http://dx.doi.org/10.4134/JKMS.j140768 pISSN: 0304-9914 / eISSN: 2234-3008
THE ∂-PROBLEM WITH SUPPORT CONDITIONS AND PSEUDOCONVEXITY OF GENERAL ORDER
IN K ¨AHLER MANIFOLDS
Sayed Saber
Abstract. Let M be an n-dimensional K¨ahler manifold with positive holomorphic bisectional curvature and let Ω ⋐ M be a pseudoconvex domain of order n − q, 1 ≤ q ≤ n, with C2 smooth boundary. Then, we study the (weighted) ∂-equation with support conditions in Ω and the closed range property of ∂ on Ω. Applications to the ∂-closed extensions from the boundary are given. In particular, for q = 1, we prove that there exists a number ℓ0>0 such that the ∂-Neumann problem and the Bergman projection are regular in the Sobolev space Wℓ(Ω) for ℓ < ℓ0.
1. Introduction
The solution of the ∂-Neumann problem has many important applications in the theory of several complex variables and in partial differential equations, particulary in the study of the ∂-problem with support conditions. Let Ω be an open subset of an n-dimensional complex manifold M . We say that Ω has C2 boundary in M if for all z ∈ bΩ there exist an open neighborhood U of z and a C2 function δ : U −→ R, called a defining function of Ω at z such that dδ(z) 6= 0 and Ω ∩ U = {z ∈ U : δ(z) < 0}. Then Ω is pseudoconvex of order n − q in M, if and only if the Levi form ∂∂δ has at least n − q non-negative eigenvalues on Tz(bΩ) for each defining function δ of Ω near z, where Tz(bΩ) is the holomorphic tangent space of bΩ at z (Called such a subset Ω a (q − 1)- pseudoconvex open subset with C2 boundary) (cf. Eastwood and Suria [11], Suria [26])). Pseudoconvex open sets in the original sense are pseudoconvex of order n − 1. If an open set Ω in an n-dimensional complex manifold M is weakly q-convex, 1 ≤ q ≤ n, then Ω is pseudoconvex of order n − q in M.
However, the converse is not valid even if M = Cn (see Diederich and Fornaess [10]) .
Received December 15, 2014.
2010 Mathematics Subject Classification. 32F10, 32W05, 32W10, 35J20, 35J60.
Key words and phrases. ∂, ∂b and ∂-Neumann operators, pseudoconvex domains, CR manifolds.
c
2016 Korean Mathematical Society 1211
The ∂-problem with support conditions was considered by Derridj [9]. Shaw [23] has obtained a solution to this problem on a pseudoconvex domain Ω with C1 boundary in Cn. If Ω is a weakly q-convex domain in Cn, Saber [20] established this problem on Ω. Also in [21], he is extended this result to a weakly q-convex domain in a Stein manifold. On a locally Stein domain of the complex projective space CPn, Cao-Shaw-Wang [6] obtained a solution to this problem (see also [5]). Saber [22] extended these results to a general pseudoconvex domain in CPn. Our main result in this paper is to extend the result of Cao, Shaw and Wang and Saber to all general pseudoconvex domain Ω in an n-dimensional K¨ahler manifold M with positive holomorphic bisectional curvature (we recall that by Siu [24], a compact K¨ahler manifold with positive holomorphic bisectional curvature is biholomorphic to CPn). We also construct
∂-closed extensions from the boundary.
The Bergman projection P denotes the orthogonal projection from L2(Ω) to ker ∂ ∩ L2(Ω), where ∂ is the Cauchy-Riemann operator. In [6], Cao, Shaw, and Wang extend Berndtsson and Charpentier’s result [3] to obtain estimates for the ∂-Neumann operator. Although they work in CPn in this paper, their proof for this result also applies in Cn (see [14]). Harrington [13] proved this result on a bounded pseudoconvex domain with Lipschitz boundary. Here, we prove this result for forms on a pseudoconvex domain Ω in an n-dimensional K¨ahler manifold M with positive holomorphic bisectional curvature.
The plan of this paper is as follows: In Section 2 we study the ∂-equation with support conditions in a pseudoconvex domain of order n−q, 1 ≤ q ≤ n−1, with C2 smooth boundary in an n-dimensional K¨ahler manifold with positive holomorphic bisectional curvature. Also the ∂-closed property of ∂ is obtained and we construct ∂-closed extensions from the boundary. In Section 3, we use the modified weight function method to study the same result of Section 3 in these spaces. In Section 4, we prove that the ∂-Neumann problem and the Bergman projection are regular in the Sobolev space Wℓ(Ω) for ℓ ∈ (0, 1) in a special case when q = 1.
