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Solving with Lp -estimates on q -convex intersections in complexmanifoldShaban Khidr aa Faculty of Science, Mathematics Department, Beni-SuefUniversity, Beni-Suef, EgyptPublished online: 15 Feb 2008.

To cite this article: Shaban Khidr (2008): Solving with Lp -estimates on q -convex intersectionsin complex manifold, Complex Variables and Elliptic Equations: An International Journal, 53:3,253-263

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Complex Variables and Elliptic EquationsVol. 53, No. 3, March 2008, 253–263

Solving �› with Lp-estimates on q-convex

intersections in complex manifold

SHABAN KHIDR*

Faculty of Science, Mathematics Department, Beni-Suef University, Beni-Suef, Egypt

Communicated by H. Begehr

(Received 21 April 2007; in final form 7 May 2007)

We construct some �@-solving bounded linear integral operators that satisfy Lp-estimates,1� p�1, on q-convex intersections with C3-boundary in Kahler manifold.

Keywords: �@-Equation; Lp-Estimates; q-Convex intersection

AMS Subject Classifications: 32F27; 32C35; 35N15

1. Introduction

Grauert and Lieb [1], Ramierz de Arellano [2], and Henkin [3,4] were the first who

constructed integral operators for solving the �@-equation for (0, s)-forms, s� 1, on

smooth strongly pseudo-convex domains. The Lp-estimates for solutions of the�@-equation were first obtained by Kerzman [5] for complex-valued �@-closed (0, 1)-

forms and by Øvrelid [6] for (0, s)-forms. Abdelkader and Khidr [7] extended the results

of Øvrelid to (r, s)-forms, 0� r� n, 1� s� n, on strongly pseudoconvex domain with

smooth C4-boundary of an n-dimensional Stein manifold. Abdelkader and Khidr [8]

extended their results in [7] to an E-valued (r, s)-forms, s� q, on strongly q-convex

domain with smooth C2-boundary of an n-dimensional Kahler manifold, where E is a

holomorphic line bundle that satisfies a certain positivity conditions. In the present

article, by using the method of multiple-weighted kernels constructed by Berndtsson [9],

we will extend our results to q-convex intersection of Kahler manifold. More precisely

we will prove the following main theorem.

Theorem 1.1 Let M be a Kahler manifold of complex dimension n and let E!M be a

holomorphic line bundle over M. Let ���M be a C3 q-convex intersection. If E is

*Email: skhidr@yahoo.com

Complex Variables and Elliptic EquationsISSN 1747-6933 print/ISSN 1747-6941 online � 2008 Taylor & Francis

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semi-positive (resp. semi-negative) of type k on ��. Then there exist bounded linear

operators

T s : L1r,sð�,EÞ ! L1

r,s�1ð�,EÞ, s � q

such that:

(i) Ts f¼ g solve the equation

�@g ¼ f ð1:1Þ

for any f 2 L1r,sð�,EÞ with �@f ¼ 0, s� q, and rþ s� nþ k (resp. s� q and

rþ s� n� k).(ii) Moreover, if f 2 Lp

r, sð�,EÞ then there exists a constant Cs such that

kgkLpr, s�1

ð�,EÞ � CskfkLpr, sð�,EÞ, 1 � p � 1: ð1:2Þ

The constant Cs is independent of f and p. If f is C1, then g is also C1.

The study of �@-equation on intersections of strongly pseudo-convex domains

was pioneered by Range and Siu [10], followed by Lieb and Range [11], Michel [12],

Michel and Perotti [13] and Menini [14]. Range and Siu [10] generalized the results of

Grauert and Lieb [1] to transversal intersection D ¼ D1 \D2 � � �DN, N� 1, of smooth

strongly pseudo-convex domains. Independently, Peters [15] extended the results of

Range and Siu [22] to nontransversal intersections of more than two nonsmooth

domains and he obtained Holder and Ck-estimates. Some years later, Chang and Lee

[16] obtained Lp-estimates for solutions of �@-equation on transversal intersection

