1. Introduction
Sobolev estimates are crucial tools in the study of complex analysis on pseudoconvex manifolds. In this paper, we will focus on the Sobolev estimates for the
operator and the
-Neumann operator on such manifolds. Consider a Hartogs pseudoconvex domain
D with a
boundary in a Kähler manifold
X of complex dimension n, and if
is a positive line bundle over
X whose curvature form satisfies
with constant
, then the operators
N,
,
and the Bergman projection
are regular in the Sobolev space
for some positive
. This result generalizes the well-known results of Berndtsson–Charpentier [
1], Boas–Straube [
2], Cao–Shaw–Wang [
3], Harrington [
4] and Saber [
5] and others in the case of the Hartogs pseudoconvex domain in a Kähler manifold for forms with values in a holomorphic line bundle. Indeed, in [
1], Berndtsson–Charpentier (see also [
6]) obtained the Sobolev regularity for
for a pseudoconvex domain
. In [
2], Boas–Straube proved that the Bergman projection
B maps the Sobolev space
to itself for all
on a smooth pseudoconvex domain in
that admits a defining function that is plurisubharmonic on the boundary
. In [
3], Cao–Shaw–Wang obtained the Sobolev regularity of the operators
N,
,
and
on a local Stein domain subset of the complex projective space. In [
4], Harrington proved this result on a bounded pseudoconvex domain in
with a Lipschitz boundary. In [
5], Saber proved that the operators
N,
and
are regular in
for some
on a smooth weakly
q-convex domain in
. Similar results can be found in [
7,
8,
9,
10,
11,
12,
13,
14,
15,
16].
This paper is organized into five sections. The introduction presents an introduction to the subject and contains the history and development of the problem.
Section 2 recalls the basic definitions and fundamental results. In
Section 3, the basic Bochner–Kodaira–Morrey–Kohn identity is proved on the Kähler manifold. In
Section 4, it is proved that the
smoothly bounded Hartogs pseudoconvex domains in the Kähler manifold admit bounded plurisubharmonic exhaustion functions.
Section 5 deals with the
estimates of the
and
-Neumann operator on the
smoothly bounded Hartogs pseudoconvex domains in the Kähler manifold.
Section 6 presents the main results.
2. Preliminaries
Assuming that X is a complex manifold of the complex dimension n, , (resp. ) is the holomorphic tangent bundle of X (resp. at ) and is a holomorphic line bundle over X. A system of local complex analytic (holomorphic) coordinates on X is a collection (for some index set J) of local complex coordinates such that:
(i) , i.e., is an open covering of X by charts with coordinate mappings satisfies .
(ii) is a system of transition functions for ; that is, the maps are biholomorphic for each pair of indices with being nonempty (i.e., (resp. ) are holomorphic maps of onto (resp. onto )).
Assume that
is the local coordinates on
. A system of functions
is a Hermitian metric along the fibers of
with
in
and
is a
positive function in
. The (1,0) form of the connection associated with the metric
ℏ is given as
,
.
is the curvature form associated with the connection
and is given by
Definition 1. Ξ is positive at if the Hermitian form is positive definite on , ∀.
Along the fibers of
,
is a Hermitian metric for which
is positive; i.e.,
. Then,
defines a Kähler metric
on
X,
Let
(resp.
) be the space of
differential forms (resp. with compact support) on
X with values in
. A form
is expressed on
as follows:
where
and
are multi-indices and
is a section of
. Define the inner product
where
Let
Let
be the Hodge star operator, which is a real operator and satisfies
For the proof, see Morrow and Kodaira [
17]. Set the volume element with respect to
as
. The inner product
and the norm
are defined by
The formal adjoint operator
of
is defined by
and
. Let
be defined locally as
; the inner product
is given by
From Stokes’ theorem,
,
, one obtains
As a result,
for
.
is the Hilbert space of the measurable E-valued forms , which are square integrable in the sense that . Let and . In , the spaces , and are the kernel, the domain and the range of , respectively. A Bergman projection operator . Let be the unbounded Laplace–Beltrami operator from to with Dom Dom Dom Dom and Dom. Let be the -Neumann operator on forms, solving for any form in . Denote by the Bergman operator, mapping a form in to its orthogonal projection in the closed subspace of -closed forms.
Let
be the Sobolev space with
and let
denote its norm. ∀
, one obtains
. Thus,
is an elliptic and
for
if and only if
For the proof, see Theorems 4.1 and 4.2 in Jersion and Kenig [
18], Lemma 2 in Charpentier [
19] and also Theorem C.4 in the Appendix in Chen and Shaw [
20].
Proposition 1 ([21,22,23]). (i) If satisfies supp. and supp., then ; i.e., in . (ii) is dense in in the sense of . (iii) is dense in (resp. in the sense of the norm ( (resp. ).
(iv) on .