2. The ∂-problem with support conditions
Let M be an n-dimensional complex manifold with the K¨ahler metric g and let Ω be a bounded domain with C2-smooth boundary bΩ in M . Let L2(Ω) denote the space of square integrable functions on Ω with respect to the Lebesgue measure in M . We use L2p,q(Ω) to denote the space of (p, q)-forms with coefficients in L2(Ω). For f, g ∈ L2p,q(Ω), denote the inner product hf, gi and the norm kfk by:
hf, gi = Z
Ωf ∧ ⋆ g and kfk2= hf, fi,
where ⋆ is the Hodge star operator. Let ∂ : L2p,q(Ω) −→ L2p,q+1(Ω) be the maximal closure of the Cauchy-Riemann operator and ∂⋆ be its Hilbert space
adjoint. One defines the ∂-Neumann operator N = Np,q: L2p,q(Ω) −→ L2p,q(Ω), as the inverse of the restriction of = p,q to (ker )⊥, where = ∂ ∂∗+ ∂∗∂ is the Laplace Beltrami operator. Define the space of the harmonic (p, q)-forms H by
H = {u ∈ Dp,q : ∂u = ∂∗u = 0}.
Let ∇ be the Levi-Civita connection of M with the K¨ahler metric g. The Levi-Civita connection, sometimes also known as the Riemannian connection or covariant derivative. Let {ei} be an orthonormal basis of vector fields. For any two vector fields u, v, we consider the curvature operator of the connection
∇
R(u, v) = ∇u∇v− ∇v∇u− ∇[u,v]
and set Rijkl = g(R(ei, ej)ek, el). We also define the Ricci tensor Rij = P
kεkRikkj which turns out to be self-adjoint with respect to g and the scalar curvature Θ as the trace of the Ricci tensor
Θ =X
i
Rii=X
i,j
εiεjRjiij.
Let d(z, bΩ) be the boundary distance function from z ∈ Ω to the boundary bΩ induced by a complete K¨ahler metric on M . If the boundary bΩ of an open subset Ω of M is a real submanifold of class C2 in M , there exists an open subset U of M such that bΩ ⊂ U and the boundary distance function d(z, bΩ) is of class C2 on Ω ∩ U (see [18]).
Proposition 2.1. Let Ω be a compact domain with C2-smooth boundary bΩ and defining function δ(x) = −d(x, bΩ). Then, for any u ∈ Cp,q∞(Ω) ∩ dom ∂∗φ
with q ≥ 1, and φ ∈ C2(Ω), we have
(2.1)
k∂uk2φ+ k∂∗φuk2φ= hΘu, uiφ+
∂uIJ
∂zk
2
φ
+ h(i∂∂φ)u, uiφ
+ Z
bΩ
((i∂∂δ)u, u)e−φdS.
Proof. This formula is known (cf. [2], [8], [16], [24] and [27]) for some special cases, although it has not been stated in the literature in the form (2.1). The boundary term had been computed in [12]; by combining the Morrey-Kohn technique on the boundary with non-trivial weight function. One combines the results of [16] and [27] with the interior formulae discussed above, one can prove that (2.1) holds for the general case with a weight function e−φ and the curvature term. For the case φ = 0, the stated formula was proved in Siu
[24].
Theorem 2.2([18, Corollary 6.5]). Let M be an n-dimensional K¨ahler mani- fold with positive holomorphic bisectional curvature and letΩ ⋐ M be a pseudo- convex domain of order n − q, 1 ≤ q ≤ n with C2 smooth boundary. Then, the
Levi form of the function− log d(z, bΩ) has at least n−q+1 positive eigenvalues at each point of Ω.
