# Global Regularity for the∂

Authors: Shaban Khidr; Osama Abdelkader
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

LetM be a C
∞ compact C
R manifold of C
R-codimension l
≥
1 and C
R-dimension n
-
l in a complex manifold X of complex dimension n
≥
3. In this paper, assuming that M satisfies condition Y
(
s
) for some s with 1
≤
s
≤
n
-
l
-
1, we prove an L
2-existence theorem and global regularity for the solutions of the tangential Cauchy-Riemann equation for (
0
,
s
)-forms on M.


---

## Body

## 1. Introduction and Basic Notations

The tangential Cauchy-Riemann complex (or∂
-
b-complex) was first introduced by Kohn and Rossi [1] for studying the holomorphic extension of C
R functions from the boundary of a complex manifold. The closed range property is related to existence and regularity theorems for ∂
-
b and for C
R manifolds to a reason of embedding. It is worth then to mention that the ∂
-
b-operator has closed range in the L
2-sense on boundaries of smooth bounded pseudoconvex domains in C
n due to Shaw [2] for all 1
≤
s
<
n
-
2 and Boas and Shaw [3] for s
=
n
-
2. Later, Kohn [4] obtained results analogue to those of [2, 3] on boundaries of smooth bounded pseudoconvex domains in a complex manifold. Nicoara [5] extended the results of Kohn [4] to compact, orientable, pseudoconvex C
R manifold of real dimension 2
n
-
1, at least five, embedded in C
N, N
≥
n, leading to global regularity for the ∂
-
b-equation on such C
R manifolds. The main tool in his proof is that of microlocalizations using a new type of weight functions called strongly C
R plurisubharmonic functions (see also [6]).In addition, Harrington and Raich [7] adapted the microlocal analysis used by Nicoara [5] to establish the closed range property for the ∂
-
b-operator on C
R manifold of hypersurface type satisfying weak Y
(
s
) condition. More precisely, by using the weighted ∂
--theory, they showed that the complex Green’s operator is continuous in the L
2-Sobolev spaces W
k, k
∈
N, and they further obtained a global solution with C
∞-regularity for solutions of the ∂
-
b-equation for (
0
,
s
)-forms.This paper is concerned with proving anL
2-existence theorem for the ∂
-
b-Neumann problem on a C
∞
C
R compact manifold M of real dimension 2
n
-
l (l
≥
1) that satisfies condition Y
(
s
) for some s with 1
≤
s
≤
n
-
l
-
1 in an n-dimensional complex manifold X and with establishing the global regularity properties of the ∂
-
b-equation. In particular, our ∂
-
b-problem is set up in the usual L
2-setting with no weights using our arguments in [8, 9]. Namely, via a partition of unity, we globalize first the local maximal L
2-Sobolev estimates obtained by [10] for □
b and patching them together to obtain global ones on M. Further, we explore an L
2-existence theorem for the ∂
-
b-equation on M. These L
2 results allow us to prove that the complex Green operator G
b and the Szegö projection operators S
s are continuous in the Sobolev spaces W
0
,
s
k
(
M
) for some s such that 1
≤
s
≤
n
-
l
-
1 and k
≥
0. Furthermore, we obtain a global smooth solution for the ∂
-
b-equation given smooth data on M. Before we proceed, we recall first some basic definitions and notations on C
R manifolds.Definition 1.
LetM be a C
∞-manifold of real dimension 2
n
-
l. Then a C
R structure on M is given by a complex subbundle T
1,0
(
M
) of the complexified tangent bundle C
T
(
M
)
=
T
(
M
)
⊗
C such that the following conditions are satisfied. (1)
dim
⁡
C
T
z
1,0
(
M
)
=
n
-
l, where T
z
1,0
(
M
) is the fiber at each z
∈
M.
(2)
If we defineT
0,1
(
M
)
=
T
1,0
(
M
)
¯, then T
1,0
(
M
)
∩
T
0,1
(
M
)
=
{
0
}.
(3)
T
1,0
(
M
) is involutive (or formally integrable); that is, if L
1 and L
2 are two smooth sections of T
1,0
(
M
), defined on an open subset U of M, then so is their Lie bracket [
L
1
,
L
2
]
=
L
1
L
2
-
L
2
L
1, for every open subset U of M. A C
∞ manifold M endowed with this C
R structure is called a C
R manifold of C
R-dimension n
-
l and C
R codimension l.LetM be a generic C
R manifold of real dimension 2
n
-
l embedded in an n-dimensional complex manifold X. Such a manifold M can be represented locally in the following form: for each z
∈
M there exists an open neighborhood U of z in X such that
(1)
M
∩
U
=
{
ζ
∈
U
∣
ρ
1
(
ζ
)
=
⋯
=
ρ
l
(
ζ
)
=
0
}
,
where {
ρ
ν
}
ν
=
1
,
…
,
l are C
∞ real-valued functions on U such that
(2)
∂
-
ρ
1
(
ζ
)
∧
⋯
∧
∂
-
ρ
l
(
ζ
)
≠
0
on
⁡
M
∩
U
.The complex subbundle which defines the inducedC
R structure on M is given by T
1,0
(
M
)
=
T
1,0
(
X
)
∩
C
T
(
M
). Denote by C
0
,
s
∞
(
M
) the space of (
0
,
s
)-forms with C
∞-coefficients on M. The involution condition (3) of Definition 1 implies that there is a restriction of the de Rham exterior derivative d to C
0
,
s
∞
(
M
), which is defined by ∂
-
b
:
C
0
,
s
∞
(
M
)
→
C
0
,
s
+
1
∞
(
M
).Let us equipX with a Hermitian metric such that T
1,0
(
X
)
⊥
T
0,1
(
X
) and consider on M the induced metric, then T
1,0
(
M
)
⊥
T
0,1
(
M
). Let D
0
,
s
(
M
) be the space of (
0
,
s
)-forms whose coefficients are C
∞ with compact support in M. We then can define a Hermitian inner product on D
0
,
s
(
M
) by
(3)
(
φ
,
ψ
)
=
∫
M
‍
〈
φ
,
ψ
〉
z
d
v
,
where d
v is the volume element associated with the induced metric on M and 〈
φ
,
ψ
〉
z is the pointwise inner product induced on C
0
,
s
∞
(
M
) by the metric on C
T
(
M
) at each z
∈
M. Let ∥
φ
∥
2
=
(
φ
,
φ
) be the corresponding norm and L
0
,
s
2
(
M
) the L
2-completion of D
0
,
s
(
M
) with respect to this norm. Let ∂
-
b
:
L
0
,
s
2
(
M
)
→
L
0
,
s
+
1
2
(
M
) be the maximal closed extension of the original ∂
-
b on C
0
,
s
∞
(
M
). A form u
∈
L
0
,
s
2
(
M
) is in the domain of ∂
-
b if ∂
-
b
u, defined in the sense of distributions, belongs to L
0
,
s
+
1
2
(
M
). In this way, ∂
-
b defines a linear, closed, densely defined operator. Let ∂
-
b
*
:
L
0
,
s
+
1
2
(
M
)
→
L
0
,
s
2
(
M
) be the L
2-Hilbert space adjoint of ∂
-
b such that (
φ
,
∂
-
b
ψ
)
=
(
∂
-
b
*
φ
,
ψ
) for all ψ in Dom
⁡
(
∂
-
b
) and φ in Dom
⁡
(
∂
-
b
*
). The Kohn-Laplacian □
b is defined by
(4)
□
b
=
∂
-
b
∂
-
b
*
+
∂
-
b
*
∂
-
b
:
Dom
⁡
(
□
b
)
⟶
L
0
,
s
2
(
M
)
,
where
(5)
Dom
⁡
(
□
b
)
=
{
φ
∈
Dom
⁡
(
∂
-
b
)
∩
Dom
⁡
(
∂
-
b
⋆
)
⊂
L
0
,
s
2
(
M
)
∣
∂
-
b
φ
∈
Dom
⁡
(
∂
-
b
⋆
)
,
∂
-
b
⋆
φ
∈
Dom
⁡
(
∂
-
b
)
}
.We recall that the Kohn-Laplacian□
b is not elliptic, so it has a characteristic set of dimension l. Let N
(
M
) be the l-dimensional bundle such that
(6)
C
T
(
M
)
=
T
1,0
(
M
)
⊕
T
0,1
(
M
)
⊕
N
(
M
)
.
Let N
*
(
M
) be the dual bundle of N
(
M
). Let γ
∈
N
*
(
M
), then γ annihillates T
1,0
(
M
)
⊕
T
0,1
(
M
). Thus N
*
(
M
) is called the characteristic bundle. The Levi form of M at a point z
∈
M is defined as the Hermitian form on T
1,0
(
M
) with values in N
(
M
) such that
(7)
L
z
(
L
1
,
L
2
)
=
i
π
z
(
[
L
1
,
L
¯
2
]
z
)
,
L
1
,
L
2
∈
T
1,0
(
M
)
,
where π
z is the projection of C
T
z
(
M
) onto N
z
(
M
).The Levi form ofM at a point z
∈
M in the direction γ
∈
N
*
(
M
) is the scalar Hermitian form denoted L
z
(
γ
) and is given by
(8)
L
z
(
γ
)
=
〈
L
z
(
L
1
,
L
2
)
,
γ
〉
=
i
〈
[
L
1
,
L
¯
2
]
,
γ
〉
z
,
L
1
,
L
2
∈
T
1,0
(
M
)
.Definition 2 (see [10, Definition 1.2]).
AC
R manifold M of real dimension 2
n
-
l and codimension l
≥
1 in a complex manifold of complex dimension n is said to satisfy condition Z
(
s
), 1
≤
s
≤
n
-
l
-
1, at a point z
∈
M in the direction γ
∈
N
*
(
M
) if the Levi form L
z
(
γ
) has at least n
-
l
-
s
+
1 positive eigenvalues or at least s
+
1 negative eigenvalues. M is said to satisfy condition Y
(
s
) at z
∈
M if it satisfies condition Z
(
s
) for all directions γ
∈
N
z
*
(
M
).Note that in the hypersurface case, that is,l
=
1, the condition Y
(
s
) defined above is equivalent to the classical Y
(
s
) condition of Kohn for hypersurfaces (see, e.g., [11] for more details). In particular, if the C
R structure is strictly pseudoconvex; that is, the Levi form of M is positive or negative definite, condition Y
(
s
) holds for all 1
≤
s
≤
n
-
2.


