Publicacions
Ma emá iques,
Vol
35
(1991),
559-569
.
AN
INTEGRAL
FORMULA
ON
SUBMANIFOLDS
OF
DOMAINS
OF
Cn
Abs ac
A
Bochne -Ma inelli-Koppelman
ype
in eg al
o mula
on
submani olds
o
pseudocon ex
domains
in
C'
is
de i ed
;
he
esul
gi es,
in
pa icula ,
in eg al
o mulas
on
S ein
mani olds
.
The
me hod
o
in eg al
ep esen a ions
in se e al
complex
a iables
has
been
p o ed
o
be
qui e
e icien
in
cons uc ing
holomo ph_ic
unc ions
and
mo e
gene al analy ic objec s
(di e en ial
o ms
sol ing
he
á-equa ion,
sec ions o
holomo phic
ec o
bundles
e c)
;
see,
o
example,
Henkin and
Lei e e
[4],
[5],
Henkin and
Polyako
[6]
and
Range
[10]
.
This
me hod
e ol es
abou
Bochne -
Ma inelli-Koppelman's
in eg al
o mula
:
i
D
C
C''
ís
a
bounded
domain
in
C'
wi h
smoo h
bounda y
óD
hen
(1
.1)

.=
uAK,-
á
.AK
q
+8(J
UAK
y
-
1)
D
D

D
o
e e y
(0,
q)- o m
u
wi h
C
1
-coe icien s
on D,
whe e
K
9
a e app op ia ely
cons uc ed
ke nels
(see
O elid
[9])
.
The
in eg al
o mula
(1
.1)
canbe
modi ied
o
p oduce
a
a ie y o
o he
o mulas
wi h
which
a ious
p oblems
(such
as
he
á-equa ion
and
in e pola ion
p oblems)
can
be
sol ed
;
hus
(1
.1) is
he
i s
main
s ep
in
se e al
cons uc ions
.
The
pu pose
o his
pape
is
o
cons uc
an
analogue
o
(1
.1)
i
D
is
eplaced
by
a
_submani old
M
o
D
.
Mo e
p ecisely
le
D
_be
a
bounded
domain
in
C'
and
D
a
pseudocon ex
one
wi h
D
C
_D
;
le
M
be
a
(closed
and
comp_lex)
submani old
o
D
and
le
M
=
:
D
l
M
.
Assume
ha
áM
=
(0D)
n
M
_
is
smoo h
.
Then
we
will
cons uc
ke nels
K,«,
z)
de ined
o
z)
E
M
x
M
wi h
(
~
z so
ha
(1
.2)