D ¼ f� < 0g\ f� < 0g, where {�<0} and {�<0} are C3 strongly pseudoconvex domains

in Cn and C

m, 1�m� n, respectively. On the other hand, Laurent-Thiebaut and

Leiterer [17] solved the �@-equation with uniform estimates on piecewise smooth

intersections of q-convex domains, 1� q� n. Their type of domains were originally

considered by Henkin [18]. However, their solution operator is not suitable for

obtaining L2 (or more generally Lp, 1� p�1) estimates. By using the idea of multiple-

weighted kernels constructed by Berndtsson [9], Ma and Vassiliadou [19] constructed a

solution operator for the �@u ¼ f with Lp-estimates, 1� p�1, when f is a �@-closedcomplex-valued (0, s)-form, s� q, defined on a C3 q-convex intersection in C

n.As to the plan of the article, in section 2, we present the multiple-weighted kernels of

Berndtsson and Andersson [20]. Section 3 is devoted to the construction of the local �@solving integral operator with Lp-estimates, 1� p�1. The global solution with

Lp-estimates is proved in section 4.We begin by fixing some notation which will be used throughout the article. We recall

that a smooth domain � �� Cn is strongly q-convex if there exist a bounded

neighbourhood U of @� and a smooth defining function � : U ! R, such that

� \U ¼ fz 2 U; �ðzÞ < 0g, and its Levi form

L�ðz; tÞ ¼X @2�ðzÞ

@z�@ �z�t� �t�; t ¼ ðt1, t2, . . . , tnÞ 2 C

n, ð1:3Þ

has at least n� qþ 1 positive (>0) eigenvalues at each point z 2 U.

Definition 1.2 [19] A domain ���Cn is called a C3 q-convex intersection if there exist

a bounded neighbourhood U of �� and a finite number of real C3 functions �1, . . . , �N

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defined on U such that � \U ¼ fz 2 U; �1ðzÞ < 0, . . . , �N < 0g and the following twoconditions are satisfied:

(i) For 1 � i1 < i2 < � � � < i‘ � N the 1-forms d�i1 , . . . , d�i‘ are R-linearly indepen-dent on \

j¼‘j¼1f�ij � 0g.

(ii) For 1 � i1 < i2 < � � � < i‘ � N, for every z 2 \j¼‘j¼1f�ij � 0g, there exists a linear

subspace TIz of C

n of complex dimension at least n� qþ 1 such that fori 2 I ¼ ði1, . . . , i‘Þ the Levi forms L�i ðz; :Þ restricted on TI

z are positive definite.

This definition can be also made for complex manifold by local holomorphic chart.

2. Berndtsson–Andersson multiple-weighted kernels

Here we introduce the multiple-weighted kernel of Berndtsson–Andersson [20] onpiecewise smooth domains in C

n. Let �¼ (�1, . . . , �n) and �¼ (�1, . . . , �n) be two C1-mapson an open set of Cn. We write

!ð�Þ ¼ d�1 ^ d�2 ^ � � � ^ d�n; !0ð�Þ ¼Xnj¼1

ð�1Þj�1�j^k6¼j

d�k;

and

h�, �i ¼Xnj¼1

�j�j, j�j2 ¼ h�, ��i:

Let � ¼ fz 2 U; �1ðzÞ < 0, . . . , �NðzÞ < 0g��Cn be a domain with piecewise smooth

boundary of Cn and let

sð�, zÞ ¼ ðs1ð�, zÞ, . . . , snð�, zÞÞ : ��� �� ! Cn

be a C1-map (Leray map) satisfying the following conditions:

(i) For every compact set K � � there exist two positive constants C1 (K) and C2(K)such that

jsð�, zÞj � C1ðKÞj� � zj,

jhsð�, zÞ, � � zij � C2ðKÞj� � zj2

uniformly for all � 2 �� and z 2 K.(ii) hsð�, zÞ, � � zi 6¼ 0, for � 6¼ z.

Let

Qið�, zÞ ¼ ðQi1ð�, zÞ, . . . ,Q

iNð�, zÞÞ :

��� �� ! Cn, i ¼ 1, . . . ,N,

be C1-maps which are holomorphic in z 2 � for fixed � 2 � and let fGigNi¼1 be

holomorphic functions, in one complex variable, on an open neighbourhood of theimage set of ��� �� under the map ð�, zÞ :! 1þ hQið�, zÞ, z� �i with Gi(1)¼ 1. With themaps s(�, z) and Qi(�, z) we associate the (1, 0)-differential forms, also denoted by s(�, z)and Qi(�, z),

sð�, zÞ ¼Xnj¼1

sjð�, zÞdð�j � zjÞ, Qið�, zÞ ¼Xnj¼1

Qijð�, zÞdð�j � zjÞ,

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and the kernels

Kð�, zÞ ¼ �Cn

P�0þ�1þ���þ�N¼n�1

ðn�1Þ!�1!...�N!