Theorem 2.3. LetM be an n-dimensional K¨ahler manifold with non-negative holomorphic bisectional curvature and let Ω ⋐ M be a pseudoconvex domain of order n − q with C2 smooth boundary. For 0 ≤ p ≤ n, 0 ≤ q ≤ n, the
∂-Neumann operator Np,q: L2p,q(Ω) −→ L2p,q(Ω) exists and the harmonic forms Hp,q(Ω) = {0} for 1 ≤ q ≤ n and we have the following properties:
(i) range (Np,q) ⊂ dom(p,q), Np,qp,q = I on dom(), (ii) for f ∈ L2p,q(Ω), we have
f = ∂ ∂∗Np,qf ⊕ ∂∗∂Np,qf, 1 ≤ q ≤ n, f = ∂∗∂Np,0f ⊕ P f, q = 0,
where P is the projection from L2p,0(Ω) onto L2p,0(Ω)∩ ker ∂ and Np,0= ∂∗Np,12 ∂,
(iii) ∂N = N ∂, and ∂∗N = N ∂∗,
(iv) for all f ∈ L2p,q(Ω), we have the estimates kNfk ≤ ckfk, k∂Nfk + k∂∗N f k ≤√
c kfk.
Proof. In fact, the boundary integral in (2.1) is non-negative for q ≥ 1 from our condition. By taking φ ≡ 0 in Proposition 2.1 and using Theorem 2.2, we find the fundamental estimate
(2.2) kuk2≤ c
k∂uk2+ k∂∗uk2 .
This means that has closed range and Hp,q(Ω) = {0}. Thus, we can establish the L2-existence theorem of the ∂-Neumann operator. As an immediate consequence of the basic estimate (2.2) is the following theorem whose proof can be found in [16].
Theorem 2.4. Let Ω and M be the same as in Theorem 2.3. If 1 ≤ q ≤ n, we have
(1) The operator ∂ has closed range in L2p,q(Ω) and L2p,q+1(Ω).
(2) The operator ∂∗ has closed range in L2p,q(Ω) and L2p,q−1(Ω).
(3) The canonical solution operators to ∂ given by ∂∗N : L2p,q(Ω) −→
L2p,q−1(Ω) and
N ∂∗: L2p,q+1(Ω) −→ L2p,q(Ω) are continuous.
(4) If f ∈ ker (∂), then ∂∗N f gives the solution u to the equation ∂u = f . (5) If f ∈ ker (∂∗), then ∂N f gives the solution u to the equation ∂∗u = f .
Using the duality relations pertaining to the ∂-Neumann problem, one solve the ∂-equation with support condition in Ω. This method was first used by Kohn-Rossi [15] for smooth forms on strongly pseudoconvex domains. More precisely, we prove the following theorem:
Theorem 2.5. Let Ω and M be the same as in Theorem 2.3. Then, for f ∈ L2p,q(M ), 1 ≤ q ≤ n − 1, satisfying ∂f = 0 in the distribution sense in M and f is supported in Ω, there exists u∈ L2p,q−1(M ) such that ∂u = f in the distribution sense in M with u is supported in Ω and
Z
Ω|u|2dV ≤ C Z
Ω|f|2dV for some C > 0.
Proof. Let f ∈ L2p,q(M ) and f is supported in Ω. Then f ∈ L2p,q(Ω). From Theorem 2.3, the ∂-Neumann operator Nn−p,n−q exists for n − q ≥ 1. Since Nn−p,n−q= −n−p,n−q1 on range n−p,n−qand range Nn−p,n−q⊂ dom n−p,n−q, then
Nn−p,n−q⋆ f ∈ dom n−p,n−q⊂ L2n−p,n−q(Ω) for 1 ≤ n − q. Thus, we can define u ∈ L2p,q−1(Ω) by
u = − ⋆ ∂ Nn−p,n−q ⋆ f .
Thus u is supported in Ω, i.e., u vanishes on bΩ. Now, we extend u to M by defining u = 0 in M \ Ω. It follows from the same arguments of Theorem 9.1.2 in [7] and Theorem 2.2 in [1] that the form u satisfies the equation ∂u = f in
the distribution sense in M .
Proposition 2.6. Under the same assumption of Theorem 2.4, for any f ∈ L2p,n(M ), 0 ≤ p ≤ n, such that f is supported in Ω and
(2.3)
Z
Ωf ∧ g = 0 for every g ∈ L2n−p,0(Ω) ∩ ker ∂.
One can find u ∈ L2p,n−1(M ) such that ∂u = f in the distribution sense in M with u is supported in Ω and
Z
Ω|u|2dV ≤ C Z
Ω|f|2dV for some C > 0.
Proof. Using Theorem 2.3, the ∂-Neumann operator Np,0 exists for any 0 ≤ p ≤ n and we have
Np,0= ∂⋆Np,12 ∂.
