VOL. LXVII 1994 FASC. 2
SOME REMARKS ON HOLOMORPHIC EXTENSION IN INFINITE DIMENSIONS
BY
P H A M K H A C B A N (HANOI)
In finite-dimensional complex analysis, the extension of holomorphic maps has been investigated by many authors. In recent years some au- thors have considered this problem in the infinite-dimensional case. The aim of the present note is to study the extension of holomorphic maps with values in some Banach complex manifolds.
Let X be a Banach complex manifold. We say that X has the holo- morphic extension property (briefly HEP) if every holomorphic map from a Riemann domain Ω over a Banach space having a Schauder basis to X can be extended holomorphically to bΩ, the envelope of holomorphy of Ω.
Now as in [10], X is said to satisfy the weak disc condition if every sequence {fn} of holomorphic maps from the unit open disc D into X, convergent in H(D∗, X), where D∗ = {λ ∈ D : λ 6= 0}, is convergent in H(D, X), the space of holomorphic maps from D into X equipped with the compact-open topology.
In this note we shall prove the following two theorems.
Theorem A. Let X be a Banach manifold satisfying the weak disc con- dition. Then X has the HEP.
Theorem B. Let X be a pseudoconvex Banach manifold having C1- partitions of unity such that every holomorphic map from D∗ into X can be extended holomorphically to D. Then X has the HEP.
Here X is called pseudoconvex if KbP := {z ∈ X : ϕ(z) ≤ sup
K
ϕ for all plurisubharmonic functions ϕ on X}
is compact for each compact subset K of X.
Cover X by a locally finite system of coordinates {(Ui, ϕi)}. Let {Vi} be another open cover of X such that Vi ⊂ Ui, dist(Vi, ∂Ui) > 0 and ϕi(Vi) is isomorphic to a ball in a Banach space for every i.
1991 Mathematics Subject Classification: Primary 58B12.
[155]
By hypothesis there exists a C1-partition of unity {hi} such that hi= 1 on Vi and supp hi⊂ Ui for every i. Let p : T X → X be the tangent bundle of X. For each u ∈ T X, put
kuk =X
hi(pu)kDϕi(pu)(u)k.
Denote by %X the integral distance on X associated with k · k. Then %X
defines the topology of X (cf. [1]).
P r o o f o f T h e o r e m A. Let f : Ω → X be a holomorphic map, where Ω is a Riemann domain over a Banach space B with a Schauder basis. By SX we denote the sheaf of germs of holomorphic maps on bΩ with values in X and by SXf the domain of existence of f . Then SXf is the component of SX containing the set {(x, fx) : x ∈ Ω}, where fx denotes the germ of f at x. We have the commutative diagram
Ω −→f X
e
y α& x ef
Ωb ←−
β SXf
in which e, α, β are canonical maps and ef is the canonical extension of f . (i) First we shall prove that SXf satisfies the weak disc condition. Given a sequence {un} ⊂ H(D, SXf) convergent to u in H(D∗, SXf). By hypothesis { ef un} converges to h in H(D, X). Take a neighbourhood U of h(0) in X which is isomorphic to an open subset of a Banach space E. Then we can assume that ef un(δD) ⊂ U for n ≥ 1.
Put K = cl conv(S
n≥1f ue n((δ/2)D)). Let F be the canonical Banach space spanned by K. Then
f ue n : ∂((δ/2)D) → F is continuous for n ≥ 1. Hence by the Cauchy formula
f ue n(z) = 1 2πi
R
|λ|=δ/2
f ue n(λ)(λ − z)−1dλ it follows that ef un: (δ/2)D → F is holomorphic for all n ≥ 1.
Now for each n ≥ 1, consider the map uen : (δ/2)D → lim ind
0∈W H∞(W, F )
where H∞(W, F ) is the Banach space of bounded holomorphic functions on W with values in F , defined by
eun(λ) = [un(λ) ◦ θγun(λ)]0
where γ = pβ,b p : bb Ω → B defining Ω as a Riemann domain over B and θv(r) = v + r.
Then {eun} converges to u in H((δ/2)De ∗, lim ind0∈WH∞(W, F )), and hence in H((δ/2)D, lim ind0∈WH∞(W, F )). Since the above inductive limit is regular there exists a neighbourhood W of zero in B such that {eun} is contained and bounded in H∞(W, F ). Extend u holomorphically to 0 ∈ D by setting u(0) =u(0) ◦ θe −1γu(0). It remains to check that {un} converges to u in H(D, SXf).