## 2.L
2-Existence Theory for ∂
-
b

LetM be a C
∞ generic C
R manifold of real dimension 2
n
-
l and codimension l
≥
1 in a complex manifold X of complex dimension n. For each point p
0
∈
M, there is then a neighborhood U of p
0 in X and a local orthonormal basis consisting of smooth vector fields L
1
,
…
,
L
n
-
l for T
1,0
(
U
) (see, e.g., [12, Section 7.2; Theorem 3]). The collection of vector fields {
L
¯
1
,
…
,
L
¯
n
-
l
} forms a local orthonormal basis for T
0,1
(
U
). Let T
1
,
…
,
T
l be real vector fields on U such that the set {
L
1
,
…
,
L
n
-
l
,
L
¯
1
,
…
,
L
¯
n
-
l
,
T
1
,
…
,
T
l
} forms a local orthonormal basis for C
T
(
U
). Denote by {
ω
1
,
…
,
ω
n
-
l, ω
-
1
,
…
,
ω
-
n
-
l, γ
1
,
…
,
γ
l
} the basis for C
T
⋆
(
U
) dual to {
L
1
,
…
,
L
¯
n
-
l
,
T
1
,
…
,
T
l
}. In terms of this basis, an element φ in C
0
,
s
∞
(
U
) can be uniquely expressed as a sum:
(9)
φ
=
∑
|
I
|
=
s
‍
φ
I
ω
¯
I
,
where I
=
(
i
1
,
i
2
,
…
,
i
s
) is an s-tuple of integers with 1
≤
i
1
<
⋯
<
i
s
≤
n
-
l and ω
¯
I
=
ω
¯
i
1
∧
⋯
∧
ω
¯
i
s.We then have(10)
∂
-
b
φ
=
∑
|
I
|
=
s
‍
∑
j
=
1
n
-
l
‍
L
¯
j
(
φ
I
)
ω
¯
j
∧
ω
¯
I
+
⋯
=
∑
|
J
|
=
s
+
1
‍
(
∑
j
,
I
‍
ε
J
j
I
L
¯
j
(
φ
I
)
)
ω
¯
J
+
⋯
,
where ε
J
j
I is zero if j
∪
{
I
}
≠
J as sets and is the sign of the permutation that reorders j
I as J if j
∪
{
I
}
=
J, and the ⋯ stands for terms of order zero. Using integration by parts, we obtain
(11)
∂
-
b
*
φ
=
-
∑
|
I
|
=
s
‍
∑
j
=
1
n
-
l
‍
L
j
(
φ
j
I
)
ω
¯
I
+
⋯
=
-
∑
|
K
|
=
s
-
1
‍
(
∑
j
,
I
‍
ε
K
j
I
L
j
(
φ
I
)
)
ω
¯
K
+
⋯
.
For φ in C
0
,
s
∞
(
U
¯
), the subspace of smooth (
0
,
s
)-forms on U that can be extended smoothly up to and including the boundary, we set
(12)
∥
φ
∥
L
(
U
)
2
=
∑
j
=
1
n
-
l
‍
∥
L
j
(
φ
)
∥
2
+
∥
φ
∥
2
,
∥
φ
∥
L
¯
(
U
)
2
=
∑
j
=
1
n
-
l
‍
∥
L
¯
j
(
φ
)
∥
2
+
∥
φ
∥
2
.
If we further assume that M satisfies condition Y
(
s
) for some s with 1
≤
s
≤
n
-
l
-
1, for each p
0
∈
M, we can find a constant C
=
C
(
p
0
)
>
0 such that
(13)
∥
φ
∥
L
(
U
)
2
+
∥
φ
∥
L
¯
(
U
)
2
≤
C
(
∥
∂
-
b
φ
∥
2
+
∥
∂
-
b
*
φ
∥
2
+
∥
φ
∥
2
)
uniformly for all φ
∈
D
0
,
s
(
U
) (see, e.g., [10]).SetL
j
=
X
2
j
-
1
+
i
X
2
j; j
=
1
,
…
,
n
-
l. The condition Y
(
s
) implies that the real vector X
1
,
…
,
X
2
n
-
2
l and their commutators of length at most two span the tangent space at each point in U. Thus X
1
,
…
,
X
2
n
-
2
l satisfy Hörmander’s finite rank condition of order two. It follows then from [13, Theorem A] (see also [14]) that there is a positive constant C
=
C
(
U
) satisfying the following 1
/
2-subelliptic estimate:
(14)
∥
φ
∥
1
/
2
(
U
)
2
≤
C
(
∑
i
=
1
2
n
-
2
l
‍
∥
X
i
φ
∥
2
+
∥
φ
∥
2
)
,
φ
∈
D
0
,
s
(
U
)
.Here and always∥
·
∥
k
(
U
) denotes the L
2 Sobolev space k-norm, ∥
·
∥
-
k is the norm of its dual space, and ∥
·
∥ is the usual L
2-norm. We may omit the subscript U from the norm notation when there is no danger of confusion.Combining the above1
/
2-subelliptic estimate with (13), as in [10], we get the following theorem.Theorem 3.
LetM be a C
∞
C
R manifold of real dimension 2
n
-
l and codimension l
≥
1 in a complex manifold X of complex dimension n. Suppose that M satisfies condition Y
(
s
) for some s with 1
≤
s
≤
n
-
l
-
1. For each point p
0
∈
M, there is then an open neighborhood U on which the Kohn Laplacian □
b satisfies the 1
/
2-subelliptic estimate
(15)
∥
φ
∥
1
/
2
(
U
)
≤
C
(
∥
∂
-
b
φ
∥
2
+
∥
∂
-
b
*
φ
∥
2
+
∥
φ
∥
2
)
uniformly for all φ in D
0
,
s
(
U
).
In addition, ifM is compact, the estimate (15) holds uniformly on M for all φ in C
0
,
s
∞
(
M
).Theorem 4 (see [10]).
LetM be given as in Theorem 3 and ϕ the unique solution of the equation (
□
b
+
I
d
)
ϕ
=
f for f
∈
L
0
,
s
2
(
M
), where I
d is the identity operator. Let U
⊂
⊂
M be a relatively compact subset of M. If the restriction of f to U is in C
0
,
s
∞
(
U
), the restriction of ϕ to U is then in C
0
,
s
∞
(
U
). In addition, suppose that η and η
1 are two cut-off functions supported in U such that η
=
1 on the support of η
1; then if the restriction of f to U is in the L
2-Sobolev space W
0
,
s
k
(
U
) for some nonnegative integer k, the restriction of η
1
ϕ to U is in W
0
,
s
k
+
1
(
U
) and there is a constant C
k
>
0 (independent of f) such that
(16)
∥
η
1
ϕ
∥
k
+
1
(
U
)
≤
C
k
(
∥
η
f
∥
k
(
U
)
+
∥
f
∥
)
.Patching the above local estimates, we obtain the following global one.Theorem 5.
LetM be a C
∞ compact C
R manifold of real dimension 2
n
-
l and codimension l
≥
1 in an n-dimensional complex manifold X. Suppose that M satisfies condition Y
(
s
) for some s with 1
≤
s
≤
n
-
l
-
1. Let ϕ
∈
Dom
⁡
(
□
b
) such that (
□
b
+
I
d
)
ϕ
=
f for f in W
0
,
s
k
(
M
), k
≥
0, then ϕ is in W
0
,
s
k
+
1
(
M
) and there exists a constant C
k
>
0 (independent of f) such that
(17)
∥
ϕ
∥
k
+
1
(
M
)
≤
C
k
∥
f
∥
k
(
M
)
.Using Theorem5 and following an induction argument on k, we get the following result.Proposition 6.
LetM be given as in Theorem 5. Then the Kohn Laplacian □
b is hypoelliptic. Moreover, if □
b
ϕ
=
f for f in W
0
,
s
k
(
M
), k
≥
0, then ϕ is in W
0
,
s
k
+
1
(
M
) and there is a constant C
k
>
0 (independent of f) such that
(18)
∥
ϕ
∥
k
+
1
(
M
)
2
≤
C
k
(
∥
f
∥
k
(
M
)
2
+
∥
ϕ
∥
2
)
.Let(19)
H
0
,
s
b
(
M
)
=
{
α
∈
Dom
⁡
(
∂
-
b
)
∩
Dom
⁡
(
∂
-
b
⋆
)
⊂
L
0
,
s
2
(
M
)
∣
∂
-
b
α
=
∂
-
b
⋆
α
=
0
}
be the closed subspace of L
0
,
s
2
(
M
) consisting of harmonic forms and
(20)
⊥
H
0
,
s
b
(
M
)
=
{
α
∈
L
0
,
s
2
(
M
)
∣
(
α
,
ϕ
)
=
0
∀
ϕ
∈
H
0
,
s
b
(
M
)
}
.
The main L
2-result is the following theorem.Theorem 7.
LetM be a C
∞ compact C
R manifold of real dimension 2
n
-
l and codimension l
≥
1 in an n-dimensional complex manifold X. Suppose that M satisfies condition Y
(
s
) for some s such that 1
≤
s
≤
n
-
l
-
1. Then the following holds. (1)
The space of harmonic(
0
,
s
)-forms H
0
,
s
b
(
M
) is of finite dimensional.
(2)
The operators∂
-
b
:
L
0
,
s
2
(
M
)
→
L
0
,
s
+
1
2
(
M
), ∂
-
b
⋆
:
L
0
,
s
+
1
2
(
M
)
→
L
0
,
s
2
(
M
), and □
b
=
∂
-
b
∂
-
b
*
+
∂
-
b
*
∂
-
b
:
Dom
⁡
(
□
b
)
→
L
0
,
s
2
(
M
) have closed ranges.
(3)
The complex Green operatorG
b
:
L
0
,
s
2
(
M
)
→
Dom
⁡
(
□
b
) exists and is a compact operator in L
0
,
s
2
(
M
).
(4)
For anyf in L
0
,
s
2
(
M
), we have
(21)
f
=
∂
-
b
∂
-
b
⋆
G
b
f
+
∂
-
b
⋆
∂
-
b
G
b
f
+
H
0
,
s
b
f
,
whereH
0
,
s
b is the orthogonal projection of L
0
,
s
2
(
M
) onto H
0
,
s
b
(
M
).
(5)
G
b
H
0
,
s
b
=
H
0
,
s
b
G
b
=
0. G
b
□
b
=
□
b
G
b
=
I
d
-
H
0
,
s
b on Dom
⁡
(
□
b
).
(6)
IfG
b is defined on L
0
,
s
+
1
2
(
M
) (resp., L
0
,
s
-
1
2
(
M
)
), ∂
-
b
G
b
=
G
b
∂
-
b on Dom
⁡
(
∂
-
b
) (resp., ∂
-
b
⋆
G
b
=
G
b
∂
-
b
⋆ on Dom
⁡
(
∂
-
b
⋆
)
).
(7)
Iff is in L
0
,
s
2
(
M
) such that ∂
-
b
f
=
0 and f
⊥
H
0
,
s
b
(
M
), then f
=
∂
-
b
∂
-
b
⋆
G
b
f and u
=
∂
-
b
⋆
G
b
f is the unique solution to the equation ∂
-
b
u
=
f which is orthogonal to Ker
⁡
(
∂
-
b
) and satisfies ∥
u
∥
2
≤
C
∥
f
∥
2.
(8)
G
b
(
C
0
,
s
∞
(
M
)
)
⊆
C
0
,
s
∞
(
M
), and for each k
∈
R there is a positive constant C
s such that the estimate ∥
G
b
f
∥
k
+
1
≤
C
s
∥
f
∥
k holds uniformly for all f in C
0
,
s
∞
(
M
).Proof.
SinceM is compact, via a partition of unity, the estimate (15) holds globally on M. Suppose that f
k is a sequence in Dom
⁡
(
∂
-
b
)
∩
Dom
⁡
(
∂
-
b
*
)
∩
L
0
,
s
2
(
M
) such that ∥
f
k
∥ is bounded, ∂
-
b
f
k
→
0 in the L
0
,
s
+
1
2
(
M
)-norm and ∂
-
b
*
f
k
→
0 in the L
0
,
s
-
1
2
(
M
)-norm as k
→
∞. Thus, we have ∥
f
k
∥
1
/
2
(
M
)
≤
c for some constant c. By Rellich’s Lemma, the inclusion map i
M
:
W
0
,
s
1
/
2
(
M
)
→
L
0
,
s
2
(
M
) is compact; we can then extract a subsequence of f
k which converges in L
0
,
s
2
(
M
). Then the hypotheses of Theorem 1.1
.
3 in Hörmander [15] are satisfied which implies that H
0
,
s
b
(
M
) is finite dimensional and the estimate
(22)
∥
f
∥
2
≤
C
(
∥
∂
-
b
f
∥
2
+
∥
∂
-
b
*
f
∥
2
)
holds for every f in Dom
⁡
(
∂
-
b
)
∩
Dom
⁡
(
∂
-
b
*
) with f
⊥
H
0
,
s
b
(
M
).
By Theorem1.1
.
2 in [15], we then conclude that the operators ∂
-
b
:
L
0
,
s
2
(
M
)
→
L
0
,
s
+
1
2
(
M
) and ∂
-
b
⋆
:
L
0
,
s
2
(
M
)
→
L
0
,
s
-
1
2
(
M
) have closed ranges. We obtain also from (22) that
(23)
∥
f
∥
≤
C
∥
□
b
f
∥
,
f
∈
Dom
⁡
(
□
b
)
,
f
⊥
H
0
,
s
b
(
M
)
.
This estimate implies that □
b is one-to-one and in view of Theorem  1.1
.
1 in [15] that the range of □
b is closed. It forces, since □
b is self-adjoint, the strong Hodge decomposition:
(24)
L
0
,
s
2
(
M
)
=
Range
(
□
b
)
⊕
H
0
,
s
b
(
M
)
=
∂
-
b
∂
-
b
*
Dom
⁡
(
□
b
)
⊕
∂
-
b
*
∂
-
b
Dom
⁡
(
□
b
)
⊕
H
0
,
s
b
(
M
)
.
Thus □
b
:
Dom
⁡
(
□
b
)
→
⊥
H
0
,
s
b
(
M
) is one-to-one and onto. This implies the existence of the complex Green operator G
b
:
L
0
,
s
2
(
M
)
→
Dom
⁡
(
□
b
) as a unique operator that inverts □
b on ⊥
H
0
,
s
b
(
M
). The operator G
b is defined as follows: if f is in Range
(
□
b
), we define G
b
f
=
ϕ, where ϕ is the unique solution of □
b
ϕ
=
f with f
⊥
H
0
,
s
b
(
M
). G
b is extended to the whole L
0
,
s
2
(
M
) space by setting G
b
=
0 on H
0
,
s
b
(
M
). The boundedness of G
b in L
0
,
s
2
(
M
) follows from (23).
To show thatG
b is compact in L
0
,
s
2
(
M
), it suffices to show compactness on ⊥
H
0
,
s
b
(
M
) (since G
b
≡
0 on H
0
,
s
b
(
M
)). When f
⊥
H
0
,
s
b
(
M
) (and hence G
b
f
⊥
H
0
,
s
b
(
M
)), the integration by parts, Cauchy-Schwarz inequality (|
(
u
,
v
)
|
≤
∥
u
∥
∥
v
∥), and (23) imply
(25)
∥
∂
-
b
G
b
f
∥
2
+
∥
∂
-
b
*
G
b
f
∥
2
=
(
∂
-
b
G
b
f
,
∂
-
b
G
b
f
)
+
(
∂
-
b
*
G
b
f
,
∂
-
b
*
G
b
f
)
=
(
∂
-
b
*
∂
-
b
G
b
f
,
G
b
f
)
+
(
∂
-
b
∂
-
b
*
G
b
f
,
G
b
f
)
=
(
f
,
G
b
f
)
≤
∥
f
∥
∥
G
b
f
∥
≤
C
∥
f
∥
2
.
By applying (15) to G
b
f and using (23), we get
(26)
∥
G
b
f
∥
1
/
2
(
M
)
2
≤
C
(
∥
∂
-
b
G
b
f
∥
2
+
∥
∂
-
b
*
G
b
f
∥
2
+
∥
G
b
f
∥
2
)
≤
K
∥
f
∥
2
,
where K is a positive constant. Thus the compactness of G
b in L
0
,
s
2
(
M
) follows from Rellich’s Lemma.
The assertions in (5) follow immediately from the definition ofG
b. For assertion (
6
), if f
∈
Dom
⁡
(
∂
-
b
) and G
b is also defined on L
0
,
s
+
1
2
(
M
), by (21) and the first assertion of (5), we have
(27)
G
b
∂
-
b
f
=
G
b
∂
-
b
∂
-
b
*
∂
-
b
G
b
f
=
G
b
(
∂
-
b
∂
-
b
⋆
+
∂
-
b
⋆
∂
-
b
)
∂
-
b
G
b
f
=
G
b
□
b
∂
-
b
G
b
f
=
∂
-
b
G
b
f
.
A similar equation holds for ∂
-
b
⋆. Assertions (1)–(6) have been established.
To show assertion(
7
), if f
⊥
H
0
,
s
b
(
M
) and ∂
-
b
f
=
0, then ∂
-
b
∂
-
b
⋆
∂
-
b
G
b
f
=
0 as well (from (21)). Consequently, ∥
∂
-
b
⋆
∂
-
b
G
b
f
∥
2
=
(
∂
-
b
∂
-
b
⋆
∂
-
b
G
b
f
,
∂
-
b
G
b
f
)
=
0, since ∂
-
b
G
b
f
∈
Dom
⁡
(
∂
-
b
⋆
), and hence ∂
-
b
⋆
∂
-
b
G
b
f
=
0. Thus f
=
∂
-
b
(
∂
-
b
⋆
G
b
f
) and u
=
∂
-
b
⋆
G
b
f is orthogonal to Ker
⁡
(
∂
-
b
). Following assertion (
3
) and the fact that G
b is bounded, u satisfies the following L
2-estimate:
(28)
∥
u
∥
2
=
∥
∂
-
b
⋆
G
b
f
∥
2
=
(
∂
-
b
⋆
G
b
f
,
∂
-
b
⋆
G
b
f
)
=
(
∂
-
b
∂
-
b
⋆
G
b
f
,
G
b
f
)
=
(
(
∂
-
b
∂
-
b
⋆
+
∂
-
b
⋆
∂
-
b
)
G
b
f
,
G
b
f
)
=
(
f
,
G
b
f
)
≤
∥
f
∥
∥
G
b
f
∥
≤
C
∥
f
∥
2
.
Finally, we show assertion (
8
); if f
∈
C
0
,
s
∞
(
M
), then f
-
H
0
,
s
b
f
∈
C
0
,
s
∞
(
M
) and, since M is compact, f
∈
Dom
⁡
(
□
b
). On other hand, from assertion (5), □
b
G
b
f
=
f
-
H
0
,
s
b
f. Since □
b is hypoelliptic, by Proposition 6, G
b
f
∈
C
0
,
s
∞
(
M
).
Again Proposition6 implies
(29)
∥
G
b
f
∥
k
+
1
(
M
)
≤
C
k
(
∥
□
b
G
b
f
∥
k
(
M
)
+
∥
G
b
f
∥
)
≤
C
k
(
∥
f
∥
k
(
M
)
+
∥
H
0
,
s
b
f
∥
k
(
M
)
+
(
const
.
)
∥
f
∥
)
≤
C
∥
f
∥
k
(
M
)
.
Here we have used the fact that H
0
,
s
b
(
M
) is of finite dimension to conclude the estimate
(30)
∥
H
0
,
s
b
f
∥
k
(
M
)
≤
C
k
∥
H
0
,
s
b
f
∥
≤
C
k
∥
f
∥
k
(
M
)
for some constant C
k. The theorem is proved.


