Gennadi Henkin Jiirgen Leiterer
Theory of Functions on Complex Manifolds
1984
Birkhiuser Verlag Basel· Boston· Stuttgart
LDnry of Congress Cataloging in Publication Data Henkin, Gennadi, 1942Theory of functions on complex manifolds. (Monographs in mathematics; v.79) Bibliography: p. Includes index. 1. Functions of several complex variables. 2. Complex manifolds. I. Leiterer, JUrgen, 194511. Title. III. Series. QA331.H4525 1984 515.9'4 83-7157
ISBN 3-7643-1477-8 Cfp-Kurztitelaufnahme der Deutschen Bibllothek Henkin, Gennadi: Theory of functions on complex manifolds I Gennadi Henkin; JUrgen Leiterer. - Basel; Boston; Stuttgart: Birkhiuser. 1984. (Monographs in mathematics; Vol. 79) ISBN 3-7643-1477-8 NE: Leiterer, JUrgen:: GT
All rights reserved. No part of this publication may be reproduced. stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without prior permission of the copyright owner. ~1983 Akademie Verlag Berlin Licensed edition for the distribution in all nonsocialistic countries by Birkhlluser Verlag. Basel 1984 Printed in GDR ISBN 3-7643-1477-8 ISBN 0-8176-1477-8
Preface
Till the early fifties the theory of functions of several complex variables was mainly developed by constructive methods of analysis. We emphasize the work of A. WElL in 1935 and of K. OKA in the period from 1936 till 1951. WEIL generalized the Oauchy integral formula. to polynomial polyhedra in Q)" and obtained an analogue of the Runge approximation theorem for such polyhedra. Equipped with the Weil formula OKA solved the so-called fundamental problems (Cousin problem, Levi problem, et a1.). In the fifties H. C.ARTAN, J. P. SERRE and H. GBAUEBT discovered that by means of the theory of sheaves introduced in 1945 by J. LEBAY the constructive methods of analysis in the theory of Oka can be reduced to a minimum and, moreover, that the theory of Oka admits far-reaching generalizations. In the sixties L. Ht>BMAND1IIR, J. J. KOHN and C. B. MORREY deduced the main results of OKA with the help of methods from the theory of partial differential equations and obtained, in addition, estimates in certain weighted La-metrics for solutions of the Cauchy-Riemann equations. During the fifties and sixties it seemed that the method of integral representations, which works so successfully in the case of one variable, is not suitable to the case of several variables, because it is troublesome and gives only very special results. However, in the seventies integral representations turned out to be the natural method for solving several problems related to Oka's theory, which are connected with the boundary behaviour of holomorphic functions. The basic tool is an integral representation formula for holomorphic functions discovered in 1955 by J. LEBAY, which contains the 'Veil formula as a special case. Certain developments of this formula made it possible to solve several of such problems that are not easily obtained with other methods. Moreover, it turned out that by means of these formulas one can build up a large part of the theory of functions of several complex variables in a new and more constructive way. It is the aim of this book to present such a new introduction to the theory of functions of several complex variables, where the main results will be obtained in a strengthened form - uniform estimates for solutions of the Cauchy-Riemann equations, uniform estimates for extensions of holomorphic functions from submanifolds, uniform approximation of holomorphic functions that are continuous on the boundary, et a1. It has been assumed that the reader has a certain knowledge of the theory of functions of one complex variable and the calculus of differential forms (Stokes' formula). Chapter 1 starts with facts concerning holomorphic functions, plurisubharmonic functions, domains of holomorphy and pseudoconvex domains. Then we deduce from Stokes' formula the l\lartinelli-Bochner formula and the Leray formula as well ss their generalizations to differential forms in (fJ1t (Koppelman formula and KoppelmanLeray formula). In Chapter 2 first the Cauchy-Riemann equations are solved by means of integral
6
Preface
formulas in pseudoconvex open sets in en. Then we prove this result on Stein manifolds, where an inductive procedure with respect to the levels of a strictly plurisubharmonio exhausting function will be used. For strictly pseudo convex open sets with OJ-boundary, solutions of the Cauchy-Riemann equations with 1/2-Holder estimates are obtained. The identity of domains of holomorphy (Stein manifolds) and pseudoconvex open sets in Cft (oomplex manifolds with strictly plurisubharmonic exhausting funotion) is proved, that is, the Levi problem is solved. Further, uniform approximation theorems are proved. Chapter 3 is devoted to strictly pseudoconvex open sets in Oft with not necessarily smooth boundary. By means of integral formulas the Cauchy-Riemann .equations are solved with uniform estimates in such sets. A uniform approximation theorem is proved for functions whioh are continuous on a strictly pseudoconvex compact set and holomorphic in the inner points. Further, for strictly pseudoconvex open sets D with not neceB88ri1y smooth boundary, an integral formula is constructed which gives bounded holomorphic extensions to D for bounded hoI om orphic functions defined on the intersection of D with a complex plane. In Chapter 4 this result will be generalized to the case of an ~terse~tion with an arbitrary closed complex submanifold in some neighbourhood of D. Chapters 1-3 are self-contained. Here we do not use without proof any result from the theory of functions of several oomplex variables. Only in Chapter 4 we use without proof some special results from the theory of coherent analytic sheaves, for the proof of which we can refer to several books devoted to this subject. In Chapter 4 our prinoipal aim is to extend the integral formulas introduced in the preceding sections to Stein manifolds. Moreover, in this chapter the Weil formula for analytic polyhedra as well as its generalization to differential forms and a more general class of polyhedra. in Stein manifolds is proved. Some applications of these formulas are given. In the Notes at the end some further applications are outlined. In our opinion Chapters 1 and 2 can be used as an elementary introduction to the theory of functions of several complex variables. Chapters 3 and 4 contain more speCial and more difficult results obtained only recently by means of complicated estimations, and references to the theory of coherent analytic sheaves. They can be used as an introduction to one of the actual fields of research in complex analysis. There is also another way to develop the theory of functions of several complex variables by means of integral formulas. This way was outlined in 1961 by E. BISHOP and is based on the concept of special analytic polyhedra. In distinction to the approach presented in this book, the way of BISHOP is suitable not only for smooth complex m.a.nifolds but also for analytic spaces with singularities. However, this way seems to. be more complicated and, above all, does not give uniform estimates, whereas in our opinion the latter is the main advantage of the method of integral formulas. Finally, we point out that in our opinion there are also further interesting possibilities for applying the method of integral formulas, for example to the theory of OR-functions and to problems of complex analysis and integral geometry on projective manifolds connected with the theory of R. PENROSE. We thank Dr. B. JOBICXll (Berlin) who helped improve parts of the manuscript. We thank also Dr. R. HOPPNER and G. REIHER from the Akademie-Verlag Berlin for support and cooperation. We are greatly indebted to Prof. H. BoAS (New York) for proof reading and removing a lot of mistakes (including the worst English ones). Berlin and Moscow, July 1981
G. M. HENKIN and J.
LlIlITBBEB
Contents
1.
Elementary properties of functions of several complex variables ......... .
9
Summary.................................................................
9
1.1. 1.2. 1.3. 1.4. 1.5. 1.6. 1. 7. 1.8. 1.9. 1.10. 1.11. 1.12.
Holomorphic functions... ...... ..... .•.•. ..... .. .•.••... .. . . ....... Application of the Cauchy-Green formula to the a-equation. . . . . . . . . . . . . . Domains of holomorphy ...........................•................ Plurisubharmonio functions ......................................... Pseudoconvex sets.. . .. . ... . . . . . ... .. . . . .. . . . .. . . . . . .. . . . .. . . .. . .. . Preliminaries concerning differential forms. . . . . . .. .. . .. . .. .. .. . . . . . . .. The differential fOnDS ro'(v) and cu(u) •...•...•.........•.............. Leray maps and the operators BaD, BD, LfD and R:D .................. The Martinelli-Bochner formula ... - . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Leray formula ............................................... The Koppelman formula. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Koppelman-Leray formula. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Notes............................................................ Exercises, remarks and problems (one oomplex variable) ................ Exeroises, remarks and problems (several oomplex variables) ............
9 17 19 23 32 '3 4:6 4:8 6' 55 67 59 61 61 64
I.
The a-equation and 'he "fundamental problems" of 'he iheory of fUDoilons on Siein manifolds .................................................
67
Summary.................................................................
67
Formula for solving the a-equation in Ot striotly oonvex open sets Holder estimates.. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . .. .. .. Solution of the a:equation with HOlder estimates in O· strictly pseudoconvex open sets in eft .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. The support function 4)(%, C) • • • • • . . . . • • . • . . • • • • • . . • • • • . • • . . . • . • • • • • • • The Oka-Hefer lemma and solution of (w(z, e), C - 2:) = fI(z. C) ....••••• Formula for solving the a-equation with Holder estimates in 0 1 strictly pseudo con vex open sets in (fJft ••••••••••••••••••••••••••••••••••••••• Oka-Weil approximation ........................................... Solution of the a-equation in pseudoconvex open sets in (Jft. . . . . . • • . • . • .• Uniform approximation.. . . . . .. . ..... ... .. . .. . . . . . . .. . . . . . . . . . . . .. .. The a-equation and Cousin problems in holomorphio veotor bundles over complex manifolds. . ... ... . .... . .. .... . . . . . .. . .. . . . .. . . ... . . ... . . .. Fredholm solvability of the a.equation on complex manifolds with strictly pseudoconvex Ol.boundary. ... ...... .. . . . .. . .. . . . . . . . . . . . . . . . ... .. .. Solvability of the a-equation on complex manifolds with striotly plurisubharmonio exhausting ai-function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solution of the Levi problem for complex manifolds. . . . . . . . . . . . . . . . . . .. Notes ............................................................ Exercises, remarks and problems .....................................
87 69
2.1. 2.2.
2.S.
2.4. 2.5. 2.6.
2.7. 2.S. 2.9. 2.10. 2.11. 2.12.
2.13.
73 77 79 82 83 85 86 87 89 92 97 101 102
8 3.
Contents Theory of functions on smc$ly pseudoconvex se$s with non-smooth boundary in q;- ...•..........•.........................................•.... 107
Summary . . . . . . . . . . . • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 107 3.1. 3.2. 3.3. 3.4. 3.5. 3.6.
The Koppelman-Leray fonnula ...................................... Unifonn estimates for the a-equation ................................. Improvem.e~t of th~ estim!,~es on the boundary ....................... De~omposltlon o.f SlD~antleS ....................................... Urufornl approxImation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. Bounded extension of holomorphio functions from compl(:'x pla.nes ....... Notes ............................................................ Exercises, remarks and problems .....................................
4.
Global integral formulas on Stein manifolds and applications ............. 158
107 116 125 131 139 144 154 155
SUlnmary ... . . . . . . • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 158 4.1. 4.2. 4.3. 4.4. 4.6. 4.6. 4.7. 4.8. 4.9. 4.10. 4.11. 4.12.
Coherent analytic sheaves .......................................... The seotion 8{Z, C) and the function q>(z, C) • • • • • • . • • . • • • . • • . . • . . • • . • • . •• The Martinelli-Boohner formula and the LCl'ay formula. . . . . . . . . . . . . . . .. The Leray-Norguet formula. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. The Koppelman formula and the Koppelman·Leray formula ............ The Koppelman-Leray-Norguet formula .............................. The Weil formula. .................................................. The support functions <J) and
159 161 163 169 173 180 182 185 188 192 195 197 198 199
Appendix 1. Estimation of some integrals ...................................... 202 Appendix J. On Banach's open mapping $heorem ............................... 209 BlbUography .............................................................. 21 2 LiB$ of symbols ............................................................ 223 Subjec$ index .............................................................. 225
I.
Elementary properties of functions of several complex variables
Summary. In Seotion 1.1 we define holomorphic functions of severaJ complex varia.bles
and prove simple properties of these functions. In Seotion 1.2, by means of a. sUriple extension of the Cauchy-Green formula to several variables, we solve the inhomogeneous Cauohy-Riemann equations for some speoial cases. As a consequence we obtain a theorem of Rartogs which gives examples of open sets in (fJn (n ~ 2) where aJl holomorphio funotions can be continued holomorphically to a larger open set. Open sets for which this is not possible are called domains of holomorphy. These are investigated in Section 1.3. Seotion 1.4 is devoted to continuous plurisubharmonio and strictly plurisubharmonio 0 1 functions. In Section 1.5, by means of these functions, pseudoconvex and strictly pseudoconvex open sets are introduced. We prove that every domain of holomorphy is pseudoconvex, but the converse (Levi's problem) is left to Section 2.7. Sections 1.6-1.12 are devoted to integral representation formulas for functions as well as for differential forms n (fJ1l. These formulas form the basic tool for the methods developed in the present book.
1.1.
Holomorphic functions
We assume that the reader knows a certain amount of the theory of functions of one complex variable. Nevertheless we begin with a proof of the Cauchy-Green formula for one complex variable, because this proof is the model for the proofs of the integral representation formulas for several complex variables which form the basis of this book. Notation. Let (jJl be the complex plane. By Xl' Xa we denote the real coordinates in (JJ1 such that q]l 3 Z = Xl ixa• For every complex-valued continuous function J in an open set in q]1 we define (in the sense of distributions)
+
~ := ~ (::1 + ~ !J 8J:= -6/ dz 8z
and
:{:= ! (~ - ~ !.).
8f8J:= -= d.Z , 8z
where Z : = Xl - ix2 ' Then the differential dJ of J ('an be expressed as dJ = 8J + 8J. We also write dJ, oj and aJ instead of dJ, aJ and 8J to make clear that the differentiation is with respect to z if f depends on other variables, too. Recall that a ai-functionJis holomorphic if and only if 8J = 0 (the Cauchy-Riemann equations). 1.1.1. Theorem (Cauchy-Green formula). Let D e c 0 1 be an open 8et with (J1oundary cD, and let f be a complex-valued continuoU8 Junction on D Buch that 8J i8
10
1. Elementary properties
alBo continuou8 on Dl). Then
~ [!(C) dC _ ~
J(z) =
C-z
2m.
2m
8D
1
8J(')
1\
C-z
dC ,
zED.
(1.1.1 )
D
Proof. Fix zED. Then, forC e D " z, d,[(f(C) dC)/(C - z)] = 8J(C) therefore, by Stokes' formula, for sufficiently small 8 0
..!...
J
J
I(C) dC = ~ jJ(C)d C- ~
>
1\
dC/(C - z) and,
a/(C) 1\ de.
2.1ti C - z 27ti C- z 27ti C- z "--I -CE3D CED, ,,-.\ >_ 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 ,,-.,<-
de
1\
de = 2ni.
2n max IJ(C) - J(z) 1 ~ 0 for
8
-+ 0 .
"-.1-8
r
_... 0...
J(C) dC = 2.1ti/(z) . • C- z
1'-'1-_
Since the first integral on the right-hand side of the Cauchy-Green formula (1.1.1) depends holomorphically on zED, and since 81 = 0 for holomorphic functions, we obtain: 1.1.!. Corollary. Let j be a cominuoUB oomplex-valued junction in an open Bel D Then J i8 hoZomorphic in D iJ and only iJ = 0 in D.
C
([}1.
8J
1.1.1. Theorem. Let D c:: c:: qp be a bounded open Bet, and let 1 be a bounded continuou8 in D. Then the continU0U8 Junction
comple~.valued Junction
u(z) := _
~
2na
1
1(C) dC 1\ dC C-z
,
(1.1.1 ')
ZED,
D
•
f,8
a 80Z· uhon oJ.~8'U 8i =
f'in D .
Proof. First we consider the case that/is continuously differentiab1e in D. Fix~ eD. It is suffioient to prove that au/81, = J in some neighbourhood of E. Choose a Coo_ funotion X on (J1 suoh that X = 1 in a neighbourhood Ve c:: c:: D of ~ and X = 0 in some neighbourhood ,of 'Ut(z) := -
(J1"
D. Then'll = 'Ut
~ 1%(C)/(C) de 1\ dC, 2111
D
C-
Z
+ U 2' where
us(z) := _
~ 1Q- - %(C»/(C) de 1\ dC. 27tl D
C-
Z
el/ez whioh is defined in the sense of distributions in D is continuous in D and admits a continuous continuation to ii.
1) This means,
1.1. Holomorphic functions
11
Since 1 - X = 0 in V f , 'lLa is holomorphic in V" that is, 8~ = 0 in V,. Therefore, it rema.ins to prove that ~/8z = / in V" Since X = 0 in a. neighbourhood of f1" ,. D. the function xl can be continued by zero to (J1, and we obtain "-1(z) = _
~ JX(C)f(l;) dE A de = C- z
_
2m
c·
~
Ta.king into account tha.t 8/8z[X(C elude that &u,.(z) = _
8z
~ JX(C + z)/(C + z) de" de. C
2nt.
+ z)I(t; + z)] de =
.
~ Jae[x(c + z)/(C + C
z)]
"de =
8~(l;
+ ")/(r: + z)], we 0011-
_ ~ J~[X(C)/(C)] " 8C.
2ni
23zi ~
r: -
t,
Application of Theorem 1.1.1 with! replaced by 'l1 and D equal to a diso oontainlDg the support of X now gives that 8~/fJi = J in V,. Now we consider the case that f is an arbitrary continuous and bounded funotion on D. Choose a sequence J~ of continuously differentia.ble functions which CODVerpi uniformly on D to f. Then the sequence . Ut{z) := _
~fJI:(C) dE" dC C -z
2nI
D
converges uniformly on D to u, a.nd therefore we have in the sense of distributioD8 8'U/ai = lim 8u"j8Z = lim!1: = J.• Now we pass to the case of several complex variables. Notation. Let on (n. = 1, 2~ .0.) be the space of all n-tuples z = (Zt, ••• , z.) of oomplex numbers zf. The components Z:t., ••• ,ZtI of z E C" will be called the O(Jft(mical (COM") . coordinates of z. BY:l1, '" , X2" we denote the real coordinates in (JtI such that Z, =- z, Wj+.' For Y ~ CA, let GO(Y) be the space of continuous complex-valued functiou on Y. If D C lJ" is open, then OI:(D) (Ie = 1,2, •.. , (0) is the space of i-timea continuously differentiable (with respect to the real coordinates ~) complex-valued functions in D, and O~(D) is the subspace of allf E Oi(D) vanishing outside a oompaot subset of D. We introduce the differential operators
+
a :=2"1(8ax, + T18ZJ+n a)
8"1
and
8 1(86z, - Taz.1+_ 1 e) •
az, :="2
a"
These operators 8/CZf and alai, will also be denoted by 8f and respectively. A ...... order is an n.tuple k = (lei' ... , le,,) of non.negative integers. For every multi-order I: """e wn'te ~~ ",,~. . . .Zi. if ~ E (fJ", k"~.l 0 ••• 0 alt. n' ~ •--1. " •• "'l.. ••• k' • ., 0"1:·- '-Jl Oft. and Q"a.U- •
z.,
N
:= ~. 0
... 0
8/:= Then
4 -
8~", If D c 0" is open andf E OO(D), then we define
• -dz, 8J " 8! - In . I: and fJ!:= I: --=-dz,
i-1 6Zf
;-1
az,
D.
dl = of + a/in D.
(Remark. In general, fJf and 61 are differential forms whose coefficients are distributions. However, in this book we need 81 (al> only if of (8j) is continuoua.) A set P c (jft is said to be an open (closed) :poZydiBc if there are open (olosed) dtace P 1, ••• , P,. in qJ1 such that P = PI X ••• X p •. If " is the center of P" then the point
12
1. Elementary properties
(~1' ••• , 'ft) is called the center of P. If the multi-radiu8 of P.
rl
is the radius of PI, then (rl' ... , rft ) is called
*
1.1.4. Lemma. Let at be complex numbers defined for all multi-orders k. Suppose that 0 Jor all j) the power serie8 ~ al:~t converge8. Then the series ~ atZ k
Jor 80me ~ E Oft (~f
t
IZfl
converges normally in tM polydisc ~
Proof. By hypothesis there is C ~
<
00
such that
sup latZi:l:S;;; 0 ~ ril+ ... +1:"
< 00 • •
lell ::arlfJI
k
k
< 00 •
sup latZll IZJI~rIM
i
< 1~II,j = 1, ... ,n, that is, for all r < 1 la.t~kl ~
C for all k. Consequently,
j;
1.1.5. Theorem. Let D c on be an open set and f a complex-valued function in D. Then the following conditiOnB are equivalent: (i) J E OO(D) and 8f = 0 in D. (ii) J E OO(D) and f is holomorphic in each variable Zt when the other variables are kept Jixed. (iii) J E OO(D) and, Jor every polydi8c P = PI X ••• X P nee D,
J
1 1(1,)=-(2ni)n
8P1 x ... x8P II
(C1
f(C) dCl " ... 1\ dCn , zt) ..... (eft - zn)
1, E P.
(1.1.2)
(iv) J admit8 local power series expansions, that is, for every point ~ ED there are complex numbers at deJined Jor every multi-order k such tnat, Jor all Z in 80me neighbour-
hood oJ~, J(z)
= I: a.t(z
- ~)" .
(1.1.3)
i
II these equivalent conditions are Julfilled, (v) J E OOO(D). (vi) For every polydi8c P
8"f(z) =
~ (2ni)"
=
then, moreover:
P l X ... X p" c c D and every multi-order k,
J
8P1 x ... xap"
(Cl
f(C) dCl -
1\ ... 1\
zt)ka+l ... (ell -
dCft ._ ,
Z
EP .
(1.1.4)
Zn)k,,+l
(vii) The coeJJicients in the power series expansion (1.1.3) are uniquely determined,
where 8"/(~) alc=--'
(1.1.5)
k!
(viii) The power serie8 expansion (1.1.3) converges uniJormly in each polydi8c Pee D centered at ~.
Proof. (i) <=* (ii) according to Corollary 1.1.2. By repeated use of (1.1.1), we obtain the implication (i) ~ (iii). We prove that (iii) ~ (iv). Let ~ ED and P = P l X ..• X P n c c D be a polydisc centered at~. Since for 1, E P and CE 8PI X ••• X 8Pn
_____ ~ _______ = (C1 -
Zt) ... (ell - Zn)
i kl •... ,k,,=O
~J)kl ... (zn ~1)~+1 .•• (Cn -
(Zl -
(C1 -
~'-'~ ~n)kll+l
1.1. Holomorphic funotions
with uniform J(z)
~onvergence
~
=
k
13
in C, it follows from (1.1.2) that
J
[1 (2ni)tI
!( C) del /\ ... /\ dC, , ) ~l )1'1+ 1 ... (en _ En)k"+l (z
('1 _
k
- E) ,
8P1 x ..• x8P.
z E P. (1.1.5')
Now we prove that (iv) 9 (ti). It follows from (1.1.3) and Lemma 1.1.4 that J is locally the uniform limit of polynomials in~, ... , Zn. Since such polynomials fulfil (ti), and since the uniform limit of continuous functions is continuous and the uniform limit of holomorphic functions of one complex variable is hololIlorphic, we conolude that J satisfies condition (ii). Now we suppose that the equivalent conditions (i) -(iv) are fulfilled. Then (v) follows from (iii) and (vi) can be obtained by differentiation under the sign of integration in (1.1.2). Property (vii) follows, ha.ving chosen a sufficiently small polydisc P = PI X ••• X P n centered at~, by the following computation, which is permissible in view of (1.1.4) and Lemma 1.1.4:
8~
~(~)
-
k!
- (2ni)"
7a,
J
a: -
~)' del /\ ..• /\ de.
(I- _ 1: \k1 +l
,I- _ 1: \i,,+l
BPI x ... x8PA
k! = --. (2nl)n
~ I
(2ni)n rot
'I
a, - - odC -~) C=E = k I at . l!
(viii) follows from (vii), (1.1.5') and Lemma 1.1.4. • 1.1.0. Definition. Let D c (/)tI be an open set. A complex-valued function I in D is said to be holomorphic (or analytic) if the equivalent conditions in Theorem 1.1.5 are fulfilled. The set of all holomorphic functions in D will be denoted by O(D). 1.1.7. Corollary. For every open set D con, O(D) i8 a ring, that i8, if J, 9 € O(D), then J 9 E O(D) and !g E O(D). Further, if f E O(D) and f{z) =f= 0 for all zED, then
+
I/J E O(D). Proof. This follows from condition (ii) in Theorem 1.1.5 and the oorresponding properties of holomorphic functions of one complex variable. 1.1.8. Corollary (Maximum principle). Let D c on be an open 8et and I E O(D). Supp08e that there exists a point ~ ED such that IJ(z)\ ~ I!(~)\ Jor all zED. Then f i8 oonstant in D if D is connected. Proof. This follows from condition (ii) in Theorem 1.1.5 and the maximum principle for holomorphic functions of one complex variable. • 1.1.9. Corollary. Let D ~ 0" be an open set and /, uniformly on every compact subset of D, then f E O(D).
E
O(D). II /, ~ I when ; ~
00,
Proof. This follows from condition (ii) in Theorem 1.1.5 and the corresponding property of holomorphic functions of one complex variable. Clearly, it can also be obtained from oondition (iii) in Theorem 1.1.5. • 1.1.10. Corollary (the uniqueness of holomorphic continuation). Let D ~ Cft be an open 8et and f E O(D). If there is a point E ED where 8J:f(~) = 0 !or all multi-order8 1:, then! = 0 in D if D i8 connected. Proof. This follows from condition (iv) and relation (1.] .5) in Theorem 1.1.5. •
14
1. Elementary properties
1.1.11. Corollary. For every open set D c 8i:O(D)
~
en
and aU rnulti-orders le, we have
O(D).
Proof. This follows from property (vi) in Theorem 1.1.5. •
1.1.12. Corollary (Cauchy's inequality). 1/ / is hoZomvrphic in the poZydisc \z,\ < r1,j = 1, ... ,n}, u'here 0 < r, < 00, j = 1, ... ,n, then lor all
P := {z e lr: multi-orders k
\61:/(0)\ ~ k! r-J: sup \/(z)\ . • EP
Proof. This follows from property (vi) in Theorem 1.1.5. •
1.1.18. Theorem. Lee D ~ (/)ft be an open set. Then, lor every compact set K c: c: D and every multi-order le, there exi.!Jts a constant 0 = O(K, k) such that max \81:/(z) I ~ 0 JI/I lEX
for all / E O(D) ,
d0'2ft
D
where d0'2. ia the Lebesgue measure in (In.
Proof. First let n = 1. Choose X E OO'{D) such that X = 1 in a neighbourhood UK of K. Then, by (1.1.1), for every / E OeD) x(z)f(z)
=
-~ fl(C) 8X(C) 1\ dC , C --
2Jtl D,Ug
Z
and differentiation leads to the required estimate. Repeated use of this gives the theorem for polydisc8 D of arbitrary dimension. The general case follows by use of a. covering of K by a. finite number of polydiscs c: c: D . •
1.1.14. Corollary (Stieltjes-Vitali). Let D C (JR be an open set and let II: be a sequen~ 0/ holomorphic functions in D which i8 uniformly bounded on every compact 8ubset 0/ D. Then there is a subsequence fkJ converging uniformly on every compact sub8et of D to
a limit in O(D). Proof. Since 8/i: = 0, it follows from Theorem 1.1.13 that all first-order derivatives (with respect to the real coordinates) of /1: are uniformly bounded on any compact subset of D. Hence the corollary follows from Ascoli's theorem and Coronary 1.1.9. •
1.1.15. Definition. Let D ~ (/)n be an open set. A map! = (fl' ... ,jm): D ~ om is said to behoZomorphic in D if!I'''' ,/m E O(D). The set of all holomorphic maps/: D ~ will be denoted by om(D). If I E om(D) and ~ E D, then the matrix
om
. (8/ (E»)J-l, ...
JJ(~)'=
-f 8EI:
,m
(j is the row index)
i-I .... ,n
is called the (complex) Jacobi matrix of I at E. / is .called regular at E e D if rank J,(E) = min {n, m}. Let D, 0 S on be open sets. A biholomorphic map/rom D onto 0 is by definition a bijective map / from D onto 0 such that both I E OR{D) and 1-1 E Oft(O). Then we shall also say that / is biholomorphic in D.
1.1.18. Proposition. Let D
~ (Jft be an open
set and f = (11' ... ,1m)
E
om(D). Then
(i) For every, E D,
(1.1.6)
11
1.1. Holomorpbio functions
(ii) If g is a comi",tw'U8 complex-tJOJued function in some MSghbotwlaootl of l(lJ), • • (in the 8ense of distributions)
8(g 0 f)
8g 8171 1-1 8J.8z f
8Z1
and 8(g 0
az,
. for J = 1, ••• , n ,
m
~ --
-- =
i
J) =
i- 1
~ 8~1e for
(l~:'l
j = 1, ... , n.
(1.1.8)
8fT: 6Z1
Proof. Since all = 0 for j = 1, ... ,ft., and d = a + 5, (1.1.6) follows from the Taylor formula. Since d = a + "8 and = aj~ = 0, we obtain
8ft
ft
~ j=l
=
(8(9 0 1) - - dz, 8zt
8(g + -,:;dZI 0 /)
(au au - ) -dft+--::-dft =
m
~
8!~
6/t
k=1
_ )
8z J
=
d(g
0
f)
(8g
n til 8/1: 8g aJ" _) 1: 1: --d.z l +-=--=-dil , j; .. 1 8/t 8zf all: ai,
j=l
which implies (1.1.7) amd (1.1.8) by comparison of the coefficients of
cka, .... _~
Jr<E) for all
I E O"'(D),
, (.~Q)
~ ED.
-(1.1.9)
Proof. That 9 0 IE Ot(D) follows from (1.1.8). (1.1.9) follows from (1.1.7)•• 1.1.IS. Theorem (Implicit function theorem). Let U be a neigkbourltood 0/ t E ,.~.,." Then f is biholomorphic in 80me neighbourhood 01 Ei/ aM only il
J E 0"( U).
det J/(~)
(l.l.~O)
=F 0 .
Proof. It follows from (1.1.9) that (1.1.10) is necessary. We prove that OOI'ldltioa (1.1.10) is sufficient. Without loss of generality we can assume that == 0,/(1. and JJ(E) is the unit ma.trix. Denote by id the identity ma.p of fJ" and define
e
i :=
id
-!
on
U.
Choose a neighbourhood V of 0
a<
00
li(z) I ~ Olzll
for
o.
Z E
E
en such tha.t Vee U. Then, by (1.1.6), fOt JOm&
V•
< ~}, ~ > 0, and choose B > 080 amal1 that
We use the notation Eo(<5) := {z E (f)n: Izi e < 1/20 and Eo(2s) C V. Then j(Eo(s/2J:) quently, the series 00_
k:=
~!k, i-I
.-,."
where fit
E o{s/2 lt +1 ) for k
C
= 0, IJ 2, ••••
CJoDae..
'
..
:=/0 ... of
J
-.i: Umel
converges uniformly on Eo(e) and h(Eo(e)) c: Eo(2s) C U. As the uniform limit at holomorphio ma.ps h is holomorphic. Further, (id h) 0 J = 1 0 (id A) .. iL,.
+
+
1.1.19. CoroJlary. If X c ()n and k E {I, 2, •.• , n - I}, 'ken 'Ae/~ coM~ are equitJalent: (i) For every point ~ E X there exis" a biholomorphio map 1= (/1' ... ,/~ .. ..., mighbourkood U of E suck that X n U = {z E U:!1I+1(Z) = ... = I.(z) .. O} •
16
1. Elementary propertieB
(ii) For every point ~ E X there exist a neighbourhood V of ~ and a regular holomorphic map g: V -7 C"-i Buch that X n V = {z E V: g(z) = O}.
Proof. The implication I i) ~ (ii) is trivial. We prove that (ii) -=} (i). Let ~ E X and let Y, g as in condition (ii). Denote by G the linear map from (J)fI into q;n-k defined by the Jacobi matrix JfI(~)' Since rank J,,(~) = n - k, G is onto and we can find a linear map A: C'" -7 ([J" such that the map A EB G: ([In -7 on defined by (A CD G) (z) := (A(z), G(z)) is invertible. Define f(z) := (A(z), g(z)) for z E V. Then det Jl(~) = det A EB G 9= 0 and, by Theorem 1.1.18, J is biholomorphie in some neighbourhood U C V of~. Since X n U = {z E V: g(z) = O}, and since (f1;+V ••• ,fn) = g, it follows that X n U = {z E U:fk+l(Z) = ... =f",(z) = O} ••
1.1.20. Definition. Let D c cn be an open set. A subset X C D is said to be a complex 8ubmanifold oj ([In if the equivalent conditions (i) and (ii) in Corollary 1.1.19 are fulfilled. If, moreover, X is a closed subset of D, then X is called a closed complex stibmanifold of D. It follows from condition (i) in Corollary 1.1.19 that every complex submanifold of C" in a natural way becomes a complex manifold in the following abstract sense:
1.1.21. Definition. A complex manifold 0/ complex dimension n is a real2n-dimensional Ooo-manifold X together with a family {( Uf' qJl)}jeI satisfying th(' fo]]owing conditions: (i) For every j E I, V f is an open subset of X, and U U f = X. jeI
tii) For all j E I, qJ, is a. GOO-diffeomorphism from U, onto some open set in (f)n. (iii) For all i,j E i, qJj 0 rpi 1 is a biholomorphic map from cp1.( U t nUl) onto qJf( U ( nU,), Then a couple (VI tp) is called a system of holomurphic coorclinat(;& in V, if the family {(UI , qJ,)}jEl U {(V, tp)} fulfils conditionR (i)-(iii). A family (Va' '1!'a)",EA is called 8 holomorphic atla8 of X if ev('ry (Va' 1"'0<) is a sYRtem of holomorphic coordinates and X = U VII. (In particular, the system {( U 1, 'P1)}jEI above is a holomorphic atlas of X.) OlEA
Every open subset of a complex manifold becomes in an obvious sense a complex manifold. A function f defined on a complex manifold X is said to be holomorphic on X if, for every system (V, tp) of hololllorphiu coordinates in V, the function f 0 1p-l is holomorphio on tp( V). If X, Yare two complex manifolds, then a map f: X ~ Y is called holomorphic if, for every sYAtcm (V, r) of holomorphic coordinates in X and every system (V, tp) of holomorphie coordinates in Y, the map lp 0 f 0 cp-l is holomorphic in q;(U nj-l(V)). If, moreover,jis a bijective map from X onto Y andf-l is also holomorphic, then f is said to be biholomm-phic f1'orn X onto Y. If there exists a. biholomorphic map from X onto Y, then X and Yare call('d biholorn01-phically equivalent. A subset Z of a complex manifold X is said to be a complex submanifold oj X if, for every system (U, q;) of holomorphic coordinates in X, Cf( U n Z) is a complex submanifold of q;( U). If, moreover, Z is a closed subset of X, then Z is called a closed complex 8ubmanijold of X. Every complex submanifold of a complex manifold becomes in an obvious sense a complex manifold. If U is an open set in a complex manifold, then we denote by O( U) the set of all holomorphic funotions in U, and by 0"( U) we denote the set of all holomorphic maps from U into OJ:.
1.2. Application of the Ca.uchy-Green formula
1.2.
17
Application of the Cauchy-Green formula to the a-equation
In this section we begin the study of the inhomogeneous Cauchy-Riemann equations for several complex variables. This is the system of differential equations
OU =j1' j 1, ... , n , (1.2.1) oil which is also called the o-equation. It is clear that the following compatibility condition is necessary for the existence of a solution:
011 _ oj"
j, k
OZt - oz,'
=
1, ... , n.
(1.2.2)
Using essentially only the Cauchy-Green formula, we shall prove in the present section that condition (1.2.2) is also sufficient for the local solvability of (1.2.1) as well as for global solvability if the functions 11' ... ,1" have compact support, where in this case, if n ;;;::: 2, the solution can also be chosen with compact support. In general, global solvability on open sets D S ([JfI. can be obtained only under additional conditions onD. 1.2.1. Theorem. Let n ~ 2, /1' ... ,1" E G~( (In), and assume that the compatibility condition (1.2.2) is/ulfilled. Then there is au E G~«(Jn) satisfying (1.2.1).
Proof. Define u(z) := _
~ fll(C, Z2' ••• , z,,) de ~dC. C- ~
(1.2.3)
2nl
CEt'l
After a change of variables we can write u(z) =
~ fil(ZJ. - C, Z2'
••• ,ZtI)
de
A
dC .
2nl C 'eel This shows that u E Gl(fCn). From Theorem 1.1.3 it follows that ou/8il = fl in P. Now let k E {2, ... ,n}. Then, by differentiating under the sign of integration we obtain
= __ 1
8u
8z Since
f (81l(C,
Z2; ••• ,
k
,eCl
Oil/BzIe
8It/c~,
=
,eCl
1\
dC.
~
and since by (1.1.1),
_~ f~~a;, 2nl
z,,)/8zt )dC
C-
2ni
Z2' ••• ,
z,,)/8C) de 1\ dC =
C-
it(Z) ,
Zl
we conclude that oU/OZt = It also for k = 2, ... , n. Hence u satisfies (1.2.1). In particular, u is holomorphic outside suppl1 u ... u supp!". On the other hand it follows from (1.2.3) that u(z) = 0 if IZ21 Iz,,\ is large enough. By the uniqueness of holomorphic continuation this implies that u(z) = 0 in the unbounded component of (Jft" (supp It u ... u supp f,,) • • Note that Theorem 1.2.1 is in contrast with the situation in the case of one complex variable. In fact, if u, f E G~( (Jl) such that 00/8"7. = f, then by Stokes' formula
+ ... +
2 BenkJn/Lelterer
18
1. Elementary properties
(/(J) di
~,
A dz =
Jd(u(J) dJ) = O. This contrast becomes even clearer in the following
C'
theorem of Bartogs, which is 8. consequence of Theorem 1.2.1. 1.1.1. Theorem (Hartop'theorem). Let n ~ 2,let D c tJ" be an open 8et, and let K be a oompact 8ubset 0/ D 8uch t1lm D '" K i8 connected. Then, Jor every k E OeD '" K)
OeD) suck tkat H = It, in D '" K. Proof. Choose X E 0o(D) such that X = 1 in some neighbourhood of K, and set
there exists H
E
11 := { -
(aox/aid k
in D '" K , outside D '" K .
Then the compatibility condition (1.2.2) is fulfilled and Theorem 1.2.1 gives U E OMD) such that au/ai, = If' By setting H : = (1 - x) 11, - 'U, we define H E OeD). Let U be the unbounded component of the complement of supp X. Then u = 0 in U and, therefore, H = 11, in D n U. Since D '" K is connected and (D '" K) n U =f= 0, we conclude that H = ~ in D '" K . • 1.1.8. Corollary (Non-existence of isolated singularities). Let n ~ 2 and let D C (/J" be an open sel. 1/1, ED, then etJery lwlomorphic Junction in D '" z can be kolomorphically continued to z.
1.1.4. Corollary. LeI n ~ 2, and let D c N(j) := {z E D:J(z) = O}, then (i) D '" NeJ) i8 connected, (ii) N(J) i8 not compact.
Q)ft
be a connected open set.
II/ E ()(D) and
Proof. (i). We only ha.ve to prove that for every open ball B ~ D the Bet B '" N(J) is connected. To do this we consider two arbitrary points z, wEB", N(f) and ihow that z and w belong to the same component of B '" N(!). Let X be the complex line which contains 1, and w. Then, by Corollary 1.1.17, the restriction ofJ to X n B can be considered as a holomorphic function of one complex variable. Since J:$ 0 on X n B(f(z) =F 0 and I(w) =1=0), X n B n N(J) is discrete in X n B. Since X n B is a disc and therefore oonnected, this implies that (B n X) '" N(f) is connected. Consequently, z and w belong to the same component of B '" N(J). (ii). Assume tha.t N(f) is compact. Since, by part (i), D "N(j) is connected, then it follows from Theorem 1.2.2 that II! can be continued holomorphically to N(J). This is a contradiotion, because! = 0 on N(f) . • To solve the inhomogeneous Cauchy-Riemann equations locally without the hypothesis on compact support is slightly more complicated than the proof of Theorem 1.2.1. In Section 2.1 we shall do this by means of an integral formula for strictly convex smooth domains. Here we prove the following result in polydiscs, which also implies local solvability:
en
1.1.1. Theorem. Let D c be an open set and let h, ... ,fft. E OOO(D) such tkat the compatibility condition (1.2.2) is Jtdfillea in D. Then, for every open polydi8c P = PI X ••• X p .. c: c: D there i8 a U E OOO(P) sati8fying (1.2.1) in P. Proof. We shall prove inductively that the theorem is true if 1.+1 = ... = I" = O. This is trivial if m = O. Assume that it has already been proved for m - 1. Let D' := D~ X ... X D~ and D":= D~ X ... X If; be open polydiscs such that P c: c: D" c: c: D' c: c: D and choose X E Oo(D~) such that X = 1 in D;. Define
1.3. Doma.ins of holomorphy
19
for z € D' . v(z) := _
~ JX(C)/m(Z]., ..• ,zm-l, C, Z",+l, ••• ,Ztt) dC
C-
2nl
~ED;'
1\
dC .
Z'"
Since X E O~(D;"), after the change of variables C- Zm ~ C we see that v E OOO(D'). Since X = 1 in D~ it follows from Theorem 1.1.3 that OV Jm In . -_-= 8zm
D" •
(1.2.4)
Since 8/m/ozl = 8/f/aZm = 0 for j the sign of integration that
o~ == 0 = If
8z,
D'
in
=
m
for j
+ 1, ... , n, we obtain by differentiation under = m + 1, ... , n .
(1.2.5)
if
Then the functions := If - av/ai" j = 1, ... , n, fulfil the compatibility condition (1.2.2) and it follows from (1.2.4) and (1.2.5) thati", = ... = itt = 0 in D". Therefore, by the inductive hypothesis, we can find W E OOO(P) such that 8w/oz., = in P and tL := v w is the required solution of (1.2.1) in P . •
if
+
1.3.
Domains 01 holomorphy
Hartogs' Theorem 1.2.2 shows that, for n ~ 2, there are examples of open sets D ~ Dc such that every holomorphic funotion in D admits a holomorphio extension to D. In this section we shall examine this phenomenon more closely. Open sets D c 0'" with the property that, roughly speaking, there is no part of the boundary across which every holomorphic function in D can be oontinued holomorphioally are of special interest in the theory of functions of several complex variables. We give a precise definition:
on
1.3.1. Definition. An open set D c en is called a domain of Iwlomorphy if the following oondition is fulfilled: If U, V c are open sets suoh that (I) 0 =F U c V n D =F V , (2) V is connected, then there exists an J E O(D) such that it is not possible to find € O( V) with = J in U.
on
i
i
Remark. We do not assume that a domain of holomorphy is conneoted. The following definition is closely connected with the notion of a domain of holomorphy: 1.8.2. Deimition. Let D we define
C
(J'"
be an open set and K a compaot subset of D. Then
K'b := {z ED: II(z) 1S; sup IJ(C)I
for all
J E O(D)}
•
tEE
The set if> 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
such that I/(w) I = 1 and
for all z E K . •
1.3.4. Corollary. For every open set D i8
(]'I
j-l
C,., and every compact 8ubset K
C
0/
D,
iiZ
bounded.
Proof. By Proposition 1.3.3, i~A is bounded. Clearly,
KE ~ Kg" .•
1.3.5. Theorem. Let D c qyn be a domain of holomorphy, K c c:: D a cO'Inpact set, and let P be an open polydi8c with centre at O. If p
+w~ D
Jor all
w
E
K ,
(1.3.1)
p
+w~ D
lor all w
E
if> .
(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,
+ < <
< <
iZ
t
max 18 f(w)1 (t r)1: ~ 0 toEK le! 0 J, where 0, := max {If(z)l: z E (Pt ,
+ w)
for Borne
Since 8.tO(D) c: O(D), and by definition of and all multi-orders Ie
KfJJ,
W E K}
.
this implies that for every f
E
O(D)
18.tf(woJI (t r)lc '5:. 0 . leI 0 J
Taking into account (1.l.5), the last inequality leads to the conclusion that, for every E O(D), the power series expansion of f at Wo converges in the polydiBc Pte + woo Since D is a domain of holomorphy, this implies that Pt. Wo ~ D . •
f
+
1.3.6. Corollary. Let D
~
8etK c c D,
dist (K, 8D)
=
dist
qyn be a domain of holomorphy. Then, for every compac' .
(KfJJ,
aD) ,
where di8t (X, aD) := inf {Iw - zl : w E K,
(1.3.3) Z E
aD}.
1.3.7. Theorem. For every open set D c cn, the following condition8 are equivalent: (i) D i8 a domain of holomorphy. (ii) There exi8t8 an f E O(D) 8uch that for every couple of open 8ets U, V ~ (/}n satisfying condition8 (1) and (2) in Definition 1.3.1, it i8 not possible to find j E O( V) with =fin U.
i
(iii) For every compact 8ub8et K c c D, if> 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
<
<
00
f := n (1 - fm)m converges uniformly on every compact subset m .... l J is not identically 0 in any of the oomponents of D.
of D,
J EO (D),
and
Now let U, V be open sets satisfying conditions (1) and (2) in Definition 1.3.1. Assume that there is an E V) such that = I in U. Let W be the component of V n D containing U. ThenJ = J in W. Since V is connected and V is not contained in D, the set oW n 8D fl V is not empty. Fix CE 8W n 8D fl V. Since X n W is dense in Wand the sequence {.;(m)} contains every point in X n W infinitely many times, we can find a subsequence {.;(m')}:'1 of points in X n W which tends to C. Choosing ~ sufficiently large, we can assume that B".,~ W and, therefore,1}(m,) E W for all 8. Since all derivatives of I of order m, vanish at n(m,) and since j = I in W, it follows that all derivatives of J-at C( = lim 1}(m,» vanish. Consequently,jis identically o in V, which is a contradiction, becausef is not identically 0 in W. The equivalence of conditions (i)-(iv) is proved. It remains to show that (iii) ~ (v). First we assume that (iii) is violated. Then there is a compact set K c: c: D such that KZ is not compact. Since iZ is bounded and relatively closed in D, we can find a sequence ~(k) E KZ tending to some point ~(oo) E 8D. This sequence is discrete in D, but for all f E O(D)
i q(
J
<
sup If(~(k))\ ~ sup 1/(z)1 k
ZEKZ
=
sup If(z)1
< 00 ,
ZEK
that is, (v) is violated. Now we assume that (iii) is fulfilled. Then we can find a sequence {Km}:... l of O(D)-convex subsets of D such that every compact subset of D is contained in some Em. Let X ~ D be an infinite set which is discrete in D. Then we can choose a sub. sequence {Kmf}~l and a sequence {E(j)}~l of points in X such that ~(i) E E mJ +1" K".J' Since all Kmf are O(D)-convex, we can find!, E O(D) such thatff(';<J») 1 but Iff I 1 on KmJ' Replacingfj by a high power off" we may arrange that 1/21 on K"'J' but J,(~(i» j. Then J : = L f1 converges uniformly on every compact Bubset of D and
Ilfl <
>
>
<
j
defines a holomorphic function f on D such that lim f(E(j) = is fulfilled. •
00.
Hence condition (v)
22
1. Elementary properties
1.I.S. Corollary. EtJery ccmtJex open see D
~ Q)A
is a domain 01 kolomorpky.
Proof. This follows from condition (iii) in Theorem 1.3.7 and Proposition 1.3.3. • I.B.9. Corollary. Let {D,}jEJ be a Bgstem oj domains oj holomorphy in interior D oj n D, is a domain oj holomorphy.
()A.
Then tke
JEJ
=
Proof. Let K be a oompact subset of D. Since by Corollary 1.3.6 <list (Kj;J' aD,) diet (K, aD,), and since Kj; c ifj,J for all j E J we conclude that diet (KZ, 8D) :2: inf dist (Kfj,J' 8Dt ) = dist (K, aD) , j
that is, condition (iv) in Theorem 1.3.7 is fulfilled . • 1.3.10. Corollary. Let D S is a domain oj kolomorpky.
()A
and G c
()m
be domaina
0/ kolomorpky.
Then D X G
Proof.LetXC D X Gbe an infinite set which is discrete inD X G. We can assume that X D := {z ED: (z, w) E X for some W E G} is infinite and disorete in D (otherwise XG is so). Then, by hypothesis, there is! E O(D) which is unbounded on X D • Setting F(z, w) : = f(z) for (z, w) ED X G, we obtain F E O(D X G) which is unbounded on X. Hence, by condition (v) in Theorem 1.3.7, D X G is a domain of holomorphy. • I.B.l1. Theorem. Let D c qyn be an open set, G c Q)m a domain 01 kolomorphy, and let F: D ~ em be a holomorphic map. Supp08e tkat at lease one oj the following conaitiona is fulJilled: (i) D is a domain oj kolomorphy. (ii) F-l(G) c c D. Then F-l(G) is a domain oj holomorphy. Proof. Let X ~ F-l(G) be an infinite set which is discrete in F-l(G). By condition (v) in Theorem 1.3.7, it is sufficient to findJ E O(F-l(G) which is unbounded on X. Since one of the conditions (i) and (ii) is fulfilled, we only have to consider the case that X is not discrete in D. In this case we can choose a sequence {~(t)} k-l of points in X which converges to some point ~ ED" F-l(G). Since F is continuous and F(~) ~ G then the set {F(E(I;»} is infinite and discrete in G. Since G is a domain of holomorphy, we can therefore find agE O(G) which is unbounded on {F(~(k»)}. It remains to set f:= go F . • 1.1.12. Definition. A bounded open set Dec: ()A is called an analytic polyhedron if there are holomorphic functions Fv ... ,FN in some neighbourhood UJj of D such that D = {z E UJj: IFI (z)1 < I for j = I, ... , N} • 1.1.13. Corollary. Every analytic polyhedron is a domain 01 holomorphy. Proof. First we observe that every open polydisc is a domain of holomorphy (this follows, for example, from Corollary 1.3.8). Let G be the open polydisc of multiradius (1, ... , 1) and with centre 0 E eN. If F = (Fl' ... ,Fs ), then D = F-l(G) and, therefore, the corollary follows from Theorem 1.3.11 (ii) . • We conolude this section with the following Remark. Every open set D in the complex plane is a domain of holomorphy. This follows from the Weierstrass product theorem. Another possibility is to check,
1.4. Plurisubharmonic functions
23
for example, condition (v) in Theorem 1.3.7: If X is an infinite disorete set in D, then X is unbounded or X n aD =4= O. In the first case the function /(z) := z is unbounded on X and in the second case, for every w EX n 8D, the function/(z) := I, (z - w) is holomorphic on D and unbounded on X.
1.4.
Plurisubharmonic functions
In the next section we introduce the concept of pseudoconvexity. Pseudooonvexity will be defined by means of plurisubharmonic functions, which will be investigated in the present section. First we recall the notion of a subharmonic function of one complex variable. 1.4.1. Def"mition. Let D ~ (J1 be an open set. A continuoU8 subkarmonic funcnon in D is a continuous1 ) function e: D ~ III such that the following condition is ful. filled: For every ~ E D and all 0 r diet (E, aD)
< <
J
1
e(~) ~ 2nr
b
/1(Z)
Idzl
2~J /!(~ + r ."') d'l' ,
=
Iz-~I-r
where
Idzl
:=
~dX2
Xt
(1.4.1)
0
=-':'rlxt =
r dcp if z
XI
=, +
Xt
+ iX2
=+ E
r elf'. (1.4.2)
Every harmonio function is Bubharmonic; then (1.4.1) is valid with equality. We give some other equivalent definitions of continuous subharmonic funotions. 1.4.2. Theorem. Let D ~ (fJl be an open 8et, and lee e: D ~ R be continuoU8. Then each oj the following conditions is nece8sary and suJficient Jor e to be subkarmonic: (i) For every ~ ED there exist8 an e 0 8uch that (1.4.1) hold8 Jor all 0 r 8. (ii) For every compact set K c: c: D and every harmonic Junction h in some neighbourhood of K such that h ~ e on aK, we have h 2:: e on K. (iii) For every E E D, all 0 r dist (~, 8D), and every harmonic function h in Bome neighbourhood oj the elisc Ie - ~I = r such that h(C) ~ e(C) Jor Ie - ~I = r, we have h(C) ~ e(C) for alllC - EI ~ '1'.
< <
>
< <
Proof. It is trivial that subharmonicity implies condition (i). Further, it is clear that (ii) ~ (iii). Since for harmonic functions (1.4.1) is valid with equality, (iii) implies subharmonicity. It remains to prove that (i) ~ (ii). Let K c c D be a compact set and h a harmonic function in some neighbourhood of X suoh that 11, ~ e on ax. We have to prove that := max (e(z) - h(z) ~ O.
o
seeK
Assume that 0> O. Set Kc := {z E K: e(z) - h(z) = O}. Then Kc is a non-empty compact subset of the interior of K. Choose a. point E E Kc with diet (E, aK) = dist (Xc, aK). Then for all sufficiently small r > 0 _1_
2nr
J
(e(z) - h(z)
Idzl
<0
.
Iz-el=r 1) Usually the notion of subharmonicity is defined for upper semioontinuou8 funotions with
values jn R u {- oo}, but in the present book we need this for continuous funotions only.
24
1. Elementary properties
This contradicts the inequality 0 =e(E) -h(E) ~ (1/2nr)
> 0,
is valid for all sufficiently small r of",. •
f
(e(z) - h(z)) Idzl, which
Iz-EI-'
because of condition (i) and harmonicity
In this book subharmonic OJ.functions are of special interest. These functions can be characterized as follows: ~ qJ1
1.4.3. Theorem. Let D
be an open aet. A Ol-function
e:
D -+ III i8 aubharmonic
;,f ana only if
:e~l ~ 0
for all
(1.4.3)
ZED.
This theorem is an immediate consequence of the following lemma: 1.4.4. Lemma. Let D ~ (/)1 be an open set, e: D -+ /R l a C2-function, and r dist (" oD), then we define (see (1.4.2) for notation)
o<
<
M(r) := _1_
2nr
J
J
~ E D.
If
2n
=
e(z) Idzl
~
e(E
2n
IE-zl-'
+ r ellp ) drp.
(1.4.4)
0
Then = e(~)
lim M(r) ' .... 0
J
and
-_!
dM(r)
dr
(1.4.5)
nr
-Sle - _ dXI 1\ dX1 •
(1.4.6)
Sz Sz
IE-zl
Proof. (1.4.5) is trivial. We prove (1.4.6). First observe that dM(r) =
dr
~
21l
J~f!(E +
J
2n
dr
o
r
ellp ) dcp = _1_ 2nr
Se
8r
Idzl .
Iz-EI ... ,
Since by (1.4.2) Se Idzl
Sr
= (~ Xl oXl
r
+ ~ X2) Idzl = r
oXl!