4. Bounded P.S.H. Functions and Hartogs Pseudoconvexity in Kähler Manifolds
Definition 2 ([28]). Ω is the smooth local Stein domain if ∀ point , and ∃ is a neighborhood U if z satisfies , which is Stein.
Definition 3 ([29]). We say that Ω is Hartogs pseudoconvex if there exists a smooth bounded function h on Ω such that for some , where ω is the Kähler form associated with the Kähler metric.
In particular, every Hartogs pseudoconvex domain admits a strictly plurisubharmonic exhaustion function and is thus a Stein manifold.
Next, we will examine several examples of Hartogs pseudoconvex domains.
Example 1. Suppose X is a complex manifold with a continuous strongly plurisubharmonic function and is a Stein domain. According to [30], there exists a Kähler metric on X such that Ω is Hartogs pseudoconvex. Example 2 ([29]). All the local Stein-domain subsets of a Stein manifold are in the Hartogs pseudoconvex domain.
Example 3 ([29]). Every pseudoconvex domain in the subset of a Stein manifold is a Hartogs pseudoconvex domain.
Example 4 ([30]). Any local Stein domain subset of a Kähler manifold with positive holomorphic bisectional curvature satisfies Equation (18) on . Example 5 ([30]). If Ω is a local Stein domain of the complex projective space , then Ω satisfies Equation (18). The canonical line bundle
K of
X is defined by transition functions
with
Hence,
determines a metric of
K. Let
be a Hermitian metric of
and
its curvature tensor. So,
determines a Hermitian metric of
and
Then, from Proposition 4,
for
,
. Using
, one obtains
With respect to the
and
, and for
, we define the global inner product
and the norm
by
As Theorem 1.1 in [
31], one obtains
Theorem 1. Suppose X is an n-dimensional complex manifold and is a Hartogs pseudoconvex. , where ζ is the Kähler metric ω on X. If , then for some constant .
Proof. Using Equation (
18) and if
,
Let
be an orthonormal basis for
near
p. In this case, near
, choose local coordinates that satisfy
,
,
. The Hermitian form for
is denoted by
. The inequality (22) gives the coordinates
Expanding (23), one obtains
for
, replacing
v by
,
The inequality’s left side can be expressed as follows:
For
, we assume that
From Equation (
24), one obtains
Take a look at
; for a small enough
,
in a neighborhood of
p. On the sphere
, inequality (25) still holds for
in a neighborhood of
, where
. This gives us
for
,
. But, when
and
, one obtains
, where
. So, by using (25),
and for
,
Recalling this one yields
□
Lemma 1. Let be a Hartogs pseudoconvex in an n-dimensional complex manifold X. Suppose is the order of plurisubharmonicity for : Then, ∀ and ; there exists with . Also, there exists , which satisfies Proof. By Equation (
21), ∃
satisfies
on
D. Since
Also, by using Equation (
18), one obtains
Therefore, from Equations (29) and (30), one obtains
Since
and
, then
Then, from Equations (31) and (32), one obtains
Then, Equation (
26) is proved.
To prove Equation (
27), choose
, and by using Equation (
28), one obtains
Then, Equation (
27) is proved. □
5. The Estimates of
As in [
21,
22,
23,
32,
33], one proves the following results:
Theorem 2. Let be a Hartogs pseudoconvex in an n-dimensional complex manifold X. Let Ξ be a positive line bundle over X whose curvature form Θ satisfies , where . Let , , a -closed form. Then, for , there exists , which satisfies and Proof. The boundary term in Equation (
20) vanishes since
. For
,
, and since the curvature form
of
satisfies
then by using Equation (
18), one obtains
Also, from the assumption of pseudoconvex on
D, one obtains
for all
. Let
, with
,
and
. Then, for every
form
u with compact support, one obtains
Using the Riesz representation theorem, the linear form
is continuous on Rang
in the
norm and has norm
, with
Following Hahn–Banach theorem, ∃ is an element that is
E valued
from
u on
D (with a smooth boundary) perpendicular to
with
,
for all
with both
and
and also
. Hence,
and
Exhaust a general pseudoconvex domain
D by a sequence
of
pseudoconvex domains:
with
for each
. On each
, ∃ a
satisfies
and
Choose a subsequent
of
, satisfying
in
weakly. Moreover,
□
Theorem 3. Let X, D and Ξ be the same as Theorem 2. Let , , with . Thus, ∃ satisfies and Proof. Since
and from Equation (
18), one obtains
∀. This completes the proof of Theorem 3. □
Following Theorem 3, as in [
34,
35], one can prove the following:
Theorem 4. Let X, D and Ξ be the same as Theorem 2. Then, □ has a closed range and . For each , there exists a bounded linear operator which satisfies
(i) Rang Dom and on Dom.
(ii) ∀,
(iii) For , one obtains (v) If and , then and .