Gð�1Þ1 ð1þ hQ1ð�, zÞ, z� �iÞ . . .

Gð�NÞN ð1þ hQNð�, zÞ, z� �iÞ

^s^ðdsÞ�0^ðdQ1Þ

�1^���^ðdQNÞ�N

hs, ��zi�0þ1 ,

Pð�, zÞ ¼P

�0þ�1þ���þ�N¼n�1

ðn�1Þ!�1!...�N!

Gð�1Þ1 ð1þ hQ1ð�, zÞ, z� �i . . .

Gð�NÞN ð1þ hQNð�, zÞ, z� �iÞ

^ðdQ1Þ�1 ^ � � � ^ ðdQNÞ

�N :

where Cn¼ (�1)n(n�1)/2(1/(2�i)n) and GðkÞi the kth-derivative of Gi.

The assumption on s imply that the coefficients of K(�, z) are integrable in � 2 ��nz

with uniformly bounded L1-norm for z 2 K and continuous off the diagonal. The kernel

P(�, z) is continuous on ��� �� and away of the diagonal it satisfies P(�, z)¼ d�,zK(�, z).Then using the proof of ([20] Theorem 1) and ([21] Theorems 5 and 6), one has the

following theorem:

Theorem 2.1 Let �, K(�, z) and P(�, z) be given as above. Then for any f 2 C1r, sð

��Þ

0� r, s� n, it holds

fðzÞ ¼

Z�2@�

fðzÞ ^ Kð�, zÞ þ ð�1Þrþsþ1

Z�2�

�@fðzÞ ^ Kð�, zÞ

þ ð�1Þrþs �@z

Z�2�

fðzÞ ^ Kð�, zÞ �

Z�2�

fðzÞ ^ Pð�, zÞ: ð2:1Þ

3. The Local �› solving operator

The main task of this section is to construct a local �@ solving operator with Lp estimates.

For this purpose, firstly, we recall with slightl modification the multi-weighted kernels

constructed in [19]. Let D � Cn be a bounded domain and let � be a C3 real-valued

function on D. Denote by F�(�, �) the Levi polynomial of � at � 2 D. For � 2 D, z 2 Cn,

F�ð�, zÞ ¼ 2Xnj¼1

@�ð�Þ

@�jð�j � zjÞ �

Xnj, k¼1

@2�

@�j@�kð�j � zjÞ � ð�k � zkÞ:

Let ���U��Cn be a C3 q-convex intersection with the defining functions �i, i¼ 1,

2, . . . ,N, and U as in Definition 1.2. Set

�I ¼ fz 2 U; �iðzÞ < 0; i 2 Ig, SI ¼ fz 2 U; �iðzÞ ¼ 0; i 2 Ig:

For � 2 SI there exists a smoothly bounded strongly pseudoconvex domain D* defined

by D� ¼ fz 2 U; ��ðzÞ < 0g such that @D� intersects real transversally

fz 2 U; �i1 ðzÞ < 0g, . . . , fz 2 U; �i‘ ðzÞ < 0g and � 2 D�. We denote by I* the multi-index

(i1, . . . , i‘, �), where I¼ (i1, . . . , i‘), 1� i1< i2< � � �< i‘, and we define a local q-convex

intersection �I� by

�I� ¼ fz 2 U; �jðzÞ < 0; j 2 I�g:

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Since �I is a q-convex intersection, then for every z 2 �I there exists an (n� qþ 1)-

linear vector subspace TIz of C

n such that the Levi forms L�iðz; �Þ are positive definite on

TIz for all i 2 I. Denote by P the orthogonal projection of C

n onto TIz and set

QIz : Id� P.Taylor’s theorem and the q-convexity of �I imply that there exist an > 0 and two

positive constants A and � such that the following estimate holds

ReF�ið�, zÞ � �ið�Þ � �iðzÞ þ�

2� � zj j2�A QI

zð� � zÞ�� ��2, ð3:1Þ

for �, z 2 � such that j� � zj � .Since �i is of class C3 on U, then we can find C1 functions ajkð�Þ, ¼ 1, . . . , ‘, j,

k¼ 1, . . . , n such that for all � 2 �

ajk

��� �@2�ð�Þ

@�j@�k<j�

2n2:

Denote by ðQIzÞkj the entries of the matrix QI

z i.e

QIz ¼ ðQI

zÞnk, j¼1 ðk ¼ column indexÞ:

We set for ð�, zÞ 2 Cn��

!ij ð�, zÞ ¼

@�ið�Þ

@�j�Xnk¼1

ajkð�k � zkÞ þ AXnk¼1

ðQIzÞkjð�k � zkÞ:

In view of Hefer’s Theorem 2.5.4 in [22] (see also Theorem 3.1 in [6]), there exist

C1-functions Hj(�, z) which are bounded away from zero such that the functions !�j ð�, zÞ

defined by

!�j ð�, zÞ ¼ Hjð�, zÞ �

@��ð�Þ

@�jþOðj� � zjÞ

are holomorphic in z 2 �.Now for �¼ i1, . . . , i‘, � and ð�, zÞ 2 ��� we set

�ð�, zÞ ¼ h!�ð�, zÞ, � � zi, F�ð�, zÞ ¼ �ð�, zÞ � ��ð�Þ:

The estimate (3.1) implies

2Re�ð�, zÞ � ��ð�Þ � ��ðzÞ þ�

2j� � zj2,

and consequently

2ReF�ð�, zÞ � ���ð�Þ � ��ðzÞ þ�

2j� � zj2 > 0, ð3:2Þ

for any ð�, zÞ 2 ��� ��n� sufficiently close to each other.Define

Q�ð�, zÞ ¼

!�ð�, zÞ

F�ð�, zÞ þ , > 0,

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then Q�ð�, zÞ 2 C1ð ��� ��Þ and the estimate (3.2) implies that

Re Q�ð�, zÞ, z� �

� �þ 1

� �¼ Re

h!�ð�, zÞ, z� �i þ F�ð�, zÞ þ

F�ð�, zÞ þ

� �

¼ Re���ð�Þ þ

F�ð�, zÞ

� �

¼ ð���ð�Þ þ ÞReF�ð�, zÞ þ

F�ð�, zÞ þ �� ��2 > 0: ð3:3Þ

Let sð�, zÞ ¼ ð�1 � z1, . . . , �n � znÞ be the section that defines the usual

Bochner–Martinelli kernels in Cn. With the two maps s(�, z) and Q�

ð�, zÞ we associate

the two (1, 0)-forms, also denoted s(�, z) and Q�ð�, zÞ,

sð�, zÞ ¼ hsð�, zÞ, d� � dzi, Q�ð�, zÞ ¼

!�ð�, zÞ

F�ð�, zÞ þ , d� � dz

� : ð3:4Þ

Estimate (3.3) together with Gi1ð�Þ ¼ � � � ¼ Gi‘ ð�Þ ¼ G�ð�Þ ¼ �n imply the existence of

the functions G�(�). Thus with respect to the multiple weights Q�ð�, zÞ, s(�, z) and

fG�ð�Þg�¼i1,..., i‘, � we define the multiple-weighted kernels for the local q-convex

intersection �I� . Now, without loss of generality, assume that I*¼ (1, 2, . . . , ‘þ 1).

Then for ð�, zÞ 2 ��� �� we define

KI� ð�, zÞ ¼ �Cn

Xa0þa1þ���þa‘þ1¼n�1

ðn� 1Þ!

a1! . . . a‘þ1!G

ða1Þ1 ð1þ hQ1

ð�, zÞ, z� �i . . .

Gða‘þ1Þ‘þ1 ð1þ hQ‘þ1

ð�, zÞ, z� �iÞ

^s ^ ðdsÞa0 ^ dQ1

� �a1^ � � � ^ dQ‘þ1

� �a‘þ1

ðþ hs, � � ziÞa0þ1,

PI� ð�, zÞ ¼

Xa0þa1þ���þa‘þ1¼n

ðn� 1Þ!

a1! . . . a‘þ1!G

ða1Þ1 1þ Q1

ð�, zÞ, z� �� �� �

. . .

