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_ It is clear that the right-hand side of this relation tends to the right-hand side of (1.1.1) when 8 ~O. Consider the left.hand side. Since de 1\ dC/2i is the Lebesgue measure on (}1, Stokes' formula gives
~ ~ J J 1'-,,-_ ~ J "-zl-_ C- z
Further
=
el
Hence
lim
z) dC =
"-'1=_
f(Ci - J(z) dC -
(t -
Z
~
82
J ,,-., is called the O(D)-hull of K. If K
=
gz, then K
is oalled O(D)-oontJez.
1.8.8. Proposition. For every compact set K c: c: I1tt, the O( oral-hull i8 conlainetl in the convu hull oj K (convex in the geometric sense oj 0" = Pl. j.
20
1. Elementary Properties
+
Proof. Let x,(z) be the real coordinates of z E qyn such that z1 = xl(z) iXj+n(z). If the point w E qyn does not belong to the convex hull of K, then we can find real num2ft
bers 111' ••• ,112ft such that n
2n
l: x,(w) Yl
= 0, but ~ x,(z) y,
j-1
j-1
< 0 if z E K. Set VI := lit + iYj+n'
Then I(z) : = exp (l: zIti,) is a holomorphic function in
If(z)1
.
(1.3.2)
then
Proof. Let r be the multi-radius of P. For 0 < t < 1 we denote by P t the polydisc of multi-radius tr with centre at O. It is sufficient to prove that P e w ~ D for all wE and 0 t 1. To do this we fix some Wo E iZ and 0 to 1. Without loss of generality we can assume that P is bounded. Then it follows from Cauchy's inequality (Corollary 1.1.12) and (1.3.1) that, for all f E O(D) and all multi-orders Ie,
+ <
i8 al80 compact. (iv) For every compact Bub8et K c c D, (1.3.3) is valid.
1.3. Domains of holomorphy
21
(v) For every inJinite set X cD, which is di8crete in D, there exi8ts anI € ()(D) which i8 unbounded on X.
Proof. The implication (ii) ~ (i) is triviaJ. (i) ~ (iv) according to Corollary l.3.6. Since, by Corollary 1.3.4, for every compact K c: c: D, is bounded, we obtain that (iv) ~ (iii). Now we prove that (iii) ~ (ii). From condition (iii) we obtain a sequence {Em}f of O(D)-convex compact sets Km C c: D such that every compact subset of D is contained in some Km. Let X be a countable dense set in D, and let {E(tft)}:_l be a sequence of points in X which conta.ins every point in X infinitely many times. Denote by B". the maximal open ball centered at ~m) which belongs to D. Fix n(m) E Bm " Xm. Since all Km are O(D)-convex, we can then find functionsJ". E O(D) such tha.t fm(n(fTI» = 1, but Ifml 1 in Km. By raising these funotions to high powers we may arrange that Im(1} (fTI» = 1 and Ifml 1/2 m in Em. Moreover, we can assume that every 1m is not identically 1 in any of the components of D. Then the infinite product
:iZ
O.
In
e. E PO(D,} . •
1.4.11. Theorem. Let G c: Q)'" and D c: ()ft be open sets, and let '1: G ---+ D be a holomorphic map. Then (i) If e E PO(D), then (! 0 J E PO(G). (ii) Suppose that I i8 biholomorphic from G cmto D. If e E CO(D), then e E PO(D) if and omy iJ (! 0 J E PO(O). If e E 02(D), then g is strictly plurisubharmonic in D if and only if (! 0 f is serictly pluriBubharmonic in G. Proof. If e E OI(D),
Z EO,
w
= f(z}
and; E C"', then by Proposition 1.1.16 (ii) (1.4.15)
Since for biholomorphicl the Jacobi matrix (8w./8z f }".J is invertible, and by Theorem 1.4.7 (ii), this implies part (ii). Further, it follows from (1.4.15) and Theorem 1.4.7 (i) that part (i) is valid for a.lle E PO(D) n 02(D). To complete the proof of part (i), we consider an arbitrary e E PO(D). By Theorem 1.4.10 we can then find a sequence of plurisubharmonic Ooo·functions fl' which converges to fl uniformly on every compact subset of D. Then all (!. 0 J E POCO) and it follows from Proposition 1.4.9 (iv) that
e J E PO(G) • • 0
Remark. In view of part (ii) of this theorem, continuous plurisubharmonic functions and strictly plurisubharmonic Ot·functions can be defined also on complex manifolds (cp. Definition 1.1.21). If Y is a complex submanifold of a complex manifold X and Q is a continuous plurisubharmonic (strictly plurisubbarmonic ot) function on X, then
1.4. Plurisu bharmonic functions
27
it is clear (by Definition 1.4.6) that the restriction of f! to Y is plurisubharmonic (strictly plurisubharmonic).
1.4.12. Tbeorem. Let G C Ill; be an open 8et, and let rp: G -»0 B1 be a OI.Junction which is CQ1l,vex, tkat i8, 1&
I:
01 (t)
~ 0 Jor all t E G and y
_C{J_ Y.Y"
. . ,,-1 ot. ot"
E
IJ.t.
(1.4.16)
S'UlppOBe tkat D C on i8 open and f!1, •.• ,f!t E PO(D) 8uch tkat(l?1(Z), ... ,f!t(Z») E G lor all zED. Define e(z) : =- f1i(et(z), ... , et(z»Jor ZED. Then (i) If min "-l ..... k
8cp(t) 8t.,
~o for all t E G ,
(1.4.17)
e
then E PO(D). (ii) IJ (1.4.17) i8 fulfilled anil, moreover,
max ocp(t)
>0
for all t
EG
,
(1.4.17')
.-1, ... ,1: 8t,
and if the fU'Mtiona el' ... , et are 0 2and 8trictly pluri8Ubharmooic in D, tken e is a 8trictly plurisubharmonic 02-junction in D. Proof. We only have to prove this for n = 1. In view of the Approximation Theorem 1.4.10 we can also assume that the functions el are O· (cp. the end of the proof of Theorem 1.4.11). Now let Z = X], + ixa ED, XI E 1R1, and t = (£'1(Z), ••• ,et{z». Then
82e(z) = o~
i
02(0) Re L --CI Ct 1.1:=1 0Cf
8Ct
=
(Rx(C), x(C) ,
30
1. Element&ry properties
where z(C) = (~(C), ... ,Z2n(C) and h,) is the scalar product in R2n. Together with (1.4.22) this implies that
e
0
+ Rz(C), x(C) + o(ICI
v(C) = (z(C)
2I
)
for C ~ 0 .
(1.4.24)
If e E ([J" suoh that x(e) is an eigenvector of the matrix R, with eigenvalue,t, then it follows from (1.4.23) that z(ie) is an eigenvector of R, with eigenvalue -,to Consequently, since B is symmetric, we can find vectors el , ..• , en E and numbers A1 , ... ,An ~ 0 such that the vectors x(e1 }, ... , x(e n ), x(i~), ... ,x(ie n) form an orthonormal basis in Il'/." and
en
(1.4.25) Then the veotors
tl
e,. form an orthonormal basis in
, ... ,
(f)n
and the complex-linear
n
ma.p u: (f)n
~
en defined by u(C) :=
I: Cfl3f, C E Oft,
is unitary. The corresponding map
j-l
of real coordinates x(C) ~ x(u(C)) is defined by the matrix
(~~(~;
Xl(~) ... xl(e,,)
U:=
.:.
x1(ie1) ... xl(ie n )
)
~~.(~~) ~.~(;",; :.. ''''~;ie~) .
Since U is orthogonal, and since by (1.4.25)
U-IRU =
C·~·A.-AI"~ -A)'
it follows from (1.4.24) that for lei () 0
V
0
u(C)
"
= I:
(1
j-l
~0
+ AI) IXf(CW· + j-1 I: (1 11.
We conolude the proof by setting t : = v
0
A.s) IZJ+n(C)I'
+ 0(1C11) •
u. •
on
1.4.18. Dermition. A subset T c eft is said to be a real (complex) plane in if, for Z E T, T - z is a real-(complex-)linear subspace of (f)n. Complex planes of complex dimension 1 are called complex lines. A real plane P in C" is said to be totally real if there is no complex line which is contained in P. A O1-submanifold of 0" is called totally real if the real tangent plane at every point of X is totally real. 1.4.17. Proposition. (i) 1/ X is a totally real O1-submanifoZd of (f)n, then the real dimension 01 X is :::;: n. (ti) II X is a Ol-submanifold 01 and the real tangent plane 01 X at some point E E X is totally real, then there exi8ts a neighbourhood U 01 ~ 8uch that U n X is totally real.
en
The simple proof is left to the reader.
en.
1.4.18. Theorem. Let X be a totaUy real Ol-submanilold 01 Then there exi8t a neighbourhood U oj X and a non-negative strictZy plurisubkarmonic 02-function e in U suck that X
=
{z E U: (}(z)
=
O}
=
{z E U: de(z)
=
O} •
(1.4.26)
31
1.4. Plurisubharmonic functions
Proof. For z
we denote by f,(X) the real tangent plane of X at z, and set
E X,
,.. tz(X) :=
{C Eon: C+ z E t.(X)}
•
Let p. be the orthogonal real-linear projection from en onto the real-orthogonal complement of the real-linear subspace t,(X). Since X is (Jl, then p. depends continuously on z E X and
PAC - z) = o(IC - zl) for C, z EX, IC - zl
~0
,
(1.4.27)
uniformly in every compact subset of X. Consequently,
(P.(C -
z), C-
Z)B
o(IC - z12) for C, z EX,
=
Ie -
zl
~0 ,
(1.4.28)
uniformly in every compact subset of X, where h ')R denotes the real scalar product in en = Rh. By Whitney's extension theorem (see, for example, MALORANGlIl [2]), it follows from (1.4.27) and (1.4.28) that there is a real-valued Ol-function e on e" such that at Z E X we have the Taylor expansion
e(C)
z), C
(P.(C -
=
- Z)B
+ o(IC -
z12) •
(1.4.29)
This implies that, for some neighbourhood U of X, (1.4.26) holds. It remains to prove that e is strictly plurisubha.rmonic in some neighbourhood. of X. By Definitions 1.4.6 (ii) and 1.4.5 we have to prove that for every point z E X and all 0 =1= 10 E on
+
8le(z AW) 8A 81
Let it = Yl
=
+ iys,
(1/4) (81/8y~ 8le(z
+
I
>0,
it
(1.4.30)
E ()1 •
AmoO
YI E
R. Then it fonowa from
(1.4.29) and the relation 81/8).
aI
2
8 /81/i) that
+ ).w) I
8). 8A
= -1 4
).-0
[(P.(1o),
w)s
. . + (P,(lW), lW)R] •
Since PI is a real-orthogonal projection, it follows that 8'e(z
+ ).w) I
8A 8),
=
1=0
~ [IP.(w)11 + IP,(iw)l'] .
(1.4.31)
4
Since X is totally real and W 9= 0, at least one of the following relations is valid: w El t:(X) or iw El t,(X), that is, IP.(w)11 + IP,(iwWl 9= o. Together with (1.4.31) this implies (1.4.30) . • 1.4.19. Lemma. Let e be a strictly pluri8ubharmonio Ol-function in 80me neighbourhood of a compact set K c: c: en. Then there exi8t8 an e 0 with the following property: If fP i8 a real-valued O~-function in a neighbourhood of K such tkat
>
2
8 o(D) is the set of all continuous plurisubharmonic functions in D. called the PO(D)-hull oj K. If K~ = K, then K is said to be PO(D)-convex.
Kb
is
1.5.4. Proposition. Let D c (/)n be an open set and K a compact 8ub8et 01 D. Then
i~ c:= .if> . Proof. If f E OeD), then, by Proposition 1.4.8, IfI E PO(D) • •
(1.5.1)
Remark. In Section 2.7 we shall prove that, for pseudoconvex D, (1.5.1) is va.lid with equality. This fact is called the solution of the Levi problem.
1.S.S. Theorem. If D ~ (i) D i8 p8eudoconvex.
is an open 8et, then the following conditions are equivaienl: ,., (ii) For every compact sub8et K of D, Kb is also a compact sub8et oj D. (iii) There exist8 a e E PO(D) such that, for every <X E R, D~ :=
(f)n
{z ED: e(z)
< o:}
c: c: D .
Proof. The case D = on is trivial. Then (i) holds by definition, (ii) follows from Proposition 1.5.4, and the function Izil satisfies condition (iii). Let D ~ If (i) is valid, then the function Izil - In dist (z, aD) fulfils condition (iii). It is clear that (iii) => (ii). So we only have to prove that (ii) => (i). Suppose that (ii) is fulfilled. We have to prove that -In dist (z, eD) is plurisubharmonic. By Theorem 1.4.2 (iii) it is sufficient to prove the following lemma:
on.
Lemma. Let {~
~
ED, 0
=F W
+ AW: A E Cl, IAI
E on and r ~ r} ~
> 080 that
D,
(1.5.2)
and let h be a harmonic function in 80me neighbourhood oj the di8c IAI h(A)
~ -In
dist (~
Then h(A) > -In dist
(~
~ r
in
Q)1
Buch tkat
+ AW, aD)
Jor
IAI
= r .
(1.5.3)
+ AU', aD)
for
IAI
~
,. •
(1.5.4)
Proof of the lemma. Let h* be a harmonic function in some neighbourhood of ~ r such that f : = h + ih* is holomorphic in a neighbourhood of IAI ~ r. Then (1.5.3) is equivalent to the inequality diet (~ + AW, aD) :2: le-1(1)1 for IAI = r, that is,
IAI
~
+ AW + ~ e-f(),) E D
for
IAI = rand CE Oft with ICI
0 on [lan. Consequently, for z E 8D and wE {T. - z},
This implies that OOt (C.(A), 8D) 0 if 0 < s < 1 and IAI ~ E. SinceC8(O) = ~ + 8rJ ED for 0 < 8 < 1, we conclude that C.(l) E D for all 0 < s < 1 and IAI :5: E. Consequently a passage to the limit shows that C1(A) E jj for alllAI < E. Since ~ E U aD' after shrinking E, we can assume that C1 (A) ED n U aD for IAI ~ 8. Therefore, e(C1 (A» = -dist (C1 ().), 8D) for IAI ~ E, and by (1.5.14) (
-e(C1(A»)
~ le(E)1 (e"IAII/2 -
1) leDlJ.+""'1
IAI ~
for
E •
The function on the right-hand side is strictly convex and ~ 0 in some neighbourhood of A = O. Since e(C1 (0») = 0, it follows that -e 0 C1 is strictly convex at 0 and . d(e 0 C1 ) (0) = O. In particular,
5I1e(C1(A»
I = 1prp vanish at E. This is not possible, because e is strictly plurisubharmonic at ~ (cp. Theorem 1.4.7 (ii)).l).
>
We shall now examine when an open set with C2-boundary is strictly pseudoconvex.
1.5.17. Theorem. Let Dec (f)1I be an open eet with Ci-boundary and (> a real-valued C2-function in a neighbo'urhood U aD of oD 8uch tha.t U aD n D = {z E U aD : (>(z) O} and d(>(z) =1= 0 f01' Z E 8D. Then D is sl·rictly pseudoCDnvex iJ and only if
0 ~n K. Then, by Lemma 1.4.19 and Proposition 1.5.16, for sufficiently small e properties. •
> 0, D
:= {z E U D : e(z) -
EX(Z)
< O}
has the required
1.5.21. Theorem. For every strictly pseudoconvex open set Dec QJ" and every neighbourhood U D of D, there exi8ts a 0 00 st'rictly pseudoconvex open set jj c c q}ft Buck that DC D~ UD'
Proof. By Theorem 1.5.19, after shrinking UD, we can find a strictly plurisubharmonic 02-function (! in UD sa.tisfying (1.5.18) and (1.5.19). Then there exists eo 0 so small that De. : = {z E UD : e(z) 3eo} is a relatively compact subset of UD• In view of the Approximation Theorem 1.4.10, we can choose a plurisubharmonic 0 00 • function cp in some neighbourhood of Deo which is so close to r! on D,. that, for some ~ 0, the strictly plurisubharmonic function e(z) := cp(z) ~ Izl2 satisfies the condition D C DB :=-: {z ED•• : e(z) e} ~ D.. for all eo ~ e ~ 2Eo •
>
0 if max (x, Y - £) > 0, cp(z, y) < 0 if :l1 < 0 and y ~ O. Then tp(z) := tp(e(z) , el(Z) is a strictly convex Ok.function in U1 • Define D := := {z E U 1 : tp(z) O}. Then.Df. n G_ ~ D ~ Di n G_, and therefore K C jj C UK n G_. Sinoe tp is strictly convex, and since no point in eD is a local minimum of tp, we have dtp(z) =1= 0 for z E aD. Consequently, D is a 01: strictly convex open set. (ii) Sinoe K has a basis of pseudoconvex neighbourhoods, by Corollary 1.5.11, we can find a 0 00 striotly pseudoconvex open set Of. such that K c c U/. c c: UK n G. In view of Theorem 1.5.19, there is a strictly plurisubharmonic OOO·function el in some neighbourhood U1 c:C UK n G of D~ such that D~ = {z E U1 : el(z) < O}. Choose £ 0 so small that Di := {z E Ut : ~(z) < 8} c: c: UI • (This is possible, because, by Proposition 1.5.16, dl!1(z) =F 0 for all Z E 8D~.) Define tp(z) := q;(e(z), (!1(Z), z E U1 , where tp(:.r:, y) = f(x) + g(y) is the same function as in the proof of part (i). Then, by Theorem 1.4.12 (ii), tp is strictly plurisubharmonic in UI , and D := {z E U1 : tp(z) < O} is strictly pseudoconvex, where D~ n G_ ~ n ~ Dt n G_ and, therefore, K ~ D C UK n-G_. Now we assume that ~(z) =1= 0 for all z Elr. If z Ern aD, then e(z) = 0 and tp(z) = I(e(z» + g(~(z» = O. Since 1(0) = 0 and get) 0 for t > 0, this implies that e(z) = 0 and el(Z) ~ 0 for z Ern eD. Since get) = 0 if t ~ 0 and, therefore, dg(t)/dt = 0 if t ~ 0, we oonclude that
.ii:.
>
dtp(z) = d/(t)
>
dt
I
'_Q(~)
d(!(z)
for all
Z
Ern
oD .
r,
Since df(t)/dt 0 for all t E H, and since d(!(z) =1= 0 for all Z E it follows that d1p(z) =t= 0 if z Ern aD. Choose a neighbourhood J" c c U1 of r n aD such that d1p(z) =t= 0 for all Z E V. Let X ~ 0 be a GOO. function on fen snch that X = 1 in fen" V and X = 0 in some neighbourhood W c: c: V of r n aD. In view of Morse's Lemma 1.5.9, there is an arbitrarily small real. linear map L: q;n -+ III such that the critioal points of tp + L are isolated. If 80 := sup IL(z)/, then we oan choose 80 ~ 8 ~ 2£0 so leU.
1.6. Preliminaries concerning differential forms
43
that d(V' + L + e) (z) =l= 0 if ('I' + L + e) (z) = 0, z E Ul . Since Vee UI , X = 1 on u\ '" V and 1p + L + e is strictly plurisubharmonic in UI , by Lemma 1.4.19, we can choose L (and therefore e) so small that V' + X(L + e) is strictly plurisubharmonic in U1 .Further,forsufficientlysmallL,d(1p + X(L + e») (z) =l=Oif(V' + X(L + e))(z) =0, Z E UI • Since, moreover, X(L + e) ~ 0 on Ul (e ~ Eo, X ~ 0), we conclude that D' := {z E UI : (1p + X(L + e») (z) O} is a 0" strictly pseudo convex open set cD. To complete the proof, w~ observe that, for sufficiently small L, 11' + X(L + e) s: 0 on K, and therefore K ~ D', because X = 0 on W, 1p ~ 0 on K and 1p 0 on K" W «(1 < 0 and f!l < 0 on K" W) . •
f =
L
(1.6.4)
j(p,fJ) ,
p+g=deg!
where J(p,g) is a (p, q)-form. _ Now we want to generalize the operators 8 and 6 defined in Section 1.1 for functions to differential forms of arbitrary degree. If J is a continuous differential form represented in the form (1.6.3), then we define (in the sense of distributions) ~'
8f :=
8fIJ
1\
dzi
1\
d?
(1.6.5)
III+IJI=degJ
and
8J :=
l:'
8fIJ
1\
dz I
1\
dz J
(1.6.6)
•
III +IJI =deg!
+
Then df = 6f oj, and if J is of bidegree (p, g), then 6J is of bidegree (p 6f is of bidegree (p, q I).
+
1.6.4.1. Proposition. For every continuoU8 differential form sense of distributions) -
82f
= (88
+ (8)! =
-
02J
=
I
+ 1, q) and
on D, we have (in the
(1.6.7)
0.
Proof. In view of (1.6.4), we only have to prove this for (p, q)-forms!. Since d a = 0 and d = 0 + 0, we have -
62J -I- (60
-
-
+ 68) j + 82J = d 2f =
0.
This implies (1.6.7), because the three forms on the left-hand side are of different bidegrees. •
46
1. Elementa.ry properties
1.8.4.2. Proposition. Let h = ("'1' ••• , hm) : D ~ om be a holomorphic map and an open set such that h(D) ~ G. (If/ is a differential form on G, then we denote by "'*f the pull-back of / with respect to h, that is, the form defined on D by (1.6.9) below.) Then (i) For every (p, g)10rm f on G, h*f iB a (p, g)-form on D. (ii) For every continuous differential/ormf on G, we have (in t"'e sense oj distributions)
G~
em
8",·/ =",* 8!
and
-
-
(1.6.8)
8",*/ = ",* 8/.
Proof. If
11::::}: dzi, 1\ ••• 1\ dz"
~
/ =
1\
dZJ1
1\ ... 1\
1\ ... 1\
dh i ,
1\
I ;S;il < ... + 1, and, by definition of = 0 when degl < q + l.
Further, by degree reasons, J(C) " Df(z, C) = 0 if degl q integration with respect to C, we have JJ(C) " Dr(z, C) Consequently, 'ED
BDJ =
f J(O,deln(C) "
D(degf)-l(Z,
C)
if
1 ~ deg! ~ n ,
(l.8.7)
tED
{o
otherwise.
Similarly we obtain that
f I(O,degf)(C) " Ddegj(z, C) BanJ(z) = J'eaD 0 otherwise.
1
Since D(z ,.) , '"
=
if 0 ~ degf ~ n - 1 ,
(l.8.8)
(n - 1) I w,([ - z) " w(C) (2:rri) 11 IC - zl2n '
this implies that (1.8.4) coincides with (1.8.1) whenJ is a I-form, and (l.8.5) coincides with (1.8.2) when J is a function. Further, it follows from (1.8.7) and (l.8.8) that BDI is of bidegree (0, (degJ) - 1) and Ban/is of bidegree (0, degl). Notation. Let Y c en be a measurable set. Then, for every measurable complexvalued function I on Y, we define .
111110, Y
:=
sup II(z)1
,
(l.8.9)
ZEY
and we denote by LOO( Y) the Banach space of all measurable complex. valued func· tions! on Y with 111110, Y < 00. The space of all continuous complex. valued functions! on Y will be denoted by CO( Y). We set BOO( Y) : = OO( Y) n LOO( Y). BOO( Y) forms a closed subspace of the Banach space LOO( Y). If Y is a compact set, then BOO( Y) = OO(Y).
