$L^2$ Serre Duality on Domains in Complex Manifolds and Applications

An $L^2$ version of the Serre duality on domains in complex manifolds involving duality of Hilbert space realizations of the $\bar{\partial}$-operator is established. This duality is used to study the solution of the $\bar{\partial}$-equation with prescribed support. Applications are given to $\bar{\partial}$-closed extension of forms, as well to Bochner-Hartogs type extension of CR functions.


Introduction
A fundamental result in the theory of complex manifolds is Serre's duality theorem. This establishes a duality between the cohomology of a complex manifold Ω and the cohomology of Ω with compact supports, provided the Cauchy-Riemann operator ∂ has closed range in appropriate degrees.
More precisely, this can be stated as follows: let E be a holomorphic vector bundle on Ω, and let H p,q (Ω, E) denote the (p, q)-th Dolbeault cohomology group for E-valued forms on Ω, and let H p,q comp (Ω, E) denote the (p, q)-th Dolbeault cohomology group with compact support. Let E * denote the holomorphic vector bundle on Ω dual to the bundle E, and let n = dim C Ω. Then (we assume that all manifolds in this paper are countable at infinity): Serre Duality Theorem. Suppose that each of the two operators has closed range with respect to the natural Fréchet topology. Then the dual of the topological vector space H p,q (Ω, E) (with the quotient Fréchet topology) can be canonically identified with the space H n−p,n−q comp (Ω, E * ) with the quotient topology, where we endow spaces of compactly supported forms with the natural inductive limit topology.
In fact, condition that the two maps in (1) have closed range is also necessary for the duality theorem to hold (see [9]; also see [26,27,28] for further results of this type. ) Serre's original proof [35] is based on sheaf theory and the theory of topological vector spaces. A different approach to this result, in the case when Ω is a compact complex manifold, was given by Kodaira using Hodge theory (see [23] or [7].) In this note we extend Kodaira's method to non-compact Hermitian manifolds to obtain an L 2 analog of the Serre duality. Special cases of Serre-duality using L 2 methods have appeared before in many contexts (see [25], or [11,Theorem 5.1.7] and [19,20], for example.) The L 2 -Serre duality between the maximal and minimal realizations of the ∂-operator is also used in the study of the ∂operator on compact complex spaces (see e.g. [31,Proposition 1.3]) and more general duality results (of the type discussed in §3.6 below) are used as well in these investigations (see [33,Chapter 5].) Our treatment aims to streamline and systematize these results, with emphasis on non-compact manifolds, and point out its close relation with the choice of L 2 -realizations of the Cauchy-Riemann operator ∂, or alternatively, choice of boundary conditions for the L 2 -realizations of the formal complex Laplacian ∂ E ϑ E + ϑ E ∂ E .
The L 2 -duality can be interpreted in many ways. At one level, it is a duality between the standard -Laplacian with ∂-Neumann boundary conditions, and the c -Laplacian with dual ( "∂-Dirichlet") boundary conditions. Using another approach, results regarding solution of the ∂-equation in L 2 can be converted to statements regarding the solution of the ∂ c equation. This leads to a solution of the ∂-Cauchy problem, i.e., solution of the ∂-equation with prescribed support. At the heart of the matter lies the existence of a duality between Hilbert space realizations of the ∂-operator. This is explained in §3.6. However, for clarity of exposition, we concentrate on the classical duality between the well-known maximal and minimal realizations of ∂ in the rest of the paper.
As an application of the duality principle, we consider the problem of ∂-closed extension of forms. It is well-known that solving the ∂-equation with a weight can be interpreted as solving ∂ with bundlevalued forms (see [8].) The weight function φ corresponds to the metric for the trivial line bundle with a metric under which the length of the vector 1 at the point z is e −φ(z) . It was used by Hörmander to study the weighted ∂-Neumann operator by using weight functions which are strictly plurisubharmonic in a neighborhood of a pseudoconvex domain. When the boundary is smooth, one can also use the smooth weight functions to study the boundary regularity for pseudoconvex domains (see [24]) or pseudoconcave domains (see [36,37]) in a Stein manifold. In this paper we will use the Serre duality to study the ∂ problems with singular weight functions. The use of singular weight functions allow us to obtain the existence and regularity problem on pseudoconcave domains with Lipschitz boundary in Stein manifolds. The use of singular weights has the advantage that it only requires the boundary to be Lipschitz. Even when the boundary is smooth, the use of singular weight functions gives the regularity results much more directly (cf. the proof in [37] or [2,Chapter 9]). This method is also useful when the manifold is not Stein, as in the case of complex projective space CP n . In this case, any pseudoconconvex domain in CP n is Stein, but CP n is not Stein. In recent years these problems have been studied by many people (see [15,4,3]) which are all variants of the Serre duality results.
The plan of this paper is as follows. In §2, we recall basic definitions from complex differential geometry and functional analysis. This material can be found in standard texts, e.g. [12,43,14]. Next, in §3 we discuss several avatars of the L 2 -duality theorem: at the level of Laplacians, at the level of cohomology and for the ∂ and ∂ c problems. We discuss a general form of the duality theorem using the notion of dual realizations of the ∂ operator on vector bundles. In §4, we apply the results of §3 to trivial line bundles with singular metrics on pseudoconvex domains. This leads to results on the ∂-closed extension of forms from pseudoconcave domains. In the last section, we use the L 2 duality results to discuss the holomorphic extension of CR forms from the boundary of a Lipschitz domain in a complex manifold. We obtain a proof of the Bochner-Hartogs extension theorem using duality.