## 3. Sobolev Space Estimates

In this section, we prove that the complex Green operatorG
b, the canonical solution operators ∂
-
b
G
b and ∂
-
b
*
G
b, and the Szegö projection S
s operators enjoy some regularity properties in the L
2-Sobolev spaces W
0
,
s
k
(
M
), k
≥
0, for some s with 1
≤
s
≤
n
-
l
-
1. Furthermore, we obtain a global regularity for the solutions of the ∂
-
b-equation. By the same way for bounded pseudoconvex domains, a differential operator is said to be exactly regular if it maps all L
2-Sobolev spaces W
0
,
s
k
(
M
) (k
≥
0) to themselves and globally regular if it maps the space C
0
,
s
∞
(
M
) continuously to itself.


### 3.1. Continuity of the Complex Green Operator

We prove first the continuity of the complex Green operatorG
b on W
0
,
s
k
(
M
), k
≥
0.Theorem 8.
LetM be a C
∞ compact C
R manifold of real dimension 2
n
-
l and codimension l
≥
1 in an n-dimensional complex manifold X. Suppose that M satisfies condition Y
(
s
) for some s with 1
≤
s
≤
n
-
l
-
1. Then the complex Green operator G
b is continuous on the Sobolev space W
0
,
s
k
(
M
), k
≥
0; that is, there is a constant C
=
C
(
k
)
>
0 such that
(31)
∥
G
b
f
∥
k
(
M
)
≤
C
∥
f
∥
k
(
M
)
,
f
∈
W
0
,
s
k
(
M
)
.Proof.
We consider the special case whenk
=
0,1
,
2,3
,
…. Indeed the general case is then derived by means of interpolation of linear operators. Since M is compact, it is easy to show that C
0
,
s
∞
(
M
) is a dense subspace in W
0
,
s
k
(
M
). Further, by Theorem 7 (8), we have G
b
f
∈
C
0
,
s
∞
(
M
) for f
∈
C
0
,
s
∞
(
M
). Thus it suffices to establish (31) for f
∈
C
0
,
s
∞
(
M
). For k
=
0, (31) follows from (23).
For eachk
≥
0, let Λ
k
(
ξ
) be a pseudodifferential operator of order k with symbol (
1
+
|
ξ
|
2
)
k
/
2. Let U be an open neighborhood of ζ in M and let η and η
1 be two cutoff functions with supports in U such that η
=
1 on supp η
1; then η
Λ
k
η
1
f
∈
D
0
,
s
(
U
) whenever f
∈
D
0
,
s
(
U
).
Recall that the compactness ofG
b in L
0
,
s
2
(
U
) is equivalent to the compactness estimate: for every ϵ
>
0 there is a constant C
(
ϵ
)
>
0 such that for every φ
∈
Dom
⁡
(
∂
-
b
)
∩
Dom
⁡
(
∂
-
b
*
)
(32)
∥
φ
∥
2
≤
ϵ
Q
b
(
φ
,
φ
)
+
C
(
ϵ
)
∥
φ
∥
-
1
(
U
)
2
,
where Q
b
(
φ
,
φ
)
=
(
∂
-
b
φ
,
∂
-
b
φ
)
+
(
∂
-
b
*
φ
,
∂
-
b
*
φ
). For this estimate and further results on the compactness of the complex Green operator see, e.g., [16–19].
Applying (32) for η
Λ
k
η
1
G
b
f, we obtain
(33)
∥
η
Λ
k
η
1
G
b
f
∥
2
≤
ϵ
Q
b
(
η
Λ
k
η
1
G
b
f
,
η
Λ
k
η
1
G
b
f
)
+
C
(
ϵ
)
∥
η
Λ
k
η
1
G
b
f
∥
-
1
(
U
)
2
.
We sometimes use A for η
Λ
k
η
1 and A
* for its formal adjoint, which is also a tangential operator of order k. We estimate the first term on the right hand side in (33), it is a standard consequence of [20, Corollary 3.1] (or [11, Lemma 2.4
.
2]) that
(34)
Q
b
(
A
G
b
f
,
A
G
b
f
)
=
Re
Q
b
(
G
b
f
,
A
*
A
G
b
f
)
+
O
(
|
|
|
D
G
b
f
|
|
|
k
-
1
(
U
)
2
)
≤
Re
Q
b
(
G
b
f
,
A
*
A
G
b
f
)
+
C
∥
G
b
f
∥
k
(
U
)
2
.
Here we have used the fact that the tangential derivative D
α of order |
α
|
=
λ satisfies the tangential Sobolev estimate |
|
|
D
α
f
|
|
|
r
≤
∥
f
∥
r
+
λ.
Takingv
=
A
*
A
f in the form Q
b
(
G
b
u
,
v
)
=
(
u
,
v
), we get
(35)
Q
b
(
A
G
b
f
,
A
G
b
f
)
≤
Re
(
f
,
A
*
A
G
b
f
)
+
C
∥
G
b
f
∥
k
(
U
)
2
≤
|
(
f
,
A
*
A
G
b
f
)
|
+
C
∥
G
b
f
∥
k
(
U
)
2
.
The Cauchy-Schwarz inequality implies
(36)
Q
b
(
A
G
b
f
,
A
G
b
f
)
≤
∥
A
f
∥
∥
A
G
b
f
∥
+
C
∥
G
b
f
∥
k
(
U
)
2
.
Inequality (33) becomes
(37)
∥
η
Λ
k
η
1
G
b
f
∥
2
≤
ϵ
∥
f
∥
k
(
U
)
∥
G
b
f
∥
k
(
U
)
+
C
(
ϵ
)
∥
η
Λ
k
η
1
G
b
f
∥
-
1
(
U
)
2
.
Summing over a partition of unity subordinate to an open covering of M by patches {
U
i
}
i
=
1
m, we obtain estimate like (37) on each of these patches and using the interior regularity properties, we get
(38)
∥
G
b
f
∥
k
(
M
)
2
≤
ϵ
∥
f
∥
k
(
M
)
∥
G
b
f
∥
k
(
M
)
+
C
(
ϵ
)
∥
G
b
f
∥
k
-
1
(
M
)
2
.
The first term in the right-hand side of (38) is estimated by ϵ
(
s
.
c
.
)
∥
G
b
f
∥
k
(
M
)
2
+
ϵ
(
l
.
c
.
)
∥
f
∥
k
(
M
)
2, where s
.
c. and l
.
c. denote a small and a large constants, respectively, in the inequality |
a
b
|
≤
(
s
.
c
.
)
a
2
+
(
l
.
c
.
)
b
2. The second term is estimated by interpolation of Sobolev norms (∥
G
b
f
∥
k
-
1
(
M
)
2
≤
ε
∥
G
b
f
∥
k
(
M
)
2
+
C
(
ε
)
∥
G
b
f
∥
2) and then by using the continuity of G
b in L
0
,
s
2
(
M
) with L
2-bounded norm.
Adding up the analogues terms and absorbing, by choosingϵ and ε to be small enough, ∥
G
b
f
∥
k
(
M
)
2 into the left, this gives
(39)
∥
G
b
f
∥
k
(
M
)
2
≤
C
∥
f
∥
k
(
M
)
2
+
K
∥
f
∥
2
,
where C
=
C
(
ϵ
,
k
)
>
0 and K
=
K
(
ϵ
,
k
)
>
0. The embedding Sobolev space implies (31) for k
=
0,1
,
2,3
,
…. The general case is obtained from interpolation of linear operators. As mentioned above, the density of C
0
,
s
∞
(
M
) in W
0
,
s
k
(
M
) passes (31) to forms f in W
0
,
s
k
(
M
). This proves the continuity of G
b in W
0
,
s
k
(
M
).Corollary 9.
LetM be given as in Theorem 8, then the canonical solution operators ∂
-
b
G
b and ∂
-
b
*
G
b are continuous on W
0
,
s
k
(
M
) for all k
≥
0.Proof.
We argue by induction onk. The case when k
=
0 follows from (25). Suppose that the assertions hold for positive integers less than k and assume that ζ, U, η, and η
1 are given as in the proof of Theorem 8. By the interior elliptic regularity properties, we prove first a priori estimate for ∂
-
b
G
b
f and ∂
-
b
*
G
b
f with f
∈
D
0
,
s
(
U
) as follows:
(40)
∥
η
Λ
k
η
1
∂
-
b
G
b
f
∥
2
+
∥
η
Λ
k
η
1
∂
-
b
*
G
b
f
∥
2
=
(
η
Λ
k
η
1
∂
-
b
G
b
f
,
∂
-
b
η
Λ
k
η
1
G
b
f
)
+
(
η
Λ
k
η
1
∂
-
b
*
G
b
f
,
∂
-
b
*
η
Λ
k
η
1
G
b
f
)
+
O
(
(
∥
η
Λ
k
η
1
∂
-
b
G
b
f
∥
+
∥
η
Λ
k
η
1
∂
-
b
*
G
b
f
∥
)
∥
G
b
f
∥
k
(
U
)
)
=
(
η
Λ
k
η
1
∂
-
b
*
∂
-
b
G
b
f
,
η
Λ
k
η
1
G
b
f
)
+
(
η
Λ
k
η
1
∂
-
b
∂
-
b
*
G
b
f
,
η
Λ
k
η
1
G
b
f
)
+
O
(
(
∥
η
Λ
k
η
1
∂
-
b
G
b
f
∥
+
∥
η
Λ
k
η
1
∂
-
b
*
G
b
f
∥
)
∥
G
b
f
∥
k
(
U
)
+
∥
G
b
f
∥
k
(
U
)
2
(
∥
η
Λ
k
η
1
∂
-
b
G
b
f
∥
+
∥
η
Λ
k
η
1
∂
-
b
*
G
b
f
∥
)
)
=
(
η
Λ
k
η
1
□
b
G
b
f
,
η
Λ
k
η
1
G
b
f
)
+
O
(
(
∥
η
Λ
k
η
1
∂
-
b
G
b
f
∥
+
∥
η
Λ
k
η
1
∂
-
b
*
G
b
f
∥
)
∥
G
b
f
∥
k
(
U
)
+
∥
G
b
f
∥
k
(
U
)
2
(
∥
η
Λ
k
η
1
∂
-
b
G
b
f
∥
+
∥
η
Λ
k
η
1
∂
-
b
*
G
b
f
∥
)
∥
G
b
f
∥
k
(
U
)
)
≤
C
1
∥
f
∥
k
(
U
)
∥
G
b
f
∥
k
(
U
)
+
C
2
(
(
∥
η
Λ
k
η
1
∂
-
b
G
b
f
∥
+
∥
η
Λ
k
η
1
∂
-
b
*
G
b
f
∥
)
∥
G
b
f
∥
k
(
U
)
+
∥
G
b
f
∥
k
(
U
)
2
(
∥
η
Λ
k
η
1
∂
-
b
G
b
f
∥
+
∥
η
Λ
k
η
1
∂
-
b
*
G
b
f
∥
)
)
.
Summing over a partition of unity, using the small and large constants for the resulting terms ∥
f
∥
k
∥
G
b
f
∥
k, ∥
∂
-
b
G
b
f
∥
k
∥
G
b
f
∥
k, and ∥
∂
-
b
*
G
b
f
∥
k
∥
G
b
f
∥
k, using (31) and adding up the analogues terms, we see that the terms on the right-hand side containing ∥
∂
-
b
G
b
f
∥
k
2 and ∥
∂
-
b
*
G
b
f
∥
k
2 can be absorbed into the left hand side. We therefore obtain
(41)
∥
∂
-
b
G
b
f
∥
k
(
M
)
2
+
∥
∂
-
b
*
G
b
f
∥
k
(
M
)
2
≤
C
∥
f
∥
k
(
M
)
2
,
f
∈
D
0
,
s
(
M
)
.
This completes the induction on k for the norms of ∂
-
b
G
b and ∂
-
b
*
G
b. By the density of C
0
,
s
∞
(
M
) in W
0
,
s
k
(
M
), the estimates extend to forms in W
0
,
s
k
(
M
). As before, the general case is obtained from interpolation of linear operators. Then ∂
-
b
G
b and ∂
-
b
*
G
b are continuous on W
0
,
s
k
(
M
).