In
1.8. Lera.y maps and the opera.tors BeD' B D , L:'D and R:D
< a < 1, we define the a-Holder norm Ilflls,y by IlfllGlIY := IIfllo.¥ + sup 1/(li - /(C)I. (1.8.10) Z.'EY - Z Set HGI( Y) : = {j E CO( Y) : II/ILl, y < oo}. Endowed with the norm II·IIGI. y, HGI( Y) forms For / E OO(Y) and 0
a Banach space, which is called the 8pace oj a-HoliJ,er contin'U0'U8lunctionll (Holder 8pace).
<
B:'Df = B:Df(o, dei J) •
(1.8.17)
54
1. Elementary properties
= 0 if deg !
Further, by degree reasons, J(C) " D:(z, C, 1) finition of integration with respect to a;, 1),
J J(C)
" D:(z, C, 1)
if
= 0
deg!
CE&D
> q + 1, and, by the de-
0, Stokes' formula gives
"z.
"10«(;)
f
IC-&1-8
f(C) O(C) =
f f(C) 0«(;) - f 8.r«(;) " 0«(;) , D. zl > 6}. Therefore, we have
aD
where D. := {C ED: 1(; side of this equality tends to f(z) when
f
J(C) 0«(;)
1C'-z:P=.
=
=.O(C) +
f
J(z)
B ~
It-zl
to prove that the left-hand
O. We have
f
(J(I;) - I(z) 0«(;) .
1'-&1-.
By Proposition 1.7.2, Stokes' formula gives
f
J
O(C) --= (11. - I)! (2ni)n
6 2"
"-11-.
IC-zl-.
=
J (. - - )
1) t "(2m)"
(11. -
d
2'
6
z)" ro(C)
roaC -
J
co,(C -
z)
"ro(C)
IC-&I 0,
U a ) n (([J'" X BUPP v) = (oD X D u au.) n (on X SllPP v) .
This is a. relation between f!ets only without orientation. Taking into account the orientation, we obtain
o(D X D" V,) n ((In X supp v) = (aD
x
D - OUr:) n «(fJn X sUPp't') .
By Stokes' formula, this and (1.11.5) imply that
J
f(C) aJ)xD
1\
f
9(2:, C)
al(C)
v(z) -
1\
0(2:, C)
A
1\
(I(C)
ad.
O(z, C)
1\
J
v(z) - (-I)f
DxJJ,,-U.
A
V(Z)
I(C)
1\
8(z.
C)
1\
ov(z) •
DxD"-Ue
It is clear that the integrals on the right-hand side tend to the corresponding integralR in (1.11.3) when e ~ o. Therefore, it remains to prove that
f I(C)
lim
1\
6(z, C)
v(z)
1\
f
(-1)11 J(z) A v(z) •
=
(1.11.6)
D
8~a~
Consider the map P(~J z) := (z + E, z) from Q)fI X (In onto itself. If S. := {~E (fJn:I~1 = e}, then P(S. X ([In) = 8U,. Let P* be the pull-back of differential forms defined by T, and let /(C) = L' 11(1;) Since 1'(z) contains the factor dZt 1\ ••• 1\ dz,. a.nd,
del.
111-,
therefore, (l)z,;(z
+ E) A v(z) =
w(~) 1\ v(z),
it follows that
P*(f(~) " O(z, C) "v(z))
= ~ fJ(z
+ ~) d{z + i)i (n .- .1) I rul(~)
Since the degree of (IJ'(~) d(z
1\
+ E)1 1\ w'(E)
Moreover, dz 1
1\
ro'(E)
1\
w(~) is 2n -
ro(~)
" v(z) •
1 = dim B Be' we obtain tha.t
(l)(~) 1St x fl" = di I
1\
1\
1~12n
(2nl)n
Ill-q
w(~) = (-1)'1 w'(~)
1\
1\
ro'(~)
1\
oo(E) ls~ x e" •
00(,) A dz 1 • Therefore,
J
!(C) " 0(2, C) "v(z)
au. = (-1)'1
J
I: [(11. -. Il! !JI(Z +
111=,
(231:1)"
ZEtJ"
~) ro'(~) ~nro(~2] dZ l 1\ v(z) • 1~1
EESII
This implies (1.11.6). In fact, by the Martinelli-Bochner formula (1.9.2), the terln ill brackets is equal to 11(z)
+
(n - 1)!
(2ni)" -
J
(11(1. +~)
fES.
which tends to 11(z) when
E
~
o.•
- fz(z))
oo'(~)
1\
w(~)
--I~'
1.12. The Koppelman-Leray formula
59
1.12. The Koppelman-Leray formula In this section we prove an integral representation formula, which contains the Koppelman formula (1.11.1) as well as the Leray formula (1.10.1) as special cases. As a corollary we obtain a formula for solving the inhomogeneous Cauchy-Riemann equations for the case that there exists a Leray map w(z, C) which is holomorphic in z. 1.12.1. Theorem (Koppelman -Leray formula). Let Dec on be an open see with piecewi8e (Jl-bO'Undary D and let w(z, C) be a Leray map for D. Suppose, in addition, that all derivative8 oj w(z, C) which are 0/ order ~ 2 in z and oj order ~ 1 in Care continuous for all zED and C in 80me neighbourhood oj OD.l) Then, Jor every contin1tOU8 (0, g)-jorm J on Jj 8uch that 8j is also continuou8 on D, 0 ~ q ~ n, the forms L:Df, R:»!, BD 8J, R:Daf, BDj, 8R:Djand-8B Dfare continuous in D, and we have (-1)1/= L':Df - (R:'D
+
B D) 8j + 8(R:D
+ B D) f .
(1.12.1)
(BereL:D , R:D and Bn are the integral operators defined in Section 1.8.)
z,
L:
R:
Remark. If w(z, C) = Cthen D = BaD, D = 0 and, therefore, (1.12.1) is the Koppelman formula (1.11.1). If q = 0, then, by (1.8.7) and (1.8.18), BD! = 0 and D! = 0, and therefore (1.12.1) is the Leray fonnula (1.10.1).
R:
Proof of Theorem 1.12.1. By the remark above we can assume that 1 ~ q ~ n. It is clear that L:DJ and R:D! are continuolls in D, and, by Theorem 1.11.1, BD 81, BDJ and 8B D j are continuous in D. Since, by hypothesis of the theorem, w(z, C) is Os in z, by differentiation under the sign of integration, we obtain that 8R':nf is continuous in D. It remains to prove (1.12.1). In view of the Koppelman formula (1.11.1) we only have to prove that
eRrD/ = Ban! - L':Df + R:D 61 in D. To do this we introduce the following abbreviations:
(1.12.2) 00 :
= oo(C),
•- (n - . I)! ~ ; +1 1]jtII( z,.",. 1'01) 1\ dz.C.A l1t(z,." tI1,. 1'0~) O .~ (-1) (2nl)" J... 1
Li
v :=
,
k:t-J
I)! ~ . + 1 1];tII( z,."I'o) ,. 1 ;; d) tII( ,.~) . ~ (-1)' 1\ (vz.c+ A fJk Z,,,,1'0 (2nl)" ;-1 j;+J
(n -
•
These forms are continuous in some neighbourhood W CD X on X [0,1] of D X aD x [0, 1]. Since (1]tII{z, C, l), C - z) = 1 in W, it follows from Proposition 1.7.1 that (in the sense of distributions) d,.C.l (6 1\ (0) = 0 in W. Since 8,(6 1\ co) = 0, this implies that {Sz.e d.l 8.} (6 1\ w) = o. Hence -, (8"., d,t) (6 1\ (0) a,(6 1\ (0) + (8,_, dA Cal ((6 - 6) 1\ (0) = 0 in W.
+ +
+
+
+ +
(1.12.3) The monomials in 8 - jf (with respect to dz" dZ" del, dC~ and dl) and, therefore, in (8•. , dot a.) ((6 - 8) 1\ (0) contain at least one of the differentials ~, ... , dz" as a factor. The same is true for 8.(9 1\ co), whereas the monomials in (8.. , dA) (0 1\ (0)
+ +
+
1) The theorem is valid also without this assumption, but then its proof beoomes teohnioany more complioated. In this book we need this theorem only for the oase that w(z, C) is
even holomorphio in z.
60
1. Elementary properties
do not contain any of the differentials dz f • Therefore, it follows from (1.12.3) that {a"c + d,,) (0 " (.I)) = 0 in W, that is,
(8c + d,,) (0 "
(.I))
= -8,(0 " co)
•
By hypothesis of the theorem, both 8J and the form on the right-hand side of the last relation are continuous in W (after shrinking W if necessary). Therefore, the following computation is correct (that is, there is no problem concerning products of distributions) d",,(! " fj" co) =
(Be + d1) (J "
(j" co) = (af) "
,r" co + (-1)1 J (1.12.4)
and the forms on the right and the left-hand side of (1.12.4) are continuous in W. Consequently, we can apply Stokes' formula to the form! ,,8 "co on 8D X [0, 1] for fixed zED. This gives
J
(af)" (j "CO - 8z3D xJ[0,1]J" 0 "
8Dx[O,1]
f
co =
f" 8 ": 00
3DxO
-
1J " jj " co •
3
xl
(1.12.5) Using the proof of Proposition (1.7.3), one obtains
o" 00 I
(n -I)! -'( w(z,C) co (w(z,
_ (n -
and
o" col
e), e_ z)
= (2ni)"
1-0
,.
"00(",)
I)! m'(w(z, e)) "co(e) <w(~~
(2ni)"
= (n -
e), e-
z)"
!~ co' ( C- z ) "co(1") = IC -
(2m)"
1-1
)
zl2
...
(n - I)! oo'(C -
(2ni)n
Ie _
z) "oo(C) . zl2n
Consequently, (1.12.5) coincides with (1.12.2) . •
1.12.2. Corollary. Let Dec en be an open set with piecewise CI-boundary, and let w(z, e) be a Leray mapJor D, which depends holomorphically on zED. For q = 1, ... ,n, we introduce the abbreviation
Then, Jor every continu0'U8 {O, q)-Jorm ~ q ~ n, we have f = 8Tf / Tg+l 8/.
(1.12.6) on D such that oj is also continuous on D,
-
f
-
1
+
(1.12.7)
I/8J = 0, then u : = T of is a continuous 80lution of the equation, au = ! in D. Moreover, thi8 8olution u i8 0 4 in D for all 0 IX 1, and, if f is 0" in D, k = 1,2, ... , 00, then u is O"+t1. in D for all 0 IX 1.
<
RMANDER [1]. Corollary 1.5.6 was proved in 1910 by E. LEVI [1], who also stated the oonverse problem (the Levi problom). The Martinelli-Bochner formula was found by MARTINELLI [1, 2] (1938, 19(3) and BOOHNEB [1] (1943). Its generalization to differential forms, the Koppelman formula, was obtained in 1967 by KOPPELMAN [1]. The Leray formula for holomorphic funotions (Corollary 1.10.2) was proved in 1956 by LERAY [1] using an idea of ]'ANTAPPIE [1] (1943). Its generalization to smooth funotions (Theorem 1.10.1) was obtained in HENKIN [1] (1970) (see alBO GRAUERT/LIEB [1] (1970». The Leray-Koppelman formula for the first time appeared in LIEB [1] (1970) and 0VRELID [1] (1971) (see also POLoJAXOV [1] (1971».
Exercises, remarks and problems (one complex variable) Here we give exeroises, remarks and problems concerning the case of one complex variable which are of interest from the point of view of several complex variables. 1. Let Dec (jJ1 be an open set. Prove that, for every I E Coo(D), the equation Bu/Bi = I has a solution U E COO(D). Hint. Use Theorem 1.1.3 and the Runge approximation theorem. !. (Solvability of the Cousin problems). Let U t ~ (jJl, j E J, be open sets and D = U U J " and let 111 E O(V, n Vt), i, i e J,suoh that Iff = Itk + Il#t in V, n V t n Uk for all i, i, k E J. Prove that there exist It E O(Vj), i E J, suoh that It 1 = It - " in U, n U , for all i, i E J. Hint. Consider the case J = {l, 2}. If cf E COO(V I ), j = 1,2, suoh that III = Ct - ct in VI n V 2 , then, by setting qJ := 8ct/8z on U I, we obtain qJ e COO(UI U Ua). If 8u/8i = qJ, where U e OOO(VI u l)2)' then 1, := c1 - '11, i = 1,2, solves the problem. a. Let R 1 , HI C C (f}1 be open rectangles such that RI U Ra is also a rectangle. a) Prove that, for pvery bounded holomorphic funotion I in Rl n R I , there are bounded holomorphio functions 11 in HI such that f = 11 - It. in Rl n R B• b) Prove that, for every 0 < IX < 1 and every f E H ec (1i;n-i?-;) n O(RI n R t ) there are 11 E Hec(R t ) n O(R I ) such that f = 11 - 12 in III n R2 • 0) Prove that for every f E CO(RI n R 2 ) n O(RI n Ra) there exist 11 E CO(R ) n O(R,) I such that I = 11 - 12 in HI n Ha· Hint. Use the fact that, by Theorem 1.1.3 and Lemma 1.8.6 (i), for every bounded continuous function qJ on RI uRI! there exists a solution of 8u/8i = tp whioh belongs to HO«R1 U R z ) for all 0 < IX < 1. 4. Let D ~ (Jl be open and J1>f2 E O(D) such that Jil(Z) I Ifa(:) I 0 for all zED. Prove that there exist gil gl E O(D) such that gIll gJI == 1 011 D. 5. Let Dec QJl lJe an open set with rectifiable boundary. For f e OO(8D), we define Kd(z) := I f(C) dC/(C - z), zeD.
+
aD
+
*
62
1. Elementary properties
a) (Plemelj-Privalov theorem, see, for example, MUSCHBLISVILI [1]). Prove that, if D is the unit disc, then KD is bounded as an operator from HI¥(8D) into HI¥(D), 0 < (X < 1. b) It is easy to see that Exercise 3 b) can be solved by means of the Plemelj-Privalov theorem. Prove that this is possible also for Exercise 3c). c) (SALABV [1]). Suppose there is a constant 0 < 00 such that, for every, E aD and all d> 0, the length of the curve {C E aD: IC - ~I < d} is :;;; Od. Prove that, for every o < (X < I, KD is bounded as an operator from Ht%(8D) into H"(D). d) Remark. DYNKIN [1] oonstructed an example of a domain Dec (fJl with reotifiable boundary such that KD is not bounded as an operator from Hl/2(8D) into oo(.D). 6 (RUDIN [1], Bee also HOFFMAN [1]). Let Dec (fJl be the unit disc. Prove that there does not exist a bounded linear projection from Ht%(8D) onto Ht%(D) n OeD) ifn (X = 0, 1. _ Hi n t. Assume there exists such a projection P; prove that then PI : = (lj2n) f (PI,) -t dt -n
+
is also bounded, where le(8) : = 1(8 t) ; prove that P = KD (Exeroise 5); this is not possible, because KD is not bounded between these spaces. '1. Let Dec CI be the unit diso. Prove that there does not exist a. bounded linear projection from 0 1 (15) onto OI(D) n OeD) (OI(D) is the space of all f E OI(D) whose firstorder derivatives are continuous on D). 8. Remark. Let D ~ (fJl be the unit diso. A complex measure ft on D is called a Oarle8on measure if there is a constant a < 00 suoh that, for all d > 0 and Z E aD,
f
Idlll (C) ~ Od •
'ED.\C-.z:'
0 we denote by ~(D) the space of holomorphic functions sup f If(rC)IJ' Id~1 0 0, where
lila
fa
:=
fa
D BUCk that
l: I,~ =
i-I
1 in D.
l: I/,P·. Then there
i-I
exiBt bounded hoZomorphic /unction8 h, in
We give an outline of CARLII80N'S [2, 3] original proof as well as of WOLJrF'S recent more elementary proof (see KOOSIS [1] and GAMELIN [1]). Carleson's proof. 1. The ,,soft" part. From CABLlIlSON'S imbedding theorem above and the Hahn-Banach theorem we obtain that, for every Carleson measure ft, there is a function U E LOO( aD) such that
~ Jdll(C) + 2- JU(C) de = 2ni C - z 2ni C- Z D
0
for
Z E QJl "
D.
(*)
aD
For ZED the left-ha.nd side of (.) defines a function u in D that is bounded on aD and such that 8ujai = p. in D. 2. The uhard" part. For simplicity let n = 2. This part amounts to finding e > 0 and functions tpl' 'PI in D such that fP1 9'155 1, 0 ~ tp, ~ 1, tp,(z) = 0 if 1/,(z) I :i s, (8tp,jaz) 0, where BA;(Z) : = II -I11 .
;-1 %1
17
-
%f%
j+1:
Theorem (CARLESON [2], see also HAVINjVINOGRADOV [1]). For every bounded 8equence Wt of complex numbers, there exi8ts a bounded holomorph'Lc function h in D BUch that h(zA;) = 10k (k = 1, 2, ••• ). This theorem oan be proved by the following scheme: Let B be the Blaschke product of the sequence {ZI:}; prove that (8(1/B)/8z) di 1\ dz is a Carleson measure (Remark 8); let W(z) := 0 if % « {zt. za, ••• } and W(Zk) := Wk (k = 1,2, ... ); using the result stated in the "Boft" part of Carleson's proof of the Corona theorem (Remark 8), we obtain a solution of fJu/8z = W B that is bounded on aD; set It : = Bu. For the details of this simplified version of CABLESON'S [2] proof see AIIUB [2]. Remark also that, by a recent result of JONES (see GORIN/HRUSOEVjVINOGRADOV [I]), this theorem can be proved by means of the following explioit formula: h(::) : = ~ k-l
Wk
Bt(z) Fjt(z) , Bt(zA;) Fj;(ZA;)
where
F t ('"N) ._(I-IZI:II)2 • -exp ( I -
Z~
1 ~ 1 -z~z(l_1 ~ Zt· 2 log (e/c5')j=-i I - ZA;Z
+
\1»)
16. Let R l , RI and GL(n, QJ) be as in Exercise 11, and let A: RI n R z - GL(n, QJ) be a bounded holomorphic function such that A -1 also is bounded in HI n B •. a) Open problem. Do there exist bounded holomorphio functions AI: BI - GL(n. QJ) suoh that A = AlA. in HI n BI! b) Prove that for n = I the answer is affirmative.
Exercises, remarks and problems (several complex variables) 17. Prove that for every analytic polyhedron D CC QJn there exist an open polydisc Pee eN (for some N > n) and a uiholomorphic map h from some neighbourhood of 15 onto some closed Bubmanifold Y in some neighbourhood of P such that h(D) = P n Y. 18 (OIBKA/HENKIN [I], FORNAESS [1]). Prove that for every strictly pseudooonvex 0 1 _ domain Dec eft there exist a strictly convex O'-domain Gee eN (for some N > n), a olosed oomplex Bubmanifold Y in some neighbourhood of G such that the intersection Y n aG is transversal, and a biholomorphic ma.p h from some neighbourhood of ii onto r suoh that h(D) = G n Y. 19. Open problem. Is the open unit ball in en (n ~ 2) biholomorphioally equivalent to some closed complex submanifold in a. bounded open polydisc ? 20. Open problem. Let Dec en (n ~ 2) be a. pseudoconvex domain with Ooo-boundary. Does there exist a closed oomplex submanifold X ill Borne oonvex domain Gee QJN (for some N> n) such that D is biholomorphically equivalent to X? Remark. KOHN/ NIBlDNBERG [1] oonstructed a pseudoconvex domain Dec P with real analytic boundary and with the following property: There does not exist a. olosed Ooo-submanifold Y in
65
Exercises, remarks and problems
a neighbourhood of some convex 0 00 domain Gee eN such thM G n Y is et complex submanifold of G which is Liholornorphically equivalent to D. 21. Subharmonicity and plurisllbhannonicity ('an be defined also for upper continnous functionB with values in [ - 00, (0) (by the Bame Definitions 1.4.1 and 1.4.6). Prove that, for every holomorphic funct,ion f in the open set D ~ ()fI.,l11 IiI is plurisuLharmoni(! in D; if f =P 0, then 111 Ifl is eveu phn·iha,rrnonic . .2.2 (Poincul'e-Lplong equality). Let f be a holomorphiu function in the OpflIl S('t D ~ 0", and M J := {z ED: f(z) = O}. a) SUPPOSE', additionally, that M, is a smooth submetnifold of D. Prove thtlt: l. M J is a complex sublllun.ifoid of D; 2. dime M J :.= n - 1; a. thoro is an integer 1 ~ k < 00 such that, for every fixed W E };f" 0 < lim I/(z) /lz - u.:l k < 00; 4. for every C:JO-fliIlCtiOll rp with compact support in D, we have Z-+tJJ
+
f
tp A
In
III
f
= 2:rk
dU2n
D
rp
d0'2n-2 ,
M/
where A is the Laplacian and dU2n, da2n-2 are the Euclidean volume forms in (fJft and MIt respeotively. b) Consider the' general case (without the condition that M J is smooth)_ Let M1 be the regular components of 1~1, (for properties of zero sets of holomorphic functions (analytio sets) see, for example, GUNNTNOjRo8S1 [1), and let. k1 be the order of vanishing of I on M" Pro,,-e that then
f
({I
LI In III da2n
2n
=
l:
D
k1
J rp dU2n- 2 •
MI
.23. Let u: [O~ 1] - (fJfI. bp U continuous map, and let Dec em be an open set. Then 0 and He(c5) := {z e 8: IF~(z, E)I < c5'l} (for the definition of F(/(z. E) see (1.4.18)). Prove that, if L/ = 0 on He(c5) n aD, then f admits a CI-continuation to He(c5) n 15 that iB holomorphic in Hf(c5) n D. 30 (AGRANOVSKIJjVAL'SXIJ [1], STOUT [2]). Let Dec f/)n be an open set with 0 00 _ boundary, and let f be a continuous funotion on aD. Prove that, if. for every oomplex line X ~ en, I admits a continuous continuation to 15 n X that is holomorpbic in D n X, then I admits a continuous oontinuation to jj tha.t is holomorphio in D. a) Prove that Lfl8D
=
2.
The f)-equation and the "fundamental problems" of the theory of functions on Stein manifolds
Summary. In Seotion 2.1 we construct a formula for solving the a-equation in strictly oonvex open sets in en with OZ-boundary. In Seotion 2.2 we prove that the solution obtained by means of this formula admits 1j2-Holder estimates. In Seotion 2.3 we prove solvability of the a-equation with Ij2-Holder estimates in Os strictly pseudooonvex open sets in (In. Here the main idea is that, after a.n appropriate ohoioe of looal holomorphio coordinates, the boundary of such sets is strictly convex. This makes it possible to use locally the formula from Section 2.1 and, by means of the Holder estimates, to obtain Fredholm solvability. Some elementary additional arguments then oomplete the proof. In Sections 2.4 and 2.5, for strictly pseudoconvex open sets, we construot a Leray map w(z, C) which is holomorphic in z. In Seotion 2.6 we prove Ij2-Holder estimates for the corresponding integral operators. This makes it possible to obta.in a more explioit solution of the a-equation with Ij2-Holder estimates. In Seotion 2.7 by means of the Leray map from Seotion 2.5 we prove that, if D.~ f1Jn is a pseudoconvex open set, then every holomorphio function in a neighbourhood of a PO(D)-oonvex compaot subset K of D can be approximated uniformly onK by holomorphic functions inDo Then we show that every pseudoconvex open set in (/]tl is a domain of holomorphy (solution of the Levi problem) and, in Section 2.8, we prove that the 8-equation can be solved in arbitrary pseudoconvex open sets in on. In Section 2.9 we UBe the Leray map from Seotion 2.5 to show that, for every 0 2 strictly pseudoconvex open Bet Dec f1J", every continuous function on jj tha.t is holomorphic in D can be approximated uniformly on if by holomorphic functions in a neighbourhood of D. _ In Sections 2.10-2.13 we investigate the a-equation in a more general situation: We consider (0, q)-forms with values in holomorphic vector bundles over complex manifolds. First, in the same way as in Section 2.3, we prove Fredholm solvability (with If2-Holder estimates) of the a-equa.tion on complex manifolds with striotly pseudoconvex OZ-boundary (Section 2.11). Then, in Section 2.12, we prove that the a-equation oan be solved on complex manifolds which admit a striotly plurisubharmonio exhausting OS·funotion - if D is a 0 2 strictly pseudoconvex open set in such a manifold, then Ij2-Holder estimates are obtained. Here the main idea is an inductive procedure with respeot to the levels of a striotly plurisubharmonic exhausting OS-function whose critioal points are isolated (using the results in en). In Section 2.13 we give the definition of a Stein manifold and prove that a complex manifold is Stein if and only if it admits a striotly plurisubharmonic exhausting 0 2 function (solution of the Levi problem for manifolds).
as
2.1.
