Abstract. Let M be a real analytic manifold, X a complexification of M , Ω an open subset of M with N

CONTINUATION OF HOLOMORPHIC SOLUTIONS OF MICROHYPERBOLIC DIFFERENTIAL EQUATIONS

Andrea D’Agnolo Giuseppe Zampieri

Abstract. Let M be a real analytic manifold, X a complexification of M , Ω an open subset of M with N

(Ω) 6= T

M, x

∈ ∂Ω. Let γ (resp γ

) be an open set of Ω ×

T

X with convex conic fibers and with Ω ×

γ ⊃ Ω ×

T

X (resp γ

= Ω ×

T

X); denote by U (resp W) the Ω-tuboids in X with profile γ (resp γ

) (cf [Z]) and by S the neighborhoods of x

.

Let P = P (x, D) be a differential operator at x

with C

-coefficients which is microhyperbolic to each −θ ∈ N

(Ω)

in γ

relative to Ω (in the sense of (2.3)).

We prove that for every U, W, S there exist W

, S

such that

f ∈ O

(U ∩ S), P f ∈ O

(W ∩ S) implies f ∈ O

(W

∩ S

).

A similar result is obtained for Ω-microhyperbolic operators in the sense of [S-Z] and for semihyperbolic operators in the sense of [Kan], [Kat].

We aim to refine the above conclusions and show (cf [D’A-Z]) that in the preceding hypotheses P is an isomorphism of the sheaf (C

)

MX

(cf [S]) at any p ∈ γ

.

1. Preliminaries. Let X be a complex manifold, P a differential operator with holomorphic coefficients, and let σ(P ) be the principal symbol of P . First we introduce a lemma which will be our main tool in proving propagation theorems.

Lemma 1.1. Let {V ^α } ^α (0 ≤ α ≤ 1) and V be open sets in X such that:

(i) V ₀ ⊂ V , V ^α ⊂ V ^β , for β > α, (ii) V α = S

β<α

V β , V α = T

β

>α

V β

, (iii) ∂V α ∩ V 1 \ V ⊂⊂ V 1 ,

(iv) N _x ^∗ (V _α ) 6= T x ^∗ X for every x ∈ ∂V α ∩ V 1 \ V ,

(v) σ(P )(z, ζ) 6= 0 for every z ∈ ∂V α ∩ V 1 \ V and for every ζ conormal to V α

at z (cf. par. 1).

Then:

(1.2) f ∈ O ^X (V ), P f ∈ O ^X (V ∪ V 1 ) implies f ∈ O ^X (V ∪ V 1 ).

Proof. For f as in the left hand side of (1.2) set V = {V ∪ V ^α ; f ∈ O ^X (V ∪ V ^α ) }, endowed with the natural order relation; this is an inductive family. Let V ∪ V α

be a maximal element for V and suppose by absurd that α 0 < 1.

Note that f ∈ O X (V α

) and, by (iii), P f ∈ (O X ) z ∀z ∈ ∂V α

∩ V 1 \ V . Using (iv) and the refined version of the theorem of Cauchy-Kowalevsky-Leray given in [B-S

Appeared in: Rend. Mat. Appl. (7) 12 (1992), no. 1, 143–156.

1

1], we conclude that f extends holomorphically to a neighborhood of ∂V α

∩V 1 \ V . By (iii) (V ₁ \ V ) ∩ V α ⊂⊂ V 1 \ V , hence from (ii) we get that, in V 1 \ V , {V β } β

(β > α 0 ) is a fundamental system of neighborhood of V α ; it follows that each open set containing (∂V α

∩ V 1 \ V ) ∪ (V ∪ V ^α

) contains also V ∪ V ^β for some β > α ₀ . Hence f ∈ O X (V ∪ V β ) which is a contradiction. 

Remark 1.3. This result is a variant of a wider principle by Kashiwara concerning the “propagation of cohomology of a complex” (cf. [K-S 1,theorem 1.4.3]).

2. Statement of the results. 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 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 (cf [K-S 1]) and by N (S) the normal cone to S in X; these are objects of T X . The same notation will be used to denote the normal cone to a subset S of the manifold M , which is, of course, an object of T M .

Let Ω ⊂ M be an open set verifying for a fixed x 0 ∈ ∂Ω

(2.1) N _x ^∗

(Ω) 6= T x ^∗

M.