### 3.2. Exact and Global Regularity Theorems

We now show the expression of the complex Green operator by Szegö projections.Theorem 10.
The Szegö projectionsS
s
:
L
0
,
s
2
(
M
)
→
Ker
⁡
(
∂
-
b
) are given by the following relations:
(42)
S
s
=
I
d
-
∂
-
b
*
∂
-
b
G
b
=
I
d
-
G
b
∂
-
b
*
∂
-
b
,
s
≥
0
,
(43)
S
s
-
1
=
I
d
-
∂
-
b
*
G
b
∂
-
b
,
s
≥
1
.Proof.
We first show that∂
-
b
*
∂
-
b
G
b
=
G
b
∂
-
b
*
∂
-
b. For α
,
β
∈
⊥
H
0
,
s
b
(
M
), we observe that
(44)
∂
-
b
α
=
0
⟹
∂
-
b
*
∂
-
b
G
b
α
=
0
⟹
α
=
∂
-
b
∂
-
b
*
G
b
α
=
G
b
∂
-
b
∂
-
b
*
α
,
(45)
∂
-
b
⋆
β
=
0
⟹
∂
-
b
∂
-
b
*
G
b
β
=
0
⟹
β
=
∂
-
b
*
∂
-
b
G
b
β
=
G
b
∂
-
b
*
∂
-
b
β
.
As Range
(
∂
-
b
)
⊥
Ker
⁡
(
∂
-
b
*
) and Range
(
∂
-
b
*
)
⊥
Ker
⁡
(
∂
-
b
), one has
(46)
∂
-
b
α
=
0
⟹
∂
-
b
G
b
α
=
0
,
(47)
∂
-
b
*
β
=
0
⟹
∂
-
b
*
G
b
β
=
0
.
Anyf
⊥
H
0
,
s
b
(
M
) can then be written as f
=
α
+
β so that ∂
-
b
α
=
0 and ∂
-
b
*
β
=
0. By (45) and (46), we then have
(48)
∂
-
b
⋆
∂
-
b
G
b
f
=
∂
-
b
⋆
∂
-
b
G
b
(
α
+
β
)
=
∂
-
b
⋆
∂
-
b
G
b
β
=
G
b
∂
-
b
⋆
∂
-
b
β
=
G
b
∂
-
b
⋆
∂
-
b
f
.
This implies the second equality in (42). Now, If f
∈
Ker
⁡
(
∂
-
b
), then (
I
d
-
G
b
∂
-
b
*
∂
-
b
)
f
=
f, so the expression for S
s holds. Next, if f
⊥
Ker
⁡
(
∂
-
b
) and hence f
⊥
H
0
,
s
b
(
M
), so f
=
∂
-
b
∂
-
b
⋆
G
b
f
+
∂
-
b
⋆
∂
-
b
G
b
f and u
=
∂
-
b
⋆
∂
-
b
G
b
f is the canonical solution to the equation ∂
-
b
u
=
∂
-
b
f. Thus ∂
-
b
(
f
-
u
)
=
0, that is, f
-
u
∈
Ker
⁡
(
∂
-
b
). We claim that u
⊥
Ker
⁡
(
∂
-
b
). Indeed, for all g
∈
Ker
⁡
(
∂
-
b
) one has (
u
,
g
)
=
(
∂
-
b
⋆
∂
-
b
G
b
f
,
g
)
=
(
∂
-
b
G
b
f
,
∂
-
b
g
)
=
0. Since f
⊥
Ker
⁡
(
∂
-
b
), it turns out that f
-
u
⊥
Ker
⁡
(
∂
-
b
) so f
-
u
=
0 and then 0
=
f
-
u
=
(
I
d
-
∂
-
b
⋆
∂
-
b
G
b
f
). This proves (42). Similarly, we get (43).Theorem 11.
LetM be given as in Theorem 8. Then the Szegö projections operators S
s
-
1 and S
s are continuous in the Sobolev spaces W
0
,
s
-
1
k
(
M
) and W
0
,
s
k
(
M
) for all k
≥
0, respectively.Proof.
We investigate first the continuity ofS
s
-
1. For the case k
=
0, when f
∈
L
0
,
s
2
(
M
), we have
(49)
∥
∂
-
b
*
G
b
∂
-
b
f
∥
2
=
(
∂
-
b
*
G
b
∂
-
b
f
,
∂
-
b
*
G
b
∂
-
b
f
)
=
(
G
b
∂
-
b
f
,
∂
-
b
∂
-
b
*
G
b
∂
-
b
f
)
=
(
G
b
∂
-
b
f
,
∂
-
b
f
)
=
(
∂
-
b
*
G
b
∂
-
b
f
,
f
)
≤
∥
∂
-
b
*
G
b
∂
-
b
f
∥
∥
f
∥
.
Here we have used the fact that ∂
-
b
∂
-
b
*
G
b
∂
-
b
f
=
∂
-
b
f, because ∂
-
b
2
=
0. The relation (43) thus implies that ∥
S
s
-
1
f
∥
≤
C
∥
f
∥. This proves the continuity in L
0
,
s
-
1
2
(
M
).
The casek
≥
1. Applying (32) for φ
=
η
Λ
k
η
1
G
s
∂
-
b
f on U, we obtain
(50)
∥
η
Λ
k
η
1
G
b
∂
-
b
f
∥
2
≤
ϵ
Q
b
(
η
Λ
k
η
1
G
b
∂
-
b
f
,
η
Λ
k
η
1
G
b
∂
-
b
f
)
+
C
(
ϵ
)
∥
η
Λ
k
η
1
G
b
∂
-
b
f
∥
-
1
(
U
)
2
.
The first term on the right-hand side of (50) is estimated as
(51)
Q
b
(
A
G
b
∂
-
b
f
,
A
G
b
∂
-
b
f
)
=
∥
∂
-
b
A
G
b
∂
-
b
f
∥
2
+
∥
∂
-
*
A
G
b
∂
-
b
f
∥
2
=
(
∂
-
b
A
G
b
∂
-
b
f
,
∂
-
b
A
G
b
∂
-
b
f
)
+
(
∂
-
b
*
A
G
b
∂
-
b
f
,
∂
-
b
*
A
G
b
∂
-
b
f
)
=
(
A
∂
-
b
G
b
∂
-
b
f
,
∂
-
b
A
G
b
∂
-
b
f
)
+
(
A
∂
-
b
*
G
b
∂
-
b
f
,
∂
-
b
*
A
G
b
∂
-
b
f
)
+
(
[
∂
-
b
,
A
]
G
b
∂
-
b
f
,
∂
-
b
A
G
b
∂
-
b
f
)
+
(
[
∂
-
b
*
,
A
]
G
b
∂
-
b
f
,
∂
-
b
*
A
G
b
∂
-
b
f
)
.
The sum of the last two terms on the right-hand side of the preceding equality is estimated by
(52)
∥
[
∂
-
b
,
A
]
G
b
∂
-
b
f
∥
∥
∂
-
b
A
G
b
∂
-
b
f
∥
+
∥
[
∂
-
b
*
,
A
]
G
b
∂
-
b
f
∥
∥
∂
-
b
*
A
G
b
∂
-
b
f
∥
≤
∥
D
G
b
∂
-
b
f
∥
k
-
1
(
U
)
∥
∂
-
b
A
G
b
∂
-
b
f
∥
+
∥
D
G
b
∂
-
b
f
∥
k
-
1
(
U
)
∥
∂
-
b
*
A
G
s
∂
-
b
f
∥
≤
∥
G
b
∂
-
b
f
∥
k
(
U
)
(
∥
∂
-
b
A
G
b
∂
-
b
f
∥
+
∥
∂
-
b
*
A
G
b
∂
-
b
f
∥
)
=
O
(
(
l
.
c
.
)
∥
G
b
∂
-
b
f
∥
k
(
U
)
2
+
(
s
.
c
.
)
(
∥
∂
-
b
A
G
b
∂
-
b
f
∥
+
∥
∂
-
b
*
A
G
b
∂
-
b
f
∥
)
2
∥
G
b
∂
-
b
f
∥
k
(
U
)
2
)
=
O
(
∥
G
b
∂
-
b
f
∥
k
(
U
)
2
)
.
We then have(53)
Q
b
(
A
G
b
∂
-
b
f
,
A
G
b
∂
-
b
f
)
≤
(
∂
-
b
G
b
∂
-
b
f
,
A
*
∂
-
b
A
G
b
∂
-
b
f
)
+
(
A
∂
-
b
*
G
b
∂
-
b
f
,
∂
-
b
*
A
G
b
∂
-
b
f
)
+
O
(
∥
G
b
∂
-
b
f
∥
k
(
U
)
2
)
.
The first term on the right-hand side of (53) equals zero due to the fact that ∂
-
b
G
b
∂
-
b
f
=
∂
-
b
2
G
b
f
=
0.
We now analyze the second term as follows:(54)
(
A
∂
-
b
*
G
b
∂
-
b
f
,
∂
-
b
*
A
G
b
∂
-
b
f
)
=
(
∂
-
b
A
∂
-
b
*
G
b
∂
-
b
f
,
A
G
b
∂
-
b
f
)
=
(
A
∂
-
b
∂
-
b
*
G
b
∂
-
b
f
,
A
G
b
∂
-
b
f
)
+
(
[
∂
-
b
,
A
]
∂
-
b
*
G
b
∂
-
b
f
,
A
G
b
∂
-
b
f
)
=
(
A
∂
-
b
f
,
A
G
b
∂
-
b
f
)
+
(
[
∂
-
b
,
A
]
∂
-
b
*
G
b
∂
-
b
f
,
A
G
b
∂
-
b
f
)
=
(
∂
-
b
A
f
,
A
G
b
∂
-
b
f
)
+
(
[
A
,
∂
-
b
]
f
,
A
G
b
∂
-
b
f
)
+
(
[
∂
-
b
,
A
]
∂
-
b
*
G
b
∂
-
b
f
,
A
G
b
∂
-
b
f
)
=
(
A
f
,
∂
-
b
*
A
G
b
∂
-
b
f
)
+
⋯
=
(
A
f
,
A
∂
-
b
*
G
b
∂
-
b
f
)
+
(
A
f
,
[
∂
-
b
*
,
A
]
G
b
∂
-
b
f
)
+
(
[
A
,
∂
-
b
]
f
,
A
G
b
∂
-
b
f
)
+
(
[
∂
-
b
,
A
]
∂
-
b
*
G
b
∂
-
b
f
,
A
G
b
∂
-
b
f
)
.
Thus
(55)
Q
b
(
A
G
b
∂
-
b
f
,
A
G
b
∂
-
b
f
)
≤
(
A
f
,
A
∂
-
b
*
G
b
∂
-
b
f
)
+
E
+
O
(
∥
G
b
∂
-
f
∥
k
(
U
)
2
)
≤
|
(
A
f
,
A
∂
-
b
*
G
b
∂
-
b
f
)
|
+
|
E
|
+
O
(
∥
G
b
∂
-
b
f
∥
k
(
U
)
2
)
,
where
(56)
E
=
(
A
f
,
[
∂
-
b
*
,
A
]
G
b
∂
-
b
f
)
+
(
[
A
,
∂
-
b
]
f
,
A
G
b
∂
-
b
f
)
+
(
[
∂
-
b
,
A
]
∂
-
b
*
G
b
∂
-
b
f
,
A
G
b
∂
-
b
f
)
.
As above, the three terms on the right-hand side of (56) are estimated, respectively, by
(57)
∥
A
f
∥
∥
[
∂
-
b
*
,
A
]
G
b
∂
-
b
f
∥
≤
∥
f
∥
k
(
U
)
∥
G
b
∂
-
b
f
∥
k
(
U
)
≤
(
s
.
c
.
)
∥
f
∥
k
(
U
)
2
+
(
l
.
c
.
)
∥
G
b
∂
-
b
f
∥
k
(
U
)
2
,
∥
f
∥
k
(
U
)
∥
A
G
b
∂
-
b
f
∥
≤
(
s
.
c
.
)
∥
f
∥
k
(
U
)
2
+
(
l
.
c
.
)
∥
A
G
b
∂
-
b
f
∥
2
=
O
(
∥
G
b
∂
-
b
f
∥
k
(
U
)
2
)
,
∥
[
∂
-
b
,
A
]
∂
-
b
*
G
b
∂
-
b
f
∥
∥
A
G
b
∂
-
b
f
∥
≤
(
s
.
c
.
)
∥
∂
-
b
*
G
b
∂
-
b
f
∥
k
(
U
)
2
+
(
l
.
c
.
)
∥
A
G
b
∂
-
b
f
∥
2
.
Now we are left with the first term in the right-hand side of (55) which, by applying the Cauchy-Schwarz inequality, is estimated by ∥
f
∥
k
(
U
)
∥
∂
-
b
*
G
b
∂
-
b
f
∥
k
(
U
). By choosing the s
.
c. small enough we can absorb the first term in the right-hand side of the last inequality into ∥
f
∥
k
(
U
)
∥
∂
-
b
*
G
b
∂
-
b
f
∥
k
(
U
). This completes the estimation of the first term on the right-hand side of (50). Therefore (50) becomes
(58)
∥
η
Λ
k
η
1
G
b
∂
-
b
f
∥
2
≤
ϵ
∥
f
∥
k
(
U
)
∥
∂
-
b
*
G
b
∂
-
b
f
∥
k
(
U
)
+
ϵ
(
s
.
c
.
)
∥
f
∥
k
(
U
)
2
+
ϵ
C
∥
G
b
∂
-
b
f
∥
k
(
U
)
2
+
C
(
ϵ
)
∥
η
Λ
k
η
1
G
b
∂
-
b
f
∥
-
1
(
U
)
2
≤
ϵ
∥
f
∥
k
(
U
)
∥
∂
-
b
*
G
b
∂
-
b
f
∥
k
(
U
)
+
ϵ
(
s
.
c
.
)
∥
f
∥
k
(
U
)
2
+
ϵ
C
∥
G
b
∂
-
b
f
∥
k
(
U
)
2
+
C
′
(
ϵ
)
∥
G
b
∂
-
b
f
∥
k
-
1
(
U
)
2
.
By summing over a partition of unity subordinate to an open covering of M by patches {
U
i
}
i
=
1
m so that on each of these patches an estimate like (58) is satisfied, using the interior regularity properties, we get
(59)
∥
G
b
∂
-
b
f
∥
k
(
M
)
2
≤
ϵ
∥
f
∥
k
(
M
)
∥
∂
-
b
*
G
b
∂
-
b
f
∥
k
(
M
)
+
ϵ
s
.
c
.
∥
f
∥
k
(
M
)
2
+
ϵ
C
∥
G
b
∂
-
b
f
∥
k
(
M
)
+
C
′
(
ϵ
)
∥
G
b
∂
-
b
f
∥
k
-
1
(
M
)
2
.
By using the small and large constants, the first term on the right-hand side in (59) is estimated as
