IJMMS 32:6 (2002) 383–386
PII. S0161171202007901
http://ijmms.hindawi.com
© Hindawi Publishing Corp.
COHOMOLOGY WITH BOUNDS AND CARLEMAN ESTIMATES
¯
FOR THE ∂-OPERATOR
ON STEIN MANIFOLDS
PATRICK W. DARKO
Received 20 June 2001
Cohomology with bounds are used to globalize a result of Hörmander obtaining Carleman
estimates for the Cauchy-Riemann operator on Stein manifolds.
2000 Mathematics Subject Classification: 32C35, 32E10, 35N15.
1. Introduction. In [7] Hörmander proved the following theorems.
Theorem 1.1. Let Ω Cn be a bounded pseudoconvex domain, and let f ∈ L2(p,q) (Ω)
¯
be a ∂-closed
(p, q)-form, q ≥ 1, then there is a (p, q −1)-form u ∈ L2
(Ω) such that
(p,q−1)
¯ = f and
∂u
uL2
(p,q−1) (Ω)
≤ Kf L2
(p,q) (Ω)
,
(1.1)
where K is a constant depending on the diameter of Ω.
Actually the above theorem was contained in the following.
Theorem 1.2. Let Ω Cn be a bounded pseudoconvex domain, ϕ any plurisub¯
(p, q)-form, q ≥ 1, then there
harmonic function on Ω, and f ∈ L2(p,q) (Ω, ϕ) a ∂-closed
2
¯
is a (p, q − 1)-form u ∈ L
(Ω, ϕ) such that ∂u = f and
(p,q−1)
uL2
(p,q−1) (Ω,ϕ)
≤ Kf L2
(p,q−1) (Ω,ϕ)
,
(1.2)
where again K depends on the diameter of Ω.
These theorems turned out to be very useful in complex analysis and their applications include the Levi problem with bounds and cohomology with bounds. It is, therefore, natural to seek to generalize these theorems to manifolds. In [5], Theorem 1.1
was so generalized and we generalize Theorem 1.2 in this paper.
The term Carleman estimates refers to the estimates in Theorem 1.2 and we use
Theorem 1.2 to obtain a Leray’s isomorphism theorem with bounds on Stein manifolds, combining with weak elliptic estimates to get the generalization of Theorem 1.2
to Stein manifolds.
2. Preliminaries
2.1. Let X be an n-dimensional complex manifold with a C ∞ -Hermitian metric and
Ω X a relatively compact Stein subdomain of X. Where ϕ is any plurisubharmonic
384
PATRICK W. DARKO
function on Ω, the scalar product
(f , g) =
Ω
e−ϕ f Λ ∗ ḡ
(2.1)
makes the space L2(p,q) (Ω, ϕ) = {f measurable on Ω : Ω e−ϕ f Λ ∗ f¯ < ∞} a Hilbert
space, where ∗ is the Hodge ∗-operator associated with the metric and the orientation
on X.
Our result is as follows.
¯
in the sense of distributions. Then
Theorem 2.1. Let f ∈ L2(p,q) (Ω, ϕ) be ∂-closed
2
¯
there is a u ∈ L
(Ω, ϕ) such that ∂u = f in the sense of distributions and
(p,q−q)
uL2
(p,q−1) (Ω,ϕ)
≤ Kf L2
(p,q) (Ω,ϕ)
,
q > 0,
(2.2)
where K depends on Ω.
2.2. Let U be a bounded open set in Cn , ᏻ the structure sheaf of Cn . A section
f = (f1 , . . . , fp ) ∈ Γ (U , ᏻp ), where p is an integer, is L2ϕ -bounded if
f L2 (U ,ϕ) = f1 L2 (U ,ϕ) + · · · + fp L2 (U ,ϕ) < ∞,
(2.3)
where ϕ is a plurisubharmonic function in U. We then denote all sections of ᏻp over
U that are L2ϕ -bounded by Γϕ (U , ᏻp ).
For the definition of L2ϕ -bounded sections of coherent analytic sheaves, we require
the coherent analytic sheaf Ᏺ to be defined on a simply connected polycylinder neighborhood V of the closure of U . Then there is an ᏻ-homomorphism in another simply
connected polycylinder neighborhood V 1 of the closure of U
λ
ᏻp → Ᏺ → 0,
(2.4)
where p > 0 is some integer, and f ∈ Γ (U, Ᏺ) is L2ϕ -bounded if f ∈ Γϕ (U, Ᏺ) :=
λ(Γϕ (U , ᏻp )). It can be shown, as is done in [2], that Γϕ (U, Ᏺ) is independent of λ
and p, so that Γϕ (U , Ᏺ) is well defined.
Now, let Ω be a relatively compact Stein subdomain of an n-dimensional complex
manifold X, and ϕ a plurisubharmonic function defined on Ω. An open subset Y of Ω
is said to be admissible for the coherent analytic sheaf Ᏺ defined in a neighborhood
of the closure of Ω in X, if Y is Stein, there is a coordinate neighborhood V in X of
the closure Ȳ of Y such that V is biholomorphic to a simply connected polycylinder
V 1 in Cn . The section f ∈ Γ (Y , Ᏺ) is L2ϕ -bounded if
f ∈ Γϕ (Y , Ᏺ) := g ∈ Γ (Y , Ᏺ) : η∗ (g) ∈ Γϕ·η−1 η(Y ), η∗ (Ᏺ) ,
(2.5)
where η is the restriction of the biholomorphic map V → V 1 to Y and η∗ (Ᏺ) is the
zeroth direct image of Ᏺ on Y .
COHOMOLOGY WITH BOUNDS AND CARLEMAN ESTIMATES . . .