The Bergman projection operator Pp,0 is given by
∂⋆∂Np,1= I − Pp,0.
Define u by
u = − ⋆ ∂Nn−p,0⋆ f.
Thus, we have
(2.4) ∂u = (−1)p+n⋆ ∂⋆∂Nn−p,0⋆ f
= f − (−1)p+n⋆ Pn−p,0⋆ f.
From (2.3), for any g ∈ L2n−p,0(Ω) ∩ ker ∂, we get h⋆f, gi = (−1)p+n
Z
Ωg ∧ f = 0.
Thus Pn−p,0⋆ f = 0 and from (2.4) we have ∂u = f in Ω. Using ⋆u∈ dom ∂⋆ and extending u to be zero outside Ω, then ∂u = f in M in the distribution
sense.
The necessary and sufficient condition on f ∈ W
1
p,q2 (bΩ) to have a ∂-closed extension F on Ω is summarized as follows.
Theorem 2.7. LetΩ and M be the same as in Theorem 2.3. For f ∈ Wp,q12 (bΩ), 0 ≤ p ≤ n, 1 ≤ q ≤ n − 1, we assume that
(2.5) ∂bf = 0, if 1 ≤ q ≤ n − 2, and
(2.6) Z
bΩf ∧ h = 0 for every h ∈ L2n−p,0(Ω) ∩ ker ∂ if q = n − 1.
Then there exists F ∈ L2p,q−1(Ω) such that F = f on bΩ and ∂F = 0 in Ω.
Proof. For a form f ∈ W
1
p,q2 (bΩ), one can extend f componentwise to Ω such that each component is in W1(Ω) as follows: Let f′∈ Wp,q1 (Ω) be an arbitrary extension of f .
First we prove the case 1 ≤ q ≤ n − 2. From (2.5), we can require that f1= ∂bf′ = 0 in bΩ. If we extend f1 to be zero outside Ω, we get ∂f1= 0 in M in the distribution sense. Set
g = − ⋆ ∂Nn−p,n−q−1⋆ f1.
From Theorem 2.5 and its proof, we have g ∈ L2p,q(Ω). Set g = 0 outside Ω.
Then ∂g = f1 in the distribution sense on M with g is supported in Ω.
Setting F = f′− g in Ω, we have F = f on bΩ and ∂F = ∂f′ − ∂g = f1− f1= 0 on Ω. This proves the case 1 ≤ q ≤ n − 2.
When q = n − 1 and f satisfies (2.6), we let f1 = ∂f′. Then for any h ∈ Cn,01 (Ω) with ∂h = 0, we have
(2.7)
Z
Ω
f1∧ h = Z
Ω
∂f′∧ h = Z
bΩ
f ∧ h = 0.
When h ∈ L2n,0(Ω)∩ ker ∂, (2.7) follows from approximating h by smooth forms.
Apply Proposition 2.6 to f1, since f1 satisfies (2.3) by (2.7). Setting g =
− ⋆ ∂Nn−p,n−q−1⋆ f1and F = f′− g as above, by the proof of Proposition 2.6, we can conclude the proof for the case of q = n − 1 as well.
3. The weighted ∂-problem with support conditions
Let M be an n-dimensional complex manifold with the K¨ahler metric g.
Let Ω be a bounded domain with C2-smooth boundary bΩ in M and δ be its defining function. Let L2p,q(Ω, e−φt) = L2p,q(Ω, δt) = L2p,q(δt) be the weighted L2 space with respect to the weight function φt = −t log δ, t ≥ 0. Then for f, g ∈ L2p,q(δt), we denote the inner product hf, gitand the norm k f kt by:
hf, git= Z
Ωf ∧ ⋆tg and kfk2t = hf, fit,
where ⋆t= δt⋆ = ⋆ δt. Let ∂ : dom ∂ ⊂ L2p,q(δt) −→ L2p,q+1(δt) be the maximal closure of the Cauchy-Riemann operator and ∂∗t be its Hilbert space adjoint.
Let t= ∂ ∂∗t+ ∂∗t∂ be the weighted Laplace-Beltrami operator and Ntbe the weighted ∂-Neumann operator defined as in Section 2. The weighted space of the harmonic (p, q)-forms Htis defined by
Ht= {u ∈ Dp,q: ∂u = ∂∗tu = 0}.