(60)
ϵ
(
(
s
.
c
.
)
∥
f
∥
k
(
M
)
2
+
(
l
.
c
.
)
∥
∂
-
b
*
G
b
∂
-
b
f
∥
k
(
M
)
2
)
.
Then adding and choosing ϵ and the s
.
c. small enough we can absorb the third term on the right-hand side of (59) into the left-hand side; we obtain
(61)
∥
G
b
∂
-
b
f
∥
k
(
M
)
2
≤
ϵ
C
∥
∂
-
b
*
G
b
∂
-
b
f
∥
k
(
M
)
2
+
C
′
(
ϵ
)
(
∥
f
∥
k
(
M
)
2
+
∥
G
b
∂
-
b
f
∥
k
-
1
(
M
)
2
)
.
Applying this inequality with k replaced by k
-
1 to the last term on the right-hand side and repeating, we obtain
(62)
∥
G
b
∂
-
b
f
∥
k
(
M
)
2
≤
ϵ
C
∥
∂
-
b
*
G
b
∂
-
b
f
∥
k
(
M
)
2
+
C
′
(
ϵ
)
(
∥
f
∥
k
(
M
)
2
+
∥
G
b
∂
-
b
f
∥
2
)
.
We have
(63)
∥
η
Λ
k
η
1
∂
-
b
*
G
b
∂
-
b
f
∥
2
=
(
η
Λ
k
η
1
∂
-
b
*
G
b
∂
-
b
f
,
η
Λ
k
η
1
∂
-
b
*
G
b
∂
-
b
f
)
=
(
∂
-
b
*
η
Λ
k
η
1
G
b
∂
-
b
f
,
η
Λ
k
η
1
∂
-
b
*
G
b
∂
-
b
f
)
+
O
(
∥
G
b
∂
-
b
f
∥
k
(
U
)
∥
η
Λ
k
η
1
∂
-
b
*
G
b
∂
-
b
f
∥
)
=
(
η
Λ
k
η
1
G
b
∂
-
b
f
,
η
Λ
k
η
1
∂
-
b
∂
-
b
*
G
b
∂
-
b
f
)
+
O
(
∥
G
b
∂
-
b
f
∥
k
(
U
)
∥
η
Λ
k
η
1
∂
-
b
*
G
b
∂
-
b
f
∥
)
=
(
η
Λ
k
η
1
G
b
∂
-
b
f
,
η
Λ
k
η
1
∂
-
b
f
)
+
O
(
∥
G
b
∂
-
b
f
∥
k
(
U
)
∥
η
Λ
k
η
1
∂
-
b
*
G
b
∂
-
b
f
∥
)
=
(
η
Λ
k
η
1
G
b
∂
-
b
f
,
∂
-
b
η
Λ
k
η
1
f
)
+
O
(
∥
G
b
∂
-
b
f
∥
k
(
U
)
(
∥
f
∥
k
(
U
)
+
∥
η
Λ
k
η
1
∂
-
b
*
G
b
∂
-
b
f
∥
)
)
=
(
∂
-
b
*
η
Λ
k
η
1
G
b
∂
-
b
f
,
η
Λ
k
η
1
f
)
+
O
(
∥
G
b
∂
-
b
f
∥
k
(
U
)
(
∥
f
∥
k
(
U
)
+
∥
η
Λ
k
η
1
∂
-
b
*
G
b
∂
-
b
f
∥
)
)
=
(
η
Λ
k
η
1
∂
-
b
*
G
b
∂
-
b
f
,
η
Λ
k
η
1
f
)
+
O
(
∥
G
b
∂
-
b
f
∥
k
(
U
)
(
∥
f
∥
k
(
U
)
+
∥
η
Λ
k
η
1
∂
-
b
*
G
b
∂
-
b
f
∥
)
)
≤
∥
η
Λ
k
η
1
∂
-
b
*
G
b
∂
-
b
f
∥
∥
η
Λ
k
η
1
f
∥
+
O
(
∥
G
b
∂
-
b
f
∥
k
(
U
)
(
∥
f
∥
k
(
U
)
+
∥
η
Λ
k
η
1
∂
-
b
*
G
b
∂
-
b
f
∥
)
)
.
Again summing over a partition of unity, using the interior regularity properties and the small and large constants technique, we obtain
(64)
∥
∂
-
b
*
G
b
∂
-
b
f
∥
k
(
M
)
2
≤
C
(
∥
G
b
∂
-
b
f
∥
k
(
M
)
2
+
∥
f
∥
k
(
M
)
2
)
.
Substituting (62) into (64), we obtain
(65)
∥
∂
-
b
*
G
b
∂
-
b
f
∥
k
(
M
)
2
≤
K
ϵ
∥
∂
-
b
*
G
b
∂
-
b
f
∥
k
(
M
)
2
+
C
′
(
ϵ
)
(
∥
f
∥
k
(
M
)
2
+
∥
G
b
∂
-
b
f
∥
2
)
.
Choosing ϵ
>
0 small enough allows us to absorb the first term on the right-hand side into the left, we then get
(66)
∥
∂
-
b
*
G
b
∂
-
b
f
∥
k
(
M
)
2
≤
C
′
(
ϵ
)
(
∥
f
∥
k
(
M
)
2
+
∥
G
b
∂
-
b
f
∥
2
)
.
As the operator ∂
-
b
* has L
2
(
M
)-closed range, it follows from Theorem 1.1
.
1 in Hörmander [15] that there is a positive constant C such that
(67)
∥
G
b
∂
-
b
f
∥
≤
C
∥
∂
-
b
*
G
b
∂
-
b
f
∥
.
Then, by (49), we obtain
(68)
∥
G
b
∂
-
b
f
∥
≤
C
∥
f
∥
.
Substituting (68) into (66), we get
(69)
∥
∂
-
b
*
G
b
∂
-
b
f
∥
k
(
M
)
2
≤
C
∥
f
∥
k
(
M
)
2
.
By (43), the Szegö projection S
s
-
1 is therefore continuous on W
0
,
s
-
1
k
(
M
) for each k
=
0,1
,
2
…. The general case is obtained from interpolation of linear operators.
For the continuity of the Szegö projectionS
s, in view of (42), it suffices to show that
(70)
∥
∂
-
b
*
∂
-
b
G
b
f
∥
k
(
M
)
2
≤
C
∥
f
∥
k
(
M
)
2
,
k
≥
0
.
For k
=
0, we have
(71)
∥
∂
-
b
*
∂
-
b
G
b
f
∥
2
=
(
∂
-
b
∂
-
b
*
∂
-
b
G
b
f
,
∂
-
b
G
b
f
)
=
(
∂
-
b
f
,
∂
-
b
G
b
f
)
=
(
f
,
∂
-
b
*
∂
-
b
G
b
f
)
≤
C
∥
f
∥
∥
∂
-
b
*
∂
-
b
G
b
f
∥
.
For k
≥
1, as before, an elliptic regularity argument implies
(72)
∥
η
Λ
k
η
1
∂
-
b
*
∂
-
b
G
b
f
∥
2
=
(
η
Λ
k
η
1
∂
-
b
*
∂
-
b
G
b
f
,
η
Λ
k
η
1
∂
-
b
*
∂
-
b
G
b
f
)
=
(
η
Λ
k
η
1
∂
-
b
∂
-
b
*
∂
-
b
G
b
f
,
η
Λ
k
η
1
∂
-
b
G
b
f
)
+
(
η
Λ
k
η
1
∂
-
b
*
∂
-
b
G
b
f
,
[
η
Λ
k
η
1
,
∂
-
b
*
]
∂
-
b
G
b
f
)
+
(
[
∂
-
b
,
η
Λ
k
η
1
]
∂
-
b
*
∂
-
b
G
b
f
,
η
Λ
k
η
1
∂
-
b
G
b
f
)
=
(
η
Λ
k
η
1
∂
-
b
f
,
η
Λ
k
η
1
∂
-
b
G
b
f
)
+
O
(
∥
η
Λ
k
η
1
∂
-
b
G
b
f
∥
∥
η
Λ
k
η
1
∂
-
b
*
∂
-
b
G
b
f
∥
)
=
(
η
Λ
k
η
1
f
,
η
Λ
k
η
1
∂
-
b
*
∂
-
b
G
b
f
)
+
O
(
∥
η
Λ
k
η
1
∂
-
b
G
b
f
∥
∥
η
Λ
k
η
1
∂
-
b
*
∂
-
b
G
b
f
∥
)
+
O
(
∥
f
∥
k
(
U
)
∥
η
Λ
k
η
1
∂
-
b
G
b
f
∥
)
≤
∥
η
Λ
k
η
1
f
∥
∥
η
Λ
k
η
1
∂
-
b
*
∂
-
b
G
b
f
∥
+
O
(
(
∥
f
∥
k
(
U
)
+
∥
η
Λ
k
η
1
∂
-
b
G
b
f
∥
)
∥
η
Λ
k
η
1
∂
-
b
*
∂
-
b
G
b
f
∥
)
.
Summing over a partition of unity, using the small and large constants argument, absorbing the terms containing ∥
∂
-
b
*
∂
-
b
G
b
f
∥
k
(
M
), and finally using the fact that ∂
-
b
G
b is continuously bounded on W
0
,
s
k
(
M
), we conclude (70) which proves the continuity of S
s on W
0
,
s
k
(
M
).Corollary 12.
LetM be a C
∞ compact C
R manifold of real dimension 2
n
-
l and codimension l
≥
1 in an n-dimensional complex manifold X. Suppose that M satisfies condition Y
(
s
) for some s with 1
≤
s
≤
n
-
l
-
1. Then for any f in W
0
,
s
k
(
M
) (k
≥
0) such that ∂
-
b
f
=
0 and f
⊥
H
0
,
s
b
(
M
), there exists u in W
0
,
s
-
1
k
(
M
) which solves the equation ∂
-
b
u
=
f.Theorem 13.
LetM be a C
∞ compact C
R manifold of real dimension 2
n
-
l and codimension l
≥
1 in an n-dimensional complex manifold X. Suppose that M satisfies condition Y
(
s
) for some s with 1
≤
s
≤
n
-
l
-
1. Then for any f in C
0
,
s
∞
(
M
), with ∂
-
b
f
=
0 and f
⊥
H
0
,
s
b
(
M
), there exists a global solution u in C
0
,
s
-
1
∞
(
M
) to the equation ∂
-
b
u
=
f.Proof.
By Corollary12, for each k
≥
0, there exists some u
k
∈
W
0
,
s
-
1
k
(
M
) such that ∂
-
b
u
k
=
f. We modify each u
k by an element of Ker
⁡
(
∂
-
b
) in order to construct a telescoping series that belongs to W
0
,
s
k
(
M
) for each k
≥
1. To conclude the proof, we first claim that W
0
,
s
k
(
M
)
∩
Ker
⁡
(
∂
-
b
) is dense in W
0
,
s
m
(
M
)
∩
Ker
⁡
(
∂
-
b
) for any k
>
m
≥
0. Since C
0
,
s
∞
(
M
) is dense in W
0
,
s
m
(
M
), m
≥
0, in the W
m-norm, then for a given η
∈
W
0
,
s
m
(
M
)
∩
Ker
⁡
(
∂
-
b
) there is a sequence η
j
∈
C
0
,
s
∞
(
M
) converging to η in the W
0
,
s
m
(
M
)-norm; that is, ∥
η
j
-
η
∥
m
(
M
)
→
0 as j
→
∞. ∂
-
b
η
=
0 implies that η
-
S
s
η
=
∂
-
b
*
G
b
∂
-
b
η
=
0, so η
=
S
s
u. Let η
^
j
=
S
s
η
j. η
^
j
∈
W
0
,
s
k
(
M
)
∩
Ker
⁡
(
∂
-
b
) since the Szegö projection S
s is a bounded operator on W
0
,
s
k
(
M
). By the same reason we have ∥
η
^
j
-
η
∥
m
(
M
)
=
∥
S
s
(
η
j
-
η
)
∥
m
(
M
)
≤
C
∥
η
j
-
η
∥
m
(
M
)
→
0 as j
→
∞. This implies that η
^
j
→
η in the W
m-norm. Thus, indeed, W
0
,
s
k
(
M
)
∩
Ker
⁡
(
∂
-
b
) is dense in W
0
,
s
m
(
M
)
∩
Ker
⁡
(
∂
-
b
) for any k
>
m
≥
0.
Next, using this result and following the inductive argument due to [21, page 230], we can construct a sequence u
~
k
∈
W
0
,
s
-
1
k
(
M
), ∂
-
b
u
~
k
=
f, and ∥
u
~
k
+
1
-
u
k
∥
k
(
M
)
≤
2
-
k as follows:
(73)
u
~
1
=
u
1
,
u
~
2
=
u
2
+
v
2
,
where v
2
∈
W
0
,
s
-
1
2
(
M
)
∩
Ker
⁡
(
∂
-
b
) is such that
(74)
∥
u
~
2
-
u
1
∥
1
(
M
)
≤
2
-
1
and in general
(75)
u
~
k
+
1
=
u
k
+
1
+
v
k
+
1
,
where v
k
+
1
∈
W
0
,
s
k
+
1
(
M
)
∩
Ker
⁡
(
∂
-
b
) is such that
(76)
∥
u
~
k
+
1
-
u
k
∥
k
(
M
)
≤
2
-
k
.
Clearly∂
-
b
u
~
k
=
f, so set
(77)
u
=
u
~
j
+
∑
k
=
j
∞
‍
(
u
~
k
+
1
-
u
~
k
)
,
j
∈
N
.
It follows that u
∈
W
0
,
s
-
1
k
(
M
) for each k
∈
N, and hence u
∈
C
0
,
s
-
1
∞
(
M
) and ∂
-
b
u
=
f. The general case is obtained from interpolation of linear operators.