Notation and preliminaries
Throughout this article, Ω will denote a Hermitian manifold, and E a holomorphic vector bundle on Ω.
2.1. Differential operators on Hilbert spaces. The metrics on Ω and E induce an inner product (, ) on the space D(Ω, E) of smooth compactly supported sections of E over Ω. The inner product is given by where , is the inner product in the metric of the bundle E, and dV denotes the volume form induced by the metric of Ω. This allows us to define the Hilbert space L 2 (Ω, E) of square integrable sections of E over Ω in the usual way as the completion of the space of smooth compactly supported sections of E over Ω under the inner product (2). Let A be a differential operator acting on sections of E, i.e. A : C ∞ (Ω, E) → C ∞ (Ω, E), and let A ′ be the formal adjoint of A with respect to the inner product (2). Recall that this means that for smooth sections f, g of E over Ω, at least one of which is compactly supported, we have The well-known facts that A ′ exits, that it is also a differential operator acting on sections of E, and that A ′ has the same order as A follow from a direct computation in local coordinates using integration by parts. It is clear that (A ′ ) ′ = A, i.e. the formal adjoint of A ′ is A.
By an operator T from a Hilbert space H 1 to another Hilbert space H 2 we mean a C-linear map from a linear subspace Dom(T ) of H 1 into H 2 . We use the notation T : H 1 H 2 , to denote the fact that T is defined on a subspace of H 1 (rather than on all of H 1 , when we write T : H 1 → H 2 .) Recall that such an operator is said to be closed if its graph is closed as a subspace of the product Hilbert space H 1 × H 2 .
The differential operator A gives rise to several closed operators on the Hilbert space L 2 (Ω, E).
1. The weak maximal realization A max : we say for f, g ∈ L 2 (Ω, E) that Af = g in the weak sense if for all test sections φ ∈ D(Ω, E) we have that (This can be rephrased in terms of the action of A on distributional sections of E, but we will not need this.) The weak maximal realization A max is the densely-defined closed (cf. Lemma 1) linear operator on where Af is taken in the weak sense. On Dom(A max ), we define A max f = Af in the weak sense.
2. The strong minimal realization A min is the closure of the densely defined operator A D on L 2 (Ω, E), where A D denotes the restriction of A to the space of compactly supported sections D(Ω, E). More precisely, Dom(A min ) consists of those f ∈ L 2 (Ω, E), for which there is a g ∈ L 2 (Ω, E) and a sequence {f ν } ⊂ D(Ω, E) such that f ν → f and Af ν → g in L 2 (Ω, E). We set A min f = g. The fact that A D is closeable is a standard result in functional analysis (see [14].) More generally, a closed realization of the differential operator A is a closed operatorÃ : L 2 (Ω, E) L 2 (Ω, E) which extends the operator A min . Such an operator satisfies Note that if Ω is complete in its Hermitian metric (in particular if Ω is compact), then the space D(Ω, E) of compactly supported smooth sections of E is dense in Dom(A max ) in the graph norm, and it follows that A max = A min , and there is a unique closed realization of A as a Hilbert-space operator. We are more interested in the case when Ω is not complete, e.g., when Ω is a relatively compact domain in a larger Hermitian manifold.
We now recall the following well-known fact, which follows immediately from (4) (see [14,Lemma 4.3]): Lemma 1. As operators on L 2 (Ω, E), the weak maximal realization A max of the differential operator A and the strong minimal realization A ′ min of its formal adjoint A ′ are Hilbert space adjoints, i.e. we have A max = (A ′ min ) * (note that this implies that A max is closed) and also A ′ min = (A max ) * . Proof. Let A ′ D denote the restriction of A ′ to the compactly supported smooth sections D(Ω, E). Then A ′ D is a densely defined linear operator on L 2 (Ω, E) and its closure is . It now follows that (A ′ D ) * = A max . By taking the closure, we conclude that (A ′ min ) * = A max . Since T * * = T it follows that A ′ min = (A max ) * . We note parenthetically that all the definitions and results of this section also hold in the simpler situation when Ω is a Riemannian manifold, and E is a complex vector bundle, and are independent of the holomorphic structure of Ω and E.

2.2.
Bundle-valued forms. We recall the standard construction of forms on Ω with values in E . Recall that an E-valued (p, q)-form on Ω is a section of the bundle Λ p,q T * (Ω) ⊗ E, where Λ p,q T * (Ω) is the bundle of C-valued forms of bidegree (p, q) (see [43] for details.) We denote by C ∞ p,q (Ω, E) the space of E-valued (p, q)-forms of class C ∞ , so that if {e α } k α=1 is a local frame of E, then locally any element φ of C ∞ p,q (Ω) has a representation φ = where the φ α are (C-valued) (p, q)-forms with smooth coefficients. It is well-known that the operator ∂ gives rise to an operator See [12] for details of this construction. For each p with 0 ≤ p ≤ n, this gives rise to a complex (C ∞ p, * (Ω, E), ∂ E ) of E-valued forms on Ω.
