COHOMOLOGY WITH BOUNDS AND CARLEMAN ESTIMATES FOR THE ∂̄-OPERATOR ON STEIN MANIFOLDS

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.

Theorem 1.1.Let Ω C n be a bounded pseudoconvex domain, and let f ∈ L 2  (p,q) (Ω) be a ∂-closed (p, q)-form, q ≥ 1, then there is a (p, q −1)-form u ∈ L 2  (p,q−1) (Ω) such that ∂u = f and where K is a constant depending on the diameter of Ω.
Actually the above theorem was contained in the following.
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.

2.1.
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 function on Ω, the scalar product makes the space L 2 (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.
where K depends on Ω.

Let
where ϕ is a plurisubharmonic function in U. We then denote all sections of ᏻ p over U that are L 2 ϕ -bounded by Γ ϕ (U , ᏻ p ).For the definition of L 2 ϕ -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 where p > 0 is some integer, and It can be shown, as is done in [2], that Γ ϕ (U, Ᏺ) is independent of λ and p, so that Γ ϕ (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 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 .

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 Ω j , where each Ω j is admissible for Ᏺ.If ᐂ = {Ω j } j∈I , I = {1,...,m}, where each Ω j is as above, then ᐂ is a finite admissible cover of Ω for Ᏺ and we define the L 2 ϕ (alternate) q-cochains of ᐂ with values in Ᏺ as those cochains where , which are alternate and satisfy c α ∈ Γ ϕ (Ω α , Ᏺ) for all α ∈ I q+1 .Denote by C q ϕ (ᐂ, Ᏺ) the space of L 2 ϕ -bounded cochains.The coboundary operator and call it the L 2 ϕ -bounded cohomology of ᐂ with values in Ᏺ.We then have the following theorem.
Theorem 2.2.For any q ≥ 1, the natural map 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].
We then have the following lemma.