385
2.3. Let Ω, X, ϕ, and Ᏺ be as above. Then it is clear that Ω is a finite union Ω =
m
j=1 Ωj , where each Ωj is admissible for Ᏺ. If ᐂ = {Ωj }j∈I , I = {1, . . . , m}, where each
Ωj is as above, then ᐂ is a finite admissible cover of Ω for Ᏺ and we define the L2ϕ
(alternate) q-cochains of ᐂ with values in Ᏺ as those cochains
Γ Ωα , Ᏺ ,
c = cα ∈ C q (ᐂ, Ᏺ) =
(2.6)
α∈I q+1
where Ωα = Ωi0 ∩ · · · ∩ Ωiq , α = (i0 , . . . , iq ), which are alternate and satisfy cα ∈
q
Γϕ (Ωα , Ᏺ) for all α ∈ I q+1 . Denote by Cϕ (ᐂ, Ᏺ) the space of L2ϕ -bounded cochains.
The coboundary operator
δ : C q (ᐂ, Ᏺ) → C q+1 (ᐂ, Ᏺ)
q
q+1
q
(2.7)
q
q
maps Cϕ (ᐂ, Ᏺ) into Cϕ (ᐂ, Ᏺ). If Zϕ (ᐂ, Ᏺ) = {c ∈ Cϕ (ᐂ, Ᏺ) : δc = 0} and Bϕ (ᐂ, Ᏺ) =
q−1
q
q
δCϕ (ᐂ, Ᏺ), then as usual Bϕ (ᐂ, Ᏺ) ⊆ Zϕ (ᐂ, Ᏺ) and we define
q
q
q
Hϕ (ᐂ, Ᏺ) := Zϕ (ᐂ, Ᏺ)/Bϕ (ᐂ, Ᏺ)
(2.8)
and call it the L2ϕ -bounded cohomology of ᐂ with values in Ᏺ. We then have the following theorem.
Theorem 2.2. For any q ≥ 1, the natural map
q
Hϕ (ᐂ, Ᏺ) → H q (Ω, Ᏺ)
(2.9)
is an isomorphism.
Theorem 2.2 is used to prove Theorem 2.1, but we do not prove Theorem 2.2 here
because its proof is easier than the proof of the theorem in [3].
3. Carleman estimates
p
3.1. Let Ω, X, and ϕ be as above. If U ≠ ∅ is open in Ω̄, then BΩ (U, ϕ) is the Hilbert
space of holomorphic p-forms h on Ω ∩ U such that hL2 (U ∩Ω,ϕ) < ∞.
(p,0)
p
p
If V is open in Ω̄ with V ⊂ U , the restriction map γVU : BΩ (U, ϕ) → BΩ (V , ϕ) is
p
p
2
defined. Then Bϕ = {BΩ (u, ϕ); ᐅU
V } is then the canonical pre-sheaf of Lϕ -holomorphic
p
p-forms on Ω. The associated sheaf Ꮾϕ is the sheaf of germs of L2ϕ -holomorphic
p-forms on Ω̄.
We then have the following lemma.
p
Lemma 3.1. The cohomology group H q (Ω̄, Ꮾϕ ) = 0 for all q ≥ 1 and p ≥ 0.
Proof. Let Ᏼp be the sheaf of germs of holomorphic p-forms on X. Note that for
p
Y admissible in Ω, Γϕ (Y , Ᏼp ) = ᏮΩ (Y , ϕ), since Ᏼp is a coherent analytic sheaf in a
neighborhood of the closure of Ω in X. Now, if ᐂ is any finite admissible cover of Ω
q
q
for Ᏼp (Theorem 2.2), Hϕ (ᐂ, Ᏼp ) is isomorphic to Hϕ (Ω, Ᏼq ). But any finite cover of
Ω̄ has a refinement ᐁ = {Vj }j∈J such that ᐂΩ = {Vj ∩Ω}j∈J is a finite admissible cover
q
p
of Ω for Ᏼp . Therefore, Hϕ (Ω̄, Ꮾϕ ) is isomorphic to H q (Ω, Ᏼp ) for q ≥ 0 and p ≥ 0.
By Cartan’s theorem (Theorem 1.2), we have H(Ω, Ᏼp ) = 0 for all q ≥ 1 and p ≥ 0.
p
Therefore, H q (Ω̄, Ꮾϕ ) = 0 for q ≥ 1 and p ≥ 0.
386
PATRICK W. DARKO
3.2. By the following lemma (whose proof follows from Theorem 1.2), the proof of
Theorem 2.1 is concluded.
p
Lemma 3.2. The cohomology group H q (Ω̄, Ꮾϕ ) is isomorphic to the quotient space
2
¯ = 0 / ∂h
¯ : h ∈ L2
¯
g : g ∈ L2(p,q) (Ω, ϕ), ∂g
(p,q−1) (Ω, ϕ), ∂h ∈ L(p,q) (Ω, ϕ) .
(3.1)
p
Since H q (Ω̄, Ꮾϕ ) = 0 for q > 0 and p ≥ 0, then we get the following lemma.
¯
q > 0, then there is a u ∈ L2(p,q−1) (Ω, ϕ)
Lemma 3.3. If f ∈ L2(p,q) (Ω, ϕ) is ∂-closed,
¯
such that ∂u = f .
Now referring to [1, Theorem B, page 750], the proof of Theorem 2.1 is complete.
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Patrick W. Darko: Department of Mathematics and Computer Science, Lincoln University of the Commonwealth of Pennsylvania, Lincoln University, PA 19352, USA
E-mail address: pdarko@lu.lincoln.edu