With the holomorphic vector bundle E → Ω we can associate the dual bundle E * → Ω, which is a holomorphic vector bundle over Ω, such that over a point x ∈ Ω, the fiber (E * ) x of E * coincides with the dual vector space (E x ) * of the fiber E x of E. One then has a natural isomorphism of bundles E ∼ = (E * ) * , and we will always make this identification. If E is endowed with a Hermitian bundle metric, this induces a Hermitian bundle metric on E * in a natural way, via the identification of E and E * given by the Hermitian product on each fiber.
We can also define a wedge product of an E-valued (p, q)-form and an E * -valued (p ′ , q ′ )-form with value an ordinary (i.e. C-valued) (p + p ′ , q + q ′ )-form in the following way. Suppose that {e α } k α=1 is a local frame for the bundle E over some open set in Ω, and let {f α } k α=1 be a frame of E * . Given φ ∈ C ∞ p,q (Ω, E) and an ψ ∈ C ∞ p ′ ,q ′ (Ω, E * ), we locally write φ = α φ α ⊗ e α and ψ = β ψ β ⊗ f β , and define pointwise This extends by bilinearity to a wedge product on C ∞ * , * (Ω, E) ⊗ C ∞ * , * (Ω, E * ). If E is a holomorphic vector bundle on Ω define a linear operator σ E on C ∞ * , * (Ω, E) as follows: let φ be a form of bidegree (p, q). Then we set and extend linearly to C ∞ * , * (Ω, E). Clearly (σ E ) 2 is the identity map on C ∞ * , * (Ω, E). Further, if T is any R-linear operator from C ∞ * , * (Ω, E) to C ∞ * , * (Ω, F ) (where F is another holomorphic vector bundle on Ω) of degree d, i.e., if for a homogeneous form f we have deg(T f ) − deg(f ) = d, then we have the relation It is easy to see that the wedge product defined in (7) satisfies the Leibniz formula We note here that the Hermitian metric on Ω and the bundle metric on E have not played any role in this section.
2.3. The space L 2 * (Ω, E). We now use the facts that the manifold Ω has been endowed with a Hermitian metric which we denote by g, i.e., each tangent space T x Ω has been endowed a Hermitian inner product g x (·, ·), which depends smoothly on the base point x and also the fact the holomorphic vector bundle E has been endowed with a Hermitian metric h, i.e. for each x ∈ Ω, h x is a Hermitian product on the fiber E x of E over x. The dual bundle E * can be endowed with a Hermitian metric in the natural way.
The bundle Λ p,q T * Ω ⊗ E has a natural Hermitian inner product (cf. (10) below), so we can construct the space L 2 p,q (Ω, E) = L 2 (Ω, Λ p,q T * Ω ⊗ E) of square integrable E-valued forms using the method of §2.1. We let L 2 * (Ω, E) be the orthogonal direct sum of the Hilbert spaces L 2 p,q (Ω, E) for 0 ≤ p, q ≤ n. We write down the pointwise inner product on the space of E-valued forms. Let φ be as in (5), and let ψ be another (p, q)-form with local representation with respect to the same local frame. The pointwise inner product of the E-valued (p, q) forms φ and ψ is given by at each point x in the open set where the frame {e α } is defined, where by , on right-hand side the standard pointwise inner-product on C-valued (p, q)-forms is meant (see [2].) It is not difficult to see that this definition is independent of the choice of the local frame. We extend (10) to a pointwise inner product on C ∞ * , * (Ω, E) by declaring that forms of different bidegree are pointwise orthogonal.
2.4. The Hodge Star. The pointwise inner product (10) and the wedge product (7) can be related by the Hodge-star operator, the map where dV is the volume form on Ω induced by the Hermitian metric g. It is easy to check that (11) defines ⋆ E as an R-linear and C-antilinear map i.e., for a C-valued function f and a E-valued form φ, we have and that where σ E , σ E * are as in (8).
We recall the well-known formula for ϑ E , and take this opportunity to point out that the formula for ϑ E given in the popular reference [12, p. 152] has a typographical error. Lemma 2. The following formula holds: Proof. It is sufficient to consider the case when the smooth forms φ and ψ are of bidegree (p, q − 1) and (p, q) respectively and at least one of them has compact support and compute Corollary 1. We also have the formula Proof. Using (14), we compute The result follows on replacing E by E * .

Duality
3.1. The basic observation. According to the conventions of multidimensional complex analysis, we adopt the following notation: we write for (ϑ E ) min , the strong minimal realization of ϑ E on L 2 * (Ω, E). By Lemma 1, the operators ∂ E and ∂ * E are Hilbert space adjoints to each other, as are the operators ∂ c,E and ϑ E .
The operator σ E defined in (8) extends from the space D * (Ω, E) of compactly supported forms to give rise to an unitary operator on L 2 * (Ω, E). Similarly the Hodge-Star operator ⋆ E defined in (11) extends from D * (Ω, E) to give rise to a conjugate-linear self-isometry of L 2 * (Ω, E). We continue to denote these Hilbert space realizations by σ E and ⋆ E respectively. We are now ready to prove the main observation behind the use of the Hodge-⋆ operator in L 2 theory: Let Ω be a Hermitian manifold, and E a holomorphic vector bundle on Ω equipped with a Hermitian metric. Let ∂ E , ∂ * E , ϑ E * , ∂ c,E * be the Hilbert space realizations as defined above, and let f ∈ L 2 * (Ω, E): ( Proof. The results are obtained by taking the minimal and maximal realizations of (14) and (15) respectively.