Proposition 3.1. Let Ω and M be the same as in Theorem 2.3. Then, for t > 0 and for u ∈ dom(t) of degree q ≥ 1, we have
(3.1) tkuk2t≤ (k∂uk2t+ k∂∗tuk2t).
Proof. By Theorem 2.2, we have that φt= −t log δ, t ≥ 0 is strictly plurisub- harmonic for q ≥ 1 with i∂∂φ ≥ ω, where ω is the K¨ahler form of M. Us- ing H¨ormander’s weighted L2 estimates for the ∂-Neumann problem (see e.g.
Proposition A.4 in [6]), we have the following formula: for any (p, q)-form u ∈ dom ∂ ∩ dom ∂⋆,
tkuk2t≤ (k∂uk2t+ k∂∗tuk2t)
for t > 0.
Theorem 3.2. LetΩ and M be the same as in Theorem 2.3. For each t > 0, the ∂-Neumann operator Np,qt : L2p,q(δt) −→ L2p,q(δt) exists, where 0 ≤ p ≤ n, 0 ≤ q ≤ n, and the harmonic forms Htp,q(Ω) = {0} for 1 ≤ q ≤ n and we have the following properties:
(i) range (Nt) ⊂ dom(t), Ntt= I on dom(t), (ii) for f ∈ L2p,q(δt), we have u = ∂ ∂∗tNtf ⊕ ∂∗t∂Ntf , (iii) ∂Nt= Nt∂, and ∂∗tNt= Nt∂∗t,
(iv) for all f ∈ L2p,q(δt), we have the estimates kNtf kt≤ ckfkt,
k∂Ntf kt+ k∂∗tNtf kt≤√ ckfkt. Proof. In fact, by using Proposition 3.1, we have for t > 0
tkuk2t ≤ htu, uit ≤ ktf ktkukt, i.e.,
tkukt≤ ktukt.
Since t is a linear closed densely defined operator, then, from [16]; Theorem 1.1.1, Range(t) is closed. Thus, from (1.1.1) in [16] and the fact that t is self adjoint, we have the Hodge decomposition
L2p,q(Ω) = ∂ ∂∗tdom(t) ⊕ ∂∗t∂ dom(t).
Since t is one to one on dom(t) from (1.5.3) in [16], then there exists a unique bounded inverse operator
Nt: Ran(t) −→ dom(t) ∩ (ker(t))⊥
such that Nttf = f on dom(t). Therefore, we can establish the existence theorem of the inverse of tthe so called weighted ∂-Neumann operator Nt. Following (3.1), we immediately have the standard consequences of such L2 estimates.
Theorem 3.3. Let Ω and M be the same as in Theorem 2.3. If t > 0, 1 ≤ q ≤ n, we have
(1) The operator ∂ has closed range in L2p,q(δt) and L2p,q+1(δt).
(2) The operator ∂∗t has closed range in L2p,q(δt) and L2p,q−1(δt).
(3) The canonical solution operators to ∂ given by
∂∗tNt : L2p,q(δt) −→ L2p,q−1(δt) and Nt∂∗t : L2p,q+1(δt) −→ L2p,q(δt) are con- tinuous.
(4) If f ∈ ker (∂), then ∂∗tNtf gives the solution utto the equation∂ut= f . (5) If f ∈ ker (∂∗t), then ∂Ntf gives the solution utto the equation∂∗tut= f . Using the duality relations pertaining to the ∂-Neumann problem, one solve the weighted ∂-equation with support condition in Ω. More precisely, we prove the following theorem:
Theorem 3.4. Let Ω and M be the same as in Theorem 2.3. Then, for f ∈ L2p,q(δ−t), 1 ≤ q ≤ n − 1, satisfying ∂f = 0 in the distribution sense in M and f is supported in Ω, there exists u ∈ L2p,q−1(δ−t) such that ∂u = f in the distribution sense in M with u is supported in Ω and
Z
Ω|u|2δ−tdV ≤ C Z
Ω|f|2δ−tdV for some C > 0.
Proof. From Theorem 3.2, the ∂-Neumann operator Nt exists for forms in L2n−p,n−q(δ−t). Thus, we can define ut∈ L2p,q−1(δ−t) by
ut= − ⋆t ∂ Nn−p,n−q ⋆−tf .