,Se dX2
_
eXl
,Of! dxl 8x2
,
and by Stokes' formula, this implies that dM(r) =_1_ dr 2nr
J (ale + o~
f!)dx
Z
8
O~
1
1\
dx . 2
Iz-EI
Taking into account the relation
aZe aZe
ale
-+-=4-, Oxi
8x~
(1.4.7)
8z OZ
we obtain (1.4.6) . • 1.4.5. Definition. Let D c:
(/)1
be an open set. A C2·function f!: D -+ III is said to be
Btrictly 8ubharmonic in D if
>
0 for all ZED. 8z 8% Now we pass to the case of several complex variables. Sle(Z)
(1.4.8)
25
104. Plurisubharmonic funotions
1.4.6. Definition. Let D C (f)n be an open set. (i) A continuous plurisubkarmonicfunction in D is a continuous function e: D ~ III such that the following condition is fulfilled: For arbitrary v, w € (f)n, the funotion C-* (/(v Cw) is subharmonic in that part of 0 1 where it is defined. The set of all continuous plurisubharrnonic functions in D will be denoted by PO(D). (ii) A Ol-function (!: D -* /R1 is called strictly pluriaubharmonic if, for arbitrary z, w E Q)n with w =+ 0, the function C-*(!(z + Cw) is strictly subharmonic in that part of (f)1 where it is defined.
+
1.4.7. Theorem. Let D C (f)n be an open set, and let (i) e is plurisUbharmonic if and only if n
~ j, k=1
(ii)
82e(z)
-
- - _ ~1~t ~ 8z1 8z k
0 for all
zED
e: D ~ Rl be a Ol-function. The"" ~ E (f)n •
(l.4.9)
=+ ~ E en .
(1.4.10)
and
e is strictly pluri8ubharmonic iJ and only if 82(!(z)
"
-
~ - - _ ~1~ I: J,k=1
SZI 8z"
Proof. Let v, W 8Ie(C)
--=
8e a,
E (f)n
>0
if zED
and ~(C)
=
I
e(v
and
0
+ Cw). Then by Proposition 1.1.16
-
8'(!(z) -WfWt. i, i - I a~ t Z-lI+ttD n
~
aZ
Together with Theorem 1.4.3 and Definition 1.4.5 this proves the theorem . • Notation. The Hermitian form n
~
j,i-l
82e(z)
-
- - - ~1~1c
(1.4.11 )
aZ1 8zl:
is called thc Levi Jorm oj (! at z. 1.4.8. Proposition. Let D
C
(f)n
be an open set and f E O(D). Then
Proof. We only have to prove this for n = 1. Let Then by Cauchy's formula
f(~) = ~ 2nl
f
~
ED and 0
1/1
E
PO(D).
< r < dist (E, aD).
f(z) dz, Z -
~
Iz-et .or
which implies (1.4.1) for IfI· • In the following proposition we collect some properties of plurisubharmonic functions which are immediate consequences of Definition 1.4.1 of subharmonicity. 1.4.9. Proposition. Let D C on be an open set. (i) If~, cp E PO(D), then e + cp E PO(D). (ii) If e E PO(D) and 0 ~ c < 00, then c~ E PO(D). (iii) If (!f E PO(D) Jor all j in 80me index set J and if sup el is continuou8 in D, tken sup (!1 E PO(Dt jeJ jeJ
(iv) IJ (!Tc E PO(D) and (!I: -* e when k ~ 00, uniformly on every compact 8ub8et of D, then ~ € PO(D). (v) Every convex function from CO(D) (8trictly convex function from 02(D)) is plurisubharmonic (strictly plurisubharmonic).
26
1. Elementa.ry properties
Remark. If both e E PO(D) and -e E PO(D), then e is pI uri harmonic, tha.t is, the functions e(v Cw) in Definition 1.4.6 are harmonic. If! E O(D) and - IfI E PO{.lJ), then J is a constant.
+
1.4.10. Theorem. Let rp
f cp(C) da2tl(C) =
E 0o( en)
8uch that rp
~
0 in en, rp(z)
= 0 i/lzl
~
1,
1, and (1.4.12)
(}f&
where da2,,(C) is the Lebe.8gue measure in eft. Let D c: en be an open set and If E: 0, then we 8et D. := {z ED: diet (z, 8D) e} and
>
>
e.(z) :=
Je(z -
c·
ee) cp(C) da2ft{C)
Then e. E OOO(D.} n PO(D.) lor all e cofn1)act subset 01 D.
for
z E De .
e E PO(D). (1.4.13)
> 0 and e. ~ Q when E: ~ 0,
uniformly on every
Proof. We have
e.(z) = Je(C) rp (z - C) da2,,(~t 2n e
• E:
for
(1.4.14)
ZED ••
(Jft
Together with (1.4.12) this implies that e. ---+e uniformly on every compact subset of D. By differentia.tion under the sign of integration in (1.4.14) we see that E OOO(De ). From (1.4.13) it follows that, on every compact subset of De' (!. is the uniform limit
e.
N
of functions of the form z -+ ~
e(z -
eCI) Cf, where N
j-I
view of Proposition 1.4.9 this implies that
< 00, C, E Q)" and c, > 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
•. ,,=-1 ot. ot"
OXI
+
ox!
f
.-1
fcp(t) ole.(z) (I"
8~
for j = 1,2. Since by (1.4.16) the first sum is ~ 0, and since 81/8~ = 48 2 /8z 8z, it follows that
+ 81/8xi
f
BJe{z) :2: 8cp(t) 811(!,,(z) 8z 8z - .=1 &. 8z 6z . In view of Theorem 1.4.7 this completes the proof. •
Notation. Let D c
on be an open set and e a real-valued CI-function in D. Then we
define "
FQ(z, C) := -
[
2
n 8e(') 1: j-l
for
8Ct
(~
-
CI)
8Ie(C) + i,Ie",,! I:n - (zl 8C1 8CI:
z, CE D.
CI) (Zt - Ct)
] (1.4.18)
Remark. FQ(z, C) is only continuous in C. Later we shall replaoe the continuous derivatives 8Be(C)/oCI 8Ct by sufficiently close 01·functions. The obtained modification of 'ifQ(z, C) will be denoted by F(z, C) = F Q(z, C) then.
e.
The function FQ(z, C) is called the Levi polynomial of It plays an important role for strictly plurisubharmonic C 2·functions. The following lemma describes the connection between the Levi polynomial (1.4.18) and the Levi form (1.4.11).
28
1. Elementary properties
1.4.13. Lemma. Let D ~ Cft be an open set, and let in D. Then, Jor all CED and z ~C, '"
e(z) = e(C) - Re F,iz, C)
a2 (C) i, k-l ac, aCTe
1: ~ (zl -
+
R
(!
be a real-valued OB-function
C1) (Zt - Ct)
+ o(IC -
Z12) ,
(1.4.19)
F i8 the real part of F
where Re
Q
Q'
Proof. Letxf = xl(C) be the real coordinates ofC Then a computation gives
and
E
CRsuch that C,
=
Xl(C)
+ ixj+,,(C).
2ft aBe (C)
1
-
1: ---(Xl(Z) - Xl(C)) (XTc(Z) - Xt(C))
=
~
2 i,M-l R
i,k-l
ax, aXt a2e(C)
_
----=- (zl - CI) (Zt aCt aCk
- CTc)
+ Re [ft~
82e(C)
]
- - (Zj - el) (ZTc - ek) • J,k=l aCI ac"
Consequently, (1.4.19) is the Taylor expansion of e at
e.•
It is trivial that every strictly convex oa-function e is strictly plurisubharmonic (Proposition 1.4.9 (v)). The following theorem shows that if de =1= 0, then, after an appropriate local change of holomorphic coordinates, the converse is also true.
1.4.14. Theorem. Let e be a strictly plurisubharmonic C2-function in some Mighbourhood oj 0 E ([P'. IJ de(O) =f= 0, then there exists a bilwlomorphic map h from some neighbourhood U of 0 onto a neighbourhood W of 0 such that e 0 h -1 is lJtrictly convex in W, toot is, 2R
82(,)
0
11, -l(C)
1: -;.,.,
;.k-l
where
X, =
oXl
0Xt
CE Wand
t l tt> 0 for all
Xl(C) are the real coo'rdinates oj Cc
0
=F t
en 8uch that C, =
E
112ft,
xl(C)
(1.4.20)
+ iXj+n(C),
Proof. Since de(O) =1= 0 and (,) is real, we have 8e (0) =1= O. Assume that 8e(0)/8C1 =1= O. Then h(z) := (FQ(z, 0), Zz, ... , Zft) is biholomorphic from some neighbourhood U onto some neighbourhood V of 0 = h(O) (for the definition of FQ see (1.4.18)). Let f(C) = (/l(C), ... ,f,,(C) : = h-1 (C), C E V. Then, by Lemma 1.4.13, for, -+ 0 e 0 J(C) = e(O) - Re FrAf(C), 0)
+. i
J, k=l
882e(80J f1(C) f1r(C} Z,
+ 0(1/(')1 2) •
Zk
Since, by definition of J = h-1 , FQ(/(C), 0) = C1 , this implies that for C -+ 0
e
0
J(C) = (,)(0) - Re C1
ft
62(,)(0)
_
+ 1: --_ J'(C)f1:(C) + 0(I/(C)1 2 ) • j,k=18z 1
8zt
Let F = (F1' ... , F tal be the linear part of fat 0, that is, the linear map F: qyn such thatf(C) = F(C) + 0(ICI 2 ) for C ~ O. Then we further obtain that for' ~ 0
eo f(C) = e(O) -
Re C1
+
i
j.~-l
e(OJ F,(C)FIc(C)
(j2
8z1
8z~
+ o(ICI 2) •
~
en
1.4. Plurisubharmonic functions
29
Hence, by uniqueness of the Taylor expansion,
~
i
2
5 (! 0 f(O) xl(C) xJ;(C) = 8xf 8xt
i,k-1
82e(0} Ff(C)FJ;(C)
for all C E
en.
i.i=18%1 eZI;
Since F(~) =F 0 if C =1= 0 (I is biholomorphic at 0), and in view of Theorem 1.4.7 (i), this implies (1.4.20) for C= 0 and, therefore, for all C in some neighbourhood W ofO. • Remark. Theorem 1.4.14 is not valid without the hypothesis de(O} =1= O. Coun terexample: e(z) = 2x~ - x~, z = a; ix:a E fC1 (d(!(O) = 0 but (! does not have a local minimum at 0). In the general case there is the following result:
+
1.4.15. Theorem. Let (! be a strictly plurisubharrrwnic 02-function in some neighbourhood of 0 E eft. Then there exist a complex-linear i,'Jomorphi8m t: en ~ eft and numbers At ~ 0 (j = 1, ... ,11,) such that
1 02{! (
0
t( 0
8xt 8x.t
+A1
o
))2n
1
j.k=l
+ An 1 - Al
o
.. 1 - Aft
where XI
xl(C) are the t'eal coordinates of CE
=
en 8uch that C, =
x,(C)
+ ix; +n(C).
Proof. Let
2e L'- ( 8 (O))ft . - eCI aCt i. k=l
=
(j
row index) .
By Theorem 1.4.7 (ii), L is strictly positive-definite. Consequently, there is an invertible complex 11, X ~ matrix V = (ViI:) (j = row index) such that V+L V is the unit matrix, where V = (Vfk) and V+ = (Vtl) ti = row index}. If v is the complexlinear isomorphism of en defined by V, then this implies (cp. Proposition 1.1.16)
le v(o))n = (e8C 8Ct j, k-l
(1 ... 0) .
V+L V =
0
0
1
(1.4.21)
1
Without loss of generality we can assume that e(O) = 0 and de(O) (1.4.21) and Lemma 1.4.13, we have the Taylor expansion at C= 0
e
v(C)
0
~ ICI 2 +
£ 8C8 (!(O)oCt C1Ck + o(ICI
=
O. Then, by
2
Re
j,k-l
2) •
(1.4.22)
I
Let
+
2
= A iB, 8C1 ac" ;.k=1 where A and B are real matrices. Set (
8 (>(0))n
(A -B)
R '= . -B -A .
(1.4.23)
Then R is a real symmetrio 2n X 2n matrix and a computation shows that
ft 82(>(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
I
then
I<
e f()f' all
Z
EK ,
e + fP i8 striotly plurisubkarmonio in a neighbourhood of K.
Proot In view of Theorem 1.4.7 (ii), it is sufficient to set 8Ie(z) ~ ~ nIZEK,~EC".lel-l ;.1:-1 8z , 8ZJ:
._ 1 e .- -
.
nun
;. ~
----,1,1:.
•
32
1. Elementary properties
en
1.4.20. Corollary. Lei X c ([)ft be a totally real ()l-submanifold 0/ and K a compact subset of X. Then there exi8ts a non-negative 8trictly pluri8ubha,rmonic Ol-function e in some neighbourhood U of K 8uch that K
=
{z E U g: e(z)
=
O} •
(1.4.32)
Proof. By Theorem 1.4.18 there is a non-negative strictly plurisubharmonic 0 2 _ function ex in some neighbourhood Ux of X such that X = {z E U x: ex(z) = O}. Choose a COO-function 91 ~ 0 on eta such that K = {z E q;(z) = O} (see, for example, Lemma 1.4.13 in NARASIMHAN [2]). By Lemma 1.4.19 we then can find E 0 such that e := ex E91 is strictly plurisubharmonic in some neighbourhood U of K. Clearly, K = {z E U: e(z) = O} • •
en:
>
+
1.4.21. Theorem. Let D ~ ([)n be an open set and e a strictly plurisubharmonic 0 2_ function in D. Then,jor every e ED, there exi8t a neighbourhood U ~ D of ~ and a totally real Ol-submanifold X of U such that {z E U: de(z)
=
O}
ex.
(1.4.33)
Proof. Without loss of generality we can assume that ~ = 0, e(O) = 0 and de(O) = O. Then, by Theorem 1.4.15, we can also assume that for C~ 0 , ft ta e(C) = ~ (1 A,) IXI(C)12 ~ (1 - A() IXJ+,,(C)12 0(ICI 2) , (1.4.34)
+
;-1
+
+
j-I
whereAf ~ 0 and X1 = xl(C) are the real coordinates ofC such thatCI = X1(C) + iXj-l n(C). Set 911(') := 0e(C)/8Xj, j = 1, ... ,n. Then, by (1.4.34), d91l 1\ ••• 1\ dcpft =F 0 in some neighbourhood U of O. Consequently, X := {z E fj: 91l(Z) = ... =
1.4:.22. Corollary. Let D c ([)" be an open 8et and e a non-negative strictly plurisubharmonic 02-junction in D. Then, for every point ~ ED, there exi8t a neighbourhood U ~ D of Eand a totally real (}1-8ubmanifold X of U 8uch tkat {z E U: (!(z) = O} c {z E U: de(z) = O} ~ X •
(1.4.35)
Proof. This follows from Theorem 1.4.21, becausee is non-negative, and therefore de(z) = 0 if e{z) = O. •
1.0.
Pseudoconvex sets
If D ~ en is an open set, then we write dist (z, cD) := inf {Iz - CI : CE 8D}. It is clear that the function D ~ z ~ dist (z, 8D) is continuous and> 0 in D. Consequently, -In dist (z, 8D) is a continuous real-valued function in D.
en
1.3.1. Definition. An open set D ~ is said to be pseudoconvex if the function -In diet (z, 8D) is plurisubharmonic in D. en is called pseudoconvex.
1.5. Pseudo convex sets
I.S.2. Coronary. (i) If D C (/)" and G C D X G C (/)"+m is pseudoconvex.
(/)"
33
•
are pseudoeonvu open 8eta, tken
(ll) If {Df}jEJ is a family of pseudoccmvex open 8ets in is pseudoconvex.
(/)n,
then tke interior oj
n D, JeJ
Proof. This follows from Proposition 1.4.9 (iii) . • I.S.3. Definition. Let D C (f)n be an open set and K a compaot subset of D. Then we define Kb := {z ED: e(z) ~ maxe(C) for all e E PO(D)} , CeK
where ]>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
<1.
(1.5.5)
<1.
(1.5.6)
Relation (1.5.4) is equivalent to ~
+ AW + C
e-f(),)
ED for
To prove (1.5.6), we fix some'
Fe : = 3 Henkin/Leiterer
{~
E
IAI
~ rand
CE (Jft
with
on with 1'1 < 1 and define
+ AW + tC e-!C),) : IA\
:5: r}
for
0 ~t~ 1•
lei
34
1. Elementary properties
•
re
Let T be the set of all 0 ~ t ~ 1 such that C D. Then we have to prove that 1 E T. Since, by (1.5.2), 0 E T, it is sufficient to show that T is both open and closed in [0, 1]. That T is open is clear. To prove that T is closed, we consider the compact set
K := {E
+ AW +
t'
e-1().): IAI = r, 0 ~ t ~ I} .
By (1.5.5), K ~ D and in view of condition (ii), i£ is a compact subset of D. To prove that T is closed it is consequently sufficient to show that, for every t E T,
r,s K"p
D•
+
+
Consider some t E T. Then, for every(? E PO(D), the function A -+(?(~ AW ~ e-1().») is subharmonic in a neighbourhood of the disc IAI ~ r (cp. Theorem 1.4.11 (i)). By (1.4.1) this implies that, for every (? E PO(D) and alllAoI ~ r,
(!(E
+ Aow + tC e-1(.t.») ~ max (!(~ + ).w +
tC e-1(A»)
IAI ... ,.
that is,
~ max (!(t') , l'EK
re C K£ .•
1.5.6. Corollary. Every domain
0/
holomorphy D
C (f)rt.
is pseudoconvex.
Proof. This follows from Theorem 1.5.5 (ii), Theorem 1.3.7 (iii) and Proposition 1.5.4. • The following results shows that pseudoconvexity is a local property of the boundary.
1.5.7. Theorem. Let D C (f)ft be an open set. II, lor every E E aD, there is a neighbourhood U of Esuch that U n D i8 pse'Udoconvex, then D i8 pseudooonve:t. Proof. For every ~ E aD and every neighbourhood U of E we can find a neighbourhood V S U such that dist (z, aD) = diet (z, 8( U n D)) for z E V. Therefore, it follows from the hypothesis of the theorem that there is a neighbourhood Wof aD such that -In dist (z, aD) is plurisubharmonic in W n D. Choose an increasing convex G2. function ({J: R -+ R such that cp{lzl) -In dist (z, aD) for zED" Wand cp{t) -+ 00 when t -+ 00. By Theorem 1.4.12 (i), cp(lzl) is plurisubharmonic for all z E (jJn. By Proposition 1.4.9 (iii), the function (?{z) : = max (
>
en
1.5.8. Theorem. Let D C be a pseudoconvex open set, anil let K be a compact subsee of D Buch that K = K£. Then, Jor every neighbourhood UK oj K, there exists a function (} satisfying the following conditions: (i) (? is a 8trictly pluri8ubharmonia GOO-Junction in D. (ii) (! 0 in K and (! > 0 in D " U x. (iii) For every (X E R, {z ED: e{z) < (X} c c D.
<
Proof. Let 1jJ E PO{D) satisfy condition (iii) in Theorem 1.5.5. Adding a constant t01jJ, we can assume that'll' Oin K. Consider the compact setE' := {z ED: 1jJ(z) ~O}. Since K is PO(D)-convex, for every z E X' n (D " U x), we can find
<
>
<
<
<
>
>
and, by Proposition 1.4.9 (iii), cP E PO(D). Clearly, ({J satisfies conditions (ii) and (iii).
1.5. Pseudoconvex sets
35
e
To obtain a strictly plurisubharmonic O°O·function in D sa.tisfying (ii) and (iii), we set D f := {z ED: tp(z) j}, j = 0, 1,2, ... , and construot a sequence ef (j = 1,2, ... ) such that for j = 1,2, ... the following conditions are fulfilled:
<
e,
is a strictly plurisubharmonic O°O·function in some neighbourhood of D , . ef < 0 in K. (c) l!'~ tp in D , . (d) l!1 = (1j-l in Dj _ 2 if j ~ 2. Then, by setting e := (11 in Di - 2 , we obtain the required funotion. At first we construct el' By (1.5.7) we can choose e 0 so small that tp + 8 < 0 in K, and by Theorem 1.4.10 we can find a plurisubharmonic Goo·function ~ in some neighbourhood of Dl such that let - tpl < 8/2 in .D1 • Then, for t5 > 0, e(z) + t5lzP' is strictly plurisubharmonic in a neighbourhood of D1 , and if c5 is sufficiently small, then the function (1l(Z) := ~(z) + lJ Iz21 + e/2 has the required properties. Now suppose that the functions el' ... , ej-l are already constructed, j ~ 2. Since (1j-l is 0 00 in some neighbourhood of D;_lJ we can find a. COO·function ej-l in C" suoh that ~-1 = e,-l in a neighbourhood ofD, _ 1. Further, by Theorem 1.4.10, we can choose a plurisubharmonic Goo.function in a neighbourhood of D, such that leI - tpl 1/2 0 be so small that the strictly plurisubharmonio Coo·function e,(z) in D,. Let lJ := e,(z) + lJ Izl2 also fulfils the estimate lei - tpl < 1/2 on D,. Then < j - 3/2 in Di - 2 and €!1 j - 3/2 in jj, "Dj _ 1 • Choose a convex O°O·function X: R -+ II such that X(t) = 0 if t ~ j - 3/2 and dX(t)/dt > 0 if t > j - 3/2. Then X 0 = 0 in ~-2 and X 0 > 0 in jjj " Dj _ ll and it ~llows from Theorem 1.4.12 that X 0 ef is pluri. subharmonic in a neighbourhood of D, and strictly plurisubharmonic in a neighbour. hood of D, "D j _ l • Now we claim that for sufficiently large G E II the function l!, := ~-l + Ox 0 ~, fulfils conditions (a)-(d). Firstly observe that (d) and (b) are valid for every real O. In fact, since X 0 = 0 and ej-l = ej-l in Dj - 2 , we obtain (d), and, since K S Do C Di - 2 , this implies (b), because ej-l satisfies (b). That (c) is fulfilled for suffioiently large 0 follows from the fact that ei-l (= (1i-l) ~ tp in Di - 1 , X 0 el ~ 0 on jj" and X0 0 on D, " Di - 1 • It remains to show that (a) is true if 0 is sufficiently large. Since ej-l (= l!j-l) ~ strictly plurisubharmonic and X 0 is plurisubharmonio in a neighbourhood of Di-l~ we obtain that, for 0 > 0, l!1 is strictly plurisubharmonic in a neighbourhood ~f Dj _ 1 • Further, since X 0 l!1 is strictly plurisubharmonic in a neighbourhood of D1 '" Dj - ll it follow!. from Lemma 1.4.19 that e, is also striotly plurisubharmonic in a neighbourhood of D, " D j - 1 if G is large enough . • (a) (b)
>
e,
>
<
el
>
el
el
e,
e, >
e,
Theorem 1.4.21 and Proposition 1.4.17 (i) show that the set of critical points of a strictly plurisubharmonic 02-function is contained locally in a real Ol·submanifold of real dimension ~ n. (A point z is called a critical point of a Ol·function e if de(z) = 0.) The following lemma of Morse shows that every strictly plurisubharmonic 02.function can be uniformly approximated on compact Bets by strictly plurisub. harmonic C2·functions whose critical points are isolated. I.S.9. Lemma (Morse's lemma). Let D c ()n be an open 8et and e a 8trictly pluri8tJbharmonic OJ.JunctiO'16 in D. Then, Jor efJerye > 0, there exists a real Unear map 3*
36
L:
1. Elementary properties
en ~ III Buch tlae max IL(z)1
~ B,
(1.5.8)
.EC-.Iz;J .. 1
tke Bet Crit (e + L) := {z ED: d(fl 8trictly plurisubharmonic in D.
+ L) (z) =
O} i8 discrete in Dl), and
(!
+L
is
Proof. Since fl is.OI, by a lemma of M. MORSE (see, for example, § 2, Lemma A in [1]), for almost all real. linear maps L: QJn ~ III the critical points of fl + L are non-degenerate and therefore isolated. In particular, this is the case for arbitrarily small L. That fl + L is strictly plurisubharmonic follows from Theorem 1.4.7 (ii) and the fact that the second-order derivatives of L vanish. •
MILNOR
e
1.S.10. Corollary. Under tke hypotheses 01 Theorem 1.5.8 there exiBt8 a junction BatiBfying conditions (i)-(iii) in Theorem 1.5.8 and Buch that the 8et Crit (fl) := {z ED: d(l(z) = O} is di8crete in D.
Proof. This follows immediately from Lemma 1.5.9. • 1.5.11. Corollary. Let D c: qyn be a pseudoconvex open 8et. Then, for every compact 8ee K c c: D, there exi8ts a 0 00 8trictly pseuaoconvex 2 ) open 8et D' 8uch that K c c D' c c: D.
Proof. By Theorem 1.5.5, K£ is a compact subset of D. Therefore, by Corollary 1.5.10, we can find a strictly plurisubharmonic Ooo-function fl in D such that e 0 in i~, {z ED: e(z) a:} c c: D for all a: E ll, and the set {z ED: de(z) = O} is 0 so that dfl{z) 9= 0 for Z E aDa' and set D' := Da, . • discrete in D. Choose (x'
<
<
>
1.S.12. Proposition. Let D c: qy" be an open 8et. II there exist8 a continuous plurisubharmonic function in 80me neighbourhood 6 of aD 8uch that
e
D n 6 = {z
E 6: e(z)
< O} ,
(1.5.9)
'ken D i8 pseudoconvex. Proof. Let E € eD and e > 0 so small that the ball E := {z E (f)n: Iz - ,I < e} is relatively compact in 6. Since pseudo convexity is a local property of the boundary (Theorem 1.5.7), we have to prove only that EnD is pseudoconvex. To do so we consider the continuous plurisubharmonic function cp(z) : = max (lz - EI - e, e(z)) is some neighbourhood of D n E (cp. Propositions 1.4.8 and 1.4.9 (iii)). Then rp = 0 on 8(D n E) but rp < OinD n E. In view of Theorem 1.5.5 (ii), this implies thatD n E is pseudoconvex. • We shall now examine when an open set with 0 2_boundary is pseudo convex. Firstly we introduce the notion of the complex tangent plane: 1.5.13. The complex tangent plane. Let D ~ qy" be an open set with O~-boundary aD, and let e be a real-valued 02-function in some neighbourhood UD of D such that D = {z E U D : e(z) < O} and de(z) 9= 0 for Z E aD. For ~ E fiD we denote by te the real tangent plane of aD at E, that is,
te:=
{
wE (f)":
2n l:
ofl(E)
--Xf(W - ; ) j =1 aXf
=
}
0 ,
A subset M ~ D is called discrete in D if for every of J: such that U n M = {z}. t) See Definition 1.5.15. 1)
Z E
M there is a neighbourhood U
37
1.5. Pseudo convex sets
+
where xl(C) are the real coordinates of C E qyn suoh that Cf = x,(e) iXJ +,,(C). The maxi. mal complex plane (cp. Definition 1.4.16) which is conta.ined in If is oalled the com~ tangent plane of aD at Eand will be denoted by Te that is, Te := {E + i{te - E}} n 'f' Proposition.
Te=
w
{
n
E ([J :
n ,l: 3 -1
a!')(E)
-!:-
~ 1)
(wI -
Oeal
}
=
(1.5.10)
0 .
Proof. 8!,)(~) l:n - (u'1
1 "(a()(~) - Ei) = - l: - 2 J=1 OX,
j=1 a~1
. a!,)(~»)
-1--
8xJ+n
(XI(W -~)
+
,
lXj+n(W -
.
~»)
=
0
if and only if " a!,)(~)
l:
--Xj(w -~) j=1 OXt
ae(~)
+ l:n
--Xj+n(w -~) = 0 j=1 OXj+n
and n
l:
o!,)(~) --Xj+n
}=1
--Xj(w -~) ;~1 OXj+n
8x,
Since x,(i(w -~))
==
ae(E)
n
E) - l:
(w -
=
O.
-Xj+n(u,' -~) and xj+n(i(w - ~))
this implies that l: (8e(E)/aEt) (w, -
=
Xt(w - ~), j
= I, ... , n, EI) = 0 if and only if wE te and E+ i(w - E) E teo •
n
;=1
1.5.14. Theorem. Let D c qy" be an open Bet with Ol-boundary eD, and let (! be a real-valued 02-function in a neighbourhood U aD of aD Buch that U aD n D = {z E U aD : !,)(z) < O} and de(z) 0 for z E aD. Then D i8 pseudoconvex if and only if
*
"
l: }, k=1
a2e(z) ----wiW" ~ 0 8z1 az"
z E aD
for all
where T. i8 the com.plex tangent plane of 8D at
and
W
E {T. -
(1.5.11 )
z} ,
Z.
Proof. Let el be another 02·function on UaD such that UaD n D = {z E UaD : el(Z) O} and del(Z) =1= 0 for z E 8D. Then (!1 = qJ(!, where rp is a CI·function and> 0 on [lan. Consequently, for z E 8D and wE {T. - z},
<
" 82f!t(z) j, k-l 8z1 8z"
l: -,-_-
w,_
W" =
"
82e(z)
-
l: --_w1 W " ,
rp(z)
j,k=1
az 8z,t
•
This implies that (1.5.11) is independent of the choice of e. To prove that (1.5.11) is necessary for pseudoconvexity of D, we set - dist (z, 8D) if zED, e(z):= { dist (z, aD) if z E 0"'" D .
+
(1.5.12)
Since cD is 0 2, then e is C2 in some neighbourhood U aD of 8D. If Dis pseudoconvex, then by definition, - In ( -e) is plurisubharmonic in D. By Theorem 1.4.7 (i), this im plies that
i
j, "=1
for all
Z E
[~8e(Z) 8e(z) _ ~ 62e (z)] WIW" ~ 0
e2
8z1
CZa:
UaD n D and w
E ([J".
(!
8z1 ozt
-
This implies that for
Z E
UaD n D and
2
.:. 8 Q(z) - -0• ~ --_-w1wJ; ~ 8z f 8zk A passage to the limit shows that this is also true for z j,],;-l
E
aD.
z
+ wET•
38
1. Elementary properties
Now we prove that (1.5.11) is sufficient for pseudoconvexity of D. To do this we assume that D is not pseudoconvex. Let UaD be a neighbourhood of cD such that the function e defined by (1.5.12) is O· in U aD • By Theorem 1.5.7, then there exists a point ~ E U iJD n D such that -In (-e) is not plurisubharmonic at~. This means, by Theorem 1.4.3, that for some 0 =F W EO", 8a y := -_In (-e(~ + lw»)!.~_o O. 818A
>
Then we have by Taylor's formula (cp. also Lemma 1.4.13) that forl-o.-O (1.5.13) where IX, Pare constanta. Now we choose rJ E such that IrJl = le(E)1 and ~ + rJ E 8D. Define C.(A) : = ~ + lw + 8'1J ed +Il}.' if 0 s :5: 1. Then by (1.5.13) we can find an 8 0 such that for IAI ~ 8 and 0 < 8 ~ 1
on
<
>
dist (C.(A),
eD)
~
- e(E + AW} - 811}1 IecM +tLl'l
~ le(~)1 (el'li.I'/2 -
s) led+JJJ.'1 .
(1.5.14)
>
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 <0
8A eA
8t!(C1(A» 8A
and
J.-O
I
= 0 .
A=O
Since C1(A) is an analytic function of)" we conclude that (cp. Proposition 1.1.16) 1
8 t!(z) 1: --_f&
i,k-l
I -<
8z1 8z~ .-e+7J
VfVk
0
and
By (1.5.10), this contradicts (1.5.11) at
~
"
8e(z) 1: -I
i~l 8z1 ''"'';+'1
'V1
= 0 .
+ rJ E 8D . •
In this book, pseudoconvex open sets are of special interest which can be defined by strictly plurisubharmonic (]I·functions:
1.5.15. Definition. Let Dec 0" be an open set. D is called strictly pseudooonvex if there exists a strictly plurisubharmonic 011. function in some neighbourhood 8 of the boundary of D such that D n 8 = {z E 8: t!(z) O}. If, moreover, the boundary of D is smooth of class 01: (k = 2,3, ... ,), then D is said to be a Cle strictly pseudoconvex open ,et.
<
e
on
Remark. The boundary of a strictly pseudo convex open set D cc need not be smooth. Example: {z = ~ + i:r. E ()1: 2~ - x~ + < O}. However, it is interesting that, by Theorem 1.4.21 and Proposition 1.4.17 (i), the set of points where 8D is not smooth is not too "large", namely, it is locally contained in a real Ol-submanifold of real
xt
39
1.5. Pseudocon vex sets
dimension ~ n. Observe also that strictly pseudoconvex open sets Dec C' may consist of infinitely many components. If the function e in Definition 1.5.15 is strictly convex at some point ~ E eD, thcn de(~) =F 0, because e has not a local minimum at The example above shows that this is not the case when e is strictly plurisubharmonio only. However we have the following
e.
1.5.16. Proposition. Let Dec ([}n be a strictly pseudoconvex open set, and lee e be a strictly plurisubharnwnic Q2-function in a neighbourhood 8 of eD 8uch that D n () = {z E 8: e(z) < O}. If the boundary of D i8 of clase 0·, then d(>(z) =1= 0 Jor all z E 8D.
<
Proof. By hypothesis there is a real-valued C2-function rp in ([}n such that rp 0 in D, rp 0 in en", D, and dcp(z) =1= 0 for Z E oD. Then 1p : = (>/r is a Cl-function in 8 and de(z) = 1p(z) dtp(z) for z E 8D. Therefore, we have to prove that V' =F 0 on eD. Assume that 1p(~) = 0 for some ~ E aD. Since 1p ~ 0, then d1p(~) = 0, too. Since also cp(E) = 0, we conclude that all second-order derivatives of (> = 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
<
n
82e(z)
~ ~ Wt'Wk j,k=l
8z1 ozk
:::- 0
for all
z
E
fJD
where T, is the complex tangent plane of eD a.t (1.5.15) is fulfilled, then, fo-r sUfficip.ntly large I.
and
0 9= W
E
{T I
-
z} ,
(1.5.15)
z (see Subsection 1.5.13). If condition
> 0, the function
(1.5.16) is 8trictl11 plurieubharmonic in some neighbourhood FaD ~ U aD of oD, and D n V 8D = {z E V aD : e(z)
Proof. As in the beginning of the proof of Theorem 1.5.14 we see that condition (1.5.15) is independent of the choice of (>. If D is strictly pseudoconvex, then, by definition and Proposition 1.5.16, e can be chosen to be strictly plurisubharmonic. By Theorem 1.4.7 (ii) this implies that condition (1.5.15) is necessary. Assume now that (1.5.1fi) is falfilled. We only have to prove that, for sufficiently large A, the function defined by (1.5.16) is strictly plurisubharmonic in some neighbourhood of oD, because, clearly, D n U aD = {z E U aD : ~(z) < O}. 'Ve have
e
o2e(~
=
A a2e(~
8z, ez,t
eA!?
-+- 1.2 O(!(z)
ee!z) e A4l •
a~
6z 1 oz.
8zt
Consequently, for z E eD and all WEen, n
~ j,k=l
82e(z)
_
- - _ - U'1WA;
ez, 8zk
=:
n
o2e(z)
_
~ ;:;---=- W,Wlc j,1:=1oz, 8z k
A
+ A.• 2 1 ~n
j~l
oe(z) - - Wt
8Z1
12 •
(1.5.17)
~ R"" be an open set with Oi-boundary, and let e be a real-valued Oil-function in some neighbourhood UaD of aD such that D n UaD = {x E UeD: e(x) < O}. If, for Borne ~ E aD, de(x) = 0, then the se('on(l-order derivatives of e at x va.nish.
1) This proof shows that the following more general proposition is true: Let D
40
1. Elementary properties
Set 82 (z) 8zt 8z k
}
K:= { (w,z)E(f)"x8D:lwl=land ~ ~wiW1t.~O. n
;.1:=1
Since K is compact, and since by (1.5.15),
i
I i-I
8Q(z) 8z1
w,
1
>0
(w, z) E K,
for
we can choose A so large that
-I < A mm. 11: -,,8t!(z) W, I •
S
max (ID••)eK
I
n 8 t!(z) W,Wt ~ --_-
J.1:-1
8z1 8z"
n
(UI,z)eK ;-1
By (1.5.17) and Theorem 1.4.7 (ii), then bourhood of aD. •
oz,
eis strictly plurisubharmonic in some neigh.
1.5.1S. Corollary. Let Dec (f)n be a strictly pseudoconvex open set with Olt.·boundary (Tc = 2, 3, ... , (0). Then there exists a strictly plurisubhartnonic Olt.·function t! in some neighbourhood UaD of 8D such that D n UaD = {z E UaD : e(z) O}.
<
Proof. By hypothesis the function (! in Theorem 1.5.17 can be chosen to be Ok. Then, for sufficiently large A, the function defined by (1.5.16) has the required properties. •
e
1.5.19. Theorem. For every strictly pseudoconvex open set D c: c: ([In there exists a strictly plurisubharmonic 02·function (! in some neighbourhood U D of D such that
D
=
{z E UD : Q(z)
< O}
(1.5.18)
=
(1.5.19)
and 8D = {z
E
Ujj:e(z)
O}.
If 8D is of dass OJ: (Tc = 2, 3, ... , (0), then thi8 function
ecan be chosen to be Ok in U jj.
Proof. By definition there is a strictly plurisubharmonic 02·function t!o in some neighbourhood UaD of aD such that D n UaD = {z E UaD : t!o(z) < O}. Let t5 0 so small that Kd := {z E USD : -t5 ~ t!o(z) ~ O} is a compact subset of VaDe Choose a OOO·function X: R ~ R with the following properties: X(t) = -t5 if t ~ - t5, X(O) = 0, dIX(t)/dtl ~ 0 for all - 00 < t < 00, and dX(t)/dt 0 if -t5 < t < 00. Define t!1 : = -d in D " K~ and t!1 : = X 0 Qo in V aD • Then
>
>
D
=
{z ED u U aD : l?1(z)
(1.5.20)
and, by Theorem 1.4.12, t!1 is a plurisubharmonic 02.function in D U V aD that is strictly plurisubharmonic for all z E V aD with t!o{z) > - b. By Theorem 1.5.8 there is a strictly plurisubharmonic O°O·function e2 in D such that DtI' := {z ED: t!2(Z) a} c: c D for all /!It E ll. Choose pER so large that l?1 b/2 on D "DfJ, and let tp be a real. valued OOO·function on ([In such that 1jJ = 1 in some neighbourhood of DfJ and tp = 0 in a neighbourhood of ([J1l" D. Define := tpt!B in D and := 0 in cn " D. Then is strictly plurisubharmonic in a neigh. bourhood of DfJ. Consequently, for every c 0, t!1 cea is strictly plurisubharmonic in a neighbourhood of DfJ. Since t!1 is strictly plurisubharmonic in V aD " D fJ , and since = 0 in U aD " D, it follows from Lemma 1.4.19 that, for sufficiently small c 0, = l?1 is a. strictly plurisubharmonic Ol·function in Ujj : = D u V aD• Moreover, since ~ = 0 in 8. neighbourhood of on" D, it follows from (1.5.20) that (1.5.18) is fulfilled if 0 0 is sufficiently small.
<
>-
e.
e.
e:
et
+ cel >
e2
>
+
>
1.5. Pseudoconvex sets
41
It remains to satisfy (1.5.19). To do this, we choose a Coo.function rp ~ 0 on QJft such that jj = {z E e": rp(z) = O} (see, for example, Lemma 1.4.13 in NABASIMBAM [2]). Then, by Lemm s. 1.4.19, for sufficiently small e 0 and after shrinking UD, (! Ecp is strictly p\'\lrisubharmonic in UD, and (1.5.18) and (1.5.19) are satisfied if we replace (! by (! Ecp • Finally, observe that if aD is C1r., then, by Corollary 1.5.18, eo can be chosen to be 01:, which implies that e erp is 01:. •
+
>
+
+
1.5.20. Corollary. Let Dec ([J" be a strictly pseudoconvex open set. Then, lor every compact 8et K c aD an! every neighbourhood UK? QJ" of K, there exists a strictly pseudoconvex open set D C QJfI, Buch that D u K cD C D u UK' lJthe boundary of Dis oj class Ci (k = 2, 3, ... , 00), then this strictly pseudooonvex open 8et jj can be chosen with Ok-boundary, too. Proof. Let
e and
U D be as in Theorem 1.5.19. Choose a non-negative real.valued
X E Cgo( UK n UD) such that X > 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 •
>
<
+
>
<
In view of Morse's Lemma 1.5.9, we can moreover assume that d§(z) 9= 0 for zED.. except for not more than a finite number of points. Consequently, we can choose co ~ El ~ 2eo so that de(z) 9= 0 for all Z E ai5e, • Setting jj : = DBI we conclude the proof. • Remark. Theorem 1.5.21 is not valid without the hypothesis on striot pseudoconvexity. DIEDERICH and FORNAESS [1] constructed an example of a smooth pseudo. convex open sct Dec without a basis of pseudoconvex neighbourhoods of D. Since every strictly convex 02-function is strictly plurisubharmonic, we obtain
en
1.5.22. Proposition. E1iery 0 2 strictly convex open set Dec ([J" is 8trictly pseudoconvex. It is interesting that, after an appropriate local change of holomorphic coordinates, the converse is also true. This will be proved below (Theorem 1.5.24). To do this we need part (i) of the following lemma. (Part (ii) will be needed later.)
en
ot.
1.5.23. Lemma. (i) Let G c be an open set and (/: G ~ R a strictly convex function (k = 2, 3, ... , 00). Set G_ := {z E G: e(z)
42
1. Elementary properties
8et K ~ G_ n G and every neighbO'Urhood UK of K, there exi8t.s a 8tridly convex open 8et D with Ot.boundary BUch that X ~ jj C UK n G_. (ii) Let G ~ on be an open 8et, and let ~ be a 8trictly pluri8ubharmonic ot.junction in G (k = 2,3, ... , (0). SetG_:= {z EG: (!(z)
OsosmallthatDi:= {ZE U1 :6!J(Z)<8} c:c U1 • Now we choose GOO. functions I, g: Il -+ fl with the following properties: dJ(t)/dt > 0 and dlj(t)/dtl ~ 0 for all t E R, -1 0 if t > 0; dg(t)/dt ~ 0 and d l g(t)/dt2 ~ 0 for all t E IR, g(t) = 0 if t ~ 0, g(t) > 0 if t > 0, g(t) > 1 if t > £. Set rp(x, 1/) := I(:.r:) + g(y), x, y E fl. Then cp is a convex OOO.function on Il", 8cp(:.r:, y)/8z> 0 and ecp(x, y)/ay ~ 0 on 11,'1·, cp(x, y) > 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) . •
<
<
1.5.24. Theorem. Let G C CfI be an open 8et, and let e be a 8trictly pluri8ubharmonic G"-function in G8uch that de(z) =l=Oforallz E r:= {z E G: !!(z) = O}, k = 2,3, ... ,00. Set G_ := {z E G: e(z) O}. Then, for every point E E there exist a neighbourhood Ul of Eand a ot 8trictly p8eudoconvex open set D c: c: C" BUch that (i) U e n G_ ~ D ~ G_. (ii) There i8 a biholomorphic map h of 80me neighbourhood of D such that h(D) is strictly convex.
<
r,
Proof. By Theorem 1.4.14, there is a biholomorphic map h from Bomc neighbour· hood Veof ~ onto some open set W C Cft such that!! 0 h-l is strictly convex in W. Let U' c c W be an open ball centered at h{E). Then, by Lemma 1.5.23 (i), there exists a O~ strictly convex open set D' such that U' n h( V f n (t) ~ D'~ h( V, n it). By setting U e := h-l(U') and D := h-l(D'), we complete the proof. • 1.5.25. Corollary. An open 8et Dec q)ft with O~·boundary (k = 2, 3, ... , 00) is strictly pseudoconvex if and only if, for every point E E aD, there exist a neighbourhood Ue ~ Cft of Eand a biholomorphic map he in Bome neighbourhood of U f Buch that hf ( Uf nD) i8 a strictly convex 8et with O~-boundary. Proof. By Theorem 1.5.24, the condition is necessary. That the condition is sufficient follows from Theorem 1.5.17 a.nd the fact that condition (1.5.15) is invariant under holomorphic changes of coordinates (cp. (1.4.15) and the corresponding comft
putation for
I:
(8e 0 1(1,)/8z, ) ~1).•
j-1
We conclude this section with the following Remark. Every open set Dec q)1 with Oll-boundary is strictly pseudoconvex. This follows, for example, from Theorem 1.5.17 and the fact that, for n = 1, condition (1.5.15) is trivially fulfilled, because then Tit = 0 for all z E 8D.
1.6.
Preliminaries concerning differential forms
We assume that the reader knows the calculus of differential forms on real manifolds as it is developed in standard books. In the present section we collect certain more special material and introduce some notations. Let X be a real Ol-manifold. Then we denote by dimB X the real dimension of X. The notation of a differential form on X (form on X) will be used for differential forms with complex-valued measurable coefficients. The degree of a differential form f on X will be denoted by deg f, the 8upport of f (this is the smallest closed subset K in X such that f = 0 in X " K) will be denoted by supp f. An 8-form on X is a differential
44
1. Elementa.ry properties
form of degree 8 on X; a whose ooeffioients are k lorm on X is a Ot.form denoted by O(~)(X) (k =
O"-form (if X is a Ok+l.manifold) is a differential form on X times oontinuously differentiable (k = 0, 1, ... , 00); a Of,)of degree 8 on X. The space of all Ot)-forIns on X will be 0, 1, ... , 00; 8 = 1, ... , dims X).
1.6.1. Integration with respect to a part of the variables. Let X and Y be two real Gl-manifolds, and let I be a differential form on X X Y. If YI, ... ,y,. are local coordinates on Y, then the form f can uniquely be written ~' 11(x, y) " dyl ,
I(x, y) =
(1.6.1)
Ill:iideg!
wherejr(x, y) are differential forms of degree degf - Ilion X depending on y E Y. Here the following notations are used: 1 = (~, ... ,ir ) is a multi-index of integers 1 ~ i, ~ n, dyl := dy" 1\ ... 1\ dYir' III = r is the length of I, and ~' means that the summation is over all multi-indices with strictly increasing components. If X is oriented and if the integrals J fl(X, y) exist for all III = degf - dimB X and Y E Y, then we define I
Jf(x, y) = X
( J11(X, y)) dyl .
~
III-deg!-dlmBI
(1.6.2)
x
The result of this integration is a differential form of degree degf - dimB X on Y and is independent of the choice of the local coordinates YI' ... , y". Remark. J1(x,' Y) = 0 if' does not contain monomials of degree dimB X in x. I
1.6.2. The absolute value of a differential form of maximal degree. Let X be an oriented real Gl-manifold of real dimension m, and let/be a differential form of degree m on X. Then we denote by III the differential form on X which is defined as follows: Let ~, ... ,Xm be local ooordinates on X such that dXI 1\ ... 1\ dX m defines the orientation of X. Then f = P dX1 1\ ... 1\ dx m , where p is a complex-valued function, and we define III := l
I Jfl ~ fiJI. I
X
1.6.3. The norm of differential forms at points. Let D c: R" be an open set, and let I be a differential form of degree 8 on D. Then, for xED, we denote by II/(z)11 the Euclidean length of the vector of coefficients of 1(x) with respect to the canonica.l coordinates Xl' ... , X" in ll", that is, if
I., ...,,(x) dXi,
1:
f(x) =
1\ ... 1\
dXi, ,
xED,
1 Sil < ...
then
1/(x)1
1/~ ...i,(x)\2}1/2 ,
1:
:= (
x ED.
1 =-'a< •..
II/(z)1I is called the norm of / at x. Now let X be a real 0 1 • manifold. Then, for every differential form! on X, we introduce the norm II/(x) 1\ , X E X, as follows: Let {U1} be a locally finite open covering of X such that, for every j, we can find Gl·coordinates ~), ... , xC1) in Uf , and let {XI} be a Gl-partition of unity subordinate to {U1 }. If
J(x) =
1:
~;)
)(, ... i,(X)
1 :iii, < ... < i.:iin
-' .)
..l
J .)
d~~ 1\ ... "UXi~ ,
X E
U1 ,
46
1.6. Preliminaries oonoerning differential forms
then we define
IIf(x)1I
L
:=
I:
Xl(X) (
i
1.t1!!..i.{X) I1)1/2 ,
x EX.
1:i£,< ...
Clearly, this norm depends on the choice of the local coordinates and the partition of unity. However two such norms differ by a positive continous factor only. 1.6.4. Differential forms in C". Let D be an open set in q)". We denote by ZI' ••• , z" the canonical complex coordinates in 0". Then every differential form f on D can be written uniquely
l:'
f =
fIJ dz I
1\
diJ
(1.6.3)
,
III +IJl-degJ
where fIJ are functions on D. Here the following notations are used: I = (iI' ... , i,) and J = (jv ... ,j,) are multi-indices of integers 1 :::;: ill, jp ~ n, dzi := dz i , 1\ ••• "dz"" diJ := dii , 1\ ••• 1\ dill' 11\ is the length of I, and L' means that the summation is over all multi-indices with strictly increasing components. A differential formf on D is called a (p, q)-form (0 ~ p, q ~ n) or a form of bidegree (p, q) if in the representation (1.6.3) fIJ = 0 for alII, J with (Ill, IJI) =l= (p, g), that is
L'
j =
fIJ dz I
dZJ
1\
•
III =p.IJI =g
A (p, g)-form with coefficients in O"(D), k = 0, 1, ... , 00, is called a O~'fl)-form or a. Ok-form of bidegree (p, q) on D. The space of all O~.f)-forms on D will be denoted by otJJ,fJ)(D). We write Ofp,fl)(D) = 0 for max (p, q) n. It follows from the uniqueness of the representation (1.6.3) that every differential form f on D can be written uniquely
>
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 < ...
dZJ. '
then, by definition of h*/,
~
",./ =
/J:~~:1. h d"'i 0
l;S;i l < ...
l
~I
1\ ... 1\
d~, •
(1.6.9)
This is a (p, g)-form, because the functions hi are holomorphic, and therefore dkl = 8h1 is of bidegree (1,0), and d~ = 8k1 is of bidegree (0, I). This proves part (i). To prove part (ii), by (1.6.4), we can assume that/is a (p, g)-form. Since dh* / = h· df and d = 8 5, we get (8h* - "'·8)/= (",."8 - 8",.)f. This implies (1.6.8), because the form on the left-hand side is of bidegree (p + 1, g) whereas the form on the right-hand side is of bidegree (p, q 1). •
+
+
1.8.5. Differential forms on complex manifolds. If I is a differential form on a complex manifold, then! is called of bidegree (p, g) (a (p, g)-form), if it is of bidegree (p, g) with respect to local holomorphic coordinates. It follows from Proposition 1.6.4.2 (i) that this definition is independent of the choice of local holomorphic coordinates. The unique representation (1.6.4) is valid on complex manifolds, too. The differential operators 8 and 8 can be defined on complex manifolds with respect to local holomorphio coordinates. By Proposition 1.6.4.2 (ii) these definitions are independent of the choice of local holomorphic coordinates. Proposition 1.6.4.1 holds also for complex m:nifolds. For complex manifolds X, we denote by G~. q)(X) the space of all G(p,f)-forms on X.
1.7.
The differential forms w'(v) and w(u)
Let X be a. real Ol-manifold, and let u = (Ut, ... , u,,): X ~ en and v = (v]' ... , tift): X ~ e" be Ol-maps. Then we define the continuous differential forms w(u) := dUt 1\ ... " dUll and
"
co'(v) := ~ (_1);+1 VI dV l ;-1
where l' means that
dV1
i ...
1\
dv",
must be omitted. Further, we use the notation
" v,'Uf • (v, u) : = ~ i-I
1\ ...
(1.7.1
1. 7. The differential forms w'(v) and w(u)
47
If there is a second Ol·manifold Y, the maps 11., v are defined for (x, y) € X X' Y, and 11., v are 0 1 with respect to x, then we write ro~(u) : = d.u1 1\ ••• 1\ d.u" and
1:" (_1)1+1 v,d.v1
ro;(v) :=
1\ •••
J •..
dsv" ,
1\
i-I
where de is the exterior differentiation with respect to x. If 11. and v are 0 1 with respect to (x, y), then we also write ro~if{V) and roS.if(u) instead of ro'(v) and ro(u).
1.7.1. Proposition. Let X be a real Ooo-manifold, and let u,
f):
~
X
on
be 0 1-100'18.
Then the differential form ro'(v} 1\ ro(u)
=F O.
is closed (in the sense of distributions) Jor all x E X with
Proof. Since dro(u) = 0, we have d w'(v)
1\
ro(u)
=
dw'(v)
1\
ro(u) _ d
1\
ro'(v)
1\
w(u) •
(V,U)2A
But dw'(v) = nco(v) and ft
d
1\
w'(v)
1\
=
ro(u)
n
1: (VI du, + u, dv,)
1\
w'(v)
1\
w(u)
i-I
=
n
1\
ro(u) • •
1.7.2. Proposition. Let X, = x,(C), j = 1, ... ,2n, be the real coordinates of CEO" such that C, = x,(C) -I- ix;+,,(C). Then for every Z EO", dc(ro'(C - z)
1\
Proof. d c(!: (_1);+1
w(C)) = n(2i)" dX1
(C, -
= n
1\ (dxt
-
1\ del: 1\ de,)
£1)
k+,
j
1\ ••• 1\ dX2n •
idxk +,,) 1\ (dx,
k
l
=
l
+ idxH ,,)
n 1\ del: 1\ dC, Ie
I
= n(2i)" 1\
1.7.S. Proposition. For every Ol·function 'P on X, we have ro'(1pv) = 1p"ro'(v) ,
(1.7.2)
and consequently w'(1pv) 1\ w(u)
w'(v}
1\
w(u)
(1.7.3)
-
uses the fact that w'(v) can be written as a determinant (this interpretation will be especially useful later). Firat we need some preparations. For that purpose we consider an arbitrary algebra m(for example, the algebra of exterior differential forms). Let .A = (ai;)~.1=l be a matrix with elements atl E m. Then we define det A =
1: sgn (0') aa(l).l ... aa(ft).ft ,
(1.7.4)
(I
where the summation is over all permutations 0' of {I, ... , n}, and sgn (0') is the signature of 0'. Under this definition the usual relations between row operations (the first index is the row index) and determinants hold. Therefore, as in the commutative case it can be proved.
48
1. Elementary properties
1.7.". Proposition. LeI Z be an n X n matrix with elements in the centrel ) of 2l. Then detZA = detZ det A. Remark that, if ~ is non· commutative, then in general det AZ =1= det A det Z. Further, it is possible that some of the columns of A are equal but det A =1= 0 (cp., for example, (1.7.5) below). However, some of the properties with respect to columns which are true for usual determinants remain valid also for the non-commutative case. It is easy (and left to the reader) to prove the following 1.7.5. Proposition. (i) det A is a multi-linear map 0/ the columns with respect to linear combination8 with coefficients from the centre of ~. (ii) If, Jor some 1 ~ k l ~ 16, aic = b1z~ and ail = b,z, for all i, where ble , b, are arbitrary element8 from ~ but all z, belong to the centre of~, then det A = O.