To justify (16) (14) relating the formal adjoints, it also follows that For (17), suppose that f ∈ Dom(∂ E ). This means that f ∈ L 2 * (Ω, E) and ∂ E ∈ L 2 * (Ω, E) (where ∂ E is taken in the weak sense.) Since ⋆ E is an isometry of the Hilbert space L 2 * (Ω, E) with the Hilbert space . From (15) we see that in the weak sense, we have ) is proved the same way.

Duality of Laplacians. Recall that the ∂-Laplacian on E-valued forms on Ω is the operator
The ∂ c -Laplacian on E-valued forms is the operator . Each of and c E is a non-negative self-adjoint operator on L 2 * (Ω, E). Note that on the subspace D * (Ω, E) of compactly supported E-valued forms both E and c E coincide with the "formal Laplacian" This happens if Ω is either compact or complete. We define the spaces of E-valued ∂-Harmonic and ∂ c -Harmonic forms H p,q (Ω, E) and H c p,q (Ω, E) by The following is now easy to prove Also, the restriction of the map ⋆ E to H p,q (Ω, E) gives rise to an isomorphism Proof.
It follows that the self-adjoint operators E and c E * are isospectral: a number λ ∈ R belongs to the spectrum of E if and only if λ belongs to the spectrum of c E * . Let {E λ } λ∈R be a spectral family of orthogonal projections from L 2 * (Ω, E) to itself (cf. [32, Chapters VII,VIII]) such that we have the spectral representation then F λ is an orthogonal projection on L 2 * (Ω, E * ), and we have the spectral representation These statements are purely formal consequences of (18).

3.3.
Closed-range property. In order to apply L 2 -theory to solve the ∂-equation, we first need to show that the ∂-operator has closed range. In this section we consider the consequences of this hypothesis on the ∂ c operator.
Recall that the notation T : H 1 H 2 means that T is a linear operator defined on a linear subspace Dom(T ) of H 1 and taking values in H 2 . Further, for notational simplicity, we will use ∂ E to denote the restriction ∂ E | L 2 p,q (Ω) , when p, q are given, rather than introduce new subscripts, and adopt the same convention for ∂ c,E , ϑ E , and ∂ * E . We first note the following fact Lemma 3. If any one of operators in the following list of Hilbert space operators has closed range, it follows that all the others also have closed range: Proof. Thanks to the well-known fact that a closed densely-defined operator has closed range if and only if its adjoint has closed range (see [19, (16) shows that for f ∈ Dom(∂ * if and only if ⋆ E f ∈ ker(∂ c,E * ). This means that the inequality ∂ * E f ≥ C f holds for all f ∈ ker(∂ * E ) ⊥ if and only if the inequality ∂ c,E * g ≥ C g holds for all g ∈ ker(∂ c,E * ) ⊥ . Again by [ Similarly, the L 2 -cohomology with the minimal realization is defined to the space .
If ∂ E (resp. ∂ c,E ) has closed range, H p,q L 2 (Ω, E) (resp. H p,q c.L 2 (Ω, E)) is a Hilbert space with the quotient norm. Let [·] c : ker(∂ c.E ) ∩ L 2 p,q (Ω, E) → H p,q c.L 2 (Ω, E) denote the respective natural projections onto the quotient spaces. The following result was first observed by Kodaira: We write the proof only for the operator η. The proof for η c is similar.
(i) Note that if q = 0 this is obvious, since img ∂ E : L 2 p,q−1 (Ω, E) L 2 p,q (Ω, E) = 0. Assuming q ≥ 1, we note that ker(η) = ker(∂ E ) ∩ ker(∂ * E ) ∩ img(∂ E ), and therefore a form in ker(η) can be written as ∂g, with ∂ * (∂g) = 0. Then (ii) Since η is an isomorphism, we can identify the harmonic space H p,q (Ω, E) with the cohomology space H p,q L 2 (Ω, E). Since H p,q (Ω, E) is a closed subspace of the Hilbert space L 2 p,q (Ω, E), the space H p,q L 2 (Ω, E) also becomes a Hilbert space. We can think of the map [·] as an operator from the Hilbert space ker(∂ E ) ∩ L 2 p,q (Ω, E) to the Hilbert space H p,q L 2 (Ω, E). Since η is surjective, every element of ker(∂ E ) can be written as f + ∂g, where f ∈ H p,q (Ω, E). According to the identification of H p,q (Ω, E) and H p,q E) is closed, which was to be shown.
Theorem 2 (L 2 Serre duality on non-compact manifolds). The following are equivalent: (1) the two operators (2) the map ⋆ E : L 2 p,q (Ω, E) → L 2 n−p,n−q (Ω, E * ) induces a conjugate-linear isomorphism of Hilbert spaces Consequently, we can identify the Hilbert space dual of H p,q L 2 (Ω, E) with H n−p,n−q c,L 2 (Ω, E * ) We note here that the condition (1) is in fact the necessary and sufficient condition for the existence of the ∂-Neumann operator N E p,q , defined as the inverse (modulo kernel) of the E operator on (p, q)-forms.