Let γ be an open set of Ω × ^M T M X with convex conic fiber. 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 (cf [Z]).

Here

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

Remark 2.2. Let X ∼ = R ⁿ + √

−1 R ⁿ 3 x + √

−1 y, M ∼ = R ⁿ 3 x. We recall that U is an Ω-tuboid with profile γ iff ∀γ ⁰ ⊂⊂ γ, ∃ε = ε ^γ

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

Let q ∈ ∂Ω × M T ˙ _M ^∗ X, set x 0 = π(q) (where π is the projection T ^∗ X −→ X) and let P be a differential operator with holomorphic coefficients in a neighborhood of x 0 .

Choose a system of coordinates (x; √

−1 η) ∈ T M ^∗ X and (z, ζ) ∈ T ^∗ X (z = x + √

−1 y, ζ = ξ + √

−1 η), and assume that σ(P )(z, ζ) 6= 0 for

− c 1 |η| < hξ, θi < −c 2 [ |y||η| + |ξ − hξ, θiθ|]

∀(x, √

−1 η) ∈ (Ω ∩ S) × √

−1 Λ, ∀θ ∈ ˙ N _x ^∗

(Ω), (2.3)

where Λ is a closed cone of ˙ R ⁿ and c 1 , c 2 are constants independent of x, η, θ.

Remark 2.4. Since condition (2.3) is not C ¹ -coordinate-invariant, no propagation

theorem involving the notion of micro-support of a sheaf (as in [K-S 1]) could be

applied.

Remark 2.5. It is obvious that if (2.3) is satisfied by θ then it is even satisfied by any θ ⁰ in a neighborhood of θ. It follows that we can replace ˙ N _x ^∗

(Ω) of (2.3) by ( ˙ N _x ^∗

(Ω)) ε for a suitable ε. Here, for a cone A ⊂ ˙R ⁿ , we denote by A ε the conic ε-neighborhood of A:

A _ε = {θ ∈ ˙R ⁿ : sup

η∈A h θ

|θ| , η

|η| i > 1 − ε}.

We shall now introduce a slight modification of condition (2.3) which is coordi- nate invariant.

Assume that

(2.6) θ / ∈ C ^q

(char(P ), Ω × ^M T _M ^∗ X) ∀q ⁰ ∈ λ, ∀θ ∈ ˙ N _x ^∗

(Ω) ^a ,

where char(P ) is the characteristic variety of P , λ is a closed neighborhood of q with conic fiber and where the exponent a denotes the antipodal map. Finally note that we have used the identification

T _x ^∗

M , →

j T _x ^∗

X , → _π

T _q ^∗ T ^∗ X _−H → ^∼ T q T ^∗ X,

where j is due to the complex structure of X, π ^∗ is the map associated to the projection π : T ^∗ X → X, and H denotes the Hamiltonian isomorphism.

As in [S-Z], we shall refer to (2.6) as the condition of Ω-micro-hyperbolicity in λ with respect to each θ ∈ ˙ N _x ^∗

(Ω) ^a ; this is a weaker condition then microhyperbolic- ity.

Remark 2.7. Note that one proves that if Λ ⊂⊂ (int λ) ^x

then (2.6) implies (2.3).

(Here, for A, B cones in ˙ R ⁿ , one says that A is a proper subcone of B, and writes A ⊂⊂ B, whenever A ∩ {y : |y| = 1} ⊂⊂ int B.)

Theorem 2.8. Let Ω verify (2.1), take q ∈ ∂Ω× ^M T ˙ _M ^∗ X, and let P be a differential operator at x 0 = π(q) which verifies (2.3) in some system of coordinates (resp (2.6)). Denote by U the family of tuboids whose profile γ verifies:

(2.9) Ω × M γ ⊃ Ω × M T M X, γ _x ^∗a

⊂ Λ (resp

(2.9)’ γ _x ^∗a

⊂ (int λ) x

), and by W those with profile γ ⁰ verifying

(2.10) γ ⁰ ⊃ Ω × ^M T M X

(where the exponent ∗ denotes the polar). Let S be the family of neighborhoods of x ₀ . Then:

f ∈ _{U∈U ,S∈S} lim −→ Γ(U ∩ S, O X ),

P f ∈ _{W ∈W,S∈S} lim −→ Γ(W ∩ S, O X ) implies

f ∈ _{W ∈W,S∈S} lim −→ Γ(W ∩ S, O ^X ).