u
(z)
=

U«)
A
Kq«,
z)-
SEaM
TELEMACHOS
HATZIAFRATIS
1
.
In oduc ion
CEM

S
-bu«)
A
Kq
(S,
z)
+
az
( EM
U(S)
A
K9-1
(
S,
z))
56
0

T
.
HATZIAFRATIS
o
u
E
Cho
a)
(M)
and z
E
M
.
In
[3]
we
de i ed
an
in eg al
o mula
like (1
.2)
in
he
case
M
is
a
comple e
in e sec ion
(i .e
.,
i
he e
exis
unc ions
hl,
..
.,
h
p
,
holomo phic
on
D
_
,
so
ha
M
=
{hl
=
. . .
=
h p
=
0}
and
dhl
n
.
.
.
n
dh
p
:~
0
on
M)
;
he cons uc ion
in
he
gene al
case
(Le
.,
when
M
is
no
necessa ily a
comple e
in e sec ion)
will
be
based
on
he
esul
o
[3]
.
A
simila
cons uc ion
was
ca ied
ou
by
Be nd sson
[1]
.
As
a
ma e
o
ac
ou
ke nels
a e
equi alen
o
hose
o
Be nd sson
bu
w i en
in
a
di e en
way
.
We
hink
howe e
ha
ou
e sion
o
he
cons uc ion
as well as
he
di e en
p oo
o
he
in eg al
o mula
a e o
u he
in e es
.
He e
is
an
ou line
o
he
cons uc ion
.
Fi _s
we
co e
M
by
su icien ly
small
open
se s
{U,}
(open
in
C
-
)
so
ha
each
M
n
U
o
is
a
comple e
in e sec ion
.
Then
he
esul
in
[3]
gi es ke nels
K9
o
which
(1
.2)
holds
in
M
n
U,
.
Such
ke nels
a e
no
unique
in
he
sense
ha
he e
a e
ce ain
choices
ha
can
be
made,
in
pa icula ,
K9
depends
on
He e
decomposi ions
o
he
holomo phic
unc ions
which
de ine
M
n
U
o
(as
he
se
o
he
common
ze os)
.
Bu
using
a
esul
o
Be nd sson
[1]
we
show
ha such
He e
decomposi ions
can
be
chosen
app op ia ely
so
ha
K9
=K9
onmnu,nu,
;
hus
we
can
de ine
a
global ke nel
K
9
(by
se ing
K
9
=
:
K9
on
M
n
U,)
.
Then
we
ha e
o
show
ha
(1
.2)
holds
;
and
his
is
done
along
he
same
lines
as in
[3]]
.
He e
is
an
ou line
o
his
p oo
in
he
case
u
is
a
holomo phic
unc ion
on
M
.
In
his
case
we
ha e
o
show
ha
(1
.3)

u(z)
=
~

u«)Ko«,
z),

z
E
M
CEa~u
Fix
a z
E
M
and
pick
a
U
o
wi h
z
E
U,
.
Bu ,
as
a
compu a ion
shows,
dCK0
=
0
( o
each
T
and
(
7~
z)
and
hence
dC[u«)Kó
«,
z)]
=
0
;
hus
(1
.3) is
equi alen
( ia
S okes'
heo em)
o
u(?)
=

u«)Ko«,
z)
CEaU,
which
holds
by
he
esul
o
[3]
(o
[2]
o
ha
case)
since
Ko
=
Kó
in
U
n
M
.
Ou
esul
gi es,
in
pa icula ,
in eg al
o mulas
o
domains
D
CC
X,
i
X
is
a
S ein
mani old,
ia
he
heo em
ha
S ein
mani olds
admi
embedding
in
some
ON
.
In eg al
o mulas
on
S ein
mani olds
ha e been
cons uc ed
p e iously,
using
di e en
echniques,
by
Henkin
and
Lei e e
[4]
.
Rela ed
is
also
he
wo k
o
Be nd sson
[1],
Ho mann
[7],
Palm
[9]
and
S ou
[11]
.
AN
INTEGRAL
FORMULA
ON
SUBMANIFOLDS

561
2
.
No a ion
Le
h
=
(h1,
. . . .
h
p
)
whe e
he
his a e
holomo phic
unc ions
de ined
in
some
open
se
o
Cn
and
suppose
ha
{hij
(~, z),
j
=
1,
. . . ,
n}
is
a He e
decomposi-
ion o
h
i
,
Le
.,
h?j
«,
z)
is
holomo phic
in
bo h
(
and
z
and
n
hi(S)
-
hi(z)
=
E
hij(S,z)(Sj
-
zj),i
=
1,
. .
.,p
.
j-1
We
associa e
o
hese da a he
ollowing
di e en ial
o ms
.
Fi s
n-P-1
5D(2
.1)

1
ah«,
z
)
=
cde [hlj,
. . .
,
h
p
j,
yj,
(C
+

z)yj]
whe e
y
=
(yl,
. . . ,
yn
)
is
some
smoo h
unc ion
(we
will
be mo e
speci ic
abou
y
la e )
and
c
is
no malizing
cons an
:
c
=
(-1)
-(-2-
1)

1

1

whe e
m
=
:
(2- i
^~
'
T c!
n-
p
.
In
he
de e minan
o
(2
.1),
j uns
om
j
=
1
up
o
j
=
n
o ming
he
(gis
+
az)yl
n
ows
o
i
;
also
he
column
o
di e en ial
o ms
in
he
de e minan
(n
-p
-
1)- imes
(as
i is
de e minan s
see
[4]
.
Also
de ine
(2
.2)
whe e
(a~
+
"
C7z)yn
n
-P
1
8h
1
h
ah
¡oh(C)12
de