Proof. In the diagram the map ⋆ E is known to be an isomorphism from H p,q (Ω, E) to H c n−p,n−q (Ω, E) by Theorem 1 (see equation (19).) Therefore, the map τ will also be an isomorphism, if and only if, both η and η c are isomorphisms. Thanks to Lemma 4 this is equivalent to the two maps ∂ E : L 2 p,q−1 (Ω, E) L 2 p,q (Ω, E) and ∂ c,E * : L 2 n−p,n−q−1 (Ω, E * ) L 2 n−p,n−q (Ω, E * ) having closed range. Since by Lemma 3, the second map has closed range if and only if ∂ E : L 2 p,q (Ω, E) → L 2 p,q+1 (Ω, E) has closed range, the result follows.
3.5. Duality of the ∂-problem and the ∂ c -problem. We can use the duality principle to solve the equation ∂ c u = f , provided we know how to solve ∂u = f : If Ω is a relatively compact pseudoconvex domain in a Stein manifold and q = n − 1, it is further equivalent to the condition ∂ c,E f = 0.
3.6. Duality of realizations of the ∂ operator. We now discuss an abstract version of L 2 -duality which generalizes the duality of ∂ E and ∂ c,E * discussed in the previous sections. The proofs of the statements made below are parallel to the proofs of corresponding statements (for ∂ E and ∂ c,E * ) in the previous sections, and are omitted. Let E be a vector bundle over Ω and let D : L 2 * (Ω, E) L 2 * (Ω, E) be a realization of ∂ E , acting on E-valued forms. Then D satisfies ∂ c,E ⊆ D ⊆ ∂ E . We define an operator D ∨ on the Hilbert Space L 2 * (Ω, E * ) by setting is the Hilbert space adjoint of the operator D. Then the following is easy to prove using relations (14) and (15): (1) D ∨ is a realization of the operator ∂ E * on the Hilbert space L 2 * (Ω, E * ), and its domain is ⋆ E (Dom(D * )).
(3) The map D → D ∨ is a one-to-one correspondence of the closed realizations of ∂ E with the closed realizations of ∂ E * .
We can refer to D ∨ as the realization of ∂ E * dual to the realization D of ∂ E . From now on we will assume that the realization D of the ∂ E operator is closed. Note that then ker(D) is a closed subspace of L 2 * (Ω, E). We define the cohomology groups of the bundle E, with respect to the (closed) realization D as This becomes a Hilbert space if img(D) is closed in L 2 p,q (Ω, E) Then, we can state the following generalized version of Serre duality, with exactly the same proof:  (2) the map ⋆ E : L 2 p,q (Ω, E) → L 2 n−p,n−q (Ω, E * ) induces a conjugate-linear isomorphism of the cohomology Hilbert space H p,q L 2 (Ω, E; D) with H n−p,n−q L 2 (Ω, E * ; D ∨ ) We give an example of a closed realization of ∂ which is strictly intermediate between the maximal and minimal realizations. We consider a domain Ω in a product Hermitian manifold M 1 × M 2 , such that Ω is the product of smoothly bounded, relatively compact domains Ω 1 ⋐ M 1 and Ω 2 ⋐ M 2 . We endow Ω with the product Hermitian metric derived from M 1 and M 2 .
If H 1 and H 2 are Hilbert spaces, we denote by H 1 ⊗H 2 the Hilbert tensor product of H 1 and H 2 , i.e., the completion of the algebraic tensor product H 1 ⊗ H 2 under the norm induced by the natural inner product defined on decomposable tensors by (x ⊗ y, z ⊗ w) = (x, z) H1 (y, w) H2 , and extended linearly. For details see [42, §3.4]. An example of Hilbert tensor products is the space L 2 * (Ω) of square integrable forms on the product Hermitian manifold Ω = Ω 1 × Ω 2 . In fact, if we make the natural identification f ⊗ g = π * 1 f ∧ π * 2 g, where π j : Ω → Ω j is the natural projection. If T 1 : H 1 H ′ 1 and T 2 : H 2 H ′ 2 are closed densely-defined operators, we can define an operator H ′ 1 ⊗H ′ 2 , which on decomposable tensors takes the form (T 1 ⊗T 2 )(x⊗y) = T 1 x ⊗ T 2 y. It is well-known that provided T 1 and T 2 are closed, the operator T 1 ⊗ T 2 is closable. The closure, denoted by T 1 ⊗T 2 is a closed densely defined operator from H 1 ⊗H 2 to H ′ 1 ⊗H ′ 2 . We let ∂ j : L 2 * (Ω j ) L 2 * (Ω j ) denote the maximal realization of the ∂ operator acting of C-valued forms on Ω j . Similarly, we let ∂ j c : L 2 * (Ω j ) L 2 * (Ω j ) denote the minimal realization of the ∂ operator. Consider the operator D on L 2 * (Ω) defined by where I 2 is the identity map on L 2 * (Ω 2 ) and σ 1 is the (bounded selfadjoint) operator on L 2 * (Ω 1 ) which when restricted to L 2 p,q (Ω 1 ) is multiplication by (−1) p+q . Using the techniques of [5,6] the following properties of D can be established • D is a closed densely-defined operator on L 2 * (Ω). • D is a realization of ∂ on Ω, and it is strictly intermediate between the maximal and the minimal realization. We may think of D as being the realization which is maximal on the factor Ω 1 and minimal on the factor Ω 2 . • Suppose that the maximal realization ∂ j has closed range on L 2 * (Ω j ) for j = 1 and 2. By