Remark 2.11. Let Γ be an open convex cone of ˙ R ⁿ with Γ ^∗a ⊂ Λ, fix η ∈ ˙R ⁿ and let:

γ = (Ω × Γ) ∪ (Ω × c.h.(Γ, {−η}), γ ⁰ = Ω × c.h.(γ, {−η}).

Then the same conclusion of Theorem 2.8 holds. (here c.h. denotes the convex hull.)

In fact in subsequent Theorem 2.15 the assumption η ∈ intΓ ^∗ ∩ Γ is unessential.

(It is only used in the conclusion to get c.h.(Γ, {−η}) = ˙R ⁿ .) Theorem 2.12. Let Ω = {x = (x 1 , x ⁰ ) : x 1 > 0 } and assume that

σ(P )(z, ζ) 6= 0 when (z, ζ) satisfies the conditions in (2.3) with Λ = R × Λ ⁰ (Λ ⁰ ⊂ R ˙ ⁿ⁻¹ ), and when in addition y 1 = 0. Then the conclusion of Theorem 2.8 still holds.

Note by the way that the condition for P expressed in this statement is a refine- ment of the hypothesis of semi-hyperbolicity in the sense of [Kan].

For example in T ^∗ X 3 (z, ζ), z = (z 1 , z ⁰ ) consider σ(P )(z, ζ) = ζ ₁ ² − z 1 ζ ₂ ² − Q(z, ζ ⁰ ), Q homogeneous of degree 2 and Q | ^T

X ≤ 0. This is semihyperbolic but neither Ω-hyperbolic nor it satisfies (2.3).

The proof of Theorems 2.8, 2.12 will be given in the next section; it will follow from a statement which fully describes the shape of the sets U and V .

Let Ω ⊂ M be an open set verifying (2.1). Then we can write Ω on S, neigh- borhood of x ₀ , as Ω = {x : x 1 > ϕ(x ⁰ ) } for a Lipschitz-continuous function ϕ. We set

ρ(x) = x ₁ − ϕ(x ⁰ )

and remark that for suitable constants k ⁰ , k ⁰⁰ > 0 we have:

(2.13) k ⁰ dist(x, ∂Ω) < ρ(x) < k ⁰⁰ dist(x, ∂Ω), x ∈ Ω;

hence we will use the function ρ as a substitute of the distance to ∂Ω in our arguments. Moreover, we can find l ⁰ , l ⁰⁰ > 0 so that on S:

(2.14) |ρ(e x) − ρ(x)| ≤ l ⁰⁰ |e x − x|,

(2.14)’ inf

{v∈(N

(Ω))

:|v|=1} |ρ(x + av) − ρ(x)| ≥ l ⁰ a, 0 < a << 1.

(As for (2.14)’ we have to notice that we can choose coordinates at x ₀ so that N ˙ _x ^∗

(Ω) ⊂⊂ N x

(Ω); here we identify T x

M ∼ = T _x ^∗

M ∼ = M ∼ = R ⁿ .)

Let Λ, Γ be open convex cones of ˙ R ⁿ with Λ ⊃⊃ Γ ^∗a and take η ∈ int Γ ^∗ ∩ Γ.

Theorem 2.15. Let P verify (2.3). Let

U =



(Ω + √

−1 Γ) ∪



z : t ⁰ < ρ(x) < t, y ∈ r ρ(x) − t ⁰ t − t ⁰ η + Γ



∩

∩ {z : |y| < δ

t ρ(x) } ∩ S,

(2.16)

where δ ≥ r and S is a suitable neighborhood of x 0 . Then for every convex cone Γ ⁰ ⊂⊂ Γ (Γ ⁰ 3 η, Γ ^0∗a ⊂ Λ), there exists k = k Γ

< 1 such that if t verifies

(2.17) t < kc ⁻¹ ₂ l ⁰

and if c verifies

(2.18) crl ⁰⁰

t − t ⁰ < c ₁ , crk ⁻¹ < δ, c < 1 (l ⁰ , l ⁰⁰ being the constants of (2.14)), it follows that setting

(2.19) V = {z : 0 < ρ(x) < t, y ∈ −crρ(x) η + Γ ⁰ } ∩ {z : |y| < δ

t ρ(x) } ∩ S, then for a suitable S ⁰ ⊂ S, depending on t, l ⁰ , l ⁰⁰ and the ε of Remark 2.5, the following holds:

f ∈ O ^X (U ), P f ∈ O ^X (V ) implies f ∈ O ^X (V ∩ S ⁰ ).