Thus ut supported in Ω, i.e., utvanishes on bΩ. Now, we extend ut to M by defining ut= 0 in M \ Ω. It follows from the same arguments of Theorem 9.1.2 in [7] and Theorem 2.2 in [1] that the form u satisfies the equation ∂u = f in the distribution sense in M . Thus the proof follows. Proposition 3.5. Let Ω and M be the same as in Theorem 2.3. Put D = M \Ω. Then, for any f ∈ Wp,q1+ε(D), ∂f = 0, 0 ≤ ε < 12 there exists F ∈ Wp,qε (M ) such that F |D= f and ∂F = 0 in M .
Proof. Since Ω has C2smooth boundary, then from [25] there exists a bounded extension operator from Wp,qs (D) to Wp,qs (M ) for all s ≥ 0. Moreover, one can choose an extension ˜f ∈ Wp,q1+ε(M ) of f such that ˜f |D = f and ∂ ˜f ∈ Wε(Ω) ∩ L2(δ−2ε, Ω) with k ˜f kW1+ε(M)≤ CkfkW1+ε(D).
Define T ˜f by T ˜f = − ⋆2ε∂N2ε(⋆−2ε∂ ˜f ) in Ω. As in Theorem 3.4, T ˜f ∈ L2(δ−2ε, Ω). But for a C2-smooth domain, we have that T ˜f ∈ L2(δ−2ε, Ω) is comparable to Wε(Ω) for 0 ≤ ε < 12. This gives that T ˜f ∈ Wp,qε (Ω) and T ˜f satisfies ∂T ˜f = ∂ ˜f in M in the distribution sense if we extend T ˜f to be zero outside Ω.
Since 0 ≤ ε < 12, the extension by 0 outside Ω is a continuous operator from Wε(Ω) to Wε(M ) (see e.g. [17]). Thus we have T ˜f ∈ Wε(M ). Define
F =
(f, x ∈ D f − T ˜˜ f, x ∈ Ω.
Then F ∈ Wp,qε (M ), and F is ∂-closed extension of f to M . Corollary 1. Let Ω and M be the same as in Theorem 2.3. Then Wp,01+ε(D) ∩ Ker ∂ = {0}, 1 ≤ p ≤ n and W0,01+ε(D) ∩ Ker ∂ = C.
Proof. Using Proposition 3.5 for q = 0, we have that any holomorphic (p, 0)- form on D extends to be a holomorphic (p, 0) in M , which are zero (when
p > 0) or constants (when p = 0).
Corollary 2. Let Ω and M be the same as in Theorem 2.3. Then, for any f ∈ Wp,q1+ε(D), where 0 ≤ p ≤ n, 1 ≤ q ≤ n − 2, p 6= q, and 0 ≤ ε < 12, such that ∂f = 0 in D, there exists u ∈ Wp,q−11+ε (D) such that ∂u = f in D.
Proof. If p 6= q, we have that F = ∂U for some U ∈ Wp,q−11 (M ). Let u = U on D, we have u ∈ Wp,q−11 (D) satisfying ∂u = f in D.
4. Sobolev regularity for the Bergman projection
Let M be an n-dimensional K¨ahler manifold with positive holomorphic bi- sectional curvature and let Ω be a pseudoconvex open subset of order n − 1 in M . Moreover, suppose that the boundary bΩ of Ω is a real submanifold of class C2 in M and δ(x) = −d(x, bΩ) be a C2-defining function for Ω. Thanks to Takeuchi’s theorem, Ohsawa and Sibony [19] proved the existence of the Diederich-Fornaess exponent in C2 pseudoconvex domains in a K¨ahler mani- fold with positive holomorphic bisectional curvature:
Theorem 4.1. Let Ω ⋐ M be a pseudoconvex domain of order n − 1 with C2 boundary in a complete K¨ahler manifold with positive holomorphic bisectional curvature. Then there exists α > 0 such that ϕ = −δα is strictly plurisubhar- monic in Ω. More precisely, there is a positive constant cα such that
(4.1) i∂∂ϕ(u, u)≥ cα|δ|α|u|2 for all u ∈ Cp,q∞(Ω).
Definition 1. We call tbΩthe upper bound of the Diederich-Fornaess exponent for the distance function to the boundary of Ω, defined by
tbΩ= sup{0 < η ≤ 1 | i∂∂(−δα) > 0 on Ω}.
From Theorem 4.1, tbΩis strictly positive on C2relatively compact pseudocon- vex domains in a complete K¨ahler manifold with positive holomorphic bisec- tional curvature.