duality, ∂ j c has closed range in L 2 * (Ω j ) as well. Using either of the methods of proof used in [5, Theorem 1.1] or [6, Theorem 1.2], we can conclude that the operator D also has closed range. Further, we have the Künneth formula: • The dual realization D ∨ is the one which is minimal on Ω 1 and maximal on Ω 2 ; it can be represented as Provided ∂ has closed range in each of Ω 1 and Ω 2 , the operator D ∨ again has closed range, and the Künneth formula holds: Suppose that dim C Ω j = n j , and set n = n 1 + n 2 = dim C (Ω). We have by Serre duality, H n−p,n−q (Ω; D ∨ ) ∼ = H p,q (Ω; D) via the map ⋆. Note that this could also be deduced from the knowledge of Serre duality on the factors: indeed for each (p 1 , q 1 ), we have H p1,q1 (Ω 1 ), and for each (p 2 , q 2 ) we have H n2−p2,n2−q2 L 2 (Ω 2 ) ∼ = H p2,q2 c,L 2 (Ω 2 ). Therefore,

∂-closed extension of forms
In this section, we assume that Ω is a relatively compact domain in a Hermitian manifold X. We assume that the holomorphic vector bundle E is defined on all of X. Proof. By definition, This proves the "only if" part of the result. Assume now that both f 0 and ∂(f 0 ) are in L 2 * (Ω, E). To show that f ∈ Dom(∂ c,E ), we need to construct a sequence f ν ∈ D(Ω, E) which converges in the graph norm corresponding to ∂ to f . By a partition of unity, this is a local problem near each z ∈ bΩ, and we can assume that E is a trivial bundle near z. By the assumption on the boundary, for any point z ∈ bΩ, there is a neighborhood ω of z in X, and for ǫ ≥ 0, a continuous one parameter family t ǫ of biholomorphic maps from ω into X such that Ω ∩ ω is compactly contained in Ω, and t ǫ converges to the identity map on ω as ǫ → 0 + . In local coordinates near z, the map t ǫ is simply the translation by an amount ǫ in the inward normal direction. Then we can approximate f 0 locally by f (ǫ) , where ǫ ) * f 0 is the pullback of f 0 by the inverse t −1 ǫ of t ǫ . A partition of unity argument now gives a form f (ǫ) ∈ L 2 * (X, E) such that f (ǫ) is supported inside Ω and as ǫ → 0 + , Since bΩ is Lipschitz, we can apply Friedrichs' lemma (see [18] or Lemma 4.3.2 in [2]) to the form f (ǫ) to construct the sequence {f ν } in D(Ω, E).

4.1.
Use of singular weights. Let X be any Hermitian manifold, and let Ω ⋐ X be a domain in X. We assume that Ω is pseudoconvex, and for z ∈ Ω, let δ be a distance function on Ω. We will assume that δ satisfies the strong Oka's lemma: where c > 0 and ω is a positive (1,1)-form on X.
Such a distance function always exists on a Stein manifold. For example, if Ω is a pseudoconvex domain in C n , we can take δ(z) to be δ 0 e −t|z| 2 where δ 0 is the Euclidean distance from z to to bΩ and t > 0. The distance function δ is comparable to δ 0 . For each t > 0, let E t denote the trivial line bundle C × Ω on Ω with pointwise Hermitian inner product u, v z = (δ(z)) t uv, where u, v ∈ C are supposed to be in the fiber over the point z ∈ Ω. On a Stein manifold, we can take δ to be δ 0 e −tφ for sufficiently large t, where δ 0 is the distance function to the boundary with respect to the Hermitian metric on X and φ is a smooth strictly plurisubharmonic function on X. In classical terminology of Hörmander, this corresponds to the use of the weight function φ t = −t log δ. The dual bundle (E t ) * with dual metric can be naturally identified with E −t , i.e. the weight t log δ. We will denote in conformity with the classical notation. Note that for t > 0, the function δ −t blows up at the boundary of Ω. If t ≥ 1, a form in L 2 p,q (Ω, δ −t ) smooth up to the boundary vanishes on the boundary. We have the following: Let Ω be a relatively compact pseudoconvex domain with Lipschitz boundary in a Hermitian Stein manifold X of dimension n ≥ 2. Suppose that f ∈ L 2 (p,q) (Ω, δ −t ) for some t ≥ 0, where 0 ≤ p ≤ n and 1 ≤ q < n. Assuming that (in the sense of distributions) ∂f = 0 in X with f = 0 outside Ω, then there exists u t ∈ L 2 (p,q−1) (Ω, δ −t ) with u t = 0 outside Ω satisfying ∂u t = f in the distribution sense in X. For q = n, we assume that f satisfies Ω f ∧ g = 0 for every g ∈ ker(∂) ∩ L 2 (n−p,0) (Ω, δ t ), the same results holds.
Proof. Using the notation E t as in (25) it follows that for any t > 0, the map ∂ E * t has closed range in each degree following Hörmander's L 2 method [19] with weights since the weight function satisfies the strong Oka's lemma (see [16]) This equivalent to the ∂-problem on the pseudoconvex domain Ω in the bundle E * t = E −t , i.e., with plurisubharmonic weight −t log δ. The result now follows on combining the solution of the ∂ c problem as given by Theorem 3 and the characterization of the ∂ c operator as given by Proposition 2.