<
The differential form ro'(v) can be written ro'(v)
=
1 det (n - 1) 1
(V~ . ~v: V"
:.: ~~1).
(1.7.5)
-----dv" ... dv" ,,-1
Proof of Proposition 1.7.3. It follows from Proposition 1.7.5 that det (1pV, d(tpt'), ... , d(1pV» = det (1pV, tp dv
+ v dtp, ... , 1p dv + v d1p)
= det (V'V, tp dv, ••• , "I' dv) = 1p" det (v, dv, ... , dv) .
1.7.6. Deflnition. Let 1 ~ m ~ n, let ~, ... ,am be column vectors of length n of differentialjorms, and let Sl, ... ,8m be ·non-negative integers 8uch that ~ Sm = n. Then det" ....," (aI' ... ,am) : = det (~, ... , ~, ... , am, ... , am) ~ '1 'M
+ ... +
---
1.S.
Leray maps and the operators
BaD' B D , La'D
and BaD
In this section we give the definitions and some simple properties of the integral operators used in the integral representation formulas, which will be proved in Sections 1.9-1.12.
on:
1.8.1. Orientation of 0". In this book we use the following orientation in If Xf = x'(C), j = 1, ... ,2n, are the real coordinates of C E such that C1 = xf(C) ixJ+,,(C), then the differential form dx1 1\ ••• 1\ dx2n defines the orientation of (fJf!. For open sets D ~ q;n we use the same orientation. If D ~ on is an open set and M is a relatively open subset of aD which is smooth ()f class 0 1 , then M is oriented by the orientation which is induced from D.
on
+
Remark. In the literature the orientation of (fJ" is also defined by d~ 1\ dXl +n dx" 1\ dX2" = (_1)n("-1)/2 dX1 1\ ••• 1\ dx2". Then we obtain a corresponding change of the sign in the integral formulas given in this book. f\ ••• 1\
1.8.2. Open sets with piecewise (J1-boundary. Let Dec on be an open set. The boundary aD of D is said to be piecewise 0 1 if there is a finite number of real. valued 1)
The centre of i( is the set of all a e ~ such that ab = ba for every b E ~. If ~ is an algebra of differential fonns, then this is the subalgebra of forms of even degree.
1.8. Leray maps and the operators BeD, B D, LfD and RfD
<
49
O1-functions el' ••• ,ell: on ([)ft such that D = {z E ([)": (!I(Z) 0 for j = 1, ... ,k} and d{!jJ(z) 1\ ••• 1\ d(!j,(z) =1= 0 for every collection of indices 1 ~ jl j. ~ k and all points 1, E aD with (!iJ(z) = ... = ej.(z) = o.
< ... <
Remark. For open sets with piecewise Ol-boundary it is easy to find a sequence of open sets Dm C C D with Goo-boundary such that the following two conditions are fulfilled: (i) For every compact set K c c D there is a number.mK such that K C D", for all m ~ mK. (ii) If J and g are continuous differential forms on D of bidegree 2n and 2n - 1, respectively, then = lim and g = lim f g. In the proofs of
f!
r
f!
D
ab
D".
aD".
the integral representations in Sections 1.9-1.12 the existence of such a sequence D.". is used only. Therefore, these formulas are valid also for more general open sets. In the present book we need these formulas only for open sets with piecewise 01- boundary. I.S.3.
B D for I-forms and BaD for functions. We recall the notations
w(C) := dCl
1\ ••• 1\
dC" and w'(C -
,. z) := ~ (_I)J+l (CI - i/)
V dCI: i:+J
i-I
from Section 1.7. Since, for fixed Z E (f)", the differential form (£0'([ - z) 1\ w(C))/IC - zI 2n is smooth in (f)"" z and has an integrable singularity at C = z (the order of this singularity is 2n-I), we can define the following two integral operators: If Dec (f)" is an open set and! is a bounded I-form on D, then we define (B f) (1,) := (n - 1) I D (2,ni)"
JI(r)
1\
'"
£0,([ - z) 1\ .w(C) IC _ zl2ta '
1, ED.
(1.8.1)
CED
If Dec (f)n is an open set with piecewise 01- boundary and I is a bounded measurable function on aD, then we define
(BanJ)(z) :=
~-. I)! JI(C) w~(~_-=-.~~.~ w(C) IC - zl2n
(2,nl)"
zED.
'
(1.8.2)
CUD
Remark.
For n = I,
1
BDf(z) = (1/2ni) 1(C) dC/(C - z)
and
BaJ(z) = (1/2ni)
X f I(C) dC/(C - z) are the operators from the Cauchy-Green formula. (l.l.I). aD
I.S.4. B D and BeD for differential forms of abritrary degree. For (z, C) E (L,n X C" consider the differential form
w;,c(f - z)
1\
w(C)
IC - Z12"
i
(_I)i+1
C, -
%1 /\ (del: - dZk )
1\
w(C) .
IC - z12,. k+J
j=1
(l.8.3)
This differential form is Coo for C =f= z and has a singularity of order 2J~ - 1 at I; = z. Since this Singularity is integrable with respect to C, we can give the following definition (see Subsection 1.6.1 for the definition of integration with respect to C) : If Dec 0" is an open set and! is a bounded differential form on D, then we define (BDj)(z) : = (n
-=:!}~ II(C)
(2nl)n
CED 4:
Benkln/Lelterer
1\
£0;, c(C - z)
Ie -
A.
zl2n
~o!0
'
zED.
(1.8.4)
50
1. Elementa.ry properties
If Dec en is an open set with pieoewise Ol.boundary, andJis a bounded differential form on aD, then we define
J
(~~)~)! I(~) A "'~ '(f~-=-Z~I~ ",(~) ,
(B.J) (z) : =
zED.
(1.8.5)
'e8D
Definition (1.8.4) ooincides with (1.8.1) when J is a l.form, and definition (l.8.5) coincides with (1.8.2) when I is a function. To see this we consider the unique representation (n - I)! w;.dC (2ni)"
-
z)
1\
Ie - z1211
w(C)
=
"il D (z C) g ... o
II,
,
where DII(z, C) is of bidegree (0, q) in z and of bidegree (n, n - q - 1) in a bounded differential form on D and let
C. Let I be
>
be its unique representation as a sum of (r, s}-forms. Sincef(r,B)(C) "w(C} = 0 if 'I' 0, then BDJ = BDI(o,deg./) . (l.8.6)
> + 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).
< <
Now suppose that D ~ eft is an open set. Then for 0 a 1 we denote by OGl(D) the space of all/ E OO(D) such that IIJII.s.K 00 for every compact set K cc: D (the functions in OGl(D) need not be bounded). If k :2: 0 is an integer, then by Ok+GI(D) will be denoted the space of all k times continuously differentiable complex-valued functions on D such that all derivatives of order k belong to OGl(D). The functions in O"+GI(D) will be called OJ:+"'-functiO'M in D. If Y S eft is a measurable set, then the notations L(;,f)( Y), Y), B!JfJl.f)( Y), Hep. q)( Y) [L(:,( Y), OfB)( Y), BC?,)( Y), H(s)( Y)] will be used for the spaoes of differential forms of bidegree (p, q) [degree 8] and with coefficients in LOO( Y), OO( Y), BOO( Y), BIA( Y), respectively. The spaces L(p,9')( Y), Bef".f)( Y), L(~( Y) and BG1,)( Y) become Banach spaces if we introduce the norm
<
0&",)(
IIJllo,Y:= supllf(z)ll, ZEY
where the norm IIJ(z)1I was defined in Subsection 1.6.3. If D c eft is an open set, then we denote by O~~~(D) [O~,t"(D)] the space of differential forms of bidegree (p, q) [degree 8] in D whose ooefficients belong toOJ:+GI(D). The forms in O(~;f)(D) [of.tlA(D)] are called O~;f)-forms [of.jCl-forms] or OJ:+GI-forms of bidegree (p, q) [degree s]. 1.8.S. Lemma. Let D be a bounded open set in eft. Then (i) For every bounded difJerentialJormJ on D, BDlis a OGl-form in D for all 0 Q; 1. Moreover, Jor every 0 < a < 1, there is a cOnlltanf O. < 00 such 'hal, for every bounded differential Jorm / on D, II B DIll IA. D :5: 0 CI 11/110, D' (ii) IJ! is a bounded OJ:-form on D, 1 :5: 1c :5: 00, then BDf is a OJ:+tII_!orm in D lor all
< <
0<(1<1. 01
Proof. (i). It follows immediately from the definition of BD that, for some constant 00 and all z, E ED,
<
IIBDf(z) - BDMII :s; Cd Ifllo,D}l
JIlf_-z~
-
I~~ -=- :~
I
da... ,
CED
where do'2ft is the Lebesgue measure in C". In view of Proposition 1, Appendix 1, it follows that, for some 01 00,
<
IIBDJ{z) - BDJ{E) II :5: 0211Jllo,D Iz - EIIIn Iz - Ell. Since, for all 0 < (X < 1, sup Iz - EI 1 --IIn Iz - "I < 00, we obtain assertion (i). 8. lED
(ii). We fix ZED and prove that BDJ is O~+tII in some neighbourhood of z. Ohoose a function X E O::'(D) such that X = 1 in a neighbourhood U. of z. Then it is olear that BD(l - X)fis 0 00 in It remains to prove that BDx/is OJ:+", in U•• By (1.8.7), we can assume that J is a (0, q)-form and
a•.
BDx/(E)
,.
= JX(~)J(C) 'ED
1\
D,-l(E, C) •
52
1. Elementary properties
Let fCC) = ~' f,(C) del and III-g
Df-l(~' C) =
~'
CPJK(~' C) de K "w(~)
IJI-g-l
"diJ •
IKI-n-q-2
Then BDxf(~) = ~' kJ(~)
dEJ , where
J
f x(e)f,(C) Cf'JK(¢:~) del" deK "w(C) ~' Jx(C + ,) f,(C + ~) CPJK(~' e+ ~) del" deK
hA~) = ~' I, K
=
~ED
I,K
"w(C)·
~EC·
+ +
The functions CflJK(;, ~ E) are independent of Eand locally integrable with respect to C. Therefore, by differentiation under the sign of integration, we obtain that BDXf is a Ct-form. If " = xI iX1+ft (xI E IR), 'V = ('VI' ... , 'V2n) is an arbitrary multi-index of non-negative integers with 'VI 112ft = k, and DV:= 8"/oxi' ... 0x'2':, then D"h J (,) =
+ ... + ~' f (D"(X!I) (C + ~) q;JK(~' C+ ~) dC I "de K I,K eEl;'''
= ~' f(U(x/,)
CPJK(~'C)
(e)
I,K eED
del" dfK
1\
"w(~)
w(C)·
It follows immediately from the definition of fPJK that, for some constant 0 all" wED,
lD"h.,(E) - D"hJ(w) I ~
~ c sup ~' I(D"{X!I) (C)I 1-1 i 'ED
1
<
00
and
II '1 - ~_ - ~ - I It; -
W_ld0'2n.
~12tt
IC - wI2ft
~ED
As in the proof of part (i), this implies that the functionsD"hJare Ct¥-functions for all
O
e»
A t"-valued Cl-function w(z, C) = (w1 (z, C), ... , w,,(z, defined for C in seme neighbourhood of aD and zED is called a Leray map for D if (w(z, e), C- z) =t= 0 for all
(z,
C)
ED X aD.
(1.8.11)
Let w(z, C) be a Leray map for D and suppose that the boundary of D is piecewise (Jl. Then we define
w(z, e)
fJtI1(z, C,).) := (1 _ A)
(w(z,
e), t; -
_+ A
z__ z, e- z)
C(C -
z)
(1.8.12)
for all zED, 0 ~ A ~ I, and C in Borne neighbourhood of aD. Then, by (1.8.11), for every fixed zED, the differential form w,(w(z,
C))
"w(t;) :=
(w(z, C), C - z)-
i (_l)j+l __(w(z, C), CC)-
1\ d~wt(z, e)
w,(Z,
i-I
"w(C)
z)n k+J
is continuous for Cin some neighbourhood of aD, and the differential form
ft
WC,A(1J (Z,C, ).)) "w(C) : = ~ ID
}-1
(_1)1+ 1 fJj(z,
C, A) 1\~.'\1Jr(z, C,).)
"w(C)
i=t=j
is continuouB for Cin some neighbourhood of 8D and for 0 ~ A ~ 1. Therefore, we can define:
1.8. Leray ma.ps and the operators B 6D , B D, L~D and RfD
If f is a bounded measurable complex-valued function on
If I
(L:Df) (z) : = (n -. I)! fl(C) w~(w(z, Cn 1\ ro(C) , (2nl)" (w(z, C), C - z)" CeaD is a bounded I-form on oD, then we define
(~_- ..~) !
(RM) (z) : =
JI(~)
A
~.1 ('1"(z, ~, 1»
aD,
53 then we set (1.8.13)
zED.
A
co(~) ,
ZED.
(1.8.14)
CEaD
O~A~1
Remark. For every open set Dec 0" with piecewise ()1-boundary, w(z, C) is a Leray map. Then LfD = BaD and RfD = O.
: = C- Z
1.S.7. L:D and R:D for differential forms of arbitrary degree. Let D c::: c (f)" be an open set with piecewise ()1-boundary, let w(z, C) be a Leray map for D, and let 1]tD(z, C, A) be defined by (1.8.12). We define w'(w(z,
C)) := 1:"
(_1);+1 Wt(z,
C) 1\ 0z.CW1(Z, C) ~+j
j-l
and W'(1'}W(z,
C, ).)) :=
"
~ (_I)j+l f}j(z, ;=-1
-
C, A) 1\ (8z•c + dA) 1]:(z, C,).) . k+j
By (1.8.11) the differential forms
w'(w(z, C)L~ w(C) (w(z, C), C - z)"
and
w'(fJtD(z, C, 1)
1\
wee)
are continuous for (z, C,).) E D X aD X [0, 1]. (Moreover, for every relatively compact open set Dec D there is a neighbourhood UaD of eD such that these forms are continuous on D X UaD X [0, 1].) Therefore, we can give the following definition (for the definition of integration with respect to C and (C,1), respectively, see Subseotion 1.6.1) : If I is a bounded differential form on aD, then we define
(L'tDf) (z) := (n -.1)! ff(C) (2nl)" Ce8D and
(R:'Df) (z) := (n -.1)! (2nl)1I
J
1\
f(C)
w'(w(z, C)) 1\ ro(C) , (w(z, C), C - z)"
1\
w'(fJtD(z, C,
A»
1\
ZED,
ro(C) ,
zED.
(1.8.15)
(1.8.16)
CeaD
O~A:;;l
Now we shall prove that (1.8.15) coincides with (1.8.13) if f is a function, and (1.8.16) coincides with (1.8.14) iff is a I-form. Consider the unique representation (n -
I)! -,
- - ' - n - W (rJ
(2nl)
tD
(z, C, J.)) I\W(C)
11-1
= 1:
~(z,
C, A) ,
g-O
where .Q~(z, C, J.) is of bidegree (0, q) in z and of bidegree (n, n - q - 1) in (C, A). Let be a bounded differential form on eD and f = ~ f(r •• ) its unique decomposition in (r, s}-forms. Since l(r,8)(C) 1\ w(C) = 0 if r 0, then
f
>
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:;;A.=-1
Hence
R tIJ_, 8DJ =
f
CE8D
!(O,des!) (C) " .Q(des!)-l(Z, C, 1)
if
1 ~ degf :::;: n , (1.8.18)
{ 0;:;A.=-1 o otherwise.
Similarly, using the unique representation 1) I w'(w(z~~)) " oo(C) = (2ni)" (w(z, C), C- z)"
(n -
where that
"r,1 .QtD(z C)
.-0
g.
'
n:(z, C) is of bidegree (0, q) in z and of bidegree (n, n L:J
We have
=
J !(O,desn(C) " .o:8Il(z, C) rCe8D
l
0
~(z, C, 1)
if
- q - 1) in
C, we obtain
degJ ~ n - 1 , (1.8.19)
otherwise. =
(n -.
-!]! i
(2m)" i-I
(-I)J+I 1Jj(Z, C,l)
1\ (8, + d.t) 1Jf(z, C, i..) "oo(C) .
~+J
In view of the factor co(C) , the operator 8, on the right-hand side can be replaced by de. Hence
0:0 (z, C, l) =
1) I 00; .t(1}tII(z, C, 1) " oo(C) (2m)" ~.
(n -
and it follows from (1.8.19) that (1.8.16) coincides with (1.8.14:) if! is 8. I-form. Similarly, it follows from (1.8.18) that (1.8.15) coincides with (1.8.13) if ! is a funtion. Remark. If w(z, C) = C- i, then differential form! on aD.
1.9.
L'tIJ/ =
BaD! and
R:D!
= 0 for every bounded
The Martinelli-Boehner formula
In this section we prove the Martinelli-Bochner formula. This is a first generalization of the Cauchy-Green formula (1.1.1) to several complex variables. 1.9.1. Theorem (Martinelli-Bochner formula). Let D c c: e" be an open set with piecewise (J1.bouftda'1I, ami let ! be a continuous junction on D such that 8! i8 alBo contin'U0'U8 on D.l) Then
!=
BaD! - B D 8! in D, where BaD and BD are the imegral operators deJined in Subsection 1.8.3.
0.
al
whioh is defined in D in the sense of distributions is continuouB in D and oontinuou8 oontinuation to D.
1) This means t
admits
(1.9.1)
1.10. The Leray formula
53
Proof. Fix zED and set 0«(;) : = (11. - I)! roc(C (2m)" <(; -
-
"ro(C) .
i)
z, (; -
z)ft
Since, by Proposition 1.7.1, dO((;) = 0 in D " z, and since 8/(C) "ro(C) = 0, it follows that d(!«(;) O{(;)) = a/{(;) in D Consequently, for every sufficiently small 6 > 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<-
= -11.1 26
"n"
dxt " ...
,,
1'-&1<·
Now we consider the integral
J
(/«(;) - I(z)) O{(;)
=
(11. - I)! - - - - - --(2m)" 621&-1
I
f
"'-1:1 =11
where 0
(1«(;) - I(z)) coc(C - - -- -z)- "co(C) --- • IC - zl
IC-.I-.
1C-1:1 =8
Since the form
J
"ro(C))/IC - zl is bounded in D, this implies the estimate (f(C) - J(z)) 8«(;) I ~ a max If«(;) - !(2:)I ,
(coc(E -
z)
It-&I:;aa
< 00 is a constant which is independent of 6. This completes the proof. •
1.9.2. Corollary. Let Dec eft be an open 8et, and let I be a continuoUB/unction on that is holomorphic in, D. Then
1 = BaD!
in
D,
D
(1.9.2)
where BaD i8 the integral operator defined in Sub88ction 1.8.3.
Remark. For 11. =-= 1, ro,(C - i) Cauchy-Green formula (1.1.1).
= C- i
and the Martinelli-Bochner formula is the
1.10. The Leray formula For n ~ 2 the kernel in the Martinelli-Bochner formula does not depend holomorphically on z, because then (C - i)/IC - i1 2,. is not holomorphic in z. Therefore, in the present section we give a generalization of the Martinelli-Bochner formula whioh is
56
1. Elementary properties
obtained by replacing the map e by a more general Leray map w(z, e). This generalization was found in 1956 by J. LERAY and forms the basic idea in the constructions of formulas for solving the inhomogeneous Cauchy-Riemann equations presented in this book. For important classes of open sets (for example, for strictly pseudoconvex open sets) this Leray map can be chosen to be holomorphic in z and, moreover, to satisfy appropriate growth conditions.
z
1.10.1 Theorem (Leray formula). Let D be a bounded open set in (fJn with piecewise (J1-boundary, and let w(z, e) be a Leray map for D. Then, jor every continuous function f on D 8uch that 8f is a"lao continuou8 on jj, we have
I
-
-
= L:Df -
(1.10.1 )
BaD 8f - B D 81 in D,
where LaD, R:Da.nd B D are the integral operators defined in Subsec.lions 1.8.6 and 1.8.3. Proof. In view of the Martinelli-Bochner formula (1.9.1) we only have to prove that (1.10.2)
Fix zED. Since <1jW(z, e, A), e - z) = 1 for 0 ~ A :::; 1 and e in a neighbourhood of aD, Proposition 1.7.1 gives dc,A[w~.A(1jW(z, e, ).)) 1\ w(e)] = O. Since af(C) 1\ w(C) = 0, this implies that
dc,A[f(e) w~.,,(1'}W(z, C, A))
1\
w(C)J = 8f(e)
1\
w'.,,(1jW(z,
c, A))
1\
w(C) .
Now we apply Stokes' formula to f(e) W~,A(1jW(Z, C, ).)) 1\ w(C) on aD x [0, 1]. This implies (1.10.2) if we take into account that, by Proposition 1.7.3, w' ( W(z C A)) C.A
1j
"
1\
w(C}'
=
£0' (
1).-0,
w(z, e) _ ) <w(z, e), C- z)
1\
w(C)
= ~~(w(z, C)) _~~(C) <w(z, C),C - z)n
and
1.10.2 Corollary. Let D be a bounded open set in
with piecewise OI-boundary, and let w(z, C) be a Leray map lor D. Then, lor every continUO'US function f on D that i8 holornorphic in D, f
=
(fJn
(1.10.3)
L:Df in D,
where L~D is the integral operator defined in Sub8ection 1.8.6.
z,
L:
&mark. If w(z, C} = C then D = BaD and RfD = O. Therefore, in this case the Leray formula (1.10.1) is the Martinelli-Bochner formula (1.9.1). 1.10.3. Remark. In the proof of Theorem 1.10.1 we did not use any information about the kind of dependence of w(z, C) on z, that is, we proved the following theorem: Let D and I be as in Theorem 1.10.1, let Zo E D, and let w(C) be a Oft-valued (Jl-map in a neighbourhood of aD 8uch that <w(C), C - 2:0) 9= 0 for all CE aD. Then I(zo} = (L:Df) (Zo) - (B:'D aJ) (Zo) - (BD aJ) (zo)' (Here the numbers (L:DJ) (zo) and (R:D 8f) (Zo) are defined by (1.8.12)-(1.8.14), where instead of w(z, C) the map w(C) is used.)
57
1.11. The Koppelman formula.
1.11. The Koppelman formula In the case of one complex variable we need only integral representation formulas for functions. For several complex variables we also need such formulas for differential forms. Here we can restrict ourselves to (0, g)-forms, because every (p, g)-form can be understood as a (0, g)-form with a values in certain holomorphic vector bundle, which will be considered in Sect~ons 2.10-2.13 and 4.12. In the present section we prove the Koppelman formula, which is a generalization of the Martinelli-Bochner formula to differential forms. The Koppelman-Leray formula, which is a generalization of the Leray formula as well as of the Koppelman formula, will be proved in the next section. 1.11.1. Theorem (Koppelman formula). Let Dec (/)11 be an open set with piecewise C2-boundary, and let f be a continuous (0, g)-form on D such that 81 is auJO continuous on D, 0 < g ~ n. Then the form8 BaJ, BD 8/, B D/ and OBD! are continuous in D, and
-
we have (-1)(1 f
= B;wf - BD Sf +
SB D!
in D.
(1.11.1)
(Here BaD and BD are the integral operators defined in Subsection 1.8.4.)
°
Proof. If g = 0, then, by (1.8.7), BD! = and (1.11.1) is the Martinelli-Bochner formula (1.9.1). Therefore, we must prove the theorem for 1 ~ q ~ 11, only. It is trivial that BaD! is continuous in D, and it follows from Lemma 1.8.5 that BD o! and BD! are ~ontinuous in D. Therefore, it !emains to prove that in the sense of distributions SB D/ = (-I)f! - BaD! BD 8/ in D. That is, wo have to prove that, for every Coo-form v with compact support in D,
+
(-1)9 D
J Bn!
1\
6v
= (-1)1
Jf
J BaDI
1\ V -
D
+DJ BD 8/ 1\ v.
v
1\
D
(1.11.2)
Since, by (1.8.7) and (1.8.8), BaDland BD Slare of bidegree (0, g) and BD/is of bidegree (0, q - 1), we can assume that v is of bidegree (11" 11, - g). Using the notation I)! ~;,dC z) ~~~0. (2ni)1I IC - 1,1 211 and the definitions of BaD and B D , (1.11.2) can be written fJ(z,
C)
(-1)(1
-
:=
-
(11, -
J
I(l;,)
1\
0(1"
C)
1\
8v(z) = (-1)9
J
j(l;,)
1\
fJ(z,
C)
1\
v(1,)
(C,z}eaD x D
+
r
J I(z)
v(z)
1\
zeD
81(C)
1\
0(1"
C)
1\
V(1,) •
(1.11.3)
Set 8(1" C) : = ((11. - 1) !j(2ni)n) w;,c(f - z) 1\ wz,c(l;, - 1,)/IC - 1,1 2". Then, by Proposition 1.7.1, dz,~O(z; C) = 0 for l;, 9= 1,. Since all monomials in the difference 8(1" C) - fJ(1" C) contain at least one of the differentials dzl , ... , dZ1l and v(1,) contains the factor dzt 1\ ... 1\ dZ1h it follows that d,.c(fJ(z,
=
C)
1\
v(1,))
= d"., (8(z, l;.) "v(z»)
(_1)211-1 O(1"l;,) I\dv(z)
Moreover, dt(C) 1\ fJ(1" l;,) = 8/(C) factor dCI 1\ ... 1\ dC". Hence d.,,(/(l;.) 1\ 0(1" C) 1\ v(1,») = 8/(/;,)
1\
fJ(1" l;,)
1\
1\
= 0(1"
-O(z,C) I\av(z)
C)
for
v(z) - (-1)1 I(C)
1\
C =F 1"
0(1" C)
for
C=f=z.
(1.11.4)
because 0(1" ·C) contains the
1\
8v(z)
for
l;, =1= z.
(1.11.5)
58
1. Elementary properties
'Ve consider the open sets U. := {(C, z)
Since
BUPP
c: c D~
v
6(D X D "
X (fJfl:
E (fJfl
Ie -
zl
< e} ,
then, for sufficiently small
B
> 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.
< <
Proof. By (1.8.19), L':»f (n - I)!
i
(2ni)" i-I
< <
3£/" g;, where Q~ is the sum of all monomials in
(_1);+1
wl(Z,
(w(z,
e) _
e), e-
1\ (Sz
z)" ~+j
+ 8e) Wj:(Z, C)
"co(e)
which are of bidegree (0, q) in z. However, since w(z, C) is holomorphic in z, these monomials va.nish if q ~ 1. Therefore, L'fDJ = 0 and (1.12.7) follows from the Koppelman-Leray formula. (1.12.1). It is clear from (1.12.7) that u = Tffis a solution
61
Notes
of Bu = J if 8j = O. The assertion about the smoothness of u follows from Lemma 1.8.5 (i) and the obvious fact that BaD! is 0 00 in D . •
Notes The Cauchy-Green formula for the first time appeared in POMPEIJU [1] (1904). This formula is one of the basio tools in the theory of functions of one complex variable. Examples for profound applicatiolls of this fonnula can be found in CARLESON [1] and VITUBKI}q [1]. Theorem 1.2.1 was proved in 1951 by GROTHENDIEOK (see DOLBEAULT rl]). Theorem 1.2.2 was found in 1906 by IIARTOGS [1]. The proof given in Section 1.2 is due to EBRENPBEIS [1] (Bee also HORMANDER [1]). The characterization of domains of holomorphy in Theorem 1.3.7 is due to CARTANJTHULLEN [1] (1932). Plurisubharmonio functions were introduced by OXA [1] (1941) and LELONG [ll (1945). The presentation in Section 1.4 pa.rtially follows HORMANDER [1]. Theorem 1.4.14 is due to NARASIMHAN [1]. Theorems 1.4.15 and 1.4.18 are proved in HARVEYjWELLS [1, 2]. Theorem 1.4.21 is due to HARVEY,f\VELLS [1] and NARASIMHAN [1]. The presentation of pseudoconvex sets in Seotion 1.6 partially follows Ht>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:'<~
For p
>
0 we denote by ~(D) the space of holomorphic functions sup f If(rC)IJ' Id~1 0<,<1 "1-1
<
I in D such that
00 •
Theorem (CABLBSON [2], for a. simpler proof see HA.VINjVINOGRADOV [1]). Let ft be a oomplex measure on D and let p > O. Then ~(D) i8 continuously imbedded in LP(ld.ul, D) (by the naturaZ map) il and only il p. iB a Oarleson measure. In particular, this implies that every Carleson measure Il on D defines a continuous linear functional h - f h(C) dp.(C) on ~1(D). Conversely, it was proved by V AROPOULOS [1] that D
every continuous linear funotional on ~1(D) can be defined by a Carleson measure. One of the important results in the theory of flJllctions of one complex variable, whose generalization to the oase of several complex variables is an open problem (op. point 14 b) in Exercises, remarks and problems at the end of Chapter 2) is the following Corona theorem of CARLESON [2,3]: Corona theorem. Let 11'"'' 1ft be bounded holomorphic lunction8 in D BUch that
inf.ED I/(z) I > 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)
+
63
EXf'reises, remarks und problems
u in D whieh is bounded on uD such that -
--
(11
1
iJu _
C'.lP2 _
cz - 1/12 t; oz
12
OqJl)
h cz '
and it remains to set hl : = (rpI/fl) -!..- Uf2 and 11.2 : = (lPz//2) - 11ft . • \Volff'R proof. For simplieity let n =- 2. Set 1.J!1 : - 1~/lfI2. Thon 111f1 1- la'l'2 = 1. Set [.I 1 := {z ED: ft(z) =l= O}, qJ := o(tf't//2)/ez in U 2 awl f{J : - -- o(lp2/ll )/Sz ill VI' It remains to find a function u. in D such t,hat. 7i is bounded on S]) and (ju/(iz = cp, because then we oan set = "Pl - uf2 and h2 : = 'IJ'2 - 11u. To find u we take the solution 1l'C) of 8'Uo/ 8z = lP which is given by the Cauchy-Green formula und look for tt holornurphic function 9 in D such that 110 -;- g is boundpu 011 cD. This means, cHHcntiully, WP have to estimate the norm of UoiaD as a linear functional on ~l(D). This can ho reou('ecl to t.he proof of the fact that (1 - Izl) 1'1'1 2 uz 1\ dz and (I - :zi) IO«r/Ozl dz 1\ ciz are Carlt'8on mt:'llSlll'€,S I1ml therefore definf' eontinuous linear functiono.ls on ,\)l(D) . •
'''t :
9. (Por ..JAKOV [2J). Let D 1, D2 C C (fJl be open discs. l"lrove that, for every bounded holomorphic function f in 1)1 n D 2 , t here are bounded holomorphio functions It in DI such tha,t f = 11 - 12 in J)1 n D2 •
10. Let D ~ ([JIbe an opf'n set and let A bo an n X n matrix whoso ontries are continuous complex-valued functions in D. Consider t.he equation
ou/oz
I
..1 Ii
.
-
f,
(lit.)
where f is a given continuous en-valued function in D. Prove that (**) can be solved locally. Hin t. Use that t.he operator for Aolving e11/0=' = 1given in Theorem 1.1.3 is "small" if D is "small". II. Let R I • R2 C C (j)l be open rectangles such that RI U Ra also is a l'eotangle. We denote by L(n, 0) the space of all complex n X n matrices, and we denote by GL(n, C) the group of invertible matrices in L(n, 0). a) (Cartan's lemma, CART AN [l ]). Prove that, for every eontinuous function A: RI n R2 - GL(n, C) that is holomorphic in Rl n R 2 , there are continuous funotions AI: Rt - GL(n, e) which are holomorphic in B1 such that A = A 1 A 2 • Hin t. The group of all continuous GL(n, C)-valupd functions OIl Rl n R2 which aro holomorphic in RI n RI is connected; consequently, A = (exp .1.,,11 ) ... (cxp 1l-1 k ), whore 1.1t : Rl n R. - L(n, C) are continuous on Rl n R2 and holomorphic in R1 n R 2 ; consequently, A can be approximated uniformly on Rl n 1-:2 by hololIlorphic fUllctions in f!)l with values inOL(n, C); consequently, we have to prove the IiAsertion only for the case that A is sufficiently closo to tho unit matrix; this can be done by means of the implicit function theorem in Banach spaces and Exercise 3 c). b) Let 0 < (X < 1. Provo that, for every (X-Holder continuous funotion A: Rl n RI - GL(n, C) that is holomorphic in RI n 112 , there aro (X-Holder continuous functions Aj: Hj - GL(n, ([J) which are holornorphic in R f such that A -= A 1 A 2 • 'Yarning. The set of holomorphic funct.ions in a neighbourhood of Rl n Ra is not dense in HfJi{RI n R 1 ) n ()( RI n R 2 ). Hi n t. !J rove and use a version of the Implioi t funotion theorem in Banach spaceR , .... ith two norms. (For allotlH"r proof, see, for example, LEITERER [1].)
12 (ROHRL [I]). Let
[ij ~
C'l, j E J, be open sets, let D : = U lIf' and let Aif: U, n U t jeJ
(sec Exercise 11) he a systAm of holomorphic functions such tha.t A(1A,~ = Aik in U t n Vj n U lc for all i, j. k E J. Prove that there exists a system of holomorphio functions At: lIt - GL(n, fJ) such that Ai1 == A,Ajl in Vi n V , for aU i, i E J. Remark. This statement admits a profound generalization to the caso of 8evl'ral complex variables (GRA tJERT [I, 2, 3]. see also C \RTAN [2]). For simple direct proofs of the special case above sec, for e~alllpIP. GRAUERT/HE1tIMERT [I] and GOHH}~RO/LEITKRlm, [1,2,3]. -+
GL(n,
(fJ)
64
1. Elementary properties
13. Let L(n, e), GL(n, e) be as in Exercise 11. Prove that, for every open D ~ QJl and every continuous A: D - L(n, e), there exists a continuous solution V: D - GL(n, C) of
au
8z -
UA
in
( •• *)
D.
Hint. Solve (***) locally; if VI' V,. are two local solutions, then VIU;l is holomorphic; use Exeroise 12. 14. Prove thnt (**) oan be solved globally. Hin t. Solve (lv/a;; = UI, where V is a solution of (*. *), and set u : = V-IV. 15. Remark. Let D ~ e1 be the unit disc, and let %1: E D (k = 1, 2, ... ) be a sequence suoh that 00 'if zl - z inf IBI:(Zt)1 > 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 <,very holomorphic function in some neighbourhood of (<«0) X oD) u U (u(t) X D) ~
O
q;n. X em admits a holomorphic extension to a neighbourhoo(l of U (u(t) X D).
Hin t. Use the proof of Theorem 1.3.5. O~'~l 24 (KNES F!R [IJ). Let D 1 • D2 ~ en be pseudoconv{'x opE"n B~ts with OS-boundary. ::;;uppose that th(' intersection aD1 n 6D a is transversal. Prove that Dl U Da is pseudoconvex if ami only if flD1 n aDa is a complex sulnnanifold of f!Jft. 25. Let Dn CC QJfI,l h~ th~ Ret of all complex n X n mn,trices Z such that ZZ* - 1 is strictly negative-c1efinite. Here 1 is the unit matrix and Z* = (z1~) if Z =- (zv) (i = row index). a) Prove that Dn is convex.. b) Prove that ])tI is not strictly pselldoconvnx if n > 1. 26. Let Dec {J'n be an analytio polyhedron, and let 11 , ••• ,iN h~ holomol'phic functions in a neighbourhood U of i5 such that D = {z E U: 1/,(z)1 < J, i = 1, ... , N}. Set rn-l := {z E U: tbel'e f'xiflts 1 ~ i1 < ... < in-l ~ N sllch that 1/;1(2:)1 := ... = Ifjn_l(z)1 = I}. Prove that every holomorpbic fWlction f in some neighl)ourhood of r n - 1 admits B holoffiOl'phic extension to D. 27. Let Dec (f)fI. be an open set with Ol-boundary, and let f = (/1' ... ,In) be a oollection of holomorphic functions in some neighbourhood of D satisfying t,hs following oonditions: (i) f 9-= 0 On BV. (ii) i h, bibolornorphic in somo neighbourhood of every point zED where f(z) = O. Then
(n -_l)!_ fW'(f)
1\
w(f)
1/12ft
(2;d)" 81)
is the number of zeros of fin D . .28 (BOCHNER [1], HARVEYJLAw~ON [1], DAU1'ov/KYTMA)lOV [1], WEINSTOOK [1])- Let Dec (f)ft be a domain with Oil-boundary, a.nd lot f be B continuous function on aD. Set /+(z) : = (Bani) (z) foT' zED and L(z) : = (Bew/)(z) for::: E (fJtI. "if. (For Z E en. " i5 we (lpfine (HaDt) (z) also hy (J .8.2»). a) Let 1'(z), z ( aD, I.m the unit exterior normal v~ctor of aD at z. Prove that lim [1+(<1: - GV(z)) - 1-(z &-+0
uniformly in := u
E
Henkin/Leltel"e1'
aD.
+ ev(2:))] =
f(z) ,
66 b) H
1. Elementary propertieB
wJ I
A
fig
=
0 for every cfn,n-2)-form 9 in some neighbourhood of aD, then
_ _
f+
is
holomorphio in D and f _ is holomorphio in QJ" " D. If, moreover, ([JfI. '" D is conneoted. then 1= f+. 29 (E. LEVI [I]. H. LEWY [I, 2]). Let D CC QJ'l be a striotly pseudooonvex open set with OIl-boundary, and let ~ be a striotly plurisubharmonio Oi-funotion in some neighbourhood 8 of 15 suoh that D = {z e 8: C?(z) < O} and d~(z) 9= 0 for z e aD. Let f be a 0 1 _ funotion in some neighbourhood U of E E aD. Define Lf : =
:~
:L - :~ :~
in
U.
0 if and only if f I A 8g = 0 for every ot~. o)-form 9 with com8DnU paot support in U. (This is true also without the hypothesis that e is striotly plurisubharmonio.) b) Let c5 > 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.
< <
< <
Proof. Since wQ(C) is independent of z (and, therefore, holomorphic in z), the theorem follows from Corollary 1.12.2 and Corollary 2.1.4 . • 2.1.6. Corollary. Let D c on be an open set and f a continuoU8 function in D. II = 0, 1, ... , 00, then f is a O"+IS-function in D for all 0 eX 1.
6f is a 01:-form in D, k
< <
au
Proof. By Theorem 2.1.5, the equation = Sf can be solved locally with a O~+IS_ function u when 8j is 0". By condition (i) in Theorem 1.1.5, then u - f is holomorphic and, therefore, f is also O"+IS . •
2.2.
Hlild er estimates
In this section we prove 1/2-HOlder estimates for the solution of ou = f given by (2.1.4) in strictly convex open sets with 02-boundary. The estimation of the operator BD is simple and was already given in Lemma 1.8.5. The estimation of the operator R:~ is more complicated. Later we need such an estimate also for D , where Dec is a 0 1 strictly pseudoconvex open set and W is a speoially constructed Leray map for D. Therefore, in the following lemma, we give the needed estimate for a general Leray map satisfying some conditions (which will be fuHilled in both cases).
B:
en
2.2.1. Lemma. Let D cc (f)n be an open set with Ol-boundary, and let U aD and UJj
be neighbourhoods of oD and D, respectively. Suppose w = w(z, C) = (w1 (z, e)' ... , w.(z, e)) is a collection oj C1-functions defined for (z, C) E UD X U aD with the following properties: (i) w(z, C) depends holomorphiclilly on z E UD and (consequently) dcw(z, C) depends holomorphically on z E UD' (ii) w(z, e) is a Leray map for D (cp. Subsection 1.8.6). (iii) For every point ~ E aD there are a neighbourhood U I 0/ and real 01-functiona t1(z, C), ... , t2n - 1(Z, C) defined for z, CE U e such tha' the following conditions are fulfilled: (a) For every fixed z E U e, ~(z, .), ... , t2"_1(Z,.) are real coordinates on oD nUt. (b) For some t5 0 and all ZED n U e, CE aD nU" we have the estimate
e
>
I(w(z,
e), e - z)1
~ t5(1~(z,
e)1
+
It(z, C) II
+ dist (z,
(2.2.1)
aD) ,
2n-l
where It(z, e)1 2
= :2: Itt(z, CH Il and dist
;-1 Then there is a constant C
<
00
(z, aD) = inf {Iz - vi : v E oD}.
8uch that,for every continuous differentialfarmf on D,
IIR:Dflll/2,D ~ Ollfllo,D
(for the definitions of 11·11I/2.D and
B:
1I-Ilo,D see Section 1.8).
Proof. If n = 1, then D = O. We can therefore assume that n ~ 2. For simplicity We write tP := (w(z, e), C - z). Then by definition of R:D (see Definition 1.7.6 for the notation det l • n-1) (2ni)n(R:Dj) (z) =
f
f(e)
CEaD
J\
det 1,n_l (1]fD(Z,
C, it),
(d A + ~,c) 1]W(z,
c, it))
J\
ro (C) ,
O:iA:;i;l
(2.2.2)
70
2. The a-equation and the ,,fundamental problems"
where 'TJ
III
(z
-
,.
~
, ,
(1
=
-
=
Since 8.(l>
~ 11.)
=
0, 8.w
(d, +'ii•. ,)
~ w(z, C) 11.) - -
-
(l>(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.
\
et.
,
(2.2.11)
3DnUE
Consider (2.2.11). Let U t and t1(z, C), ..• , '2,.-1(Z, C) be as in hypothesis (iii) of the lemma, and let q;(z, C) be the function defined for z E U t, CE 8D nU, such that, for every fixed z E U e,
C) /\ .•. /\ d ct 2n - 1 (z, e) on eD n U e • Then, for some" > 0 and all z E U" CE eD n U" we have the estimate Ie - zl ~ " It(z, e)l, and it follows from (2.2.1) that, for zED nUt, d0'2n-1 = cp(z, e)
J
d0'2"-1
I cp,"-' IC -
J - t5"
dc~(Z,
zI2.+1
3DnUe
S ~
aDnU~
C) d cfl.(Z, C) /\ ..../\ d,t 2,,_1(z, C) .... __ (It1(z, C)I + It(z, C)1 2 + dist (z, 8D))n-. t(z, C)I 2I + 1 • q;(z,
(2.2.12)
72
2. The a-equation and the "funde.mental problems"
r,
After shrinking U E if necessary, we can find constants R < 00 such that Iq;(z, ~ r for z E U e and, E aD n U e, and, for every Z E U e, the surface fJD n U e is diffeomorphically mapped by, --* (~(z, '), ... , t2,.-1(Z, ')) onto an open subset of the ball with radius R centered at 0 E /R2n-l. Therefore, it follows from (2.2.12) that
')I
f
~
d0'2,.-1
zl~+l -
1<1>111.-·" -
aDn Ue
f
r c5y ZE
dx1
(Ia;1
A ••• 1\ <1.1:'211.-1
+ Ixl2 + dist (z, 8D)t-· Ix1 ,+1 . 2
8 211.-1
Iz\
Since
(Ixll
+ Ixl + dist (z, 8D))n-'lxl 2s +1 ~ (lxII + IxI2 + dist (z, 8D))2IxI2n-s , 2
estimate (2.2.11) now follows from Proposition 3 (ii), Appendix 1. In the same way we obtain estimate (2.2.10) if we use part (i) of Proposition 3, Appendix 1. •
2.2.2. Theorem. Let D c: c: there exists a con8tant 0
QJn be a
0 2 8trictly convex open set defined by (2.1.1). Then
< 00 such that, f01' every continuoU8 differential form fonD,
IIR:Af + BvflllJ2,D ::; 0 Ilfllo,D . In pa'I'ticular, iff is a continuous (0, q)-form on D such that of then the solution of 6u = f given by (2.1.4) admit."] the estimate
=
0 in D, I :::: q :::: n,
Il ulll/2,D ~ Ollfllo.D (for the definition of
11·1I1/2.D and 1I'lIo,D see Section
1.8).
<
<
Proof. By Lemma 1.8.5 (i), for every 0 < ex 1, there is a constant Oar. 00 such that IIBDfllar..D ~ OClllfllo.D' Therefore, we only have to estimatf' IIR:~fIII/2.D' To do this we prove that we(C) = 2(8e/8,v ... , 6(!/6Cn) fulfills conditions (i)-(iii) in Lemma 2.2.1. (i) is clear, because w e(,) is independent of z. (ii) is Corollary 2.1.4. It remains to show condition (iii). Let XI = xJ (C) be the real coordinates of, E QJft such that = xJ{C) iXj+n(')' Set t1(z, C) := 1m <w,,(C), C - z) (1m := imaginary part of). Then
'1
t1(z, C)
+
=
i
J .. 1
[8!?(C) Xj+n(C - z) 8x 1
8(!(C) Xj(C OX; +"
Z)]
and, therefore,
d,~(z, C)I,-~
=
1'_11
1\
i
;-1
[8(!(Z) dxj +,,(z) _ C!?(z) dXI (z)] . 8Xt 8xj+1I_
Hence dctt(z, ')
d(!(z)
=+ 0
for all
Z
EaD;
(2.2.13)
(2.2.14)
in fact, by (2.2.13), the coefficient of dXI(z) 1\ dx;+,,(z) is -((ce(z)/6Xt)2 + (oe(z)/ 8Xj+,,)B), which cannot be zero for all I ~ j ~ n, because de(z) =l= O. Fix E E aD. Then, by (2.2.14), we can find a neighbourhood U e of ~ and Ol-functions ~, ... ,t2n - 1 on U E such that, for every fixed z E U~, ~(z, .), t2 , ... ,t2n - 1 are coordinates on oD nUl' Set tl (z, C) = tl (C) - t1(z) for z, C E U e, j = 2, ... , 2n - 1. Then, for every fixed z E U e, ft(z, .), ... ,t2,._1(Z,') are coordinates on aD nU~. It remains to prove estimate (2.2.1). Since t1(z, z) = 0, after shrinking U!, it follows from Taylor'S formula that for some ~1 0 I' - zl ~ <51 11(z, C)I for C, z E U; .
>
2.3. Strictly pseudoconvex open sets
73
By Lemma 2.1.3, this implies that, after shrinking U e' for some da > 0, I
+
+
c\ >
n
2.2.3. The Holder exponent 1/2 is the best one possible. In this subseotion we present an example of E. M. STEIN which shows that for the case that D. is a ball in tJ]a, the Holder exponent 1/2 in Theorem 2.2.2 is the best one possible in the following sense: There exist continuous (0; I)-forms / on jj which are a-closed in D such that, for every Ci 1/2, there does not exist a solution of au = f satisfying the estimate Ilull(ll,D 00. Example. Let D := {(ZJ, Z2) E Q)2: 1~12 + IZ212 < I} be the unit ball in q)2. Let 1n (Zt - 1), %1 E [I, + ooL be the hranch of the logarithm with 0 argument of 1n (Zt - 1) < 2n. Define by setting
> <
<
f( Zv
%2
) :=
1
lIn
cli2 (ZI -
°
1)
if
(Zl,Z2)ED"-.{<1(I,O)},
if
(ZI' Z2) =
(I, 0)
a (0, I)-form on D. Since In (Zl - 1) *0 for Zt« [1, + (0), / is a Coo-foml on D"-.{(I,O)}. Since In(zl-l)-+oo when Zt-+I, f is continuous on D. Since I/In (ZI - 1) is holomorphic in D, we have 0/ = in D. However: Proposition. If (.\ 1/2, then there does not exist a/unction u in D such that cu = f and Ilull""D < 00. _ Proof. Suppose'11 is a solution of cu = land, for some Ci 1/2, Ilulllll,D 00. Since 8(z2/In (~ - 1)) = f, the function u - z2/1n (Zl - I) is holomorphic in D. Let 0 < 28 < 1. Then the circles {(ZI' Z2) E {}2: ZI = 1 - f, IZ21 = V';-} and ((Zt,~) E tJ]1: ~ = 1 - 2e, IZ21 = Ve) are contained in D. SinC'c u - z2/ln (Zl - 1) is holomorphic, Cauchy's formula gives Z2 dZ 2 2.ni8 "' u(1 - c:, zz) dZ 2 ::-= In (-e) = In( -=-e)
°
>
<
>
Iz,1
and
rr
=
f
f
IZal =
8-
u(1 - 2e,
Z2)
dZ2
=
Va 2.nie
In (-2e)
.
\z.I=V; Since I lui IIll, D
°
< 00, this implies that for some constant C < 00 and all < 28 < 1
Ih0 ~') -1~I~2') I< C. -,,2. o
Hence, for all 0 < 2e This is impossible. •
2.3.
< I, ]n 2 =
lIn (-2e) - In (-e)1 ~ Ce(ll-l/'1. lIn (-e) In (-28)1.
Solution of the a-equation with Holder estimat~s in 0 2 strictly pseudoconvex open sets in t!JR
First we prove Fredholm solvability (Lemma 2.3.1 below). To givc the precise formulation we need some notations.
74
2. The a.equation and the "fundamental problems"
Notation. Let D c: c: I)" be an open set, and let 1 ::::; q ~ n. We denote by FM;q-I)(D) the domain of definition of "8 as an operator from HU~g-l)(ii) into Oro,g)(D) (for the definitions of HM,2g _ 1 /D) and oro.fiD) see Section 1.8), that is, Flti.2g_1iD) is the space of all / E HU.2'_1)(D) such that 8/ (which is defined in D in the sense of distri. butions) is continuous in D and admits a continuous continuation to D. Set EM~fJijj) := 8FtJ~g-liD). We denote by Z?o,q)(D) the space of allf E O?o,fJ)(D) such that 8J = 0 in D. z?o.g)(D) is a closed subspace of the Banach space O?o.tJ)(D). 2.3.1. Lemma. Let D c: c: on be a strictly pseudoconvex open set with 02-boundary, and let 1 ~ q ~ n. Then _ (i) there exi8t& a bounded limar operatOT S Jrom the Banach space Z&,g)(D) (endowed with the norm 11·llo.D) into the Banach 8paceHt6~tJ-I)(i5) (endowed with the norm 1I·lIl/2.D) 8uch that S(Z?o.tJ)(D)) s Fl6;g-I)(D) and 80 8 = id + K, where id is the identity operator in Z?o.f)(D) and K i8 a compact operator in Zro.a)(D).l) (ii) EM~fJ)(D) i8 a clOBed (with re8pect to the norm 11·lIo.D) 8ubspace oj z~.g)(ii) oj finite codimenBion . Proof. By Corollary 1.5.25, there is a finite covering of D by relatively open sets VI' ... , Urn ~ D such that, for every j, there is a biholomorphic map h1 in a neighbourhood of V, such that h,(U, n D) is a strictly convex set with Oi-boundary. Therefore, by Theorem 2.2.2, we can find bounded linear operators S1 from Zro.g)(D) into Hlb~lJ-l/U,) such that 881 / = J in U1 for all J E Z?o,g)(D). Choose real-valued 0 00 _ tit
functions fPl on 0" such that D n (supp fP1) c U1 and tit
~
rpf = 1 on D, and define
j=1
_
Sf := l: rp1 Sd
for
J E Z?o,tl)(D) .
j-l
Then, for every / E Z?o.!l)(D), Sf E FlJ;g-l)(D), and _
8S / =
rn
f
_
+ l: C
Since the operator K defined by the sum on the right-hand side is bounded as an operator from Z?o.tl)(D) into H16~,,)(D), it follows from Ascoli's theorem that K is compact as an operator from z?o.g)(D) into itself. This proves part (i) of the lemma. Since K is compact, (id + K)(Z?o,g) (D)) is a subspace of Zro,q)(D) of finite co· dimension. Since "8 0 S = id + K and, therefore, E~~!l)(.D) contains this subspace, it follows that E~b7g)(D) has finite codimension in Zro,g)(D). Since, moreover, Elb:fJ)(D) is the image of a closed operator (namely, of 8" as an operator with the domain of definition FM~'l)(D)), this implies that Etb7riD) is a closed subspace of Z?o.g)(D) (cp. Proposition 4, Appendix 2) . • The remainder of this section is devoted to the proof that the finite codimension of EM~ fI)(D) stated in Lemma 2.3.1 is zero. 1) Reoall that a linear operator between Banach spaces is called compact if evcry bounded set is mapped into a relatively compact set.
2.3. Striotly pseudoconvex open sets
75
2.3.2. Lemma. Let Dec ef/, be a 0 2 atrictly pseudoconvex open set, and let Zt, ••• , Zn be the canonical coordinates oj Z E q)n. Then there exi8ts a holomorphic polynomial p 'Which
depends only on Zt and wht·ch i8 not identically ze1·O such that, Jar every f 1 ~ q ~ '1£,
pJ
E E~6~ f)(D)
E Z?o.g)(D),
.
Proof. By Lemma 2.3.1, there is a finite number of forms such that linear span (Elb~g){jj) u {/l' ... ,IN}) =Z?O,f)(D).
h' ... ,IN E Z?o.f){D) (2.3.1)
For j = I, ... , N, the system J1' zJ1, ... ,zff1 is linearly independent (we can assume that every 11 is =F 0) and of length N + I. Since the codimension of Elb~f)(D) is ~ N, it follows that, for every j = 1, ... ,N, we can find a vector (c~), .•• , cW) 4= 0 of complex numbers such that N
_
~ (j) J;.f El/2 (D) .. cJ; z1J1 E (0. f) •
(2.3.2)
k=O
Set N
P1(Z) :=
!: cV)z~
N
and
p:= []PI: •
k-O
i-l
Now let f be an arbitrary form in Z~. g)(1). By (2.3.1), f can be written as _
J=
Clv
N
+ ;=1 !: Ad1 ,
where ~ are complex numbers and v E Ht6~fJ-l)(ii). Further, by (2.3.2), we can find _ _ forms U1 E H{J;q-l)(D) such that 8U1 _ _
obtain u E H('J,2fJ _ 1 }(D) such that properties. •
N
=
pJI'
Setting u : =
p'O
+i-I 1: AI (II pJj) uf, 1c+J
we
au = pJ. Hence p is a polynomial with the required
2.3.3. Lemma. Let PI and P2 be two polynomials in one complex variable without joint zeros. Then there exist polynomials ql and q2 in one complex variable such that q1P1 + Q2P2 == 1. Proof. The assertion is trivial when PI is of degree zero. Assume that the lemma is already proved for the case that the degree of PI is k. Consider the case that the degree of PI is k + 1. Let Zo be a zero of Pl' Then, by inductive hypothesis, there are polynomials Q1 and Q2 snch that Q!Pl/(Z - %0) + Q2P2 e 1, and, therefore,
[~!~) -
Ql(ZO) P2(Z)/P2(ZO)]Pl(Z) z - Zo
+ [Q2(Z) +
Pl(Z) Ql(ZO) (z - zo) P2(ZO)
]P2 (z)
==
1..