Formula for solving the a-equation in c a strictly convex open sets
Let Dec en be a strictly convex open set with Ol-boundary, that is, let D 5·
=
{z E en:e(z)
< O},
(2.1.1)
68
2. The a-equation and the "fundamental problems"
where
(1
is a real-valued Ot-function in tl" such that, for some 2" 01 (z) ~ _e_ 'ftt ~ «/tl' for all z
J,17-10Xt OXi
where x,
= x,(z) ar~
E aD
t
and
> 0,
(X
(2.1.2)
E /l2n ,
the real coordinates of z E Oft Buch that zl = xl(z)
2.1.1. Remark. Since no point of that df!(z) =f= 0 for all z E cD
+ iXJ+"(z),
aD is a local minimum of e, it follows from (2.1.2)
2.1.2. Definition w,,(C) : = 2 (8 e(C) , ... , 0(1«(;)) •
8C1
8C"
2.1.8. Lemma. There exi8t a neighbo-urhood UaD oj oD and number8 that Jor all CE UaD and z E f)ft with IC - z\ ~ B Re <wQ(C), C - z) ;;; e(C) - (1(z)
+ PIC -
B,
fJ
>0
zll
8uch
(2.1.3)
(Re : = real part oJ). Proof. Let x, Then
=
xI(C) be the real coordinates ofC E f)"such thatC,
Re (w,,(C), C- z)
= Re
r.
(O(1(C) - i Be(C))
i-I 2"
0e( C)
j-l
8x,
BXf
(Xl(' -
OXJ+"
z)
=
X,(C)
+ iXJ+,,(C)'
+ ixJ+,,(C -
z))
l: --x,(C - z}.
=
Hence, Taylor's theorem gives e(z)
= e(C)
- Re <WQ(C) , C- z) 2
2" 6 «(;) + -21 j,k-l ~ ~ x,«(; oXf OX~
z} Xi(C - z}
+ 0(/(; -
z12) •
By (2.1.2) this implies that, for a sufficiently small neighbourhood U aD of cD and sufficiently small B 0,
>
Re (w,,(C), C - z) ;;; e(C) - (1(z)
IC - z\
+ : Ie - zl2
if
CE U aD and
~
e .• 2.1.4. Corollary. wQ(C) i8 a Leray mapJor D (cp. Subsection 1.8.6). Proof. Let zED and (; E aD, and let B then, by (2.1.3), Re (wC/(C), C - z) ~ -e(z)
Z, := Since
> 0 as in Lemma 2.1.3. If Ie > O. If Ie - zl > e, then we set
zi
~
e,
(I -"~ ZI)' + iiZi Z.
IC - z.1 = e, (2.1.3) gives Re (wQ(C), C - z)
C - zl <wC/(C), C - z.) ~
= Re 1
e
-e(z.) .
Since D is convex and, therefore, z. ED, this completes the proof. •
2.1.3. Theorem. Lee D c:c:: e" be a O· 8trictly convex open set defined by (2.1.1), and let J be a continuou.s (0, g)-form on ii Buch that 8J = 0 in D, 1 ~ g ~ n. Then u := (-I)' (R:!JJ
+ BDJ)
(2.1.4)
69
2.2. Holder estimates
au
is a solution of = J in D (for the definitions of B:,g and BD 8ee Seotion 1.8). Thi8 solution belongs to C(O,f- 1 )(D) for all 0 cl 1 (for the definition of C(O,.-l)(D) Bee Subsection 1.8.4). If, moreover,for some 1c = 1,2, ... , 00,/ E Ofo,g)(D), then U E oto::-l)(D) for all 0 cl 1.
<
(z,C)
z + ~ ICC-- zl2 II. -
(2.2.3)
----.
a --;)dA
0, we have
'1"(z. C,).) =
~ ~.
+ (1 -
A)
(6:; _; a~)
dz _ C- z ~!d~ ZI2). + ~(~C IC-zI2 IC-zI2 IC-zI II.
(2.2.4)
2
Now we want to expand the determinant in (2.2.2). Observe that in addition to the rules given in Proposition 1.7.5 we also have the following rule: If in a matrix of differential forms every column contains only forms of the same degree, then after interchanging two columns the determinant of this matrix changes only by a factor ±l. Using these rules and taking into account that only monomials of degree 1 in l contribute to the integral (2.2.2), we obtain
(R:Df) (z)
J
n-2
J " ,:;:0 P. det l , l,n-.-2,.
=
(w
(l>'
f-
5ew
i
I(~-z12' q;'
d~ - dZ)
iC-= zll
1\
d). " 00(") , (2.2.5)
liD x [0, 1]
where P. are polynomials in l. Further, by multi-linearity of the determinant, (RIIIJ) (z) 8
=
J
J 1\ n~2
~etl,~.n-'-2,. (w,C
.-0 P.
-
(l>,..-.-1
z, Sew, dC-dz)"dl"oo(C). IC _ z12l+2
8Dx[O, I]
Integrating with respect to A, we obtain (Rill
J) (z) = n~2 A
aD
.-0 •
Jf"
det l ,l. n-.-2,. (w, C- Z, Sew, dC - dz) q,n-.-l IC _ zI2.+2
A
~~Q
'
aD (2.2.6) 1
where A, =
J
p,().) d)'. It follows that the coefficients of the form R':Df are linear o combinations of integrals of the following type
E(z) =
J4>"-.-1 f;"P-
1 + 2 1\ def
z 28
;.t.m
A
oo(C) ,
(2.2.7)
aD
where 0 ~ 8 ~ n - 2, 1 ~ m ~ n, II is some coefficient of the form f, and 1p is the product of some of the functions WI' "it and aWI/OCk (j, k = 1, ... ,n). Since 1p contains at least one of the functions C~ - z1 as a factor, we have for some C1 < 00
'1 -
(2.2.8)
To estimate the integrals (2.2.7) we apply Proposition 2 in Appendix 1. In view of this Proposition, it is sufficient to prove that, for j = 1, ... , n,
I8:~Z) I, Ia:~z) I:;;; 0 IIIII•. D [diBt (z, 8DW '
/2 ,
(2.2.9)
71
2.2. Holder estimates
where 0
< 00 is a constant which is independent of zED andJ. We have 8 1jJ 81jJ/8z1 (n - 8 - I) (8fP/8z/) 1jJ 8z 1 ~-'-l IC - zji~'+2 = cpn-.-l IC - zI2.+2 CP"-' IC _ zI2.+2
+ (8 + I) (/),,-.-1
(C1 -
%1) 1jJ
IC - zl2.t+'
and, since 8CP/8z1 = 0, 81jJ/8Z1
tp
(3
OZI (/)ft-8-1
I' - z12l+2 = cpn-.-1 IC -
Z,21+2
Since 81jJ/8z1, 8(/)/8z1 and 01jJ/8"iq are bounded for (z, C) implies that for some Os < 00 8
and
1
tp
I
Hence we can find 0 3
E
D
Os
I
oz, (/)ft-'-fl,-- zl2.t+2 ~ 14>1,,-·-1 Ie 8 1p CZt CPft-.-lIC _·zI2.+2
(8 + I) (CI - zl) 1p + (/),,-.-1 IC - zI2'+' •
zI2.+2
X
aD and (2.2.8) holds, this
+ 14>1"-'"
O.
- zI2;+1
I~ ICPI,,-.-IICO ..:....-zI2l+·2· 2
< 00 such that
Ia::;l I'I"!;Zll ~ O.lIfllu [ J aD
da... _.
!
+a
M2n-l
~T"-='1C
- zl 2.t+l] '
where d0'2n-1 is the volume form on aD. Therefore, it remains to prove that, for every ~ E 8D, there are a neighbourhood U~ of E and a constant 0E < 00 such that
J
4>
d~2n-l
I \"-'- IC - zl
2.t+2
~ O~[dist (z, oD)]-1/2 ..
(2.2.10)
3DnUe
and
J
d0'2n-1
S o·rdist (z OD)]-1/2.
\ 0 so small that Dec G. c cUD' and the form f.(z) : = f(zt - E, Z2, ••• ,Zft) belongs to Z?o,g)(G). By Lemma 2.3.2, we can find a polynomial pin Zt and
>
76
2. The a-equation and the "fundamental problems"
forms U o e HU~f-l)(G), u. E HlJ~f-1)(G) such that p(~) f(z) = 8uo{z) and p(~) f.(z) = 8u.(z)
for
z
EG
.
Then p(zt + e) f(z) = &u.(Zt + e, Zs, ... , zn) for Z E Ga. Choose e > 0 so small that the polynomials p(Zt) andp(Zt + e) do not have joint zeros. Then, by Lemma 2.3.3, we can find polynomials q and q. in ~ such that q(~) p(Zt) + q.(Zt) P(ZI + e) == I. Setting u(z) : = q(Zt) uo{z) + q.(zt) u.(zt + 8, Z2' ... , z,,) we complete the proof ••
2.8.5. Theorem. Let Dec en be a 0 2 strictly paeudoconvex open set, and let 1 ~ q ~ n. Then there exiM8 a constant 0 00 8uch that, for every continuous (0, q)-Iorm I on D such that 8/ = 0 in D, there exists a form u E H1J.2g_1)(D) such that
uo) E ZrO.f)(G) and! - 6(q>Uo) = 0 in E,(e) n G. Hence, ! - 8(cpUo) can be continued by zero to (j u E,(e) :::> ii u (G u E,(e»). Since G u E,(e), is a neighbourhood of K and, by hypothesis, (X) is true, it follows thatj - 8(cp-u,,) = &""1
in D !or some ~ E Hf/';'l-l)(D). Setting u that au = f in D . •
2.4.
The support function
= ~
+ lPuo, we obtain u E H16~'-l)(jj) such
fJ)(~, ')
It is the aim of the present and the next section to construct a Leray map w(z, C) for
strictly pseudoconvex open sets in qJft which depends holomorphically on z and satisfies appropriate growth conditions. The corresponding integral representation formulas will be used in Sections 2.6-2.9. The first step of this construction is the construction of the function (z, C) = (w(z, C), C - z) and will be carried out in the present section. Locally the Levi polynomial (cp. (1.4.18)) can be used as ([)(z, C). To obtain (z, C) globally, we have to solve some a-equation which depends continuously differentiably on a parameter. This can be done by means of the bounded linear operator T for solving the a-equation stated in Theorem 2.3.5. However, in our opinion, it is interesting to show that for our purpose it is sufficient to use only Lemma 2.3.4 and certa.in general arguments which follow from Banach's open mapping theorem. Therefore, we begin with Lemma 2.4.1 below. First we give some notations. Notation. If D is an open Bet in Ott, then we denote by COO(D) the Frechet space of all complex-valued COO-functions in D endowed with the topology of uniform convergence on compact sets together with all derivatives. By Z('O.l)(D) will be denoted the Frechet space (endowed with the same topology) of all Cfo.1)-formsj in D such that
8/= o.
.
2.4.1. Lemma. Let Dec qJn be a strictly pseudoconvex open set, and let Un be a neighbourhood 0/ D. Then there exists a continu0U8 linear operator T: Z('O,l) (U'D) -+ Ooo(D) 8uck that
aT J = f in D
for all
J E Z(8.1)( Un)
.
Proof. Choose a strictly pseudoconvex neighbourhood Vn of jj such that Vn c c U"D. Consider the commutative diagram Ooo( V n) - - + Coo(D}
Ie
Z(8,l)( U jj) -+ Z('O.l)( v1») -
18
Z('O.I)(D)
78
2. The a.equation and the "fundamental problems"
where the horizontal arrows denote the canonical restriction maps. We would like to apply Proposition 1 from Appendix 2. According to Lemma 2.3.4 and Corollary 2.1.6, Z
0, such
aft
and e exist.
2.5. The Oka.-Hefer lemma
Let £,
79
/3, and F(z,~) be aB in Lemma 2.4.2, where, moreover, £ iB ch08en so small that {z E ([)":
IC -
~ E aD.
for every
zl ::;: 2£} c (J
Then there exists a Ol-function c.P(z, C) defined for of aD and z E U D := D u U aD such that: (i) 4'>(z, C) depends holomorphically on z E U D•
,
Cin
(2.4.6) some neighbourhood U aD ~ 8
(ti) 4>(z, C) =l= 0 for all Z E UD, CE U aD with IC - zi ~ £ • (2.4.7) (iii) There is a Ol-function M(z, C) =1= 0 defined Jor all points z E U D, CE U aD with - zl ::;: E such that
4>(z, C) = F(z, C) M(z, C) for
Z
E UD'
C E UaD, I' - zl
~
E •
(2.4.8)
~ 2£ •
(2.4.9)
Proof. If follows from (2.4.5) that Re F(z, C) ~ e(C) - e(z)
+ pes
z, C E 8, E ~
for
IC - zl
Since e = 0 on aD and by (2.4.6), we can choose a neighbourhood VaD ~ (J of aD so small that lei ~ {le i j3 on VaD and, for every CE VaD, the ball Ie - zl ~ 26 is contained in fJ. Set V D := D u VaDe Then, for every (z, C) E Vi) X VaD with Ie - zl ~ 2£, both Cand z belong to (} and it follows from (2.4.9) that Re F(z, C} ~ {l£t/3 for all z E V:v and CE VaD with e ~ IC - zl ~ 2e. Therefore, we can define In F(z, C) for z E Vli and CE VaD with E ~ Ie - zl ~ 2E. Choose a Goo·function X on (!J" such that X(~) = 1 for I~I ::;: e £/4 and X(~) = 0 for 1,1 ~ 2£ - 6/4. For z E Vli and CE VaD' we define
+
f(z,
C)
:=
j6 z[X(C -
z) In F(z,
C)] if
£
~ IC - zl ~
2£ ,
otherwIse.
\0
Then the map V aD :3 C~ f(', C) is continuously differentiable with values in the Frechet space Zro,l)(Vi». Now we choose a neighbourhood U aD c c Van suoh that Uli : = D u UaD is strictly pseudoconvex. Then, by Lemma 2.4.1) there is a continuouB linear operator T: Zro,l)( Vjj) -+OOO( UD) such that aT cp = rp on Ujj for all cp E Z~.l)( Vjj). For z E UJj and CE UaD, we define u(z,
C)
:=
(T!hC»)
(z),
M(z,C) := exp
(-u(z,C»)
and c.P(z, C) : =
{F(Z, C) M(z, C)
exp [X(C - z) In F(z, C)
- u(z, C)]
if if
IC - zl IC - zl
~ £ , ~ £.
This completes the proof. • Remark. \Ve point out that in Theorem 2.4.3 we do not assume that aD is smooth. Remark. It follows from Theorem 2.4.3 that every strictly pseudoconvex open set Dec ([)" is a domain of holomorphy. In fact, for every fixed Co E aD, the function f(z) := 1/c.P(z,Co) is holomorphic inD (this follows from properties (i)-(iii) of c.P and estimate (2.4.5)) and fez) ~ 00 for D 3 z ~ Co.
2.5.
The Oks-Hefer lemDla and solution of <w(z, ;),; -z) = tP(z,;)
In this section we construct a Leray map w(z, C) for strictly pseudoconvex open sets in ([)n which is holomorphic in z and such that (w(z, C), C - z) = 4'>(z, C), where 4'>(z, C) is the function from Theorem 2.4.3. Locally this is simple. To do this globally, we use again the a.equa tion.
2. The a-equation and the "fundamental problems"
80
2.4.1. Lemma. Let Dec QJrt be a strictly 1Jseudoconvex open set, let Ml = {z E en: Zt = O}, antllet U D be a neighbourhood 01 D. _Then, for every ~olomorphic function 1 in Ml fl UD, there exiBt8 a holomorpkic function fin D 8uch tkat f = fin Ml n D. Proof. If U1 is a sufficiently small neighbourhood of Ml n D, then, by setting F(z) : = 1(0, Zj, ••• ,z,,), we obtain a holomorphic continuation of I to U l • Choose neighbourhoods U~, U~ of MI n jj such that U~ c c U~ c CUI' Let X be a 0 00 • function on such that X = 1 in U~ and X = 0 in U';. Define
en
en"
F(Z) 8X(z) cp(z) : =
if
ZEU
{
o
if
z
,
I
Zt
UI • Then cp is a Cfo,l)-form on QJn such that 8cp = O. Therefore, by Lemma 2.3.4, there exists a continuous function u on D such that = cp in D, that is, 8(FX - Ztu) = 0 in D. Hence j := FX - ZtU is a holomorphic function in D. Since X = 1 in D n M I , j= F =jin D fl M t • • E QJn "
au
2.5.2. Lemma. Let Dec Att
:
=
en be a strictly paeudocorwex open set, let
{z E QJn: Zt = ... =
Zt
= O} ,
1
~
k ::;: n ,
and let U D be a neighbourhood of D. Then, Jor every holomorphic Junction f in UJj J = 0 on .W" fl UD' there exist holomorphic lunctions Iv ... ,f" in D such that
with
.t
I(z)
=
~
z11,(z) Jor all zED.
j-1
Proof. For k = 1, we can set 11(z) = J(z)/Zt. Suppose the lemma is already proved for k - 1, and let/be a holomorphic func.!.ion in Ui) such thatj(z) = 0 for z EM!. n U D. Choose a strictly pseudoconvex open set D suC'h that Dec Dec UJj. Then D n M 1 is a strictly pseudoconvex open set in Ml (= q)"-1), and, by hypothesis, there are holomorphic functions]1(z2, ... ,z,,) in jj n Ml such that I;
_
_
I(z) = ~ z1!t(za, •.. ,Ztl)
;-2
for
zED n Ml .
ByLemma2.5.1, we oan find holomorphic functions!1 inD such thatJ1(z) =11(z2' ... ,zra) for zED fl MI' Setting 1 I; II(z) := - (/(z) - ~ ZtI1(Z)) for zED, ZI
;-2
we conclude the proof•• We need an extension of Lemma 2.5.2 to the Case when 1 depends continuously differentiably on a parameter. Then we want that the coefficients If also depend continuously differentiably on this parameter. To obtain this, we prove that the coeffioients If can be given by continuous linear operators. First we introduce some notations. Notation. If U ~ QJ" is an open set, then we denote by O( U) the Frechet spaco of holomorphio funotions on U endowed with the topology of uniform convergence on compact sets in U. Set Oi( U) := O( U) E9 ... EB O( U) (k times). If Y c en, then we denote by Oy(U) the 8ubspace of alII E O(U) 8uch thatJ = 0 on Y n U.
81
2.5. The Oka-Hefer lemma.
2.5.3. Lemma. Under the hypotheBe& 01 Lemma 2.5.2 there are continuous linear operators P j : 0M,JUIi) ~ O(D) such tkal,Jor every! E 0Ml(U"ii), i
!(z)
L
=
Jor all
z1(TJ) (z)
zED.
j~l
Proof. Choose a strictly pseudoconvex open set D' such that Dec D' c cUD' Denote by A the continuous lincar map from Ot(D'} into 0Ml(D') and OJ:(D) into oMJ;(D) which is defined by i
L
A(Jl' ... ,J,,) :=
Zt!1 .
j-l
Consider the commutative diagram 'Ok(D') _
Ok(D)
1"'
°
M.( UJj) ---+ ()M,,(D')
--to
° 1"'
Mk(D) ,
where the horizontal arrows denote the canonical restriction maps. By Lemma 2.5.2, OMJ:(UD)~A(Ok(D')). Further, if L~(D) is the Hilbert space of square integrable vector functions in O"(D), then the natural maps Ot(D') ~L~(D)
and L~(D) ~ 01:(D)
are continuous (see Theorem 1.1.13 for the second map). Therefore, the statement follows from Propositions 1 and 2, Appendix 2. • 2.5.4. Theorem (the Oka-Hefer lemma). Let Dec C." be a strictly pcseudoconvex open set, and let U D be a neighbourhood o! D. Then there are continuous l1:near map8 T , : O(UfJ) ~ O(D X D) such that, for every f E O(Ujj) ~ n
J(C) - f(z) = L
('1 - z1) (Ttl) (z,
C) for all z, CED.
j=1
Proof. Choose a strictly pseudoconvex open set D' c c q]2n such that D X D e c D' c c UJj X UJj. SetE1:= C, - z, and E,+n := z, for j = 1, ... , n. Define h(z, C) : =
~
for z, C E UD' Then h(D') also is strictly pseudoconvex. Define
SJ(E) := f(E l
+ En +1 , ... ,En + E2t1) -
!(EtI +1 ,
... , e2n) •
S is a continuous linear map from O(UJj) into 0M,,(h(UJj X UJj)), where M,. := := {¢ E (f)2f1: El = ... = = O}. In view of Lemma 2.5.3, there are continuous linear
eft
»~
operators PI: ()M,,(h(UJj X U D n
S!(~)
=
O(h(D')) such that
....,
~ ~1(TISf) (~)
for all
~ E h(D')
and
!
E O( UD) .
j-l
Hence
,.
....,
f(') - !(z) = (Sf) (h(z, C)) = L (C, - zl) (T,S!) (h(z, j-l
Setting (Td) (z, C) :=
(T ,Sf)
C»
for
z"
ED.
(h(z, C», we conclude the proof. •
2.5.5. Theorem. Let Dec C n be a strictly 1Jseudoconvex open ael and lee UaD, U jj, 4>(z, C) be as in Theorem 2.4.3. Suppose VaD is a neighbourhood oj aD Buch that VaD c c U aD and V.D : = VaD u D iB strictly pseuaoOOfl,vex. Then there exist continuous 6 Henkin/Leiterer
82
2. The a.oquation a.nd the "fun
To complete the proof, it is sufficient to show that, for every e 0, there is a. sequence gl E O(Dj+1),j = 0, 1,2, ... , such that go := h (by (iii), Dl ~ UK) and BUP Ig1(z) -
gj+l(z)1
max Ih(z)1 .
(2.7.1)
ZEK
It follows from Corollary 1.5.10 (or from Theorem 1.5.8 and Lemma 1.4.19) that there is a strictly pseudoconvex open set 0 c c D such that K ceO, ~ E 8G and ii is PO(D)-convex. By Theorem 2.4.3 and estimate (2.4.5), there is a OI-function (z, C) is holomorphic in z E U Q, (/)(C: C) = 0, and 080 small
_
E DtJo+l!~
"Dtlo- IS; '
(2.12.2)
To complete the proof, we shall prove that for EO := min (B~, P/30) condition (4) is fulfilled. Let 0 ~ El, Ell ~ Eo such that d(!(z) =f: 0 Define t!o := (!
z
for
E
8DtJo -
1S1
U
8DtJo +••.