Proof. We note that if
then by using Theorem 4, ∃ a
satisfies
If
is also in
, one obtains
Thus,
and Equation (
35) is proved. We shall show that
is closed. Following Theorem 4, ∀
with
and ∃ a
satisfies
and
where
. Thus,
is closed in every degree. Thus,
for
and
. Thus, from (36),
for
Dom
Dom
. Thus, ∀
Dom
,
Thus,
i.e.,
is closed. Therefore,
Also, from Equation (
37),
is 1-1 and
is the whole space
. Thus, there exists a unique inverse
which satisfies
and
∀
. Also, by (ii),
Now, we show that
on Dom
. Using (ii),
Then,
Similarly, one can prove
on Dom
. From (ii),
Thus,
implies
and
Since Thus, and is the solution which is unique and orthogonal to . □
Corollary 1. Let X, D and Ξ be the same as Theorem 2. Then, for all that satisfies , the canonical solution satisfies the estimate Proof. From (iv), one obtains
. Since
Thus, the proof follows. □
Let
Set
Since
, then
is a closed subspace of
. Let
be the Bergman projection operator.
Lemma 2 ([16]). Let X, D and Ξ be the same as Theorem 2. Then, satisfies
(i) , .
(ii) ∀; one obtains
(iii) on Dom, on Dom.
(iv) if .
(v) ∀, Proof. Let
. Since
is closed in every degree,
is closed. Thus,
and
. Let
; then,
since
. Using (v) in Theorem 5,
is the solution of
, which is unique and
. Thus,
. By using Equation (
36), one obtains
Thus,
is bounded below on
and
has a closed range and (i) and (ii) is proved. Then, from the strong Hodge decomposition,
for all
, there is a unique
that satisfies
. Extending
to
by requiring
,
satisfies (i) and (ii). (iii) is proved as before. If
,
Thus, (iv) holds on
. From (iii) in Theorem 5,
for all
,
On the other hand, one obtains
Combining Equation (
38) and Equation (
39), one obtains
and
Then, the proof follows. □
6. Sobolev Estimates
As in Cao–Shaw–Wang [
3,
35], one prove the following results:
Proof. In fact, for
and
, one obtains
Since
is of type
, then
Then, by Stokes theorem, one obtains
i.e.,
i.e.,
Theorem 5. Let X, D and Ξ be the same as Theorem 2. Let , . Then, where is an independent constant of ψ.
Proof. Therefore, for
, and by using Equation (
18), one obtains
Also, by using Equation (
40), one obtains
and since for all
,
and since
Then, by using Equations (43)–(45), the identity (42) becomes
Then the proof follows from the density of in in the sense of □
Corollary 2. Let X, D and Ξ be the same as Theorem 2. Then, Proof. Since
,
. Then, substituting
into Equation (
41), for
, one obtains
Then, by using the fact that
,
and
, one obtains
Then, the first equation of Equation (
46) is proved by choosing
. Similarly, for
,
. Then, substituting
into Equation (
41), for
, one obtains
Then, by using the fact that
,
and
, one obtains
Then, Equation (
48) is proved by choosing
. □
Theorem 6. Let X, D and Ξ be the same as Theorem 2. Let , , a -closed form. Then, for , ∃ satisfies and Proof. Let
,
. Then,
is orthogonal to all
-closed forms of
. Equation (
33) gives
For
, one obtains
Then,
for every
. Since
by choosing
, which satisfies
(i.e.,
),
It follows that
and
□
Theorem 7. Let X, D and Ξ be the same as Theorem 2. The Bergman projection is bounded from to , where .
Proof. From Lemma 2,
. Then, by using Equation (
47),
is bounded on
with
for
,
. The Bergman projection with respect to the weighted space
is denoted by
. ∀
with
, and one obtains
This implies that
because
. ∀
,
With Equations (49) to (51), one obtains
We note that
. From Equation (
52), one obtains
Using Equation (
52), one obtains that the Bergman projection satisfies
Then, the Theorem is proved. □
In the following, the Sobolev boundary regularity for N, and is studied.
Theorem 8. Let X, D and Ξ be the same as Theorem 2. Then, ∀, N is bounded from to and . Also, ∀, and one obtains the following estimates: where C depends only on .
Proof. Since
, then
. Let
be another projection operator into
. Then,
. It follows that
. The self-adjoint property of
and
gives
Thus, by using Equation (
54), and for
, one obtains
and for
, one obtains
Since for all
, one obtains
Thus, for all
, one obtains
Since
and
. Use Equations (56) and (57), and by choosing
, the second and third inequality of Equation (
55) follows. Since
Equations (56) and (57) give
□
Theorem 9. Let X, D and Ξ be the same as Theorem 2. Then, ∀ and N is bounded from to , where Also, ∀ , and one obtains the following estimates: Proof. With respect to the
norm, if
is the adjoint map of
, one obtains
Then, by using Theorem 9 and Equation (
58), the proof follows. □