Remark 2.20. Since η ∈ Γ ⁰ then for a suitable c ⁰ = c ⁰ _Γ

,η : V ⊃ {z : ρ(x) < t, |y| <

c ⁰ crρ(x) } ∩ S ⁰ .

To handle also the case when P f does not extend to a convex set we introduce the following

Theorem 2.21. Let P verify (2.3) and let c, t verify (2.17), (2.18), let U be defined by (2.16). For every g 1 (x) > 0 with inf

{x;ρ(x)=t}

g 1 (x) = r, there exists h(s), s ∈ R with h(0) = 0, h(t) = c r, h ⁰ increasing and 0 < h ⁰ ≤ cr/(t − t ⁰ ) for s > 0, such that if we set g ₂ (x) = h(ρ(x)) and

V 1 = {z : y ∈ −g 1 (x) η + Γ, |y| < δ

t ρ(x), 0 < ρ(x) < t }, (resp

V ₂ = {z : y ∈ −g 2 (x) η + Γ ⁰ , |y| < δ

t ρ(x), 0 < ρ(x) < t }), we get

f ∈ O X (U ), P f ∈ O X (V 1 ∩ S) implies f ∈ O X (V 2 ∩ S ⁰ ).

Remark 2.22. Let g ₁ (x) = h ₁ (ρ(x)) for a C ¹ -function h ₁ with h ⁰ ₁ increasing. Then one can show that the function h of Theorem 2.21 verifies h ⁰ ₁ ∧cr/t ≤ h ⁰ ≤ cr/(t−t ⁰ ).

In particular for g ₁ (x) = r ρ(x) one recovers Theorem 2.15.

3. Proofs. We will divide the proof of the theorems in some lemmas.

Lemma 3.1. Let U be as in (2.16). For every open convex cone Γ ⁰ ⊂⊂ Γ, Γ ⁰ 3 η, there exists k = k _Γ

< 1 such that if one sets for 0 ≤ α ≤ 1 and for ρ(x) < t:

(3.2) Φ α (x) = cr

t − t ⁰ (1 − α) (ρ(x) − t ⁰ (1 − α)), and

(3.3) U α = {z : ρ(x) < t, y ∈ −Φ ^α (x) η + Γ ⁰ } ∩ {z : |y| < δ

t ρ(x) } ∩ S, then:

(3.4) U ₀ ⊂ U,

(3.5) ∅ 6= (U ^α ) x ∩ {y : k ⁻¹ Φ α (x) < |y| < δ ⁰

t ρ(x) } ⊂⊂ Γ, ∀δ ⁰ < δ, and moreover the following holds. Whenever

(3.6)

 z ∈ ∂U α ∩ {z : t ⁰ (1 − α) ≤ ρ(x) < t, |y| < k ⁻¹ Φ α (x) } ζ ∈ N z ^∗ (U α ),

we have

(i) ξ ∈ (N x ^∗

(Ω) ^a ) ε

(ii) |ξ|

|η| < c ₁ (iii) |ξ|

|η||y| > c ₂ (iv) η ∈ Γ ^0∗a .

Proof. The relation in (3.4) is obvious.

For proving (3.5) let us first remark that there exists k = k _Γ

such that for a ∈ R:

(3.7) ( −aη + Γ ⁰ ) ∩ {y : k ⁻¹ a < |y| < d} ⊂⊂ Γ ∀d > 0.

Putting a = Φ α (x) in (3.7) and observing that we have

Φ α (x) ≤ cr

t ρ(x) < δ t ρ(x) (owing to the second inequality of (2.18) ) (3.5) follows.

As for (3.6), the point (i) is an easy consequence of the upper semicontinuity of

the map x 7→ N x ^∗ (Ω).

As for (ii),(iii) we first note that, on account of (2.14), Φ α (x) is a Lipschitz- continuous function with:

|Φ α (e x) − Φ α (x) | ≤ l ⁰⁰ cr

t |e x − x|,

{v∈(N

inf (Ω))

:|v|=1} |Φ ^α (x + av) − Φ ^α (x) | ≥ l ⁰ cr

t a 0 < a << 1.