(2.12.3)
+ El - (Xo, and, for j = 1, ... , M, we set (! + Bl - (Xo - (Bl + e2) 1: X•• j
el :=
• =1
Then it follows from (2.12.2) that for j = 0, 1, ... , M d(!1(Z)
=f: 0
for
Z
E
(DtJo + lSo ' " D l1oo -
(2.12.4)
Crit «(!) .
a,) "
Further, Crit «(!) n (8Do u ... u 8DM )
= 0.
(2.12.5)
In fact, assume that z E Crit (e) n 8D j • Then d(!(z) = 0 and, by (2.12.1), either o = (!1(z) = (!(z) - (Xo - 82 or 0 = (!1(Z) = (!(z) El - (Xo, that is, d(!(z) = 0 and z E 8nCl ,_1S1 u 8DClo + e . ' which is not possible because of (2.12.3). Since DtI,-Bo S Dl ~ D l1oo +e.' it follows from (2.12.4) and (2.12.5) that d(!l(z) =f: 0 for z E 8DI for j = 0, 1, ... , M. Further, it is clear from the definition of 0 and P as well as from the relations (!/ = e El - Q:o on X " (VI U ... U V M) that eo, ... ,(!M are strictly plurisubharmonic onX. Finally, it follows immediately from the definition of fb that Do = DOlo - BI ' DM = DOl,+B.and D j _ 1 ~ Dl ~ Dj - I U supp XI ~ Dj - 1 U
+
+
V".
Proof of Theorem 2.12.3. (i). By Lemma 2.12.2, we can assume that Crit (e) is discrete in X. By Theorem 2.11.3 and Proposition 3, Appendix 2, it is sufficient to prove that for all Q: E R the following statement holds: (i)(J1. If d()(z) =f: 0 for z E 8D/J1., then EU.2g{D/J1.' B) = Z'fo.g)(D/J1.' B) for all 1 ~ q $; n. Set (Xmin := min Q(z). 'Ve denote by Q:o the supremum of all (X :2= Q:mln such that (i)fJ := {z EX: d(!(z) = O}
ZEX
and open sets U l , ... , UN such that conditions (1)-(4) in Lemma 2.12.4 are fulfilled. We can find a number 0 ~ El ~ min (Eo, exo - fJ) such that de(z) =t= 0 for Z E aD"'_'l and (ii)"'_'l is true. In fact, if (xo = fJ, then we setEl:= 0. If exo p, then since Crit (e) is discrete we can choose 0 El ~ min (EO) ex o - p) so that d(!(z) =t= for Z E aD,.._81' Further, let 0 E2 eo be such that de(z) =1= 0 for z E aDIX,+B.' and Do, ... , DM the corresponding sets from condition (4) in Lemma2.12.4. SinceDo= DIX,-8.' then I can be approximated uniformly on D(J by continuous sections over which are holomorphic in Do. By Theorem 2.12.3 (i), we can apply .11f timcs Lemma 2.11.4 ~i) and obtain thatl can be approximated uniformly on Dp by continuous section sover DM which are holomorphic in D M • Since DII = D,..+B.' this is in contradiction to the definition of CX o' To complete the proof, we choose a sequence fJ exl CX2 tending to infinity such that d(!(z) 9= 0 for Z E aDell' This is possible since Crit «(!) is disc~ete. By what was proved, for every B 0 we then can find continuous sections 11 : DIXJ -+ B which are holomorphic in D(}lJ such that
°
<
E
11/1 -/IIO.Dp on DfA + 1 " DfA (see, for example, Lemma 1.4.13 in NARASIMHAN [2]). Then by Lemma 1.4.19, for sufficiently small E > 0, el := (! + EX is also a. strictly plurisubex} and, in harmonic exhausting Ol-function for X. We have Dt)& = {z EX: el(Z) aadition, f!l ~ on X " D/IC' (Observe that since we do not assume that d(!(z) =t= 0 for Z E aD,,) it is possible that e(z) = (X for some points Z EX" Dt)&') Therefore, by Lemma 2.12.2, we can find a strictly plurisubharmonic exhausting Oll-function (!2 for X such that Crit (e2) is discrete in X and which is so close to & that, for an appropriate ~> 0,
°
°
DfA
C:C:
G4 := {z EX: e2(Z)
< ex<S}
c c UJ
•
Since Crit (()a) is discrete, we can assume that, moreover, d(!2(Z) =f= 0 for Z E 8G4• Now we can apply Theorem 2.12.3 (ii). (iv) By Lemma 2.12.2 we can assume that Crit (e) is discrete in X. Then we ca.n find a. sequence exl ex. tending to infinity such that d(!(z) =1= for Z E 8D/lCJ. By
< < ...
°
2.13. Solution of the Levi problem for manifolds
97
Theorem 2.12.3 (i), then there is a sequence of continuous B.valued (0, q - I).forms on D~J such that 8U1 = J in D IIIJ • Now we distinguish the cases q = 1 and 2 =::; q ~ n. First let q = 1. Then'Uf - '"J+1 is holomorphic in Det.J and, by Theorem 2.13.3 (ii), we can find holomorphio seotions V,: X ~B such that
lI u l
1
-
Uj+l -
Then the sequence
u1 := u,.
vlllo.DIIIJ
< 2J '
"1 defined by
J
and "J+1:=
Uj+l
+ i-1 :E Vt
for j
= 1,2, ...
tends uniformly on each compact subset of X to Bome oontinuous seotion,,: X -+ B such that' = J in X. _ Now we consider the case 2 ~ q ~ n. Then ut - 'UJ+l E Zro,,-l)(D.J, B) and, by Theorem 2.12.3 (i), we oan find WI E Ht6~f-2)(jj~J' B) suoh that U, - 'UJ+l = in D fIlJ • Choose Ooo·functions X1 on X such that XI = 1 in D IIIJ _ 1 and supp XI c:: c:: Dill,' Then by setting
au
aw,
J _
U := Uj+l
+
~ 8(Xtw l:)
in
D~_l
(j = 2,3, ... )
1:::01
we obtain a continuous B-valued (0, q - I).form u over X such that
au =
Ion X ••
2.12.S. Remark. Parts (i) and (iv) of Theorem 2.12.3 oan be completed, respectively, by the following statements: (i)' If k E {O, 1, ... , oo} and! is in D., then PI is OI:+GI in Dill for all 0 01: 1. (iv)' If k E {O, 1, ... , 00) and f is 01: on X, then U oan be ohosen to be O"+tI on X for all 0 01: 1. For q = 1 this follows from Corollary 2.1.6. For 2 ~ q ~ n this can be proved, for example, in the same way as the corresponding statements in Theorem 2.6.1 (ii) and 2.8.1 by means of the global integral formula for solving the a.equation on (JIstriotly pseudo convex open sets in Stein manifolds which will be oonstruoted in Chapter 4.
<
UK. Proof. SinceKis O(X)-convex, and since X is holomorphicallyconvex and countable at infinity, we ean find a sequence of O(X)-convex sets K I , j = 1, 2, ... , such that oc
K1
= K, K1 c int K;+l (int
:= interior of), and X
= U K 1 • Set U1 := UK and
;=1 UI := int Ki+I for j = 2,3, .... Since all KI are O(X)-convex, for every j we can find a finite number of functions!;, ... ,!fU) E O(X) such that N(J)
~
11,(z)1 2
k-1
I
< 23
z E K, ,
for
(2.13.1)
N(i)
:E IJ,(z)11 > j
for
i-I
z E K i +2
"
Uf
(2.13.2)
•
By condition (ti) in Definition 2.13.3, we can assume that, moreover,
rank
[(8f:)i-l. , ' N(j)] 8z,
= n
for
z € K, .
(2.13.3)
1-1 •...• ,.
By (2.13.1) we can define 00
(}(z) := -1
NU)
+:E 1: 11:(z)l1 J-1k-l
for
Z
EX.
(2.13.4)
99
2.13. Solution of the Levi problem for manifolds
It remains to prove that this function e has the required properties. By (2.13.1) and (2.13.2), (! 0 on K and (! 0 in X " UK' The series
L jf(z) ff(C)
r(z, C) :=
z, CEX,
,
(2.13.5)
j,1e
converges uniformly on each compact subset in X X X. Therefore, r(z, C) is holomorphic in z and r(z, C) is holomorphic in C. Consequently, (!(z) = (z, z) - 1 is 0 00 on X. It follows from (2.13.2) that {z EX: e(z) (l} is relatively compact in X for all IX E R. Clearly, e is plurisubharmonic. To prove strict plurisubharmonicity, we a.ssume that for some ZO E X and E E Cft
1e, by setting J: : xi e"" - v", we obtain continuous functions i~ on .o +., which are holomorphic in D +•. Since, by (2.13.6) and (2.13.IO),J(~) = = 0 =
u(E)
CI
CI
and X = 1 in W" we have ajt(E) -
6J(E) - aVI:(E) •
By Theorem 2.12.3 (ii), we can find therefore, lim 116JjI;(E) - ajt(E) II = O.
(2.13.14)
il;
E O(X)
such that
11/1; - iI:IIDCI+a
"-+00
According to (2.13.13) and (2.13.14) this completes the proof.
< l/le
and,
Notes
101
(iii) Let V l' W, and ube as in the proof of Lemma 2.13.6 (i). Then it follows from (2.13.7) that, for some ~ 0,
>
Re u(z)
< -~
for all z E D ts + iJ n (Ve "
(2.13.15)
W,) •
Consequently, we can define In 1£ E O(Dts +, n (VI \. W,»). By Theorem 2.12.3 (iv) (cp. also the remark following Definition 2.12.1), the a-equation can be solved in Dts +cJ • By Theorem 2.10.3, this implies that every holomorphio Cousin problem over Dts +cJ has a solution. Hence In u = VI - til for some til E O( V, n D ca +iJ ) and til E O(Dca +cJ ""- We). Define f:= 1£ e-e, in Dca +, n V, and!:= e ...• in DII +iJ "- W,. Then I is holomorphio inDCI+,,/(E) = 0, and it follows from (2.13.7) that/(z) =1=0 for all E =l=z E DIS•• 2.13.7. Corollary. Let X be a Stein manifold, and let z, Then there eziBt8 J E O(X) 8uck /(z) =I=/(E).
E be
dilferent points in X.
Proof. By Theorem 2.13.5, there is a strictly plurisubharmonio exhausting 0 1for X. Without loss of generality we can assume that ~(z) ~ l!(E). Then the required function / E O(X) can be obtained from Lemma 2.13.6 (iii) and the approximation Theorem 2.12.3 (ii) . •
function~
Proof of Theorem 2.13.5. That the oondition is necessary follows from Theorem 2. 13.4. Conversely, suppose that X admits a striotly plurisubharmonio exhausting 0 1• function e. Then X is countable at infinity and oondition (ii) in Definition 2.13.3 follows from Lemma 2.13.6 (il). Let DIS := {z EX: l!(Z) < IX}, IX E R. It remains to prove that, for every IX E H, the set DIS is O(X)-convex. Let E EX" DIS. By Lemma 2.12.2 we can find a striotly plurlsubharmonio exhausting OI-function rp for X such that Crit (9J) is discrete and which is so close to l! that DIS c: c: GfI(f)' where GfJ := {z EX: 91(z) < {J} for fJ E B. After adding a small oonstant to 91, we oan assume that, moreover, drp(z) =f: 0 for Z E aOf'{f). Then, by Lemma 2.13.6 (i), there exists f E O(X) such that/(E) = 1 but IfI < 1 on DIS•• 2.13.8. Proposition. Let X be a Stein manifold. Then every 8trictly p8eudocontl~ open 8et D c: c: X i8 a Stein manifold. Proof. Let (J, l! be as in Definition 2.11.1 of strictly pseudo convex open sets, and let (ll be a strictly plurisubharmonic exhausting OI-function for X (Theorem 2.13.4). Choose 6 > 0 so small tha.t {z E (J: - 6 < ()(z) < O} c:c: (J, a.nd choose a realvalued COO-function X on (-00, 0) with the following properties: X(t) = 0 for t 6, X(t) -+ 00 for t ~ 0, X is strictly convex on ( -6, 0). Then, by Theorem 1.4.12 (ii), X 0 (! is strictly plurisubharmonic on {z E (): - 8 < e(z) < O} and el + X 0 e is a striotly plurisubharmonio exhausting Ol-function for D. By Theorem 2.13.5, this implies that D is a Stein manifold. •
III ~ max.x III for every cont.inuous funotion f on D that is holomorphic in D). Hint. The integration in the Leray formula is pssentia.lly only over the points of strict pseudooonvexity. b) Remark. Thif! is true also for arbitrary pseudoconvex domains with smooth houndary (BASENER [1], DEBIART/GAVEAU [1], HAKIM/SIBONY [1], ROSSI [1]). c) (BYOKOV [1], ARENSON [1]). Prove that the Silov boundary of an arbitrary convex domain Dec Q!fI. consists of all poin ts which are not contained in some complex disc ~ aD. d) Open problem (VITUSKIN [2]). Has the Silov boundary of an arbitrary domain C C en positive Hausdorff measw'e of order n t Remark that VITUi§KIN [2] constl'ucted an example of a domain of holomorphy in QJ2, whose Silov boundary is of topological dimension zero. 2. Prove that the Silov boundary of Dn (see point 25 at p. 65) is equal to {Z:ZZ* = I}. 3. Let Dec X be a G2 strictly pseudoconvex open set in a Stein manifold X, and let B bc a holomorphic vector bundle over D which is continuous over jj (this means, by defini tion, that there exists a covering {U f} of D by relati voly open sets U f ~. jj suoh that B is defined by transition functioIls which are continuous on Vi n V 1 and holomorphio in U, nUt). As in Subseotion 2.11.2, then we can define the spaoes Z?o,g)(D, B) and
Hb'il,2g -1..)(D, B), 1 ~1 ~ ~ 11..:.. Prove that th~e exists a bounded linear ~per~tor T from Z(o,q)(D, B) into H(J,-q-l)(D, B) suoh that aT I = finD for every f E Z(o,g)(D, B). Hint. Use a modification of the proof of Theorem 2.12.3 (i) (in this proof it is not important that the bundle B admits a holomorphic continuation to X). For another proof see LEITERl!IB[2], where a more general situation of sheaves over D is considered whioh are analytic over D and satisfy some coherence condition. 4. Let Dec X be a O~ strictly pseudoconvex open set in a Stein manifold X and let A(D) be the algebra of continuous functions on 15 which are holomorphio in D. Let II' ... , IN E A(D) such that, for every zED, I/l(z) I + 1/.Nt:) I =1= O. Prove that there are Yl' ... , YN E A(D) such that 11g1 INUN = 1 on D. Hint. Use Exercise 3. Remark. This is equivalent to the fact that the space of maximal idea.ls in A(15) is equal to ii. By means of results of KOHN [3], HAKDI/SIBONY [2] proved that this is true also for arbitrary pseudoconvex domains D cc (Jf' with Ooo-boundary.
+ ... +
+ ...
5. Let X be a Stein manifold, and let L(N, tJ) be the space of oomplex N X N matrices. a) Let M: X ~ L(N, (J) be a holomorphic map such that rank M(z) = constant for Z E X. Prove that there exists a holomorphic map V: X ~ L(N, tJ) such that MVM = M and VMV = Von X. Hint. Use Theorem 2.12.3 (iv). _ b) Let D CC X be a G2 strictly pseudooonvex open set and let M: D - L(N, C} be continuous on fj and holomorphic in D suoh that rank M(z} = constant for zED. Prove that there exists a continuous map V: D ~ L(N, C) whioh is holomorphic in D such that VMV = V and MVM = M on ii. Hint. Use Exercise 3. 6 (0VRELID [2]). Let Dec X be a 0 1 strictly pseudoconvex open set in a Stein manifold X and let A(D) be the algebra of oontinuous functions on i5 which are holomorphic in D. Let II' ... , f N E A(D) such that Ift(z)1 + ... II N(Z)! =1= 0 for all zED. Prove that then for every collection (/1' ... , gN E A(D) the following two conditions are equi vulen t: (i) Ylft YNIN = 0 on D. (ii) There exist lpjk E A(D), i, k = 1, ... ,N, such that lpJj: = -tptj and y" == lp101/t + ... lpj:NIN on D. Hint. Find continuous functions lpjJ; and use }jxercise 3.
+
+ ... +
+
7. Open problem. Let D CC (fJn be a pseudooonvex open set with smooth real-analytio boundary. Do there exist uniform estimates for the a-equation! Remark. RANGE [1] proved that this is the case for oonvex D with smooth real-analytic boundary in CI. SIBONY [1] constructed a pseudooonvex GOO-domain Dec (JI without unifonn estimates for the a-equation.
104
2. The a-equation and the "fundamental problems"
8. OpeD problem. Does there exist a convex open set Dec C- without uniform estimates for the a-equation' In particular, this problem is open for the domain of all complex N X N matrices Z such that ZZ· - I is strict1y negative-definite (cp. Exercise 25 in Chapter 1). . 9. Open problem. Let Dec C" be a pseudoconvex domain with Ooo-boundary. Let k e {O, 1,2, ... }, < IX < 1, and let I be a a-closed c1o,1)-form in D whose derivatives of order:S; Ie belong to HtJI(D). Does there exist a continuous function u on 15 such that = I in D? (For Ie + eX ~ N(n) Bee KOHN [3].)
°
au
10. We denote by GL(N, C) the group of all invertible complex N X N matrices. Let X be a Stein manifold of complex dimension n, and let A be an N X N matrix of continuous (0, I)-forms on X such that aA A A A = on X. (The product A A A is defined by the rules of matrix multiplication, where the product of elements in A is the exterior product.) a) For every continuous (0, q)-form f in an open set D CC X, ~ q ~ n, we define
+
a.J := al +o A -
A
°
°
f,
-
and we denote by Z(o,g)(D, A) the Banach space of all continuous (0, q)-forms on D such that a.J = in D. Prove that, for every poin tEE X and all e> 0, there exist a neigh bourhood V of E and bounded linear operators R q : Z~'f)(V, A) - O?o,g-l)(V), 1 ~ q ~ n, such that ad 0 B f = id (:= identity map) and IIB,II ~ e. Hint. It follows from Theorems 2.2.1 and 2.2.2 that, for every b> 0, we can find a neighbourhood V of , such that there exist bounded linear operators T.: O?o.,)(V) - O?O•• -l)(V), 1 ~ q ~ n, such that IIT.II ~ f5 0 and aT. / = / for all / e Z(O,,)(~. Define by~etting~f := + T.+IA A /) a bounded linear operator S from )(V, A) into O(O.,-l)(V), Then aA 0 S = id + M, where Mf := T.+IA A f A A T.(I + T,+lA A /). If d is sufficiently small, then IIMII ~ 1/2 and 11811 ~ 8/2. Set B := 8 0 (id + M)-I. b) Prove that for every point, e X there exist a neighbourhood Y of, and a continuous map U: V .. GL(N, C) such that
°
+
au = UA
T.(L
z?o.•
V.
in
Hin t. Use part a). (For another proof Bee MA.LGRANGE [1].) _ c) Prove that for every oontinuous (O,q)-form f on X,I ~ q ~ n, such that 8/ there exists a continuous (0, q - 1}-form u on X such that
au + A
A U
=
f
on
+A AI
=
°
X.
Hint. By part b) we can find an open oovering {VI} of X and continuous maps U I : V, -+ GL(N, C) suoh that 8U1 = U,A in V,. Then the maps U,Ur 1 are holomorphic in V, n V" and the system {U1f,} defines a a-olosed (0, q)-form with values in the corresponding holomorphic vector bundle. Use Theorem 2.12.3 (iv). 11. Let X be a complex manifold of complex dhnension n, and let GL(N, C) be the group of invertible complex N X N matrioes. Prove that the following two conditions are equivalent: _ (i) For every N X N matrix A of continuous (0, I)-forms on X such that aA + .A A A = 0, there exists a continuous map U: X -+ GL(N, tJ) such that
au =
UA
on
X.
(ii) Let {V 1} be an open covering of X and let U I : V, - GL(N, (fJ) be continuous maps Buch that UiUr l is holomorphic in Vi n v 1• Then there exist holomorphic maps HI: VI - GL(N, C) such that U(Ur1 = H(HT l in V( n V 1• _ Hint. Use Exercise lOb) to prove that (li) =9 (i). Set A := Ufl aUf in V 1 for the proof of the implication (i) =9 (ii). Remark. For Stein manifolds X, condition (ii) is fulfilled. This is a profound theorem of GRA.UERT [1, 2, 3] (see also CA.B.TA.N [2] and CORNA.LBA./ GBlJ'J'ITHS [1]).
11. Open problem. LetX ~ ()" be an open set such that the equivalent conditions (i) and (il) in Exercise 11 are fulfilled. Is then X a domain of holomorphy? Remark. LEITlllBJIB [3]
105
Exercises, remarks and problems
proved that the answer is affirmative if we suppose that, in addition, for every a-closed O{o.l)-form I on X, the equation = f has a continuous solution u on X.
au
13. Let Dec C· be a OS strictly pseudoconvex open set, and let (! be a strictly pi uri· subharmonic Oil-function in some neighbourhood 6 of D such that, D = {z E 6: e(t) < O}. For ~ E aD we set iie(6) := {z e 6: IFQ(z, E)I < 61 } (see (1.4.18) for the definition of F(l)' Let I be an (n, n)-form in D such that
,.. J Ifl H~8)nD
=
0(62")
for
6 -+ 0 ,
uniformly in ; E aD. a) (HORKANDEB [3], see also HENKIN [3]). Prove that I defines a continuous linear functional on ~l(D), where ~l(D) is defined analogously as in Remark 8 in the Exercises, remarks and problems at the end of Chapter 1. b) Prove that there is an (n, n - I)-form in D that is bounded on aD such that = /. Hin t. Use part a) and formulas for solving the a-equation. c) Remark. HENKIN [3] and V AROl'OlTLOS [1] proved the following theorem: III is a (p, q)-Iorm in D which ltilfi18 the condition ,.. J (III + IQI-l}2 III A aell> dO'2f& = O(62ft ) for 6 -+ 0,
au
H!(z, C), cP(z, C), M(z, C), and M(z, C) for CE U and z E U u D8uch tkat the following conditions are fulfilled: (i) ~(z, C) and ~(z, C) depend holomorphically on z E U u D. (ii)
~(z, C)
*
0
and
cP(z, C) =F 0 for CE U, ZED u U with , , - zl ~ B; (3.1.9)
M(z,C)=f=O and M(z,C) 9=0 lor CEU,ZEDuU;
= F(z, C) N(z, C) ana cP(z, C) = (F(z, C) - 2e(C») M(z, C) CE U, ZED u U with IC - zl ~ e •
(3.1.10)
tP(z, C) for
(3.1.11 )
(iii) ",(z, C)' (z, C) be the function from Lemma A, and let VI' Vo be neighbourhoods oj N(e) such that Vo u D i8 atrictly p8eudoconvex and VI C C Vo c c U. Then there exi8t continU0'U8 linear operator8 oj Frechet
3.1. The Koppelman·Leray formula.
III
8pace8 (cp. Section 2.5)
PI: O(U u D) ~ O«(Vo u D) X (Vo u D) with the following propertie8: (a) The OI.map W = (WI' ••• ,wn ) defined by Z E Vo u D) i8 holomorphic in Z E Yo u D and (w(z,
e), I;. -
z) = 4>(z, e)
(b) If ~m(z, 1;.) are Junctions wm = (wT, ... , w:') df'Jined by
wj(z,l;.) = T1(C/>mh
(C
a8
E
Yo, Z
E
WI(Z,
e)
:= T I (4)(·,
e))
(z, C) (C E Yo,
Yo u D) •
in condition (iv) oj Lemma A, then the Ol.map
C) (z, C)
(I;. E Vo, z E Vo u D)
i8 holomorphic in Z E Vo u D, <wm(z, 1;.), I;. - z) = (1)m(z, C) (I;. E Vo, z E Vo u D), lim w"'(z, C) = w(z, C) uniJormly in C E Vl' Z E VI U D together with the Jirst.order
and de.
m-+oo
rivatives. Proof. The statement follows by using the proof of Theorem 2.5.5, the uniform convergence C/>m -+ CP, together with the first· order derivatives, and the fact that VI C C Vo and the linear operators TI are continuous with respeot to the topology of uniform convergence on compact sets. • 3.1.2. Definition of LD and RD' Let Dec: on, C/>, ti, W, VI C c:: Vo c c U as in Construction 3.1.1. Further, we choose a neighbourhood Va of N(e) such that VI C c:: VI and a c= -function X on ([J'n such that
QJ'I"
X = 0 on VI and X = 1 on VI' (3.1.24) It follows from Construction 3.1.1, Lemma A (cp. (3.1.8)-(3.1.11» that, for every fixed zED, there is a neighbourhood V. of N(e) such that ~(z, C) =1= 0 for all C E (D n U) u Va. Together with (3.1.24) this implies that, for every fixed zED, X(C) w(z, I;.)/ii(z, 1;.), I;. E D u Vz, is a OI. map on D u V •. Consequently, for every fixed zED, the differential form _
n
w,(X(I;.) w(z, C)/rP(z, _
_
Cn := 1\ d,(X(C) w,(z, C)/fP(z, cn j=l
J on
D, we can
zED.