For real s, denote by W s (Ω) the Sobolev space of functions on Ω with s derivatives in L 2 . Let W s 0 (Ω) be the space of completion of C ∞ 0 (Ω) functions under W s (Ω)-norm. Lemma 6. Let Ω be a bounded domain with Lipschitz boundary in R n and let ρ be a distance function. For any s ≥ 0, if f ∈ W s (Ω) and ρ −s+α D α f ∈ L 2 (Ω) for every multi-integer α with |α| ≤ s, then f ∈ W s 0 (Ω) and f 0 ∈ W s (R n ) where f 0 is the extension of f to be zero outside Ω.
The lemma holds for Lipschitz domains also since we can exhaust any Lipschitz domain Ω by smooth subdomains Ω ν (see Lemma 0.3 in [38]). This is clear when the domain is star-shaped and the general case follows from using a partition of unity (see [13] for the corresponding properties for Sobolev spaces on Lipschitz domains).
Combining Proposition 3 and Lemma 6, we have the following regularity results on solving ∂ with prescribed support.
Theorem 5. Let X be a Stein manifold and let Ω ⊂⊂ X be a relatively compact pseudoconvex domain with Lipschitz boundary. Let Ω + = X \ Ω.
We remark that Corollary 2 allows us to solve ∂ smoothly up to the boundary on pseudoconcave domains with only Lipschitz boundary provided the compatibility conditions are satisfied. Results of this kind was obtained in [36] for pseudoconcave domains with smooth boundary. For Lipschitz boundary, see [30] or [15] using integral kernel methods. This is in sharp contrast of pseudoconvex domains, where solving ∂ smoothly up to the boundary is known only for pseudoconvex domains with smooth boundary (see [24]) or domains with Stein neighborhood basis (see [10]). If the boundary bΩ is smooth, Theorem 5 and Corollary 2 also hold for s = 0 (see [37,38]).

Holomorphic extension of CR forms from the boundary of a complex manifold
In this section we study holomorphic extension of CR forms from the boundary of a domain in a complex manifold X using our L 2 -duality. The use of duality in the study of holomorphic extension of CR functions with smooth or continuous data is classical (see [34]), and has been studied by many authors (see [35,25,17].) In what follows, X is a complex manifold, and Ω is a relatively compact domain in X with Lipschitz boundary (see [38] for a general discussion of partial differential equations on Lipschitz domains, and [39] for a discussion of the tangential Cauchy-Riemann equations.) We will assume that X has been endowed with a Hermitian metric, and the spaces L 2 p,q (Ω) = L 2 p,q (Ω, C) of square integrable forms are defined with respect to the metric of X restricted to Ω. Observe that the spaces L 2 p,q (Ω) as well as the Sobolev spaces of forms W k p,q (Ω) are defined independently of the particular choice of metric on X. Further, it is possible to define Sobolev spaces on the boundary bΩ in such a way that the usual results on existence of a trace still holds, e.g. functions in Ω of class W 1 (Ω) have traces on bΩ of class W 1 2 (bΩ) (see [21,22].) The main observation, which follows from the duality results in §3 is the following: Proposition 5. For any p, with 0 ≤ p ≤ n, the map ∂ c : L 2 p,0 (Ω) L 2 p,1 (Ω) has closed range.
Proof. Thanks to Lemma 3 this is equivalent to the map ∂ : L 2 n−p,n−1 (Ω) L 2 n−p,n (Ω) having closed range. But it is well-known that ∂ has closed range in this top degree on smooth domains, a fact that is equivalent to the solvability of the Dirichlet problem for the Laplace-Beltrami operator on such domains (see [11].) For a proof of the solvability of the Dirichlet problem for domains with Lipschitz boundary, see [21,22].
Recall that a holomorphic p-form is a ∂-closed (p, 0)-form. We denote the space of holomorphic p-forms on Ω by O p (Ω). We deduce a necessary condition for a (p, 0)-form on bΩ to be the boundary value of a holomorphic p-form on Ω: Theorem 6. Let f ∈ W 1 2 p,0 (bΩ) be a (p, 0) form on bΩ with coefficients in the Sobolev space W 1 2 . Then the following are equivalent: (1) There is a holomorphic p-form F ∈ O p (Ω) ∩ W 1 (Ω) such that f = F | bΩ (2) For all g ∈ L 2 n−p,n−1 (Ω) ∩ ker(∂), we have (Note that it is easy to show that a ∂-closed form with L 2 coefficients has a trace of class W − 1 2 , and hence the integral above is well defined.) (3) For any extensionf ∈ W 1 p,0 (Ω) of f to Ω as a (p, 0)-form with coefficients in W 1 , the form ∂f ∈ L 2 p,1 (Ω) belongs to the range of ∂ c on Ω. Proof.