## 3.1. Continuity of the Complex Green Operator

We prove first the continuity of the complex Green operatorG
b on W
0
,
s
k
(
M
), k
≥
0.Theorem 8.
LetM be a C
∞ compact C
R manifold of real dimension 2
n
-
l and codimension l
≥
1 in an n-dimensional complex manifold X. Suppose that M satisfies condition Y
(
s
) for some s with 1
≤
s
≤
n
-
l
-
1. Then the complex Green operator G
b is continuous on the Sobolev space W
0
,
s
k
(
M
), k
≥
0; that is, there is a constant C
=
C
(
k
)
>
0 such that
(31)
∥
G
b
f
∥
k
(
M
)
≤
C
∥
f
∥
k
(
M
)
,
f
∈
W
0
,
s
k
(
M
)
.Proof.
We consider the special case whenk
=
0,1
,
2,3
,
…. Indeed the general case is then derived by means of interpolation of linear operators. Since M is compact, it is easy to show that C
0
,
s
∞
(
M
) is a dense subspace in W
0
,
s
k
(
M
). Further, by Theorem 7 (8), we have G
b
f
∈
C
0
,
s
∞
(
M
) for f
∈
C
0
,
s
∞
(
M
). Thus it suffices to establish (31) for f
∈
C
0
,
s
∞
(
M
). For k
=
0, (31) follows from (23).
For eachk
≥
0, let Λ
k
(
ξ
) be a pseudodifferential operator of order k with symbol (
1
+
|
ξ
|
2
)
k
/
2. Let U be an open neighborhood of ζ in M and let η and η
1 be two cutoff functions with supports in U such that η
=
1 on supp η
1; then η
Λ
k
η
1
f
∈
D
0
,
s
(
U
) whenever f
∈
D
0
,
s
(
U
).
Recall that the compactness ofG
b in L
0
,
s
2
(
U
) is equivalent to the compactness estimate: for every ϵ
>
0 there is a constant C
(
ϵ
)
>
0 such that for every φ
∈
Dom
⁡
(
∂
-
b
)
∩
Dom
⁡
(
∂
-
b
*
)
(32)
∥
φ
∥
2
≤
ϵ
Q
b
(
φ
,
φ
)
+
C
(
ϵ
)
∥
φ
∥
-
1
(
U
)
2
,
where Q
b
(
φ
,
φ
)
=
(
∂
-
b
φ
,
∂
-
b
φ
)
+
(
∂
-
b
*
φ
,
∂
-
b
*
φ
). For this estimate and further results on the compactness of the complex Green operator see, e.g., [16–19].
Applying (32) for η
Λ
k
η
1
G
b
f, we obtain
(33)
∥
η
Λ
k
η
1
G
b
f
∥
2
≤
ϵ
Q
b
(
η
Λ
k
η
1
G
b
f
,
η
Λ
k
η
1
G
b
f
)
+
C
(
ϵ
)
∥
η
Λ
k
η
1
G
b
f
∥
-
1
(
U
)
2
.
We sometimes use A for η
Λ
k
η
1 and A
* for its formal adjoint, which is also a tangential operator of order k. We estimate the first term on the right hand side in (33), it is a standard consequence of [20, Corollary 3.1] (or [11, Lemma 2.4
.
2]) that
(34)
Q
b
(
A
G
b
f
,
A
G
b
f
)
=
Re
Q
b
(
G
b
f
,
A
*
A
G
b
f
)
+
O
(
|
|
|
D
G
b
f
|
|
|
k
-
1
(
U
)
2
)
≤
Re
Q
b
(
G
b
f
,
A
*
A
G
b
f
)
+
C
∥
G
b
f
∥
k
(
U
)
2
.
Here we have used the fact that the tangential derivative D
α of order |
α
|
=
λ satisfies the tangential Sobolev estimate |
|
|
D
α
f
|
|
|
r
≤
∥
f
∥
r
+
λ.
Takingv
=
A
*
A
f in the form Q
b
(
G
b
u
,
v
)
=
(
u
,
v
), we get
(35)
Q
b
(
A
G
b
f
,
A
G
b
f
)
≤
Re
(
f
,
A
*
A
G
b
f
)
+
C
∥
G
b
f
∥
k
(
U
)
2
≤
|
(
f
,
A
*
A
G
b
f
)
|
+
C
∥
G
b
f
∥
k
(
U
)
2
.
The Cauchy-Schwarz inequality implies
(36)
Q
b
(
A
G
b
f
,
A
G
b
f
)
≤
∥
A
f
∥
∥
A
G
b
f
∥
+
C
∥
G
b
f
∥
k
(
U
)
2
.
Inequality (33) becomes
(37)
∥
η
Λ
k
η
1
G
b
f
∥
2
≤
ϵ
∥
f
∥
k
(
U
)
∥
G
b
f
∥
k
(
U
)
+
C
(
ϵ
)
∥
η
Λ
k
η
1
G
b
f
∥
-
1
(
U
)
2
.
Summing over a partition of unity subordinate to an open covering of M by patches {
U
i
}
i
=
1
m, we obtain estimate like (37) on each of these patches and using the interior regularity properties, we get
(38)
∥
G
b
f
∥
k
(
M
)
2
≤
ϵ
∥
f
∥
k
(
M
)
∥
G
b
f
∥
k
(
M
)
+
C
(
ϵ
)
∥
G
b
f
∥
k
-
1
(
M
)
2
.
The first term in the right-hand side of (38) is estimated by ϵ
(
s
.
c
.
)
∥
G
b
f
∥
k
(
M
)
2
+
ϵ
(
l
.
c
.
)
∥
f
∥
k
(
M
)
2, where s
.
c. and l
.
c. denote a small and a large constants, respectively, in the inequality |
a
b
|
≤
(
s
.
c
.
)
a
2
+
(
l
.
c
.
)
b
2. The second term is estimated by interpolation of Sobolev norms (∥
G
b
f
∥
k
-
1
(
M
)
2
≤
ε
∥
G
b
f
∥
k
(
M
)
2
+
C
(
ε
)
∥
G
b
f
∥
2) and then by using the continuity of G
b in L
0
,
s
2
(
M
) with L
2-bounded norm.
Adding up the analogues terms and absorbing, by choosingϵ and ε to be small enough, ∥
G
b
f
∥
k
(
M
)
2 into the left, this gives
(39)
∥
G
b
f
∥
k
(
M
)
2
≤
C
∥
f
∥
k
(
M
)
2
+
K
∥
f
∥
2
,
where C
=
C
(
ϵ
,
k
)
>
0 and K
=
K
(
ϵ
,
k
)
>
0. The embedding Sobolev space implies (31) for k
=
0,1
,
2,3
,
…. The general case is obtained from interpolation of linear operators. As mentioned above, the density of C
0
,
s
∞
(
M
) in W
0
,
s
k
(
M
) passes (31) to forms f in W
0
,
s
k
(
M
). This proves the continuity of G
b in W
0
,
s
k
(
M
).Corollary 9.
LetM be given as in Theorem 8, then the canonical solution operators ∂
-
b
G
b and ∂
-
b
*
G
b are continuous on W
0
,
s
k
(
M
) for all k
≥
0.Proof.
We argue by induction onk. The case when k
=
0 follows from (25). Suppose that the assertions hold for positive integers less than k and assume that ζ, U, η, and η
1 are given as in the proof of Theorem 8. By the interior elliptic regularity properties, we prove first a priori estimate for ∂
-
b
G
b
f and ∂
-
b
*
G
b
f with f
∈
D
0
,
s
(
U
) as follows:
(40)
∥
η
Λ
k
η
1
∂
-
b
G
b
f
∥
2
+
∥
η
Λ
k
η
1
∂
-
b
*
G
b
f
∥
2
=
(
η
Λ
k
η
1
∂
-
b
G
b
f
,
∂
-
b
η
Λ
k
η
1
G
b
f
)
+
(
η
Λ
k
η
1
∂
-
b
*
G
b
f
,
∂
-
b
*
η
Λ
k
η
1
G
b
f
)
+
O
(
(
∥
η
Λ
k
η
1
∂
-
b
G
b
f
∥
+
∥
η
Λ
k
η
1
∂
-
b
*
G
b
f
∥
)
∥
G
b
f
∥
k
(
U
)
)
=
(
η
Λ
k
η
1
∂
-
b
*
∂
-
b
G
b
f
,
η
Λ
k
η
1
G
b
f
)
+
(
η
Λ
k
η
1
∂
-
b
∂
-
b
*
G
b
f
,
η
Λ
k
η
1
G
b
f
)
+
O
(
(
∥
η
Λ
k
η
1
∂
-
b
G
b
f
∥
+
∥
η
Λ
k
η
1
∂
-
b
*
G
b
f
∥
)
∥
G
b
f
∥
k
(
U
)
+
∥
G
b
f
∥
k
(
U
)
2
(
∥
η
Λ
k
η
1
∂
-
b
G
b
f
∥
+
∥
η
Λ
k
η
1
∂
-
b
*
G
b
f
∥
)
)
=
(
η
Λ
k
η
1
□
b
G
b
f
,
η
Λ
k
η
1
G
b
f
)
+
O
(
(
∥
η
Λ
k
η
1
∂
-
b
G
b
f
∥
+
∥
η
Λ
k
η
1
∂
-
b
*
G
b
f
∥
)
∥
G
b
f
∥
k
(
U
)
+
∥
G
b
f
∥
k
(
U
)
2
(
∥
η
Λ
k
η
1
∂
-
b
G
b
f
∥
+
∥
η
Λ
k
η
1
∂
-
b
*
G
b
f
∥
)
∥
G
b
f
∥
k
(
U
)
)
≤
C
1
∥
f
∥
k
(
U
)
∥
G
b
f
∥
k
(
U
)
+
C
2
(
(
∥
η
Λ
k
η
1
∂
-
b
G
b
f
∥
+
∥
η
Λ
k
η
1
∂
-
b
*
G
b
f
∥
)
∥
G
b
f
∥
k
(
U
)
+
∥
G
b
f
∥
k
(
U
)
2
(
∥
η
Λ
k
η
1
∂
-
b
G
b
f
∥
+
∥
η
Λ
k
η
1
∂
-
b
*
G
b
f
∥
)
)
.
Summing over a partition of unity, using the small and large constants for the resulting terms ∥
f
∥
k
∥
G
b
f
∥
k, ∥
∂
-
b
G
b
f
∥
k
∥
G
b
f
∥
k, and ∥
∂
-
b
*
G
b
f
∥
k
∥
G
b
f
∥
k, using (31) and adding up the analogues terms, we see that the terms on the right-hand side containing ∥
∂
-
b
G
b
f
∥
k
2 and ∥
∂
-
b
*
G
b
f
∥
k
2 can be absorbed into the left hand side. We therefore obtain
(41)
∥
∂
-
b
G
b
f
∥
k
(
M
)
2
+
∥
∂
-
b
*
G
b
f
∥
k
(
M
)
2
≤
C
∥
f
∥
k
(
M
)
2
,
f
∈
D
0
,
s
(
M
)
.
This completes the induction on k for the norms of ∂
-
b
G
b and ∂
-
b
*
G
b. By the density of C
0
,
s
∞
(
M
) in W
0
,
s
k
(
M
), the estimates extend to forms in W
0
,
s
k
(
M
). As before, the general case is obtained from interpolation of linear operators. Then ∂
-
b
G
b and ∂
-
b
*
G
b are continuous on W
0
,
s
k
(
M
).