(3.1.25)
is continuous for' ED. For every measurable bounded function therefore define
n'
LD!(z) := - ' (2.ni)n
J
-
J(C) w~(X(C) w(z, C)/(1)(z, C)
/I.
w(C) ,
D
Since w(z, C) and tP(z, C) are holomorphic in z, then LDJ is holomorphic in D. Further, setting _
l1t(z, t, l) : = (1 - A) X(C~ w1(z, C)' + 1 C1 - Z1 , rP(z, C) Ie - zlz n
ro(~(z,C,l)) :=
_
1\ (8z" j=l
+ d A)
~ := (~1l ... '~fI) ,
17,(z,C,l) ,
we obtain a continuous d~ferential form iii(ij(z, C, A)) defined for zED, I;. ED" z,
CI - z1 o ::; A ::; 1. The forms -8(C,z) - 2 have a singularity of order 2 at I;. = z, and the forms
(z C) II;. - zl 8,. z w-l-!-are continuous at C=
-
rP(z, C)
z.
112
3. St.riot.ly pseudooonvex sets with non-smooth boundary
Hence the monomials in w(ij(z, C, A)) which are of degree 1 in.4 have a singularity of order ~ 2n - 1 at C= z. Since only such monomials contribute to the integral (3.1.26) below, for every measurable bounded differential form! on D, we can therefore define the differential form RDJ(Z) :=
J
~ (2ni)"
/(C)
A
w(ij(z, C,l))
A
(3.1.26)
zED.
oo(C) ,
(C. A)eD x [0.1]
Then it is easy to see that RDI is continuous in D, and RDf is a (0, q - I)-form if f is a (0, q)-form (RDJ = 0 if I is a function). 8.1.1. Theorem (Koppelman-Leray formula). We use the notations from DefiniCion 3.1.2. Then (i) JOf' every oontin'U0'U8 bounded junction Ion D such that 5! is alBo continuous and bounded on D, we have (3.1.27)
(il) Jor every contin'U0'U8 bounded (0, q)-Jorm on D Buch that bounded in D, ] s: q s: n, we have
-
1= 8R D !
+ RD 81
in
51 is also contin'U0'U8 anAl
D.
(3.1.28)
Proof. Let (! be the strictly plurisubharmonic function from Construction 3.1.1 which defines D. First we prove the theorem for the special case that de(c) =t= 0 for CE 8D and I and are continuous on D.
aJ
For zED,
CED" z, 0 s: l s: 1, consider the differential form
"
w'(~(z, C, ;.) := ~ (_I)J+l i-I
ii1(Z, C,;.) ". (8- c"
+ d,J 1it(z, C,;.) .
J:cf:3
°
This form is continuous for zED, C E j j " z, s: A~ 1. The monomials in w'(1j(z, C, A)) which are of degree 1 in A have a singularity of order s: 2n - 2 at C = z. Therefore, for every continuous bounded differential form I on D, the integral
f
I(C)
T,:=
{C
E on:
i
8e(E)
i-I 8E1
ii~(~) := E~(~) n {C
E
(C, - E,)
=
E~(~)
O},
en: 11d(!(E)1I diet (C, T~)
{~>
0: There exists a
{~ E on: I~ -
< ~2},
The Bet Hf(~) will be called the Hormander ball of radius then the number diamB W:= inf
:=
~
CE He(~) n () ,
< O~S
for
(3.2.1)
zED, diet (z, He(~)) ~ ~
and
CE H,(~) n 8.
(3.2.2)
Proof. Since q(E) = 0, it follows from Ta.ylor's formula. that for some 0 1
allCE8 I(!(C)\
~ 21 -£
8e(;) (C1 -
Et)1 + C1 \C - ~\2
SEt ~ 0I[\1d(!(E)1I dist (C, Te) i-I
+ Ie -
Ell] .
< 00 and
117
3.2. Uniform estimates for the a-equation
For C E (J n HeC,6) this implies (3.2.1) in view of the definition of H,(6). By definition of F, and since (} is a OS.function, for some O. 00
< zl + IC -
IF(z, C) - F(z, E)I =5: 01(IC - Elle -
zit
+ IE -
zll)
+ 2 i-I l:" -~(E) (CI - EI) 1 , " E (}, zED. 8Et I If diet (z, He(6» ::;;; 0 and therefore Iz - EI < 2«5, and if 'E H,((z, e)1 ~ eX(IP(z, C)I + IQ(z, (;')1 + IC -
(iv) For 80me con&tant
-I") Jor 1:, Z
E
(3.2.10')
V••
eX
zll)
for
Z E
Va n D,
CE aD, (3.2.11)
I
(ii)
f ...
~!?(C)III d~2_ft -
(iii)
CEV.nHf{I)
_..
~ Oll'min {I, d 2n + 2 -
x} ,
14>(z, C)IIIC - zI2n-2-x
where a := bjdist (z, He(b») and dO'h i8 the Lebe8gue measure in on, Proof. In this proof we denote all "large" constants by G, G', ... and all "small" constants by <x, <x', .... An expression of the form a(x) ~ Ob(x) must be read as follows: There exists a constant 0 00 such that a(x) ~ Ob(x) for all x which are of interest in the corresponding situation. (i). Since He(tJ) ~ Ee(d), it follows from Proposition 5 (i), Appendix 1, that
0,1 E Ll:'r)(D), and zED
011/110,» «5. min {I, [~/dist
(z, He(~))]2"-1} •
(3.2.20)
Choose a neighbourhood V of aD such that V c:: c:: VB' The monomials in the differential form in l(z) which are of degree 1 in A can be estimated by l/lC - z123-1 uniformly for CED and z E jj" V (cp. the arguments given in Definition 3.1.2). In view of Lemma 3.2.6 (i), this implies that IBe(~)(z) ~ 0 11/llo.D ~ min {I, [~/dist (z, He(~»)]2"-1},
zED" V . (3.2.21)
By Construction 3.1.1, Lemma A, condition (ii), and estimate (3.1.8), inf
I~(z, C)I
>0 .
tED, Va •• ED
Further, since V c: c: Va,
inf
IC - zl > O. Consequently,
teD, V •• ,eV
SUp I Be'V.(z) ~ Oll/llo.D
leDn V
J
E,(z, e)
l) ~z,
+ d,.) [(I _
II
cj)(z,
=
123
0, one has
e) + A Ce) Ie -
z]
zll
f - Z _ ~z, (,'») d). + (I -,i) (8~W(~: e) _ w(z, ~ 8,~(z, C»)
= (
zl2
1(,' -
cj)(z, e)
cj)(z, (,')
cj)1(Z, C)
(.'-z
. + lo,... 1""1(.' _ zl2 Expanding the determinant (u~g the same arguments as in the proof of Lemma 2.2.1), we thus obtain for C, z E Va n D, (,' 9= z,
ro
((1 _,i) X((,'~w(z, C) +,i C- z)
tI.~2
=
~P. ..... 0
(,i) d)' " det
f&~1
1 dl
+ ' ~... 0 q.()
Ie - zll
1,1••,n-.-2
zII' w(z,-C) 8c~~z, (.') ,8cw(z, C) a C- z ) , c,8 ,. I'
( , -
IJ-1:0
cj)1(Z,
Z
-
(C - z
d t
"e 1,.,n-.-l 1-'::--1 1' ~ - z
+ n-l .-0~ r,(A.) dA. " det
I , •• n-.-l
(I)(z,
C)
1.. -
z
a
a,w(z, e) C- z ) ;;. '"z IJ-, - Z II .,..,(z, C)
(W(Z, C)
a,w(z, C)
4>(z,(.')
4>(z, C)
----,
C)
_
,
C-
i )
1:0 -
z
o"z 1-"--11
'
(3.2.23)
where P., q., r. are some polynomiaJs in A.. Observe that
a,,•1(,'C_z12 - z II ~ - 0/1(,' -
II
zll for C, z
E VI
and, by Construction 3.1.1, Lemma A, condition (ii), and estimate (3.1.8),
+
I«1)(z, (,')1 ~ cx(1C - zl2 I()(C)I) for C, z E VI n D • Together with (3.2.14), (3.2.15) in Lemma 3.2.5, and (3.2.12) in Lemma 3.2.4 this implies that for C, Z E Va () D
(C - z
det 1,1••,n-.-2
II ~
1(.' -
w(z, C) 8,q;(z~f) a,w(z, C) ~2(Z, C) , $(z, C)
I~(z, (,')I·+ 2
~ 0'
-
2
II d()((,') II .. I~(z, C)1 2 1(,' - z12n-8
f-
det 1,.,n-.-l ([C
_
i
-zI2'
IC -
z\2n.-2.-8
+ 0'
II de(C) II \4)(z, (,')\ 1(,' - z\2n-2
f - z )11
a,w(z, C) -
~(z, (,')
o
~
,a'.1l 1(- zll
dct 1••,ta_,_1
~a
-
W(Z,
+ ---...£_~ 1(,' -
zI2t1.-1' (3.2.23')
A'
- = - - - - - - - - ~ -- - ---,-- 14)(z, (,')1'1(.' - zl2n-2.-1 - It; - zl2n-l
II
e f - z )11
, c"1C - zP'
a (\lde(C)}1 + \C - zl) (llde(C)11 + IC - zl + I()(C)I)
-
II
Z\2'
C)
(~-(-z,-C)'
a,w(z, (,') -
'
f - z )11 - zli
&(z-,C-) ,a,.• "
_ Ilde(C)II + IC - zl _ ~ a' _;.;, IIde(t;) II ___ + ~~ . . 1(I)(z, C)\,+IIC - zI2n-2.-2 14>(z, C)IIC - z12ta-2 Ie - Z\2"-1
124
3. Striotly pseudoconvex sets with non·smooth boundary
It follows from (3.2.23) and the last three estimates that for zED () V.
IBII.l"V, (o) !S: OIl/Il •. D
J -__
J Ie ~:7"'-'
,eBe(d)
II_~(C)lIdO'Zta + 011/110 D 1c;J)(z, C)IIC - zI2"-2 •
+ 0 Ilfllo D ,
CEV.nH~d)
J-; ;.
Ilde(~)IIZ d0'2"__ . 1c;J)(z, C)1 2 1C - zI2,,-8
CEV.nH~d)
Now estimate (3.2.22) (and therefore (3.2.20)) follows in view of Lemma 3.2.6. II. End of the proof of Theorem 3.2.2. In view of (3.2.20), the integral RDJ(Z) converges for all z E jj and the estimates (3.2.3) and (3.2.4) hold. Therefore, it remains to prove that RDJis continuous on ii and that the operator R D: L(D) ~ oro,f-l)(D) is compact. For every l' 0 we choose a real O°O·function X~ in C" such that 0 ~ X~(C) ~ 1 for all CECA, X~(C) = 1 for ICI < 1'/2, and X~(C) = 0 for lei ~ 1'. Define for! E Lw,g)(D), l' 0, and zED
>
>
J
T RDI(z) :=
-(
w(z, C) z)f(C) co (1 - A) X(C) -;..,.---
X~(C -
q>(z r) , ~
(C,A)eD x [0, 1)
> 0 the kernel of the integral operator Ri> -
Then for every l'
z) + Ie, -- zl2
1\
oo(C).
(2ni)" RD is continuous
n!
for C, zED. Therefore, this operator is compact from Lfllo,D
sup
(3.2.24)
T-+O ! ELl'O,fJ)(D).1I1110,D-1
It follows from the definition of Be(/) that for all ~ E aD and 0
< 0 sup
in the differential form
zl2
0,,, -
lim
>0
sup II Ri>!\z) II = 0 .
'1-+0 !EL('O,f)(D), ,eD, U(a)
1I1110,D-l
Together with (3.2.26) this implies (3.2.24) . •
eE ii,
zED" U(O') by
3.3. Improvement of the estimates on the boundary
3.3.
125
Improvement of the estimates on the boundary
In the previous section we saw that the solution of 8u = J given by Corollary 3.1.4 admits uniform estimates and can be continued continuously to the boundary (Corollary 3.2.3). In the present section we give a more precise estimate for the boundary values of this solution. These estimates will be used in the next section to prove a theorem on decomposition of. singularities. For this purpose it is sufficient to consider (0, I)-forms. In the present section we restrict ourselves to this case. Notation. Throughout this section let D, e, t/J, and Definition 3.1.2. We define t/J*(z, C) := fP(C, z) ,
w*(z,
C) :=
t$, w,l" Va be as in Construction 3.1.1
-w(C, z)
for
z E Va
and
CE J". u D.
In view of Construction 3.1.1, Lemma A, condition (ii), and estimate (3.1.8), t/J*(z,
C) =F 0 and
iP(z, C) =1= 0
Therefore, for every fixed point z
E
for
z E 8D ,
CED.
(3.3.1)
8D the differential form
ro ((1 _l) X(C) w(z, C) + A w*(z, C)) $(z, C)
:= ; i-I
t/J*(z, C)
(ac + d1) ((1 _A) X(C~ w1(z, C) + A W~(z, C») t/J(z, C)
t/J (z,C)
is continuous for (C, l) E D X [0, 1]. 3.3.1. Lemma. (i) For e'lJery J E Lro,l)(D) anti all z E aD the integral
R~/(z)
:=
~
(2nl)"
J
f(C)
1\
co
((I - l) X(C~w(z,C)C) + l t/J(z,
*(z.C)1}
inf
ae6D.tED'\. V.
> O.
Hence, by (3.3.3), ID'\.V.(Z) ~ 0
II/Ho,D
J do-2n ,
ZE
aD.
BE(")
Since by definition of H ~((z, C)II
~' (lI de(C)1I + IC -
+ ~2 (IC -
(3.3.10)
.
In view of (3.3.3)-(3.3.5) we thus obtain for z Iw(z,
zlllde(C)1I
zllld()(C)II1I
+ IC -
+ IC -
+ IC -
zl
+ IC -
z12)
2(Z, C) , -fP(z',-C)
~ 0 IC - zllw(z, C)I !If(C) -
(3.3.9)
+111. zI2,,-' u I(1)(z, CHIC - zI2"-8
Since w*(z, z) = -w(z, z) and therefore, w*(z, e) = - w(z, C) for z E oD, ~ E V 2 n D
I
zl ) 1(1)(z, C)I"
I(]>(z, C)I"
cwa Ie -
z12n-3
II de(C) II + 1 ] I 0 (for the definition of the determinant cpo Definition 1.7.6)
ro(~ x~ + All w* + (1 q,
=
(/J*
~detn[(W
_
cw + 11.1 (8-;;;:;-
W -_-
~
n!
C-
_ Al _ A2 )
)dA
i
Ie - zll
a,~) +
1
(/)
l
(1 -
(/)2
,-
i )
IC - zit
-I- (w*
c;P*
-1Z )d A IC - zl2 9
-
l: - z. ] Al - Aa) ae ------Ie - zl2
(w w* Sew -- C- z ) = n-2 l: P.()·) d).l 1\ cU.s 1\ det 1. 1,.,n-2_. -;;, m* ' ~. 8,-..- -
.-0
(/)
+ n~2q.(A) d2 1-0
'"1
1\
d~
A2
1\
det 11
2 , .I.n- -I
'¥
(w ~ I"C(/J
,,-
(/J
Ii,. - zlJ
z acw 8,.-[ -_~)
- '''I''" zII' (/J
-
z 12
129
3.3. Improvement of the estimates on the boundary (A) d ~ dA d t (W* E+ "~2 ,~o r, "1. A 2 A e 1.1.",,-2-, (/)*' Ie _
_ ft-3tAd
I-!: .() .-0
A
11.1 A
dA
(*
2 A
"j. d t W ~ e 1,1,1".,,-8-. 'V At.* ' Ie
-
i z12'
8ew a t i ' CIC _
~ a-eW W a-,'V II ' - , --.;-, Z (/)1 (/) -z
i ) zll
a 1"--1 .~, - -) z t
, -
z1
'
(3.3.13) where P., q" r" t, are some polynomials. Since le(C)1 S; 0 Ie - zl for z
Il
ac IeC-- zll z II ~ Olle -
E
aD,
zll ,
and w*(z, C) = -w(z, C) + O(le - zl), and in view of estimates (3.2.14), (3.2.15) in Lemma 3.2.5, we have d
II
et 1,I, •• ft-2-,
(UI
w*
Bcw
q;' tP*' ~
a c· -
z )11 s; 0
, e Ie - zlll
-
+
II de(z) II Ie - zl 1(/)*llil'+lIC _ zlh-II-6' (3.3.14)
Ii
dt e
1,1,.,,,-2-.
(wi>' IeC-i Sew 8 C-Z)II=::;:o Ilde(z)II+Ie-zl - z12' i> ' 'Ie - zll - 1-~~+ljC· _ ~;-2'-8' (3.3.15)
Ii
(W*
det 1,I, •• n - 2 - . , f(/)* IC -
i )11 ~ IC - zl J _0. .Ilde(z)11 .. Ie - zll 1(/)*II(1)IIIC - zl .,,-11-8
z , 8....cw , ac t -
zp'
tP
+
(3.3.16)
II
d
(w*
C-
et l , 1,1, •• ,.-8-, (/)*' IC _
~ 0
-
i W 8,q) 8 W a t - i )11 -zfi ' ~B ' i> ' CIe - zll t
+
1 Ie - zI2,,-8
zi
= 6. If de(z)
Oll/llo.D
J
"-'1-'
=f= 0, then these
!a:"-1
1(1)*II(J)111C -
Zll,,-O
.
(3.3.18)
Further, it follows from Construction 3.1.1, Lemma A, condition (il), and estimate (3.1.8) that
I~(z, C)I ~ (;\(\Im F(z, C)I + le(')1 + Ie - zll) for C, z E VI n D ,
(3.3.19)
and
ItP*(z, C)I 9
Henkin/Lciwn'r
~ a:(IIm F(C, z)1 + le(C)1 + I~ - zll) for z E aD,
CE V In D. (3.3.20)
130
3. Strictly pseudoconvex sets with non-smooth bounda.ry
Since F(C, z)
+ F(z, C) =
O(IC - ZI2), (3.3.20) implies that
ItP·(z, ')1 ~ (l(IIm F(z,
')I +
le(C)1
+ IC - Z12)
z E 8D, CE VB n D . (3.3.21)
for
Finally in view of Lemma 3.2.4 (iii), for some 60 > 0 we can find real coordinates t,.(C)' ... ,t2ft (C) in the neighbourhood IC - zl < 0 16 ~ Cllfllo,D d~:~_~l OllflLo,D da~n-l ( )b2n - 8 I~(C)I + b2 +
~ OT"-l ,
0"2n-1 (8U x(T))
(3.3.23)
where 0'2n-l(8U K(T)} denotes the (2n -I)-dimensional Euclidean volume of aUg(T). Further, for f5 0 let E.( 0, then by d(E1 , ••• ,E N; d) we denote the order of the system of sets {HfJ(~)}l~11), that is, thenumberdsuch that at least one point belongs to a of the sets HfI(~)' but no point belongs to d + 1 of these sets. 3.4.1. Theorem. There exi8ts a connan' 0 < 00 with the following propeny: II ENE fJD and " > 0, Ihen every boundea hoZomorphic function f on D admit8
'1' ... ,
N
a decomposition I (i) For j =
D" (aD
= l: fl 8uch that tke following conditions are fulfilled : J-O 1, ... , N the Junction If i8 holomorphic in 80me neighbourlwod
of n HfJ{d») and admits the eBtimate (for the definition of IHlo,D see
Section 1.8)
11!ll1o.D ~ Od(E1 , ••• ,EN; d) IIlIlo.D. (ii)
(3.4.2)
10 i8 holO'mlWphic in some neighbourhood of N
D u [aD n U HIJ(d/2)] •
(3.4.3)
i-I
(iii) If, moreover, J admits a confinU0'U8 continuation to D, then the functions fo, alBo are continU0'U8 on D.
... ,f14
For the proof of this theorem we use the following 8.4.3. Lemma. There exists a con8tant 0 < 00 with tke following property: For all E E aD and ~ > 0 ,here i8 a Ooo-Junction X on 0"8uch that (for the definition of ii,(d) see Subsection 3.2.1)
o~ X~ 1 X= 1
on ~ti, on Bel-d), BUPP X c: c: He(2d);
IIdX(C)l1 ~ O(l/~
1 j'
+ Ilde(C)ll/d
(3.4.4) l
)
for CE H,(2d) n fJ,
(3.4.5)
(3.4.6)
Proof. If Ilde(E)1I ~ 2d, then He{2d) = E f {2d), and it is enough to choose X so that X = 1 on Ee{d) (~ He(d)) , X = 0 on C"" E,(2d) and for Borne 0 1 < 00, II dX(C) II ~ 0I/d 1) Ht(C)~ I/d
CE 8
and j = 1, ... , N .