(3 =⇒ 1) By Proposition 5, ∂ c has closed range in degree (p, 1), and by hypothesis ∂f is in the range of ∂ c . By Theorem 3, we can solve the equation with L 2 estimates for a (p, 0)-form u. Then F =f − u is holomorphic in Ω. Also, by Proposition 2 we have that ∂(u 0 ) = (∂u) 0 = (∂f ) 0 , where the g 0 denotes the extension of the form g on Ω to all of X by setting it equal to 0 on X \ Ω. Since (∂f ) 0 ∈ L 2 p,1 (X), by elliptic regularity, u 0 ∈ W 1 p,0 (X). It follows that u 0 has a trace (of class W 1 2 (bΩ)) on the Lipschitz hypersurface bΩ. Since u 0 vanishes identically on X \ Ω, it follows that this trace is 0. Consequently, F ∈ W 1 p,0 (Ω) and satisfies F | bΩ = f . Let f be a p-forms with coefficients in L 1 (bΩ) which is the boundary value of a holomorphic p-form F ∈ O p (Ω), then f must be CR, i.e, it must satisfy in the homogeneous tangential Cauchy-Riemann equations on bΩ in the weak sense, i.e., for each compactly supported smooth (n − p, n − 2)-form φ ∈ D n−p,n−2 (X), we have bΩ f ∧ ∂φ = 0.
(See [40] for details.) It is easy to see that (30) implies (32). But in general, the two conditions are not equivalent. One condition under which they are equivalent is the following: Let Ω be a domain with Lipschitz boundary in a complex manifold X of complex dimension n ≥ 2. Suppose that H n−p,n−1 L 2 (Ω) = 0. Then every CR form in f ∈ W 1 2 p,0 (bΩ) has a holomorphic extension F to Ω with F ∈ O p (Ω) ∩ W 1 (Ω) and F = f on bΩ.
Since Ω is Lipschitz, by Friedrich's lemma, we can find a sequence {u ν } ⊂ C ∞ n−p,n−2 (Ω) such that u ν → u in L 2 n−p,n−2 (Ω), and ∂u ν → g in L 2 n−p,n−1 (Ω) as ν → ∞. Let φ ν ∈ D n−p,n−2 (X) be a smooth compactly supported extension of the form u ν to X. Then we have The result now follows by Theorem 6.
Another extension result that can be deduced from Theorem 6 : Let Ω ⋐ X be a domain with connected Lipschitz boundary in a non-compact connected complex manifold X of complex dimension n ≥ 2. Suppose that there exists a relatively compact domain Ω ′ with Lipschitz boundary such that Ω ⋐ Ω ′ ⋐ X and H n−p,n−1 Then every CR form of degree (p, 0) on bΩ of Sobolev class W 1 2 (bΩ) has a holomorphic extension to Ω (of class W 1 (Ω).) Proof. Letf be an extension of f to Ω (of class W 1 (Ω)) and let g = ∂f on Ω 0 on Ω ′ \ Ω We claim that ∂g = 0 on Ω ′ . Indeed, let u ∈ D p,1 (Ω ′ ) be a smooth (p, 1) form of compact support in Ω ′ . We have (since ∂ = ⋆ϑ⋆ on compactly supported forms, see (15)) = 0, (since f is CR, see (32)).
As g vanishes near bΩ ′ and ∂g = 0, it follows that g ∈ Dom(∂ c ) on Ω ′ and ∂ c g = 0. Since ∂ has closed range in Ω for bidegrees (n − p, n − 1) as well as (n − p, n) it follows by duality from (33) that H p,1 c,L 2 (Ω ′ ) = 0. There is then a u ∈ Dom(∂ c ) such that ∂ c u = g. By Proposition 2, the extensions by 0 satisfy ∂(u 0 ) = (∂u) 0 = g 0 . Since g 0 is in L 2 (X) it follows that u 0 ∈ W 1 p,0 (X). Further, u 0 is holomorphic on X \ Ω and u 0 ≡ 0 on X \ Ω ′ . By analytic continuation, u 0 ≡ 0 on X \ Ω. Therefore, the trace of u bΩ vanishes, and the form F =f − u on Ω is holomorphic, of class W 1 and satisfies F = f on bΩ.

Corollary 5.
Let Ω be domain with Lipschitz boundary in a Stein manifold X of complex dimension n ≥ 2. Suppose that bΩ is connected. Then for every CR function on bΩ of class W 1 2 (bΩ) has a holomorphic extension to Ω.
Proof. In the proof of Corollary 4, we let Ω ′ be some strongly pseudoconvex domain in X and Ω ⋐ Ω ′ . Then H n,n−1 L 2 (Ω ′ ) = H 0,1 c,L 2 (Ω ′ ) = 0. The corollary follows. When X = C n and p = 0, this gives the usual Bochner-Hartogs' extension theorem. In this case, the extension function can be written explicitly as where B is the Bochner-Martinelli kernel. The function F has boundary value f as z approaches the boundary (see [41] for a proof when the boundary is smooth; in this case we can allow more singular boundary values than possible in our results with Lipschitz boundaries.) This is very different from holomorphic extension of CR functions in complex manifolds which are not Stein. We will give an example to show that the extension results on Lipschitz domain is maximal in the sense that the results might not hold if the Lipschitz condition is dropped.
We will analyze the holomorphic extension of functions on a non-Lipschitz domain. Let Ω be the Hartogs' triangle in CP 2 defined by where [z 0 , z 1 , z 2 ] denotes the homogeneous coordinates of a point in CP 2 . As usual we endow Ω with the restriction of the Fubini-Study metric of CP 2 .

Proposition 6.
Let Ω ⊂ CP 2 be the Hartogs' triangle. Then we have the following: Remark: Statements (1) and (3) above have already been proved in [15]. Regarding (2), we would like to point out a misleading statement made in that paper, where it is claimed that W 1 (Ω) ∩ O(Ω) consists of constants only (see item 5 in Example 12.1 in [15]).