2.3.4. Lemma. Let Dec q)f/, be a strictly pseudoconvex open set, and let I be a continuous (0, q)-form in some neighbourhood, UD of D such that of = 0 in UD, 1 ~ Q ~ n. Then
there exists a u E HU.2g_1)(D) such that
au = I in D.
Proof. By Theorem 1.5.21 we can find a strictly pseudoconvex open set G with Ooo-boundary such t.hat Dec Gee Ujj. Set G. := {z E en: (zt + e, %2' ••• ,Zft) € G}, e O. Choose e > 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
<
8u = /
in
D
and
Ilulll/2.D ~ Olllllo,D
(for the definitions of IHI1/2.D and 11'llo,D see Section 1.8). Moreover, there is a bounded linear operator T Irom z?o,f)(D) into HM~fJ-liD) such that T(Zfo.f)(D)) ~ FU~f-l)(D) and 0 T = id, where id is the identity operator.
a
Remark. In this section we give a first proof of this theorem. This proof can be bypassed without loss of continuity, because, by means of Lemma 2.3.4 only, in Sections 2.4-2.6 we construct a global integral representation formula, which gives (among other things) solutions of the a-equation as stated in Theorem 2.3.5 (cp. Theorem 2.6.1). Proof of Theorem 2.3.5. We' only have to prove that, for every f E Zro,f)(D), there is u E Hl6;,_l)(D) with = finD, because thc rest of the theorem then follows from Banach's open mapping theorem (which gives the estimates) and Lemma 2.3.1 (which gives the operator T - cpo Proposition 3, Appendix 2). If K c aD is a compact set, then we denote by (K) the following assertion: If UK is a neighbourhood of K and f is a continuous (0, q)-form on D u (] K such that 8j = 0
au
D u UK, then! E Etf,~ f)(D). Let (! be a strictly plurisubharmonic 02-function in some neighbourhood UaD of 8D such that U aD n D = {z E U aD: e(z) < O}. For r < 0 and ~ E on we denote by Ee(r) the open ball of radius r centered at ~. By Proposition 1.5.16, de(z) =t= 0 for Z E aD. Therefore, by Theorem 1.4.14, we can find e 0 such that the following condition is fulfilled: For every ~ e 8D, there exist a neighbourhood U~ c UaD of Ee(2e) and a biholom~rphic map he in some neighbourhood of U ~ such that e 0 h -1 is strictly convex: in hf ( U e) and h e( f:J ,) is p. ball. By Lemma 2.3.4, (8D) is true. We have to prove that (0) is true. To do this it is enough to show that, for every compact set K c aD and every point ~ E aD, the implication (K) ~ (K" Ee(e)) is valid. Suppose K ~ aD is a compact set such that (K) is true. Let ~ E aD, let 0 c U 6D be a neighbourhood of K " E,(e), and let f be a continuous (0, q}-form in jj u 8 such that 8j = 0 in DuO. We have to prove that f E Choose a real-valued COO-function X ~ 0 in on with the following properties: (i) X 0 on K " Ee(e), (ii) supp X ceO, (iii) X and the first- and second-order derivatives of X are so small that (e - X) 0 11,-1 is strictly convex on h e{iff), in
>
Elf.,/D).
>
77
2.4. The support funotion 4>(z, C)
DefineG := D
ii c
U
{z
E
U aD : ~(z)
< X(z)}. Then by (ii)
jj u 6.
Since (~ - X) 0 11,-1 is strictly convex on the ball h,(Ue), and since the set h,{G n U,) is defined in ke(U e) by (e -X) 011,-1 < 0, it follows from Lemma 1.5.23 (i) that there is a strictly convex open set W with (JI- boundary such that he(Ee(2e) n G) c W c he(Ue n G) • (2.3.3) Since h;l( W) c jj u 6, it follows from Theorem 2.2.2 that there exists Uo E HU.I,-I) (hil(W)) such that f = 8u,. in h,l(W). Choose q> E Ogo(E,(2e») such that cp = I on E,(e). Since, by (2.3.3), E,(2e) n ii c hi"l(W), then, after continuation by zero, fP'Uo belongs to HlJ~fJ-1)«(j),J - 8(q>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 <1>(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 <1>(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
-
fa o'e(~) ~ - - _ efek' 3 Cei. tEtJ",IEI-l 1,1:-1 8~18~k
p: = -1
min
(2.4.1)
Let a, k be Ol-JunctionB in a neighbourhood oj jj such that
I
max a;k(C) - 02e(C)
8C, 8Ck Let 6 > 0 be 80 8mall that Ce6
max
I< tn'
1 8Be (c)
C,lei, IC-'I S;. oXf
for
82e(z) I
8Xt - 8Xt8x.t
j, Ie = 1, ... , n .
(2.4.2)
P
(2.4.3)
where ~1 = XI(') are the real coordinates oj e z, CE 8 we deJine F(z, e) := -
[
2
fa
l:
8e(C}
-0" (Z1 - C,)
j-l
Jor
< 2n Z
1:01
E
j, Ie = 1, ... , 2110 ,
on such that e1 = Xl(e)
+ ;. l: aJk(C} (ZI k=1 fa
+ iXJ+n(e).I) For ]
C1) (Zk - Ct).
(2.4.4)
Then, lor all z, C€ "0 with IC -
Re F(z, C) ~ e(C) -
zl ::;; E we have the estimate e(z) + PIC - zI2
(2.4.5)
(Re : = real part 0/). Proof. Let C, Z E jj and IC - zl Taylor'S theorem that '"
Re F(!(z, C) = e(C) - e(z)
S; E. n
+
Then it follows from Lemma 1.4.13 and 8Se(C)
1: - - - (z1 - Cf) j,t ... 1
aC1 8C.t
(z" -
Ck)
+ R(z,~) ,
where, by (2.4.3), IR(z, C)I ~ fJ Ie - z12. By (2.4.1) this implies that Re FC/(z, C) ~ e(C) - e(z) + 2fJ Ie - zll. Since, by (2.4.2), IFC/(z, C) - F(z, C)I ~ f3 IC - Z12, we obtain (2.4.5) . • 2.4.3. Theorem. Lee D c: c: (f)n be a strictly pseudoconvex open set, let 8 be a neighbourhood of aD, and let be a striotly.,plurisubharmonic 02-function deJined in some neighbourhood oj 8 such that
e
D I) Sinoe
n (J
= {z €
8: e(z)
< O}
•
e is a Ol.funotion, and sinoe P >
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
linear maps P1: O( Ujj) ~ O( V:n X ¥:n) such that the Gl.map w = (Wl' ... , w n ) defined by
wl(z, C) := T 1(cJJ(.,
en (z, C) ,
CE V aD ,
has the following properties: (i) (w(z, C), C - z) = tP(z, C) for all (ii) w(z, C) is holomorphic in Z E V D, (iii) w(z, C) i,~ a Leray map for D.
Z
E
Vi)
Z E Vi) ,
and
(2.5.1)
'E V aD
(2.5.2)
,
Proof. By Theorem 2.5.4 there are continuous linear operators T , : O(UJj) -, O( Vjj X V:v)
such that tP(~,
n
C) -
q,(z,
C) = - l: i-1
(~,
-
zl) (T1«([)(-,
en) (z,~)
for
z, ~ E V]), CE V aD •
Define w,(z, C) ::- (T1(t1>(·,C))) (z,C). Since (l)("C) = 0, then we obtain (2.5.2), It is clear that 'w(z, C) is holomorphic in z. That w(z, C) is a Leray map for D follows from (2.5.2) and thE' fact that, by (2.4.5), tP(z, C) =1= 0 for z ( D and C E 8D . •
2.5.6. Remark. The map w(z, C) from Theorem 2.5.5 is a Leray map not only for D but for each of the strictly pseudo('onvex open sets (D " Van) U {z E V aD : e(z} c5} if ~ 0 is sufficiently small. This follows from (2.5.2) and the fact that, by (2.4.5), «P(z, C) =F 0 If e(=) ('(C). We remark also that in Theorem 2.5.5 we do not assume that aD is smooth.
<-
>
2.6.
<
Formula for solving the a-equation with Holder estinlates in 0 3 strictly pseudoconvex open sets in QJR
In the prel'eding :;ectioll, for strictly pscudoconvex open sets D c: c: (j)'fI., we COIlstructed a Leray map 'U'(z. C) which is holomorphic in z. If oD is smooth, then, by Corollary 1.12.2, this Leray map gives an integral operator for sohdng the a-equation in D. In the }lresent sedion, we prove that this integral operator admits 1/2-Ho1rler ostimates.
2.6.1. Theorem. L(~t D c: c:: 0'" be a strictly pseudo convex open set with O'~-boundary, let 'll'(Z, ') be tke Lcray map from Theorem2.5.5for D. Then there exist8 a constantG 00 $u.ch that (for notation st'e Section 1.8): _ (i) For cvery continuou-8 dilferentia,l form J on D,
<
i1 R :Dflh/2,D ~ 0 Ilfllo,D' (ii) For every continuou.q {O, q)-form Jon D .r;uch that of = 0 in, D, 1 ~ q
<
'11"
(2.6.1 ) is a contin'UOU8 .solution of Du = f in D 8'uch that Il u lll/2,J>::::; C 11/110,». Moreover, E C'(o,fJ-l)(D)for all 0 0; I and, 1'1 Jis at 'in D (k = 1, 2, ... , 00), then U E 0f(,-;;-l)(D) for all 0 ~, 1.
U
< <
< <
Proof. By Lemma 1.8.0 and Corollary 1.12.2 we only have to prove part (i). To do this, we show that 'w(z, C) satisfies condition (iii) in Lemma 2.2.1. Fi~ ~ E D. By
83
2.7. Oka-Weil approximation
e-
e)
Theorem 2.5.5 (i) and Theorem 2.4.3, <w(z, e), z) = F(z, M(z, C), and we ca.n choose a neighbourhood U~ of E so small that, for some tSl 0, I<w(z,
e)' C -
~ t51 IF(z,
z)1
C)I
for all
>
z, CE U e•
(2.6.2)
Set ~(z, C) := 1m F(z, C) (1m := imaginary part of). Let XI = XI(C) be the real coordinates of CE ([)n such that C, = X,(C) + iXj+,,(C). Then
[8
ta oe(C)] e(e) t1 (z,C) = l: --xi+_(C - z) - --XI(C - z) j-1 8xt 8xj+ta
- 1m
l:"
aj1:(C)
(e, -
Zf) (Cot - Zt)
I
i,k=l
and, consequently, d,~(z,
e)l,-z =
~
e(Z) [8--dxi+n(z)
;=1
8X1
- -oe(z)] - d x1(z) . OXj+_
(2.6.3)
Hence nlld,~(z,
e}I, .." A dQ(z)1I
~11d(!(z)112
for all
z
E
aD ,
(2.6.4)
because, by (2.6.3), the coefficient of dx ,(z) A dXi+n(z) in this form is - [(8e(z)/OXtra (oe(z)/8Xj+ta)2]. Since de(z) =f= 0 for z E oD (see Proposition 1.5.16), it follows that, after shrinking U~, we can find Ol-functions ~, ... , ~n-1 on U~ such that, for every z E U e, t1 (z, .), 2 , ... ,t2n - 1 are coordinates on U e n aD. Set tf(z, C) := t,(C) - t,(z) for z, CE U e, j = 2, ... , 2n - 1. Then, for every fixed z E Ue, the functions It(z, .), ... , t2n - 1 (Z, .) are also coordinates on aD n U e• To complete the proof, we show estimate (2.2.1) for these coordinates. Since tf(z, z) = 0, after shrinking Ue, Taylor's formula gives that there is a t5. 0 such that Ie - zl :c: t52 It(z, C)I for all z, CE U e. Therefore, it follows from estimate (2.4.5) in Lemma 2.4.2 that, after shrinking U E' we can find a t5s > 0 such that IF(z, e)1 ~ t53(1~(z, C)I e(C) - e(z) It(z, C)1 2 ) for z, C E U e •
+
t
>
+
+
Since e(e) = 0 for CE 8D, and since, after shrinking U e. there is tS, -g(z) :c: t54 dist (z, uD) for zED n Ue, this implies that IF(z,
e)1
~ t5 3tS, (It1(z,
for zED n U e and
~ E
C)I
>0
such that
+ It(z, C)\2 + dist (z, aD»)
cD n U~. According to (2.6.2) this gives (2.2.1) . •
Remark. We point out that Theorem 2.6.1 (ii) contains more information than Theorem 2.3.5, because formula (2.6.1) depends more explicitly on D than the operator T from Theorem 2.3.5. For example, one can prove that the constant G in Theorem 2.6.1 (ii) depends continuously (in some sense) on D.
2.7.
Oka- Weil approximation
Lct D c en be a pseudoconvex open set. In this section we prove that every holomorphic function in a neighbourhood of a PO(D)-convex compact subset K of D can be approximated uniformly on K by holomorphic functions in D (Theorem 2.7.1). By means of the support function 4'(z, C) from Theorem 2.4.3, this makes it possible to prove that every pseudoconvex open set in ([)n is a domain of holomorphy (Corol. lary 2.7.3). In Section 2.8 we Flhall use this approximation theorem to solve the 6-
84
2. The a-equation and the "fundamental problems"
on.
a-equation in arbitrary pseudoconvex open sets in The basic tool in the proof of Theorem 2.7.1 is the Leray formula. for holomorphic functions which is defined by the Leray map from Theorem 2.5.5.
2.7.1. Theorem. Let D c: CfI be a pseudocontJex open set, and let K c cD be a PO(D)convex compact set (cp. Definition 1.5.3). Then every holomorphic function in a neighbourhood of K can be approximated uniformly on K by holomorphic junctions in D. Proof. Let h be a holomorphic function in a neighbourhood UK of K. By Corollary 1.5.10, there is a strictly plurisubharmonic COO-function e: D ->- II Buch that (i) D~ := {z ED: e(z) < ,,} cc D for all" E ll. (ii) The Bet Crit (e) := {z ED: de(z) = O} is discrete in D. (iii) e < 0 on K and e 1 on D "- UK' Denote by "0 the supremum of all " E IR such that h can be approximated uniformly on K by functions in (}(D~). Since h E O( UK) and by (iii), (\0 O. First we prove that "0 = 00. Assume that "0 < 00. By Theorems 2.4.3 and 2.5.5 and by estimate (2.4.5), then there are a numberb 0, a OI-function C/)(z, C) and a O"-valued CI-map w(z, C) defined for z in Bome neighbourhood of Do. +4 and Cin a neighbourhood of ~.+fJ "- D fIoo - iJ with the following properties: (a) 4)(z, C) and w(z, C) are holomorphic in z. (b) (,1)(z, C) = (w(z, C), C - z). (c) (j)(z, C) =t= 0 if "0 - t5 ~ e(C) ~ "0 + t5 and e(z) e(C). (d) w(z, C) is a Leray map for D fJ • By definition of (xo' k can be approximated uniformly on K by holomorphic functions in a neighbourhood of 150.-" To obtain a contradiction (with (xo 00), it is ther~ore sufficient to prove that every holomorphic functionj in some neighbourhood of Do.-a can be approximated uniformly on K by functions in O(D/JIo+fJ)' In view of property (ii) of e, wc can assume that de(z) =t= 0 for Z E 8DfIoo - iJ • Then, in view of the Leray formula (1.10.3), we have
>
>
>
<
tJ. -
<
J(z)
f
I)! (2ni)ft
(n -
=
J(C) ~,(w(z, C)) _" C/)n(z, C)
~~~
,
z E Du..-fJ .
aDo.-iS Since w(z, C) is holomorphic for Z E Do.+ d ' it is therefore sufficient to prove that, for every fixed ~ E 8Doct - fJ the function l/~(z,~) can be approximated uniformly on K by functions in O(DCII.+ d). To do this we fix Eo E 8Du..-d and choose a finite number of points Elt ... , E" such that (xo - t5 < e(E1 ) < ... < e(~It) = (Xo + t5 and
sup IEK
Then for j
=
11
- 1) - C/)(z, - - Ej --
~(z, E1)
I < 1for '
J = 1, ... , k .
I, ... , k I
_
4>(z, -EJ -1)
-
I
00
iii(~--:~~-) ';:0
~(z, ~j-l)]'
[
1 -
c.P(z,
EI)
,
where the convergence is uniform in Z E K. Therefore, for j = 1, ... , k, the function l/
85
2.8. Solution of the a-equation in pS6udooonvex open sets
>
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
<
ZEDJ
because then g1
-+ g (
.:1
2'
for
j
=
0, 1, ... ,
O(D) uniformly on every compact Bubset of D and sup Ih - gl K
~ £.
To find this sequence, we suppose that the functions go, ... , g" are already constructed (for k = 0 this is trivial). Since D" is PO(D)-convex, then by what is proved above, there is gt+l E O(Di +2) such that Ig,,(z) - gi+l(Z)1 £/2"+1 for all
<
zED" . • 2.7.2. Theorem. Let D ~ 0'" be a pseudoconvex open set. Then, lor every compact set K c c D, i~ = K'h (cp. Definitions 1.3.2 and 1.5.3). Proof. By Proposition 1.5.4, Kb c .K~. To prove the converse we fix a point ~ ED", XI; and construct a holomorphic function h in D such that Ih(~)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
Co f UaG '" G so
close to ; that
Iif>(~,1 Co) I
> 1 + max I
(2.7.2)
1:EK
Since 0 is PO(D)-convex, by Theorem 2.7.1 we can find a holomorphic function h in D such that Ih(z) - l/
on
is a domain of holomorphy if and only if it is
Proof. This fol1ows from Corollary 1.5.6 and Theorems 2.7.2, 1.3.7 (iii) and 1.5.5 (ii).
2.8.
Solution of the a-equation in pseudo convex open sets in
en
2.8.1. Theorem. Let D c on be a pseudoconvex open set, 1 ~ q ~ 11, and Ie = 0, 1,2, ••• , 00. Then, for every 0fo. g)-form fin D such that 8f = 0 in D, there exists a solution u of 8u
=f
in
D,
86
2. The a-equation and the "fundamental problems"
which belong8 to CfO;:-l)(D) for all 0 tion l.8) .1)
< <1 L\
(for the definition of C~o-:-gCC_l)(D) see Sec-
Proof. By Corollary 1.5.10. therc exists a strictly plurisubharmonie COO-function (! in D such that (i) DIY. := {z ED: e(z) < a} c: c: D for all a E IR, (ii) the set Crit (e) := {z ED: de(z) =-= O} is dis<."rete in D. In view of (ii), we can find a sequence of numbers L\8 E IR, 8 = 1,2, ... , tending to infinity such that de(z) =f= 0 for Z E 6DIJI." that is, DIJI., iA a Coo strictly pseudoconvcx open set. Then, by Theorem 2.6.1 (ii), there are solutions u, of au· s = f in DOlo., which belong to Cfo;:-l)(D .. ,) for all 0 < 1. To construct a global solution in D, we first consider the ease q ~ 2. It is sufficient to find a new sequence {t'8}~3 of solutions of = f in DOl., such that v, E C~o;:_l)(D/lI') for all 0 1 and, in addition, t's = V,+l in D OIo._ 2 • To do this we set Va :-= ~ and suppose that, for some k ~ 3. th(' funetions va ..... Vic are already constructed. Then Vt - UJ:+1 is a-close in Da.1; and Theorem 2.6.11 (ii) gives a solution rp of acp = VI: - uk+l in D CCk _ 1 which belongs to ctO::-l) (DOIok _ 1 ) for all 0 < a < 1. Choose a Coo-functi~n X with compaC't support in DlJI.k_l such that X = 1 in DOI.t_2· Ret ~'k+l := U1:+1 + o(Xcp). This completes the proof for q ~ 2. Now we consider the case q = 1. \Ve will construct a Requence {V.}:'2 of solutions of 8v. = f in D",. which belong to Ck-t o&(Da.,) for all 0 1 such that, morcover, Iv.(z) - v,+l(z)1 < 2-' for Z E Dc<._l' Then this sequenee tends uniformly on the compact sets in D to some function u in D, whi<,'h has the required propertieA. because the differences'U - v, are holomorphic. To construct this sequence, we set t'2 : = 'U 2 and SUppO!'if' that the functions v 2 , ... , VI: are already constructed. Then the difference Vic - Uk-l-! i!'i holomorphic in DOlo,. and, by Theorem 2.7.1, we can find W E O(D) such that IVk(Z) - UJ:+l(Z) - w(z)1 2-.e for z E DtfoJ:_l' Setting Vk+l := '1tl'+l + ·w we completc the proof. •
< (.\
< (.\ <
av,
< (.\ <
<
2.9.
Uniform approximation
en
Let K c: c be a compact set and let f be a continuous fundion on K that is holomorphic in the inner points of K. Consider the following problem: Can f be approximated uniformly on K by functions holomorphic in some ndghbourhood of K ~ In distinction to the case of one complex variabl(', for n ~ 2, this problem is solved only in special cases. DIEDERIOJI and FORNAESS [1] constructed a pAcudocoIlvex open sct Dec (/Jft with smooth boundary such that, for K = D, the anRwer is negative. Tn Section 3.5 we shall prove that the answer is positive if K = {z E UK: e(z) :::;: O}, where e is a. strictly plurisubharmonic C2-function in some neighbourhood UK of K. (By Corollary 1.4.20 and Theorem 1.5.19, this is the case for every compact subset of a totally real Cl-submanifold of en and for the closure of every strictly pseudoconvex open set in (/JR.) In the present section we prove this result for the special case that K is the closure of a strictly pseudoconvex open set D c: c: eft with smooth boundary. In this case the proof is a simple application of the Lera~v formula which correRponds 1) For q = I, by Corollary 2.1.6, €'v{>ry continuous solution of in D for all 0 < <X < 1. For q ~ 2 this is not true.
au = f is a Gk+t¥-function
87
2.10. The a-equation and Cousin problems
R:
to the Leray map w from Theorem 2.5.5 as well as of the Holder estimates for D given in Theorem 2.6.1 (i). First we prove the following theorem on decomposition of singularities: 2.9.1. Theorem. Let Dec qyn be a 8trictly pseudoconvex open 8et with Ol-boundary and let U1, ••. , UN ~ qy'fl be open 8ets 8'UCh that D c: U1 U ••• U UN' Then every contin'UOUB function f on jj that i8 holomorphic in D admit8 a decomp08ition 1i
1= l.:h
_
on
D,
(2.9.1)
; ... 1
where, for j
=
1, .... N, it i8 continuous on D and holomorphic in 80me neighbourhood
of D "" (oD nUl)'
Proof. Choose XI
N
E
0o(U1) such that
:E XI =
j=1
_
1 on D. Define!, := L':DXf!, where to
L:
is the Leray map from Theorem 2.5.5 and D is the corresponding operator introduced in Section 1.8.6. Since, in view of the Leray formula. (1.10.3) for holomorphic functions, f = L:Df, then (2.9.1) holds. Since w(z, C) is holomorphic in z and X, = 0 in Cft" U f , if is holomorphic in some neighbourhood of D" (aD n U,). Since, by the Leray formula (1.10.1) for smooth functions, if = XII R'tD/ 8X, BJ 8XJ, and since, by Theorem 2.6.1 (i) and Lemma 1.8.5 (i), R:Df8xl a.nd BJ8X1 a.re continuous on jj, it follows that the functionsh are continuous on D.•
+
+
2.9.2. Theorem. Let Dec qyn be a strictZy pseudoconvex open 8et with Ol-boundary. Then every continuous function on D that i8 holomorphic in D can be approximated uniformly on jj by functions holomorphic in a neighbourhood of D. Proof. In the special case that f is holomorphic in some neighbourhood of D " S, where S c aD is a set of tmfficiently small diameter, the required approximation can be obtained by a shift in the direction of the normal vector of aD at some point in 8 (aD is smooth!). In view of Theorem 2.9.1, the general case follows from this special case. •
2.10. The a-equation and Cousin problems in holomorphic vector bundles over complex manifolds "Gnti! this section we have studied the theory of functions in open subsets of ({)fI only. The remainder of this chapter is devoted to a more general situation: Instead of open sets in qyn we consider complex manifolds, and instead of functions we consider sections in holomorphic vector bundles. In the present section we give definitions and prove a simple special case of Dolbeault's theorem on the connection between the a-equation and Cousin problems. 2.10.1. Definition. Let X be a complex manifold (Definition 1.1.21), and let GL(N, 0) be the group of invertible complex N X N matrices. (i) A holomorphic vector bundle of complex dimension N over X is a 0 00 vector bundle B over X with the characteristic fibre qyN together with a holomO'1'phic atlas of local trivializations of B, where a holomorphic atl.as 0/ local trivialization8 0/ B is a family {U1, hi} such that the following conditions are fulfilled: {U,} is an open
88
2. The a-equation and the "fundamental problems"
covering of X; for every j, k1 is a Coo·bundle isomorphism from BI Us (the restriction of B to U,) onto the produot bundle U1 X ON; the corresponding transition functions giJ: U, n Uf -+GL(N, 0) (defined by (z, g,,(z) v) = h(hj- 1 (z, v), z E U, n Uf' v EON) are holomorphic. Equipped with the atlas {BI Us' "'1}, B becomes a complex manifold. (ii) A COO. bundle homomorphism between holomorphic vector bundles Bl and B2 is called holomorphic if it is holomorphic as a map between the complex manifolds Bl and B 2• Similar we introduce the concept of a holomorphic section of a holomorphic vector bundle. (iii) A holomorphic vector bundle B over X is called holomorphically trivial if there exists a holomorphic bundle isomorphism from B onto X X (fJN. B is called holomorphically trhial over an open set U C X if the restriction BI u is holomorphically trivial. A holomorphic trivialization of B (over U) is a holomorphic bundle isomorphism from B onto X X CN (BI u onto U X CN). (iv) Recall that a complex-valued differential form of degree r over X can be defined as a section of the vector bundle I\'P*(X)(J, where T*(X)(J is the complexified cotangent bundle of X. A differential form of degree r with value8 in a holomorpkic vector bundle B (or a B-valued differential form) ()f)er X is a section of the bundle 1\" (T*(X)c) ® B. In other words, if {U,}jEI is an open covering of X such that B is holomorphically
c
trivial over each Uf, and if {gij}i,jEl is the corresponding system of transition functions, then a difJerentialJorm with value8 in B can be identified with a system {f1} of N-tuples of differential forms on U I such that It. = gill, over U, n U I for aJl i, j E 1. A differential form I with values in B is called a (0, q)-form, OrO,'l)-jorm etc. if for every open set U ~ X, where B is holomorphically trivial, the corresponding N-tuple of differential forms on U consists of (0, q)-forms, Cfo,g)-forms etc_
Remark. 'Ve do not consider (p, q)-forms, because every (p, q)-form with values in a holomorphic vector bundle can be identified with some (0, q)-form with values in some other holomorphic vector bundle. By means of local holomorphic trivializations the 8-operator can be defined locally for differential forms with values in a holomorphic vector bundle_ Since the transition functions are holomorphic, this definition is independent of the choice of local trivializations. Therefore, the a-operator is well-defined for differential forms with values in holomorphic vector bundles. 2.10.2. Def'mition. Let X be a complex manifold, B a holomorphic vector bundle over X, and {Ut};EI an open covering of X. Holomorphic Cou8in data in B means a system {jij},,;El of holomorphic sections ii;: U 1 n U, -+ B such that fiJ
+ fJk = fil:
in
U t n U j n Uk
for all
i, j, k E 1 .
The corresponding Cousin problem consists in finding a system {f1}jEI of holomorphic sections 1,: U, ~ B such that
it,; = fi -
11
in
U i n UI
for all i, j
E
I .
There is a theorem of DOLBEAULT [1,2] (see also, for example, GUNNING/RosSI [1] and WELLS [1]), which connects the solvability of the a-equation with the solvability of Cousin problems (for (0, I)-forms) and "higher" Cousin problems (for (0, q)-forms l q ~ 2). We need only the following special case of Dolbeault's theorem:
89
2.11. Fredholm solvability of the a-equation
2.10.3. Theorem. Let X be a complex manifold, and B a holomorphic vector bundle over X. Then the following two conditions are equivalent: (i) Every holomorphic Cousin problem in B has a 8olution. (ii) llor every B.valued Cfo.l)-jorm j on X 8uch that = 0 in X, there exist8 a (}oo_
8t
section u: X ~ B such that
au = f
in X.
Proof of (i) ~ (ii). Since B is locally holomorphical1y trivial, and since the 8· equation is locally solvable (see Theorem ~2.5), we can find an open covering {U t } of X and Coo -sectionR UI: Uf ~ B such that BUt = j in U t • Then the differences u( - 'ILl are holomorphic in U, () U I and form holomorphic Cousin data in B. According to (i), there are holomorphic sections h,: UI ~ B such that u, = h, - h, in U t () U1• Setting u :0= Ut - h, in U(, we conclude the proof. (ii) =9 (i). Let {U t } be an open covering of X, alldjij: U t () U , -+ B a holomorphic Cousin data. Choosing a Coo-partition of unity {xd subordinate to {U,} and setting
u,
c1 :
= - L
Xk!kj
in
U1 ,
Ie
we obtain a "Coo-solution': of the Cousin problem: iij =
:E Xk(f~J.; + f/&j)
oc, = -ec, in U,
--
8u
=
OCj
= c, -
cl
in
U, () U , .
k
() U j • Therefore, by (ii), we can find a Coo-section u: X
~
B such that
in U f • Setting hi := c, - u in U f , we complete the proof. •
2.11. Fredholnl solvability of the e-equation on complex manifolds with strictly pseudoconvex c 2 -boundary The notion of strict plurisubharmonicity is well-defined for functions on complex manifolds since it is invariant under holomorphic changes of coordinates (see Theorem 1.4.11 (ii)). 2.11.1. Definition. Let X be a complex manifold, and Dec X an open set. We shall say that D is strictly ps€udooonvex (or that D has a strictly pseudoco1ltvex boundary) if there exists a strictly plurisubharmonic Cla-function !! in some neighbourhood ()aD of 8D such that D () ()aD = {z E ()aD: e(z) O}.
<
en
In Section 2.8 wo saw that in strictly pseudoconvex open scts in the a-equation can be solved. This is not true for strictly pseudoconvex open sets in general complex manifolds. However, Lemma 2.3.1 together with its proof holds also for strictly pseudoconvex open sets in general complex manifolds as well as for forms with values in holomorphic vector bundles. To give the precise formulation of this result we introduce some notations. 2.11.2. The spaces Zro.g)(D, B), EM~g)(D, B), H(o,'liV, B) and FM~g)(D, B). Let X be a complex manifold of complex dimension n, and let B be 8. holomorphic vectOlbundle of dimension N over X. Choose a locally finite open covering {Vf } of X such that, for every j, there are holomorphic coordinates CPI: UI ~ on 8.8 well as a holomorphic trivialization "PI: BluJ ~ U , X q)N. Further, let {XI} be a Coo.partition of unity subordinate to {U , }. Then every B-valued form J over a set Y ~ X can be identified with a. system {fJ)} of vectors fW = (fl, ... ,fj») of differential forms f/)
90
2. The a-equation and the "fundamental problems"
over fPt(U I n Y)
~
0". We define N
IIf(z)l1
:=
1: XI(Z) 1: IIf1)(9'I(z)}11
.-1
J
for
Z
E Y,
where 1f,1)(fPt(z)) I is the Euclidean length of the vector of coefficients of ~)(9'1(Z») (cp. Subsection 1.6.3). _ Let Dec X be an open set. We denote b! oro,g)(D, B), 0
Ilfllo,D
:= max IIf(z) II
.
SED
The norm IHlo.D depends on the choice of the local coordin~tes Cf1, the local trivializations "1'1 and the partition of unity {XI}. However, since D is compact, differe~t choices give equivalent norms. For 0 ex 1 and 0 ~ q ~ n, we den~te by BCo.g)(D, B) the Banach space of ex-Holder continuous B-valued (0, q)-forms on D endowed with the norm
< <
N
Ilflla;,D: = 1: 1: IIf.J )lItx.9"J
where IlfJ>II~9"J
BtJ.
2.11.S. Theorem. Let B be a holomorphic vector bundle over a complex manifold X, let Dec X be a 8trictly pseudoconvex open 8et with CI-boundary, and let 1 ~ q ~ n. Then (i) the're exi8t.s a bounded linear operator S from Z&.f)(D, B} into Hlf.q-l)(D, B) 8uch that
S(Zro.(1)(D, B») 00 S
=
id
c Fl6~t-l)(D, B) and
+ K,
where id i8 the identity operator in Z?o.aiD, B) and K i8 a compact operator in Z~o.!l)(D, B), _ _ (ii) EU~(1)(D, B) is a clo8ed 8ubspace of zro.q)(D, B) of finite codimension. Proof. Repeat the proof of Lemma 2.3.1. • In the next section we show that E{J.2q)(D, B) = Zro. g)(jj, B) if Dec X is a 0 2 strictly pseudoconvex open set with the additional property that the strictly plurisubharmonic function (! in Definition 2.11.1 can be chosen to be strictly plurisubharmonic in a neighbourhood of jj (and not on1y of 8D). This result win be proved by means of an inductive procedure with respect to the levels of e. The basic step in this proof is given by the following lemma.
2.11. Fredholm solvability of the a-equation
91
2.11.4. Lemma. Let B be a holomorphic vector bundle over a complex manifold X, and let Dec X be a strictly pseudooonvex open set with 02.boundary such that Etb:g)CD, B)
=
Z?o,f)(D, B)
for
1 ~q~n .
Further supp08e that U c c X i8 an open 8et with the following properties: (1) U is biholomO'f'phically equivalent to the open unit ball in (f}ft. (2) B i8 holomorphically trivial over U. (3) There is a 8trictly pluriaubharmonic 02·function e in U such that D n U = {z E U: e(z) O} and de(z) =F 0 Jor z E aD n U. Then, Jor every strictly pseudoconvex open 8et Gee X with Ol.boundary 8uch that
<
D
C
G ~ D u V,
where
Vee U ,
(2.11.1)
the Jollowing two statements hold: (i) EU,~q)(G, B) = Z?o.g)(G, B) for 1 ~ ~ ~ n. (ii) Every continuous section of B over D that i8 holomorphic in D can be approximated
uniformly on D by continuous 8ections of B over G which are holomorphic in G. Proof of (i). By hypothesis there is a biholomorphic map T from U onto the unit ball in en. In view of (2.11.1), we can choose open sets U', U" such that U" c c U' c c U, D~ G ~ D u U" and T( U'), T( U") are also baBs. Then it follows from property (3) of U that T( V' n D) is PO(T( U)}-convex (Definition 1.5.3). By Theorem 2.7.1 this implies the following statement: Every holomorphic function in some neighbonrhood of V' n j) can be approximated uniformly on U' n D by holomorphic functions in U.
(2.11.2) 2 • boundary
By Lemma 1.5.23 (ii) we can find a strictly pselldoeonvex open set W with 0 such that GnU' ~ W ~ GnU. Then W n D has 02.boundary, and since, for 0 2 open sets, strict pseudoconvexity is a local property of the boundary (this follows from Theorem 1.5.17 or from Theorem 1.5.25), we obtain that W n D is a 02- strictly pseudoconvex open set. Now let! E ZrO,g)(G, B). \\Te have to prove~hat f = a~ for Borne u E HU;q_l)(G, B). Since B is holomorphically trivial over U 2 W, and sinC'e T( W) is a 0 2 strictly pseudo. convex open set in (/)fI., we obtain from Theorem 2.3.5 (or Theorem 2.6.1 (ii)) a form Uw
E Hl6~'l-1)( W, B) such that oUw
=f
in
W.
Moreover, by hypothesis of the lemma, there is OUD
=f
in
D.
(2.11.3) UD
E
HU,2g _ 1 )(D, B) such that (2.11.4)
N ow we distinguish the cases q = I and 2 ~ q ~ n. First let q = 1. Then Uw - UD is continuous on W n D and holomorphic in TV n D. Since B is holomorphically trivial over U, it follows from Theorem 2.9.2 that Uw - UD can be approximated uniformly on W n D by hololUorphic sections of B over some neighbourhood of W n D. In view of (2.11.2), this implies that there are holomorphio sections gyk: U -+ B such that
lim II!pI: i-+oo
+
Uw -
uDllo, U'nD = 0 .
(2.11.5)
92
2. The a-equation and the "fundamental problems"
Choose a Coo-function X on X such that X = 1 in some neighbourhood of U" and X = 0 in some neighbourhood of X " U', and define UD Uk:
Then Urc
=
{
(1 - X)
E Hl/2(G,
lim
B),
UD
aUk E
+ X(Pt + uw)
Etlh(G, B) and it follows from (2.11.3)-(2.11.5) that
-
118u): - IlIo,G
in G" U, in GnU.
lim
=
1:-+00
k-+oo
- -
11- UD 8X + (Pk
+ 'Uw) exllo,u'nD = 0 .
Since, by Theorem 2.11.3, EU~l)(G, B) is a closed subspace of Z?O,l)(G, B), this implies that I E EM~l)(G, B). _ _ Now consider the case 2 ~ q ::;: n. Then 'Uw - 'UD E Z?o, 11-1)( W n D, B}. Since T( W n D) is a CS strictly pseudo convex open set in on, it follows from Theorem 2.3.5 (or Theorem 2.6.1 (ii)) that there is v E HM~q-2)( W n D, B) such that Uw - 'UD = 8v in W n D. If X is as above, then the required solution U E HU,;'1-1)(G, B) of a'U = I can be obtained by setting 'U := 'UD in G" U' and 'U:= 'Uw - 5«(1_- x) v) in G nU'. (ii). Let U', U", W andx beasin the proof of part (i), and let/: D -+B be a continuous section that is holomorphic in D. Since B is holomorphically trivial over U, and since U' n D ~ W n D, it follows from (2.11.2) and Theorem 2.9.2 that there are holo. morphic sections Uk: U -+ B such that lim Ilf - 'Ukllo, U'nD = 0 .
(2.11.6)
1:-+00
Then 8(x(f - Uk)) = (! - Urc) aX E ZrO.l)(G, B), and, by part (i) of the lemma, we can find a sequence Pic E Hl/2(G, B) such that 8cprc = a(X(! - 'Uk))' In view of (2.11.6) and Banach's open mapping theorem, this sequence can be chosen so that (2.11.7)
lim Ilcptlll/2,G = 0 . 1:-+00
Setting
!" := Jf - xU -
+ rrc
on D, on G"D, we obtain continuous sections It: G -+ B which are holomorphic in G. It follows from (2.11.6) and (2.11.7) that lim III - fkllo.D = 0. • Uk)
lUk+ffk
~-+oo
2.12. Solvability of the ~·eqnation on complex manifolds with strictly plurisubharmonic exhausting C'A_ function In the preceding section we saw that, for strictly pseudoconvex open sets D with 0 2 _ boundary in complex manifolds, the operator a has a finitely codimensional closed image as an operator from the space of IJ2-Holder continuous (0, q - I)-forms on D into the space of a-closed continuous (0, q)-forms on D. In -the present section we shall prove that this operator is onto if D can be defined by a strictly plurisubharmonic CS·function which is defined not only in a neighbourhood of 8D but in a neighbourhood of D. Moreover, we shall prove Oka·Weil approximation and solvability of the 8equation on complex manifolds which admit a strictly plurisubharmonic exhausting Oi-function.
2.12. Solvability of the a·equation
93
2.12.1. Definition. Let X be a complex manifold. A 8trictly pluri8ubharmonic exhaU8ting Ck·fu~ction for X, k = 2, 3, ... , 00, is a strictly plurisubharmonic Ot·function (! on X such that, for every IX E R, the set {z EX: (!(z) < IX} is relatively compact in X. Remark. Let X be a complex manifold, a o E R, and let (! be a strictly plurisubharmo. nic Ot·function on X with values in the interval (- 00, ao) such that, for every - 00 a ao, the set {z EX: e(z) a} is relatively compact in X, k = 2,3, ... , 00. Then X admits a strictly plurisubharmonic exhausting Ok.function. In fact, choose a strictly convex and strictly increasing Coo·function X on (- 00, IXo) such that X(X) ~ 00 for IX ~ (Xo. Then, by Theorem 1.4.12 (ii), X 0 e is a strictly plurisubharmonic exhausting Ok.function for X. In particular, the open sets {z EX: e(z) a} in Definition 2.12.1 are also manifolds with strictly plurisubharmonic exhausting Ok.function. By Theorem 1.5.8, every pseudoconvex open set in 0 11 admits a strictly plurisub. harmonic exhausting Coo·function. Observe that a strictly pseudoconvex open Bet in a general complex manifold (Definition 2.11.1) does not necessarily admit a strictly plurisubharmonic exhausting C2·function. Finally we observe that every complex manifold with strictly plurisubharmonic exhausting 02.function admits a strictly plurisubharmonic exhausting Coo·function. We do not use this fact and its simple direct proof is left to the reader. In the next section, we obtain this as a corollary of Theorcms 2.13.4 and 2.13.5.
< <
<
<
2.12.2. Lemma (Morse's lemma). Let X be a complex manifold which is countable at infinity, and let (! be a strictly pluri8ubkarmonic 03·function on X. Further, let K c: c: X be a compact set 8uch that d(!(z) 0 for all Z E K. Then, for every 8 0, there exists a strictly pluri8ubharmonic (]I·function fl. on X 8uch that the following conditiOn8 are fulfilled: (i) e - etJ together with its first. and second-order derivatives is ~ 8 on X. (ii) The set arit (ee) := {z EX: detJ(z) = O} is discrete in X. (iii) (!8 = (! on K.
*
>
Proof. Let UK be a neighbourhood of K such that de =F 0 in UK' Since X is countable at infinity, we can find a sequence of open sets U t c: c: VI c: c: X, j = 1,2, ... , such that the following eonditions are fuMilled: 00
(1) X '" UK ~
U U l'
j=l
(2) For every j, V, is biholomorphically equivalcnt to some bounded open set
in Oft. (3) V f n K = 0 for allj. (4) For every compact set Y c: c: X, there is only a finite number of indicesj such that Y n V, =F 0. Now, by means of Lemma 1.4.19 and a lemma of Morse (cp. the proof of Lem· rna 1.5.9), we can inductively construct a sequence of O°O·functions XI on X suoh that, for every j, the following conditions are satisfied: (a) X1 = 0 in a neighbourhood of X '" V t , and consequently, X1 = 0 on K. (b) The function e Xl + ... + XI is strictly plurisubharmonic on X and has only non· degenerate critical points in K u U l U ... U U t • (c) The function XI together with its first· and second.order derivatives is ~ 8/2 1 onX. Then e, := e :E XI (this sum is locally finite) has the required properties. •
+
+
94
2. The a-equation and the "fundamental problems"
The main result of the present section is the following 2.12.3. Theorem. Let X be a complex manifold with strictly pluri8ubkarmonic exhausting Oi-function e, and let B be a holomorphic vector bundle over X. Set DG¥ := {z EX: e(z) < o:} for (X E B. Then (i) Let (X E R such that de(z) =f= 0 for all Z E eDG¥' and let 1 ~ q ~ n. Then there exist8 a bounded linear operator T fromZ/.o.'l)(Dt!" B) into HtJ~lJ-l)(DG¥' B) (see Subsection 2.11.2) such thaI aT f = f in Dt!' for every f E Zfo.'l)(Dt!" B) • (ii) Let 0: E Bsuch that de(z) =f= 0 for Z E 8DG¥. Then every continuous sectionf: DG¥ -+ B that i8 holomorphic in D t!' can be approximated uniformly on jjt!' by holomorphic sections of B over x. (iii) Let 0: E II (not necessarily ~e(z) =f= 0 for Z E 8DG¥). Then every ko~morphic section of B over some neighbourhood of Dt!' can be approximated uniformly on D IX by holomorphic sections of B over x. (iv) Let 1 ~ q ~ n. Then, for every B-valued continuous (0, q)-form f on X such that 8J = 0 on X, there exiBt8 a B-valued continuou8 (0: q - I)-form u on X such that
8u=f on
X.
This theorem will be proved by means of Lemma 2.11.4 and the follOwing 2.12.4:. Lemma. Let X be a complex manifold with strictly plurisubkarmonic exhausting Ol-function (! 8uch that Crit (e) := {z EX: de(z} = O} is discrete in X, and let B be a holomorphic vector bundle O'Ver X. Set DG¥ := {z EX: . (!(z) <~} Jor (X E R. Then, for every 0:0 E ll, tkere exi8t Eo> 0 and a Jinite number oj open set8 Ut , •.• , U AI c: c: X such that the following conditions are fulfilled: (1) For j = 1, ... , M, UI is biholomorphically equivalent to the unit ball in en. (2) For j = 1, ... , M, B i8 holomorphicalZy trivial over U1 • (3) D aI, +,. ~ U1 U ••• U U M. (4) IJ 0 ~ El' Ea =::; Eo 8uch that de(z)
=f= 0 Jor Z E 8DIII,-8J u 8DL¥o+8a' then there. are strictly pluri8ubharmonic exhausting 02-functions (10' el' ... , (! M Jor X such that d(lI(Z) =f= 0 Jor el(z) = 0, j = 0, 1, ... , M, ana, Jor the 0 2 strictly pseudoconvex open set8 D1 := {z EX: (l/(Z) < O}, we have the relation8 Do = D IIID - 8a , DM = D~+8. and Dj _ 1 ~ DI ~ Dj _ 1 U VI, where V1 c c U j = 0, 1, ... , M. " U , ••• , U M c: c: X such that (1), (2) are Proof. Clearly, we can find open sets 1 fuHilled and D"t+l ~ Ut U ••• U U M. It remains to prove that there is 0 < Eo ~ 1 so small that condition (4) is satisfied. Choose GOO-functions Xv, .• , XM on X such that M
o ~ XI
~ 1,
BUPP XI c:c U I and
we can choose these functions
l: XI i-I
80
_
_
= 1 on D CIo +1 • Since Grit (Q) n D.:x.+l is finite,
that, moreover, the following condition is fuliilled:
There exists a neighbourhood U of Crit ((!) n 8Dcc, such that, for all Z E U, there is j(z) E {I, ... , M} such that Xj(~)(z) = 1 and XI(Z) = 0 for j =i=j(z).
(2.12.1)
Choose open sets VI c: c: U, such that supp XI c: C VI. In view of condition (1), we can find holomorphic coordinates zlj, ... ,Znj on U , . Set
o := ~ sup
i
j=l ZEYJ,•• - l
I
2
8 Xf(Z)
8z,j 8z,j
I.
2.12. Solvability of the a-equation
95
Since (! is strictly plurisubharmonic, there is j= 1, ... ,M,
p> 0
such that, for all
qyn and
i
8:a()(~) ~.~r ~ P1~12 . 8zr; 8z.; Since Crit «(!) n DClo + 1 is finite, and since (2.12.1) holds, we can choose 8~ inf
~ E
zE VJ r,'= 1
that Crit «(!) n (.DGI'+I!~ " Dtlo-e~) ~ 8Dtlo and M
Ild(!(z)11 ~ 3E~
1:
;-1
IldXf(z)11
for
Z
> 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
<
is true for all (Xmln ~ {3 Q:. We have to prove that Q: o = 00. Since the relation e(z) = (Xmtn implies that z E Crit (e), there is only a finite number of points z E X with (!(z) = Q:mln' Therefore, if e 0 is sufficiently small and df!(z) 9= 0 for Z E 8DGCmlll + B, then Dl1omill+B is biholomorphically equivalent to some strictly pseudoconvex open set with 02-boundary in qy"'. By Theorem 2.3.5 (or Theorem 2.6.1 (ii)), this implies that (i)"mlD+8 is true for all sufficiently small e 0, that is, Q:o Q:mln' Assume that Q:o < 00. Then, by Lemma 2.12.4, there are a number eo 0 and open sets UI , ... , U M such that conditions (1)-(4) in Lemma 2.12.4 are fulfilled. Since Q:o "'mln, and since Crit (e) is discrete, we can find 0 El min (Eo, Q: o - (Xmln) such that d(!(z) =F 0 for z E 8DtJo - E1 • Now let E2 be an arbitrary number such that 0 ~ ell ~ Eo and de(z) =f: 0 for z E 8Dor.+ ••• Let Do, ... , DM be the corresponding sets from condition (4) in Lemma 2.12.4. Since
>
>
>
(i) tl e- 1I1 is true and Do
>
>
< <
=
DGle-£I' then Et/'~'I/jjo, B)
= Zro.fl)(Do, B)
for 1 ~ q ~ n.
96
2. The a-equa.tion and the "fundamental problems"
Applying M times Lemma 2.11.4 (i), we obtain that EU,2g)(i5M , R) = z?o.f)(DM , B) for 1 ~ q ~ n. Since DM = D~,+8.' this is in contradiction to the definition of exo' (ii) By Lemma 2.12.2, we can assume that Crit (e) := {z EX: d(!(z) = O} is discreteinX. LetfJ E Rsuch thatde(z) =l=0forz E aD p, and letj: Dp -+Bbeacontinuous section that is holomorphic in D. We first prove that for all fJ ~ ~ < 00 the following statement holds: (ii). If d(!(z) =1= 0 for z.! aD., then J can be approximated uniformly on DfJ by continuous sections over D. which are holomorphic in D IX • Clearly, (ii)p is true. Denote by exo the supremum of all ex ~ fJ such that (ii)1X is true. We must prove that (xo = 00. Assume that exo < 00. Then, by Lemma 2.12.4, there are EO > 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
°
< <
>
<
°
Do
< < < ...
> E
11/1 -/IIO.Dp <"2
a.nd
Iljj+1 - 1111o,DaJ
< 2 B+ j
l
for j = 1,2, ....
Then the sequence II tends to so~e holomorphic section!: X ~ B, uniformly on each compact subset of X, and III - fllo.Dp < E. (iii) Let (X E R and let J be a holomorphic section of B over some neighbourhood U, of ~. Choose a OOCI-function X on X such that X = on Da. U (X" DtlI + 1 ) and X> 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.
< <
ot
< <
2.13. Solution of the Levi problem for complex manifolds In this section we introduce the notion of a Stein manifold, whose definition is modeled on the properties of domains of holomorphy in C". It is the aim of this section to prove that 80 complex manifold is Stein if and only if it admits a striotly plurisub. harmonic exhausting Ola.function (solution of the Levi problem for manifolds). It is not difficult to construct a strictly plurisubharmonic exhausting O°O·funotion on Stein manifolds. As in the case of domains in Q}ft, the proof of the opposite direotion is more difficult. This proof is based on the solvability of the a.equation on complex manifolds with striotly plurisubharmonic exhausting OI-function proved in the preceding section. 2.13.1. Definition. Let X be we define
k~
:= {z EX: If(z)1
&
complex manifold. For every compact set K c: c: X
=::; sup If(C)I Cel{
Xi is called the O(X).hull of K. If K = 7
Henkin/Letterer
for all
J E O(X)}
.
.i~, then K is called O(X}.convez.
98
2. The a-equation and the ,,fundamental problems"
2.18.1. Def"mition. A oomplex manifold X is oalled holomorphically convex if, for every compact Bet K c c x,.i~ is also compact. 2.13.3. Definition. A complex manifold X of complex dimension n which is oountable at infinity is said to be a Stein manifold if (i) X is holomorphically oonvex, (il) for every point Z E X, one can find n functions f1' ... ,i,. E O(X) which form holomorphic coordinates at z (that is, there is a neighbourhood U of z sueh that the map U 3 e ~ (/1(C), ... ,f,,(e)) E eft is biholomorphic). Remark. Sometimes the following condition is included in the definition of a Stein manifold: (iii) If z and C are different points in X, then there exists an J E O(X) such that fez) =1= fCC). We shall Bee (Corollary 2.13.7) that (iii) is a consequence of (i) and (ii). . Examples. 1. By Theorem 1.3.7 (iii), every domain of holomorphy in ([In is a Stein manifold. 2. Every closed complex submanifold of a Stein manifold is a Stein manifold. 3. If X, Yare Stein manifolds, then X X Y is a Stein manifold. The simple proofs of 2. and 3. are left to the reader. Remark. In particular, every closed complex submanifold of en is a Stein manifold. Conversely, every Stein manifold is biholomorphieally equivalent to some closed complex submanifold of some Q)ft (see, for example, HORMAND1!lR [1] and GUNNING! ROSSI [1]). We do not use this result and omit the proof. 2.13.4. Theorem. Let X be a Stein manifold. Then, lor every O(X)-convex compact set K c c X and every neighbourhood UK oj X, there exists a strictly plurisubharm.onic 0 on exha'U8ting COO-function (} for X (Definition 2.12.1) such that e 0 on K and (!
<
X"
>
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
<
n
02e(zO)
-
L --_ ~~. =0, ".=1 oz,. 8z. where allj
z}) ••• , Zft N(j)
l:
are holomorphic coordinates in some neighbourhood of zoo Then for
I Ln
OJ"(ZO)
-i-r'-
i=1 r-l
CZ r
~,
12
=
By (2.13.3) this implies that
NO>
L
j
OZ,. oz,
i=1 r,.-1
~ =
'r¢.
01 /j"(z)/I -
"
L
= 0•
O• •
2.13.5. Theorem. A complex manifold X is a Stein manifold if and only if there exists a 8trictly plurisubharmonic exhausting C 2 -function e for X (Definition 2.12.1). Then, Jor every (l E ill, the set {z EX: e(z) ~ (l} is O(X)-oonvex. That the condition is necessary we know from Theorem 2.13.4. The remainder of the theorem will be obtained as a consequence of the following 2.13.6. Lemma. Let X be a complex manifold with strictly plurisubkarmonic exhausting C2-Junction Then (i) Let ~ EX, C4 := e(,), Da := {z EX: e(z) C4} and suppose that de(z) =f= 0 for all z E oDa' Then there exist a sequence fie E O(X), k = 1,2, ... , and a constant 0 00 Buch that the Jollowing conditions are fulfilled: a) flc(~) = 1 for k = 1, 2, ... , b) IIJ,cllo,Da ~ C for k = 1,2, :.. , c) for every compact set K ~ Da ,,~, lim IIJlcllo,K = o.
e.
<
<
1' ....... 00
(ii) For every point , ~ X and each holorrwrphic Junction J in some neighbourhood of " there exi8ts a sequence {flcrf of holomorphic functions on X such that lim oflc(')
=
af(~).
1:-+00
(iii) For every
E
C4
IR and all,
E 8Da, there exists a kolomorphic Junction J in some
neighbourhood oj DOl such tkatf(~) = 0 andj(z)
Proof. (i). Let ).
-
=f= 0 for all, =F z E Da.