)
(3.4.14)
1, ... , N ,
8e(C)11 ;;;i Oad(~l' ... , ~ N; d)
2
IIde(C)II d
(3.4.15)
Define 11 = LDx,1 for j = 0, 1, ... , N, where LD was defined in Definition 3.1.2. Since the kernel of the integral LD depends holomorphicaJIy on z, it follows from (3.4.13)
135
3.4. Decomposition of singularities _
N
1: X1 =
that 11 is holomorphic in some neighbourhood of D" (aD n H~i~). Since it follows from the Leray formula stated in Theorem 3.1.3 (i) that I =
N
1,
;-0
1: 11'
;-0
Now we prove estimate (3.4.2). In view of Corollary 3.2.3, RDI 8X1 is continuous on jj and, in view of Theorem 3.3.4, it follows from estimates (3.4.14) and (3.4.15) that for some 0 3 00
(C) We (X(Cl w(z, C») " w(C)
tJ)ft(z, C)
4>(z, C)
CED.
for
If aD is smooth, then (3.4.17) immediately follows from Stokes' formula. In the general case we can similarly proceed as in the proof of Lemma 3.3.3 using the fact that the set K := {C E aD: de(C) = O} is locally contained in a smooth real n-dimensional submanifold (Theorem 1.4.21) . • 8.4.1. Lemma. There i8 a constant 0
~
> 0, and! E L co.l)(D) 8uck that II/(C)II ~ lid + Ildq(C)ll/dl
for
< 00 with the CE H~(~)
following 'Property: If EE aD,
n Va,
(3.4.18)
then Jor all z E jj
Ji'(C) "w,(X(C) ~(•• C»" w(e) I~ 0 min {I, [d/dist (z, Hf(~»]2"-1} • (3.4.19) 0 the function
j
I(e)
1\ w~(x(,") ~(z,
e»
1\
w(e)
CPft(z, C)
D,HI
+ 11m F(z, e)1 + IC -
zll)
for z E VI n D"" CE aDm, IC - zl ~ 8, m ~ mo' Together with (3.2.6) and (3.2.8) this implies that for some (xs 0, IfP",(z, e)1 ~ (Xa(1 P(z, C)I + IQ(z, e)1 + Ie - zll) for z E VI n D",; CE aD",; IC - zl s: e; m ~ mo. Since q)",(z, C) =F 0 if C, z E Va and IC - zl ~ B, and in view of the uniform convergence 4>", ~ tP, the last estimate holds for zl ~ e, too, that is, (3.5.1) is proved. (3.5.2). From Construction 3.1.1, Lemma A it follows that
>
Ie -
~",(z, C) = (F(z, C) - 2e",(C» Mm(z, e) for m ~
e, z E VI;
Ie -
zl ~ e;
mo.
(3.5.5)
>
In view of the uniform convergence.Mm ~.ii it follows that for some (Xl 0, ItP",(z, C) I ~ (XIIF(z, C) - 2em(C)\ for e, Z E VI; IC - zl ~ e; m ~ mo. Since by (3.1.8), for all e, z E VI with \C - zl s: B,
Re F(z, C) :::: e(C) - e(z)
+ PIC - zP' =
em(C) - e",(z)
+ fJ IC -
< 0 in Dm, thus we obtain that for some (XI > 0 I~m(z, C)I ~ "'s(IIm F(z, ')1 + leCC) - e(z)1 + " - zll)
zll ,
and since e",
e,
Z
E
Va n Dm;
Ie - zl
s: e;
m
~
for
me,.
Together with (3.2.6) and (3.2.8) this implies that (3.5.2) holds for \C - zl ~ B. Taking into account that ~",(z, e) =F 0 for C, z E Vi with Ie - zl ~ e and that ~'" ~ i, we conclude the proof of (3.5.2). z) = F(z, C) Mm(z, for C, z e VI with IC - zl ~ B, (3.5.3). Since (w"'(z, e), we have
e-
e)
'"(z, C) = aF(z, e),. aMm(z, C) 0" -Mm(z,~) + F(z, e) - -- ---- + (I; - z\) aCt aCf for C, z E V 2 , 1 s: j < n, m ~ mo' where O(IC - zl) is uniform in m, in view of the wJ
uniform convergence w'"
~w
together with the first-order derivatives. Taking into
142
3. Strictly pseudoconvex sets with non-smooth boundary
account the uniform convergence M m ~ M together with the first-order derivatives, and taking into account that F(z, C) = O(IC - zl) and 8cF (z, C) = 28e(C) + O(lC - 1,1), we conclude that (3.5.3) holds. (3.5.4). We obtain (3.5.4) from (3.5.5) if we take into account the relations F(z, C) = O(IC - 1,1), 8cF(z, C) = O(lC - 1,1), and the uniform convergence ifm -+ if together with the first-order derivatives. • 3.3.4. Lemma. For every Ooo-function f in Oft that i8 holomorpkic in some neighbourhood 0/ D, lim 11/- LDJllo,.K(Q) = O. "' ... 00
Proof. In view of the Leray formula stated in Theorem 3.1.3 (i),f = LD"J + RD. 61 in D".. Therefore, we have to prove that lim \I RD. 8/1Io,K(Q) = O. By hypothesis, Z
81 =
E K(e)
J
~
RD 81(z) =
•
0 in some neighbourhood UD of D. Consequently, for all
(2m)n
8/(C) " OJ
((1 - it) x(:!c:Pm(z, wm(z, C) + it =~ - z ) "w(C) • C) IC - Zl2
(3.5.6)
(D ..'\. uj» x [0,1]
As in the proof of Theorem 3.2.2, we now obtain that for 1, E K(e) and C E Dm " UJj relation (3.2.23) remains valid if we replace w by Wm and 4) by q)m. Further, from (3.5.2) and (3.2.8) it follows that for some "1 0
>
e
r~m(z, C)I ~ "l(le(C)1 + Ie - ZPI) for z E K(e) , E Dm " UD, m ~ mo. Using (3.5.3) and (3.5.4), we can therefore proceed further as in the proof of Theorem 3.2.2, and we conclude that, for 1, E K(e), C E Dm " UD, m ~ mo, estimates (3.2.23') remain valid if we replace w by w m and q, by ~m. Thus we obtain that for Borne 0 1 < 00 and all z E K(e), m ~ mOl IRD. 8/(z)1
+
J
~ 0 1181110 D. - 1 •
(J
IC -
D..'\.UD
Ilde(C)l1 d0'2n I(/)".(z, C)IIC - z12n-2
+
n.'\.~
d0'2;
zl
n-
1
J __
2
llde(C)1I d0'2n_) l4)m(z, C)Il"IC - z12n-8 '
n.'\.~
where d0'2n is the Lebesgue measure in on. In view of (3.5.2) in Lemma 3.5.3, (3.2.9) and (3.2.10') in Lemma 3.2.4, and in view of the relations Q(z, C) = O(IC - zl) and P(z, C) = O(le - zl) (which follow from (3.2.6) and (3.2.8)), this implies that for some 0 1 < 00 and all z E K(e), m ~ mo,
IRD"a/(z)1 ;;;; o,lIall10'D.(
J
II:
~:f·-l
DfII'\.UD
+
J
(IQI
+ (lC -
II dcQl1 d0'2n
zl
+ IQI )1) (lC -
zl
+ IQI )2,,-2
D'"'\.UD
+
J
D,"'\.Uj)
(IQI
+ IPI + (IC -
Ild~ " d,PII d0'2n
zl
+ IQI + IPI)-)I (Ie -
zl
+ IPI + IQI)2n-8
) •
14:3
3.5. Uniform approximation
In view of Proposition 4, Appendix 1, it follows that for some 0,
J
m~mo,
t Z I < C 118t ll ( IR Df/II 8:I( ) = 3 :I O,D",
~E T,CD", "
+
J
d~
U j), ,)
dtl " ... "dt2n
~ET.(Df/II'\.Ujj,z)
(I~I + It1 2 ) It1 2n - 2 +
a" X~ "detl n-2 (*W' , a" X~'). tP q,* tP
- (n - 1)
(3.6.2)
I
A computation gives that for j = I, ... , n - 1,
-0";: . XWt - XW') *w1 (n -;;- /\ detl ,,-2 (*w' - , 8" --;;;- = (j>
I
t:p *
f/J
*
-
CE X
n D,
t:p
Together with (3.6.2) this implies that for
2)! wc'
(XW') rl-. • 'V
*W - XW) ii> (
z,,(-1)"det1,tI_l f/J*' 8"
= (n - I)!
w,,(x;,) -
(n -
1)
5"x:
"det"._2 (::' ,
8C'Z;').
To complete the proof therefore it is sufficient to prove that
I(z) : =
f J a" X~ t:p
InD
/\ det}
,
n-2
(*W' , S', X~') "ro,,(~) = tP* tP
0.
14:7
3.6. Bounded extension of holomorphic functions
To do this we would like to apply Stokes' formula. Since 8D n X is not necessa.rily smooth, we replace X n D by open sets D~ ~ X n D with piecewise smooth boundary: Let S'(e) c 8D n X be the set of points CE 8D n X with dre(C) = O. In view of Theorem 1.4.21, S'(e) is locally contained in a smooth real (n - I).dimensional submanifold of X. Therefore, we can find neighbourhoods U(t5) C X of S'(e), t5 0, such that for Borne constant 0 00 and all t5 0
>
dist (C, 8D n X) ~ Ot5
CE U(t5) ,
for
(3.6.3)
0'2n_2(U(t5)) ~ OtJ,,-l ~ OIJ ,
(3.6.4)
0"2n_s(8U(tJ») ~ OtJ,,-2 ~ 0 ,
(3.6.5)
where 0"2,,-2 and 0'2,,-3 denote the (2n - 2).dimensional and the (2n - 3)-dimen. sional Euclidean volume, respectively. Set D~ := (D n X) " U(t5). Then by (3.6.4)
J(z) = O(t5)
+
J 4> /8"
XrJ>
1\
(*WI - XW')
det 1, ,,-2 rJ>*' 8,,~
1\
w,,(C) •
Dd
Since/(C), *w(z, C), and rJ>*(z, C) are holomorphic in C, we have
r
dc, ~ det 1.,,_2 (*WI rJ>* ,8- c' XW') ~
1\
wc,(C) ]
=0•
(3.6.6)
Therefore, it follows from Stokes' formula that
J(z)
= O(tJ)
J
J XrJ> 'iP
+
1\
(*W' - XW')
det1• n _ 2 (/J*' 8" ~
1\
wc'(~) •
aDd
Since X(e) (/J(z, C)/ii(z, C) = 1 for CE 8D, implies that
J(z)
= O(tJ)
+
J
C =F z,
(*W'
J det 1,,,_2 (/J*'
and in view of (3.6.3) and (3.6.5), this
XW' ) 8c' ~
1\
wc,(C) •
aDd
Therefore by (3.6.6) and Stokes' formula we can conclude that J(z)
J(z)
=
0(15), that is,
= 0. •
For tJ > 0 and z E 8D m we denote by H~(tJ) the Hormander ball wit.h respect to e"., Dm introduced in Section 3.2.1. 3.6.S. Lemma. There exi8t8 a number tJ
CE H': ( tJ ~zn z" 11/2) n V 2 n X, mo ~ m ~ 8 (z)
1
IIdz'e(z) 1I IC' - z'l IId"P(z, C)
>0 00,
8uch that for all
Z
E
8D m " X and
the following estimate8 hold:
21 8e I'
d"Q(z, C)I I
(Z) z"
(3.6.7)
oZn
~ ,/ 1
!!dz'e(z)1 12 , r 2n. where P, Q are the polynomial8 from Lemma 3.2.4. 1\
Proof. By definition of the Hormander ball and by (3.2.2), for
~ EH:, ( "
Ia:!:) z_1"2) n
V. , "'0 ,;; '" ,;;
IC 10·
Zl2
~ ~218e(Z) z" I oZn
(3.6.8)
Z
E 8D",
" X,
00, we have the ineq ua]i tieB (3.6.9)
148
S. Strictly pseudoconvex sets with non-smooth boundary
and IF(z, C)I ;;;;
06'16:!:) "I·
Since
i:, 8(>(z) (,, -
2
z,) = F(z, C)
i-I 8z,
this implies that for some 0'
< 00
+ O(IC -
Z12) ,
and all
cDm '" X, i;
Z E
E H':
(~ lo~;:) Zn 11/2)
n V2 n X _
8e(z) z,.
1
If 6
+ n~l oe(Z) (;1 i-1
OZn
1
-
%1)
OZ,
1
~ C'~216~(Z) Zn I. eZ n
< 1/20',
we obtain therefore that for all n. Va n X, mo ~ m ~ 00,
n~1 oe(z) 1 i-I
(;1 - Z1)
I
~ ~ 1 ce(z)
8z t
2
8z n
zn
Z E
8D m
'"
8 ("') 11/2) X, 'E H': ( b ~ zn 1
8zn
I. I
This implies (3.6.7), because of the Schwarz inequality and the relation
1Id.-e(z)1I = 2 Cl:T~(Z), }-1
IT'·
8z
From (3.6.7) and (3.6.9) we obtain that for all z n VI n X, ?no ~ m ~ 00,
E 8D m '"
X,,
E H":
(
0 (z)
~ ~ zn 1
11/2)
OZn
(3.6.10)
In the Bame way as in the proof of (3.2.10') in Lemma 3.2.4, one can prove that for some 0" < oc and all C, Z E Vi IId"P(z, C)
J\
d"Q(z, ')11 ~
lin1
11dz-('?(Z) 112 - C" (lIdz'e(z)IIIC - zi
Together with (3.6.10) this implies (3.6.8) if ~ ~
3.6.6. Lemma. There is a constant 0 m S; 00 the following estimate.! hold: (i)
(ii)
(iii)
(iv)
+ IC -
z12) •
> 0 is sufficiently small. •
< 00 8uck that
for all z E oDm '" X and mo
3.6. Bounded extension of holoDlorphic functions
14:9
(v)
f CeXnD",nVI
where
d0'2n-2
Ild,'e(C)II18e(C)/8C,,1 d0'2n-2 ~ O/lz.1 , I!(z, C)IIC ~ zl28-5
CE(ZnDt.nV.>,B:'
Since by (3.2.8), IQ(z, C)I ::; C IC - zl, in the integral on the right-hand side, Ie - zl can be replaced by Ie' - z'l + IQ(z, C)I. Therefore~ we can apply Propositions 4 and 7, Appendix 1, and obtain (3.6.15). It remains to prove that for all z E eD". " X,
mo ~ m
~ 00,
•
f
tEB. nXnV.
_JLdc'e(C)l11 ~e(c)/dC~~~0'2"_~ 14>".(z, CWI I4>!(z, e)11C - z12A-6
~ C/lz,,1 .
(3.6.16)
Since de(~) = de(z) + O(lC - zl), the integral on the left-hand side in (3.6.16) can be estimated by the integrals in parts (i), (ii), (iv) of the lemma and the integral which is obtained from the integral in (3.6.16) after replacing IIdc,~(C)lII8e(C)/8Ctll by IId..e(z)III8e(z)/8z"l, that is, we only have to prove that for all z E aD". "X,
3.6. Bounded extension of holomorphic functions
mo < m ::;:
J
00,
Jm(z) :=
CeH:'nXnV.
~-
153
1I~z'()(~~!~~~2ft-2
::;:
l!Pm(z, ')1 1 1!P!(z, C)I IC - z12n-6
a _~z'e(z)H_. I(8e(z)/8z ..) z..1 (3.6.17)
In view of (3.2.11), (3.2.12), (3.5.1), (3.5.2), and (3.6.8), for all z mo ::;: m < 00, Jm(z)
a
~
f
IId"P(z, C)
(IP(z, e)l
1\
aDm " X,
E
dc,Q(z, ')II daz..-2 zll)3 IC - ZI2.. -5 •
+ IQ(z, C)I + Ie -
~'ClI:'nXn VI
f - - z·l/"d"e(z)I'
Together with (3.6.7) this implies that for all z J m(z)
~a
-
(I~~:)
IId"P
2
E
1\
Dm " X,
mo ~ m
dc.QIJ d0'2n-2
~ 00,
8----- .
+ iPl + IQI + Ie - Oil)
IC - zl"-'
Cer.nXnV.
+
Since by (3.2.6), and (3.2.8) IP(z, ')1 IQ(z, ')1 ~ C I' - zl, in the last integral, we can replace IC - z\ by IP(z, C)I IQ(z, e)1 + Ie' - z'l. Therefore, we can apply Propositions 4 and 7, Appendix 1, and we obtain (3.6.17) . •
+
O. 9. Let B be a holomorphio veotor bundle over a complex manifold X, Jet Dec X be a striotly pseudoconvex open set (with not neoessarily smooth boundary), and let 1 ~ q ~ n. Prove that then the following two statements hold: (i) There exists a bounded linear operator S from Z?o.!l)(D, B) into O~o.f_l)(D, B) such that 08 = id K, where K is compact. (ii) The image of "8 as an operator between O?O.f-l)(D, B) and Z(~.g)(D, B) is a closed and finitely codimensioDal subspace of Z?o.f)(D, B). Hint. Use Lemma 1.6.23 (ii), Corollary 3.2.3, Theorem 3.2.2 (iii) and the scheme of the proof of Theorem 2.11.3. 10. Prove that under the hypotheses of Theorem 2.12.3, for every (X E R (without the hypothesis d(!(z) =1= 0, Z E aD), the following two statements hold: _ (i)' For every 1 ::;; q ~ n, there exists a bounded linear operator T: ZfO.f)(D~, B) o - O(o.r-l)(D", B) such that a 0 T = id. (ii)' Every continuous section f: fj~ - B tha.t is hQlomorphic in D(% can be approximated uniformly on D" by holomorphic sections of B over X. Hin t. Use Exeroise 9 and the soheme of the proof of Theorem 2.12.3. 11. (The uniqueness theorem of PIN~UX [1]). Let Dec q]n be an open Bet with 01·boun· dary and let M ~ aD be an n-dimensional totally real Ol·submanifold of aD. Let f be a continuous function on 15 that is holomorphio in D such that f = 0 on M. Prove that I !!5 0 on 15. Hint. Choose a bordered real Ol·submanifold N ~ jj such tha.t: 1. aD n aN S;; ltJ, 2. aN n D is also a smooth totally real submanifold and "sufficiently close" to aD naN; then we only have to prove that, for every Ooo·function qJ with oompact support in D, I fez) qJ(z) dZ1 A ... A dz. = 0;
e
a
+
8E8NnD
157
Exercises, remarks a.nd problems
in view of the Approximation Theorems 2.7.1 and 3.5.1 (cp. the remark following Theo· rem 3.5.1 and Corollary 1.4.20) and since aN n D is "sufficiently close" to aN n aD, 'P can be approximated on aN n D uniformly by holomorphic funotions ~ in a neighbourhood of N; since f = 0 on aD n aN, it follows from Stokes' formula that
I fez) aNnD
~(z) dZ 1 1\ ••• 1\
dz
=I
d(J(z) ~(z) dZt
N
1\ ••• 1\
dz,,)
=
0 .
12. (The uniqueness theorem of TUMANov.) Let Dec qJft Le a. strictly pSE"udoconvex domain with real-analytic boundary. Denote by Ak(D), k = 0, 1, 2, ... , the space of all holomorphic functions in D such that the derivatives of order k are continuous on D. Let Al/2(D) =: OeD) n J{ltt(D) (cp. Section 1.8 for the definition of Hl/'(D», and let! E Attt(D) (cp. point 10) such that 1m! = 0 on a set E ~ aD of positive (2n - I)-dimensional Hausdorff measure. a) Prove that I = const in D. Remark. TUJ,UNOV [1] proved this for f E AI(D) and SlBONY [3] for I E A leD). _ b) Prove that this is not true for f E...4(D}. Hint: Use the recent results of ALEXSANDBOV [1] on the existence of inner functions in strictly pseudoconvex domains. 13 (NAGELjRUDIN [1]). Let Dec qJft be a strictly pseudoconvex Ol-domain, and let r be a rea.l Ol-curve on aD which is transversal to the complex tangent plane of aD at each point in Prove that every bounded holomorphic funotion in D has non-tangential boundary values almost everywhere on 1". Open problem. Let Dec qJn be a polydisc, let X ~ en be a. oomplex plane of complex dimension ~ n - 1, and let f be a bounded holomorpbic function on X n D. Does there exist a bounded holomorpbic function F in D such that F = I on X n D ? Remark. POLJA.XOV [1] proved this for n = 2.
r.
r.
4.
Global integral formulas on Stein manifolds and applications
Summary. In this chapter we generalize the integral formulas prosentflct in Chapters 1- 3 to Stein manifolds (Sections 4.3, 4.5, 4.8, and 4.10). :Moreovcr, in Sections 4.4 and 4.6 a formula will be construoted where for different pieces of the boundary different Leray maps are used. (For space-saving reasons we did not separately prove this formula for the case of (}n.) This formula will be called the Leray-Norguet formula (for the case of functiolls) and the Koppelman-Leray-Norguet formula (for the general case of (0, q)-forms). The KoppelmanLeray-Norguet formula is especially useful for so-called real non-degenerate strictly pseudoconvex polyhedra (Section 4.9). This class of polyhedra contains the real nondegenerate analytio polyhedra as well as the real transversal intersectiolls of 0 2 strictly pseudoconvex open sots. For such polyhedra, by means of the Koppelman-Leray-Norguet formula, a formula for solving the a.:equation is obtained. (Undpr certain additional conditions the solution given by this formula admits uniform estimates - cpo tho Notes at the end of this chapter. Unfortunately we could not include these estimates in this book.) For real non-degenerate analyt,io polyhedra, the Leray-Norguet formula implies a generalization of the Weil fonnula (mentioned in the preface to this book and in the Notes at the end of Chapter 2) to Stein manifolds (Section 4.7). In Section 4.10 we generalize the results from Sectiolls 3.1 and 3.2 to the case of strictly pselldoconvex open sets (with not necessarily smooth boundary) in Stein manifolds. (The corresponding generalizations of Sections 3.3 - 3.5 can be obtained similarly and are left to the reader.) In Section 4.11, by means of the formulas and estimates presented in Section 4.10, we generalize the Extension Theorem 3.6.8 to the case of the interse('tion of a strictly pseudoconvex open set D (with not nE"ceBSarily smooth boundary) in a Stein manifold with an arbitrary cloRed complex submanifold of some neighbourhood of i5. In Section 4.12 we show how to extend the integral formulas pres(>uted in this book to (0, q)-forms with values in holomorphic vector bundles. Now let us f'xplain the idea of the construction of the integral formulas ou Stein manifolds pr(lsented in this ohapter. Let X be a Stein manifold of complex dimension n, and let Dec X be an open set with smooth boundary. Consider, for example, the MartinelliBochner formula. (1.9.2) for holomorphic functions. Then the problem is, by what can the map C - z be replaced? Fu'st consider the case that there exists a holomorphic ma.p u(z, C) from X X X into en such that the following two conditions are fulfilled: (I) u(z, C) 0 if C Z; (2) for every fixed point Z E X tho map u(z, .) is biholoJllorphic in some neighbourhood of z, where u(z, z) = O. Then u can be used instead of C - =, and, as in the case of (1.9.2), one can prove that, for every holomorphio function f in SOUle neighbourhood of ii,
*
f(z}
=
*
(n -
I~
(2ni)n
ff(C)
aD
ru'~~~.(u(z, ')) , lu(z,C)12n
ZED.
However, such a map u need not exist. Moreover, if X is not parallelizable, then it is even impossible to satisfy only condition (2) (this follows from results of SCHNEIDER [I], for more details cpo also HENKI"NjLEITERER [1]). In order to avoid this difficulty, in 1974 A. DYNIN proposed to use Iio holomorphic ma.p 8(Z, C) with values in the complex tangent bundle T(X) of X (cp. Subsection 4.2.3) such t.hat 8(:::, C) E T,(X) for all (z, t) E X X X an(1 such thliot tho following conditions 0.1'0 ful-
159
4.1. Coherent analytic sheaves
filled: (1) 8(Z, C) =1= 0 for Z =1= C; (2) for every fixed Z EX, 8(Z, .) is biholomorphio in some neighbourhood of z, where 8(Z, z) = O. It is easy to find a fibre preserving Ooo-map a from the complex tangent bundle T(X) onto the complex cotangent bundle T*(X) of X suoh that /Iall := (aa, a)1/2, a e T(X) defines a norm in the fibres of T(X), where (b, a> is the value of the covector b E 2'1(X) at a e T,(X) (cp. Subseotions 4.2.3 and 4.3.1). Further, it turns out (cp. Subsection 4.3.1) that the form ro,,(a8(z, roc(8(Z, C») has an invariant meaning (whereas the forms ro,(a8(z, C») a.nd ro,(8(Z, C)) oan not be defined independently of the choice of local coordinates). Now one can prove that, for every holomorphic function in some neighbourhood of ii,
e» "
/(z)
=
-.-.!l.!