á~P
,
d(j
Ioh(S)1
2
=

y
:

1
a(h1
. . . .
.
,
hP)
(~)
2
.
1<j
l
<
. .
.<jp<n

a(Sjl,
. . . ,
(j )
indica ed)
;
o
p ope ies
o
such
o
cou se
Q'«)
is
de ined
o
(
wi h
Joh(«
:yÉ
0
.
Wi h
his
no a ion
se
K
h
(~,
z)
=
ah«~
z)
n
~3h(~)
.
Le
ce,
be
he
pa
o
a
h
which
is
o
ype
(0,
q)
in
z
and
(0,
n
-p-
q
-
1)
in
~,
i
.
e
.,
Also
le
Kq
(~,
z)
=
a9
«,
z)
nQ
h
(~)
.
-
)

_~-
_
c q
=
c
C
n

q
-
1

de [hlj,
. . . .
h
p
j,
yj,
azyj,

aCyj
is
epea ed
O
cou se
Kq
depends
on
he
choice
o
he
He e
decomposi ions
{hij}
o
hi,
al hough
we
do no
indíca e
his
in
he
no a ion
;
bu
i
will
be
clea
om
he
con ex
which
choice
we
will
be
using
in
each
case
.
Also
we
ha e
been
agueabou
he
domains
in
which
he
abo e
unc ions
and
di e en ial
o ms
a e
de ined,
because
he e,
we
simply in oduced
no a ion
and
we
will
be
e y
speci ic
abou
his
la e
.
56
2

T
.
HATZIAFRATIS
3
.
Cons uc ion
o
he
ke nels
_
Fi s
we
desc ibe
he
se ing
.
Le_
D,
D
be
domains
in
C"
wi h
D
bound_ed,
D
pseudocon ex
and
D
C
D
.
Le
M
be
a
closed
complex
su_bmani old
o
D
o
(complex)
dimension
m
._
Se
M
=
DnM
and
aM
=
(OD)nM
and assume
ha
aD
is
smoo h
and
ha
M
mee s
aD
ans e sally
so
ha
aM
is
also
smoo h
.
Le
,y
=
(-y,,
. .
.,
-yn)
:
D
x
D
-
{~
=
z}
-
C'
be
a
smoo h
map
sa is ying
(3
.1)
((j

zj)
=
1
and

(-
z'
o
0
<
j(
-
zi
<
6
o
some
small
6
>
0
j=1

zl2
n

(j
7j

^y
.,
I-
(an
examPle
o

ls
g
.
l en
bY

-z'
all
(,
z
wi h
(

z)
.
7

hj
=

Z
In
his
se ing
we
will
now
con
_
s uc
he
ke nels
.
Le
{U
a
}
be a
se
o
small
con ex
open
se s
o
en
so
ha
M
C
U
a
U
aand
mo eo e
le
ha
=
(he,',
...
.
ha)
:
D
~
CP,
p=
:
n-
m,
be
holomo phic
maps
so
ha
MnU,
=
{z
E
U
a
:
ha
(z)
=
0} and
_

17h
a
l
~
0on
M
_
nU
a
;
ha
such
unc ions
há
exis
ollows
om
Ca an's
Theo em
A
since
D
is
assumed
o
be
pseudocon ex
.
Fu he mo e
he e
exis
p x p
ma ices
A,T
=
[(A ,)ik]1<i,k<p
o
unc ions
hi
~

í
hi
(A,T)ik,
holomo phic
in
U
a
n
UT,
so
ha
A,
T
Le
.,
(3
.2)
The
exis ence
o
such
ma ices
A
.T
ollows
om
Ca an's
Theo em
B
since
U
a
n
U
T
is
con ex
.
Lemma
1
.
ah
;
ah,
;
_
A
uT

~eu
a
nuTnM
.
ahp ahp
ac
;
aS
;
P oo
::
Di e en ia ing
(3
.2)
we
ob ain
ahá
a5j
ahk
(A,T)ik
-
+
k=1
a(j
k=1
p
há
=
E(A,T)ikhk,
i
=
1,
. .
.
,
p
.
k=1
hT a(AoT)ik
.
k
a(j
on
u,
nu
T
,
Since
hk
=
0
o
C
E
U
Q
n
U
T
n
M,
he
o mula
o
he
lemma
ollows
immedi-
a ely
.
The
ollowing
lemma
is
p o ed
by
Be nd sson
[1,
p
.414]
;
i_ s
P oo
is
based
on
Ca an's
Theo em
B,
using
again
he
pseudocon exi y
o
D
.
Lemma
2
.