Gða‘þ1Þ

‘þ1 1þ Q‘þ1 ð�, zÞ, z� �

� �� �^ dQ1

� �a1^ � � � ^ dQ‘þ1

� �a‘þ1,

where Cn¼ (�1)n(n�1)/2 (1/(2�i)n).For ð�, zÞ 2 ��I� ^

��I� , we have

dQ�ð�, zÞ ¼

dP

nj¼1!

�j ð�, zÞðd� � dzÞ

F�ð�, zÞ þ þdF�ð�, zÞ ^

Pnj¼1!

�j ð�, zÞðd� � dzÞ

ðF�ð�, zÞ þ Þ2

,

and

ðdQ�Þ

k¼

dP

nj¼1!

�j ðd� � dzÞ

�KðF� þ Þ

kþ k

dF� ^ dP

nj¼1!

�j ðd� � dzÞ

�k�1

ðF� þ Þkþ1

¼ O1

ðF� þ Þ2

� �:

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This together with estimates (3.2) and (3.4) imply that the kernel KI� ð�, zÞ satisfies the

estimate:

KI� ð�, zÞ

�� �� ¼ O ���ð�Þþ

ðj��ð�Þjþj��ðzÞjÞnþ1

j��zj2n�1

�, �, z 2 ��I� : ð3:5Þ

Furthermore, we have

KI� ð�, zÞ

�� �� ¼ O j��ðzÞj

nþ1j��zj2n�1

�, � 2 @�I� , z 2

��I� : ð3:6Þ

Estimate (3.5) together with the assumption on s show that, by the Lebesgue-dominatedconvergence theorem, we can pass to the limit ! 0 in the formula (2.1) for themultiple weights Q�

ð�, zÞ, s(�, z) and fG�ð�Þg�¼i1,..., i‘, �. The boundary integration isvanishing according to (3.6)

If we set

KI�0 ð�, zÞ ¼ lim

!0KI�

0 ð�, zÞ �, z 2 �I� , � 6¼ z,

then the above results together with the fact that ðPI� ð�,zÞÞr, s � 0, 0� r� n and s� q,

where ðPI� ð�, zÞÞr,s are the components of PI�

ð�, zÞ of type (r, s) in z (cf. [21] Lemma 2),enable us to have the following theorem:

Theorem 3.1 For any f 2 C1r, sð

��I� Þ, s� q, there is a positive constant cn,s such that

fðzÞ ¼ cn, s �@z

Z�2�

fð�Þ ^ KI�0 ð�, zÞ �

Z�2�

�@fð�Þ ^ KI�0 ð�, zÞ

� ,

In particular, if �@f ¼ 0, then

uðzÞ ¼ ð�1ÞsZ�2�

fð�Þ ^ KI�0 ð�, zÞ

is a continuous solution to the equation �@u ¼ f on �I� .

By using the standard regularization methods we obtain the following lemma:

Lemma 3.2 For any f 2 L10,sð�I� Þ with

�@f 2 L10,sþ1ð�I� Þs � q, we have:

fðzÞ ¼ cn, s �@z

Z�2�

fð�Þ ^ KI�0 ð�, zÞ �

Z�2�

�@fð�Þ ^ KI�0 ð�, zÞ

� :

The operator T s : L10,sð�I� Þ ! L1

0,s�1ð�I� Þ can be defined by

f ! ð�1ÞsZ�2�I�

fð�Þ ^ K1�0 ð�, zÞ, s � q: ð3:7Þ

The local Lp-boundedness follows from the classical theorem on integral operatorsonce the following lemma holds:

Lemma 3.3 There exists a constant C such that the kernel KI�0 ð�, zÞ satisfies the following

estimates: R�2�I�

KI�0 ð�, zÞ

�� ��dVð�Þ � C, for almost all z 2 �I� ,R�2�I�

KI�0 ð�, zÞ

�� ��dVðzÞ � C, for almost all z 2 �I� ,

Proof Similar to the proof of Lemma 2.6.1 in [5]. g

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From now on, �: E!M denotes a holomorphic line bundle over an n-dimensionalcomplex manifold M and ���M denotes a q-convex C3 intersection with �i,i¼ 1, . . . ,N, and U of Definition 1.2. Let � 2 @� be an arbitrary fixed point. Thenthere is a multi-index I �� of maximal length such that � 2 S��. By virtue of Lemma 2.4in [17], there exists a finite number of open sets u1, u2, . . . , um 2 M such that�� u1 [ u2 [ � � � [ um and each uj \�, 1 � j � m, is a local q-convex intersection.Moreover, we may assume that E is trivial over some coordinates neighbourhoodszj ¼ ðz1j , z