(n (2ni)2n
jl(C) ro,(as(z, C») " rod 8(Z, n
+1
and
Rsg = 0
if
IKI
~
n.
(4.4.10)
This follows from the fact that dimB SK = 2n - IKI and the form under the sign of integration in (4.4.7) and (4.4.9), respectively, is of degree ~ ft and ~ ft + 1, re. spectively, in C. Remark. For N = 1, Definitions (4.4.7) and (4.4.9) agree with (4.3.10) and (4:.8.11), respectively. 4.4.S. Theorem (Leray-Norguet formula). We 'U8e the notation8 from Sub8eo1ion 4.4.1. Le' (at, ... , 8;', ~*) be a Leray-Norguet section /or (D, 8, 9') (cp. Subsection 4.4.2), GftCI le, v ~ max (2n", n,,*) be an integer. Phe"" for every con,inuo'UB function J Oft jj ...
172
4. Global integra.l formulas on Stein manifolds
that 8f i8 a180 continuoU8 on D, we have
=
I
-
1:
~ LSIlI -
RSII oj - B D
IKI:an-l
IKI~n
-
81
in
(4.4.11 )
D,
where the 8ummatio7&8 are over all strictly increaBing collections K = (k1 , ••• ,Ic,) of integer8 1 ~ kl k, ~ N with l ~ n and l ~ 16 - 1, respectively, and Lsg = LslI(ql', ,8;,8), RSII = Rsll(cP", ,8;,8,8) (cp. Subsection 4.4.2) and BD = BD(cp", 8,8) (cp. Subsection 4.3.2).
st, ...
< ...
O.
>
Since Dec X, for sufficiently small e 0, oU. n (D X D) is smooth. Since supp vee D, for sufficiently small e 0, we have
>
v(D X D '" U,J n (supp v) X Taking into
ac(~ount
X)
= (D X aD u au.) n (supp v) X
X) .
also the orientations, this can be written
o(D X D '" U.) n (supp v) X X)
=
(D X aD - au.) n ((BUPP vl X X) .
Therefore, it follows from Stokes' formula and (4.5.13) that
Jx aD f(C)
1\
6(z, C) "v(z) -
f
f(C)
1\
O(z, C)
1\
v(z)
(z.C)eau 8
(z.C)eD
J
8/(C)
1\
O(z, C)
1\
v(z) - (-1)(1
(z, C)eD x D'\" U,
J
(z.t)ED x D'\" u 8
f(C)
1\
O(z,
C)
1\
8v(z) •
It is clear that the integrals on the right-hand side tend to the corresponding integrals in (4.5.11) when e
lim
J
--7
O. Therefore, it remains to prove that
I(C)
1-+0 (•• C)eau.
1\
O(z,
C)
1\
v(z) = (-1)(1
J I(z)
.ED
1\
v(z) •
(4.5.14)
176
4. Global integral formulas on Stein manifolds
Let {(U hI)} be the holomorphic atlas of X from Subsection 4.3.1, and let {XI} be a Oeo-partition of unity subordinate to {Vf }. We write VI := XIV and choose open sets " VI c c.:: V f such that supp X1 ~ V,. Then it is sufficient to prove that, for allj, lim
f
f(C)
O(z,
1\
C)
1\
=
"1(z)
(-I)f
.-+0 ('.C)e8U.
J fez)
(4.5.15)
Vj(z) •
1\
,eVJ
From now on let jbefixed. If e is sufficiently small, then, for all (z, C) E U B with we have CE U I • Therefore, (4.5.15) can be written
f
lim
f(C)
1\
O(z, C)
t\
=
"1(Z)
J !(z)
(-1)'
.-+0 (•• C)E(VJxUJ)n6U.
1\
Z
E VI'
(4.5.16)
v1(z) •
:IEVJ
To prove (4.5.16) we need some preparations. Let u(z, C) be the expression of 8(Z, C) with respect to (VI' 11,,) (cp. Subsection 4.3.1), and let w(z, C) be the expression of 8(Z, C) with respect to (U I , 11,1)' By condition (il) in Lemma 4.2.4, we can find EO 0 such that the map T: (U, XU,) n U .. -+ X defined by T(z, C) : = (h,(z), u(z, C»
en
>
en
en
cn.
is biholomorphic from (U f X U 1) n U.. onto some open set WBt ~ X Let p-l: W .. -+ (Vf X V,) n V •• be the inverse of T. Let a('f}, ~): W .. ~ (U1 be the holomorphic map which is defined by ~(h1(z), u(z, for all (z, E U:71 x U , ) n U ... Then (4.5.17) T-l(TJJ~) = (hj"1(1]), ~(1], ~» for all (TJ'~) E W ••.
e»
=,
e)
Further, it is clear that U(T-l(1],~» = ~
for all
(1],~) E
(4.5.18)
W"
and T-l(TJ,O) = (hj"1(TJ), h;-l(TJ))
for all TJ E hl (U I )
\Ve set Z. := T(Vf X U , ) n 8U.) for 0
o < E1 ~ Eo such that
it follows from (4.5.17) and (4.5.18) that for (1],;) E Z.
(I
1)1 . a·f('I},~) (2nl)n
(n -
(1], ;) =
1\ () 1\ VI)
rp"(T_-l(1], ;)) w~.~W(T-l('I}, ~»)
.
1\
we(~)
1\
(hjl). v,('I}) .
(4.5.30)
2 8 "
Since the volume of Ze is of ordere 2n - 1, it follows from (4.5.22), (4.5.28), (4.5.29) and (4.5.30) that the proof of (4.5.16) will be complete if we show that lim (n. - 1) 1 (2m')1& .~o
f
a */( J
1:) 'I}, ~
1\
~~(W(T-l('I}, ;))) e2"
I\we(;) 1\ (11,-1). V (11) j
1
°1
(Pl.e)eZ.
= (-I)'
f
(h;-') *1(71)
A
(hi')· vf(7J) •
(4.5.31)
'1 ehj(YJ)
Since, for every 17 f h,( VI), the set of all ~ E ([)n with (1j,;) E Z. is of real dimension 2n - I, and sinee the form ro~(W(T-l(1], E))) 1\ we(;) is of degree 2n - 1 in~, in (4.5.31) the form 0;·1 can be replaced by a:l. In \iew of (4.5.27) this inlplies that in (4.5.31) the form 0;·1 can be replaced by (hjl). I. Therefore, it is sufficient to prove that, for 12
HenkiD/Leiterer
178
4. Global integral formulas on Stein manifolds
every fixed 1'] E hl ( V I),
J
(n - 1)1 (2ni)"
we(w(T-l(1'], En)
A
WE(E)
= 1.
eh
{lEfJtI:('1.f)EZ.}
This follows from the Leray formula (cp. Remark 1.10.3), because, for w,(W(T-l(1'], E)))
A
wl(E)
=
We(w(T-l(1'], E»)
1\
(f},~) E Z.,
we(~)
and, by (4.5.18), (W(T-I(f}, ~)), ~>
=
(W(T-l(f}, ~)), U(T-l(1'], ~))>
= 118(T-l(f}, ~))II~ = e2 • •
4.5.3. Theorem (Koppelman-Leray formula). We use tke notation8 from Sub8ection 4.3.l. Let (8*, ~*) be a Leray section Jor (D, 8, cp) (Subsection 4.3.2), an,a let 'V ~ max (2n~, n,,*) be an integer. SUPP08e, in addition, that all derivatives oj cp"(z, C) 8*(Z, C)/(s·(z, C)' s(z, C) 'which are of order ~ 2 in z and of order ~ 1 in Care continuous Jor all (z, C) in some neighbourhood W C D X X of D X aD.l) Then, for every continuous (0, q)-Jorm f on D 8uch that 5f i8 also continuous on D, 0 < q ~ n, BD aj, R aD!, afRaDfand aBDi are continuous in D, and we have thejorrns LaDj, RaD
W,
(-I)
In this section we construct the support functions 4>(z, C) and ~(z, C) (cp. Theorem 2.4.3 and Construction 3.1.1) for strictly pseudoconvex open sets in Stein manifolds. 4.S.1. Notation and preliminaries. Let X be a Stein ma.nifold of complex dimension ?I, let Do C C X be a strictly pseudoconvex open set, let 9 c c:: X be a neighbourhood of oDo, and let eo be a strictly plurisubharmonic GI·function in a neighbourhood of 9 such that Do n 0 = {z EO: eo(Z) < o}. Remark. We do not assume tha.t deo(z) =1= 0 for all % E oDo. Further, the case Do = 0 is also admitted. _ Set N(eo) := {z E (): eo{z) = o} and suppose that N(eo) c c () .
(4.8.1)
We choose a metric dist (-, .) on X as follows: Consider X as a O°O·submanifold of some flm and denote by dist (0, .) the restriction of the Euclidean metric in B'" to X. If Y, Z C X, then we write dist (Y, Z) : = inf {dist(z, C) : z E Z, CE Y}. Let A, c:: c BI c:: c X (j = 1, ... , L) be finitely many open sets such that 8 ~ Al U ••• u A L and, for every 1 ~ j ~ L, there are holomorphic coordina.tes b, = (bjb ••• , bin): B1 ~ en. For % E BI we write %1 = (zil' ••• , %jA) := b,(z) = (bil{z), ••• ,bJta(z)), and we denote by Xii = Xji(Z), k = 1, ... ,2n, the corresponding real coordina.tes such that ZJk = x;,,(z) i%ji+A(Z), k = 1, ... , n. We can assume that the
+
186
4. Global integral formulas on Stein manifolds
sets b,.(B1 ), ••• ,bL(BL ) are convex. For z, CE B, we set [z, Cll := {e E B f : ~1 = AZf + (1- A) CI' 0 ~ A ~ I}, j = 1, ... ,L. Since (10 is strictly plurisubharmonic on (J, since Af C C B f , and since '8 C Al U ••• U A L , we can find numbers 8, {J 0 and OI·functions ajW on B, (j = 1, ... ,L; le, l = 1, ... , n) such that the following con· ditions are fulfilled:
>
> 28
dist (AI, 8Bf )
i 1:.1-1
_~le~C]
for j = I, ... , L •
(zJI: - CJI:) (zi' - Cil)
8Cji 8Cjl
for CE AI n 8,
i
Z
E (j
0
with
> 3{J[dist (z, C)]2
< dist (z, C) :::; 28
I
(8 (1o(C) - a.1w(C») (Zjk - Cji) (Zjl - eft) I .t. I -1 8'Ji 8ej l for CE A, n 8,
~ ~
1 2 k.I-l
for
(4.8.2)
Z
EO
I
(
8 (1o(C) aXji aXil
dist(z, C) ~ 2e
with
8B(1o(~») (xii(Z) aXjk
1
and j = 1, ... , L . (4.8.3)
< P[dist (z, C)]2 and j = 1, ... , L .
- Xjk(C» (X;I(Z) -
8xil
CE.A, n 8; e, Z E 8; diet (z, C) ~ 28; e E [z, C]1;j
(4.8.4)
Xjl(C») 1 < P[dist (z, e)]2 = 1, ... ,L.
(4.8.5)
Since N«(1o) c: C (J, after shrinking e, we can find open sets W, U such that Do u U is strictly pseudoconvex and, moreover, the following conditions are fulfilled: N«(1o)
C
c: U c: c: W c: c: (J ,
(4.8.6)
elp
0 so small that, for every e EID12(O) with 11(10 - (1112.8 ~~, (1o(z) (1(z)
>0
for
Z E
6"
U
(4.8.11)
and conditions (4.8.3)-(4.8.7) are fulfilled with (1 instead of (10' where (4.8.12)
e
We denote by U~«(1o) the set of all E ID'l2(0) with 11(10 - el12, fj :::;~. Then every (1 E U.,(eo) is strictly plurisubharmonic on 0 (thiS follows from (4.8.3) with e instead of eo) and it follows from (4.8.11) that, for every E ~«(1o),
e
Do := (Do" U) u {z E U: (1(z)
< O}
187
4.8. The support functions t1J and ;)
is a strictly pseudo convex open set such that
DQ
U
U = Do
SDe C U and D(} n U
U,
U
=
{z
E
U:
~(z)
< O} • (4.8.13)
Let O1((Do u U) X U) be the Frechet space of all O1-functioDs in (Do U U) X U endowed with the topology of uniform convergence together with the first-order derivatives on compact sets. For ~ E llc,({)o) we define L ~(C)" ~ rl(C) [2" ~ -SJ(Zjk - C1k) ~ aiI:Z(C) (z,;1: - Cik) (Zil -
FQ(z, C) := -
;-1
for
+
~p
k-1
dist(z, C) ~ 28 •
z, C E6 with
ejl)
]
~l-1
(4.8.14)
This definition is correct, because, by (4.8.2), for all z, CE (1 with rl(e) dist (z, C) ~ 28, z belongs to B 1•
=t= 0
and
4.8.2. Lemma. We use the notations from Subsection 4.8.1. Th.en, lor every ~ E U~(eo),
Re F,lz, C) ~ e(C) - e(z)
for
z, CEO
+ {J [diet (z, C)]2
dist (z, C)
with
~
28 •
(4.8.15)
Proof. Let C EAt, Z E 0 and diet (z, C) ~ 28. Then it follows from Lemma 1.4.13 and Taylor's theorem that
"ce(C)
- Re [ 2 ~ - - (Zjj; 1:=1 SC}k
= €l(C)
- {)(z)
+
"
Cjk)
-
+
82e(C)
--- -
~
1:, Z-1
oejl: BCjl
"
~
e(C) -"B2--
(Zjj: -
Cjk)
(ZjI -
k.'-1 oe';l: SC;l _ -;: , (Zjk - Cjk) (Zjl - ~jl) T R{z,
ep)
]
C) ,
where, by (4.8.5) (which holds, by definition of Uc7 (eo), with €l instead of ()o), IR{z, ')1 ~ {J[dist (z, e)]I. By (4.8.3) and (4.8.4), this implies (4.8.15) . •
4.8.3. Theorem. We use the notations 0/ Subsection 4.8.1. Then there exi8t continuous maps a, a, b, b from Uc7 (eo) (endowed with the metric induced by the norm 11-112,8) into the Frechet 8pace 01(Do U U) X U) 'With the following propertie8: If e E Uc7 (eo) and qJQ := a(e), $(} := a(e), .1lfQ := b(e) and MQ := b({)), then the following conditions are fulfilled: (i) Wiz, C) and (PQ(z, C) are holomorphic in Z E D(} u U (= Do U U, cpo (4.8.13)). (ii) qJQ(z, C) =F 0 and $Q(z, C) =F 0 for C E U, Z E D(1 u U with dist (z, C)
~
e•
=t= 0 for C E U, Z E D(1 U U • (4.8.17) qJ(}(z, C) = F,/z, C) MQ(z, C) ,and iP,,(z, C) = (FQ(z, C) - 2e(C» MQ(z, C) for C E V, Z E DQ U U with dist (z, C) ~ 8 • (4.8.18) $(J(Z' C) = qJQ(Z' C) for CE N(€l), Z E Do U U . (4.8.19) Miz, C) =F 0
(iii)
(4.8.16)
and
Me(z, C)
Proof. Conditions (4.8.3)-(4.8.7) with f! instead of (4.8.7)Q' First we prove that, for everYf! E Uc7 (eo),
> -83P
and
ReFQ(z, C)
W
U DIl
with e S; dist (z, C)
2
Re FQ(z, C) for
eo will be denoted by (4.8.3)(1 to
CE W,
Z E
-
ei{J 3
-2t?(C) > -
~ 28 •
(4.8.20)
188
4. Global integral formulas on Stein manifolds
To do this we fix Bome points, E Wand z E W u DQ with 8 ~ dist (z, C) ~ 28. Then, by (4.8.8), z E 0 and it follows from (4.8.15) that Re F Q(z, C) ~ ()(C) - ()(z) + {J8 2 • Together with (4.8.7)Q this implies
> -e(z) + j-{J8
Re FQ(z, C)
2
(4.8.21)
and (4.8.22) Further, since z E () and z E W U D Q, at least one of the following relations holds: z E W or z E () n D Q• If z E W, then it follows from (4.8.7)Q that ()(z) 8 2{J/3. If z E () n D Q, then it follows from (4.8.13) and (4.8.11) that e(z) < O. In both cases -()(z) > - 8 2f3/3. Together with (4.8.21) and (4.8.22) this implies (4.8.20). In view of (4.8.20), for CE Wand z E W u DQ with 8 s dist (z, C) ~ 28, we can define In F,(z, C) and In (FQ(z, ') - 2()(C)) for all (! E ll.,«()o)' Choose a COO. function X on X X X such tha.t X(z, C) = 1 if dist (z, C) ~ 8 + 8/4 and X(z, C) = 0 if dist (z, C) ;;;::: 28 - 8/4. For CE Wand Z E W u DQ we define
k(Z, C)I ~ ~((!,,(t) - e.t(z) + [dist (z, e)]2) for zED" U 8", C E 8" with dist (z, C) ;;;; 2e • ~1c(Z,
~ e.
(4.9.1)
(4.9.2) (iv) (/>,,(z, z) = 0 for all z E Ok . (4.9.3) Set 'P,,(z, C) := (/Jk(F,,(z), FI:(e» for Z E Fj;l(D k U Ok), CE Fi"l(O,,). Then, by Corollary 4.9.4, after shrinking 8", we can find T*(X)-valued Ol-ma.ps h:(z, C) defined for z E F'kl(DIe U Ok) and ~ E Fi"l(()k) such that the following conditions are fulfilled: (1) hf(z, C) E T:(X) for z E F];I(Dre U Ot), CE F];l(OIl) • (2)
ht(z, C) is holomorphic in z E Fi 1(Dt
(3)
cp(z, C) ~,,(z, C)
U
Ok) .
= 0 B'UCk that supp! ~ HT("), tken/or all zED
[
"
"RDf(z)1I :::;; 0 dist (z, Hr(~))
]2"-1 ~ IIfllo.D .
(4.10.20)
(iii) By parts (i) and (ti) tke integral RD deJines a bounded linear operator from Lco.,)(D)
into O?o,!l-I)(D) (cp. Subsection 2.11.2). This operator is compact.
Proof. Repetition of tha.t of Theorem 3.2.2. •
4.10.7. Corollary. Let X be a Stein manifold and let D c::: c::: X be a strictly pseudocootJeX open set (witk rwt necessarily smootk boundary). Furtker, kt U1, ... , U N ~ X be open sets suck that D C U1 U ••• U U N' We denote by HOO(D) the Banach space of bounded kolomorpkic Junctions in D endowed with the BUP-Mrm. Then there eziBt bounded linear operators L,: HOO(D) ~HOO(D), j = 1, ... , N, with tke/ollowing properties: (i) For every f E HOO(D), f =
f
N
1: LJ. J=1
HOO(D) and j = 1, ... ,N, tke function L,f is bounded and holomorphic t:n 80me neighbourhood oJ j j " (aD nU,), _ (iii) Iff E HOO(D) admits a contin'UOUB continuation to D, tkenthe!uootionsLj, ••• , LNf admit continuous continuations to jj also. (ti) For every
E
Proof. Choose XI E O({'(Uf ) such that 1: X, = 1 on 15, a.nd define L,f := LDXJ, where LD is as in Theorem 4.10.4 (see the proof of Proposition 3.4.1 for further details) . • Remark. Corollary 4.10.7 can be proved also without use of global integral formulas on Stein manifolds by means of a construction given in the proof of Theorem 3 in HENKIN [5].
4.11. Bounded extension of holomorphic functions from complex submanifolds In this section we prove 4.11.1. Theorem. Let X be a Stein manifold, let D c::: c X be a strictly p8euaocontJtx open set (with not necessarily smooth boundary), and let Y be a closed compk~ submanifold of 80me neighbourhood of D. Then (i) For every bounded holomorphic function f on Y n D, there e~i8ts a bounded holomorphic Junction F on D such that F = f on Y n D. (ii) For every continuous function f on Y n D that is holomorpkic in Y n D, tltere e~i8t8 a continuou8 function F on 15 that is holomorphic in D such that F = / on Y n D. Proof. We prove parts (i) and (ti) simultaneously. Letf be a bounded holomorphic function on Y n D . { continuous function on Y n D that is holomorphic in Y n D .
>
< "},
For ~ E aD and l> 0, we set E,{l» := {z EX: dist (z, E) whe~ dist (., .) is the metric introduced at the beginning of Subsection 4.8.1. Since Y n D is compact, we 13·
196
4. Global integral formulas on Stein manifolds
can choose lJ > 0 so small that for every ~ E Y n D there is a biholomorphic map hl : E~(lJ) ~ en such that he(Y n Ee(lJ)) is the intersection of he(EE(lJ» with a complex plane in e". In view of Corollary 4.10.7, it is sufficient to prove Theorem 4.11.1 for the case when I has the following property: There are a point ~ E Y n aD and an open set Do ~ X such that (4.11.1)
and
I {I
is bounded and holomorphic in Y n Do . is continuous on Y n Do and holomorphic Y n Do .