=
(AI' ..• ,Aft) be holomorphic coordinates in some neighbourhood
n
0
Ve of ,. Set u(z) := 2
(~)
I: ~ (AJ(z)
811.1 Then 'U is holomorphic in V~, u(,) = 0 and, by Lemma 1.4.13,
-
~(,))
j =1
Re u(z) = Q(z) - !?(~) -
fa
82
(~)
(Aj(z) - At(')) (AI:(z) - AI:")) •
1l.1:
(2.13.6)
02
(,)
e (Af{z) L --_ fa
J. k ... l
oAf OAIe
-]- 0(1).(,) - ).(z)1 2) • 7'
+
+ i,k-l L OAl o
-
).,(~)) (AI;(z) -AlcU))
100
2. The a-equation and the "fundamental problems"
Since e is strictly plurlsubharmonic, this implies that, after shrinking V E' we ca.n find p 0 such that Be u(z) e(z) - e(E) - PI).(z) - ).(E)I' for all Z E Ve' (2.13.7) Then ett<.) = 1 and Ieue.) I 1 for all ~ =1= zeD;. n Ve' (2.13.8)
>
<
<
Choose a neighbourhood We cc V, of ~ and a Coo·function X on X such that X = 1 on W, and supp X C C V,. Then e"" aX, k = 1, 2, ... , is a sequence of a-closed Oc':'l)· forms on X and lim Ilei " aXllolD"
i-+oo
= 0.
Since d(!(z) =\= 0 for Z E 8Dts , we can apply Theorem 2.12.3 (i) and obtain continuous functions VI: on such that aVt = eh aX in DIS and
n.
lim IlvtllolDGI
=
(2.13.9)
O.
i-+oo
it :
it
+
Setting = X e"· - VI: v,,(E), one obtains continuous functions on D., which are holomorphic in D •. Since BUPP X C C VI' it follows from (2.13.8) and (2.13.9) that il:(E) = 1, k = 1,2, ... , SUPt 11.~llo.Dts < 00 and, for every compact set K C De& ,,~, lim" lIil:llo.K = O. The required functions /1: E O(X) we now obtain by means of the Approxima.tion Theorem 2.12.3 (ii). (ii) We can assume that J(E) = 0 . (2.13.10) Set cl : = e(E) and suppose that Wf, V e, u, X are as in the proof of part (i). After shrinking W f and V I' we can assume that J is holomorphio on V,. Define by f/>" : = J en aX, k = I, 2, ... , a sequence of a-closed cro,l)-forms on X with supp qi1e C C V E ' " W" Then it follows from (2.13.7) that, for some 6 > 0, liml: 1If/>I:IIO.Da+" = O. By Lemma 2.12.2 we can assume that Crit (e) := {z EX: de(z) = O} is discrete in X. Choose 0 < B lJ such that de(z) ~ 0 for z e 8DGI +8 • Then, by Theorem 2.12.3 (i), we can find continuous functions Vt on D GI +8 such that
<
(2.13.11)
and (2.13.12)
lim IIvtllo.D.+8 = 0 .
} ... oo
Since f/JII = 0 in We' the functions this implies tha.t lim 6v1e(E)
VI:
are holomorphic in W,. Together with (2.13.12)
=0.
(2.13.13)
Since 8v1e = f/>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. •
<-
Notes The Oka-Hefer theorem 2.5.4 was obtained in OKA [8J and HEFER [1]. Solvability of Cousin problems in domains of holomorphy in (Jft was first proved in 1937 by OKA [2]. Together with the solvability of the a-equation in polydiscs obtained by DOLBEAULT and GBOTHENDIEOK (see CARTAN [3], DOLBEAl1LT [1, 2]. MALGBANGE [1]. HORXANDEB [11, ANDBEIAN CAZAOU [I]) this gives solvability of the a-equation in domains of holomorphy (DOLBEAULT [1, 2]). The identity of pseudoconvex open sets and domains of holomorphy in C" was proved first for n = 2 in 1942 by OKA [4] and then in 1963/64 for the general case by OKA [6], BBBMEBMANN [1] and NOBGl1ET [1]. The Approximation Theorem 2.7.1 (for O(D)-oonvex compact subsets of domains of holomorphy D) was obtained in 1935136 by
102
2. The a-equation and the "fundamental problems"
WElL [1] and OKA [7]. Observe that. in the pioneering works of OKA and WXIL, the integral representation formula for holomorphic functions in polynomial polyhedra found by BERGMAN [1] and WElL [1] (see Chapter 4) served as the basic tool. OKA'S results on solvability of Cousin problems as well as the Oka-Weil approximation theorem in domains of holomorphy in en were generalized to Stein manifolds in the seminars of H. CABTAN in 1960/51 (OARTAN [3]). In these seminars the more general theory of coherent analytic sheaves on Stein manifolds was developed (see also OKA [6], GUNNING/RosSI [1], GRAUERT/ REM1U£BT [I]. ANDBEIAN OAZACU [1] and H6RMANDER [1]). Some special results of this theory will be given without proof in Chapter 4. The solution of the Lovi problem for manifolds (Theorem 2.13.6) is due to GRAUERT [4]. The approximation theorem for pseudoconvex compact sets in Stein manifolds (Theorem 2.12.3 (iii» was proved by DOCQUIER/ GRAUERT [1]. Solvability of the a-equation direotly (that is, without use of the solution of the Levi problem) in pseudoconvex open sets and complex manifolds with strictly plurisubharmonic exhausting function for the first time was obtainod by means of L 2 estimates in the Neumann problem by HORMANDER [1, 2]. KOHN [1] and MORBEY [1]. This method made it possible to obtain the main results with appropriate L 2 -estimates on the boundary in the strictly pseudoconvex case (KOHN, MORREY) and in the domain in the general case (HORMANDER). Solvability of the a-equation with uniform estimates in 0 3 strictly pseudoconvex open sets on Stein manifolds was obtained in 1970 by GRAUERT/ LlxD [J], HENKIN [1], KEltZMAN [1, 2], LIEB [2] and 0VRELID [1] by means of integral formulas constructed by HENXIN [4] and RAMIREZ [1]. HOlder estimates were proved by HENKINfROMANOV [1], KERZMAN [1, 2] and LIEB [2]. Observe the following result of KRANTZ [1]: If D cc en is a domain with smooth boundary such that the a-equation can be solved with 1/2-H<>lder estimates in D, then D is strictly pseudoconvex. SIDONY [1] constructed a pseudoconvex domain with smooth boundary without uniform estimates for the a-equation. The Approximation Theorem 2.9.2 was proved by HENKIN [4] (other proofs were obtained by KERZMAN [2] and LIED [1]). DIEDERICH/FoRNAESS [1] constructed a pseudoconvex domain with smooth boundary, where the Approximation Theorem 2.9.2 is not valid. The solution of = I (for CO, I)-forms I) given by Theorem 2.6.1 (ii) is the unique solution defined by the oondition LfD'U = O. (lie)
a
au
The solution of
Bu
=
au = f obtained by the method of KOHN [2] is defined by the condition
(**)
O.
where B is the orthogonal projection from Li(8D) onto the subspaC'e spanned by the holomorphic functions on a neighbourhood of D (the Szego projection). Aftm' the work of FElI'J'l!IB.lI4AN [1], where asymptotics for the Bergman and Szego kernf'ls are given, it was proved by FOLLAND/STEIN [1] and GRXINER/STEIN [1] that the solution of au = f satisfying (. *) in strictly pseudoconvex C3-domains admits the same estimates as the solution defined by (*). Moreover, fo1' the ball, the solution of = I satisfying (*) coincides with the solution satisfying C·*). In the case of an arbitrary strictly pseudoconvex domain with smooth boundary, the solution satisfying (*) gives a "parametrix" for the solution satisfying (lit lit) and conversely. KEBZMAN and STEIN [J] proved the formula S = L':D(I - A)-I, where A = K* - K is compact. For peeudoconvex domains with Ooo·boundary, by means of the method of LI-estima.tes, it was proved by KOHN [3] that the a-equation can be solved with Coo-estimates on the boundary. Using results of DIEDERICH/FoBNAES8 [2] •. for pseudoconvex domains with real-analytic boundary, KOHN [4] also obtained subelliptic La-estimates on the boundary for the a-Neumann problem. The a-equation in domuins with piecewise smooth bounda.ry will be considered in Chapter 4.
au
Exercises, remarks and problems 1 a) Prove that the SiJov boundary of a bounded oonvex domain with Ooo-boundary in
e" is the closure of the points of strict pseudoconvexity
(the Silov boundary of a domain
103
Exercises, remarks and problems
Dec (fJn is by definition the smallest closed set K ~ aD such that maxi> 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!<8}nD
au
uniformly in E E aD (dO'2f& iB the Lebesgue measure), 'hen there is a solution 01 = /, whoss boundary'lJal'Ue8 belong ro each V(8D), 8 < 00. Moreover, V ABOl'OlTLOS [1] proved that this solution oan be ohosen to have boundary values in BMO( 8D); here BMO( aD) is the space of aJJ. measurable functions f on aD with sup lE8D.I>O
J
-~-
-IHe(6) n aDI ....
1/ -
1,.8I da2ft-l
<
00 ,
H,
where datft-l is the volume measure on aD,
IHe(6) n aDI
:=
,..
J
da2n-l,
11.,:=....
1 IHl(6) n aDI
HeC')n8D
<
da2ft-l .
BMO(8D) $; n L·(8D). V ABOPOUL08 [1] proved also that, for '<00 n, a solution with bounded boundary values on aD need not exist.
Observe that D:XJ(8D)
q
JI B,<')naD
~
14. Let Dec Cft be the unit ball, and let 11' ... , IN be bounded holomorphio funotions in D such that inf (1/11 + ... + lIND> O. _
D
_
a) Let A(D) be tho space of oontinuous functions on D that are holomorphic in D. For fP E Loo(8D) we write lP -L A(D) if, for all h e A(D). J fPC') h(C) ru'(C) A ru(') = O. Prove that.
_
aD
for every 9' e Loo(8D) with 9' -L A(D), there exist 9'1' ... 'fPN E Loo(8D) such that 'Pit .... fPN...LA(D) and P=tlP1+'" +INPN' Hint. Use Exeroise 13b) and an appropriate analogue of Carleson's construction in the proof of the Corona theorem (CARLESON [3]). b) Open problem (the Corona problem for the ball in (]ft, n ~ 2). Do there exist bounded holomorphic functions ~, .... hN in D such that 11k,. + ... + INhN = I in D t Remark. HENKIN [3] and VAROPOlTL08 [1] (see also AllAR [2]) proved that there exist holomorphic functions h1 E n ~(D) such that ~ tlhl = 1 in D, where ~(D) is defined ,,<00
analogously as in Remark 8 in the Exeroises, remarks and problems at the end of Chapter 1. Observe that tho "hard" part in Carleson's proof of the Corona theorem (op. Remark 8 in the Exercises, remarks and problems at the end of Chapter 1) works also in this situation. However. the corresponding a-equation (the "soft" part) oan be solved with boundary values in BMO(oD) only. (Note that for the unit diso D. for every function« in D with «laD e BMO(aD),. there exists a holomorphic funotion gin D such that u + g is bounded. In fact, FEFFERMAN/STEIN [1] proved that such functions« Oan be written u =! + h, where /laD, hlaD e Loo(aD) and h is the Hilbert transform of h (that is h + ih E O(D) and h(O) = 0). Set g : = i(h ih).)
+
106
2. The a-equation and the "fundamental problems"
13 (LELONG [2], BREMBBMANN [2]). Prove tha.t, for every continuous plurisubharmonio funotion'U in the pseudoconvex open set D ~ (fJf', there exist holomorphic functions im in D such that u(z)
=
lim
BUP
2.. In Ifm(z) I ,
;-+00 m~j m
zED,
where the oonvergence is uniform on each compact set. Hin t. Prove that the set D* := {(z, w): zeD, W E (J1, lu(z) I < - In Iwl} is pseudoconvex; consequently (since the Levi problem is solved - Theorem 2.7.2). there is a holomorphic fUIlction f(z, w) in D* whioh oannot be continued holomorphically to a larger open setj take for fm(z) the 00effioients of the power series I(z, w) = ~ fm(z) w m with respect to w (see also RONKIN [1]). 16 (Dolbeauh's theorem). Let X be a complex manifold of oomplex dimension n, let U = {UI }JEI be an open oovering of X suoh that every U 1 is a Stein manifold, and let B be a holomorphic vector bundle over X. For 0 ~ q ~ n we denote by Of(U, B) the group of all collections {fi •. :.J,}/U...... J,)EIg + 1• where !Jo ...;r is a holomorphic section of B over Ui. n ... n Uir' By settmg g+1 f
(f5 /)J'···;9+1 : =
}~O ( - l)k Ijo •••"k... jg+1
we define a. homomorphism ~I: 0 1(11, B) -- 01+1(U, B). 'Vc denote by al the 8-operator oonsidered as an operator from the spaco of all B-valued O~q)-forms over X into the space of all B-valued 0(~f+1)-forms over X. Prove the following generalization of Theorem 2.10.3: For q = 1, ... ,?I, the factor group8 kernel aI/image c:5f-l and kernel af/image 8f are isomorphic.
3.
Theory of functions on strictly pseudoconvex sets with non-smooth boundary in en
Summary. In this chapter we consider strictly pseudoconvex open sets Dec (Jft (Definition 1.5.15) whose boundary need not be smooth. In Section 3.1 we construct a Koppelman-Leray formula. for such open sets, where the Leray map from Theorf'm 2.5.5 is used. (Since the boundary of D need not be smooth, the boundary integrals in the KoppelmanLeray formula (1.12.1) will be replaoed by volume integrals.) In Sections 3.2 and 3.3 we prove that the solution of (fu = f given by this formula admits uniform cstimates. (Holder estimates are not obtained - however, see point 2 in the Exercises, remarks and problems at the end of this chapter.) Moreover, these estimates contain an estimation of u by the Hormander diameter (Subsection 3.2.1) of the support of /. In Scction 3.4 we prove a theorem on decomposition of singularities with uniform estimates for bounded holomorphio functions in D as well as for continuous functions on D whioh are holomorphio in D. In Section 3.5 we show that every continuous function on i5 that is holomorphio in D can be approximated uniformly on jj by holomorphio functions in some neighbourhood of D. Seotion 3.6 oontains the following theorem: Let X ~ (}ft be a complex plane. Then every bounded holomorphic fUllction / in X n D admits a bounded holomorphio extension to D. If f is continuous on D, then this extension can be chosen to be oontinuous on D. In Chapter 4 this theorem will be generalized to the case that X is an arbitrary complex Rubmanifold of some neighbourhood of jj and D is a strictly pseudoconvex open set in a Stein manifold.
3.1.
The Koppelman-Leray formula
In the theory developed in Chapter 2 for smooth strictly pseudoconvex open sets, the Koppelman-Leray formula (1.12.1) plays an important role. We would like to use this formula also in the present chapter, where the case of not necessarily smooth strictly pseudoconvex open sets is considered. Recall that the Leray map in Section 2.5 was already constructed for the general case of strictly pseudoconvex open sets with not necessarily smooth boundary. However, in view of the boundary integrals, we cannot immediately write the corresponding Koppelman-Leray formula. Therefore, in the present section, we give some modifications of this fomlula replacing the boundary integrals by volume integrals. The first step, which will be carried out in the following Subsection 3.1.1, consists in a modification of the support function cJ>(z, C). Roughly speaking, we will replace the Levi polynomial FQ(z, C) by FQ(z, C) - 2e(C), and so we obtain a function 4)(z, C) such that iP(z, C) = tP(z, C) for CE oD and (in distinction to tP(z, C)) ~(z, C) =F 0 for all z, CED. Using Stokes' formula, we can then replace all boundary integrals by volume integrals. In the present chapter we also need information about the dependence of the Koppelman-Leray formula on "small" changes of D. Therefore, together with D we
108
3. Strictly pseudoconvex sets with non-smooth boundary
also consider sets D", which are "close" to D, and we will do all constructions "uniformly in m". a.l.l. Construction of (I), . , tv. Let Dec Cft be a strictly pseudo convex open set, let 6 ceQ]" be a. neighbourhood of 8D, and let e be a strictly plurisubharmonic 0"· function in a neighbourhood of 8 such that D n 6 = {z E 6: e(z)
< O} •
(3.1.1)
Remark. We do not assume that ~(C) =F 0 ifC E 8D, and we also do not assume that D *8. Define N(e) := {z E 8: e(z) = O} and suppose that e =F 0 on 88, that is, N(e) c c (J.
(3.1.2)
By X, = x,(C) (j = 1, ... , 2n), we denote the real coordinates of CE C" so that C, = X, + ixJ +ft (j = 1, ... , n). Since N(e) c c (J and since e is strictly plurisubharmonic and 0 1 , we can choose numbers 8, fJ 0 and Ol.functions ail: on "8 such that the following estimates hold
>
dist (N(e), 86)
> 28 ,
(3.1.3)
i ac,8Ie(~eel! E,i) > 3 fJ I~II sup I ale (C) - aJI(e) I< P... , 'EO ac, aCIl nl
for all 0 =F~ e lJ" ,
inf
(3.1.4)
,eO J,II-l
ax,8Ie(z) I< 2nfJ
al!,)(c)
lax,
aXt -
8Xt
r C, z EO
f l
(3.1.5)
with
IC - zl ~ 28 •
(3.1.6)
0
We define ft 8e(C) F(z, C) := 2 ~ - - (C, - zl) -
j-l
a~
ft
1:
~J:-l
aJJ:(C)
(C, - zl) (CI: - z.) •
(3.1. 7)
(c, z E 0, IC - zl ~ 28) ,
(3.1.8)
Lemma A. We have the eatimate
and there exist a neighbourhood U c c (J of N(e) and Ol-/unotionB 4>(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)
=
Jar
C E N(e)
,
zeUuD.
(3.1.12)
109
3.1. The Koppebnan-Leray formula.
(iv) Let {~111}:-1 be a 8equence oj strictly pluri8'Ubharmcmic OZ-Junctions in a neighbO'Urhood of (j tending, together with the first and second-order derivatives, uniformly on 6 to e. Set F m(Z,
e) = 2
" I: i-1
8~ (C) " _111- I: aik(C) (Cf - z,) (CI: - Z.t) . 8C1 i,1-1
Then there exist an integer mo and (J1·function8 4>",(z, C)' by (P"" e by em, etc., (5) (P",(z, C), ~m(z, C)' M m(Z, e), and Mm(Z, e), together with thefir8t-order derivative8, uniformly tend in CE U, z E U u D to 4>(z, e),
<
Remark. The functions tP and tP m agree with the functions 4> given by Theorem2.4.3 for D and D m , respectively. The construction of these functions given below coincides with the construction in the proof of Theorem 2.4.3. However, in distinction to Theorem 2.4.3, we give a parallel construction of ,p'" and, moreover, we have to look at the dependence on m of both constructions. Proof of Lemma A. Estimate (3.1.8) follows from Lemma 2.4.2. In view of (3.1.3), we can choose a neighbourhood W of N(e) 80 small that
> 2s max 1()(e)1 < s2fJj3 .
dist (W, 80) and
(3.1.13) (3.1.14)
CeW
Now we choose open sets Uo and U such that D u Uo is strictly pseudoconvex and N(e) c c U c c Uo cc: W . (3.1.15) Next we prove that Re F(z, e) for
> s2fJj3
eE W,
Z E
W uD
C) - 2e(C) > s'P/3 ~ It - zl ~ 2e.
Re F(z,
and
with
s
(3.1.I6)
To do this we fix points eE if and z E W u D with e ~ It - zi ~ 2e. Then, by (3.I.I3), Z EO and it follows from (3.1.8) that Re F(z, C) ~ e(e) - e(z) {Jel . To· gether with (3.I.14) this implies
+
> -e(z) + ifJs? Re F(z, C) - 2e(C) > -e(z) + ipS2 •
Re F(z, C~ and
(3.1.17) (3.1.18)
Further, since z E () and z E W u D, at least one of the following relations holds: z E W or z E 6 n D. If z E W, it follows from (3.1.14) that e(z) el p/3, and if z E 6 nD, it follows from (3.1.1) that ~(z) O. In both cases -e(z) 1- SI{J. Together with (3.1.17) and (3.1.18) this implies (3.1.16).
<
< >-
110
3. Striotly pseudooonvex sets with non-smooth boundary
Now we shall construct the functions iP, iJ, M, M. In view of (3.1.16), for z E W u D with 6 ~ IC - zl ~ 26, we can define In F(z,~) and In (F(z, C) Choose a Goo.function X on ([Jl such that X(1]) = 1 if 11]1 ~ 6 6/4 and X(1]) I'll ~ 26 - 6/4. For CE W, z E W u D we define
+
f(z,C)
:=
if 6 ~ IC - zl otherwIse,
{8.[x(C - z} In (F(z, C»)J
o
~
C E W, 2e(C}). = 0 if
(3.1.19)
26,
and
]('1., C) := {B.[X(C -
z) In (F(z, C)
- 2e(C»)]
o
if 6 ~.IC - zl otherWIse .
~ 26,
(3.1.20)
Since F(z, C} is holomorphio in z, 1(z,~) andj(z, ~) are B.-closed G~l)-forms on W u D, and the maps W 3 C-+- f(-, C) and W ~ C-+- j(-, C), considered as maps with values in the Freohet space Z
T:
Z~. 1)( W
such that aT rp and Z E Uo u D
= rp
u D)
C)
cJ>(z, C)
cn
(z) ,
:= exp [-u(z,
(j)(z, C) : = {
:=
{
u D)
on Uo u D for all rp E Z(O,l)(W u D). Now we define for all CE Uo
u(z,~) := (Tf(-, M(z,
-+- COO( U0
C)] ,
u(z,
C)
:=
M(z,
C}
(Tj(., C» (z)
:= exp [-u(z,
F(z, e) M(z, C) if exp [X(C - z) In F(z, C) - u(z, C)] if (F(z,
e) -
2e(C)) M(z, e)
exp [x(C - z) In (F(z, C) -
,
2e(C» -
(3.1.21)
C)] ,
IC - zl ~ 6 , (3.1.22) IC - zl ~ 6 , if IC - zl ::;; 8 ,
u(z, e)]
if
IC - zl
~6 •
(3.1.23)
It is easy to check that cJ> and ~ fulfil conditions (i) -(iii). Consider condition (iv). In view of the uniform convergence em -+ e together with the first· and second· order derivatives, we can ohoose mo so large that the estimates (3.1.4) -(3.1.6) remain valid if we replaoe, for m ~ 'lng, (! by em. Then it follows from Lemma 2.4.2 that (3.1.8) also remains valid if we replace e by em and F by F m. It is clear that for sufficiently large m the statements (1), (2), (3) in condition (iv) hold. Finally, we choose mo so large that, for m ~ m o, (3.1.16) remains valid if we replace e by em and F by F m. For m ::2: mo we can then define Jm, j m, Urn, um, . li . m, Mm , t:P m, and (Pili replacing {} by {}"', F by F m etc. in (3.1.19) - (3.1.23). It is easy to check that the funotions so defined have the required properties. We only remark that for the proof of the uniform convergence 'U m -+- u and ftm -+ U on (U u D) x U together with the first· order derivatives, we have to use that U c c Uo, because the linear operator T: Z(O, 1)( W u D) -+ GOO( Uo u D) is only continuous with respect to the Frechet topology of these spaces, which implies uniform convergence (with respect to z) on compact Bubsets only. •
Lemma B. Let U be the neighbourhood from Lemma A, let 4>(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)
A
w'(ij(z, C, A)) " oo(C) ,
zED,
converges and depends continuously differentiably on z ED, where differentiation and integration can be interchanged. In particular, since dimB (D X [0, I]) is odd,
8, f I(C)
A
ro'(ij(z, C, A))
A
f
oo(C) = -
D x (0. 11
8,[J(C)
A
ro'(ij(z,
D x [0. 1]
c, l)
A
zED.
oo(C)] ,
(3.1.29)
Now let f be a continuous (0, g)-form on D such that
o ~ q ~ n. Since
(8c•• + d1 ) w'(1j(z, C, ;.) = nw(ij(z, C, it)
81 is
also continuous on
D,
,
we then obtain the relation d'.AU(C)
"m'(ti(z, C, A) "oo(C)]
=
8c!(C) 1\ w'(ii(z, C, l)
+ (-I)' n/(C) A w(ij(z, C, A» A oo(C) -
8,[f(C)
A
A
w'(ij(z, C'
oo(C)
it»
A
oo(C)] •
Since the singularities at C= z of the monomials in ro'(~(z, C, A) which are of degree 1 in A are of order ~ 2n - 2, we can apply Stokes' formula and, taking into account
113
3.1. The Koppe-lman-Leray formula
-= J -J -J
that 8(D X [0, 1])
ocf(C)
oD X [0, 1] - D X 1\
w'(ij(z, C, J.»)
1\
w(C)
°+
D X 1 (dimB D is even I), we obtain
+ (-I)f (n(2ni)fI RD/(z) - I)!
J
Dx[O,lJ
U.[/(I;)
A
ro'(ij(z, 1:, A))
I(!;)
A
1(1:)
ro(I;)] =
A
D x [0, 1]
A
ro'(ij(z, t A)
A
ro(!;)
3D x [O,IJ
£0' (fj(z, 1:, A)
A
ro(C)
J
+
1(1:)
A
ro'(fj(z, /; , AI)
A
zED.
ro(l;)'
Dxl
DxO
(3.1.30)
If C E 8D, then, by Construction 3.1.1, tP(z, C) = ~(z, C) = (w(z, C), C - z) and, therefore,
~(z, C, J.) =
~~z,_C)
(1 _ J.). .
+ J. C- z_.
Consequently,
Jf
1)
J-'('" (,,) 1\ w t'}(z, ",II.)
1\
'" w(",)
(2ni)ft = --- _.-
BCD.f
ZED,
8DJ(Z),
I)!
(n -
(3.1.31)
aDx[O,l]
where B:Dfw&s introduced in Section 1.8. Taking into account that ij(z, C, 1) = (C - i)1 Ie - '1,12 and using the same arguments as in the proof of Proposition 1.7.3, we obtain
1
f (C)
1\
ro'(~(z, C, A)) 1\ w{,)
zED,
(2ni)ft. BD/(z) , (n - 1)
=
(3.1.32)
Dxl
where BDf was introduced in Section 1.8. Since ij(z, C, 0) = X(C) w(z, C)/~(z, C) is holomorphic in z, we further obtain that
Jf(l;)
A
£0' (ij(z, /;, Al)
A
ro(l;) =
DxO
J
1(1;)
A
roc (x(~ :(z,
D
(
1;))
A
z =D.
rotC) ,
,e) (3.1.33)
+
Using the abbreviation Tq := (-I)f (R:D B D ), we obtain from (3.1.29)-(3.1.33) that, for every continuous (0, q)-form f on D such that 8J is also continuous on D, o ~ q ~ n, the following relation holds: RDf(z)
= Tqf(z)
+ (_I)fl+l (n(2nl)ft -.1)! [a z JI(C) " w'(ij(z, C, A»)
1\
co(C)
Dx[O,l]
+JI(C) 1\ (J)~ (~(~. w(z, C») 1\ w(C) + J Be/(C) 1\ w'(ti(z, C, A» C/J(z, e) Dx[O,lJ
1\
CO(C)] ,
J)
zED. Applying this to -
RD 8/('1,)
(3.1.34)
e/, it follows =
that
--
Tg+l 8f(z)
+ (-l)f (n.~-
..1)' _ . [-8z
J-
8r:/
1\
ro'(1i(z,C,l» "w(C)
Dx(O.l]
+ JaJ(C) D
S
HenJdn/Leiterer
1\
w, (X(C2 W(Z,C») "W(C)] , C) ~
'I,
ED,
(3.1.36)
114
3. Strictly pseudoconvex sets with nOD-smooth boundary
for every continuous (0, q)-formJ on D such that 8f is also continuous on D, 0 < q Further, by degree reasons for every such form,
1
!(C)
A
ro,(X(~ w(z, C») A cu(C) f/>(z,
D
and, since w(z,
= 0
E
D)
(z
E
if
q
=F I
n.
,
(3.1.36)
= 1.
(3.1.37)
C)
C) and i(z, C) are holomorphic
8, fl(l;) A (Ot(X(~ w(z, 1:)) A cu(C)
in z,
0
=
D)
if
q
C)
. f/>(z,
D
(z
~
If follows from (3.1.34), (3.1.36), and (3.1.37) that, for every continuous (0, q)-form 1 on jj such that is also continuous on D, 0 ~ q S n, -8R D !(z) = 8T'l!(z)
8.t
1
+ (_I)f+ ~-(21U)n -.I)! 8~
ad(C) A W'(i7(z,C,J.») Aro(C) ,
1
Dx[O.l]
ZED.
(3.1.38)
For q ~ I it follows from (3.1.36) that the last integral on the right-hand side of (3.1.35) is zero. Therefore, for I S q ~ 17., from (3.1.35) and (3.1.38) we obtain the relation RD 8J + 8R D ! = aT'l! + Tg-I-l 8!. In view of the Koppelman-Leray formula (1.12.7), this implies (3.1.28). Consider the case q = 0, that is, let I be a continuous function on jj such that is also continuous on D. Then, in view of the Koppelman-Leray formula (1.12.1),
I = L:Di + TI 81 , where L':D! was introduoed in Section 1.8. Since, for CE 8D, X(C) = <1)(z, C) = (w(z, C), C - z), from Proposition 1.7.3 we obtain L':J(z) = (n -. 1) J jl(C) (2nl)"
rot (X(~ w(z, C») C)
8D
A
81
(3.1.39) =
I and iP(z, C)
ro(C) .
(3.1.40)
'
Since
dcWe (X(~ w(z, C») = nco, (X(~ W(Z,C») , f/>(z,
C)
tJ>(z, C)
it follows from Stokes' formula that
L':of(z) =
(71(;:i)~ I [
J
(V(I;) A
(O~ (X(~(:(~; 1;1) A (0(1;)] + LDI(z) ,
zED.
D
(3.1.41 ) Further, since q
=
J ad(C)
0, by degree reasons, A
00' (~(z, C, J.»)
A
cu(C) = 0 ,
zED.
(3.1.42)
Dx[O.l]
From (3.1.35), (3.1.42), (3.1.41), and (3.1.39) follows (3.1.27). Now we pass to the general case, when not necessarily de(z) =F 0 for z E 8D, and! is an arbitrary continuous bounded (0, q)-form in D such that 81 is also continuouB and bounded in D.
3.1. The Koppelman-Leray formula
115
In view of Morse's lemma. (see, for example, MILNOR [1], § 2, Lemma A), there are real-valued real-linear functions p". in (J" such that /pml 11m on 8 (8 is the sa.me neighbourhood of N(e) as in Construction 3.1.1), and for each of the functions (/ p", there is no more than a finite number of points E 8 such that d(e p"') (e) = o. Then for every m we can choose em so that 11m em 21m a.nd dee Pm) (e) =f: 0 for all E"O with e(C') Pm(C') = -e",. Define em(C) = e(e) 9/m(e) Em. Then the functions em together with the first and second-order derivatives tend to uniformly on 0. (The second-order derivatives of em and e even coincide.) Therefore, we have the situation which is described in Construction 3.1.1, Lemma A (iv) and Lemma B (b). In view of the special choice of em, moreover:
<
e
e
< <
+
dem(z)
Dm
+
+ + + e
=f= 0 for z E aD",;
c:c:
+
(3.1.43)
D;
(3.1.44)
for every compact set K ~ D there is an integer mK such that} K C Dm for m ~ mx.
(3.1.45)
Let LDfII and RD ". be the operators defined by (3.1.25) and (3.1.26) after repla.cing w by wm and (p by ~m. Since w m ~ w and ~m ~ ~ together with the first-order derivatives uniformly in z E VI U D, and since ~m(z, C) =f: 0 (C E U n D, zED) a.nd X(C) = 0 (C' ED" Vv zED), it follows that for every fixed zED
e,
11m
w~
(X(')_ wm(z, e») -_ W, (X(e). .:----w(z, e»)
m-+oo
(3.1.46)
-~
cJ>m(z, e)
uniformly in CED. In view of (3.1.45), this implies that for every continuous bounded function f on D and each zED
lim LD..J(Z) = LDf(z) •
(3.1.47)
m-+oo
Similarly as in the case of (3.1.46) we obtain that for every fixed zED lim
Ie -
zl2n-l
£0
(1 _ X(e! A)
m-+oo
=
IC - zI2"-1
wm(z, C) 4)m(z, C)
+ A ( - Z2) Ie - zl
£0(1- A)X(~ w(z, C) + A C- z)
(3.1.48)
IC - zit uniformly in CED and 0 ~ A ~ 1. Since the singularity 1/IC - zl2A-l is integrable and
independent of m, this together with (3.1.44) and (3.1.45) implies
lim RD.f(z)
=
RDf(z)
(3.1.49)
m-+oo
for every continuous bounded differential form f on D and every fixed zED. Since for every Dm the theorem is already proved, from (3.1.47) and (3.1.49) part (i) of the theorem follows. To prove (ii) we first observe that (3.1.48) not only holds for every fixed zED but uniformly on every fixed compact subset of D. Consequently, (3.1.49) also holds uniformly on every compact subset of D. Since the theorem is already proved for every D m , this implies - lim 8RDm f =/ - lim R D", 8/ = f - RD 8/ (3.1.50) " ' ..... 00
8-
m-+oo
116
3. Striotly pseudooonvex sets with non-smooth boundary
uniformly on every compact subset of D and for every continuous bounded (0, q)form I on D such that 81 is also continuous and bounded on D. Taking into account tha.t (3.1.49) and (3.1.50) hold uniformly on every compact subset of D, we oonolude that 8RDf is continuous and (3.1.28) holds. • 3.1.4. Corollary. U8ing the notatiof&8 introduced in Definition 3.1.2, we have: (i) For every bounded holomorphic function f on D the following representation formula (Leray Jormula) holda:
I = LrJ in
D.
(3.. 1.51)
(ii) If 1 ~ q ~ n, then for every bouMed contin'lW'UB (0, q)-Jorm fonD, Buck that
8J=
°
in D, u : = RDJ
(3.1.52)
is a continuO'UB (0, q - I)-form on D which 80lves the equation Su =
3.2.
f.
Uniform estimates for the a-equation
In this section we prove that the solution of 8u = f given by Corollary 3.1.4 (ii) admits uniform estimates. Moreover, we shall give more preoise estimates by taking into account the HOrmander diameter of supple 8.!.1. The Bormander diameter. Let D, (!, 8 be as in the beginning of Construction 3.1.1. For every E E 8D and ~ 0 we define (cp. also Subsection 1.5.13)
>
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
:=
~
<~} ,
H~~):= .iiE(~) n D •
centered at
E E 8D with W
EI
~.
If W
~ D
c He(~)}
will be called the H armander diameter of W. Proposition. Let ~(z, C) and F( z, C) be tke functions introdueea in Oonstruction 3.1.1. Then there are constant8 0 00 and (X 0 BUCk tkat for all E E aD and ~ 0
<
e(C)
< O~I
for
IF(z, C) - F(z, E)I
>
>
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,(<<5) and I' - EI < 6, this implies that for some O. < 00 IF(z, C) - F(z, E)I ~ Oa(c5 + IId(}(E)IIIC - EI) •
therefore
1
This implies (3.2.2), because I' - EI ::;;; dist (e, Te) and CE He(<<5) (cp. the definition of He(6»
(2 i
Ilde(E)11 =
j8(!(E)
12)1/2 .
8E1 3.2.2. Theorem. Let Dec en be a 8trictly pseudoconvea: open 8et (with not neces8arily smooth boundary), let X, w, ~ be as in Definition 3.1.2, and let 1 =5: q =5: n. Then (for the definitions of the norm 1I·llo.D and the Banach spaoes Lw,,)(D) and C'fo.,-l)(D) J-l
see Section 1.8): (i) For every f E Lto:g)(D) and each point z E i5, the integral
RDf(z) :=
~ (2m)"
J
ICC)
A
W
(1 -
,l)
X(~ w(z, C) + Ii C4>(z, C)
(C.1)eDX[O.I]
Ie -
i)
AW(C)
zll
converge81 ), and the 80 defined (0, q - I)-form RDf i8 continuoU8 on D. (ii) ThR,re is a constant 0 < 00 such that for every f E Lro,g)(D)
IIRDJllo.D ~ 0 diamB (suppf) IIJllo,D , and, moreover, if E E 8D, c5 > 0 such that supp f c H e(c5), then for all z E jj IIR D f(z)1I
~
0 [o/diet (z, H,(0»]2"-1 c5llfllo,D .
(3.2.3)
(3.2.4)
(iii) By pam (i) and (ii) the integral RD define8 a bounded linear operator from Lro.,)(D) into £:10.,-I)(D). Thi8 operator i8 compactS).
3.2.3. Corollary. Let Dec (Dn be a strictly paeudoconvex open set (with not necusarily smooth boundary). Then there is a constant 0 < 00 8uch that lor every continuous bounded and a-closed (0, q)-form/ on D, the solution oj au = lin D, given by the formula u := RDj (cp. Corollary 3.1.4), i8 continuous on D and admit& the estimate lIullo,D ~ C diamH (supp/) II/Ho,D .
Moreover, if E E 8D, 0 > 0 Buch that Ilu(z)11 ~ C[~/dist
suppf~ Ht(~),
(z, H e(O)]2f&-1 c5ll/l1o,D.
(3.2.5)
then/or all zED (3.2.5')
Before proving Theorem 3.2.2, we give three lemmas which are also important for the subsequent sections of this chapter. 1) For ZED this integral was introduced in Definition 3.1.2, and in this case it is easy to
see that it converges. Here we sta.te the convergence also for Z E aDl II) Recall that a linear opera.tor between Banaoh spaces is called compact if every bounded set is mapped into a relatively oompaot set.
118
3. Striotly pseudooonvex: sets with non-smooth boundary
e,
3.1.4. Lemma. Let D, F, tP, j, and V. be as in Oonstruction 3.1.1 and Definition 3.1.2. Then there exist real-valued quadratic polynomial8 P(z, C) and Q(z, C) in the real coordinates oj C, w'h08e coeJJicient8 are continuous Junctions in z E Vs, such that the following e8timates hold: (i)
P(z, C) = 1m F(z, deP(z, C) (ii)
Q(z,
C)
+ 0(1(;' -
de 1m F(z, C)
=
e) = e(C)
- e(z)
+ O(IC -
+ 0(1C -
+ O(le -
de(e)
deQ(z, C) =
z12)
(iii)
IId,F(z, e) 1, __ "de(z)11 ~
zl)
vn 1
z12)
e,
for
Z E
zl) Jor
(3.2.6)
Vs ,
(;" z
E
(3.2.7)
Va.
(3.2.8)
C, z E VI'
for
for C, z E VI 11d(!(z)11 2 Jor
(3.2.9) (3.2.10)
z E VI'
< 00,
and Jor 80me oon8tant 0
IIdeP(z, e) "d,Q(z, (;')11
~ y~ Ilde(I:)II" -
0<11 de(l:) II II: - -I
+ II: -
> 0, 14>(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
+ IQ(z, e)1 + 1(;' -
z, CE VI n D .
(3.2.12)
Proof. Let XI = xf") be the real coordinates of (;' E (J" so that e1 = xl(e) Then
+ ixJ+n «(;,).
eX(lP(z, e}1
i
1m F(z, C) =
Oe «(;')
aXI
i-I
z12) for
(xJ+,.(e) - xJ+"(z)) -
Be«(;') (Xf(C) - Xf(Z» 8xJ+ft
2,.
+
l:
'Ujt«(;') (XI«(;') - Xt(Z» (XI:(C) - Xt(Z» ,
i.i-l
where
U,11:
are Cl-functions in a neighbourhood of
- { Be(z) 8Xi+"
+
+
Va.
l: Bx;+ft BI(!(Z) (x,(e) ax,
,-1
Define
X,(Z»)} (xI(e) - X1(Z»)]
2ft
l:
'U;k(Z) (X1(C) - Xt(Z}) (XI: (C) - Xj;(Z»)
1,1:-1
a.nd Q(z, C) :=
2,.
Be(z}
j-l
8Xt
l: 1
2,.
(X1(C) -
8Ie(z)
X1(Z»)
+ - 1: - - (X,(C) 2 J.i-l 8x, 8Xi
- Xt(z» (XI:(C) - Xt(z}) .
Then parts (i) and (ii) easily follow from Taylor's theorem.
(3.2.13)
119
3.2. Uniform estima.tes for the a-equa.tion
We prove part (iii). By definition of IldF /\ dell, we have IId,F(z,.C) 1,=. /\ de(z)11 2
!:
[8 Im"F(~, C)
l~j~k:i2"
oX1
2:
In view of (3.2.13), for j
=
I
,_.
8e(z) _ 8 1m F(z, C) 8xt 8xt
I
c-,
8e (Z)]2 8x1
.
1, ... , n ,
I
o1m F(z, C) I
o1m F(z, C) = _ °e(z) • = 8e(z) and ,_. OXt 8x;+" 8x1 c... 8x;+" Taking the terms with k = j + 110 only, this implies (3.2.10): Ild,F(z, C)I,-.
Ade(z)l12 ~ -£ [( 88e(Z) )2 + (8e (Z»)2]2 >
0XI
xJ+"
j-l
~ [~ (8 e(Z»)1]2 = .!.-llde(Z)II' . 110
8x1
j-l
n
Together with relations (3.2.7), (3.2.9), and de(z) = <.Ie(C) + O(IC - '1,1), this implies (3.2.10'). Finally observe that part (iv) follows from (3.2.6) and (3.2.8) and the estimates
+ 11m F(z, C)I + IC - '1,11) ~ a:(\e(C)1 + 1()(z)1 + 11m F(z, C)I + IC -
14)('1" C)I ~ a(I~(z)1 Ia>(z, C)I
('I,
E VI
'1,12)
n D, CE aD) , ('1" C E Va n D) ,
which hold in view of Construction 3.1.1, Lemma A, relations (3.1.8)-(3.1.11) •• 3.2.5. Lemma. Let D, (!, $, w, and VB be a.9 in Construction 3.1.1 and DeJini,gion 3.1.2. Then there is a constant C 00 such that
<
1U'(z, e)1 ~ O(llde(C)II
+ IC -
I~4>~t) I~ o(I~~) 1+
'1,1) for C,
IC - -I
'I,
(3.2.14)
E VI'
+
III(C)I) !or
c, z EV.;
j = 1, ...
,n,
(3.2.15)
and for every differential form f
IIf(C) for
A
8,q)(z, C)II ~ O[II!(C)
A
8e(C)1I
+ 11!(C)l1 (Ie -
'1,1
+ le(C)!)]
C, 'I, E VI'
(3.2.15')
Proof. In view of Construction 3.1.1, Lemma A, condition (ti), and Lemma B, property (a)
(w(z, e), C - '1,) = CP(z, C) = F(z, C) M(z, C) Consequently, for j = I, ... , nand C,
w/(z, C)
+ O(lC - zll
=
'I,
for
e,
'I,
E Va
with
Ie - '1,1
~ 8.
E VI with Ie - '1,1 ~ 8,
8F~~: C) M(z, CH F(z, C) 8~~: C) .
Since 8F(z, C)/oC1 = 28e(e)/OCI + O(IC - '1,1), this implies (3.2.14). In view of Construction 3.1.1, Lemma A, condition (ii),
~(z, C) = (F(z, C) - 2~(e)) M(z, C)
(e, 'I,
E VI;
Ie - '1,1
~
Since 8cF(z, C) = O(IC - '1,1), thif( implies (3.2.15) and (3.2.15') . •
8) •
120
8. Strictly pseudoconvex sets with non-smooth boundary
8.1.8. Lemma. ut D, e,~, and VI be a8 in OOflt8tructicm 3.1.1 and Definition3.1.2. 00 8uck that Jor all ~ E 8D, ~ 0, z E jj n VI' and Phen tkere 6zi8tB a constant 0 x = 1,2, the following eetimate8 are fulfilled:
<
f
(i)
>
~ Od" min {I
d0'2"
Ie - zI2"-" -
,
d2tl -
H}
,
'EBe(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
<
f
d0'2ft
IC -
:s; Gb" .
zI2n-" -
H~I)
Moreover, it is clear that
f
~h:s;
Ie - zI2n-" - [dist
(z,
1
fd
H,(b»]2ft-K
CEBe(c1)
~G
0'2n -
~
[dist---:(z-,-H-,-(b-»-]2n---x'
Be(")
(ti). Since for CE He(b) n VI' IIde(C)1I ~ de(~) zlll , we have
+ GtJ,
and, by (3.2.l2), I~(z, C)\
~ <x Ie -
Since
fda•• ~ 0{::+9l11de(~)II.
if if
Ilde(~)l1 ~b,
IIde(EHI
~
(3.2.16 )
tJ ,
BI<")
this implies that
f
Ilde(C)1I
dU2ft
~0
li(z, e)11C - zI2n-l-H -
~2n+l. [dist
_
(z, H,(tJ»1 2ft + 1 -
• x
'EY. nHe(c1)
This proves part (ii) for the case the case
a < 1, that is,
diet
(z, He(6))
> tJ. Now we consider (3.2.17)
3.2. Uniform estimates for the a-equa.tion
121
From (3.2.9) and (3.2.12) 'it follows
f
Ilde(e)1I dO'Ita
CeV.nBe(lJ)
~0 -
I~(z, e)lle - zrm- 1-"
f
(IQ(z, C)I
Ild,Q(z, C)II d0'2ft zll) Ie - zI2ft-1-"
+ IC -
+
0
V,nB~lJ)
f
~~----
Ie -
zI2"-,, .
teBfC.J)
The second integral on the right. hand side can be estimated in view of part (i). Since IQ(z, e)1 ~ 0 IC - zl, the first integral on the right.hand side can be estimated by the integral \\d,Q(z, em d0'2" (IQ(z, C)I (IC - zl IQ(z, C)1)2) (Ie - zl IQ(z, ')I)2n-l-H'
f
+
+
+
V,nH!<'>
In view of Proposition 4, Appenduix 1, this integral can further be estimated by
-_ _ (I~I f + ItF~) d_~
dt2" _ ItI 2ft - 1-" ,
(3218 .. )
A •••. A
te BI".ltl < R. t1EQ(Z. Be<'»
where R : =
sup
(Ie - zl
CeHe(")nV.
+ IQ(z, e)1) ~ 06.
In view of (3.2.17), it follows from
(3.2.1) and (3.2.8) that for all CE VI n H,(6) IQ(z, e) - Q(z, ~)I ~ I~(C)I
+ 0 Ie -
~pa ~
061
(3.2.19)
•
Consequently, (3.2.18) can be estimated by
~
f
te flI", ItI < OlJ, It1-Q(z, e)1 :;;oeS'
(1t,.1
". .
A
+ It12) ItI
dt2ft 2tl
-
1
-
N
'
which is ~ Ollc as follows from Proposition 5 (ii), Appendix 1. (iii). For diet (z, He(ll» ~ 6 we can usc similar arguments as in the case of part (ii). 'Ve omit these arguments and consider the case (3.2.17). It follows from (3.2.10') and (3.2.12) that
f
teV.nHE<")
~0 -
Ilde(e)112 d0'2ft I~(z, C)1 2 1e - zI2"-2-,,
f f ;.;.
Ild,P(z, e)
(lP(Z, C)I
A
d,Q(z, C)II d0'2" zll)II' - ZI2"-2-~
+ IQ(z, e)1 + Ie -
CeV.nHe<")
+0
Ilde(e)1I d0'2ft I(P(Z, e)1 Ie - zI2"-1-,,
V.nGHe(lJ)
+0
J_ teH!<")
d0'2ft_. IC - zI2"-"
The last two integrals can be estimated in view of parts (i) and (ii). Since IP(z, e)1
+ IQ(z, C)I
~ 0 Ie -
zl ,
the first integral on the right-hand side can be estimated by the integral
f teV.nBe(lJ)
(IPI
+ IQI + (IPI +
Ild,P A d,QII dO'z" IQI Ie - zl)l)1 (lPI
+
+ IQI + IC -
zI)2"-2-'"
122
3. Strictly pseudoconvex sets with non-smooth boundary
In view of (3.2.19) and Proposition 4, Appendix 1, this integral can further be estimated by ~ 1\ ... 1\ dt21l
f
'Ell''', 1'1<01. II, -Q(~ f)\ :a C~·
(I~I
+ Ital + Itl l )lltl 2n - 2 -
N
'
The required estimate now follows from Proposition 5 (iii), Appendix 1. • 3.2.7. Proof of Theorem 3.2.2. We shall use the notations 0 and lX for "large" and "small" constants, respectively, as was explained in the beginning of the proof of Lemma 3.2.6. I. Estimation of RD/(Z). For W c jj and! E Lto.r)(D) we write
Il J
1\
co
[(1 - A)X(C2{P(z, C)C) + AIeC-- zI2z ] 1\00(C)11 '
We want to prove that for all ~
E
aD, «5
1w(z) :=
I(C)
x(O.l]
1H,<')(z) ~
w(z,
ZED.
> 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<4>'V.(z) ~ Oll/llo.D
leDn V
J
E,<~)nD
do-2ft
•
Since, clearly, [ do-23 ~ O~ min {I, [~/dist (z, He(~))J2A-l} ,
ZenD
this together with (3.2.21) implies that for the proof of (3.2.20) it is sufficient to show that zEDnV. (3.2.22)
Taking into account that x(C) = I for (cp. Definition 1.7.6 for det A )
CE V2 , we obtain for C, z E V2 n D with C=1= z
3.2. Uniform estimates for the a-equation
Since
8zw(z, e) = (8
c,
0 and 8.4>(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 L
lim
=0.
IIRi>fllo,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
< <5 < 1
.DnEf(~)~ He(Vr~) ,
where r := max Then for all
te8D 1', 0'
Ilde(~)l1
+ 1.
(3.2.25)
For
0'
>0
we set U(O') := {z ED: dist (z, aD)
< O'}.
> O,! E L('O,f)(D), and Z E U(O') there is some ~ E aD such that
supp X~(C - z) fCC)
C
E,(.,;)
C
E t (.,;
+ 0') .
In view of (3.2.25) and (3.2.20) this implies that for every! E L(8,g)(D) and all T, 0' with T + 0' < 1
On the other
co
11/11 0, D Vi
+ hand, for every fixed 0' > 0, the Ii) X(C2 w(z, C) + A C- z ),
sup II R1/(z) II ~ 0 .zEU(a)
(1 _
c;J)(z, C)
Ie -
(3.2.26)
(1 •
~onomial8
which are of degree 1 in A, can be estimated uniformly in zIZta-l. Hence for every fixed 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,
<
00
1\
co(C) (3.3.2)
converge8. (ti) There exists a con8tant 0 f E L(8.l)(D) 8uch that
w*(z, C») t/J*(z, C)
with the. following property:
suppJe He(~)l) ,
II, E 8D, ~ > 0, anti (3.3.3)
II/(C) 11~ I/~ + Ilde(C)II/~2 Jor CE H~(t5) n Vs , 11/((;) 1\ oe(C)l1 ~ Ilde(C)II/~ for CE He({) n VI'
(3.3.4) (3.3.5)
then Jor all z E 8D IIR~/(z)1I ~
o min {I, [t5/dist (z,He(~»)]2ft-I}.
(3.3.6)
Proof. We shall use the notations 0, 0', ... and /x, /X', ... for "large" and "small" constants, respectively, in the sense explained at the beginning of the proof of Lemma. 3.2.6. It is sufficient to show that for every form J E Lro,l)(D) with the properties (3.3.3) to (3.3.5) (for some ~ E aD and t5 0) the integral (3.3.2) converges and satisfies estimate (3.3.6). In the general case the convergence of (3.3.2) then follows by choosing t5 80 large that D c H,.(t5). For this purpose, let t5 0, ~ E aD, and J E Leo. l)(D) so that
>
>
1) H,(d) was defined in Subsection 3.2.1.
126
3. Striotly pseudooonvex Bets with non-smooth boundary
(3.3.3)-(3.3.5) hold. For every W
J
I w(Z) : =
1J(l;) It.
C
D and Z
E
an we shall write
((1 - J.) ~(l;~lP(z,w(z,C)C) + J. w*(z, C)) lP*(z, C)
W
(C, A)e W x [0,1]
1\
w(C)
I.
We ha.ve to prove that
z
E
oD.
(3.3.7)
At first oonsider ID'\. v.(z). Since fP*(z, C) = lP(C, z); it follows from Construction 3.1.1, Lemma A, condition (ii), and estimate (3.1.8) that min {14)(z,C)I, 14>*(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 ~(<5)
f
dO'2n
Iit<~)
~ 0 min {I. <5211 , !52n+2/1Ide(;)112} ,
this, together with (3.3.4), implies that for all Z ID'\.v.(z) ~ 0 min {I, ~2n-l}
E
oD
< 0' min {I, [t5Jdist (z, H;(6))]2n-l} .
To complete the proof of (3.3.7), therefore, it remains to estimate the integral IDnv.(z). Using determinants (cp. Definition 1.7.6) and taking into account that X = I on V2 , we obtain for CE VI n D and z E aD
ro ((1 -1) x(C~ w(z, C) + 1 w*(z, C)) lP*(z, C)
4»(z, C)
= .!... det" ((d). + 8 )[(1 _ c
n!
-
-
Since a,w*(z, C} = 0 and o,lP*(z, C) (d,\
+ 8 ) [(1
_ 1) ~z, C) (j)(z, C)
C
1) ~z, C) 4»(z, C)
=
lP*(z, C)
0, we have forC E Via n D and Z E aD
+ J. ~*(z, C2] 4)*(z,(;)
= (w*(Z, C) _ w~z, C)) d). + (1 _ (l}*(z, C)
+ 1 w*(z, C)]) .
4)(z, C)
1) (
8~w(z, C) lP(z, C)
_
~(z'i) 8cq,(z, C)) . ~2(Z, ~)
Expanding the determinant (using the same arguments as in the proof of Lemma 2.2.1), we thus obta.in for z E aD and C E V 2 n D
ro ((1 _ J.) w~z, C} + A w*(z, C)) = lP(z, C)
lP*(z, C)
p(l) dJ..
1\
det 1•n _ 1 (~z' C) 4J(z,C)
, ~:..w(Z, C)) 4){z, C)
(3.3.8)
127
3.3. Improvement of the estimates on the boundary
where p(l), q(l) are some polynomials in l. From (3.2.14)1 (3.2.12), (3.3.3), and (3.3.4) it follows that for z E aD and ~ E Va n D
II
J(C)
A
det1,"_1 (~z' e) , e.:.w(z'~»)11 (]>(z, e) (1)(z, ~)
~ 0' ~Jlde~~~_"_ ~ I(]>(z, C)lle - zl2ta-2
II de(e) II 2
0'
+Tll u l4>(z, CW' IC -
~ O(~+ !.I.~()(Cl!!)(II~(C)1I + ~ ~
~lI
+ 0' _ _1
~ Ie - zI2"-1
IIde(C)1I
0'
J(~) tHlet
1,1,,,
+ O(lC -
Since le(C)! z
E
aD, ~
E
A
8~tP(z, C) II
1
Iw(z,
C)I Ilf(~)
~ O(lI de(C)l1
A
E
C)III!(C)
aD and in view of (3.2.14) and (3.2.15') we obtain for
8ctP(z, C) II
+ Ie -
~ O(lIde(C)1I -
<
A
zl) (lIf(C)
1\
+ IC -
zl) ("de(C)!1 ~
2
G'
+ II!(C)IIIC -
oe(C)11
E
zl) •
aD, C E Va n D
8c4>(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)
<5
Zlllde(C)II) ~ll
zlllllde(C)II) .