2j , . . . , z

nj Þ of each uj \�, 1 � j � m. A Hermitian metric along the fibers of E is

a system of positive C1-functions h¼ {hj}, each defined on uj, such that hj ¼ jeijj2hi on

ui \ uj, where eij is the system of transition functions of E. The curvature formassociated to the metric h is defined by �¼ {�j} such that�j ¼

ffiffiffiffiffiffiffi�1

p�@@ log hj ¼

ffiffiffiffiffiffiffi�1

p Pn�, �¼1 �j� ��dz

�j d �z

�j , where �j� �� ¼ @2 log hj=@z

�j @ �z

�j :

Definition 3.4 Let �: E!M be given as above. Then

(i) E is k-positive (resp. k-negative) at x 2 uj \�, if the formXn�,�¼1

�j� �� ð3:8Þ

is a Hermitian form on Tx(M) having at least n� kþ 1 positive (resp. negative)eigenvalues.

(ii) E is semi-positive (resp. semi-negative) at x 2 uj \�, if the form (3.8) is semi-definite Hermitian form on Tx(M).

(iii) E is semi-positive (resp. semi-negative) of type k at x 2 uj \�, if E is both semi-positive and k-positive (resp. semi-negative and k-negative) at x.

Let ds2 ¼Pn

�,�¼1 gj� ��dzaj d �z

�j be the Kahler metric defined on M. Denote by �r,sð�,EÞ

the space of E-valued (r, s)-forms and of class C1. For ’, 2 �r, sðM,EÞ we define alocal inner product at z 2 uj by

1

hj’jðzÞ ^ ? jðzÞ ¼ ð’ðzÞ, ðzÞÞdv,

with the Hodge star operator ? and the volume element dv are defined by ds2, (�, ) aC1 function on M independent of j. Let fðzÞ

�� �� ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðfðzÞ, fðzÞÞ

pfor f 2 �r, sð�,EÞ

and Lpr, sð�,EÞ denotes the Banach space of forms in �r,s(�,E) normed

kfðzÞkLpr, sð�,EÞ ¼ ð

R� jfðzÞjpdvÞ1=p <1, 1 � p <1, and kfðzÞkL1

r, sð�,EÞ ¼ ess subz2�jfðzÞjfor p¼1.

Let Tsj be the operators induced in L1

0, sð�Þ by the operator (3.7) for each � \ uj. Thatis there are bounded linear operators

T sf ¼Xmj¼1

jTsj f �\uj ,�� s � q, ð3:9Þ

where j are C1-functions with compact support in uj such thatPm

1 j ¼ 1.

Theorem 3.5 Let Ts be the linear operators defined by (3.9) and let f 2 L10, sð� \ uj,EÞ

with �@f ¼ 0, q� s, then there exists a form g ¼ Tsf 2 L10, s�1ð� \ uj,EÞ solving �@g ¼ f.

Moreover, if f 2 Lp0, sð� \ uj,EÞ there exists a positive constant Cs such that

kgkLp0, s�1

ð�\uj,EÞ � CskfkLp0, s

ð�\uj,EÞ, 1 � p � 1:

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4. Globalization

In this section, a global �@ solving operator with Lp-estimates, 1 p�1, will be obtained.

This will be done by means of pushing out technique of Kerzman [5] which involves the

following three main steps:Firstly, the local result, Theorem 3.5, enables us to make quantitatively a well-known

extension trick, Lemma 4.1, which in turn implies that one can solve �@� ¼ f (with Lp-

bounds) in a C3 q-convex intersection � such that �� �.