## 3.2. Exact and Global Regularity Theorems

We now show the expression of the complex Green operator by Szegö projections.Theorem 10.
The Szegö projectionsS
s
:
L
0
,
s
2
(
M
)
→
Ker
⁡
(
∂
-
b
) are given by the following relations:
(42)
S
s
=
I
d
-
∂
-
b
*
∂
-
b
G
b
=
I
d
-
G
b
∂
-
b
*
∂
-
b
,
s
≥
0
,
(43)
S
s
-
1
=
I
d
-
∂
-
b
*
G
b
∂
-
b
,
s
≥
1
.Proof.
We first show that∂
-
b
*
∂
-
b
G
b
=
G
b
∂
-
b
*
∂
-
b. For α
,
β
∈
⊥
H
0
,
s
b
(
M
), we observe that
(44)
∂
-
b
α
=
0
⟹
∂
-
b
*
∂
-
b
G
b
α
=
0
⟹
α
=
∂
-
b
∂
-
b
*
G
b
α
=
G
b
∂
-
b
∂
-
b
*
α
,
(45)
∂
-
b
⋆
β
=
0
⟹
∂
-
b
∂
-
b
*
G
b
β
=
0
⟹
β
=
∂
-
b
*
∂
-
b
G
b
β
=
G
b
∂
-
b
*
∂
-
b
β
.
As Range
(
∂
-
b
)
⊥
Ker
⁡
(
∂
-
b
*
) and Range
(
∂
-
b
*
)
⊥
Ker
⁡
(
∂
-
b
), one has
(46)
∂
-
b
α
=
0
⟹
∂
-
b
G
b
α
=
0
,
(47)
∂
-
b
*
β
=
0
⟹
∂
-
b
*
G
b
β
=
0
.
Anyf
⊥
H
0
,
s
b
(
M
) can then be written as f
=
α
+
β so that ∂
-
b
α
=
0 and ∂
-
b
*
β
=
0. By (45) and (46), we then have
(48)
∂
-
b
⋆
∂
-
b
G
b
f
=
∂
-
b
⋆
∂
-
b
G
b
(
α
+
β
)
=
∂
-
b
⋆
∂
-
b
G
b
β
=
G
b
∂
-
b
⋆
∂
-
b
β
=
G
b
∂
-
b
⋆
∂
-
b
f
.
This implies the second equality in (42). Now, If f
∈
Ker
⁡
(
∂
-
b
), then (
I
d
-
G
b
∂
-
b
*
∂
-
b
)
f
=
f, so the expression for S
s holds. Next, if f
⊥
Ker
⁡
(
∂
-
b
) and hence f
⊥
H
0
,
s
b
(
M
), so f
=
∂
-
b
∂
-
b
⋆
G
b
f
+
∂
-
b
⋆
∂
-
b
G
b
f and u
=
∂
-
b
⋆
∂
-
b
G
b
f is the canonical solution to the equation ∂
-
b
u
=
∂
-
b
f. Thus ∂
-
b
(
f
-
u
)
=
0, that is, f
-
u
∈
Ker
⁡
(
∂
-
b
). We claim that u
⊥
Ker
⁡
(
∂
-
b
). Indeed, for all g
∈
Ker
⁡
(
∂
-
b
) one has (
u
,
g
)
=
(
∂
-
b
⋆
∂
-
b
G
b
f
,
g
)
=
(
∂
-
b
G
b
f
,
∂
-
b
g
)
=
0. Since f
⊥
Ker
⁡
(
∂
-
b
), it turns out that f
-
u
⊥
Ker
⁡
(
∂
-
b
) so f
-
u
=
0 and then 0
=
f
-
u
=
(
I
d
-
∂
-
b
⋆
∂
-
b
G
b
f
). This proves (42). Similarly, we get (43).Theorem 11.
LetM be given as in Theorem 8. Then the Szegö projections operators S
s
-
1 and S
s are continuous in the Sobolev spaces W
0
,
s
-
1
k
(
M
) and W
0
,
s
k
(
M
) for all k
≥
0, respectively.Proof.
We investigate first the continuity ofS
s
-
1. For the case k
=
0, when f
∈
L
0
,
s
2
(
M
), we have
(49)
∥
∂
-
b
*
G
b
∂
-
b
f
∥
2
=
(
∂
-
b
*
G
b
∂
-
b
f
,
∂
-
b
*
G
b
∂
-
b
f
)
=
(
G
b
∂
-
b
f
,
∂
-
b
∂
-
b
*
G
b
∂
-
b
f
)
=
(
G
b
∂
-
b
f
,
∂
-
b
f
)
=
(
∂
-
b
*
G
b
∂
-
b
f
,
f
)
≤
∥
∂
-
b
*
G
b
∂
-
b
f
∥
∥
f
∥
.
Here we have used the fact that ∂
-
b
∂
-
b
*
G
b
∂
-
b
f
=
∂
-
b
f, because ∂
-
b
2
=
0. The relation (43) thus implies that ∥
S
s
-
1
f
∥
≤
C
∥
f
∥. This proves the continuity in L
0
,
s
-
1
2
(
M
).
The casek
≥
1. Applying (32) for φ
=
η
Λ
k
η
1
G
s
∂
-
b
f on U, we obtain
(50)
∥
η
Λ
k
η
1
G
b
∂
-
b
f
∥
2
≤
ϵ
Q
b
(
η
Λ
k
η
1
G
b
∂
-
b
f
,
η
Λ
k
η
1
G
b
∂
-
b
f
)
+
C
(
ϵ
)
∥
η
Λ
k
η
1
G
b
∂
-
b
f
∥
-
1
(
U
)
2
.
The first term on the right-hand side of (50) is estimated as
(51)
Q
b
(
A
G
b
∂
-
b
f
,
A
G
b
∂
-
b
f
)
=
∥
∂
-
b
A
G
b
∂
-
b
f
∥
2
+
∥
∂
-
*
A
G
b
∂
-
b
f
∥
2
=
(
∂
-
b
A
G
b
∂
-
b
f
,
∂
-
b
A
G
b
∂
-
b
f
)
+
(
∂
-
b
*
A
G
b
∂
-
b
f
,
∂
-
b
*
A
G
b
∂
-
b
f
)
=
(
A
∂
-
b
G
b
∂
-
b
f
,
∂
-
b
A
G
b
∂
-
b
f
)
+
(
A
∂
-
b
*
G
b
∂
-
b
f
,
∂
-
b
*
A
G
b
∂
-
b
f
)
+
(
[
∂
-
b
,
A
]
G
b
∂
-
b
f
,
∂
-
b
A
G
b
∂
-
b
f
)
+
(
[
∂
-
b
*
,
A
]
G
b
∂
-
b
f
,
∂
-
b
*
A
G
b
∂
-
b
f
)
.
The sum of the last two terms on the right-hand side of the preceding equality is estimated by
(52)
∥
[
∂
-
b
,
A
]
G
b
∂
-
b
f
∥
∥
∂
-
b
A
G
b
∂
-
b
f
∥
+
∥
[
∂
-
b
*
,
A
]
G
b
∂
-
b
f
∥
∥
∂
-
b
*
A
G
b
∂
-
b
f
∥
≤
∥
D
G
b
∂
-
b
f
∥
k
-
1
(
U
)
∥
∂
-
b
A
G
b
∂
-
b
f
∥
+
∥
D
G
b
∂
-
b
f
∥
k
-
1
(
U
)
∥
∂
-
b
*
A
G
s
∂
-
b
f
∥
≤
∥
G
b
∂
-
b
f
∥
k
(
U
)
(
∥
∂
-
b
A
G
b
∂
-
b
f
∥
+
∥
∂
-
b
*
A
G
b
∂
-
b
f
∥
)
=
O
(
(
l
.
c
.
)
∥
G
b
∂
-
b
f
∥
k
(
U
)
2
+
(
s
.
c
.
)
(
∥
∂
-
b
A
G
b
∂
-
b
f
∥
+
∥
∂
-
b
*
A
G
b
∂
-
b
f
∥
)
2
∥
G
b
∂
-
b
f
∥
k
(
U
)
2
)
=
O
(
∥
G
b
∂
-
b
f
∥
k
(
U
)
2
)
.
We then have(53)
Q
b
(
A
G
b
∂
-
b
f
,
A
G
b
∂
-
b
f
)
≤
(
∂
-
b
G
b
∂
-
b
f
,
A
*
∂
-
b
A
G
b
∂
-
b
f
)
+
(
A
∂
-
b
*
G
b
∂
-
b
f
,
∂
-
b
*
A
G
b
∂
-
b
f
)
+
O
(
∥
G
b
∂
-
b
f
∥
k
(
U
)
2
)
.
The first term on the right-hand side of (53) equals zero due to the fact that ∂
-
b
G
b
∂
-
b
f
=
∂
-
b
2
G
b
f
=
0.
We now analyze the second term as follows:(54)
(
A
∂
-
b
*
G
b
∂
-
b
f
,
∂
-
b
*
A
G
b
∂
-
b
f
)
=
(
∂
-
b
A
∂
-
b
*
G
b
∂
-
b
f
,
A
G
b
∂
-
b
f
)
=
(
A
∂
-
b
∂
-
b
*
G
b
∂
-
b
f
,
A
G
b
∂
-
b
f
)
+
(
[
∂
-
b
,
A
]
∂
-
b
*
G
b
∂
-
b
f
,
A
G
b
∂
-
b
f
)
=
(
A
∂
-
b
f
,
A
G
b
∂
-
b
f
)
+
(
[
∂
-
b
,
A
]
∂
-
b
*
G
b
∂
-
b
f
,
A
G
b
∂
-
b
f
)
=
(
∂
-
b
A
f
,
A
G
b
∂
-
b
f
)
+
(
[
A
,
∂
-
b
]
f
,
A
G
b
∂
-
b
f
)
+
(
[
∂
-
b
,
A
]
∂
-
b
*
G
b
∂
-
b
f
,
A
G
b
∂
-
b
f
)
=
(
A
f
,
∂
-
b
*
A
G
b
∂
-
b
f
)
+
⋯
=
(
A
f
,
A
∂
-
b
*
G
b
∂
-
b
f
)
+
(
A
f
,
[
∂
-
b
*
,
A
]
G
b
∂
-
b
f
)
+
(
[
A
,
∂
-
b
]
f
,
A
G
b
∂
-
b
f
)
+
(
[
∂
-
b
,
A
]
∂
-
b
*
G
b
∂
-
b
f
,
A
G
b
∂
-
b
f
)
.
Thus
(55)
Q
b
(
A
G
b
∂
-
b
f
,
A
G
b
∂
-
b
f
)
≤
(
A
f
,
A
∂
-
b
*
G
b
∂
-
b
f
)
+
E
+
O
(
∥
G
b
∂
-
f
∥
k
(
U
)
2
)
≤
|
(
A
f
,
A
∂
-
b
*
G
b
∂
-
b
f
)
|
+
|
E
|
+
O
(
∥
G
b
∂
-
b
f
∥
k
(
U
)
2
)
,
where
(56)
E
=
(
A
f
,
[
∂
-
b
*
,
A
]
G
b
∂
-
b
f
)
+
(
[
A
,
∂
-
b
]
f
,
A
G
b
∂
-
b
f
)
+
(
[
∂
-
b
,
A
]
∂
-
b
*
G
b
∂
-
b
f
,
A
G
b
∂
-
b
f
)
.
As above, the three terms on the right-hand side of (56) are estimated, respectively, by
(57)
∥
A
f
∥
∥
[
∂
-
b
*
,
A
]
G
b
∂
-
b
f
∥
≤
∥
f
∥
k
(
U
)
∥
G
b
∂
-
b
f
∥
k
(
U
)
≤
(
s
.
c
.
)
∥
f
∥
k
(
U
)
2
+
(
l
.
c
.
)
∥
G
b
∂
-
b
f
∥
k
(
U
)
2
,
∥
f
∥
k
(
U
)
∥
A
G
b
∂
-
b
f
∥
≤
(
s
.
c
.
)
∥
f
∥
k
(
U
)
2
+
(
l
.
c
.
)
∥
A
G
b
∂
-
b
f
∥
2
=
O
(
∥
G
b
∂
-
b
f
∥
k
(
U
)
2
)
,
∥
[
∂
-
b
,
A
]
∂
-
b
*
G
b
∂
-
b
f
∥
∥
A
G
b
∂
-
b
f
∥
≤
(
s
.
c
.
)
∥
∂
-
b
*
G
b
∂
-
b
f
∥
k
(
U
)
2
+
(
l
.
c
.
)
∥
A
G
b
∂
-
b
f
∥
2
.
Now we are left with the first term in the right-hand side of (55) which, by applying the Cauchy-Schwarz inequality, is estimated by ∥
f
∥
k
(
U
)
∥
∂
-
b
*
G
b
∂
-
b
f
∥
k
(
U
). By choosing the s
.
c. small enough we can absorb the first term in the right-hand side of the last inequality into ∥
f
∥
k
(
U
)
∥
∂
-
b
*
G
b
∂
-
b
f
∥
k
(
U
). This completes the estimation of the first term on the right-hand side of (50). Therefore (50) becomes
(58)
∥
η
Λ
k
η
1
G
b
∂
-
b
f
∥
2
≤
ϵ
∥
f
∥
k
(
U
)
∥
∂
-
b
*
G
b
∂
-
b
f
∥
k
(
U
)
+
ϵ
(
s
.
c
.
)
∥
f
∥
k
(
U
)
2
+
ϵ
C
∥
G
b
∂
-
b
f
∥
k
(
U
)
2
+
C
(
ϵ
)
∥
η
Λ
k
η
1
G
b
∂
-
b
f
∥
-
1
(
U
)
2
≤
ϵ
∥
f
∥
k
(
U
)
∥
∂
-
b
*
G
b
∂
-
b
f
∥
k
(
U
)
+
ϵ
(
s
.
c
.
)
∥
f
∥
k
(
U
)
2
+
ϵ
C
∥
G
b
∂
-
b
f
∥
k
(
U
)
2
+
C
′
(
ϵ
)
∥
G
b
∂
-
b
f
∥
k
-
1
(
U
)
2
.
By summing over a partition of unity subordinate to an open covering of M by patches {
U
i
}
i
=
1
m so that on each of these patches an estimate like (58) is satisfied, using the interior regularity properties, we get
(59)
∥
G
b
∂
-
b
f
∥
k
(
M
)
2
≤
ϵ
∥
f
∥
k
(
M
)
∥
∂
-
b
*
G
b
∂
-
b
f
∥
k
(
M
)
+
ϵ
s
.
c
.
∥
f
∥
k
(
M
)
2
+
ϵ
C
∥
G
b
∂
-
b
f
∥
k
(
M
)
+
C
′
(
ϵ
)
∥
G
b
∂
-
b
f
∥
k
-
1
(
M
)
2
.
By using the small and large constants, the first term on the right-hand side in (59) is estimated as
(60)