By Corollary 1.5.20 (which Can be easily proved also for strictly pseudo convex open sets in Stein manifolds), we can assume that, moreover, Do is strictly pseudo convex and Y is a closed complex submanifold in some neighbourhood of Do. Further, by Lemma 1.5.23 we can choose a striotly pseudo convex open set DE C X such that
E,(lJj3) n Do C DE ~ Ee(lJ/2) •
(4.11.2)
It follows from (4.11.1) that D C Do U EE(lJ/3). Therefore, we can choose a strictly pseudoconvex open set Dl such that D c c Dl CC Do U E~(bj3). Then the sets U~ := Ee(lJj3) n Dl and Uo := Do n D] form an open covering of Dl . Since lJ is chosen so small that with respect to appropriate local holomorphic coordinates on Ee(lJ) the manifold Y n Ee(lJ) ,is a complex plane, and since De c c EE(lJ), it follows from Theorem 3.6.8 that there exists a
bounded holomorphic function fe on DE { continuous functionfe on De that is holomorphic in
D~
suoh that Ie = Ion Y n DE' Further, since Do is a Stein manifold (Proposition 2.13.8), we obtain from Corollary 4.1.8 a holomorphic function 10 on Do such that 10 = I in Y n Do. Thenlo - fis holomorphic in De n Do andfo - f = 0 on Y n DE n Do, that is, 10 - IE 3' y{De n Do) (cp. Example 4.1.5). Since, by (4.11.2), U E n Uo C DE n Do n D l , we can therefore apply Theorem 4.1.3 (ii) to the open covering {U e, Uo} of Dl and the function 10 -Jrestrioted to U e n U o. Thus we obtainjE E $y(UE),J~ E $y(Uo) with fo - I = on U E n Uo• Setting F : = 10 in Uo n Do and F : = f~ in Ut n DE' we ('omplete the proof. • Remark. Recall that by Lemmas 3.6.2 and 3.6.7 the extension of bounded holomorphic functions stated in Theorem 3.6.8 (i) can be given by a bounded linear operator. Moreover, the decomposition of singularities stated in Corollary 4.10.7 is obtained by bounded linear operators. Combining this with similar arguments as in the proof of Lemma 4.9.3 (connected with Propositions 1 and 2 in Appendix 2), the proof of Theorem 4.11.1 given above can be easily modified to a proof of the following theorem: Under the hypothe8es 01 Theorem 4.11.1 there exists a bounded linear operator E: HOO( Y n D) ~ BOO (D) such that EI = J on Y n D Jor all f E Boo( Y n D). Here Hoo(D) and HOO( Y n D) are the spaces of bounded kolomorphic lunctions on D and Y n D, respectively. It is not clear whether such a bounded linear operator exists for part (ii) of Theorem 4.11.1 (cp. also point 3 in the Exe~cises, remarks and problems at the end of Chapter 3). If 8D is smooth and intersects Y transversally, then the answer is affirmative (HENKIN [5]). .
io - h
io
ie
197
4.12. FOl'mulas in holomorphic vector bundles
4.12. Formulas in holomorphic vector bundles In this section we show that the integral formulas presented in this book can be generalized to (0, q)-forms with values in holomorphic vector bundles. The idea is as follows: 1. This is trivial for product bundles. 2. It can be proved that, for every holomorphic vector bundle B over a Stein manifold X, there exists an injective holomorphic homomorphism of vector bundles eX: B ~ X X ([)N (for certain N) as well as a surjective holomorphic homomorphism of vector bundles {3: X X (f)N ~ B such that p 0 eX = id (id: identity map). 3. Since the a-operator commutes with IX and p, the formulas in B follow from the corresponding formulas in X X ([)N. Let us explain this in detail. 4.12.1. Lemma. Let B be a holomorpkic vector bundle over a Stein manifold X. Suppose, additionally, that X is a relatively compact open sub8et of 80me larger Stein manifold. Then there exi8t an integer N and a 8urjective holomorphic homomorphi8m of tJector bundles p: X X f/)N ~ B. Proof. In view of the hypothesis that X is a relatively compact open subset of
some larger Stein manifold, this follows immediately from Cartan's Theorem A (Theorem 4.1.3 (i)) . • Remark. For our purpose (formulas on compact sets) it can be assumed without loss of generality that X is a relatively compact open subset of some larger Stein manifold. However, we remark that Lemma 4.12.1 is valid also without this assumption. FORSTER/RA:M8POTT [1] proved that this is possible for N ~ dime B + [(dime X)/2]. 4.11.1. Lemma. Let B, X, fJ, N be as in Lemma 4.12.1. Then there exi8t8 a holomorphic homomorphism of vector bundle8 eX: B ~ X X ON such that {301. = ide
Proof. The proof is similar to the proof of Lemma. 4.2.1 and is left to the rea.der. • Now it is easy to generalize the integral formulas of the preceding section to (0, g)forms with values in holomorphic vector bundles. Consider, for example, the Koppelman-Lerayformula stated in Theorem 4.10.4. Let X, B, «, p, N be as in Lemmas 4.12.1 and 4.12.2. Let Dec X be a strictly pseudoconvex open set (with not necessa.rily smooth boundary) and let L D , RD be as in Theorem 4.10.4. We denote by LZ and R~ the corresponding operators for forms with values in the product bundle X X CN. Set L~ : =
{3
0
LZ
0 eX
and
R~: =
{3
0
R:Z
0 eX •
-
-
-
-
Since a., pare holomorphic and, therefore, a 0 a:. = IX 0 a and a 0 {3 = {J 0 eX = id, then we obtain from Theorem 4.10.4 the following
0
(4.12.1)
a, and since
p
_ 4.11.3. Corollary. (i) For every continuous bounded 8ection /: D af is alBo continuo1U and bounded in D, we have f = L!Jf
+ RB 8f
in
D.
~
B Buch that (4.12.2)
(ii) Let 1 ~ q ~ n, and let f be a B-valued continuoU8 and bounded (0, g)-form on D
such that af is also continuous and b"Oundea on D. Then
f
=
8R~f + RB 8f in D.
In particular, if
8f =
(4.12.3)
0 in D, then
u:= RBf is a continuo1U 80lution of
(4.12.4)
au = J in D.
198
4. Global integral formulas on Stein manifolds
Finally, we remark that, since D e c X and, therefore, Theorem 4.10.6 is valid also for the operator R~.
IX,
{3 are bounded on D,
Notes Integral formulas for solving the a-equation on Stein manifolds X were first obtained in 1975/76 by PALM [I] and STOUT [1], however under the additional condition that X is a complete intersection in lJ". The formulas on general Stein manifolds presented in this ohapter (without those of Seotions 4.10-4.12) were obtained in 1979 by HENKIN/ LlUTJlBlIB [1] by means of ideas of DYNIN (personal oommunioation, 1974) and BISHOP [1] (1961). HOBTIIANN[I](1979) used aoonstruotion of TOLEDOjTONG [1] and obtained a formula for solving the a-equation on striotly pseudoconvex Coo-domains in Stein manifolds. The Weil formula in (In was obtained in BERGMAN [1] (1934-36) and A. WElL [1] (1935) (see also SOIllMII8 [1], GLEASON [1], HABVEY [2]). The Koppelman-Leray-Norguet formula for analytic polyhedra in (In was obtained in 1971 by HENKIN [6] for (0, I)-forms and then by POLJAKOV [1] for (0, q)-forms by means of ideas of HENKIN [1], LERAY [1], LIEB [2], KOPPBLMAN [1]. and NOBGUET [2]. Then RANGE/SIU [1] and POLJAKOV [3] proved this formula for domains with piecewise smooth strictly pseudoconvex boundary in en. For the general case of striotly pseudoconvex polyhedra in this formula was obtained by HENKIN (see CIBKA/HBNKIN [1]), SBBGBEV [1] and HENKINjLEITEBEB [1], see also 0VBELID [3]. _ Under oertain additional "complex" non-degeneration conditions the solution of the a-equation on real non-degenerate striotly pseudoconvex polyhedra D in a Stein manifold X which is given by the Koppelman-Leray-Norguet formula (4.9.7) admits uniform estima.tes. The first such estimates were obtained in 1971 by HENKIN [6] for the case of analytio polyhedra in (In and (0, I)-forms. POLJAKOV generalized this result to (0, q)-forms (see FUKs [1]). For domains with piecewise smooth strictly pseudoconvex boundaries in (In such estimates were obtained in 1973 by RANGE/SIU [1] and POLJAKOV [3]. HENKIN/SEBOEEY [1] proved uniform estimates for the a-equation in striotly pseudoconvex polyhedra in en of a more general olass oontaining both classes montioned above. The class of striotly pseudooonvex polyhedra considered in HENKDJ/SERGEEV [1] is defined by two oonditions (0) and (OR) which can be formulated as follows: Let X, D, UD, X~, Ft. 61 , ei. N be as in Definition 4.9.1. For every collection K = (kl , ••• ,Ie,) of integers 1 ~ ~ < ... < Ie, :;; N we set
en
SIC := {z
E
aD: l?i,(Fis(z»
= ...
= (?lc,(FTc,(Z»)
= O} •
Condi tion (C). For every collection K = (~, ... , lei) 0/ integer8 1 ~ kl < ... < k, ~ N ehe complex Jacobi matrix (with respect to locaZ holomorphic coordinates) 0/ the map (FTc" •.. , Fta): U'D - XtJ X ••• X Xi, has constant rank on S K, which is ~ min {dime Xi" ... , dimo Xi,}. We denote this rank by rIC. Condition (OR). Let K = (~, ... , k,) and M = (~, ...• m,) be two coUection8 oj integer8 1 :i kt < ... < le, ~ N and 1 :;;; m 1 < ... < me :;;;; N 8uch thai, rKu{m,} > rK for every s = I, ... , t. Then the rank 0/ the real Jacobi matrix (with respect to local real coordinates) 0/ the map (Qma 0 F M" ••• ,~, 0 F M " Fi" ... , Fi,): U'D - Be X Xi, X .,. X Xi,
+
V equal to t 2'1' K on SKulL. Observe that oondition (OR) implies that D is real non-degenerate. HDKDfjPBTBOBJAlf [1] (1978) obtained uniform estimates for the a-equation in real non-degenerate analytio polyhedra Dec tJI satisfying the condition that U SIC is the ailov boundary of D. 11lI-2 FBl7JIm [1] (1981) obtained an appropriate generalization of this result to the oase of tJn. By means of the integral formulas on Stein manifolds presented in this chapter, all these estimates for the a-equation can be generalized to Stein manifolds. Observe also the following recent result of HXUNEJUNN [1] which gives another (very simple) poBSibility to generalize the results of HENED'/SlIBGEEV [1] to Stein manifolds: Let X be CJ closed complez submani/old 0/ e"', and let Dec X be a 8trictly p8eu.doccnwex
199
Exercises, remarks and problems
poZyhedron sati8/ying conditions (C) and (CR). Then there exi8t8 a strictly p8eudooonve:J: polyhedron Dl C C eM sati8/ying conditions (C) and (CR) as welZ as a neighbourhood UI ~ Co/ 151 and a holomorpkic map k from Ul0nto Ul n X BUck that h(z) == z lor z E UI n X and h(D1 ) = D. Remark that, for the case of an analytic polyhedronD, the corresponding striotly pseudoconvex polyhedron Dl need not be also an analytio polyhedron. Thus, in this way, estimates for the a-equation in analytio polyhedra in Stein manifolds are obtained by means of estimates for a more general class of polyhedra in em. The Extension Theorem 4.11.1 (i) was obtained in 1980 by HENlWi/LIIITIIBD [2]. Part (i) of thisj theorem was proved independently by AKa [I] (AllAR a.ssumed that aD is of class 0 00 whereas Y is allowed to be a Ooo-submanifold of some neighbourhood of D suoh that Y n D is complex). For the case that aD is (]I and the interseotion aD n Y is transversal, Theorem 3.11.1 was proved in 1972 by HENKIN [5]. Observe also that OuMENGE [1] (for the case of transversal interseotion Y n aD) and AMAB [1] (for the general case) obtained a version of Theorem 4.11.1 for functions in Hardy ola.sses. "Partially", Theorem 4.11.1 (ii) is a generalization of some of the known results on interpolation sets (cp. points 6 and 7 in the Exercises, remarks and problems at the end of Chapter 3 - a set N ~ aD is called an interpolation set if every oontinuous function on N extends to a oontinuous function on 15 that is holomorphic in D).
Exercises, remarks and problems 1. Let X be a Stein manifold of complex dimension n. Suppose that there is a holomorphio vector bundle B of complex dimension n over X as well as a holomorphic map h: X X X - B suoh that the following conditions are fulfilled: (I) h(z, C) e B. for all z, C E X (B. is the fibre of B over z). (2) h(z, C) 9= 0 for z =f: C. (3) For every fixed z e X, the map h(z, C) (considered as a B.-valued map) is biholomorphic for C in some neighbourhood of z, where h(z, z) = O. Then it is easily seen that in the construction of the integral formulas presented in this chapter, Band h can be used instead of T(X) and 8, where the factor ql can be omitted. Open problem. Let X be an arbitrary Stein manifold of complex dimension n. Do there exist a holomorphic vector bundle B of oomplex dimension n over X and a holomorphic map h: X X X -. B such that conditions (1)-(3) are fulfilled? I. Prove the Extension Theorem 4.11.1 for the case that Y is a closed analytio subset of some neighbourhood of i5 whose singularities are isolated and contained in D (for the definition of analytic sets see, for example, GBAUEBT/FmTZ80HE [1] and G11N1UlfGjRoa81 [1]). 3. (A counterexample to the extension Theorem 4.11.1 for the case that Y has a singularity on aD.) Let D:= {z E 0 1 : IZtlt IZ8 - 111 < I}, Y l := {z ED: Zt = O}, Y. := {z ED: ~ = z:}. Then Yl U Yt is a olosed complex submanifold of D (whioh cannot be oontinued smoothly into a neighbourhood of i5 at the point (0,0»). Then there does not exist a bounded holomorphio function F on D such that F(z) = 1 for z E Y1 and F(z) = 0 for z E Ys. Proof. If F is a bounded holomorphio funotion in D, then it follows from the Cauchy formula applied to the discs G. := {z ED: %1 = e}, e > 0, that there is a oonstant 0 < 00 such that
+
IaF(C, e) I~ Oe
for all
8zl
This is impossible if F(O, e)
=
B
>
0.
1 and F(el , 8)
=
0 for all
B>
O.
4. Open problem. Does there exist a (not necessarily real non-degenerate) analytio polyhedron Dec CI without uniform estimates for the a-equation? 5. a} (HENKIN [7], LIED [2], 0VBBLID [2]). Let X be a Stein manifold, and let D CC X be a strictly pseudoconvex polyhedron satisfying conditions CO) and (OR) mentioned in
200
4. Global integral formulas on Stein manifolds
the Notes above. Let A(D) be the algebra of all continuous funotions on D that are holomorphio in D. Prove that for every point zED the ideal {f E A(D) : fez) = O} is finitely generated. b) Open problem. Does there exist a domain of holomorphy D cc (!J2 such that, for some zED, the ideal {f E A(D}, 1(1,) = O} is not finitely generated? 8 (GRolllOVjELIA.SBERG [I]). Prove that every Stein manifold of complex dimension n is biholomorphioally equivalent to a olosed oomplex submanifold of C[Sn}2l+2. Remark. A proof of the Gromov-EliaAberg theorem is not yet published. It was proved by REMMERT, BISHOP [1] and NAIU,SWRAN [3] (see HORMANDER [1]) that every Stein manifold of complex dimension n is biholomorphically equivalent to some closed complex submanifold of 1]2"+1.
'1. Let D .. = {Z E en': ZZ· < I} (cp. point 25 in the Exeroises, remarks and problems at the end of Chapter 1), and let SD.. = {Z:ZZ* = I} be theSilovboundary of Dft. a) Prove the Bochner formula (BOOHNER [2]) :
feZ) =
e..
f ~(~J~_~ ___ , [det(l - G*Z)]fI
ZED.. ,
8D,.
where f is a continuous function on 15" that is holomorphic in D". _ b) Find a formula for solving 8g = /, where f is a continuous (0, I)-form on 81 = 0, such that Sf :=
f [det - __ g(G) dGO*Z)]ft __ = 0 (1 -
for all
V ..
with
ZED".
SD"
0) (DAuTov/HENKIN (unpublished». Prove that, for some satisfies the estimate sup Ig(Z) I [dist (Z, SD,,)]n'-2 ZED,.
~
a
1. e) Open problem. Do there exist uniform estimates for the a-equation in D" if n > I ? f) Open problem. Let zED... Is the ideal {f E A(Dn): 1(1.) = O} finitely generated? Here .4(15,,) denotes the algebra of continuous functions on 15ft that are holomorphic in Dn'
8. Let X be a Stein manifold, let Dec X be a strictly pseudoconvex polyhedron satisfying conditions (C) and (CR) mentioned in the Notes above. Let SD bo the BergmanSiIov boundary of D, and let K be the Cauchy-Leray-Norguet projection defined by the first sum in (4.9.5) for every continuous funotion on SD. For 0 < CIt. < I we denote by HtII(SD) the space of CIt.-Ht>lder continuous functions on SD and by AG¥(D) the space of CIt.-Holder continuous functions on 15 that are holomorphic in D. a) (AJRAPlIITJAN [I], JORIOKlII [I]). Prove that, for every 0 < CIt. < 1 and I E HG¥(SD), XI E AP(D) for all 0 < (J < tx. _ b) (JORICKE [1]). Let D be a polydisc and 0 < <X < ]. Find f e Hf%(R) suoh that K/ Ef AtII(D). c) (AHERN/SCHNEIDER [I], PRONG/STEIN [1]). Let D be a smooth strictly pseudooonvex domain and 0 < <X < 1. Prove that, for every f E H tII(8D}, HI E AtII(D). d) Let D .., SD.. be as in point 10, and let S be the Cauohy-Szego projection for D", that is, the orthogonal projeotion from Ls(Dn) onto the subspace spanned by the holomorphio functions in a neighbourhood of D,.. Theorem (JOBIOKE [2]). Then, for each 0 I and IE H(JI. (SD,,), Sf(z) = 0 (~tJj-W'/2)')(ln!5)a) where ~ = dist (r., 8 Dn) and C1 = - I for n odd, C1 = 0 for n even. This estimate cannot be improved.
011 X and oach Z E P Remar k. If X =
-(
rp (z ) 1p z,
Q]2
1
z) = (2:r:i}n
J -(
IPtp ., z)
diI" ... 1\ di",
(it _ Jl(Z)) ... (f"'----=...c.c_'--j-"'(-z» ,
oS
Hint. Apply the formula in d) to the function tF1Ph z). 11. For som.e further results in the theory of functions on eomplex manifolds obtained by the method of integral formulas let us reff'r to ANDEBSONjBERNDTSSON [1], AJRAPl£TJANjHENXIN [1], AJSENBERG/JulAxov [1], BERNDTSSON [2]. BISIIOP [2], DAuTOVjHENKIN [1], GINDIKIN/HENXIN [lJ, GRIFFITIISjH.ARBIS fl], HARVEyjroLKING [1], HE-:ofKIN [:l], KRANTZ
[2],
RUDIN
[3J,
SKODA
[2],
VLADDIJROV
[1].
Appendix I. Estimation of some integrals
In this appendix we give estimates for BOrne integrals in B,n, which are used in the book. Notation. By n we denote a positive integer. If x E 11/', then we denote the canonical coordinates of x by Xt, .•• , x"' a.nd we assume that the orientation of ll" is defined by the form dX1 " ••• " dx", x E 11!'. We will also write dx1 ... dx" and daft instead of dXt A ... A dx". If 8 ~ IR" is a smooth surface of real dimension k ~ 71., then we denote the Euclidean volume form on S by dOt. For x E R,n we set fi
x" := (x3 ,
...
Ixl:= (~ IX112)112.
and
,x,,)
i-I
In the proofs we denote all "large" constants by 0, 0', ... , and all "small" consta.nts will be denoted by ex, ex', •••• An expression of the form a(x) ~ Obex) (a(x) ~ exb(x») must be read as follows: There exists a constant 0 < 00 (~ 0) such that a(x) S;;; Obex) (a(x) ~ exb(x») for all x considered in the corresponding situation.
>
Proposition 1. Let n ~ 1 and 0 jOf' all t, 8 € Il" with It\, \8\ ~ R
Jl
< R < 00. Then there i8 a constant 0 < 00 8uch that
xt-Btl daft 8\"
XI-it ---- - - - -
sEfltI.lsl E 2 , A': EI -* H 8uch that A21 = A" A'. Proof. Set P 21 := A"PA', where P is the orthogonal projection from H onto Ker (AuA ") . • A bounded linear operator A in a Banach space E is called a Fredholm operator, if dim Ker A < 00 and dim (E/lm A) < 00 (then 1m A is a topologically closed subspace of E - cpo Proposition 4 below). A linear operator Yin E is called compact if, for every bounded Bubset U of E, V( U) is a relatively compact Bubset of E. Recall that for every compact operator V in E, the operator id + V is a Fredholm operator, where dim Ker (id + V) = dim (E/lm (id V)) (id := identity opera.tor).
+
Proposition 3. Let E, F be Banach spaces and let A be a closed linear operator between E and F with the domain of definition D(A) ~ E. Suppo8e that the following conditions aTe fulfilled:
Appendix 2. On Ba.nach's open mapping theorem
211
(i) A(D(A)) = F. (ii) There exists a bounded linear operator B:F -+E BUch that B(F)
id - AB i8 compact. Then there exiBts a bO'UMet linear operator and AA(-l)= ide
A(-l):
S D(A) and
F -+ E 8UCht that A(-l)F ~ D(A)
Proof. Since id - AB is compact, we can find closed linear subspaces M and N of F such that M n KerAB = 0, AB(M) n N = 0, M KerAB = AB(M) + N = F. Since A(D(A)) = F, and since dim N = dim Ker AB, we can find a linear operator S: Ker AB -+D(A) such that AS(Ker AB) = Nand ASx 9= 0 for all o 9= x E Ker AB. Let B': F -+ E be the bounded linear operator defined by B'x = Bx for x E M and B'x = Sx for x E Ker AB. Then AB' is invertible. Set A(-l)
+
:=
B'(AB')-l . •
Proposition 4. Let A be a closed operator between Banach spaces E and F with the domain of definition D( A) C E. If A (D( A») is finitely codimensional in F, then A (D( A) ) is a topologically closed subspace oj F. Proof. Let n be the codimension of A(D(A)) in F. Choose a linear operator B: on -+ F such that 1m B + A (D(A)) = F. Let A' be the operator with the domain of definition D(A') := D(A) EEl QJI& which is defined by A'x = Ax for x E D(A) ffiO and A'x : = Bx for x EO ffi en. Then A' is a closed linear operator between E EB (Jft and F such thatA'(D(A')) = F. Set KerA':= {x ED(A'): A'x = O}. Then KerA' is a topologically closed subspace of E EB Oft. Let A' be the closed linear operator between E EEl en/Ker A' and F with the domain of definition D{A ')/Ker A' which is induced by A'. This operator is bijective from D(A')/Ker A' onto F. By Banach's open mapping theorem its inverse (.A»-l is a bounded liDEiB,r opera.tor from F into E EB en/Ker A'. Since A(D(A») = [(..4')-1]-1 (E Ef) O/Ker A'), this implies that A(D(A)) is topologically closed . •
14·
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List of symbols
real, complex number~ real, complex }~llelidean space of dimension 11, topological closure of the set X X z : = (ZI' ... , Z,.) for z = (zl' ... , Z,.) E (]'I := (lz112 + IZnI2)1/2 for Z = (Zl' ... , zn) E (In Izl means that Y is contained in a compact subset of X YccX dimBX, dimcX real, complex dimension of X degree of the differential form I deg I where I is a differential form; 42 1'1 where I is a differential form; 42,88 11/(x)11 length of the multi.index I; 43 III the smallest closed set outside of which I vanishes supp I summation over strictly increasing multi.indices; 43 E' boundary of D aD Euclidean distance if X, Y ~ llm; see p. 185 if X, Y ~ manifold dist (X, Y) p PO(D)-hull of K; 31 J) if) O(D).hull of K; 17, 97 ",D I . .evi polynomial of (!; 25 F,,(z, C) 76,106,187 F(z, C) 75--77,106,107,187 r/J(z, C) 106,107,187 ~(z, C) 116 H~(b) Hormander, Euclidean ball; 116 Il~(b), E~(()) Hormander ball; 194 H:(t5) Hormander diameter; 116 diamH d(~l' ... ,eN; b) order of the system H~J(b); 132 Lebesgue measure in gc or Euclidean volume form on k-dimensional derk surfaces of R,m er~( U) := dert u for matrices of differential forms; 45,46 det, det, ..... ,'''' 44 w,w' sup-norm; 48, 88 1I'l\o,Y (l-Holder norm; 49, 88 IHla,y T(X), T*(X) complex tangent, cotangent bundle of X: 162 Bz fiber of the vcctor bundle B over z (b,a) := blat bnan if b, a E q;n; see p. 164 if bE T:(X). a E T.(X)
Il,C ll'l'l., q;n
+ ...
i
J
+ ... +
224
List of symbols
t'lhcaf of germs of holomorphie sections in the holomorphic vector bundl{' B; 160 := x@xxc N ;260 smooth pif~r('s of a pie