Lemma 4.1 Let � be given as above. Then, there exists another slightly larger C3

q-convex intersection ���M such that ����� and for any f 2 L10, sð�,EÞ with �@f ¼ 0,

s� q, there exist two bounded linear operators L1, L2, a form f ¼ L1f 2 L10, sð�,EÞ and a

form u ¼ L2f 2 L10, s�1ð�,EÞ such that:

(i) �@f ¼ 0 in �.(ii) f ¼ f� �@u in �(iii) If f 2 L

p0, sð�,EÞ, then f 2 L

p0,sð�,EÞ and u 2 L

p0,s�1ð�,EÞ with the estimates

kfkLp0, s

ðb�,EÞ� C1kfkLp

0, sð�,EÞ, ð4:1Þ

kukLp0, s�1

ð�,EÞ � C2kfkLp0, s

ð�,EÞ 1 � p � 1, ð4:2Þ

where the constants C1 and C2 are independent of f and p. If f is C1 in �, then

f is C1 in � and u is C1 in �.

Proof Let � 2 @�. Then there is a multi-index I�� of maximal length such that � 2 S��.

Let �I� denote the local q-convex intersection on which the �@-equation can be locally

solved with Lp-estimates. We may assume that �I� ¼ f��i1,..., ��i‘, ��? < 0g:

Then

@���[�2@�

f��? < 0g:

Since @� is compact, then there are finitely many neighbourhoods Wi ¼ f��i? < 0g of

�i 2 @�, i¼ 1, 2, . . . ,m covering @� such that for each �i 2 @� we have

Wi Vi��uji � U for a certain ji 2 I�, zjiðWiÞ��Cn. Let �i, i¼ 1, 2, . . . ,m, be a

partition of unity subordinate to {uji} such that �i 2 C10 ðf��i? þ �i < 0gÞ, where �i > 0 is

sufficiently small and 0 � �i �Pm

1 �i ¼ 1 in a neighbourhood Vi of @�. We enlarge � to

� in m step, defining

��i ¼ fz 2 � [ Vi : ð�1 � �

Xi

k¼1

�kÞðzÞ < 0, . . . , ð�N � �Xi

k¼1

�kÞðzÞ < 0g, i ¼ 1, 2, . . . ,m,

where � is a positive constant to be fixed later. We set ��0 ¼ � and � ¼ ��

m. Obviously

� ��i � ��

iþ1 � � � � � ��m ¼ �. In order to terminate the proof we need to the

following lemma:

Lemma 4.2 For any fi 2 L10, sð�

�i ,EÞ with �@fi ¼ 0, i¼ 1, 2, . . . ,m� 1, there exist two

forms fiþ1 2 L10, sð�

�iþ1,EÞ and ui 2 L1

0, s�1ð��i ,EÞ such that (i), (ii) and (iii) of Lemma 3.1

hold when f, f, u, � and � are replaced by fi, fiþ1, ui, ��i and ��

iþ1 respectively.

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Ending the proof of Lemma 4.1 Applying Lemma 4.2 m-times beginning with ��0 ¼ �,

f0¼ f and ending with ��m ¼ �, fm ¼ f we get:

f ¼ fm�1 � �@um�1

¼ fm�2 � �@um�2 � �@um�1

¼ f0 � �@u0 � �@u1 � � � � � �@um�1

¼ f� �@ðu0 þ u1 þ � � � þ um�1Þ

¼ f� �@u,

where we set u¼ u0þ u1þ � � � þ um�1. Collecting the estimates for fiþ1 and ui in each

step, we obtain the estimates (4.1) and (4.2).Clearly f and u are linear in f, and if f is C1 in � then f and u are C1 in � and �,

respectively. Since �i 2 C10 ðf��i? þ "

�i < 0gÞ, i¼ 1, 2, . . . ,m then there is a positive

constant b� such that jPm

k¼1 D��kðzÞj � b� for all z 2 U and for all multi-indices � with

j�j � 3. Thus, choosing � sufficiently small such that all ��i , i¼ 1, 2, . . . ,m, in particular

��m ¼ �, are C3 q-convex intersections. g

Secondly we will fit into � a strongly q-convex domain �1 with C3-boundary such

that ����1���. That is we have to prove the following lemma:

Lemma 4.3 For each neighbourhood Wi � M, i¼ 1, 2, . . . ,m, of @� and for �>0, there

exist a neighbourhood W0i Wi of @� and a C3 strongly q-convex domain �1 such that