ϵ
(
(
s
.
c
.
)
∥
f
∥
k
(
M
)
2
+
(
l
.
c
.
)
∥
∂
-
b
*
G
b
∂
-
b
f
∥
k
(
M
)
2
)
.
Then adding and choosing ϵ and the s
.
c. small enough we can absorb the third term on the right-hand side of (59) into the left-hand side; we obtain
(61)
∥
G
b
∂
-
b
f
∥
k
(
M
)
2
≤
ϵ
C
∥
∂
-
b
*
G
b
∂
-
b
f
∥
k
(
M
)
2
+
C
′
(
ϵ
)
(
∥
f
∥
k
(
M
)
2
+
∥
G
b
∂
-
b
f
∥
k
-
1
(
M
)
2
)
.
Applying this inequality with k replaced by k
-
1 to the last term on the right-hand side and repeating, we obtain
(62)
∥
G
b
∂
-
b
f
∥
k
(
M
)
2
≤
ϵ
C
∥
∂
-
b
*
G
b
∂
-
b
f
∥
k
(
M
)
2
+
C
′
(
ϵ
)
(
∥
f
∥
k
(
M
)
2
+
∥
G
b
∂
-
b
f
∥
2
)
.
We have
(63)
∥
η
Λ
k
η
1
∂
-
b
*
G
b
∂
-
b
f
∥
2
=
(
η
Λ
k
η
1
∂
-
b
*
G
b
∂
-
b
f
,
η
Λ
k
η
1
∂
-
b
*
G
b
∂
-
b
f
)
=
(
∂
-
b
*
η
Λ
k
η
1
G
b
∂
-
b
f
,
η
Λ
k
η
1
∂
-
b
*
G
b
∂
-
b
f
)
+
O
(
∥
G
b
∂
-
b
f
∥
k
(
U
)
∥
η
Λ
k
η
1
∂
-
b
*
G
b
∂
-
b
f
∥
)
=
(
η
Λ
k
η
1
G
b
∂
-
b
f
,
η
Λ
k
η
1
∂
-
b
∂
-
b
*
G
b
∂
-
b
f
)
+
O
(
∥
G
b
∂
-
b
f
∥
k
(
U
)
∥
η
Λ
k
η
1
∂
-
b
*
G
b
∂
-
b
f
∥
)
=
(
η
Λ
k
η
1
G
b
∂
-
b
f
,
η
Λ
k
η
1
∂
-
b
f
)
+
O
(
∥
G
b
∂
-
b
f
∥
k
(
U
)
∥
η
Λ
k
η
1
∂
-
b
*
G
b
∂
-
b
f
∥
)
=
(
η
Λ
k
η
1
G
b
∂
-
b
f
,
∂
-
b
η
Λ
k
η
1
f
)
+
O
(
∥
G
b
∂
-
b
f
∥
k
(
U
)
(
∥
f
∥
k
(
U
)
+
∥
η
Λ
k
η
1
∂
-
b
*
G
b
∂
-
b
f
∥
)
)
=
(
∂
-
b
*
η
Λ
k
η
1
G
b
∂
-
b
f
,
η
Λ
k
η
1
f
)
+
O
(
∥
G
b
∂
-
b
f
∥
k
(
U
)
(
∥
f
∥
k
(
U
)
+
∥
η
Λ
k
η
1
∂
-
b
*
G
b
∂
-
b
f
∥
)
)
=
(
η
Λ
k
η
1
∂
-
b
*
G
b
∂
-
b
f
,
η
Λ
k
η
1
f
)
+
O
(
∥
G
b
∂
-
b
f
∥
k
(
U
)
(
∥
f
∥
k
(
U
)
+
∥
η
Λ
k
η
1
∂
-
b
*
G
b
∂
-
b
f
∥
)
)
≤
∥
η
Λ
k
η
1
∂
-
b
*
G
b
∂
-
b
f
∥
∥
η
Λ
k
η
1
f
∥
+
O
(
∥
G
b
∂
-
b
f
∥
k
(
U
)
(
∥
f
∥
k
(
U
)
+
∥
η
Λ
k
η
1
∂
-
b
*
G
b
∂
-
b
f
∥
)
)
.
Again summing over a partition of unity, using the interior regularity properties and the small and large constants technique, we obtain
(64)
∥
∂
-
b
*
G
b
∂
-
b
f
∥
k
(
M
)
2
≤
C
(
∥
G
b
∂
-
b
f
∥
k
(
M
)
2
+
∥
f
∥
k
(
M
)
2
)
.
Substituting (62) into (64), we obtain
(65)
∥
∂
-
b
*
G
b
∂
-
b
f
∥
k
(
M
)
2
≤
K
ϵ
∥
∂
-
b
*
G
b
∂
-
b
f
∥
k
(
M
)
2
+
C
′
(
ϵ
)
(
∥
f
∥
k
(
M
)
2
+
∥
G
b
∂
-
b
f
∥
2
)
.
Choosing ϵ
>
0 small enough allows us to absorb the first term on the right-hand side into the left, we then get
(66)
∥
∂
-
b
*
G
b
∂
-
b
f
∥
k
(
M
)
2
≤
C
′
(
ϵ
)
(
∥
f
∥
k
(
M
)
2
+
∥
G
b
∂
-
b
f
∥
2
)
.
As the operator ∂
-
b
* has L
2
(
M
)-closed range, it follows from Theorem 1.1
.
1 in Hörmander [15] that there is a positive constant C such that
(67)
∥
G
b
∂
-
b
f
∥
≤
C
∥
∂
-
b
*
G
b
∂
-
b
f
∥
.
Then, by (49), we obtain
(68)
∥
G
b
∂
-
b
f
∥
≤
C
∥
f
∥
.
Substituting (68) into (66), we get
(69)
∥
∂
-
b
*
G
b
∂
-
b
f
∥
k
(
M
)
2
≤
C
∥
f
∥
k
(
M
)
2
.
By (43), the Szegö projection S
s
-
1 is therefore continuous on W
0
,
s
-
1
k
(
M
) for each k
=
0,1
,
2
…. The general case is obtained from interpolation of linear operators.
For the continuity of the Szegö projectionS
s, in view of (42), it suffices to show that
(70)
∥
∂
-
b
*
∂
-
b
G
b
f
∥
k
(
M
)
2
≤
C
∥
f
∥
k
(
M
)
2
,
k
≥
0
.
For k
=
0, we have
(71)
∥
∂
-
b
*
∂
-
b
G
b
f
∥
2
=
(
∂
-
b
∂
-
b
*
∂
-
b
G
b
f
,
∂
-
b
G
b
f
)
=
(
∂
-
b
f
,
∂
-
b
G
b
f
)
=
(
f
,
∂
-
b
*
∂
-
b
G
b
f
)
≤
C
∥
f
∥
∥
∂
-
b
*
∂
-
b
G
b
f
∥
.
For k
≥
1, as before, an elliptic regularity argument implies
(72)
∥
η
Λ
k
η
1
∂
-
b
*
∂
-
b
G
b
f
∥
2
=
(
η
Λ
k
η
1
∂
-
b
*
∂
-
b
G
b
f
,
η
Λ
k
η
1
∂
-
b
*
∂
-
b
G
b
f
)
=
(
η
Λ
k
η
1
∂
-
b
∂
-
b
*
∂
-
b
G
b
f
,
η
Λ
k
η
1
∂
-
b
G
b
f
)
+
(
η
Λ
k
η
1
∂
-
b
*
∂
-
b
G
b
f
,
[
η
Λ
k
η
1
,
∂
-
b
*
]
∂
-
b
G
b
f
)
+
(
[
∂
-
b
,
η
Λ
k
η
1
]
∂
-
b
*
∂
-
b
G
b
f
,
η
Λ
k
η
1
∂
-
b
G
b
f
)
=
(
η
Λ
k
η
1
∂
-
b
f
,
η
Λ
k
η
1
∂
-
b
G
b
f
)
+
O
(
∥
η
Λ
k
η
1
∂
-
b
G
b
f
∥
∥
η
Λ
k
η
1
∂
-
b
*
∂
-
b
G
b
f
∥
)
=
(
η
Λ
k
η
1
f
,
η
Λ
k
η
1
∂
-
b
*
∂
-
b
G
b
f
)
+
O
(
∥
η
Λ
k
η
1
∂
-
b
G
b
f
∥
∥
η
Λ
k
η
1
∂
-
b
*
∂
-
b
G
b
f
∥
)
+
O
(
∥
f
∥
k
(
U
)
∥
η
Λ
k
η
1
∂
-
b
G
b
f
∥
)
≤
∥
η
Λ
k
η
1
f
∥
∥
η
Λ
k
η
1
∂
-
b
*
∂
-
b
G
b
f
∥
+
O
(
(
∥
f
∥
k
(
U
)
+
∥
η
Λ
k
η
1
∂
-
b
G
b
f
∥
)
∥
η
Λ
k
η
1
∂
-
b
*
∂
-
b
G
b
f
∥
)
.
Summing over a partition of unity, using the small and large constants argument, absorbing the terms containing ∥
∂
-
b
*
∂
-
b
G
b
f
∥
k
(
M
), and finally using the fact that ∂
-
b
G
b is continuously bounded on W
0
,
s
k
(
M
), we conclude (70) which proves the continuity of S
s on W
0
,
s
k
(
M
).Corollary 12.
LetM be a C
∞ compact C
R manifold of real dimension 2
n
-
l and codimension l
≥
1 in an n-dimensional complex manifold X. Suppose that M satisfies condition Y
(
s
) for some s with 1
≤
s
≤
n
-
l
-
1. Then for any f in W
0
,
s
k
(
M
) (k
≥
0) such that ∂
-
b
f
=
0 and f
⊥
H
0
,
s
b
(
M
), there exists u in W
0
,
s
-
1
k
(
M
) which solves the equation ∂
-
b
u
=
f.Theorem 13.
LetM be a C
∞ compact C
R manifold of real dimension 2
n
-
l and codimension l
≥
1 in an n-dimensional complex manifold X. Suppose that M satisfies condition Y
(
s
) for some s with 1
≤
s
≤
n
-
l
-
1. Then for any f in C
0
,
s
∞
(
M
), with ∂
-
b
f
=
0 and f
⊥
H
0
,
s
b
(
M
), there exists a global solution u in C
0
,
s
-
1
∞
(
M
) to the equation ∂
-
b
u
=
f.Proof.
By Corollary12, for each k
≥
0, there exists some u
k
∈
W
0
,
s
-
1
k
(
M
) such that ∂
-
b
u
k
=
f. We modify each u
k by an element of Ker
⁡
(
∂
-
b
) in order to construct a telescoping series that belongs to W
0
,
s
k
(
M
) for each k
≥
1. To conclude the proof, we first claim that W
0
,
s
k
(
M
)
∩
Ker
⁡
(
∂
-
b
) is dense in W
0
,
s
m
(
M
)
∩
Ker
⁡
(
∂
-
b
) for any k
>
m
≥
0. Since C
0
,
s
∞
(
M
) is dense in W
0
,
s
m
(
M
), m
≥
0, in the W
m-norm, then for a given η
∈
W
0
,
s
m
(
M
)
∩
Ker
⁡
(
∂
-
b
) there is a sequence η
j
∈
C
0
,
s
∞
(
M
) converging to η in the W
0
,
s
m
(
M
)-norm; that is, ∥
η
j
-
η
∥
m
(
M
)
→
0 as j
→
∞. ∂
-
b
η
=
0 implies that η
-
S
s
η
=
∂
-
b
*
G
b
∂
-
b
η
=
0, so η
=
S
s
u. Let η
^
j
=
S
s
η
j. η
^
j
∈
W
0
,
s
k
(
M
)
∩
Ker
⁡
(
∂
-
b
) since the Szegö projection S
s is a bounded operator on W
0
,
s
k
(
M
). By the same reason we have ∥
η
^
j
-
η
∥
m
(
M
)
=
∥
S
s
(
η
j
-
η
)
∥
m
(
M
)
≤
C
∥
η
j
-
η
∥
m
(
M
)
→
0 as j
→
∞. This implies that η
^
j
→
η in the W
m-norm. Thus, indeed, W
0
,
s
k
(
M
)
∩
Ker
⁡
(
∂
-
b
) is dense in W
0
,
s
m
(
M
)
∩
Ker
⁡
(
∂
-
b
) for any k
>
m
≥
0.
Next, using this result and following the inductive argument due to [21, page 230], we can construct a sequence u
~
k
∈
W
0
,
s
-
1
k
(
M
), ∂
-
b
u
~
k
=
f, and ∥
u
~
k
+
1
-
u
k
∥
k
(
M
)
≤
2
-
k as follows:
(73)
u
~
1
=
u
1
,
u
~
2
=
u
2
+
v
2
,
where v
2
∈
W
0
,
s
-
1
2
(
M
)
∩
Ker
⁡
(
∂
-
b
) is such that
(74)
∥
u
~
2
-
u
1
∥
1
(
M
)
≤
2
-
1
and in general
(75)
u
~
k
+
1
=
u
k
+
1
+
v
k
+
1
,
where v
k
+
1
∈
W
0
,
s
k
+
1
(
M
)
∩
Ker
⁡
(
∂
-
b
) is such that
(76)
∥
u
~
k
+
1
-
u
k
∥
k
(
M
)
≤
2
-
k
.
Clearly∂
-
b
u
~
k
=
f, so set
(77)
u
=
u
~
j
+
∑
k
=
j
∞
‍
(
u
~
k
+
1
-
u
~
k
)
,
j
∈
N
.
It follows that u
∈
W
0
,
s
-
1
k
(
M
) for each k
∈
N, and hence u
∈
C
0
,
s
-
1
∞
(
M
) and ∂
-
b
u
=
f. The general case is obtained from interpolation of linear operators.



---

*Source: 326434-2014-06-12.xml*