The e
exis
unc ions
h
~
«,
z),
i
=
1,
. . . .
p,
j
=
1,
. . . ,
n,
holo-
mo phic
in
(C,
z)
E
(U,
n
M)
x
M
so
ha
hij
i

ihlj
(3
.3)
1
h-
j
j

h-
pi
Fu he mo e
h
~
«,
z)
Na e
holomo phic
ex ensions
in
C
in
a
neighbou hood
(in
en)
o
U
a
n
M,
sa is ying
Then
T
P oo
.
De ining
AN
INTEGRAL
FORMULA
ON
SUBMANIFOLDS

563
n
o j=1,
_,
n,CEu,nu,nM,zEM
(3
.4)

1
:
h
íj
(C,
z)
(SJ
-
zj)
=
h°
«)
o
zE
M
.
j=1
Lemma
3
.
Le
A
E
CPXP
and
B,
C
E
CnxP
so
ha
A
-
BT
=
C
T
(T
deno es
anspose)
.
and,
hence,
a
hh
'
=
de (A,)
de [C,
* *
*]
=
de (A)
de [B,
* *
*]
whe e
*
*
*
deno e
app op ia e
di e en ial
o ms
( he
same
on
bo h
sides
o
he
equa ion)
so
ha
he
ma ices
[B,
* *
*]
and
[C,
* *
*]
a e
n
x
n
.
P oo
:
This
is
a
s aigh o wa d
compu a ion
based
on
he
mul ilinea i y
o
he
de e minan s
(as
unc ions
o
hei
columns)
and
he
de ini ion
o
he de-
e minan s
.
(This
gene alizes
he
ac
ha
he
de e minan
o
he
p oduc
o
squa e
ma ices
is
equal
o
he
p oduc
o
he
de e minan s
o
hese
ma i-
ces)
.
Lemma
4
.
I
aho
is
he
o m
(2
.1)
associa ed
o
he
He e
decomposi ions
{h9'j
ande¿
h
is
de ned
simila ly,
hen
(3
.5)

ah'
=
de (A,
T
)
.
a
h
'
o (
E
U
o
n
U
T
n
M,
z
E
M
and
simila ly
H
T
,
we
ob ain
om
Lemma
2
ha
A
oT
-
(HT)
T
=
(Ha
)
T
o
(E
U,
n
U,
n
ú,
z
E
M
;
hus
Lemma
3
applies
and
gi es
(3
.5)
.

564

T
.
HATZIAFRATIS
Lemma
5
.
We
ha e
(3
.6)

ph
=
P oo
..
By
Lemma
1
we
ob ain
The e o e
1
de (

ph
°

E
U
Q
n
U
Tn
M
.
ah°

ahi
=A
aT

]
~N9k
1<i,k<p

N
.9k
1<i,k<p
.
a(hi,
. .
.,hP)
=
de (AQT)

a(hi
...
.,hP)
,
~
;
p
)

a«il
,
.. . ,
~
;p)
and
(3
.7)

j7h
o
l
2
=
1
de (A,)1
2
jVhT1
2
( his
also
shows
ha
de (A,
T
) ,-~
0
on
U
Qn
U
T
n
M)
.
Also
se ing
and
simila ly
o
BT
we
see
ha
Lemma
1
gi es
Hence,
by
Lemma
3,
8h1
aC,
. . .
acá
&h,'
ah,
aC