(ii) is then a consequence of the first inequality of (2.18). As for (iii) we have, if

|y| < k ⁻¹ Φ α (x) and ρ(x) < t, then clearly |y| < k ⁻¹ cr and therefore

|ξ|

|η||y| > l ⁰ cr t

1

k ⁻¹ cr > c ₂ (due to (2.17) ).

Last, (iv) is obvious. 

The family {U ^α } ^α can be modified as follows. Let T ⁰ = R × {x ⁰ : |x ⁰ | < σ}, T ⁰⁰ = R × {x ⁰ : |x ⁰ | < σ + σ ⁰ }, let e N be an open cone in ˙ R ⁿ , and set

Ω = e e Ω _{N ,T e}

,T

= [

x∈∂Ω∩T

(x + e N ) ∩ T ⁰⁰ .

For a suitable choice of e N , T ⁰ , T ⁰⁰ we have

(i) Ω e ∩ S ⊂ Ω, ∂ e Ω ∩ T ⁰ = ∂Ω ∩ T ⁰ , (ii) ∅ 6= e Ω ∩ {x : ρ(x) = t} ⊂⊂ T ⁰⁰ ,

(iii) N _x ^∗ (e Ω) ⊂ (N x ^∗

(e Ω)) ε , ∀x ∈ ∂ e Ω ∩ T ⁰⁰ . (3.8)

Similarly to Ω, such an e Ω can be represented as e Ω = {x : x 1 > e ϕ(x ⁰ ) } for a Lipschitz-continuous function e ϕ so that the corresponding conditions to (i)-(iii)’s of (3.8) hold, i.e.:

(i)’ ϕ(x e ⁰ ) ≤ ϕ(x ⁰ ) and ϕ(x e ⁰ ) = ϕ(x ⁰ ), for |x ⁰ | < σ, (ii)’ ϕ(x e ⁰ ) < ϕ(x ⁰ ) + t, for x ⁰ ≥ σ + σ ⁰

(iii)’ -the same as in (iii)-.

(3.8)’

Let e ρ(x) = x ₁ − e ϕ(x ⁰ ) and observe that we could choose e ϕ so that e ρ still verifies the assumptions (2.14) with new constants l ⁰ , l ⁰⁰ . Let U be as in (2.16) on T ⁰⁰ , let Γ ⁰ ⊂⊂ Γ, let t, c verify (2.17),(2.18). Define

Φ e α (x) = cr ρ(x) e − t ⁰ (1 − α) ρ(x) e − ρ(x) + t − t ⁰ (1 − α) , and

(3.9) U e _α = {z : ρ(x) < t, y ∈ −e Φ _α (x) η + Γ ⁰ } ∩ {z : |y| < δ ⁰

t ρ(x) } ∩ S for some k ⁻¹ cr < δ ⁰ < δ.

We then have the following

Lemma 3.10. For a P verifying (2.3) the sets { e U α } α and U verify the hypotheses of Lemma 1.1.

Proof. (i) and (ii) of Lemma 1.1 are obvious. As for (iii) it is enough to show that for every x ∈ π( e U _α ) 

= π  U e _α 

we have ( e U _α ) _x ⊂ U x ∪ ( e U ₁ ) _x . To prove it, we will distinguish three cases.

If e ρ(x) < t ⁰ (1 − α) we get, for some a > 0, ( e U α ) x = ( √

−1 aη + √

−1 Γ ⁰ ) ∩ {z :

|y| ≤ δ ⁰ /t ρ(x) } ⊂⊂ U x .

If e ρ(x) = t ⁰ (1 − α) then ( e U α ) x = √

−1 Γ ⁰ ∩ {z : |y| ≤ δ ⁰ /t ρ(x) } ⊂⊂ ( e U ₁ ) x ∪ U ^x . If e ρ(x) > t ⁰ (1 − α), since e Φ _α (x) ≤ Φ α (x) we have e U _α ⊂ U α and (3.5) holds with U α replaced by e U α , hence

∅ 6= ( e U α ) x ∩ {z : |y| ≥ k ⁻¹ Φ α (x) } ⊂⊂ U x , and moreover it is easily seen that

( e U α ) x ∩ {z : |y| < k ⁻¹ Φ α (x) } ⊂⊂ (U 1 ) x . Last, for ρ(x) near t we have

( e U α ) x ⊂ −cr η + √

−1 Γ ⁰ ∩ {z : |y| ≤ δ ⁰ } ⊂⊂ U x , since c < 1 and Γ ⁰ ⊂⊂ Γ (in the sense of Remark 2.7).

