M ))  

V .

To this end one only needs to write 1 = P · P ⁻¹ at any t ^∗ ∈ V and use (2.1) which is valid since P ⁻¹ is of negative order.

We remark that it would have been possible to get (2.2) without using (2.1).

Assume p 6= 0 in {z : |z− x ^◦ | < r}×{ζ : |ζ i | ≤ k 1 |ζ n |}. Take k < inf{k 0 /3, k ₁ } and let S, Γ, U , U x , . . . , be defined as in section 1. Recall in particular that U x,ν and U x,ν ∩U x are k −Σ _x,

_η,ν −plat, and recall that N(∪ ν U x,ν ) ∩σ((S ⁰ ∩Ω x ) × M T M ) ˙ 6= ∅.

(i) For every F ∈ O X (U ) there exists G _x,ν ∈ O X (U _x ∩U x,ν ) so that P _Σ

x,◦ η,ν

G _x,ν = F . In fact this solution exists in a neighborhood of Σ

x, η,ν

∩ U ^x by Cauchy- Kowalevsky’s theorem and then it extends to U x ∩ U ^x,ν by [B-S,Th´eor`eme 2.5.4].

(ii) By a similar argument P _Σ

x,◦

η,ν

G x,ν − P Σ

x,◦

η,ν

G x,ν

= 0 implies G _x,ν − G x,ν

∈ O X (U _x,ν ∩ U x,ν

∩ B ⁰ ).

Reasoning as in the first section we can find a function G ∈ O ^X (U ∩ B ⁰ ∩ {z : |y| <

b ⁰ ρ(x) }) such that G − G ^x,ν ∈ O ^X ( e U x,ν ∩ B ⁰ ). In particular P _Σ

x,◦

η

G − F ∈ O ^X 

∪ ^ν U e x,ν

 ∩ B ⁰  , with e U x,ν defined as in Remark 1.4.

(iii) If G ∈ O ^X (U ), P _Σ

x,◦

η

G ∈ O ^X (U _x ⁰ ), U _x ⁰ ⊃ ((S ⁰ ∩ Ω ^x ) + √

−1 Γ ⁰ ) ∩ {z : |y| <

b ⁰ ρ(x) } with Γ ⁰ ⊃ Γ, then G ∈ O ^X (U _x ⁰ ). (Once more U _x ⁰ is chosen, without loss of generality, so that U _x ⁰ ∩ U x,ν is k − Σ _x,

_η,ν −plat.)

Collecting these results (for different S, Γ), one gets (2.2).

In fact let g ∈ π ∗ Γ _γ

(S, α(π ⁻¹ Γ _Ω ( B M ))) (with γ = Ω + √

−1Γ; S, Γ as above), and let Pg=0.

Hence g = b(G), G ∈ O X (U ⁰ ) (U ⁰ as in (1.12) for S ⁰ ⊂⊂ S, Γ ⁰ ⊂⊂ Γ), P g = P

j b(F j ), F j ∈ O X (U _j ⁰ ) (U _j ⁰ as in (1.12) with S ⁰ ⊂⊂ S, Γ ⁰ j ⊂⊂ Γ j , Γ ^∗a _j ∩ I = ∅).

One solves P _Σ

η

G j = F j , G j ∈ O ^X (U _j ⁰ ∩ {z : |y| < b ⁰ j ρ(x) } ∩ B ⁰ ) by (i)-(ii). This gives b(P Σ

η

(G − P

j G j )) = 0; hence by (iii) (applied with Γ ⁰ = R ⁿ ) we get:

b(G − X

j

G j ) = 0 thus (2.3) is injective.

By (i)-(ii) the surjectivity follows at once.

References

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´

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