. . .
aCn
1T
.
(
B
T
)
T
=
(Bu
)T
(3
.8)

de

ah1
,
. . . ,
ah
P
C
;

a(
;
,
d

=
de (A,)
-
de
Now
(3
.6)
ollows
om
(2
.2),
(3
.7)
and
(3
.8)
.
We
a e
now
eady
o de ine
he
ke nels
:
n-p
ahi

ahT
~
,
a(,
. . .
,

,
dj
a(j
K«,
z)
=
:
K
h
'
(~,
z)
=
ah «,
z)
A
Qho
(~)
o
(
E
U
Qn
M,
z
E
M,
~
,~
z
.
By
Lemmms
4and
5,
i
ollows
ha
K(~,
z)
is
well-de ined
o

E
M,
z
E
M,
z
.
Simila ly
we
de ine
K
9
=
aq

,Q
h
o
q
>
0 and
K_1
=
0
.
AN
INTEGRAL
FORMULA
ON
SUBMANIFOLDS

565
4
.
The
in eg al
o mula
Wi h
he
ke nels
jus
cons uc ed
we
will
p o e
he
ollowing
heo em
.
Theo em
1
.
I
u
E
Cho
e)
(M)
(0
<
q
<
m)
and
z
E
M
hen
u(z)
=
~

u«)
n
Kq(C,
z)-
(EaM
Fi s
a
lemma
:
Lemma
6
.
We
ha e
(
,9
c
+
c
9z)K(S,
z)
=
0
.
P oo
.
:
I
su ices
o
show
(4
.1)

(8S
+
á
z
)
K
ho
(C,
z)
=
0
o
cE
U
o
l
M,
zE
M,
c
=~
z
.
Now
obse e
ha
(4
.2)

«I
-
zj)a
ho
(S,
z)
=
de
P oo
o
Theo em
1
:
I
su ices
o
show
ha
o
W
E
(Có
(M))(m,
m
-
q)
.
CEM
-
bu«)
A
K9
(S,
z)
+
áz
IEM
u«)
A
K
9
-1
(C,
z)]
.
(4
.3)

J

U(Z)
A
W(z)
=
J

EBM
u
n
K
q
A
W-
zEM

zEM
~
n,-
-1
hay

.
.
.

ha
y
7j

(ac
+
az)yi
J
2<j<n
(in
he
abo e
de e minan
j
uns
om
j
=
2 o
j
=
n
o ming
he
2nd
up o
he
n h
ow
o
i )
.
We
ob ained
(4
.2)
in
he
ollowing
way
:
(SI
-
z1) mul iplied
he
i s
ow
o
he
de e minan
which
de ines
aho
«,
z)
and
hen
we
added
o
ha
i s
ow
he
j h- ow
mul iplied
by
(Si
-
zj)
(j
=
2,
.
.
. ,
n)
.
Then
(4
.2)
ollows in
iew
o
he
i s
o
he
assump ions
in (3
.1)
and
(3
.4)
.
Now
(4
.2)
easily
implies
ha
(á
C
+
Ó
z
)
a
ho
=
0
;
since,
mo eo e ,
5CQ«)
=
0
(by
[2,
Co olla y
1,
p
.
76]),
we
ob ain
(4
.1)
which
comple es
he
p oo
o
he
lemma
.
áuAK
y
Acp+
J

(az(]

uAK
9
-1))A9
zemCEM

zEM

(EM
56
6

T
.
HATZIAFRATIS
Le
us
poin
ou
he
way
in
which
he
a ious
o ms
in
he
igh -hand
side
o (4
.3)
dependon
he
a iables
~
and
z
:
u_

«5u«)

u«),
K
9
=
K,«,
z),
K
9
-1
=
Kq-1
«,
z),
cp
=
;o(z)
and
8u
=

.
By
deg ee
easons,
(4
.3)
is
equi alen
. o
(4
.4)