Together with (3.3.10), (3.2.11), and (3.2.12) this implies that for z
l'f(C)
il
S;;
1\
E oD,
CE Va n D
det l ,l,n_2 (w*(z, C)., w(z, ~ a;(]>(z, C) , \W(z, C2)11 (1)*(z, C) (z, C)
~ [-;;.; __ "de(C~_~2. _
- b I(]>(z,
+ 0 [_ ~2
zl), we have
_ (w*(Z, C) w(z, C) 8~~(z, C) 8~w(z, e»)11 2 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
+ _
IIdQ(~)1l2 _ _ + ___. IIde(C)1I ] zI2n-' I0(z, ~)/ IC _ zI2,,-8 .
14>(z, C)1 2 1C -
(3.3.11)
128
3. Striotly pseudooonvex sets with non-smooth boundary
It follows from (3.3.3), (3.3.8), (3.3.9), and (3.3.11) that for z
~~
ID y(z) n.
J J --
IC - z12n-1
6
CeH!<')nV.
+0 6-
11
1
CEB!<')n v.
ItP(z, CHIC _ z12n-2
t5
CEBe(d)
+~
aD
J ~ Ilde~C~ll~~ + J_
+~
d0'2n
~
-
E
0 ~s
CeHe(d)n V.
11
I
CeB ')nV.
11
The required estimate now follows from Lemma 3.2.6. • Notation. Let LI := {A = (AI' As) : 0 ~~, As =:; 1, At + As = I} endowed with an arbitrary orientation. In view of estimates (3.2.11) and (3.2.12) in Lemma 3.2.4, for every fixed z E aD the differential form
co (Al X(C~ w(z, C) + AI W*(Z, 9 + (1 (/)(z, C)
A(8
:= i-I
As) E-
i )
IC -
+ d A) (AI X(C~ Wf(Z, C) + As wj(z, C) + (1
C;P(z, C) is continuous for (C, A) E (D" z) X LI. '
_ At _
(/)*(z, C)
zl-
8.S.I. Lemma. For etJery z E aD there ezi8ts a comtane 0 bO'UMet11-Jorm fonD and aU 6 > 0
J
1(~) :=-
<
it)
As) ~ -
- Al _
/c - zit
C;P*(z, C)
00
8uch tkat Jor every
I/(C)
CED.IC-.JI-',AEA 1\
ro (AI ~(z, C) + As w*(z, C) + (I C;P(z, C)
_ Al _ As)
C;P*(z, C)
C-
Ie -
i )
1\
w(C)
zl2
I
~ 0 IIJllo.D d lIn tSl • (3.3.12) Proof. We shall use the notations 0,0', ... and ee, ee', ••• for "large" and "small" oonstants, respeotively, in the sense explained at the beginning of the proof of Lemma 3.2.6. Since a,w*(z, C) = 0 and a,c;p*lz, C) = 0, and since X(C) = 1 for IC - zl ~ ~ if z E an andtS is small enough, we have for all sufficiently small d > 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
+
(3.3.17)
If de(z) = 0, then it follows from estimates (3.3.13) -(3.3.17) and estimates (3.2.11), (3.2.12) in Lemma 3.2.4 that
1(6) ;,,; 011/110.»
J I' ~:;:_. :s:
O'l! liillo,D ,
lC-zl-'
where da~ft-l is the volume form of the sphere IC estimates give 1(6)
< Ollfllo,D
J-- d~!!~ +
IC-zl-'
14>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 < <50 of z such that ~(C) = e(C) and t2 (C) = 1m F(z, C). Now it follows from (3.3.18), (3.3.19), and (3.3.21) that for <50 > 6 > 0 16 ~ Cllfllo,D d~:~_~l OllflLo,D da~n-l ( )b2n - 8 I~(C)I + b2 + <52n - 6 (I~(C)I + It2(C)I + 62)3 •
J_
J
1~-zl-eS
I'-zl="
In view of Proposition 6, Appendix 1, this implies the required estimate. •
3.3.3. Lemma. Let f be a continuous bounded (0, I)-form on D such that 8f = 0 in D. Then Jor all z E 8D R~j(z) = RDj{z) ,
where R~ is defined in Lemma 3.3.1 and RD i8 defined in Theorem 3.2.2 (i).
Proof. Fix z E aD. We would like to apply Stokes' formula to the differential form f(C)
1\
W(AI X(~ w(z, C) tP(z, C)
+ Ai w·(z, C) +
(1 _
tP·(z,C)
Al _ A C-= Z ) 1\ w(C) 2
)
IC -
(3.3.22)
Zl2
on D X A. However, this form has a singularity ate = z. Therefore, we introduce open sets DeS as follows: Let K c aD be the set of points CE aD with de(C) = o. In view of Theorem 1.4.21, K is locally contained in a smooth real n-dimensional submanifold. 0 we can find a neighbourhood U K(-r) of K with piecewise Therefore, for every T smooth boundary au K(T) such that
>
~ 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.(<5) be the open Euclidean ball of radius tS centered at z. Since for every fixed c5 0 the form (3.3.22) is bounded on (D"-.E.II(c5)) X A, and in view of (3.3.23), for every c5 0 we can find TeS 0 so small that
>
>
>
J
>
f
[Dn(8UK('r.m B.("»]x.d
I
1\
W (~x~ + Ai w· t/J tP·
+ (1
-
~-
Ai)
~= Z ) 1\ W(C)I :::;: f5 • I~
-
zlB
_
(3.3.24)
8f
Define D" := D '- (E.(c5) u U K(TeS)). Since = 0, the form (3.3.22) is closed on D" X L1. Consequently, it follows from Stokes' formula and estimates (3.3.24) and (3.3.12) that
J
lim [ ' ... 0
f
OJ (AI
1\
X~ + (1 (/J
("AJJ)ED'X(~I~ W (A2 w· + (1 _ tP· (~,A.)eDeS
+
AI)
A) z
C- z ) /\ w(C)
IC -
Z[i
C- z ) 1\ OJ(C) IC - zl2
x[O, 1]
J
(c. A.)eD, x [0.1]
(1 - '-,) ~ + 2.;:) "W(C)] ~ O.
J" W
(3.3.25)
131
3.4. Decomposition of singularities
<-w(C, z), C- z) = C/J(C, z) = cJJ*(z, ,) and, therefore, w* C- z ) O~AaS:I, ( ~ cp* + (1 -~) IC _-~I!' C - z = 1 for CED,
Since <w*(z, C), C- z) =
we obtain that (J)
( cp* + w* Aa-
(1 -
-) IC - zl2
-C - Z As) -----
=
tI
;=1
+ at) (. Aa- + (1 cP* -
1\ (d" I
Wj
Cf - zf --) Ie - zlS
- A2 ) - - =0.
Consequently, the second integral in (3.3.25) is zero. Hence (3.3.25) implies the required equality R~f(z) = RDf(z) ••
en
3.3.4. Theorem. Let D c: c: be a strictly pBeudoconvex open set (with not nece8sarily smooth boundary), and let (! be a strictly pluri8ttbkarmonic C'-function in some neighbourhood (J of an Buck that n n (J = {z E (J : e(z) < O}. Then there ezms a constant C 00 with the following property: Let E E aD, <5 0, and let f be a continuous, bounded, and a-closed (0, I)-form on D such that the following estimates are fulfiller], (for the definition of H!(<5) see Subsection 3.2.1):
<
>
suppf~
(3.3.26)
He{<5) ,
Ilf(C)l1
~ ; + II~~C)II
Ilf(C)
8e(C)11 ~ II de,,(C) 11 for CE H!(~)
1\
for
CE HE(~) ,
(3.3.27) •
(3.3.28)
Then the solution of 8u = f given by the formula u := RDf is continuous em D (op. Corollary 3.1.4) and admits tke estimate lu(z)l
s
C min {I,
[~/dist
(z, HE(~))J2n-l}
Proof. Since by Lemma 3.3.3, for z from Lemma 3.3.1. •
E
aD,
,
z E aD.
(3.3.29)
RDJ(Z) = Ri,J(z), the statement follows
Remark. Estimate (3.3.29) was proved for Z E aD only. It is not clear whether it also holds for zED.
3.4.
Deeolnposition of singularities
First we observe the following proposition, which is a simple consequence of the Leray formula proved in Theorem 3.1.3 (i) and the estimates given for the operator RD in Theorem 3.2.2 (i):
en
3.4.1. Proposition. Let D c: c: be a strictly pBeudoconvex open set (with not -necessarily smooth boundary), and let U t ~ en (j = 1, ... , N) be open 8ets such _
tha,t D
N
U U j • Then every bounded holomorphic function Jon D admits a decompo8ition
~
i-I N
f
=
'Lf1,
(3.4.1)
j=l
where every 11 is bounded and kolomcwphic in some neighbourhood oj D " (aD n U1). 1/, nwreover, f a.dmits a continuous continuation to D, then the functions fl also admit ('ontinuoua continuations to D. 9·
3. Strictly pseudooonvelt sets with non-smooth boundary
132
Proof. Choose Ooo-functions XI on 0" such that l: XI = 1 on D and XI = 0 on U f , and definelt := LDXJ, where LD is the operator defined in Definition 3.1.2. ~ince XI = 0 on 0"" U then It is holomorphic in some neighbourhood of " D" (aD n U1). Since, in view of the Leray formula stated in Theorem 3.1.3 (i), J = LDf, (3.4.1) follows. Iff is continuous on D, then all 1, are continuous on jj since, by Theorem 3.1.3 (i), 0" "
11 = x1I - R D!aXI, and sinoe, by Theorem 3.2.2 (i), the functions RDJ eXt are continuous on D . • The aim of the present section is the proof of a more precise result. Notation. Throughout this section D c: c: 0" is a strictly pseudoconvex open set and U is a strictly plurisubharmonic OS.function in a neighbourhood 8 of aD such that
'1' ...
D n 6 = {z E 6: e(z)
< O} •
If ,ENE aD and" > 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<.f5) waB defined in Subsection 3.2.1.
3.4. Decomposit.ion of singularities
133
for CE C". Therefore, we can assume that
>
li~(E)l1 2d. Now we choose O°O·functions 9'(A) on
o~
~
- 9' -
1
on
,
9'
on,
S;
Oall d e(E)II/ll 2
Ildtp(l)l\
~
08/d •
1}(l), ••• ,.o(n-l)
2
- IId()(,)1I
0
for for
I
for
< d1lllde(E)11 , > 2d1lllde(E)l1 , 11]1 < d , 11]1 > d3/2 ,
IAI IAI
(3.4.8) (3.4.9)
(3.4.10)
,
(3.4.11)
{C - E: CE T t }. Define 1}(n)-
{I
() = { tp'YJ o for
IId9'(A)1I Further, let
and tp('YJ) on fIJ"-l such that
< 00,
and for Borne constant 0 8
:=
(,1,) =
(J1
on
O~V'~1
(3.4.7) (J1
be an orthonormal basis of the linea.r space T t - E
{8o(l:)/8~}n .. ~
~1 i-I·
Since (! is real and, therefore,
Ilde(E)II
=
~ 2(l: 1 8e(E)/8E,1 2 )1 / 2 ,
j-l
then
{}(l), ••• , f)(n)
is an orthonormal basis in C". Let PI
coordinates in (/)'" such that C=
= P,(C), CE C",~ be the complex
l:" P,(C) {}(j), CE on. Then J-l
2 n P,,(C) = - - l:" 8e(E) C, , CEO. (3.4.12) II de (E) II j - I 8E I Define X(C) = rp(P,,(C - E)) V'(Pl(C - E), ••• , Pn-l(C - E)), and check the conditions
(3.4.4)-(3.4.6). _ n-l Let CE He(ll). Then (I: IPI(C - E)lI)I/1 i-I
s: IC - EI
~
II and
IPn(C - E)I = dist (C, T , ) ~ dlllllde(E)11 . Consequently, by (3.4.8) and (3.4.9), X(C)
= 1. n-l
on and x(C) =f= O. Then by (3.4.8), (3.4.9), and (3.4.7), (1: IP,(C J-l and IPn(C - E)I ~ 2b /llde(E)11 < d. It follows that It - EI = (l:" IPI(C - E)II)1/2 < (9d /4 + tJl)1/1 < 2tJ Let C E
E)II)1/2 < tJ3/2
l
l
3-1
and IIrle(E)II dist (C' T~) = II d e(E)IIIP.(C - E)I
< 2tJ
lI
= (y'2t5) I
,
that is,
CE H,(y2d)
c:: c:
14(2d) .
It remains to prove estimates (3.4.5) and (3.4.6). Since D(l), ... , DC.) is an orthonormal
basis in U', we have 18Pf(C)/8Ckl ~ 1 for CE (J'f& and j,1c = 1, ... ,n. Together with (3.4.8)-(3.4.11) this implies tha.t for some constant 0, 00 d II x(C)II ~ 0 4(1/<5 IIde(E)ll/tJ' ) .
+
<
134
3. Strictly pseudooonvex eets with non·smooth boundary
Taking into account tha.t for CE H~(2d) n 8 Ilde(~)11 ~
+ Ie -
Ilde(C)1I
~I ~
Ilde(C)1I
+ 2~ ,
we thus obtain (3.4.5). From (3.4.12) it follows that
8~(fJ..(C - ~)) = 8cp~,i) I
dJf,.(C -~ ~)
A-jJ,.('-~)
8,i
8
=
i
2
"-fJ,,(C-l) IIde(~)1I j - l
8,i = 8cp(,i)
-
I
2
"-jJ,,('-~)
8,i
11,,1.n(I:)11 ~ ~
8()~) de1 8~1
(8 (~)
0(1'" -
~I).
P.. -l(C -
~»
+
(!"
"
I:
)
Consequently, for CE 8, 8X(C)
A
+ 8
8()(C) = q;(P..(C - ~»
I
6C1jJ({31(C -
~), •.. ,
'P(tJ,.CC - ,), ••• ,fJ.. -l(C - ~»
O(IC -
~I)
A
A
8e(C)
8e(C) •
II de(~)11
l-JJ,.(C-t)
In view of (3.4.8)-(3.4.11), we can therefore conclude that for some constant C6 and all CE 8 -
-
< 00
Ca(llde(C)ll/d + IC - Elllde(C)II/t5 I ) • This implies (3.4.6), because IC - EI < 2~ for C E He(2d) . •
118X(C) A Be(C)1I
~
Proof of Theorem 3.4.2. In view of Lemma 3.4.3 there are COO·functions if (j = 1, ... , N) on (J" such that 0 ~ Xt ~ 1 on QJ", Xl = 1 on iiEJ(~/2), sUPP %1 c: c: jj~J(~)' and for some constant C1 00
<
118Xf(C)l1 ~ Cl(l/~ 118X1(C) Define Then
Xo:=
A
+ IldaCC)II/d
l
)
Se(C)" ~ 0 1 IId()(C)"/~
for
CE 8 ,
for CE f)
(I - Xl) ••• (1 - iN) and XI =
•
Xt/CXo + ...
+ iN)
for j = 0, 1, ... , N.
N
l: X,
= 1
on
Xo = 0
on
QJ" , N
J-O
U H el(d/2) ,
(3.4.13)
i-1
supp X1 c: C ffeJ(~) and for some constant O2
for j = 1, ... , N ,
< 00
II 8Xt(C) II ;;;i OBd(~I' ... , ~ N; d) (lId
CE 8 -
for
118X1(C)
for
A
and j
=
+ Ild(>(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
<
sup IRDI aXtl ~ Osd('l' ... , EN; <5) 11/11o.D for j = 1, ... , N • zEitD
Taking into account that, by Theorem 3.1.3 (i),
11 = xl1 - RDJ 8X1 ,
(3.4.16)
,e
+
we thus obtain that I/f(z) I ~ (1 20ad('1' •.• N; <5)) Iln/o,D for all zED which are sufficiently close to aD (/1 need not be continuous on D). By the maximum prin_ ciple for holomorphic functions this implies (3.4.2). Finally, we consider the case that I is continuous on D. Since, by Corollary 3.2.3, for all j = 0, ... , N the function RDf 8X1 is continuous on D, and since (3.4.16) holds for j = 0, ... , N, we then conclude that the functions Jo, ... ,fN are continuous on D, too . • For functions which are continuous on D, now we shall give a more precise result, which will be used in the next section to prove an approximation theorem for holomorphic functions in D which are continuous on D. First we introduce SODle notations. Notation. Let D, Q, F, $, tfJ, w, X' and N«(}) c: C VI c: c: 0 be as in Construction 3.1.1 and Definition 3.1.2. Choose a OOO·function 1} on (fJ'fl such that 1) = 1 in some neighbourhood of D and {} = in some neighbourhood of qyn", (D u V s). By Construction 3.1.1, Lemma A, condition (ii) and estimate (3.1.8L
°
eE Va " D . Consequently, for every Ol-function ffJ on en and all zED, we can define tfJ(z,
C) =F 0 for zED and
L
(z): =
aDflJ
- (n - I)! (2xi)n
J
8({}(J-) I"
q.>
(C))" w~(w(z,
en "
w(C) •
(3.4.16')
V."D
Remark that, by Stokes' formula, in the case of smooth eD, LaDflJ = L:DflJ, where L:D is the boundary integral introduced in Subsection 1.8.6. Further it follows from Stokes' formula that the definition of LaDffJ is independent of the choice of {} (also for the general case when aD is not necessarily smooth). 3.4.4. Lemma. Fm< every Oi-function ffJ on qyn we have LaDflJ(z)
= LDflJ(Z)
+ (11. -.1)! J8flJ(~) " (J)~(X(e) ~(zJlt ~!l}_(e) Jor
(2m)n D
'
C)
where LJ) is the operator introduced in Definition 3.1.2.
Proof. Consider the differential form 'Y on Cf(C) w~(X(C) ~(z,
C)) " w(~!
en defined by if C ED,
~n(z, C)
y(C) : =
I
P(C) 'I'(C)
Wt(W~.C)~ C~ w(C)
if CE V, " D •
zED,
(3.4.17)
186
3. Strictly pseudoconvex sets with non-smooth boundary
Since X(C) = 1 for CE VI a.nd ~(z, C) = 4)(z, C) for CE aD, the form i' is continuous on eft. Taking into account that d wC(w(z, C))
,
" w(C)
for CE
= 0
VI
<1»"(z, C)
(cp. Proposition 1.7.1) and
d, w,(%(C)
~(z, Cn "w(C)
=
nw,(X(~ w(z, C») "w(C)
<1»"(z, C)
for C ED
U
VB
<1»(z, C)
(cp. Proposition 1.7.3), we see that dy is continuous on eft" aD and admits continuous continuations from eft" D to aD and from D to aD, where dy{C)
= 8(D(C) tp(C)) " waw(z, C)) "w(C)
for CE
4>"(z, C)
e" " D
and dy(C) = 8tp(C)
A
w~(X(C) ~(z, Cn "w(C)
+ nq>(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) <1»ft(z, C)
81<")
Proof. In view of Construction 3.1.1, Lemma A, condition (ii), and estimate (3.1.8)
inf I~(z, c)1 > 0 and inf I$(z, C)I > O. "D. V. .ED, V,. 'ED Further, dah < 0 0 min {din, ~1ta+2/11dq(E)IIZ} for some 0 0
f
'ED,
BI<")
< 00 such that IIf(C) II ~ 01(1/d + Ilde(E)lI/~I)
<
00
and, by (3.4.18),
there is a constant 0 1
I,v.
This implies that for some constant 0" sup IE])
~
a.nd
Be('
O.(l/d
for
CE Hf(~)
(3.4.20)
.
< 00
If(C) " ~i(x(C) ~(z, C))" wee) 4>"(z, C)
I
+ IIdq(E)II/t5I ) min {t52ft, t52ft+2/lIde(E)IIZ}
~
20zb2n - 1
137
3.4. Decomposition of singularities
Therefore it remains to prove that for some 0 8
j If(C) /\ co,(x(C) ~(z, cnC) /\
sup
w(C)
< 00
I ~ 0, min {I, [c5/dist (z, H,(c5»12.. -1} .
(I)"(z,
ZEV.
HeC/J)nV.
This follows from Lemma 3.2.6 because, by (3.4.18), (3.2.12), and (3.2.14), for 00 some 04
<
Il
f(C)
~i(~(e) ~(z, en" co(C) I s;;. 0, ~n(z, C)
+ 04 _____ llde(C)l1
+ 0,
2
1
i
+ 04
II dt!(C) II
Ie -
zl2ft-2
Ilde(c)11
c5
_1--::--"""7"
IC - zl2ft-l
.•
61 IcP(z, C)II' - zI2,.-8
6 ItP(z, C)l 2 IC - zI2,.-, 3.4.0. Corollary. For every
_
c5 I~(z, C)I
f E L(8,l)(D) the function
f (C) " ~(x( e) ;:( z, e» 0)
" co(C) ,
cpn(z, C)
zED,
(3.4.21)
i8 holomorphic in D and continuous on D. Proof. The holomorphicity follows from the fact that ~(z, C) and w(z, C) holomorphically depend on zED and ~(z, C) =1= 0 for C, zED. Further it is clear that for all , E aD and c5 > 0 the function
j
I(e)
1\ w~(x(,") ~(z,
e»
1\
w(e)
CPft(z, C)
D,HI
depends continuously on z E H,(6), and it follows from Lemma 3.4.5 that for all
E E aD
lim sup /J-+O ZED
I ff(C) /\ Web;(C) ~(z, cnC) " I= w(e)
0.
(l)n(z,
He(/J)
This implies the continuity of (3.4.21) on D.
•
The right. hand side in (3.4.17) is defined not only for (]1·functions P on en but, by Corollary 3.4.6, for all bounded continuous functions rp on D such that 8p E L('O, l)(D). 'Ve use this to extend the definition of the operator LaD: Definition. For every bounded continuous function rp on D, such that we define LaDP(Z) = LDP(Z)
+ (n/~, ~)! jap(c) coc(X(c) ~(z, e»" co(c) 1\
m)
D
(j)n(z, C)
,
&p E L('O,l)(D) zED. (3.4.22)
3.4.7. Proposition. (i) For every bounded continuous junction fP on D, 8uch that ap E L W•1 )(D), the Junction LaDqJ is continuous onD- and holomorphic in D. (li) There exi8t8 a constant 0 00 with the following properly: If ~ E aD, d 0, ana qJ is a bounded continuou8 function on 15 8uch that 8'1' E L(O'.1)(D) and the following
-
<
>
138
3. Strictly pseudoconvex sets with non-smooth boundary
conditions are fulfilled supp
Hf(b) ,
8qJ C
116q;(C)1I ~ 1/6 + Ilde(C)II/b2 -
-
lIaqJ(C) " a~(C)l1 ~
for CE H~(b) n V 2
,
IIdg(C)II/b for CE H;(~) n V s ,
then Jor all z E aD IL 8DCP(Z) I ~ 1q;(z)1
+ 0 min {I, [~/dist (z, H t (b))J 2n - 1} •
Proof. In view of the Leray formula stated in Theorem 3.1.3 (i), LpqJ = cp - RD aqJ on D. By Theorem 3.3.4 this implies that LDfP is continuous on D and admits the estimate ILDcp(z)l ~ 1q;(z)1 0 min {I, [~/dist (z, H~{(5))]2n-l} , z E 8D .
+
Together with Lemma 3.4.5 and Corollary 3.4.6 this proves the proposition. • 3.4.8. Theorem. Let Dec q;n be a strictly pseudoconvex open set. Then there exi8ts a oonstant 0 < 00 wieh the following property: If ~v ... , ~ N E aD, b > 0, and if f is a continuou8 function on D that i8 holomorphic 'i11, D, then J admit.s a decomposition N
f = fo
+ LSD qJ +i-I "£ f1 ,
(3.4.23)
where the following condition.s are lulJilled : _ (i) For j = 1, ... , N the function J1 ia continuous on D and holomorphic in aome neighbourhood of jj " (aD n Htib)). Further, for j = I, ... , N and all z E aD we have the estimate If1(z)1 ~ Od(~l' ... , EN; c5) w(t5,f) min {I, [b/diat (z, H~I(t5))]2n-l} ,
where w(t5,J) :=
sup c.'1El>;
(3.4.24)
If(C) - f(r;)l.
le-'1I:5t~
on
c-
(ii) qJ i8 a COO-function on and llcpllo. ~ II/lIo.D' (iii) Jo is continuou8 on jj and holomorphic in some neighbourhood of N
D
u
(6D
n
U Hei~/2)) .
(3.4.25)
j-l
Proof. Let XI (j = 0, I, ... ,N) be the same functions as in thc proof of TheoN
rem 3.4.2. Definefo := L aDxo/,f1 := L aD X1(J - 1(~1)) for j = 1. ... , N, rp := "£ X1J(~1). _
Since
j-l
81 =
0, it foHows from the Leray formula stated in Theorem 3.1.3 (i) and the
definition of LaD that LaDI = LDJ =
N
J. Together with 1: X1 =
1 this gives (3.4.23).
j=O
Now we prove condition (i). In view of Proposition 3.4.7 (i), the functions/l , ... ,IN are continuous on i5. Since BUPP XI c c H,it5), it fohows thatJ1 is holomorphic in some neighbourhood of j j " (aD n HfJ{b)). Since supp 6X1(f - 1(~1)) ~ H fJ (c5), and since by (3.4.14) and (3.4.15), for CE HfilJ),
118X1(C) (j(C) - J(~I))1I ~ Od(E1, .. · '~N; t5} (1/t5 and
+ IId~(C)II/~2) ().)(~,J)
139
3.5. Uniform approximation
it follows from Proposition 3.4.7 (ii) that for Borne 0'
< 00 and all z E aD
+ min {I, [~/dist (z, H~/a))]2n-l}] d(~l' ... , ~ N; t5) w(~,J) . This implies (3.4.24), because X1(Z) = 0 for Z H eJ (t5) and, therefore, for some 0" < 00
111(Z)1 ~ O'[lxf(z)(l(z) - f(~1)) I
(f
and all
Z
E
cD,
IXt(z) (f(z) - f(EI))1 ~ 0" min {I, [~/dist (z, Hel(~))]2"-1} •
Condition (ii) immediately follows from the definition of rp. Finally we prove condition (iii). By Proposition 3.4.7 (i) the functionfo is continuouB _
N
on D. Since Xo (3.4.25). •
3.5.
=
0 on
...,
U Hej(~/2): fo
is holomoI'J2hic in Borne neighbourhood of the set
Js:l
Uniform approximation
Notation. Throughout this section D c c fen is a strictly pseudooonvex, open set (not necessarily D 9= 8), and (! is a strictly plurisubharmonic OI.function in a. neigh. bourhood () of aD such that D n()= {ZEO:e(Z)
and
N(e):= {ZEO:e(Z) =O} ccO.
Set K(e) := D u N(e). In the present section we prove the following 3.5.1. Theorem. Every continuous function f on K(e) that i8 holotnorphic in D can be approximaled uniformly on K((!) by Junction8 which are holomorphic in, 80me neigh. bourhood oj K((!). Remark. This theorem contains two special cases. If D = 0, then K(e) = N(e) is the zero set of a non-negative strictly plurisubharmonic function defined in a neigh. bourhood of K((!). In this ca.se the theorem asserts that every continuous function on K(e) can be approximated uniformly on K((!) by functions which are holomorphic in a neighbourhood of K(e). If D 9= 0 and N((}) = cD, then K(e) = jj is the closure of a strictly pseudoconvex open set D c: c fen. Recall (Theorem 1.5.19) that for every strictly pseudo convex open set D, the function e can be chosen so that N((!) = aD. Before starting the proof of Theorem 3.5.1 we give two lemmas. Notation. Further on in this section we use the notations introduced in Section 3.4.
>
0 and every Goo-function fP on 3.5.2. Lemma. For sufficiently small t5 U~(jj) := D u {z E ([)n: diet (z, D) a} and for all z E U~(ii)
<
L (z):= _ (n - I)! dC(J
(2m)"
f
8{{}(l")
(l")) A ~,(w(z,
~ f(J ~
eff
we define
en C) w(C) • A
(/)ff(Z,
V.",Ud(D)
Then L~rp is kolomorphic in U{J(D) and
lim IIL{Jf(J - LaDC(Jllo.D = 0 .
"-..0
Proof. The conclusion that LdC(J is holomorphic in U~(D) follows from the fact that w(z, C) and qJ(z, C) are holomorphic in z and (()(z, C) =1=0 for Z E U~(jj) andC E VB " u~(jj) (cp. Construction 3.1.1, Lemma A, conditions (i), (ii), and estimate (3.1.8)). Further,
140
3. Striotly pseudooonvex sets with non-smooth boundary
from (3.4.16') it follows that for some 0 1
J
< 00
')I
Iw(z, d0'2" I~(z, C)I" '
zED,
UcJ(D)'\.D
where do.. is the Lebesgue measure in 0". From Construction 3.1.1, condition (ii), and estimate (3.1-.8) one obtains that for some (X 0
>
I~(z, ~)I ~ (X(I€!(') - €!(z)l
+ IC -
for
z12)
CE U,(D) " D •
zED,
Together with Lemma 3.2.4 (ii) and estimate (3.2.14) in Lemma. 3.2.5 this implies that for Borne O. < 00 IL,cp(z) - LaDCP(z)1
~ -
J
a I
(IQ(z,
J
')I +
Ild,Q(z, CHI dO's" (1 Ie - z12) IC - zI2"-2 + fa
UcJ(D)'\.D
d0'2"
IC - z12"-1 '
Ud(D)'\.D
where dO'sti is the Lebesgue measure in P. The second integral on the right-hand side tends to zero for d ~ O. Therefore, we only have to consider the first integral. Since, by (3.2.8), Q(z, C) = Onc - zl), for some Oa 00 we have
<
J
Ild,Q(z, CHI d0'2" (IQ(z, C)I + IC - zll) Ie - zI2"-2
U,(D)'\.D
~~
J
IId,Q(z,
(IQ(z, ')1
+ (IQ(z, C)I + IC -
em dO's" zl)2) (IQ(z, C)I
+ Ie -
zI)2t1-2
.
UI(D)'\.D
In view of Proposition 4, Appendix 1, we thus obtain for some 0,
J
(IQ(z,
U,(D)'\.D
IId,Q(z, Clil d0'2t1 ~0 I' - zll) If - zl2n-2 - f
')I +
J
< 00, dt,. " ... "dtln (I~I + Itli) Itlln-2'
IE Ta(UcJ(D)'\.D)
zED,
where, H ZI(') are the real coordinates of CE _
en,
2"
_
P.(U,(D) "D) := U {(IQ(z, C)I, IZt(C - z)l, ... ~ ... , IZ2"(C - z)l): C E U,(D) "D} . i-I
J
In view of Proposition 5 (ii), Appendix 1, the integral on the right-hand side converges for every" > O. Since the integrand does not depend on t5 and since the measure of p.(U,(D) " D) tends to zero for t5 ~ 0, uniformly in z, this completes the proof. • Notation. Set €!m(C) := €!(C) - 11m for m = 1,2, ... , and let mo, N(f!".), ~m' tP"., w"', D". be as in Construction 3.1.1, Lemma A, condition (iv), and Lemma B, property (b). Let Va and X be as in Definition 3.1.2, suppose that Dm C D U VI and N(€!",) C V2 for m ~ mo, and let L D., RD. (m ~ mol be the operators introduced in Definition 3.1.2. Further on, we need the estimates given in Lemmas 3.2.4 and 3.2.5 "uniformly in m".
141
3.5. Uniform approximation
Therefore, we prove 3.5.3. Lemma Iq)",(z, e)l ~ (X(IP(z, e)1
+ IQ(z, e)1 + IC -
zll)
for
z e VI n Dm,
eE aD", ,
(3.5.1)
Ii)",(z, e)1 ~ (X(IP(z, C)I Iw"'(z, e)1
+ IQ(z, e)1 + Ie -
s: O(lIde(e)1I + IC -
zl)
zll)
for
z, CE VI n Dm ,
(3.5.3)
for C, Z E VI ,
64l;t 1:) 1:s; 0 (I ~~~) 1+ II: - -I + le.«(;)I)
1
for
(3.5.2)
C, -
E
Vi;
j = 1, ... ,n. (3.5.4)
>
Proof. Let e 0, M, M"" Ai, if", be as in Construction 3.1.1. (3.5.1). It follows from Construction 3.1.1, Lemma A that 4J",(z, C) = F(z, e) M.(z,e) for C, z E VI; IC - zl ~ e; m > me, (remark that de", = de and therefore Fm = F). In view of the uniform convergence M '" ~ M it follows that for some (Xl 0,
>
Iq)",(z, e)1 ~ (Xl IF(z, C)I
for e, z E Vi;
Ie - zl
s: e;
m ~ mo.
<
If z E VI n D"" C E aD"" then e(z) 11m, eCC) = 11m and therefore e(e) - e(z) = Ie (e) - e(z)l. Taking into account (3.1.8), thus we obtain that for some (XI 0,
Iq)",(z, ')1 ~ (Xa(le(C) - e(z)1
>
+ 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 +
<
00
and all z
E K(e),
"It\2n-l ... ". dt 2n
J
teT.
d~ " ... " dt!"
(I~I + It:al +1t1 2)2ItI 2"":'S
) '
where, if x1(C) are the real coordinates of CE eft,
T 1(D m" Un, z) := {(lx1(C - z)l, ... , IX2ft(C - :::)1): CE Dm " Ujj} , 2,.
Vjj, z) := U {(IQ(z, C)I, Ixl(C - z)\, "'A"" IX2n(C - z)/): CE Dm " Un} , ;=1 ; Ta(Dm" Ujj, z):= U {(\Q(z, C)\, IP{z, C)I, Ixt(C - z)l,
T 2(D m "
1~j
"'",,'" , I X
j,k
2n(C - z)l): CE Dm "
Vii} .
In view of Proposition 5 (i) -(iii), Appendix I, the integrals on the right-hand side of this estimate converge for all m ~ mo and z E K(e). Since the integrands are independent of m, and since the measures of the sets T 1(D m " UD, z) tend uniformly in z E K{e) to zero if m -+ 00,1) this completes the proof of Lemma. 3.5.4. • 3.5.5. Proof of Theorem 3.5.1. For n = 1 this theorem follows from VITUSKIN'S theorem (VITUSKIN [I]). Therefore, we can assume that n ~ 2. Further, by Lemma 3.5.4 we only have to prove the following proposition: Every continuous function! on jj that is holomorphic in D, can be approximated uniformly on jj by holomorphic functions in some neighbourhood of D. In fact, combining this proposition and the Weierstrass approximation theorem, we obtain that every continuous function on K(e) that js holomorphic in D can be approximated uniformly on K(e) by functions which are 0 00 on Oft and holomorphic in some neighbourhood of D, and by Lemma 3.5.4 for such functions the theorem is _ a1ready proved. And so, we consider a continuous function Jon D that is holomorphic in D. Set Crit (€!) := {~
E
oD:
de(~)
= O} .
Since, by Theorem 1.4.21, Crit (q) is locally contained in a smooth real n-dimensional manifold, there is a constant 01 00 depending on e only such that: For every c5 0 there exists a finite set SIJ ~ arit (e) with the following properties:
<
(1)
>
Crit ((!) c U H,(lJ/2) 2)
eeS"
(2) For every point
= £} if k =i=j (8:. k
(3.5.7) 00
z E f!J n , there is a decomposition SIJ = U 8=.k, where 8~.1; n 8;,j
= {} for k large enough) such that for k
card (8:.1;) ~ min {OJ, Olk"-l} ,
k-O
= 0,1,2, ...
(3.5.8)
1) This follows from the fact that the measure of Dm "\. UD tends to zero, which follows from the fact that the set K(e) "\. UJj is locally contained in a smooth real n-dimensional Bubmanifold (Theorem 1.4:.21). 2) Recall that, for; E Crit (e), He(~) = Ee(d) is the Euclidean ball of radius d centered at ~ (op. Subseotion 3.2.1).
144
3. Strictly pseudo convex sets with non'Bmooth boundary
where card (S:.~) denotes the number of points in we have the estimate diet (3)
d(S~,
(z, ile( 6» ~
0)
01
S:. k' and for Ie =
1,2, ... and ~ E
S=.k
(3.5.9)
:2: k<5 •
(3.5.10)
,
where d(S." <5) is the order of the system of sets {.iie(<5)}fEs~, that is, at least one point belongs to d(S." 6) of the sets Hl (<5) (~ E S.,), but no point belongs to d(S~, 6) 1 of these sets. In view of (3.5.7) and (3.5.10) and Theorem 3.4.8, one obtains the following 00 such that for every 0 0 the function J ad. proposition: There is a constant Oa mits a decomposition J + LaDer} + I: with the following properties:
+
<
=.ro
n
feS~
>
n
__
(i) For every ~ € S." the function is continuous on D and holomorphic in some neighbourhood of D" (aD n He(o», and for every Z E aD
111(z)1 (ti)
~ 010.w(lJ,J) min {I, [ojdist
(z, He(<5»)]2n-l} •
(3.5.11)
rp6 is a O°O·function on q;n.
.ro is continuous on i5 and holomorphic in some neighbourhood of D u Crit (e). We already know that the functions LaDrpiJ and to can be approximated uniformly (iii)
on D. For L 8Drp" this follows from Lemma 3.5.2. }"or.ro we can use the following argument: Let U be a neighbourhood of Crit (e) such thatfo is holomorphic in D u V. !hen ~e can choose a strictly pseudo convex open set D' with 02.boundary such that D ~ D' ~ 15 u V, and the assertion follows from Theorem 2.9.2. To complete the proof it is therefore sufficient to show that
lim lDax 11: n(z)1 6~O
lED
=
0.
(3.5.12)
fES.,
From (3.5.11), (3.5.8), and (3.5.9) it follows that l: II:(z)1 ~ O~02W(O,J) for Z E aD, and Ees:.o for Ie = 1,2, ... , 1: 1,n(z)1 ~ ~Oaw(o,f) 11k" for Z € aD. Consequently,
eES:.k
11: r,(z)1 feS6
S;
G10aro(O, f)
(1 + .t=11e i 1,,)
for
Z E
aD.
This implies (3.5.12), because n ~ 2, w(o,f) ~O for «5 ~O, and the functions l: 1I(z) are holomorphic (maximum principle!). • eeS"
3.6.
Bounded extension of holomorphic functions from complex planes
In this section we prove that every bounded holomorphio function J defined on the intersection of a strictly pseudoconvex open set D with a complex plane X admits a bounded holomorphio continuation to D. If, moreover, I is continuous on D n X, then this continuation can be chosen to be continuous on ii. In Section 4.11 we shall generalize this result to the case that X is an arbitrary closed complex sub· manifold in some neighbourhood of Do
1M
3.6. Bounded ("xteosion of holomorphic fWlctions
3.6.1. Notation and preliminaries. Let Dec (f)ft be a striotly pseUdooonTex open set (with not necessarily smooth boundary). Let (}, F, i, f./J, w, X, and N(e) ~ V. be as in Construction 3.1.1 and Definition 3.l.2. Set em(C) := e(C) - 11m for m = 1,2, ••• J and let me, N(em), ~m' f/J"., w m, D", be as in Construction 3.1.1, Lemma A, condition (t.) and Lemma. B, property (b). Suppose tha.t Dm ~ D U VI and N(O,.) ~ VI for m m.Set X := {z E on : z,. = O} •
sr:
For C = (C1 ,
...
-
on we write C' : = (Cv ... , eA- 1), a A-I 8
"A) E n-l
Be' :=- l: -=- dCf, j=1
8,,:= l: -dtf'
6C1
}-I
ac,
d" := 8" + 8~, , c.oc.(t) = del" ... 1\ dCft-l . Sometimes we shall write eoo := (!, (/J 00 : = (j) etc. For m" S; 'It" :s: 00 we introduce the notation (w"'(z, e))' := (wr(z, C), ... , W:'_l(Z, C)) and the differential fonn
roc' (X(C'jwm(Z, C))'):=-= ~A18"X(~ wj(z,t). tPm(z, C) 3=1 (])tlI(Z, C)
(3.6.1)
From Construction 3.1.1, Lemma A, condition (ii), and estimate (3.1.8) it follows that for all mo < m < 00 there exists a. neighbourhood U aD... of aD", " x Buoh that
,Z
,z.
~m(Z,C)=f=O for tEXnDm, zED".uUaD... Consequently, for every fix~ Z E Dm u UaD,.,x, mo ~ 'In ~ 00, the differential form (3.6.1) is continuous on X n Dm. For every bounded holomorphic function! on X n D,. and
Z E
Dm
U
U aD."I, therefore we can define
E"J(z) :=
(~
jf(t) wc' (X(C) i~m(z~ ~lL) ,,~(C)
_ I)! (2nl)n-l
.
(/)",(z, ')
XnD ...
3.6.2. Lemma. For every bounded, holomorphic Junction! on X n D"" EJ i. AoID· morpkio on Dm U Uw".,x ana J(z) = E"J(z) for z E X n Dm (mo ~ rill ~ 00) • Proof. That Emf is holomorphic follows from the fact that (P,.(z, ") and w-(z, C) are holomorphic in z. The equality f = Ef IXnD... follows from the Leray formula stated in Theorem 3.L3 (i) if we take into account that Construction 3.1.1 applied to X n DtlI gives the restriction of ~m and (wtll)' to X . • 3.8.3. Further notation and preliminaries. For z
E
V2 , ~
E
V 2 n D,., and 11'0 s: m
~ 00,
we define
tP!(z, C) := ~m(C, z) ,
(*wm(z, C)'
:=
*wm(z, C) := -w"'(C, z) ,
(*wr(z, C), ... , ·W:_l(Z, C» .
In view of (3.5.1) and (3.5.2) in Lemma 3.5.3, 4):(z, C) =F 0 and q)",(z, C) =t= 0 for Z E aD", " x, CE X n D"" 1nt :s: m ~ 00. Consequently, for every fixed z E aD .. " X, mo ::; m ~ 00, the differential form (for det1 • tI _ 1 cpo Definition 1.7.6) d t (*tOm(Z, C) e 1,7&-1 cp* ,. , .
m(z,,)
is oontinuous on D", n X. 10 HenklnJLeiterer
a,c xa:)_w"'(z, C)) tPm(z,C)
146
3. Strictly pseudoconvex sets with non-smooth boundary
3.8.4. Lemma. For every bounded holomorphic Junction J on X n D"" and all " X, me, ~ m ~ 00, we have E t(z) rrv
=
Z
f
i - I )"
tI (2 ')"-1
"'*( " ) " , . i
1,11-1
I"
'.I'm
CEXnDM
Proof. In this proof we omit the index m. Fix z
*W - XW) ( *W ~det"_l (-0... XW') ~ + (n (/>* .. (/J
a, x(~) w""(z, ~)) /\
(*w"'(z) C)
J(") det
1n
E
Z,"
'Vm(Z,
aD" X.
C)
Z E
W
C'
8D m
(,.) • I"
Then
(_1)" det 1,tI_l l/J*' 8" ~ =- -
-.. xw(/>
1) 8,.,~ /\ det 1
,
,,-2
(*w' - XW') -,0,., (/J*.
--N-
(/>
•
Since
= ( 1 - n-l .l: (C, -
Z1)
J ... l
+ (n
-
1)
*w) (n --J t:p
n-l
l: (C, -
i~l
(xw')
I)! We' ~ f/J
XW , f/J
--
Z1) 5" -:;- /\
(*w' -
xu/)
det 1,n-2 ~,8,,--z-
'.1'*
(j>
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)lIC - z1Zft-6
is the Euclidean volume form on X.
Proof. In this proof we use the notations 0, 0', ... and «, «', ... for "large" and "small" constants, respectively, as explained at the beginning of the proof of Lemma. 3.2.6. (i)
f
<
d0'2n-2
IC - zl2ta-l =
XnD.
f ---+ (lz.P'
d0'2.--:"_----::2----,-_ _=---_=_ IC' - z'II)1 Ie' - z'1 2. - 6
XnD.
<
f(l~ ~·It~)~~~~. ·
te R2n-2
ItI
where diarn (Dm) is the Euclidean diameter of Dm. In view of Proposition 7, Appendix 1, this implies the required estimate. (ii). In view of (3.2.11), (3.2.12), (3.5.1), (3.5.2), and (3.2.9) the integral in (ii) can be estimated by the integral in (i) and the following integral: Ild"Q(z, e)11 d0'2.-_2-----,:~ (IQ(z, C)I IC - zll)IIC - zl2a-'·
f
+
XoD",oV.
Since by (3.2.8), IQ(z, C)I ~ 0 IC - zl, and, consequently, for C E X n Dm n VI' Z E aDm " X, mo ~ m ~ 00 ,
(IQ(z, C)I ~ «(l znI
2
+ IC - zll)1 IC - zl2n-' + IQ(z, C)I + (IQ(z, C)I + IC' -
z'1)2)' (IQ(z, C)I
+ IC' -
z'I)2"-' ,
we can apply Proposition 4, and obtain that the last integral can be estimated by
f
dtl (lznl2
1\ ••• 1\
dt2a _ 2
+ I~I + Itll)1 ItI 2n -' ' that IQ(z, C)I + IC' - z'l < R
,eR2.-2,,'I
<
+
V;;
Together with (3.2.11), (3.2.12), (3.5.1), (3.5.2) this implies that the integral (iii) can be estimated by the integrals in (i), (ii), and the following integral:
f
Ild"P(z, C) /\ dc-Q(z, C)II d0'2n-2 (IP(z, ell + IQ(z, e)1 Ie - zll)8lC - z12n-5 .
+
XnD.nVI
+
Since by (3.2.6) and (3.2.8), IP(z, C)I IQ(z, e)1 ~ 0 IC - zl and, consequently, for all z E aD m " X, CEX n D", n V 2 , mo ::;;: m ~ 00,
+
(lPI IQI + Ie 2 ~ (l znl + IPI
-
+
zll)8lC - zI2.-5 IQI (IPI IQI
+
+
+ IC' -
%'1)1)8 (lPI
+ IQI + \C' -
z'1)2n-6,
150
3. Strictly pseudoconvex sets with non-smooth boundary
we can apply Proposition 4, Appendix 1, and obtain that the last integral can be estimated by
< 00
+
+
<
is so large that IP(z, C)I IQ(z, C)I IC - zl R for all z E 8D m , ~ 00. The required estimate now follows from Proposi tion 7 , Appendix 1. (iv). First we remark that, in view of (3.2.11), (3.2.12), (3.5.1), (3.5.2), and the estimate 18e(C)/8C,,1 ~ 18e(z)/8z,,1 + c IC - zl, the integral in part (iv) can be estimated by the integral in part (i) and the integral which is obtained from the integral in part (iv) after replacing 18e(C)/8C,,1 by 18e(z)/8z"l, that is, we only have to prove that for all z E 8D m"X, mo :s;;: m ~ 00, where R
CEX n Dm n Vi' mo:S;;: 'In
J-
CeZ'nD.n v.
Let 6
:s;;: c 1 8e (Z) z" 14),.,(z, C) 4):(z, C)IIC - zI2"-' 8z" d0'2,,-2
1-
1
(3.6.11)
•
> 0 be the number from Lemma 3.6.5. For simplicity we write
H~ := H~(d I~~ "I'/} In view of Construction 3.1.1, Lemma A, we have 14)",(z, C)I, Ir/J!(z, C)I ~ a; lem(C)1 for Z E 8D,." CE Dm n V 2 , mo ~ m ~ 00. Together with (3.2.1) (it is clear from the proof of (3.2.1) that (3.2.1) holds uniformly in m), this implies that ;;; ,. ""'. l'Vm(z,~)I,I'Vm(z,C)I ~cX~ 21
for
Z
E
8e
(Z) --z" 8z"
I
aD""
(3.6.12)
Together with (3.2.11), (3.2.12), (3.5.1), (3.5.2) this implies that for all mo ~ m ~ 00,
Z E
oDm '" X,
In view of Proposition 7, Appendix I, the integral on the right-hand side can be 8e(Z) 1-1 . Therefore to prove (3.6.11) it is sufficient to prove that estimated by --Z"
1 8z"
for all z
E
oDm " X, mo ~ m ~
I m(Z) :=
J
CeIr,'nV.nZ'
00,
d0'2,,_2
< C 10Q(Z) --z"
I~",(z, C) 4)!(z, ')1 IC - zI2"-4 -
OZ"
\-1 .
(3.6.13)
151
3.6. Bounded extension of holomorphic functions
It follows from estimate (3.6.8) in Lemma 3.6.5 and the estimates (3.2.11), (3.2.12), (3.5.1), (3.5.2) that for all z E fJD". " X, mo ~ m ~ 00,
f
-2
~ 0 IIaz,{?(z) II
Im(z)
IIdc-P(z, e) /\ d"Q(z, em d0'2,._2 (IP(;,- e)l- TQ(z, e)-~ff-~----;ji)21~---'::":-
ZI2"~'-
+
Ce.u:'nV,nZ
f
.,;; C II d,,!,>(z)II-1
II ~P(z,.;)
A
dc4(z, elll du•• _.
'eH!'nV,nX
In view of estimate (3.6.7) in Lemma 3.6.5, this implies that for all
mo
~
m
~ 00,
1
< oI8{?(Z) z 1".( ) = 8z" ft
f
2
Z
E 8D m "
Z
X,
IIdc-P(z, e) /\ dc-Q(z, em d0'2tt-2_______ _
(I P(z, ~)I
+ IQ(z, e)1 + Ie -
zll)8/lle - z12ft-5 .
CeH:'nV.nX
= 2
We now distinguish the cases n by definition of H': for ~ E H':
I~ _ zi ~ 61 fJe(z)
and n
~
3. If n = 2, then we use the fact that
zft 11/2 .
fJz ft
+
+
For n ~ 3 we use the inequality IP(z, C)I IQ(z,~)1 Ie' - z'l ~ 0 IC - zl, which follows from (3.2.6) and (3.2.8). Then we obtain that for all z E fJD m " X, mo ~ m ~ 00,
I (z) m
f
82 ~ 0 1 8e (Z) z 1- / fJZ2
-
2
Ildc,~(z, C) /\ dc-Q(z, C)II d0'2tt-2_ (I P(Z, C)I
n = 2
if
+ IQ(z, C)I )3/2
'
H:'nV,nX
and 1".(z)
<
C 18~(Z) Zttl-
f
2
CZ"
(IPI
Ildc,P /\ d"QII d0'2tt-2 _ -----z'1)2tt-6
if n
+ IQl)s/2 (IPI + IQI + Ie' -
~
3.
H!'nva nX
In view of Proposition 4, Appendix I, this implies that for all z E 8D". " X,
mo
~
m
~ 00,
I (z) ~ C 8e (Z) z ". CZ 2 2 '1
< oI8e(z) 8z ft
".(z) =
~~ ~ (I~I + It21 )3/2
if
n
=
2
'e RI, Itl <'m(')
and
I
f
I-S/2
f
1Ztt
2
'e R2" - 2. Itl < Itll
dtl
/\ ... /\
dt2t1 -
2
(I~\ ;. It21)3/2ItI IA - 5
'
'f 1
~3
n -
,
B",(z). +I~I <'.(%)
where rm(Z):= sup {IP(z, C)I
+
IQ(z, C)I}
~H!'
and
R",(z):= sup {IP(z, C)I
+ Ie' - z'I}. In view of (3.2.1), (3.2.2), and the definition of C E H':, mo ~ m ~ 00, we have the estimates Ie - zl S 61 8e (Z) z" 11/2
8zft
and le".(C)I, IF(z, e)1
~ 06218e(z) z".1 8z"
+ IQ(z, e)1
~~
H':, for all z
E fJD". "
X,
152
S. Strictly pseudoconvex sets with non-smooth boundary
Further, by (3.2.6) and (3.2.8), for all z E eD.. " X, ~ E H':, mo ~ m ::; 00, IP(z, C)f ~ IF(z, e)1 + C Ie - zll and IQ(z, C)I ~ le(C) - e(z)1 + C IC - zl2 = lem(C)! + C Ie - zll. This implies that for all z E aD". " X, mo ~ m ::; 00, rm(z) ::; C 18e(Z) z"
8z"
I
and
Rm(z)
~ C 18e(Z) Z" 11/2 . 8z"
Now estimate (3.6.13) follows from Proposition 8, Appendix 1. (v). Let <5 > 0 be the number from Lemma 3.6.5. For short, we write
H:,
:=
H:,
(a I~!:) "1''").
First we prove that
f
(3.6.14)
Observe that 18e(e)/8C,,1 ~ loe(z)/8z,,1 + C IC - zl and, by (3.2.9), II de'!! (C) II ~ Ild"Q(z, C)II + C IC - zl. Together with (3.2.11), (3.2.12), (3.5.1), (3.5.2) this implies that the integral on the left-hand side of (3.6.14) can be estimated by the integrals in parts (i), (ti), (iv) of the lemma and the integral which is obtained from the integral in (3.6.14) after replacing IIdcre(C)1I18e(C)/8~AI by Ild,.Q(z, C)II 18e(z)/8z"l, that is, we only have to prove that for all z E 8D". " X, mo ~ m ~ 00, (3.6.15)
From (3.6.12) and (3.2.11), (3.2.12), (3.5.1), (3.5.2) it follows that for all z
me,
E
8D", " X,
~m ~ 00,
f
IIdc-Q(z, C)II d0'2A-2 Iq)m(z, ~)IIlI4>!(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) . •
+
<
3.6.7. Lemma. There is a con8tant 0 00 suck that lor all mo ~ m S;;; bounded kolomorphic function f on X n Dm the following estimate hold8:
IE"J(z) I
~
00,
and every
z E Dm " (8D", n X) .