By using the previous theorem, Berndtsson and Charpentier [3] generalized H¨ormander’s weighted method and they established an L2-estimate theorem for ∂. Since
i∂∂(−δα) = α δα
(1 − α)i∂δ ∧ ∂δ
δ2 +i∂∂(−δ) δ
, then from (4.1),
(4.2) (1 − α)i∂δ ∧ ∂δ
δ2 +i∂∂(−δ)
δ > 0 on Ω.
But
i∂∂(− log δ) = i∂δ ∧ ∂δ
δ2 +i∂∂(−δ)
δ .
It follows, from (4.2), that
(4.3) i∂∂(− log δ) > αi∂δ ∧ ∂δ δ2 .
By taking Ψ = −ℓ log δ, where ℓ is a positive constant, we obtain i∂Ψ ∧ ∂Ψ = ℓ2i∂δ ∧ ∂δ
δ2 . Thus, from (4.3), we obtain
i∂∂(− log δ) >α ℓ2
i ∂Ψ ∧ ∂Ψ.
Then, there exists α ∈ (0, 1) small enough such that
(4.4) i∂Ψ ∧ ∂Ψ < ℓ
α
i∂∂Ψ, on Ω.
Thus by using (4.4), as Berndtsson and Charpentier [3], we can prove the following theorem:
Theorem 4.2. Let Ω and M be the same as in Theorem 4.1. Let f ∈ L2p,q(Ω, δβ−ℓ) ⊂ L2p,q(Ω, δβ), 1 ≤ q ≤ n, with ∂f = 0 and u = ∂∗βNβf be the solution to the equation ∂u = f in L2p,q(Ω, δβ). Thus, there exists a positive constant c such that
Z
Ω|u|2δβ−ℓdV ≤ c Z
Ω|f|2δβ−ℓdV.
Following Proposition 2.1, using Theorem 2.2, and by taking φ = −β log δ, β ∈ (0, 1), we have for u ∈ dom(t) of degree q ≥ 1,
kuk2φ≤ k∂ uk2φ+ k∂∗φuk2φ
for any u ∈ Cp,q∞(Ω) ∩ dom ∂∗φ.
Thus, by the same argument of Theorem 4.3.4 in [7], we can prove the following theorem:
Theorem 4.3. Let Ω and M be the same as in Theorem 4.1. For every f ∈ L2p,q(Ω, e−φ), with ∂ u = 0, one can find u ∈ L2p,q−1(Ω, e−φ) such that
∂ u = f and
Z
Ω|u|2e−φdV ≤ c Z
Ω|f|2e−φdV.
One can always select the solution u of Theorem 4.3 satisfying the additional property u ∈ L2p,q−1(Ω, e−φ) ∩ ker ∂⊥
, i.e., satisfies the following:
Z
Ω
e−φ u ∧ ⋆ v = 0
for any ∂-closed form v ∈ L2p,q−1(Ω, e−φ). Hence the theorem implies that u satisfies
Z
Ω|u|2e−φdV ≤ c Z
Ω|∂u|2e−φdV.
Remark 4.1. It is verify that, if T⋆is the adjoint map of T with respect to the L2-norm, then
kT fkWp,qk/2(Ω)= sup
g∈L2
hT f, giL2
kgkWp,qk/2(Ω) = sup
g∈L2
hf, T⋆giL2
kgkWp,q−k/2(Ω) (4.5) ≤ kT⋆kWp,q−k/2(Ω)kgkWp,qk/2(Ω).
Thus, by applying the same technique of Berndtsson-Charpentier [3], we can prove the following main theorem:
Theorem 4.4. Let Ω and M be the same as in Theorem 4.1. Then, there exists a number k0 ∈ (0, 1) such that the operators N, ∂∗N and P are exact regular in the Sobolev space Wp,q±k(Ω) for 0 < k < k0/2.
In fact, the proof of the regularity in the Sobolev space Wp,qk (Ω) of the Bergman projection P and the canonical solution operator ∂∗N to ∂ is the same as in [3]. By a result of Boas-Straube [4], the ∂-Neumann operator N is regular if and only if the Bergman projection P is. Thus the exact regularity of N follows. Then by using (4.5), the proof is complete.
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Sayed Saber
Mathematics Department Faculty of Science Beni-Suef University Egypt
E-mail address: [email protected]