L

u(z)
n
w(z)
=

u
n
K
A
cp-
EM
ó(MxM)
(in
ob aining
(4
.4)
we
used
also
he
ac
ha
J
lS~,z)

uAKAW=

uAKAW
E(óM)xM

8
(M
x
M)
which
holds
since
u
has
compac
suppo
in
M)
.
By
S okes'
o mula
(4
.5)
a(mxm)uAKAW=Ld[UAKAW]+L
uAKA<p
whe e
C
E
=
{((,
z)
E
M
x
M
:
j(
-
zi
=
E}
.
By
Lemma
6
and
deg ee
easons
L
buAKA(p+~
(áz(J
uAK))AW
MxM

M M
1
~~

d[uAKAW]
=

auAKAW-(-1)9J

uAKn77cp
;
MxM-{~~-z~<E}
Mxm

MxM
hence
(4
.4)
will
ollow
om
(4
.5)
as
soon
as
we
es ablish
he
ollowing
(4
.6)

lim

u
A
K
A
;o
=

u(z)
n
W(z)
.
E-0
£,z)EQ

.
EM
Bu
we
may
assume
wi hou
loss
o
gene ali y
ha
supp(u)
C
Ua
o
some
u
.
Then
o
E
small
enough
uAKAcw=
uAK
h
'n
;o
C
E

C
E
and
(4
.6)
ollows
om
(4
.7)

lim

uA
Kho
A
cp
=

u
(z)
1
w(z)
E-0
c
E

IzEM
Bu
(4
.7)
is
exac ly
wha
is
p o ed
in
[3,
p
.
339
341]
(in
ha pa
o he
p oo ,
(3
.4)
plays
an
impo an
ole)
.
This comple es he p oo
o
he
heo em
.
his
is
an
analogue
o
Cauchy's
in eg al
o mula
on
M
.
Con inuing
o
assume
D
o
be
s ic ly
pseudocon ex
and
he
gj's
as
abo e,
le
us
se
and
AN
INTEGRAL
FORMULA
ON
SUBMANIFOLDS

567
As
we
men ioned
in
he
in oduc ion
he
in eg al
o mula
ha
we
p o ed
can be
used
o
de i e
a ious
o he
in eg al
o mulas
.
He e
we
discuss
wo
examples
.
Wi h
no a ion
as
be o e
assume
u he mo e
ha
D
is
a
s ic ly
pseudocon ex
domain
.
Then,
by a
classical
cons uc ion
o
Henkin and Rami ez
(see
[51),
he e
exis
unc ions
g
j
(~, z),
j
=
1,
...
,
n
de ined
o
«,
z)
E
(aD)
x
D,
which
a e
smoo h
in (
and
holomo phic
in
z,
so
ha
G(~,
z)
_
:
~~
i
(~
j
-z
j
)gj
«,
z)
0
.
Thus
i
we
se
n-p-1
C«,
z)
_
[C(~,
z)1
-p
de [hi ,
. .
. ;
h'i,
g~,
DZgj
1
A
aho
(C)
o
(
E
U
naM
and
z
E
M
hen
C«,
z)
is
w
ell-de ined
o

E
aM
and
z
E
M,
holomo phic
in z
and
(as
i
ollows' om
heo em
1)
i
ep oduces
holomo phic
unc ions
on
M,
Le,
o
E
O(M)
l
C(M)we
Na e
?7,(S,z,~)=(1-
.1)
i'-
zI2+aG(S'z)
o (EaD,zeD,
.1E[0,11
L9(~,z,, )=CCn-p-1/
4
5
.
Applica ions
(z)
=
CEaM
(0C«,
z),
9
n-P-9-1
de
[hi ,
. . . ,
hpi,
l

a

;,
(aS~+
da

n
ah~

E
U
o
n
aM,
z
E
M
;
hen
L
.«,
z,
.
) is
well-de ined
o
-C
E
8M,
z
E
M
.
Also
se
B,«,
z)
=
:
L«,
z,
)1>,-o
;
zeM
;
hen
i is
clea
ha
B,(~,
z)
is
de ined
o
~,
z
E
M,
(
:,A
z
.

Using
he
abo e
ke nels
we
de ine
he
ollowing
ope a o s
:
T
9
u(z)
=

u(S)
A
Le
-
1( S,
z, )
+

u
(S)
n
BQ-1(S,z),z
E
M
(C,a)E(aM)X[0,1]

CEM