For this consider the neighbourhood fW of u(∂((δ/2)D)) given by W =f [
{(γu(λ) + x, [u(λ)]γu(λ)+x) : |λ| = δ/2, x ∈ W }.
Take n0 such that un(∂((δ/2)D)) ⊂ fW for all n > n0. Now for each |λ| = δ/2 and n > n0there exists a neighbourhood W (n, λ) of γu(λ) in γu(λ)+W such that un(λ)x = u(λ)x for all x ∈ W (n, λ). Hence un(λ)x = u(λ)x for all x ∈ γu(λ) + W and |λ| = δ/2. This shows that the above equality holds for all |λ| ≤ δ/2. Hence un→ u in H(D, SXf ).
(ii) Now by (i), SXf is pseudoconvex.
(iii) Since B has a Schauder basis, SXf is the domain of existence of a holomorphic function [5]. This implies that the canonical map α : Ω → SXf can be extended holomorphically to bΩ. Hence β : SXf → bΩ is a biholo- morphism and ef β−1 is a holomorphic extension of f to bΩ. The theorem is proved.
R e m a r k. In the finite-dimensional case Theorem A was proved by Shiffman in [10]. Our proof is different.
P r o o f o f T h e o r e m B. By Theorem A it suffices to show that X satisfies the weak disc condition. Consider a sequence {fn} ⊂ H(D, X) converging to f in H(D∗, X). By hypothesis f ∈ H(D, X). Put
K = [
n≥1
fn(∂((1/2)D)) and Z = (K)∧P.
Since K is compact and X is pseudoconvex, Z is also compact. Moreover, by the maximum principle for plurisubharmonic functions we have
[
n≥1
fn((1/2)∂D)∧ P ⊇ [
n≥1
fn((δ/2)D).
Hence
Z ⊇ [
n≥1
fn((1/2)D).
To complete the proof of Theorem B it remains to show that Z has a hyperbolic neighbourhood W . Indeed, let dW denote the hyperbolic distance of W . Then
dW(fn(0), f (0)) ≤ dW(fn(0), fn(z)) + dW(fn(z), f (z)) + dW(f (z), f (0))
≤ 2dW(0, z) + dW(fn(z), f (z)) and these inequalities imply that fn → f in H(D, X).
By [1] it suffices to show that there exists a neighbourhood W of Z in X for which sup{kσ0(0)k : σ ∈ H(D, W )} < ∞. Otherwise we can find a decreasing neighbourhood basis {Wn} of Z such that for each n ≥ 1 there exists a holomorphic map σn from D to Wn such that kσ0(0)k ≥ n. As in [3] there exists a sequence {βn} of holomorphic maps from nD to X such that βn(nD) ⊆ σn(D), kβn0(0)k = 1 for n ≥ 1 and {βn} is equicontinuous on every compact set in C. Since Z =T
n≥1Wn, by the compactness of Z it follows that {βn} contains a subsequence {αn} converging to a holomorphic map α : C → Z. Obviously α 6= const because kα0(0)k = 1.
By hypothesis α can be extended to a holomorphic mapα from CPb 1into X. Now take a holomorphic function σ on D∗which cannot be extended to a holomorphic map from D into CP1. By hypothesis γ =ασ extends to ab holomorphic mapbγ : D → X. Since α is nonconstant, it follows thatb α is ab finite map. Hence we can find a neighbourhood U ofbγ(0) such thatbγ−1(U ) is a bounded domain in C. Thus σ extends holomorphically to 0 ∈ D. This is impossible so the theorem is proved.
R e m a r k. In the case when X is a holomorphically convex space with dim X < ∞ such that every holomorphic map from D∗ into X extends holomorphically to D, Theorem B was proved by D. D. Thai in [12].
Acknowledgements. The author would like to thank Professor N. V. Khue and Dr. B. D. Tac for suggestions that led to an improvement of the presentation of this note.
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DEPARTMENT OF MATHEMATICS PEDAGOGICAL INSTITUTE HANOI I HANOI, VIETNAM
Re¸cu par la R´edaction le 5.8.1992