����1���� ¼ fz 2 � [W0i : �1 < �, . . . , �N < �g

Proof Recall that � is defined by the functions �1, . . . , �N. For each �>0, let � be afixed nonnegative real C1 function on R such that, for all x 2 R, �(x)¼ �(� x)

jxj � �ðxÞ � jxj þ �, j 0�j � 1, 00� � 0 and �ðxÞ ¼ jxj if jxj � �/2. Moreover, we assume

that 0�ðxÞ > 0 if x>0 and 0�ðxÞ < 0 if x<0. We define as in Definition 4.2 in [22]

max�ðt, sÞ ¼ ððtþ sÞ=2Þ þ �ððt� sÞ=2Þ, t, s 2 R, and ’1¼ �1, ’2 ¼ max�ð�1, �2Þ, . . . , ’N ¼

max�ð’N�1, �NÞ. For �>0 we can choose positive numbers � ¼ �=ð2ðNþ 1ÞÞ, � ¼ �=2small enough and W0

i��W0i such that

����1fz 2 � [W00i : ð’N � �ÞðzÞ < 0g����:

According to Corollary 4.14 of [22], �1 is a strongly q-convex domain with C3 boundary

and by choosing � sufficiently small we can get ��������. g

Lastly, we apply our below global solvability of �@ with Lp-estimates on strongly

q-convex domains ofKahlermanifold [8] in conjunctionwith the results of the Lemma 4.1

to obtain the required global solution with Lp-estimates mentioned in Theorem 1.1.

Theorem 4.4 [8] Let E be a holomorphic line bundle over an n-dimensional Kahler

manifold M. Let D be a C3 strongly q-convex domains of M. If E is a semi-positive (resp.

semi-negative) of type k on �D, then for any f 2 L1r, sðD,EÞ with �@f ¼ 0, s� q and

rþ s� nþ k (resp. s� q and rþ s� n� k), there exist bounded linear operators Ts such

that � ¼ Tsf 2 L1r, s�1ðD,EÞ solving �@u ¼ f. Moreover, if f 2 Lp

r, sðD,EÞ, 1� p�1, there

exists a positive constant Cs such that k�kLpr, s�1

ðD,EÞ � kfkLpr, sðD,EÞ. The constant Cs is

independent of f and p. If f is C1, then so is �.

Proof of Theorem 1.1 Let � �� be the C3 q-convex intersection furnished by

Lemma 4.1. If f 2 L1r,sð�,EÞ with �@f ¼ 0, s� q, then Lemma 4.1 yields a form

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f ¼ L1f 2 L1r, sð�,EÞ and a form u ¼ L2f 2 L1

r,s�1ð�,EÞ such that: �@f ¼ 0; f ¼ f� �@u in �,

and (i), (ii), (iii), (4.1), (4.2) in that lemma are valid. We solve �@� ¼ f using Theorem 4.4.

Hence, � 2 L10, s�1ð�,EÞ and

�@� ¼ f ¼ f� �@u in �

the desired solution is g¼ �þ u. The Lp-estimates in Theorem 1.1 follows from those in

Lemma 4.1 and Theorem 4.4. � and u are linear in f and they are C1 if f is C1. g

References

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[2] Ramirez de Arellano, E., 1970, Ein Divisionproblem und Randintegraldarstellungen in der komplexenAnalysis. Mathematicsche Annalen, 184, 172–187.

[3] Henkin, G. M., 1969, Integral representations of functions which are holomorphic in strictlypseudoconvex domains and some applications (In Russian). Mathematicheskii Sbornik, 78, 611–632.

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[5] Kerzman, N., 1971, Holder and Lp-estimates for solutions of �@u ¼ f in strongly pseudo-convex domains.Communicatuions Pure and Applied Mathematics, 24, 301–380.

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im Cn. Mathematische Annalen, 280, 45–68.

[13] Michel, J. and Perotti, A., 1990, Ck-regularity for the �@-equation on strictly pseudoconvex domains withpiecewise smooth boundaries. Mathematische Zeitscheift, 203, 415–428.

[14] Menini, C., 1997, Estimations pour la reselution du �@ sur une intersection d’ouvert strictementpseudoconvexes. Mathematische Zeitscheift, 225, 87–93.

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