Ollfllo.D.nx for all
(3.6.18)
Proof. Since, by Lemma 3.6.2, Em/is holomorphic in Dm U U 8D.. ,X, it is sufficient to consider the case z E aD", "X, because then the general case follows by the maximum principle for holomorphic functions. It follows from Lemma 3.6.4 that for some constant 0 1 < 00 and all z E aD"" " X, mo S;;; m ~ 00,
IE"J(zl I ::;; 01 1...1iIlJllo.».nz
JIldet C;;, 6C'x;'....) 1•• - ,
A
w,,(Cl II d<1.. _ ••
XnD.. Therefore, since X = 1 on Vi and since, by Construction 3.1.1, for some all Z E aDm, CE Dm " VI, mo ~ m ::;: 00,
It$",,(z, e)l, l.p!(z, C)I ~
I
~ det
> 0 and
(X ,
it is, sufficient to estimate for all z X n Dm n V2 0f
(X
E
aD"" "X, mo
~
m
~ 00,
the integral over
m (*wm 8CW)11 <1>::' -a,. XW"' ,p.. )II :;; 0'1II det •. __ (*W'" <1>::' -,pM t
1 • __ 1
1
wm8 ~m 8 w'" )11 . +03 II det 1• 1• n _ 2 ( *W'" !P* ,---~,~ I
m!Pm!Pm
(3.6.19)
Since *wm(z, z) = _wm(z, z) and since w m -+ w uniform1y together with the first-order derivatives, there is a constant 0, < 00 such that for all C~ z E VB' mo ~ m :s; 00,
I*w(z, e)
+ w"'(z, "I
~
0, Ie - zl • Further it is clear that for some 0, < 00 and all z E 8D m, CE V 2 , lem(C)1 ~ O"IC - zl· We recall also that for some (X > 0 and all z E aD., eE Va n Dm , mo ~ m ~ 00,
1M
3. Strictly pseudoconvex sets with non-smooth boundary
(cp. (3.2.11), (3.2.12), (3.5.1), (3.5.2)) I~m(z, ')1, I
I
detl ,,-1 (*wm(z, ~), e,~m(z, ~»)II ~ ~e_ + _.... Gellde(C)11 •
and
I
d etl
a
a"~m(Z, C) "w m (::l2) I , .... I ~(Z, C)
*w"'(Z, C) w"'(z, C)
1 "-2,,.,.* (
' •
'Vm(Z,
,....
C)
~ ~e_+ .... Oellde(C)11 + _ Gellde(~)lllld,'e(C)l1 2 Ie - zI "-1 l
(3.6.21) By taking into acoount that Ilde(C)11 ~ Ild"e(C)1I + 2Iee(C)/e~"I, now the required estimate (3.6.18) follows from (3.6.19)-(3.6.21) and Lemma 3.6.6. •
3.6.S. Theorem. Let D c: c: (fJ" be a 8trictly p8eudooonvex open set (with not necusarily 8mooth boundary), and let X = {z E f/)n: zk . j-l = ... = z" = O}, 1 ~ Ie ~ n - 1. PMn (i) For every bounded holornorphic function morphic function FonD such that
f
on D n X there exists a bounded holo-
F=JonXnD. (ii) For every continuous Junction f on X n D that is holomorphic in X n D there exists a continuous/unction F on D that i8 holomorphic in D 8uch that
F=f on
XnD.
Proof. First we consider the case that Ie = n - 1. By setting F = Eoof, then part (i) immediately follows from Lemmas 3.6.2 and 3.6.7. Now we prove part (ii) for Ie = n - 1. In view of the Approximation Theorem 3.5.1, we can find holomorphic functions /t (j = 1,2, ... ) in some neighbourhoods of X n D such that 00
I: Ilf11lo,DnX }-1
For every j choose
<
00
00
mf ~
and
f =
~
/1
J=l me, so large that
_
J1
on
X n D.
is holomorphic in a neighbourhood of 00
X n D fAJ • Then by Lemmas 3.6.2 and 3.6.7, F := l: Emit gives the required exJ =- 1 tension.
In the general case, when 1 ~ k the special case proved above . •
~
n - I, the theorem follows by applying k times
Notes Without the factor diamH (supp I) in estimate (3.2.3) and without estimate (3.2.4), Theorem 3.2.2 was proved in 1974 by HENKIN (unpublished). In its formulation which is given here, Theorem 3.2.2 was obtained in 1980 by HENKIN/LEITERER [2]. In its proof the construotions given by HBliXIN [2, 3] and SKODA [1, 2] for solving the ail-equations play an important role. For the O8.8e of smooth aD, see the Notes at the end of Chapter 2. For
Exercises, remarks and problems
155
smooth aD, Theorem 3.4.2 was obtained in 1969 by HENKIN [4]. where instead of estimate (3.4.2) the estimate 11I111o,D ~ a In (0 + a/(J) Ilfllo,D was obtained. In its formulation given here, Theorem 3.4.2 as well as Theorem 3.4.8 was proved in HBNXIN/LEITERER [2]. These theorems form a. natural generalization of VITtTSKIN'S [1] theorem on decomposition of singularities for one complex variable to the case of several complex variables. The Approximation Theorem 3.5.1 was announced by HENXIN in CIBKA!HENKIN [1]. Its proof was published in HENXINjLEITERBR [2]. For the case that K(~) = 15, where D is a striotly pseudoconvex open set with ai-boundary, this theorem was proved in 1969 by HENKIN[4] and, independently, by KERZ1\UN [1] and LIBB [1]. For the case t~at K«(J) is a compact subset of a totally real Ol-submanifold X (cp. Corollary 1.4.20). Theorem 3.5.1 was proved in 1971 by HA.BVEYfWELLS [1] and, for X E 0 00 , in 1967 by HORMANDEBfWERMER [1] and NmENBERG/WELLS [1]. For non-negative e the proof of Theorem 3.5.1 is essentially simpler and was published by HENKIN in CmxA/HENKIN[I] (1975) (see also SAXAI[I] (1979». For the Extension Theorem 3.6.8 see the Notes at the end of Chapter 4. To obtain the estimates in this chapter, we essentially used an idea of RANGE/SIU [1] which consists in replacing F(z, C) and e(?;) - O(z) by polynomials (Lemma 3.2.4) and applying Proposition 4, Appendix 1.
Exercises, remarks and problems 1. Lct X ~ en be a complex plane of complex dimension ;:;;;;; n - 1, and let Dec X be a strictly pseudo convex open set (with not necessarily smooth boundary). Prove that there exists a strictly pseudoconvex open set jj C C Cn with at-boundary such that X n i5 = D. 2. Conjeeiure (with an idea how to prove). Let D cc en be a strictly pseudoconvex open set (with not necessarily smooth boundary). Then, for every continuous (0, q)-form f on jj such that 81 = 0 in D, 1 ~ q ~ n, there exists a 1/2-Holder continuous (0, q - 1)form on jj such that = f in D. Wt' propose to prove this in the following way: By point 1 the statement follows from the corresponding assertion for the case of smooth aD if an appropriate generalization of the Extension Theorem 3.6.8 to 8-closed (0, g)-forms is proved. 3. Open problem. Let Dec (fJn be a strictly pseudoconvex open set (with not necessarily smooth boundary), and let X ~ fCn be a complex plane of complex dimension:;; n - 1. By Lemmas 3.6.2 a.nd 3.6.7 the extension of bounded holomorphic functions stated in Theorem 3.6.8 (i) can be given by a bounded linear operator from the Banach space of bounded holomorphic functions in X n D into the Banach space of bounded holomorphic funetions in D. Problem. Does there exist a bounded linear operator E: A(X n D) - A(D) such that EJ = I on X n jj for all f E A(X n 15)! Here A(X n 15) (A(D» is the Banach space of continuous functions on X n i5 (D) which are hoI om orphic in X n D (D). Remark. For the case of smooth aD, such an operator E exists (HENKIN [5]). 4. Let Dec (fJn be a 0 2 strictly con vex open set, 0 E D, and let ~ be a strictly convex OB_ function in a neighbourhood (J of D such that D = {z E 6: e(z) < O}. Denote by N(D) the class of holomorphic functions / in D satisfying the condition
au
sup f lin ,,<1 aD
If(?\~)il d0'2n-l
<
00 ,
where d0'2n-l is the Euclidean volume form on aD. Let f E N(D) and let M I , k1' dO'2ta-2 be as in point 22 in the ExercisE'ls, remarks and problems at the end of Chapter 1. Prove that
:E j
kf
Ilel d0'2n-2 <
00 •
MJ
Remark. HENKIN [2,3] and SKODA [1,2] proved that also the converse is true: If M is a complex submanifold of D with 1101 d0'2,.-2 < 00, then there exists an I E N(D) suoh tha t M = {z ED: /( z) = O}. M 5. Let Dec (fJB bo the unit hall. Prove that there exists a complex submanifold M, dimcM = I, of D with f (1 - ICI)(II d0'2 < 00 (d0'2 is the Euolidean volume form on M) for aU M
156
3. Striotly pseudoconvex sets with non-smooth boundary
(J> 0 and such that the following condition is fulfilled: If f is 8 bounded holomorphio function in D suoh that f = 0 on M, then f = 0 in D. Remark. BlDRNDT8S0N [1] proved the following theorem: If D cc q]1 is a 0 1 strictly pseudooonvex open set, and M is a complex submanifold of D such ·that / dal < 00, M
then there exists a bounded holomorphio function / in D such that M = {z e D: f(z) = O}. 6. (HENKIN/CmXA [1], V AL'SXIJ [1]). Let g be a continuous function on 15 that is holo· morphio and =t= 0 in D, and let K := {z e aD: g(z) = O}. Then, for every continuous funotion Ion K, there exists 8 continuous function F on 15 that is holomorphio in D such that F = I on K. 7. Let D CC q]n be a 0 1 strictly pseudoconvex open set, and let M ~ aD be a real Ol·submanifold such that, for every ~ E M, the real tangent plane of M at ~ is contained in the complex tangent plane of aD at ~ (cp. Subsection 1.5.13). a) (HENXINJTUMANOV [1]). Prove that then M is totally real (cp. Definition 1.4.16). b) (HENXIN/TuMANOV [1]). Construct a Ol·function 9 in a neighbourhood of jj such that g -+= 0 in D, M = {z E aD: g(z) = O}, dg(z) =1= 0 for z e M but 8g(z) = 0 for z e M. 0) (RUDIN [1] and HENXINJTuMA.NOV [1] (for the case that M is of class 0 1 )). Prove that, for every continuous function f on M, there exists a continuous function F on fj that is holomorphio in D such that F = f on M. Hin t. Use an appropriate solution of8u = 8(I/g), where 9 is the funotion from part b), and prove that g := g/(1 - ug) has the properties in point 6. For further similar results see BURNS/STOUT [1], CHAUMAT/CHOLLET [1], NAGEL [1], TUMANOV [2], HENRIKSEN [1]. 8. Let Dec q]ft be a (]I strictly pseudoconvex open set, and let M ~ aD be a real 0 1 • submanifold with the following property: For every continuous function f on M there exists a continuous funotion F on 15 that is holomorphio in D such that F = f on M. Open problem. Does it follow that, for every E M, the real tangent pJane of M at ~ is oontained in the complex tangent plane of aD at ; (cp. Subsection 1.5.13) ! Remark. HENXINJTUMANOV [1] (see also NA.GEL/RuDIN [I]) proved that the answer is affirmative if M is of class 01+ 8 , e > 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,
en ,
zeD ."
(*)
aD
Unfortunately, this elegant approaoh of DYNIN does not work in general, beoause suoh a map 8(Z, C) also need not exist (This follows also from the results of SOHNEIDER [1]. Observe that the obstructions for the existence of such a ma.p 8(Z, e) are purely topological - for more details see HENKINJLEITERER [I] and LEITERER [4]). The construction presented in this chapter was found by HENKINJLxITERER [1] and consists in a combination of DYNIN'S idea with an idea of BISHOP (cp. Theorem 7 in BISHOP [1] and point 10 in the Exercises, remarks and problems at the end of this chapter). We construct a map 8(Z, C) satisfying condition (2) (whereas condition (1) can be violated), and, by means of Cartan's Theorem R from the theory of coherent analytic sheaves, we find a holomorphic function
e.
fez)
=
(n _~~ (2m)2n
f
f(C)
C) "
en "
w,(as(z, roc(s(z, C») , (as(z, C), 8(Z, C»1&
;; ED.
aD
The function
~)
l'emoves the singularities of the kernel ro,(a8(z,
C») "
rod8 (:, e)) •
Hemark that, if such singularities do not exist, then, since
4.1.
Coherent analytic sheaves
We know from Sections 2.10, 2.12, and 2.13 that every holomorphic Cousin problem in a holomorphic vector bundle over a Stein manifold can be solved. In this chapter we meet Cousin data which satisfy some additional conditions and we are interested in solutions which fulfil the same conditions. Consider an example. Let X be a Stein manifold, let {U1} be an open covering of X, and let hif E O( U t n Uf) be Cousin data. Then we know that hi1 = h, - h1 in U1, n U 1 for some system of functions h1 E 0(U1). Now suppose that, moreover, for some closed complex submanifold y C X, hij = 0 in Y n U, n U 1• Can we choose then the solution {hI} of the Cousin problem so that also hj = 0 in Y n U 1 ~ The answer is affirmative. This follows from Cartan's Theorem B from the theory of coherent analytic sheaves. In the present section we give the formulations of some special results of this theory, whieh will be used in the subsequent sections. 'Ve omit the proofs. The proofs can be found, for example, in CARTAN [3], ANDREIAN CAZAOU [1], GRAUERT/REMMEBT [2], GUNNING/RosSI [1] and HORMANDER [1]. Remark that the methods devel. oped in the present book do not give any new aspects for the proofs of these results.
160
4. GloLnl int,pgrfll fOrll1ulRS on Stein nlanifolds
In order to avoid the abstract language of general sheaves, we consider subsheaves of sheaves of gcrols of holomorphic sertions in holomorphic vector bundles only:
4.1.1. Definition. Let X be a con1plex manifold, and let B be a holomorphic vector hundle ov('r X. \Ve denote' by X()B the map \vhich attaches to every open set U c: X the space X(>R(U) of holomorphic section~ of B o'\'er U. 'VC' writ£' also O(U, B) := XOB( U). x(lB is raIled the sheaf of germ.8 of holom,orphic scct·ions of B. For the product bundle ...Y X (1N \ve 'write x ON : - Xq]XxCN and ON(U) := xOXXCN(lJ). 1\ lnap cF ,,·hich attaches to every open set U c: X a subset 3"( U) of O( U, B) is calJed an analyt'ic 8u,bsheaf of xOB if the following conditions are fulfilled: (i) li'or every open set TJ ~ X. /1'([1) is an O(U)-module, that is, if f, y E J(lI) and (l, fJ E O( (7), tht'n a,f -1- f3g E c7( [i). (ii) If V C lJ C X are open sets andf E 3"( U), then thef I V E :7( V). (iii) If Ui,ex areopensets,jElV(U U t • B)with/EF(lli)' thenfE91i'(U U i ). An analytic subshcaf c7 of X(}B iA called coherent if: for eVf>ry point Z E X, there exist a neighbourhood U" of z and finih_ ly lllany J;, ... ,f N E t7( [7 z) sueh that the following condition is fulfilled: If C E lJz, l"c is a neighbourhood of C, and f E c'T( T',) , then there exist a neighbourhood We C Jt" n lIz of C and 1jJ] ••••• 1/-, N E O( W d such that f = "PIfl -I- 1jJ ~~fN over W. l
+ ...
4.1.2. Theorem (The Oka theorem). Let X be a c01nple~l: 'inani/old and let A be an r X 8 matrix 'wh08e entries are holomorphic junctions on X. D6note by :J'{' A the analytic sub8heal of x O' 'U,h£rlt is defined by :/C A( U) : = {f E OS( U) : Af = 0 in [I}, U c X open. Then ~ A is coherent.
4.1.3. Theorem. Let X be a, Stein manifold, let B be a holornorph1:c 'vecto1' b'llndle ove,' X, and let J be a coherent analytic 8Ubs}tea! of XOB. (i) (Cartan's 'fheorem A) If f E cJ"( U), u,here U C X 'is open, then for et'cry z E ~r u'e can find f1:nit~ly 1nany 11' ... ,IN E 3'(X) as well as holom,orph£c .f11nction8 "PI' ... , 1jJ N ~:n some neighbourhood J' C U of Z 8'uch that I = "PJl + ... -I- 11' Ni N over IT. (ii) (Cartan's Theorem B) For eVC1'Y open covering {Uf}jEJ and eVl'1'Y sY8tern If-I E 3'(Uf, nUl), 'i.j E J, B'uch that fii iji: ==- Iii: in [T1. n U 1 n U t for all i,j, k E J, there e.xista a systcrn i1 E <-7( (11), j E J, .Cl'uch tl,at Iii = If. - 11 in U t n U j lor all i.j (; J.
+
4.1.4. Theorem. Let X be a co'nlplex manifold, let B be a holonlorphic vector b'undle over X, and let /F be a coherent ana,lytic 8ub8heaf of XOB. Then,jor every open set TJ c X, :-F( ~7) is a cl08cd 8ubspace of the F'l'echet space t?( [J, B) endowed with tlte topology of unijor'fn conL'etgence on cmnpact s'u,bsetB. Examples of coherent al1al)~tic Sh(,8 ves 4.1.5. Let Y be a closed corJlplcx submanifold of the conlplex Inanifold X. Define by 3" y( 1]) := {/ E O( U) :f{z) = 0 for z E U n Y} , U C X open, (4.1.1) an analytic su bsheaf C.r y of XOl. (J' y is eoherent. The simple proof of this fact is left to the reader. Hin t. lJsc appropriate local hololnorphic coordinates (Corollary 1.1.19 (i)) and local po,v{'r Heri{'s expansion of hololnorphic functions. 4.1.6. Let X be' a cOluplex Inanifold and! = (/1' ... ,IN) E ON(X). If U C X is open, then we denote by c'F,( U) the space of all functions g E O( U) such that the following condition is fulfilled: For every point Z E U, there are a neighbourhood V C U of z and holomorphic functions g, E O( V) snch that g = gIfl + ... + gNI]of' in V. It is clear from the definition that the so defined analytic subsheaf :1", of xOl is coherent. 1", is called tlte sheaf u,hich is generated by f.
4.2. The section 8(2:, ') and the function 9'(z, C)
161
4.1.7. Let X be a complex manifold, let B be a" holomorphic vector bundle over X, and let J: X ~ B be a holomorphic section. Since B is locally holomorphically triviaJ, the section f can be locally represented by a vector of holomorphic functions, where two different representations are connected by an invertible holomorphic matrix. Therefore, as in Example 4.1.6, the sectionJ generates a coherent analytic Bubsheaf rfl of zf!J1. 4.1.8. Corollary. Let X be a Stein manifold and let Y be a closed complex submanifold of X. Then, for every holomorphic function f on Y, there exi8t8 a holomorphic function F on X sueh that F = f on Y. Proof. It is trivial that f admits looal holomorphic extensions, that is, we can find an open covering {U I } of X and funotions F f E f!J( U f ) such that F f = f on Y n Ufo Then F, - FI E 3'y(U, nUl) (Example 4.1.5) and from Theorem 4.1.3 (li) weobtainHf E3'y(Uf ) such that F, - FI = H, - HI on U, n Ufo Setting F := F f - H f on UI , we obtain the required extension F . •
4.2.
The section
8(~J
') and the function
9'(~,
')
In this section we construct the map s(z, C) and the function rp(z, C) mentioned in the summary at the beginning of this chapter. We need the following 4.2.1. Lemma. Let X be a Stein manifold, let B, B' be holomorphic veclor bundlu over X, and let A: B -+ B' be an injective1 ) holomorphic homomorphism oj vector buMlu. Then there exi8t8 a holomorphic homomorphism A (-1): B' ~ B of vector bundles 8UCh ,hal A(-l)A = id (id := identity map). First we prove the following looal result: 4.2.2. Lemma. Let L(N, M, QJ) be the space 0/ complex N X M matrices (N = row index), and let GL(N, QJ) be the group of invertible complex N X N matrices. If U is a neighbourhood of 0 E P and A: U -+L(N, M, C) is a holomorphic map such that rank A(O) = M, then there exist a neighbourhood V ~ U oj 0 and holomorphic ma:JM A(-l): V -+L(M, N, 0),
T: V -+GL(N, 0)
such that, Jor all z E V, (i) A(-l)(Z) A(z) = id M (id M := unit matrix of order M), (ii) T(z) A(z) = A(O).
Proof. Since rank A(O) = M, there exists a complex M
X
N matrix Bo suoh that
BoA(O) = id M • Choose a neighbourhood V of 0 so small that, for all z E V, the matrix BoA(z) is invertible. Set A(-l)(z):= (BoA(z»-l Bo and T(z):= ida (A(O) - A(z)) A( -1)(Z). Then (i) and (ti) are fulfilled and, after shrinking V, T(z) is invertible
+
for all z E V . • Proof of Lemma 4.2.1. From Lemma 4.2.2 (i) we obtain an open oovering {Uf } of X and holomorphic homomorphisms A}-l): B'luJ ~ BluJsuoh that A}-l)A = id over Ufo For every open set U C X, we denote by rf( U) the spaoe of all holomorphic homomorphisms F: B'I u ~ BI u such that FA = 0 over U. It follows from Lemma 4.2.2 (ii) that there is a holomorphic vector bundle over X suoh that rf is the sheaf of germs of holomorphic sections of this bundle (Definition 4.1.1). Clearly, A}-l) - A~-l) E c'F(U, n Uf ). 1)
11
Injeotivity means that, for every Z E X, the kernel of the induced linear ma.p between the fibres of B an{l B' over z is trivial. Henkin/Lelterer
162
4. Global integra.l formulas on Stein manifolds
Consequently, by Theorem 4.1.3 (ii) or by Theorems 2.13.4, 2.12.3 (jv) and 2.10.3, we can find H f E 3'{Uf ) such that .A~-1)_ A~-l) = H, - H1 on U, n U 1• Setting .04(-1) := Aj-1) H t , we complete the proof. •
+
4.2.3. The complex tangent and cotangent bundles. Let X be a complex manifold of complex dimension n, and let {( UI' hf )} be a holomorphic atlas of X. Suppose U1 c: c: X. Set G(1(z) := JA,01&;l(Z), Z E Ufo n Uf , where JAiohjl(Z) is the Jacobi matrix of k,o kT1 at z (Definition 1.1.15). Then GijGjk = GiJ: on U f n U 1 n Uk (Corollary 1.1.17). The holomorphia vector bundle over X which is defined by the transition functions {Gij } will be denoted by T(X) and is called the complex tangent bundle of X. The holomorphic vector bundle which is defined by the transition functions {(G~)-l}, where is the transpose of Gij, will be denoted by T*(X) and is oalled the complex cotangent bundle of X. The fibres of T(X) and T*(X) over z E X will be denoted by Tz(X) and T:(X).
Gh
4.2.4. Lemma. Let X be a Stein manifold and let T(X) be the complex tangent bundle of X (Subsection 4.2.3). Suppose, in addition, that X is a relatively compact open subset of 80me larger Stein manifold (however, cpo the remark following this lemma). Then there exist a kolomorphic map 8: X X X ~ T(X) and a holomorphic junction rp on X X X such that the following CO'I'Mliticms are fulfilled: (i) 8(Z, C) E T,(X) for all z, CE X (that i8, 8(Z, C) is a section in the pull-back of the bundle T(X) with re8pect to the map X X X ~ (z, C) -+z EX). (ii) For every fixed Z EX, 8(Z, z) = 0 and the map 8(ZI C) from X into Tz(X) is biholomorphic in 80me neighbourhood oj C= z. (iii) cp(z, z) = 1 for all Z E X. (iv) If (f, i& the analytic subsheaf of xxx01 generated by 8 (}~xample 4.1.7), then
rp E 3.«(X X X}" {(z, z): z EX}). (v) There i8 an integer ~ ~ 0 8WJh that, for every norm 11'11 1 ) in T(X), the Junction rpK1I811-t i8 O' on (X X X) " {(z, z) : z EX}. (It i& sufficient to choOBe ~ ;;?: 9.) Remark. Lemma 4.2.4 is valid also without the hypothesis that X is a relatively compact open subset of Bome larger Stein manifold. However, then the proof is more complicated. In fact, in the following proof we use a holomorphic map f: X ~ eN (for some large but finite N) such that, for all z E X, the Jacobi matrix (with respect to local coordinates) of f has maximal rank. In the special case that X is a relatively compact open subset of some larger Stein manifold, the existence of such a map f follows immediately from the definition of Stein manifolds. In the general case suchf also exists, but a proof is necessary; remark that, moreover,j can be chosen to be biholomorphic from X onto some closed complex submanifold of CN (see, for example, GUNNING/ROSSI [1] and HORMANDER [1]). We do not need this generalization of Lemma 4.2.4, because we shall use the section 8(Z, C) and the function cp(z, C) from Lemma 4.2.4 in some neighbourhood of jj X 15 only, where D is a relatively compact open subset of a Stein manifold. Proof of Lemma 4.2.4. Let f: X ~ (f)N be as in the preceding remark. Denote by F the holomorphic homomorphism from T(X) into the product bundle X X (f)N which is locally defined by the Jacobi matrix of f. Then F is injective, and we obtain from Lemma4.2.1 a holomorphic homomorphism of vector bundles F(-l): X X Q)N ~ T(X) such that F(-l)F = id on X. Define 8(Z, C) := F<-l)(Z,J(C) - f(z)) for z, CEX. Then it is
+
1) A norm in T(X) is a Coo·map /1·11: T(X) - [0, 00) such that 1/·11 is a norm in each fibre T,(X), Z E X. By means of local trivializations of T(X) and a partition of unity~ it is ea.sy to construct such a norm (cp. Subsection 2.11.2).
163
4.3. The Martinelli-Bochner formula and the Leray formula
clear that condition (i) is fulfilled, and it follows from the Taylor formula. (1.1.6) that (ii) is satisfied. Now we construct the function rp. Set Ll := {(z, z) : z E X} and
y := {(z, C)
E X X X: 8(Z, C) =
O} "Ll~.
Then, by condition (H), Y is a closed subset of X X X, and the open sets U~ := (X X X)" Y, U y := (X X X)" Ll form an open covering of X X X. Since 3', is coherent and the constant function with value 1 belongs to 3".( U ~ n U y), from Theorem 4.1.3 (i) we obtain functions rpy E 3',( U y) and rp ~ E (F,( U~) such that 1 = rpy - rp~ in U~ n Uy• Setting rp := rpy in Uy and rp := 1 rp,j in U~, we obtain rp E O(X X X) satisfying conditions (iii) and (iv). It remains to prove condition (v). Consider some point Z E (X X X) "Ll and choose a neighbourhood U c (X X X) "Ll of Z so small that T(X) is holomorphically trivial over U. Then the map 8 can be represented by a veotor (St, ••• ,8,,) (n = dim" X) of rp.a" holomorphic functions in U. After shrinking U, we can assume that ffJ = 9'lBt for some ffJt E O( U) (this follows from the definition of 3',). After shrinking U, this implies that there is a oonstant 00 such that Irp(z, e)1 :::: 0 118(z, e)11 for all (z, C) E U. Hence rp9 11sll-2 is a Q2-function in U . •
+
+ ... +
a<
4.3.
The Martinelli-Bochner formula and the Leray formula
In this section we generalize the Martinelli-Bochner formula. (1.9.1) and the Leray formula (1.10.1) to Stein manifolds. 4.3.1. Notation and preliminaries. X is a Stein manifold, and n is the complex dimension of X. If Dec X is an open set, then the boundary aD of D will be called piecewise (J1 if there exist open sets V1, •.. , V N C X and Ol-functions (!t: Vt ~ R, k = 1, ... ,N, such that the follOwing conditions are fuHilled: (1) oD C V 1 U ••• U V N. (2) A point z E VI U ••• U V N belongs to D if and only if, for every 1 ~ k :::: N with z E VJ:l (!J:(z) O. (3) For every collection of indices 1 ~ ~ k, ~ N, we have d(!t,(z) /\ ... ... /\ d(!I:,(z) =F 0 for all Z E V k1 n ..• n Vii' Further on we suppose that D c c X is an open set with piecewise Ol-boundary. We choose the following orientation in X: If ~, ... , z" are local holomorphio coordinates in X and XI are the corresponding real coordinates such that zf = XI iXJ+" then the form dXI /\ .•. /\
<
< ... <
+
(z, (Gij(z)) ~) = H, 0
HTl(Z,~)
and
(z,
«G~,)-l (z))~) = Ht
0
(Hj)-l (z,~)
for
Z
E
U, n U t
and ~
E (/)'I.
E U , and a E Tz(X) (a E T:(X), then the vector at E (fJn with H,(a) = (z, at) (Hj(a), = (z, at») is called the expres8ion of a with re8pect to (UI, ht). If z E U, nUt, a E T,(X),
If Z 11-
164
4. Global int.egral formulas on Stein manifolds
b E ~(X) and a(, at, be, b1 are the expressions of a, b with respect to (U~, ht ) and (U1, h, ), respectively, then
(4.3.1) Therefore, the following definition is correct: If Z E X, a E T.(X) and b E. T:(X}, then we choose an arbitrary j such that Z E Uland define (4.3.2)
(b, a) := (b a,) , "
where b, and a1 are the expressions of b and a with respect to (U1, h1), and (b at) is defined a8 in Section 1.7. " Now let U C X be an open set, let Y be a real Ot-manifold, and let a: U X Y ~ T(X) and b: U X Y ~ P*(X) be OI.maps such that a(z, y) E T.(X) and b(z, y) e P:(X) for all (z,y) E U X Y. Let a1: (U nUt) X Y ~C'l& and bl : (U n U f ) X Y ~cn be the expressions of a and b with respect to (U hi)' Then it follows from (4.3.1), (1.7.5) and " Proposition 1.7.4 that
y»
£O;(b,(z,
£0., (a,(z,
1\
y» = £0; (bt(z, y))
1\
£OV(af(z,
y»
for z E U n U, n U , (4.3.3)
and y E Y. Therefore, by setting £O;(b(z,
y»
and 11
E
1\
£O.,(a(z,
y»
:= m;(b1(z,
1/»)
Y,
"m,(a,(z,
y») for
zEU
n U, (4.3.4)
y»
a continuous differential form £0; (b(z, y» 1\ w,,(a(z, on U X Y is correctly defined. We are interested in integral formulas for functions on D. Therefore, without 1088 of generality, we can assume that X is a relatively compact open subset of Bome larger Stein manifold and apply Lemma 4.2.4 (cp. also the remark following Lemma4.2.4). Let 8(Z, C), tp(z, C) and" be as in Lemma 4.2.4. Further, we fix a fibre preserving O°O.map a: P(X) ~ P*(X) such that the following conditions are fulfilled: (aa, a) ~ 0 for all a E P(X) and the map
lIall" := «aa, a»lJ2 ,
a
E P(X),
(4.3.5)
defines a norm in each fibre of P(X). Such a map a can be defined as follows: If Z E U f , a E T,(X) and a, is the expression of a with respect to (U hi), then we denote by a,a the vector in T:(X) whose expression with respect to (U " I , hI) is partition of unity XI subordinate to {U, } and define aa :=
We write
l: x,(z) a,a for
a E T,(X)
and z EX.
af.
Choose a 0 00 _ (4.3.6)
i
a(z, C) := a8(z, C)
•
(i(z, C) will be used instead of the map t
(4.3.7)
- zin 0".)
4.1.2. Leray sections, BaD and LaD for functions, and B D, RaD for I-forms. We use the notations from Subsection 4.3.1. By Lemma 4.2.4 (v), cp"/11811 1 is a Ol.function for all z, CE X with z =F C. Therefore, for every integer ." :;;::: n,,, and every fixed z E X, the differential form (cp. (4.3.4»
Cn
q;·(z, C) £O~(8(Z, 1\ £0,(8(2:, C» ---- ----!l8(z,CHl!"
4.3. The Martinelli-Bochner formula. and the Leray formula
165
is (J1 forC EX" z, and the singularity at C= Z is of order 2n - 1. Therefore, for every integer 11 ~ n,.x and every bounded I-form I on D, we can define
( B ( " '8 8) f) (z) := (n - I)! DCP, , (2ni) 11
II(C) " cp"!~ C) w~Q(z, C»)" Wt(!iz, C»
zED.
118(Z,C)lI!'"
tED
(4.3.8) Further, for every integer 11 ~ nx and every bounded measurable function! on aD, we define " '8 8) f) (z) := (n - I)! f/(1-) cp"(~'_') w,(8(z, A Wt(S(Z, £) ZED. ( B aD (cP, , (2m)" r, ...
en L
...
0_
,
tEaD
(4.3.9) A Leray section for (D, 8, rp) is by definition a. couple (8*, x*), where x* ~ 0 is an integer and 8*(Z, C) is a Cl-map with values in T*(X) defined for C in some neighbourhood of aD and zED such that the following conditions are fulfilled: (i)* 8*(Z, C) E T:(X) for all zED and C in some neighbourhood of aD. (ii)* (8*(Z, C), 8(Z, C» =!= 0 for ZED, C E aD with cp(z, C) 9= 0, and the function cp~·(z,
-_.
(8*(Z,
C),
C)
--
8(Z, C»
is Cl in some neighbourhood ~ D X X of D X aD. (This means, for every K c c D, there is a neighbourhood V K of aD such that this function is Cl for all (z, C) E K X V g.)
Example. (8, x) is a Leray section for (D, 8, cpl. If (8*, x*) is a Leray section for (D, s, cp), then, for every integer" fixed zED, the differential form (cp. (4.3.4»
~
nx* and every
cp"(z, C) W~(8*(Z, Cn " (1)(8(Z, Cn (8*(Z,C), 8(Z, C)"
is continuous forC in some neighbourhood of aD. Therefore, for every bounded measurable function I on aD, we can define (L
( .. s*
aD Cf,
8)
I)
,
(z) : = (n,. -1) t (2m)"
fl(") cp"(z, C) WC(8*(Z, C» r,
A (1),(8(Z,,"») zED. (8*(Z, C), 8(Z, C»n '
CeaD
(4.3.10) If (8*, x*) is a Leray section for (D, 8, cp), then we set
tc*. - )(z ,. A) I
.1.8
'= (1 _ J.) ___
,t",.
~.~~z,i_)___
. (8*(Z, C), 8(Z, C»
+A
8(%, C)
118(z, C)II: .
By condition (v) in Lemma 4.2.4 and condition (ii) in the definition of a Leray seotion, then, for every fixed zED, the map cpm.u(Je. "·)(z,
C) t~." .•)(z, C, A)
is Cl forO ~ A ::; 1 andC in some neighbourhood of aD. Therefore, by Proposition 1.7.3, for every integer 'P ~ max (me, 11,x*) and every fixed z € D, the differential form (ep. (4-.3.4-»
cp"(z,~) W~,A(t(~.,.,.)(z, C, A») is continuous for 0
~
A
(()t(S(Z,
C))
A ::; 1 and C in Borne neighbourhood of
aD.
166
4. Global integral formulas on Stein manifoJds"
If (8*, ~*) is a Leray section for (D, 8, rp), then, for every integer 11 and every bounded I-form I on 8D, we define
~
max
(n~, n~*)
(RaD(fP·, 8*, 8, 8) I) (z)
:=
!
("(~i~~ I
f(C)
A 'P'(O, C) 00,••('*(0, C, A») A 00(8(0, C»),
(t, A)EaD x [0. 11
0 ED, (4.3.11)
where t* = t(~ ••••• ). 4.3.1. Theorem (Martinelli-Bochner formula). We uae the notations from Sub8ection 4.3.1. Let 11 ~ 2~ be an integer. Then, for every continuou8 function f on jj 8uck that al i8 alBo ccmtinuoUB on D, we have
I
= BaD! -
BD
al
(4.3.12)
D,
in
BaD(ql, 8, 8) and BD = BD(q?, 8,8) (Subsection 4.3.2). Proof. First we prove the formula for the constant function 1= l. Fix zED. Choose UJ.( { (U" hi)} is the holomorphic atlas from Subsection 4.3.1) so tha.t z E U;. Let v(C) be the expression of 8(Z, C) and let u(C) be the expression of 8(Z, C) with respect to (UJ., hJ.) (cp. Subsection 4.3.1). Then
where BaD
=
w,(i(z, C» "WC(8(Z, C») = w,(v(C)) "wc(u(C» ,
and we have to prove that I
I : = !fP'(Z C) ,
=
(2ni)"/(n - 1) 1, where
00, (v(C))
(v(C),
"wc(u(C» •
u(C»"
CUD
For Cin a neighbourhood of aD and 0
'1(C, A) : = 'P"(', C)
s: A ::;: 1, we define
[(1 - A) <1I(C)j;~~:; v(C) +AV(C)]
and
D(C A):= '(z C) roc. A(1}(C, A») "roc(u(C») . ,
(1}(C, A), u(C»"
cp,
Since, for some" > 0, lu(C)1 ~ ~ 118(Z, C)II", the map 1'](C, A) is ()1 in a neighbourhood of aD. Since (1'](C, A), u(C» =
J D(C, A) =
8Dxl
1D(C, A) .
8
(4.3.13)
xO
Taking into account the relations 1}(C, 0) = cp"(z, C) «v(C), u(C»/lu(C)lI) u(C) and 1}(C, 1) = rp"(z, C) v('), we obtain from Proposition 1.7.3 that
D(" A)/ 1"
and
'\-0
=
.(z") (.tJ'~
rp, I,
"(.tJ~(u(C»)
lu(C)1 2ft
4.3. The Martinelli-Bochner formula and the Leray formula
167
Hence (4.3.13) can be written
1=
J
""~) .~.~(U(C)) •
(4.3.14)
'E3D
By condition (ii) in Lemma 4.2.4, we can choose a neighbourhood W c: c: D of Z with smooth boundary 8 W such that u is biholomorphic in some neighbourhood of W. Since rp is holomorphic and by Proposition 1.7.1, the form under the sign of integration on the right-hand side of (4.3.14) is closed for all C€ X with u(C) =F O. Since v ~ n" and, therefore, this form is 0 1 in X we conclude that this form is closed in X " z. Hence, by Stokes' formula, it follows from (4.3.14) that
"z,
1=
J
"(z
rp,
C) oo~~)) /\oo,(u(C») •
(4.3.15)
lu(C)l2n
cuw Since u is biholomorphic in a neighbourhood of W, we obtain I =
J
tEa(u(W»
cp'(z, u -l(E») oo'(E) ~ oo(E) • lEI
Consequently, since the function u( W) ~ ~ ~ rp(z, u-1(E») is holomorphic, and since = rp(z, z) = I (condition (iii) in Lemma 4.2.4), it follows from the Mar_ tinelli-Bochner formula (1.9.2) in QJft that 1 = (2ni)tI/(n - I)!. Now we consider the general case of a continuous function Ion D such that al is also continuous on D. Fix z € D. Since rp is holomorphic, then it follows from Proposition 1.7.1 that, for every CE jj such that 8(Z, C} =F 0, we have the relation
rp(z, u-1(0»)
a [/(C) f/!."(z, C) oo~(s(z, C)L/\ ~t(8(Z, C))] = c
118(z, C)II!"
8/(C) /\ rp"(z,C)oo~(8(Z,C») /\W,(8(Z,C» • 1\8(Z, C)II:-
(4.3.16) 2n 1 Wt(8)/1I811 is 0 for C EX" z, (4.3.16)
Since 11 :2: n" and, therefore, the form rp"oo,(s) /\ is valid for allC ED" z. Fix some local coordinates in a neighbourhood of z and denote by E. the open ball of radius B centered at Z with respect to these coordinates. Then, for all sufficiently small B 0, it follows from (4.3.16) and Stokes' formula that
>
(BD'Ee 8/) (z) = (BaD!) (z) - (B aB,!) (z) •
(4.3.17)
Since, for constant functions, the theorem is already proved, we obtain
(BaEJ) (z) = /(z)
+
/(z») (z) • 1\ oo,(8(Z, C))/1I8(z, C)II2ft has (BaE,(f -
Since the form w,(s(z, ~)) a singularity of order 2n - 1 at C= z, this implies that (BaH./) (z) tends to /(z) for B ~ O. Hence (4.3.17) tends to (4.3.12) when B ~ O. • 4.3.4. Theorem (Leray formula ). We 'USe the notations from Subsection 4.3.1. Let 8, cp) (Subsection 4.3.2), and let v ::2: max (2"", n,,*)
(8*, ,,*) be a Leray 8ection Jor (D,
be an integer. Then, Jor every contiwu0'U8 lunction I on on D, we have I = LaDI - RaD
8f - B D 8J
in
D,
ii BUch tkat 8j is also c01lotinuous (4.3.18)
168
4. Glob8} integr81 formulas on Stein m8l1ifolds
where LaD = LaD(cp·, 8*, 8). RaD = RaD(cp·, 8*,8,8) and BD = BD(cp", 8, 8) (cp. Subsection 4.3.2). To prove this theorem we need the following lemma. 4.1.5. Lemma. U'TUkr the hypotheses
dc,A[j(e) cp"(z, e) roe,A(t*(Z, e, A))
8/(C)
=
1\
0/ Theorem 4.3.4, lor every fixed zED, we have 1\
cp"(z, C) ro~.A(t*(Z, C, A»)
Jor 0 ~ A ~ 1 and C in section 4.3.2).-
80me
oo,(8(Z, e))] 1\
oo,(8(Z, e»)
eD,
neighbO'Urhood oj
where t* = t~ •• ;'6) (cp. Sub-
Proof. Since the form in brackets contains a factor of bidegree (71,,0), we only have to prove that + d A) .Q(C, A) = 0, where
(a,
D(C, A) := cp"(z, C) ro~.,\(t*(z, C, A»)
1\
0),(8(Z,
en .
Choose UJ. ({(U1, hI)} is the holomorphic atlas from Subsection 4.3.1) so that z E Ui •. Let I-' : = max (:Ie, :Ie*), let ~(C, A) be the expression of rpP(z, e) t*(z, e, A), and let u(C) be the expression of 8(Z, C) with respect to (Ui" hi.) (cp. Subsection 4.3.1). Then by Proposition 1.7.3 .Q(C,;')
= qJ"-"P(z,C)
1\ 00'
(v(e, A)) I\ro(u(C»).
Since co (u(C)) is of bidegree (n, 0), we have fa
_
D(C,A) = qJ"-"P(z, C) l: (_1)1:+1 Vta:,A) 1\ (8 c + d,t) v,(C, A) i-I
1\ 00
(u(C)) •
1+J:
Taking into account that qJ"-flP is holomorphic, we thus obtain (4.3.19)
Since fa
l: va(C, A) uj:(C)
=
rpP(z, C) (t*(z, C, A), 8(Z, C»
=
cpI'(z, C) ,
):-1
fa
_
l: Ut(C) (8, + d A) VI:(C,,t)
= 0.
J:-l
This implies that fa
_
1\ (8, + d 1)
~J:(C,,t) =
0,
(4.3.20)
1-1
because the form on the left-hand side of (4.3.20) is continuous and the set {C EX: u(C) =FO} is dense inX. From (4.3.20) and (4.3.19) it follows that(8 c+ d A) .Q(C, A) =0. • Proof of Theorem 4.3.4. By the Martinelli-Bochner formula (4.3.12), it is sufficient to prove that
RaD 8J = LaD! - BaD! in D.
169
4.4. The Leray.Norguet formula
This follows from Stokes' formula and Lemma 4.3.5 if we take into account that by Proposition 1.7.3 w' (t*(z ,. A») A W (8(Z "))1 = W~(8*(Z, C)) t\ We(8(Z, C»)
e.A
,I"
<8*(Z, C), 8(Z, C»ft
"A-O
"
and w'
(t*{z,. A))
,,1
W (8(Z
t\
,.») I1-1 =
C,"
,,,,
w~ (8(Z, C»)
t\
W,(8(Z,
118(z, C)II!ft
C»)
'
where t* = t~ •• I,.) (cp. Subsection 4.3.2). Remark. For 8* = 8 we have RaD = 0, LaD = BaD and the Leray formula (4.3.18) is the Martinelli-Bochner formula (4.3.12).
4.4.
The Leray-Norguet formula
For many purposes it is appropriate to use different Leray maps for different pieces of the boundary. This is the case, for example, for real non-degenerate analytic polyhedra (Section 4.7) as well as in the more general situation of real non· de· generate strictly pseudoconvex polyhedra (Section 4.9). In the present section we prove a corresponding generalization of the Leray formula (4.3.18), which will bo called the Lera.y.Norguet formula.. 4.4.1. Notation and preliminaries. We use the notations from Subsection 4.3.1. Set S" := {z E aD n V,,: e,,(z) = O} for k = 1, ... ,N, where VI: and el: are as in the definition of a piecewise O1.boundary at the beginning of Subsection 4.3.1. For every ordered collection K = (kl' ... , k,) of integers 1 ~ kl' ... , k, ~ N, we define Sg :=
{Si1 n ... n 81:.
if the integers kl' ... , k, are different in pairs , otherwise .
o
'Ve choose the orientation on 8 K so that N
aD
=
E S" 1:=-1
(4.4.1)
and N
aSK = :E SKI'
(4.4.2)
i=1
where 8D and 8S K carry the orientations induced from the orientations of D and SIC, respectively, and Kj := (~, ... , k"j) if K = (~, ... , k,). Then the orientation of SIC is skew.symmetric in the components of K. We denote by L1 the subset of all points (Ao, A,., ... , AN) E IlN +1 such that A1 2; 0 N
for j = 0, 1, ... ,N and :E A.1 = 1. We orient L1 by the form J-O
strictly increasing collection I = (iI' ... ,i,) of integers 0
~
d~ t\ ... t\
il
dAN' For every
< ... < i, ~ N, we Bet
I
L1 1 := {A
E
L1: :EAir = I} r=l
and choose the orientation of L11 so that J
8L1, = ~ (_I)r-l L1. r=1
19
••••
ir..... i"
(4.4.3)
170
4:. Global integral formulas on Stein manifolds
where i, means that i, is omitted. It is easy to see that then, for every strictly increasing oollection K = (~, ... ,k,) of integers 1 ~ ~ < ... < k, ~ N, AOK carries the orientation defined by the form dAi, 1\ ••• 1\ dlle, (OK := (0, kl' ... , k,). This implies that, for k = 0, 1, ... , N, the one-point set At is oriented by +1. For every collection I = (iI' ... ,i,) of integers we write III := l.
Lemma. 6(~ (_I)IKI 8 K X K
Ll OK )
~ 8 x X AK K
=
aD X L1o,
(4.4.4)
where the 8'Ummation i8 over aZZ8trictly increaaing collecticm8 K = (kI , 1~~ k, ~ N.
< ... <
••• ,
k,) oj integer8
Proof. Since dimB Bx = 2n - IKI, we obtain that the left-hand side of (4.4.4) is equal to
1: (_I)IKI (68 K
X AOK
+ (_1)2.- IKl 8 K
II
X 6A oK )
•
By (4.4.2) and (4.4.3) this implies that the left-hand side of (4.4.4) is equal to N
1: (_I)IKI 1: BKf i-I
K
if K = (kI ,
•••
that
X AOK
+ 1:K BK X
,kIKI ). Therefore, since N
~ (_I)IKI ~ ~ ~ K
i-I
8 JIj X
AOK
~
IKI=1
+ ~ 8K ~
IKIiii=2
Ax + ~ BK
lEI X ~ (-I)" AO.k, •...• £.•...• kIKl
K
,-1
8 x X Ao = aD X Ao, it remains to prove I~
X ~ (-1)" AO.kb ...• k, ~ ..... kIKI -- 0 • ~
,-1
(4.4.5)
For l = 1, ... , N, we denote by P, the set of all strictly increasing collections K = (kl' ... , k,) of integers 1 ~ ~ k, ~ N. If K = (kl' .•• , k,) E P" then we denote by KC the collection in P N - l with K u KC = {I, ... ,N}. If K' = (k~, ... , ki+l) E P'+l and j E K', that is, j = k; for some 1 ~ r ~ l 1, then we write ,,(K',j) := r and L(K',j) := (k~, •.. , k;, ... ,k;+l)' Since SK; = 0 for j E K, now (4.4.5) can be written
< ... <
+
N-1
~ (~ (_1)1 8JIj X
AOK
1-1 KePI jEl!·
+ 1:
(_I)II(K'.i) 8 K , X .x'e P l+l
AOL(K'•• » =
0•
(4.4.6)
lEK'
For every couple (K, j), where K E P, and j E KC, there exists one and only one couple (K', i), where K' E P'+1 and i E K', such that L(K', i) = K and i = j. Moreover ea.ch suoh couple (K', i) can be obtained in this way. Therefore, it is sufficient to prove that for alII ~ l ;;;; N - 1, K' E P'+l andj E K', (-I)' 8L(K'.J)i X AOL(l!'.J)
which follows from the relation
+ (_I)II(K'.J) 8.x'
X AOL(K'.;)
= 0,
SL(K'.J)J = (_I)tt(K'·J)-l-l SK' • •
4.4.2. Leray-Norguet sections, Ls. for functions and Rs. for I-forms. We use the notations from Subsections 4.3.1 and 4.4.1. A Leray-Norg'Uet 8eotion for (D, 8, qJ) is by definition a collection (81 ,,,,, 8N, ,,*), where ~ 0 is an integer and, for every 1 ~ k ;;;; N, 8t(Z, C) is a Ol-map with values in T*(X) (cp. Subsection 4.2.3) defined for Cin some neighbourhood of 81: and ZED such that the following conditions are fulfilled:
,,*
I'll
4.4. The Leray-Norguet formula
(i) * st(z, C) € Tt(X) for all zED and Cin some neighbourhood of 8. (ii)* <8:(Z, C), 8(Z, C» :::J= 0 if zED, CE 8 t and q;(z, C) =F OJ and the funotion
g1'*(z, C) (,*(z, C), 8(Z, C» is 0 1 for (.z, C) in some neighbourhood ~ D X X of D X Sil' Let (-'l., •• - , 8 N, ~*) be a Leray-Norguet section for (D, 8, rp), and let K = (kt, ••• ; ',) be a strictly increasing collection of integers I ~ kl < ... kl ~ N. By oonditiOl1 (h1 and Proposition 1.7.3, then, for every fixed zED and every integer" ~ ft~*, the clif. ferential form (cp. (4.3.4»)
<
" ' (1
rp (z, C) CO,.A I; At < *( 8:(Z, C) C)( keK
8A: Z,
, 8 Z,
)
C»
1\
COt (8(Z, C) )
is continuous for A E LJ K and Cin some neighbourhood of SIC' If I is 80 bounded measur· able function on S K. and'll ~ nx* is an integer, then we define (Lsg(rp·, st, .•• ,8;',8) f) (z) :=
J · '(
(n-l)!
s:(Z,C)) 1\ co~ ('(1, C))
I(C) cp (z, C) CO,.1 J;~All (st(%, "), 8(%, C»
(~.l)ESJ[X..dK
For all (z, C, 1) with cp(z, C) X LJ oK , we set
t(.r* .....•~. _
=F 0 in some neighbourhood C D
z CA' =
8 ••)( )
,
ZED.
).
l: 1
8(Z, C)
2
no 118(Z, C)II! + iEK
Then, for every integer v > max (n",
n~*)
rp'(Z, C) CO"A(t~r, ...••~.i.B)(Z, C, A))
1\
t
t
(".'.7)
X X X AOK of D X 8«
8:(Z, C)
(.I(z, C), 8(Z, C) •
(4.'.8)
and every fixed zED, the differential form
m,(8(z, C»)
is continuous for 1 E LlOK and Cin some neighbourhood of Sf(' If I is a bounded I·form on Sf( and f,I ~ max (n,,*, 'n"), then we define (Rs.(cp', 8~, ... ,
: = (-I)K
8;, 8, 8) f)
(,.(~)~ !
(z)
J
fCC) A'I"(z, C) ooc.• (I*(z, C, A») A 00,('(1, C») , a ED,
(C, a)E8. x .dOg
(4.4.9)
h J.* -- t** •- . were. (ll''''''N. 8 ••)
Remark Lsg
= 0 if IKI > 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, ...
< ... <
8t, ...
Remark. If (8*, ~*) is a Leray section for (D, 8, cp), then by setting 8: := s* we obtain a Leray-Norguet section (at, ... , 8~, ,,*) for (D, 8, cpl. Then Li:lg = 0, RSII = 0 if IKI ~ 2 and LSI LSN = LaD' RSa RSN = R aD , that is, in this case, the Leray-Norguet formula (4.4.11) is the Leray formula (4.3.18). To prove Theorem 4.4.3 we need the following lemma.
+ ... +
+ ... +
4.4.4. Lemma. Under the hypothel3es of Theorem 4.4.3, for every fixed ZED and every k, ~ N, we have 8trictly increasing collection K = (klJ ... , k l ) of integers 1 ~ ~
< ... <
d,..t[!(C) cp"(z, C) COe.A(t*(Z, C, A)) 1\
(OC(8(Z,
A
CO,(8(Z, C)J = 81(C)
1\
q;V(z, C) CO~.A (t*(z, C, A))
C))
!01'
all A E LlOE and section 4.4.2).
C in
some neighbourhood of 8 K , where t* = t*('1' * .. ··'N,8 JII-) (cp. Sub.•
Proof. Repetition of that of Lemma 4.3.5 . • Proof of Theorem 4.4.3. According to the Martinelli-Bochner formula (4.3.12), it is sufficient to prove that ~
IEI=aA-l
RSII
8J =
~ LSIl! -
BaD! .
IKI=aft
This follows from relations (4.4.4), (4.4.10) and Lemma 4.4.4 if we take into account that £1 0 carries the orientation +1 and, by Proposition 1.7.3,
co'
C.A
(t*(z" A)) Aco ,,-,
(8(Z
"))1
,,~
8DxA.
=co,(s(z,C)) I\CO,(8(Z,J)) •• Ils(z,C)II:"
4.4.5. Corollary. 1/ under the hypotheses 0/ Theorem 4.4.3, lor every fixed zED, tke 8ections 8t(Z, 8':(Z, are holomorphic !or in 80me neighbourhood of 8 1, ... , S N, respectively, then LSIl! = 0 if IKI =F nand, there!ore,
e), ... ,
!= ~
e)
LSIl! -
e
1:
Rsg
IKI~n-l
IEI-n
6! -
BD
8!
in
D.
(4.4.12)
In particular, if! is holomorphic in D, then
J = I:
LSIl! in
D.
(4.4.13)
IEI-n
e), ...
K
e)
e, then for every collection < ... < lei ~ N, the form
Proof. If 8t(z, ,8;(Z, are holomorphie in = (leI' ... , k,) of strictly increasing integers 1 :5: ~
is of degree n in C. Since dims 8 K
= 2n
- IKI, this implies tha.t LSIl! = 0 if IKI
=F n . •
173
4.5. The Koppelman formula. and the Koppelman-Leray formula.
4.5.
The Koppelman formula and the Koppelman-Leray formula
In this section we generalize the Koppelman formula (1.11.1) and the KoppelmanLeray formula (1.12.1) to Stein manifolds. 4.5.1. The operators BaD' B D, LaD and BaD for differential forms of arbitrary degree. 'Ve use the notations from Subsection 4.3.1. First we consider the following general situation. Let V, W~ X be open sets, let A be some real Ol.manifold, and let a*: V X W X A ~ P*(X) be a Ol-map such that a*(z,~, A) € P:(X) for all z € V, CE W, 1 EA. For z € U I n V, CE W, A E A ({ (U" htn is the holomorphic atlas from Subsection 4.3.1) we denote by at(z,~, A) the expression of a*(z, C, it) with respect to (U1, k 1). For (z, C, it) E (U I n V) X W X A, we define (for det1.ft_t see Definition 1.7.6)
-,•• e.,\ (* . _£O aJ (Z,~,A) ) .