Concerning (iv), first note that for every z ∈ ∂ e U α ∩ e U ₁ \ U we have ( |y| < k ⁻¹ Φ e _α (x) ≤ k ⁻¹ Φ ₁ (x)

-the solution of Φ e α (u) = 0 for u ⁰ = x ⁰ verifies ρ(u) < t e ⁰ -.

If one follows the lines of the proof of Lemma 3.1 it is easy to check that for such z and for ζ ∈ N z ^∗ ( e U α ) we have

|ξ|

|η| < c 1 , |ξ|

|η||y| > c 2 .

It is clear that η ∈ Γ ^0∗a and ξ ∈ (N x ^∗

(Ω)) ε due to (3.8)-(iii). Since σ(P ) verifies (2.3) (even replacing ˙ N _x ^∗

(Ω) by ( ˙ N _x ^∗

(Ω)) ε according to Remark 2.5), (i)-(iv) imply σ(P )(z, ζ) 6= 0. 

Proof of Theorem 2.15. Let be given f ∈ O X (U ), P f ∈ O X (V ) as in the statement.

The family { e U α } ^α of (3.9) has been so defined that one can find S ⁰ , depending on T ⁰ of (3.8)-(i), with

U e ₁ ∩ S ⁰ = V ∩ S ⁰ .

Using Lemma 3.10, the proof of the theorem follows immediately from Lemma

1.1. 

Proof of Theorem 2.21. The proof is the the same as the one of Theorem 2.15.

One only needs to replace in the definition of e U α the functions e φ α by k α φ e α with k α so chosen that k α φ e α < g 1 (x). Note that it is not restrictive to assume the map α → k α to be a continuous one. Thus the family V _α = S

β<α

U e _β satisfies the conditions of Lemma 1.1 and hence f extends to S

α

V α . Note that, on a small S ⁰ ⊂ S, the function sup

α k α φ e α is in the form h(s) (s = ρ(x)) for a C ¹ -function h satisfying all requirements in the statement. 

Proof of Theorem 2.8. Let f ∈ O X (U ∩ S) and P f ∈ O X (W ∩ S) where U (resp W ) is a tuboid whose fiber verifies (2.9) (resp (2.9)’), (2.10). Then for every t, t ⁰ and for suitable δ and r = r _t,t

, we can write U ∩ S as in (2.16) (possibly with a new S). Moreover for a suitable c, (W ∪ U) ∩ S contains a set V as in (2.19).

Applying Theorem 2.15, we get f ∈ O X (V ∩ S ⁰ ); then the conclusion follows from Remark 2.20. 

Proof of Theorem 2.12. As in the proof of Theorem 2.8 we can assume that f is analytic in U ∩ S and P f in V ∩ S, where U, V are defined by (2.16) and (2.19) respectively. On account of (2.3), z ₁ = 0 is non characteristic for σ(P ) at x ₀ and then there exist C so that:

(3.11) σ(P ) 6= 0 if |ζ 1 | > C|ζ ⁰ |.

We then set

˜ ˜

U α = {z ∈ X : |z ⁰ − ez ⁰ | < C|z 1 − ez 1 |, x 1 = e x ₁ ⇒ ez ∈ e U α ∩ {z : y 1 = 0 } ∩ S}.

According to (3.11) we get

f ∈ O ^X ( e U α ∩ {z : y 1 = 0 } ∩ S), P f ∈ O ^X ( ˜ U ˜ α ) implies f ∈ O ^X ( ˜ U ˜ α ).

(3.12)

On the other hand we have

σ(P )(z, ζ) 6= 0 for

( z ∈ ∂ e U α ∩ e U 1 \ U ∩ S ∩ {z : y 1 = 0 } ζ ∈ N z ^∗ ( ˜ U ˜ α )

and then

f ∈ O X ( ˜ U ˜ α ∩ S), P f ∈ (O X ) z implies f ∈ (O X ) z .

The conclusion then follows from (3.11),(3.12), via Lemma 1.1, in the same way as

it was for Theorem 2.8. 

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