1
I..
i det l •
ft _
, \
1
(* '2 ai (z, t",. IL), (8- •• ,
+ d,\) aj*,. (z", A) ) •
}"or (z,~) E Uf X X, we denote by 8t(Z,~) the expression of 8(Z, C) with respect to (U f , k , ).Since,foreveryholomorphicfunction1p(z), 8.. (tp(z} a1(z,C,1») =tp(z) 8.a1(z,C,A), then it follows from Proposition 1.7.4 and relation (4.3.1) tha.t W;.C,1(a!(z,~, 1»
for all
(z,~,
1\
en
£O,(81(Z,
A) E (U1 n U, n V) X W X
W;.C.l(a*(z, C, A»)
1\
CO,(8(Z,
e,
W;,c,A(af(z, A)) 1\ W,(8(Z, A. Therefore, by setting
=
e» := w~C.l(a!(z, C, A»
1\
C»
£O,(81(Z, C»)
for
(Z,C,A) E (UI n V) X W X A,
(4.5.1)
e,
e»
a continuous differential form W~.C.l(a*(z, J.») 1\ £Oc(8(Z, on V X W X A is correctly defined. In the same way as in the case of Proposition 1.7.3 it can be proved that, for every Ol.function tp(z, C, A) on V X W X A,
OO;.,.1(V'(z, C, 1) a*(z,~, A» 1\ (.O,(8(Z, C» = V'''~z, C, A) w;..t.c(a*(z, ~,A)) 1\ £Oc(8(Z, C» .
(4.5.2)
Now we define the operators BD and BaD' By condition (v) in Lemma 4.2.4, for every integer" ~ n~, the differential form (cp. 4.5.1)
rp·(z, C) 00;., (8(Z,
e»
1\
£O,(8(Z, C»)
118(z, C)\I~" is (Jl on (X X X) "" {(z, z) : Z E X} and the singularity at C= z is of order 21& - 1. Therefore, for every integer" ~ n~ and every bounded differential form/ on D, we can define (BD(rp·, 8, 8) J) (z) := (n - ~ (2ni)"
fl(C)
1\ cp·(z,
e»
C) 00;.,(8(%, 1\ £O,(8(Z, C», 1\8(Z, C)II:"
zED.
(4.5.3)
'ED
Further, for every integer , define (B3D(rptl, 8, 8) f) (z) := (n - I)! f!(~) (2ni)"
,e3D
~ n~
1\
and every bounded differential form / on 8D, we
~,tI(_z, e) W;.,(8(Z,C))~~~(8~: Cll, \l8(Z, e)l I;"
zED.
(4.5.4)
174
4. Global integral formulas on Stein manifolds
Now we define the operator LaD' Let (8*, x*) be a Leray section for (D, 8, cp) (Subsection 4.3.2). By condition (ii) in the definition of a Leray section, for every integer" ~ nx*, the differential form (cp. (4.5.1)) gl(z, C)
oo;,,(8*(Z, Cn 1\ W,(8(Z~) <8*(Z, C), 8(Z, C)fl
is continuous in some neighbourhood ~ D X X of D X aD. Therefore, for every integer" ;;?: "x* and every bounded differential form I on aD, we can define
(LitD(q/', 8*, 8) I) (z)
jl(C}
I)! (2ni)fl
:= (n -
1\
cp"(Z, C) W;,,(8*(Z, Cn I\W,(8(Z, e)) , <8*(Z, C), 8(Z, C»fl
zED.
CEitD
(4.5.5)
°
Finally we define R aD • Let (8*, x*) be a Leray section for (D, 8, cp). For all ~ ii ~ 1, zED and Cin some neighbourhood of such that <8*(Z, C), 8(Z, C» =F 0, we define
oD
tc*. - )(z a
,~,'
J-
A) . = (1 _ A) ____ ~*(z,~L_
<s*(Z, C),8(Z, C»
,\.0,.
+ ii
8(Z,
C)
118(Z,C)lI!'
(4.5.6)
By condition (v) in Lemma 4.2.4 and condition (ii)* in the definition of a Leray section (Subsection 4.3.2), the map tpmU(H,H·)(z,
C) t~ •. i .•)(Z, C, ii)
(4.5.7)
is for all (z, C, A) in some neighbourhood C D X X X [0, 1] of D X aD X [0, 1]. By (4.5.2) this implies that, for every integer" ~ max (nx, nx*), the differential form (cp. (4.5.1)) 01
cp'(z, C) oo;."A(e~ •• i,.)(Z, C, A))
1\
W,(8(Z, C))
is continuous in some neighbourhood ~ D X X X [0, 1] of D X aD x [0,1]. Therefore, for every integer" ~ max (»x, nx*) and every bounded differential form I on aD we can define (RaD(cp" 8*, 8, '= (n -
.
1)1
(2n:i)fl
8)1)
f
(C.A)EaD
where e*
=
(z)
I(C)
x [0,1)
I\ql(z,C) W;,C,A(t*(Z,C,
A»)
1\
W,(8(Z, C») ,
ZED,
(4.5.8)
t(~ •• i ••) .
Remark. In the case of functions or I-forms respectively, the definitions of B D , BaD' LaD and RaD given in the present subsection agree with the corresponding definitions in Subsection 4.3.2. 4.5.2. Theorem (Koppelman formula). We use the notations from Subsection 4.3.1. Let " ~ 2n", be an integer, and let f be a continuous (0, q)-form on jj 8uch that 8f is also continuou8 on i5, 0 ~ q ~ n. Then the/o·rms BaDi, BD 8f, BD/and 6B DIare continuous in D, and we have
(4.5.9)
= BaD(cp", 8, 8) and B D = B D(rp·, 8, 8} (Subsection 4.5.1). Proof. If q = 0, then, by degree reasons BDI = 0 and (4.5.9) is the Martinelli-
where BaD
Bochner formula (4.3.12). Therefore, we only have to prove the theorem for 1 ~ q ~ n.
175
4.5. The Koppehnan formula and the Koppelman-Leray formula
It is easy to see that BaDi, BD of and BDf are continuous. Therefore, it remains to
-
-
prove that in the sense of distributions OBDI = (-1)11 - BaD! + BD 8! in D. That is, we have to prove that, for every Goo·form v with compact support in D, (-1)11
J (BDj) 8v =
(-1)11
1\
D
Jf
J(BaD!) v +DJ(BD ai)
1\ V -
1\
D
1\
v.
(4.5.10)
D
Sincef, BaDland BD Sf are of bidegree (0, q) and BDjis of bidegree (0, q - 1), we can assume that v is of bidegree (11., n - q). Using the notation 8(z C) : = (n =~ ql'(z, C)
,
w;"(8(Z, C»)
w,(s(z, C))
1\
118(Z,C)lI;n
(21rl)n
'
(4.5.10) can be written
J
(-1)1
f(C)
1\
8(z, C)
1\
8v(z)
f
= (-1)' J j(z)
1\
v(z)
zeD
(t,t)eD x D
f(C)
1\
8(z,
C)
1\
+
v(z)
(t, t)eD x aD
J
81(C)
1\
6(z, C) "v(z) •
(4.5.11)
(t, t)eD x D
It follows from Proposition 1.7.1 that all monomials in ~t6(z, C) oontain at least one of the differentials dzt, ... , dZn (when zt, ... ,Zn are local holomorphic coordinates). Since v(z) is of bidegree (n, 11. - q), we obtain
d z,t(6(z, C)
1\
v(z)) = -8(z, C)
1\
for C =f: Z
8v(z)
(4.5.12)
•
:Moreover, since all monomials in 8(z, C) contain the factor dCI " ••• "den (when Cl , •.• ,Cn are local holomorphic coordinates), we get dfCC) 1\ 8(z, C) = 81(C) " 6(z, C) for C =f: z. Hence d..t(!(C)
1\
6(z, C)
1\
v(z»)
= 8f(C)
1\
6(z, C)
1\
v(z) - (-1)11 I(C)
1\
O(z, C) ,,8v(z)
C =f: z.
for
(4.5.13)
By condition (ii) in Lemma 4.2.4 we can find a neighbourhood U 4~ X X X of the diagonal L1 := {(z, z) : z E X} such that for every fixed z E X, the map s(z, C) is bi· holomorphic for all C E X with (z, C) E U.d' Consider the open sets
U.:= {(z,C)
E
U.d
X
U 4 : 118(z,C)lla <e} ,
e> 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
<
E
C
c.::
U
we can find
< E ~ E1 ,
(4.5.20)
<
and, for some constant 0 00, lu(z, e)1 ~ 0 (z, C) E (VI X U 1) n au., that is, Os
Since VI
Eo'
"
Z. c c W.. for 0
I~I ~
~
(4.5.19)
•
0
for
< s ~ SI
118(z, e)lIa ~
(TJ,~) E Z,
and
CE
for 0
< E ~ El
.
and
(4.5.21)
From (4.5.19) we obtain W(P-l(f}, 0») = 0 for all TJ E nt(U I ). Since w 0 T-lis oeo on WIS.' together with (4.5.20) and (4.5.21) this implies that there is a constant 0 00 such that Iw (T-l(TJ, ~» I ~ Os and IId'lw(T-l(f}, E») II ~ Os
<
for all 0
< s ::;;; e
1
and (1], E) E Z •. Therefore, we can find a constant 0
IIw;,,(w(T-l(TJI ~))) - wf(w(T-l(TJ, ~)))II ~ for all 0
<
e
~ SI
and
(1], E)
E Z.
< 00 such that
OS2
(4.5.22)
,
where
w'fl...,,(w)
1
-
:= (n -----_ --1) I det 1,ta- 1 (w ' 1(81
+ -a~) w) ..
and
-, co,(w) : =
1 (n -
1) I
det! .-1 (w, UeW) ~ '
(cp. Definition 1.7.6).
Let (hi 1 )* f andcx*fbe the pull-backs off defined by 11,.1 1 and a, respectively, and let (h j 1)*'(TJ)
=
1:: !'l""'('Yj) dij'l 1:a;,,< ... <,.:a;.
1\ ... 1\
dij., for TJ E ht ( U I )
•
(4.5.23)
4.5. The Koppelman formula and the Koppelman-Leray formula
177
If hjl' ... , h;" are the components of h" then it follows that I(C)
~
=
Jil ...i,(hf(C)fdhi~(C) ...-dhi..(C)
l~iJ< ...
for C E U f
(4.6.24)
,
and a */('YJ, ~)
=
~
li
l;;;iJ<".<"~n
J .....
(h1(a('YJ,
;») d'1.E~iJ (a('YJ,;))
for
(TJ, E) E
~.fh~,(a(TJ' ;))
1\ ... 1\
W...
(4.5.25)
Define
~
a:I(;, TJ) :=-
!1t...lfJ(h1(a(rJ~;))) d'lhiia (a(1], ;»)
~hJic(o;(1], E»)
1\ ... 1\
l~IJ< ...
for
('YJ, E) E W...
(4.5.26)
From (4.5.19), (4.5.20) and (4.5.21) we obtain a constant 0 such that
;» - rJl ~ Oe
Ih1(a(7}, for all 0
<8
~ fl
lId"hf(a('YJ,~)) - d1]11 ~ 08
and
and (TJ, ;) E Z,. According to (4.5.23) and (4.5.26), this implies that
lim sup IIC\:I(~, 7}) - (hjl)* 1(1])11 = O.
(4.5.27)
..... 0 ('1. E)eZE
=
1, we can find
('I},;) E Z..
(4.5.28)
Since, by (4.5.19) alld condition (iii) in Lemma 4.2.4, rp(T-l('I}, 0)) 00 such that
o<
Icp (T-l(1], ;») - 11
~ Oe
for
< e ~ el
0
and
Now we are ready to prove (4.5.16). We denote by (T-l). (j of f 1\ () 1\ vI with rc~pect to T-l. Then
I
I(C)
O(z,
1\
C)
1\
V1(Z)
(a. C)e (V j x U j)naf; e
=
I
(T-l).
1\
8
1\
v1) the pull-back
(I 1\ 8 1\ vI)
(7},~).
(4.5.29)
('1. E)EZ.
Since 8(z,
for (z, C)
C)
E (U1 X
(T-l)* 1\
Vt(z)
1\
=
I)! rp"(z, (2m)"
(n -
C) w;.c(w(z,C»
1\
w,.c(u(z,
C))
s2"
1\
v,(z)
U , ) n CU'l> 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)
+
B D) a! + 8(R3D
+ B D)!
where LaD = LaD(cp", 8*,8), RaD section 4.5.1).
=
RaD(cp", 8*,8,8) and BD
in D, =
(4.5.32)
BD(q;", 8, 8) (cp. Sub-
Remark. If 8*(Z, C) = 8(Z, C), then LaD = BaD, RaD = 0 and, therefore, (4.5.32) agrees with the Koppelman formula (4.5.9). If q = 0, then BDf = 0 and RaD! = 0 and, therefore, (4.5.32) agrees with the Leray formula (4.3.18). To prove Theorem 4.5.3 we need the following lemma. 4.5.4. Lemma. Let the hypotheses of Theorem 4.5.3 be fulfilled. Set
nez, c, A) := cp"(z, C) W;,C.A(t·(Z, C, A)) "W,(8(Z, C)) , where t· = t~ •• ".) (cp. Subsection 4.5.1). Then Jor (z, C, A) (all
+ ~,.l) (f(C) 1\ ii(z, C, A»)
= 8f(C)
A
E
W
X
[0, 1]
ii(z, /;,: A) •
(4.5.33)
Proof. Since the form ii(z, C, A) contains the factor dCI A ••• A dC" (when Cl , ••• , C" are local holomorphic coordinates), we only have to prove that (8~,c + dA) Q(z, C,A) =0. Let {(UI' hI)} be the holomorphic atlas from Subsection 4.3.1, set I-' := max ('" ,,*), let f)f(z, C, A) be the expression of ~(z, C) t·(z, C, A) with respect to (U f , hI) (cp. Subsection 4.3.1), and let 8,(z, C) be the expression of s(z,~) with respect to (Uj , hI)' Then, by (4.5.2) and (4.5.7), " D(z, C, A) = cp"-"P(z, C) ~ (-1 )k+l Vjk(Z, C, A) /\ (cz,c + d A) Vjl(Z, C, A) '~k
k-l
"wc(s,(z, C)) where VjV
••• , Vjn
(8~.,
for
Z
E
U" (z,
C, A)
E
W
X
[0, 1] ,
are the components of Vj. Therefore,
+ d A)
-
D(z,
e, A) =
"
-
nql'-nP(z, e) 1\ (8z.,
+ d l ) Vjk(Z, e, A)
1\ WC(8j{Z,
e))
J;=1
for
Z E
VI' (z, C, A)
E
W
X
[0, 1] .
(4.5.34)
1) The theorem is valid also without this additional assumption, but then the proof be-
comes teohnioally more complioated.
179
4.5. The Koppelman formula and the Koppelman-Leray formula
" f);~jk l:
We have
(cpPt*, s) = rp"', where .5j1,
=
are the components of
••• ,8JfI
81'
Since
i=l
the functions 8;1: and fP are holomorphic, this implies that n
_
l: 8ji(Z, C) (8z"
+ d A) viTc(z, C, l) =
0
z
for
U (z, C, l) "
E
i-I
E
W
X
[0, 1] .
Therefore fI
_
1\ (8z,c
i .... l
+ d,,) Vjl:(z, C, l) =
0
z E Uf, (z, C, l)
for
E
W X [0, 1] ,
(4.5.35)
because the form on the left-hand side of (4.5.35) is continuous and the set {(z, C, l) E W X [0, 1] : 8(Z, C) =t= O} is dense in W X [0, 1]. It follows from (4.5.34) and (4.5.35) that (8ZtC d;.) Q(z, C,A) = 0. •
+
Proof of Theorem 4.5.3. By the remark following Theorem 4.5.3 we can assume that 1 ~ q ~ n. It is clear tha.t LaD!, BaD 8! and BaD! are continuous in D. By Theorem 4.5.2, BD 81, BD! and 8BD! are continuous in D. By differentiation under the sign of integration (this is admissible in view of the hypothesis of the theorem on smoothness of ql8*/(8*, 8»), we obtain that 8R eD l is continuous in D. It remains to prove (4.5.32). In view of the Koppelman formula (4.5.9), the proof will be complete if we show that -
8B aD ! = BaD! - LaD! + RaD
-
81
in
(4.5.36)
D.
By Stokes' formula we obtain from Lemma 4.5.4
J
8f(C)
1\
Q(z, C, A) -
(C.l)EaD x [0,1]
=
f
8.
f(C)
1\
D(z,
(t',A)e&D x [0,1]
JI(C) 1\ :O(z, c, A)/" ... O - Jf(C)
aD
e, A)
ii(z, e, A)I;'_l .
1\
3D
This implies (4.5.36), because, by (4.5.6), Q(z ,. A)I , ~, )' ... 0
=
"(z ") w;,,~_s*(z,~))
rp
,"
(8*(Z,
1\
oo,(8(Z, C))
C), 8(Z, C)"
and Q(Z, C, A)IA=l = rp"(z,
C) W~C(8(Z, C))
1\
4.5.5. Coro]]ary. 1/, under the hypotheses morphically on zED and ill ~ q :::; n, then
f =
8(-1)11 (RaD
wc(s~z, C))
118(z, C)II~"
0/
(RaD
is a continuous solution of
+ BD)f + (_I)f+ 1 (RaD + B D ) 8f
in
if 8uch that 8f =
depends 11,010-
D.
(4.5.37)
0 in D,
+ B D )!
(4.5.38)
au = ! in D.
Proof. If 8*(Z, C) is holomorphic in z, then the form Cii;,,(s*(z, degree zero in z. Therefore, LaD! = 0 if q ~ 1 . • 12·
C)
Theorem 4.5.3, s*(z,
In particular, for every continuous (0, q)-form 1 on u := (-1)11
.•
en
1\
w,(s(z, C)) is of
180
4. Global integral formulas on Stein manifolds
4.6.
The Koppelman-Leray-Norguet formula
The Koppelman-Leray-Norguet formula, which will be proved in the present section, is a generalization of the Koppelman-Leray formula (4.5.32) as well as of the LcrayNorguet formula (4.4.11). 4.6.1. The operators LSE and Rs.a- for differential forms of arbitrary degree. We use the notations from Subsections 4.3.1 and 4.4.1. Further, let (st, ... , s;, u*) be a Leray-Norguet section for (D, 8, rp) (Subsection 4.4.2), and let K = (kl' ... , kz) be a strictly increasing collection of integers 1 ~ ~ k, ~ N. Then, by (4.5.2) and condition (ti)* in the definition of a. Leray-Norguet section, for every integer " ~ nu·, the differential form (cp. (4.5.1))
< ... <
, ~) -,
rp (z, ~
COz,t,.l
(~A 4..
st(z,
t( *( C) 8" Z,
kEK
C) ) ( ,.) ( C) "coc 8(Z, ~)
, 8 Z,
is continuous for all (z, C, A)in8omeneighbourhood~ D X X X LlKofD X BK X Ll K • For every bounded differential form f on BE and every integer l' ~ nu*,
r
(Ls.a-(rp', st, •.. , 8;,8) f) (z) :=
(n - 1) I (2m)n
..
fC ( )
~ -, (~;., st(z, C) ) (( ~)) q;'(z~~) COz,e,.l t~ I: (st(Z,C),8(Z,C) "We 8 Z, , zED.
<e,,a)eSxx.1,a-
(4.6.1)
Let t* = t~~, .... ,~" .•) (see (4.4.8)). Then, for every integer" ~ max (nu, n,,*), the differential form (cp. (4.5.1))
rp'(z, C) W;,C,.l(t*(z, C, A))
"w,(s(z,
e))
is continuous for all (z, e,,1) in some neighbourhood ~ D X X X LlOK of D X S K X Ll og• For every bounded differential form f on S K and every integer v ~ max (n", n,,*)
we can therefore define (Rs.a-(q!, 8t, ...
,8;,8, 8}f) (z)
:= (_l)IKI (n -
I)!
(2m)n
f
f(C)
1\
rp"(z, e)
w;,c..t(t*(z, e, }.)) "wc(s(z, C)) zED.
<e,,t)eSxx Llo.l'
,
(4.6.2)
Remark. If I is of degree q, then
= 0 and
= 0 if lEI
~ n
+1-
(4.6.3) q. This follows from the fact that then the form under the sign of integration in (4.6.1) and (4.6.2) is of degree :2:-n + q in C, whereas dimB S K = 2n - IKI ~ n + q - 1. Ls.a-i
RS.I'i
Remark. For N = 1, Definitions (4.6.1) and (4.6.2) agree with (4.5.5) and (4.5.8), respecti vely. Remark. If I is a function, then Definition (4.6.1) agrees with (4.4.7) and ESJlI = Iff is of degree 1, then Definition (4.6.2) agrees with (4.4.9). 4~~.2.
o.
Theorem (Koppelman-Leray-Norguet formula). We use the notations from ,8;, u*) be a Leray-Norguet section for (D, s,)rp) (cp. Subsection 4.4.2), and let" ~ max (2n;e, nu*) be an integer. Suppose, addittonally, that, for k = 1, ... ,N. all derivative8 of rp·(z~ e) s*(z, C}/(st(z, e), s(z, e)
Bub~ctiO'M 4.3.1 and 4.4.1. Further, let (8t, ...
4.6. The Koppelm8oll-Leray-Norguet fonnuls.
181
which are of oraer ~ 2 in z and oj order ~ 1 in Care continuous Jor all (z, C) in 80me neighbourhood W~ cD X X oj D X 8".1) Let J be a continuous (0, q)-Jorm on jj such that 8j is alBo continuous on 15, 0 ~ q ~ n. Then, for every collection K = (kt, ... J lei) oj 8trictly ~rea8ing £ntegers I ~ leI < ... < le, ~ N, the Jorms LsKJ, BSK 8/, B D BJ, Bsllf, Bn/, oRsllf and oBDJ are continuous in D, and we have
I: L S1cf-(BD +
(-I)'J=
IKI :a;,,-g
I:
R Sll )8J+8(BD
IKJ:a;,.-g-l
+ I:
B SK )!
IKI:a;ft-g
in D, (4.6.4) where the summation. is over all8trictly increasing collections K = (i1., ... , le,) oj integers I < lel < ... < le, ~ N with l ~ n - q - I and l ~ 11, - q, respectively, and LSll = LSll(ql, 8t, ... ,8;,8), RSll = RSll(q/', 8t, ... ,s;, 8, 8) (cp. Subsection 4.6.1) and BD = BD(q;·, 8, 8) (cp. Subsection 4.5.1). Remark. If q = 0, then BDf = 0 and Rsgf = 0 and, therefore, (4.6.4) agrees with the Leray-Norguet formula (4.4.11). If (s*, x*) is a Leray section for (D, s, qJ), then by setting = 8* we obtain a Leray-Norguet section (st, ... ,8;, x*) for (D, 8, qJ). Then LSll = 0, RSI( = 0 if IKI ~ 2 andLsl + ... + LSN = LaD, RSI + ... RSN = R aD , that is, in this case (4.6.4) agrees with the Koppelman-Leray formula (4.5.32).
s:
+
To prove Theorem 4.6.2 we need the following lemma. 4.6.3. Lemma. Let the hypotheses of Theorem 4.6.2 be fulfilled. For every collection K = (leJ , ... ,le,) of strictly increasing integer8 1 s: leI < ... le, ~ N, we write W K : = W kl U ... U W tr' Set
<
U WK
W :=
X L1 oK ,
(4.6.5)
K
where
U is over all collections K
= (leI' ... ,k,) oj strictly increasing integer8
K
1 ~ kl
< ... < le, <
N, and define
D(z, C, l) : = qJ·(z,
C) Q)~ "l('*(Z, C, l»)
1\
(O,(8(Z, C»)
for
(z, C, l) € W,
(4.6.6)
where t* = t~f, .....~,i,8) (see (4.4.8) and (4.5.1)). Then
(8% +
de,A)
(J(C)
1\
D(z, C, A)) = 6f(C)
1\
ii(z, C, l) for
(z, C, l) E W •
(4.6.7)
Proof. Repetition of that of Lemma 4.5.4. • Proof of Theorem 4.6.2. By the remark following Theorem 4.6.2, we can assume that 1 ~ q s: n. It is clear that LSKI, RSK 01 and RSlll are continuous in D. By Theorem 4.5.2, BD ai, BDI and 8BD J are continuous in D. By differentiation under the sign of integration we obtain that ORSK! is continuous in D. It rema.ins to prove (4.6.4). In view of the Koppelman formula (4.5.9), the proof will be complete if we show that
I:
IKI~"-f
oRsKJ = BaD! -
I: LSKj +
IKI~,.-g
I:
IKI:a;ft-g-l
RSJ! oj in D.
This follows by Stokes' formula from (4.4.4), (4.6.3) and Lemma. 4.6.3 . • 1) The theorem is valid also without this additional assumption, but then the proof
becomes teohnically more complicated.
182
4. Global integral formulas on Stein manifolds
4.6.4. Corollary. 1/, under tke hypotke8ea 01 Theorem 4.6.2, the sectiona 8r(Z, C), ... ,
s;(z, C) are lwlomorphic in. zED an.d if 1 ~ q :::;; n., then.
1= 8( -1)0 (BD +
~
RsK)f
lKI~A-!l
+ (_1)1+
1
(BD
+
-
~
+
~
81
in. D. (4.6.8)
-
In. particular, 101' every cominu0U8 (0, q)-form Ion D 8uch that of u := (-1)1 (BD
R sJr )
IK1~A-f-l
=
0 in D,
(4.6.9)
RsK)1
IKI~A-f
is a continuous 80lution. of 8u
=
f in D.
Proof. If st(z, C), ... , 8;(Z, C) are holomorphic in z, then, for every collection K = (1;, ... , k,) of strictly increasing integers 1 ~ ~ < ... k, ~ N, the form
<
tI( ") -, ( ~ A 8:(Z, C) ) rp Z,1:t (0•• ",\ ~E.K ~ t( 8.*(Z,.,") , 8 (Z, .,")
is of degree zero in z. Therefore, LSKI = 0 if q
4.7.
(" )
/\(0,8 Z,I:t)
~
1. •
The Weillormula
In this section we construct a Leray-Norguet section (cp. Subsection 4.4.2) for analytic polyhedra in Stein manifolds. This Leray-Norguet section is holomorphic in both variables z and C. Therefore, if such a polyhedron is real non-degenerate, then we obtain a formula for solving the a-equation (cp. Corollary 4.6.4) as well as a version of formula (4.4.13), which is a generalization of the classical Weil formula (mentioned in the Preface to this book and in the Notes following Chapter 2) to Stein manifolds. Remark that in the construction given in the present section, results from the theory of coherent analytic sheaves are used. This is not necessary for the case of qJn (cp. Remark 4.7.5 below). 4.7.1. Definition. Let X be a Stein manifold. An open set D c c: X is called an analytic poly~dron if there are holomorphic functions F 1 , ... , F N in some neighbourhood UJj of D such that
D
=
{z E UJj: IF~(z)1
<1
for
k = 1, ... , N} •
D is called real non-degenerate if, moreover, for every collection of integers 1 k, ~ N and all Z E UJj with IFk.(z)1 = ... = IFi,(z)1 = 1,
< ... <
d IFt1(z)1 /\ ...
1\
d IFil(z)1 =1= 0 .
~
kl
(4.7.1)
Remark. If D is a real non-degenerate analytio polyhedron in a Stein manifold, then the boundary of D is piecewise ()l in the sense defined at the beginning of Section4.3.1 (set Vrc:= Ujjand()rc:= IF.I-I). 4.7.2. Lemma. Let X be a Stein manifold, and. let 8(Z, C), t:p(z, C) be a8 in. Lemma 4.2.4. Suppo8e that X is a relatively compact open 8ub8et of some larger Stein manifoZtl. (However, cpo also the remark following Lemma 4.2.4. ) Then, for every holomorpkic function F on X there ezist8 a kolomorpkic map "'*: X X X ~ P*(X) (cp. Subsection 4.2.3) Buch that h *(z, C) E T:(X) and
rp(z, C) (F(C) - F(z)) = (h*(z, C), 8(Z, C»
Jor aU z, CE X
(for the definition of (., .) see (4.3.2)).
(4.7.2)
183
4.7. The Weil formula
Proof. It follows from conditions (ii)-(iv) in Lemma 4.2.4 tha.t (4.7.2) can be solved locally. That is, we can find an open covering {Wf } of X X X and holomorphic maps h1: WI ~ T*(X) such that hj(z, C) E T:(X) and ep(z, C) (F(C) - F(z))
= (ht(z, e), s(z, e)
(4.7.3)
for all (z, C) E WI' If U C;; X X X is open, then we denote by :K,( U) the set of all holomorphic maps s*: U --7 T*(X) such that 8*(Z, C) E T~(X) and (8*(Z, 8(Z, C) = 0 for all (z, C) E u. f , is an analytic subsheaf of xxX(}B, where B is the pull-back of T*(X) with respect to the map X X X ;I (z, C) ~ z E X. By the Oka Theorem 4.1.2, .Ye, is coherent. By (4.7.3), h1 - ht E .Ye,(WI n We). Therefore, we obtain from Cartan's Theorem B (Theorem 4.1.3 (li)) a collection E :K.( WI) such that hi - ht = t; - tt in W1 () Wi' Setting h* := hl - t! in W f , we complete the proof. •
e)'
t;
4.7.3. A KoppeIman-Leray-Norguet formula. Let X be a Stein manifold, and let Dec X be a really non-degenerate special analytic polyhedron. Let UD, F1, ••• , F Il be as in Definition 4.7.1. Since Dec UD, it follows immediately from the definition of Stein manifolds that, for sufficiently small E 0, the set {z E U D : IFt(z)1 1 e for k = 1, ... , N} is a Stein manifold. Therefore, after shrinking UIi, we can assume that U D is a Stein manifold. Without lOBS of generality we can also assume that X is
< +
>
a relatively compact subset of some larger Stein manifold and we can apply Lemma 4.2.4 (cp. also the remark following Lemma 4.2.4). Let 8(Z, C), cp(z, C) and ~ be as in Lemma 4.2.4. By Lemma 4.7.2, wecanfindholomorphicmapsht, ... , h;: Ujj ~P*(X) such that h:(z, C) E T:(X) and rp(z,
C) (F,,(e) -
F,,(z)) = (ht(z,
e), 8(Z, e)
(4.7.4)
for all z, C E U D and k = I, ... ,N. Set SA: := {z E aD: IF,,(z)1 = I}, k = I, ... , N, and define SK and LtK as in Subsection 4.4.1. Since, by (4.7.4), cp(z,
C) h:(z,~)
_
h't(z,
C)
(h:(z, e), s(z, C» - F 1c(e)- - FJ:(~-)
for
and
zED
CE SJ: ,
(h~, ... ,h~, I) is a Leray-Norguet section for(D, s,ep) (cp. Subsection 4.4.2). Therefore, from Corollaries 4.4.5 and 4.6.4 we obtain the following
Theorem. Let v ~ max (2n", n) be an integer, LsK. = Ls~(rp', ht, ... , h;',8), Rsg = RSg(ql', ht, ... , h;, 8, s) (cp. Subsections 4.4.2 and 4.6.1), and let BD = BD(cp', 8, 8) (cp. Subsections 4.3.2 and 4.5.1). Then (i) For eve1'y continuous function Jon D such that 8f i8 also continuous on 15, we have
f
=
L
LSlIf -
IKI=n
~
RsK.
81 -
BD
81
In particular, for every continuous function f on
f =
~
in
D.
(4.7.5)
IKI:;i;n-1
D that i8 holomorphic in D, we have
LSlIf in D.
(4.7.6)
IKI=n
(ii) Let 1 ~ q ::;: n. Then, for every continuous (0, q)10rm fonD 8uch that continUO'l.t8 on D, we have
f
=
6( -1)11 (BD
+
~
11£1 ;$;ft-I
Rsg)! + (_I)f+ 1 (Bo
+
~
Bsg)
IKI ~ft-f-l
8J
aJ i8 cmo in
D.
(4.7.7)
1M
•. Global integral formulas on Stein manifolds
I'll particular, Jor every contin'UOUB (0, q)-Jorm Jon D Buch that
81 =
0 in D,
+ 1: RSK)J IKI =-ft-f is a oontinu0'U8 Bolution 0/ au' = J in D.
(4.7.8)
u:=(-!)f(B D
4.7.4. Theorem (Weil formula). We U8e the notations Irom Subsection 4.7.3. Then, Jor every continuoU8 Junction J on jj that iB kolomorphic in D and fOT every integer .., ~ max (2n~ - n, 0), we have (2ni)tl J(z)
Cn
= 1: I£I-ft
ff(C) rp"(z, C) det (hts(z, ~~ h:,,(z~ w,(s(z, C)L (F k1 (C) - F k1 (Z)) ••• (Fk,,(C) - Fk,.(z))
,
~ESK
zED,
(4.7.9)
where the summation i8 over all collection8 K = (~, ... ,kft) of strictly increasing integers! ~ kl kft ~ N, and the Jorm det (h~, ... , hk~) W,(8) is defined analogoU8ly as in (4.3.4).
< ... <
Proof. Consider a collection K = (kl' ... ,kft) of strictly increasing integers 1 ~ kt kft ::;: N. Since the sections ht(z, C) are holomorphic in C, we obtain (cp. (4.3.4), (1.7.5), Definition 1.7.6 and Proposition 1.7.5) that on Sg X Ltx
< ... <
where the Bummation is over all permutations a of {I, ... , n} and sgn (a) is the signature of a. By (4.7.4), this implies that on SK X L1K
Hence, by (4.7.6) and (4.4.7), (2.ni)tI fez) =
1.:
J[ J
IKI-n teSK
1: sgn (a) Aka(l) dAta(2)
AE~K fI
Af(C) ql'(z, C) det (h~(z, C), ... ,ht,.(z,
A ... A
dAka(ft)]
en "wc(s{z, C)) , zED.
(FII.(C) - Fi,(z)) ... (Fi,,(e) -
Fk,,(z))
Therefore, it remains to prove that
J 1: agn (a) Aka(l) cU
AEA~
t D(2) " ... AdAlia( )
tI
fI
= 1 •
(4.7.10)
Sinoe dl,. " ... A dAN defines the orientation of L1 (cp. Subsection 4.4.1), it follows from (4.4.3) that the orientation of L10K is defined by the form dAta
1\ ...
"dAk,. = sgn (a) dAta(l)
A ...
"dAIla(tI)'
4.8. The support functions
(I)
185
a.nd fP
Since on L1 OK -dAo 1\ dAto(2) 1\ ••• 1\
dAka(n)
= dlka(l) 1\ dl ko(2)
1\ ••• 1\ dAka(n) ,
together with (4.4.3) this implies that the orientation of L1K is defined by sgn ((1) dl k a(2)
/\ ••• /\
dAtCJ(n) •
Therefore, since Aka(l)
=
1 -
Ata(2) -
••• -
Ato(n)
on LI K
,
(4.7.10) follows from the relation
j
.
1 (1 - t2 -
••• -
tn ) dta ••• dtn
= -
n!
,
, •••••• t,,~O
'.+ ... +tn~l
which can be easily verified. • 4.7.5. Remark. For the case X = q;tI., we can set 8(Z, C) = C - Z, rp == 1, " = 0, and ... , k'; are q;"·valued holomorphic functions. Then the Weil formula (4.7.9) can be written (2ni)~(z) = ~ Jf(C) det (kt,(z,C)' ... , ht,.{%,C)) del /\ ••• /\ ~C". (4.7.11)
h~,
(FI;.(C) - F1:.(z)) .•• (F1:,.(C) - FIt,.{z)
IK\=n CeSg
Remark that in this case the VV" eil formula can be proved without use of results from the theory of coherent analytic sheaves, beca.use the vector functions Ar, ... , A; then can be obtained from Theorem 2.5.4.
4.8.
The support functions
fP
and 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
<- , CElf' 3 dist (lV, (8) > 28 . max l(1o(C)1
(4.8.7) (4.8.8)
Since 8 ~ Al U ••• U A L , we can find functions ')'1 E 0o(A 1) such that ')'1 + ... + ')'L = 1 on O. If 1p is a Ol·function in some neighbourhood of a point Z E 8, then we define 111p(z)llt := 11p(z)1
+ i£ "1(Z) [~ 188V'(Z) 1+ l: la-8 (a ) I]· -I k-l xJI: k,l=1 xJi XiI 2tp Z
If 11' is a CI·function in some neighbourhood of a set Y C
0,
(4.8.9)
then we write (4.8.10)
11V'1!2,Y := sup 111p(z)III' ,eY
We denote by ID1 2 (0) the normed space of all C2·functions in a neighbourhood of 0 endowed with the norm 11·112.8' Choose ~ > 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
<
1 (z, C)
:= {Ss[X(Z,
C) In FQ(z, C)] if 8 ~ dist 0
Q
and
J(z, C) := {8 [X(Z, C) In (Fiz, C) 1
2()(C»)J
0
II
(z,
C) ~ 28 .
otherwlBe ,
<.
if 8 dist (z, C) otherwise.
~ 28,
We denote by Z
aT
uQ(z, C) : =
uQ(z, C) := a(e) (z,
C)
(TiQh e)) (z) , (Thh C») (z) ,
C) : = exp [ -u,,(z, C)] ,
;;(e) (z,
C)
:= exp [-uQ(z,
C)
if
exp [X(z, C) In Fiz, C) - u,(z, C)]
if
:= {FQ(Z,
C) b(e)
b(l!) (z,
(z,
a-( e) (z, '") " ,_ ,- {(FQ(Z, C) - 2g(C)) b(l!)
(z, C)
exp [x(z, C) In (F,,(z, C) -
2e(C») -
_
C)] ,
dist (z, C) ~ dist (z, C) ~
u(z, C)]
if if
8 • 8 ,
dist (z, C) ~ E. dist (z, C) ~ 8 .
It is easy to show that the maps a, a, b, b fulfil conditions (i) -(iii) . •
4.9.
Formulas lor real non-degenerate strictly pseudoconvex polyhedra
In this section we introduce the concept of a strictly pseudoconvex polyhedron in & Stein manifold. Special cases of such polyhedra. are analytic polyhedra (cp. Section 4.7) as well as intersections of finitely many strictly pseudoconvex open sets.
4.9. Real non-degenerate strictly pseudoconvex polyhedra.
189
By means of the support function q, from the preceding section, for such polyhedra, wc construct a Leray-Norguet section, which is holomorphic in z. By Corollary 4.6.4, this gives a formula for solving the a-equation in real non-degenerate strictly pseudoconvex polyhedra. 4.9.1. Definition. Let X be a Stein manifold of complex dimension n. An open set Dec X is called a st·rictly pseudoconvex polyhedron if there are a neighbourhood UJ) of D, finitely many Stein manifolds Xl' ... , X N of complex dimension ~ 11., holomor. phic maps F,,: Un ~ X k , k = 1, ... ,N, as well as strictly pseudoconvex open sets Die c c XJ;, k = 1, ... , b"', such that
D If
(h, ... ,
01 ,
••• ,
=
eN
F 1 1(D1) n ... n F"N 1(DN) . are strictly plurisubharmonic Ol.functions in some neighbourhoods ... , [D N , respectively, such that
ON of oDI ,
Dil n 0" = {z E Ok: e.t(Z)
< O} ,
k
= 1, ... ,
N
J
then aD c: F 1 l(81 ) U ... U F'Nl(O N) and a point z E Fl1(81) U ... U FBI (8 N) belongs to D if and only if, for every 1 ::;; k ~ N with Z E F;l(Ot}, eJ:(F~(z)) O. D is called real non-degenerate if these functions Fie and (!k can be chosen so that the following condition is fulfilled: For every collection of indices 1 ~ kl k, ~ N, we have
<
< ... <
d(ek1
o
F TcJ ) (z)
1\ ... 1\
for all z E aD with ek1(FkJ(z))
d(ekl
0
= ... =
F kl ) (z) 9= 0 ek,(Fk,(z)) = O.
Remark. The boundary of a real non-degenerate strictly pseudoconvex polyhedron is piecewise 0 1 in the sense defined at the beginning of Subsection 4.3.1. 4.9.2. Proposition. Let X be a Stein manifold. Then every strictly pBeudoconvex polyhedron D e c X is a Stein manifold. Proof. Let X k , F k , Dk be as in Definition 4.9.1. By Proposition 2.13.8, every DJ: is a Stein manifold. In the same way as the equivalence of conditions (iii) and (v) in Theorem 1.3.7 was proved, the following proposition can be proved: An open subset W of a Stein mani. fold is a Stein manifolrl if and only if, for every infinite set Y c: W which is discrete in W, there exists f E O( W) which is unbounded on Y. By means of this proposition, we can repeat the proof of Theorem 1.3.11 (ti) and obtain that the sets Fl1(DI)' ... , Fi 1 (DN) are Stein manifolds. This implies that D is a Stein manifold . • Remark. Every analytic polyhedron in a Stein manifold is a strictly pseudoconvex polyhedron. Then, in Definition 4.9.1, we have to set Xl = ... = X N = qJ1 and el(Z) = ... = (!N(Z) = Izit - 1. Further, if D 1 , ... , DN C C X are strictly pseudoconvex open sets in a Stein manifold X, then DI n ... n D N is a. strictly pseudoconvex polyhedron. Then in Definition 4.9.1 we have to set Xl = ... = X N = X and l'I(Z) = ... = FN(z) = z, Z E X. In the construction of a Leray-Norguet section for strictly pseudoconvex poly. hedra, we use the following Lemma, which shows that in Lemma 4.7.2, after shrinking X, the section h* can be given by a continuous linear operator (cp. Theorem 2.5.4, which is used in the proof of Theorem 2.5.5 for the corresponding construction in the case of strictly pscudoconvex open sets in eft). 4.9.3. Lemma. Let X be a Stein manifold, and let 8(Z, C), fP(z, C) be Q,JJ in Lemrna 4.2.4. (Suppose that X is a relatively compact open 8ublJet of 80me larger Stein manifold -
190
4. Global integral formulas on Stein manifolds
cpo a.lso the remark following Lemma 4.2.4.) Let Y c c X be an open 8et. We denote by r*(X X X) [r*( Y X Y)] the Fricket 8pace of aLL holomorphio maps
[h*: Y X Y -* T*(X)]
h*: X X X -* P*(X)
(cp. Subsection 4.2.3) suoh that h*(z, C) E P:(X) for all z, CEX [z, CE Y] endowed with the topology of uniform oonvergenoe on compact 8ets. Then there exists a oontinuo'U8 linear map H: O(X) ~ r*( Y X Y) (here the topology of O(X) is aZ80 that of uniform convergence on compad Bet8), such that, for every F E O(X),
cp(z, C) (F(C) - F(z»)
=
«(HF) (z, C), s(z,
e»
for all
z, CE Y .
Proof. Let (F, be the subsheaf of IXXOI generated by s (cp. Example 4.1.7). Then by Theorem 4.1.4, (F,(X X X) and (F,( Y X Y) are Frechet spaces endowed with the topology of uniform convergence on compact sets. Consider the commutative diagram '1 ~ r*(Y X Y) r*(x X X)
O(X)
-
191 8
) 91(', B)
(,,8>
(F,(X X X)
'.
I
(F.( Y X Y)
where rl , r2 are the restriction maps and'l?(F) (z, C) := cp(z, C) (F(e) - F(z»), z, CEX. By Lemma 4.7.2, 'l?(O(X») c cp(r*(X X X), s). Therefore, by Propositions 1 and 2, Appendix 2, the proof will be complete if we show that there are a Hilbert space Sj and continuous linear maps A': r*(X X X) -* Sj, A": Sj -* r*( Y X Y) such that r 1 = A"A'. To do this we introduce a Hermitean metric in the fibres of T*(X) as follows: Choose a locally finite open covering {WI} of X and a Goo-partition of unity {Xt} subordinate to {W 1} such that, for every j, there is a GOO-bundle isomorphism G1 : P*( W t ) -* WI X en (T*(W t ) is the restriction of T*(X) to W t ). Let g1: T*(W f ) -* (f)n be the corresponding Goo-maps for whichGf(a) = (z, gl(a)), a E T:(W I L Z E WI' For all z EX and a, b E Tt(X), we define
(a, b) : =
:E Xl(Z)
(gl(a), g1(b))c- ,
j
+
where (v, w)t',,:= V 1W1 + ... vnw" for v, wEen. By means of this Hermitean metric, we define the required Hilbert space Sj to be the space of all h* E r*( Y X Y) such that
J
(Z.C)EY X Y
(h*(z, e), h*(z,
cn d0'4n <
00 ,
where d0'4" is a volume form on X X X, that is a rea14n-form which defines the orientation of X X X. Then the linear maps A' and A" can be chosen to be the natural maps. • 4.9.4. Coronary. Let the hypotheses of Lemma 4.9.3 be fulfilled, and let V c Y be an open set. We denote by Gl(X X V) tke Frechet space of all (J1-functions on X X V endowed with the topology of uniform convergence on compact Bet8. Let ml( X X V) be the sub8pace of all V'(Z, e) E Ol(X X V) that are holomorphic in z E X. Further, we denote by 9l~( Y X V) the Frechet space (endowed with the same topology) of all OI-maps h*: Y X V ~ T*(X) Buch that h*(z, C) E T:(X) for all (z, e) E Y X V and BUok that, for every fixed CE V, h*(z, C) is holomorphio in z E Y. Then there exi8ts a continuous linear map jj: 9l1 (X X V) ~ 9l~( Y X V) Buoh that, for every 'IjJ E 9l1 (X X V) with
191
4.9. Real non-degenerate striotly pseudooonvex polyhedra
1p(Z, z) = 0 for all Z E V, we hafJe q;(z,
e) 1p(Z, e) =
for all Z E Y,
eE V •
Proof. Set (ii1p) (z, C) := -(H("I'(.' e))) (z, C), where H is the map from Lemma 4.9.3 . • 4.9.5. A Koppelman-Leray-Norguet formula. Let X be a Stein manifold, and let Dec X be a real non-degenerate strictly pseudo convex polyhedron. Let U jj, Xt, F", D", 8re, (!Ic be as in Definition 4.9.1. By Proposition 4.9.2, for sufficiently small e 0, the set D. := Fll(Dt) n ..• n Fi 1(Div}, where DZ := DII U {z EO,,: e,,(z) < e}, is a Stein manifold. Therefore, after shrinking U'D, we can assume that U'D is a Stein manifold. Moreover, without loss of generality, we can assume that X is a relatively compact open subset of some larger Stein manifold and apply Lemma 4.2.4 (cp. also the remark following Lemma 4.2.4). Let 8(Z, e), cp(z, be as in Lemma 4.2.4. By Theorem 4.8.3 and Lemma 4.8.2, after shrinking 0", we can find numbers e, a > 0 and Ol-functions f.1>1:(Z, e) defined for z E Die U 81: and eE 8" such that the following conditions are fulfilled: (/>,,(z, C) is holomorphic in zED" U 8" . (i)
>
e)' "
(ii) (iii)
C) =F 0 for zED" u 8", CEO" with dist (z, e) I(/>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) .
=
Set St := {z section 4.4.1. Since, by (4.9.4),
cp(z, C) h:(z, C) _ .ht(z, C) for zED and CE Sk
= RSg(cptl, ht, ... , h~, 8, s)
IEI;:;;n
IKI:;;n-l
(ii) Let 1 ;;;; q ~ n. Then Jor every continuous (0, q)-form f on continuous 011, D, 'We have
f =
8( -1)1 (BD +
~
RSIC)f
lEI ~n-g
+ (_1)9+ 1 (BD +
D 8uch that
~
Rsg) -8f
8f is also in D.
lEI ~,,-g-l
(4.9.6)
192
4. Global integral formulas on Stein manifolds
In particular,JO'I' every contin'U0'U8 (0, q)-JO'I"mJ on jj 8uch that 8J = 0 in D, u := (-l}f (BD
+
i8 a con'inuou8 80lution oj
~ Rsx)f IKI ='A-fl
(4.9.7)
au = J in D.
Remark. It follows from Theorem 4.8.3 and Corollary 4.9.4 that the formulas (4.9.5)-(4.9.7) depend continuously on D in the following sense: If the functions and F t from Definition 4.9.1 converge uniformly, together with the first and secondorder derivatives, then, for every fixed Z ED (or Z in some fixed compact subset of D), the kernels of the corresponding integrals in (4.9.5)-(4.9.7) converge uniformly.
et
4.10. Formulas for strictly pseudoconvex open sets with non-smooth boundary In this section we generalize the Koppelman-Leray formula constructed in Section3.1 (Theorem 3.1.3) to strictly pseudo convex open sets with not necessarily smooth boundary in Stein manifolds.
4.10.1. Notation and preliminaries. Let X be a Stein manifold, let Dec X be a strictly pseudoconvex open set, and let e be a strictly plurisubharmonic Oll-function in some neighbourhood 0 of aD such that D n (J = {z E (J: e(z) < O}. We set N(e) := {z E if: (>(z) = O} and suppose that N(e) c c (J. (We do Dot assume that d(>(z) =t= 0 for Z E aD.) Since Dec X, without loss of generality, we can assume that X is a relatively compact open subset of some larger Stein manifold and apply Lemma. 4.2.4 (cp. also the remark following Lemma 4.2.4). Let 8(Z, '), cp(z, C), " be as in Lemma. 4.2.4, and let 8(Z, C} = 0'8(Z, C) be as in Subsection 4.3.1. By Theorem 4.8.3 and Lemma 4.8.2, after shrinking 0, we can find numbers e, a > 0 as well as OI-functions cJ>(z, C} and ~(z, C) defined for zED u 0, CEO such that the following conditions are fulfilled: (i) cJ>(z, C) and tP(z, C) are holomorphic in zED u O. (ii) <1>(z, C} =F 0 and (1)(z, C) =F 0 for zED u 8, CE (j with dist (z, C) ~ e; (4.10.1) 1<1>(z,C)I ~ a (q(C) - (>(z) + [dist (Z,C)]I) (4.10.2) for ZED u 0, CE 8 with dist (z, C) ~ e;
(iii)
li(z, ')1 ~ "'( -e(C} - e(z) + [dist (z, C)]I) for zED u 8, CE 8 with dist (z, C) ~ e; (z, z) = 0 for all Z EO. ~(z, C) = <1>(z, C) for CE N((», zED u 0 .
(4.10.3)
(4.10.4) (4.10.5)
By Corollary 4.9.4, after shrinking 0, we can find a T*(X)-valued Ol-map 8*(Z, C) defined for zED u 6, CE 8 such that the following conditions are fulfilled: (1) 8*(Z, C} E T:(X) for zED u 8, CEO. (2) 8*(Z, C) is holomorphic in zED u 6.
(3) qJ(z, C) (z, C)
=
(8*(Z, e), 8(Z,
C»
for
ZED u 0, CE (j
•
(4.10.6)
4.10. Strictly pseudoconvex sets with non· smooth boundary
193
4.10.2. Remark. If zED and CE 8D, then, by (4.10.6), cp(z, C') 8*(Z, C') 8*(Z, C') = , (8*(Z, C'), 8(Z, C» 4)(z, C) where, by (4.10.1) and (4.10.2), 4)(z, e> =F O. Therefore, (8*, 1) is a Leray section for (D, 8, fP). In the case of smooth 8D this gives the corresponding formulas stated in Corollary 4.5.5 (with l' ~ max (2nx, n), LaD = LaD(cp·, 8*, 8), RaD = RaD(fP", 8*, 8, 8) and BD = BD(cp", 8, 8»). 4.10.3. The operators LD and RD. Choose X E O(f(O) suoh that X = 1 in some neighbourhood of N(e). It follows from (4.10.1) and (4.10.3) that, for every zED, there is a neighbourhood V.~ 0 of N«(}) such that ($(z, 0 for E (D n 0) u V" Since supp X ceO, this implies that, for every fixed zED, the map X(C) a*(z, C)/~(z, e) is (J1 for CE D u V;,. Consequently, for every fixed zED, the differential form
e)
*
e
e»
-co,..., (X(C) 8*(Z, C) ) 1\ COC(8(Z, (4.10.7) ~(z, C) is continuouB for CE 15, where this form is defined as follows: Let {( UI , hI)} be the
V,
holomorphic atlas from Subsection 4.3.1, let = (V;l, .•. , v;,,) be the expression of x8*1$ with respect to (U t , hI), and let 8, = (8;1, •.• ,8;,,) be the expression of 8 with respect to (U t , h1). Then we define
-co,,..., (X(C) 8*(Z, C») ~(z, C)
1\
W,(8(Z,
C))
:=
• -8,V;i(Z, C') A
" dc8;k(Z, e) A
1\
i-I
i-1
z E U I n D, CED.
for
(4.10.8)
By (4.3.1), this definition is independent of j. Therefore, for every integer v every measurable bounded function f on D, we can define LDj(z) := -n! .(2JfI)ft
Jf ,. " ,. -co, --,...,---e») (.,) fP (z,
(X{C) 8*(Z, ~(z, C)
~)
1\
,. ) co, (8(Z,~),
'ED
~
0 and
zED. (4.10.9)
Since 8*(Z, C) and ~(z, C') are holomorphic in z, then LDfis a holomorphic function in D. Set
t*(z, c, A) : = (1 _ ).) X(CL8*(Z, C) 4J{z,
e)
+A
8(Z,
e)
(4.10.10)
118(Z, Cm! .
Then, for 0 ~ A :::;; 1, zED and Cin some neighbourhood of D " z, the map cp"(z, C) X t*(z, C, A) has continuous derivatives of order 1 in C and of order ~ 2 in Z (Lemma 4.2.4 (v)). Therefore, the differential form cp""(z, C) OOz.,.A(t*(Z,
e, A»
1\
CO,(8(Z,
C»
(4.10.11)
is continuous for zED, CED" z, 0 ~ A ~ 1 (and (Jl with respect to z), where this form is defined as follows: If {(U I , h,)} is the holomorphic atlas from Subseotion 4.3.1, 'it = (tjt ... , j!) is the expression of ;* and = (8J11 ••• ,8J") is the expression of 8 with respect to (U h hI), then, for Z E U , n i5, CE j j " Z, 0 ~ A ~ 1, we define
t
8,
_
wz.c.J.(t*(z, C, A») "CO,(8(Z,
,,_
_
e» := k-1 1\ (az•e + d A) tJt(z, e, A)
fa
A
A dc81~(Z, C) • i-1
(4.10.12) 13
Henkio/I.. eiterer
194
4. Global integral formulas on Stein manifolds
By (4.3.1) this definition is independent ofj. If zED, then the monomials in the form (4.10.11) which are of degree 1 in ,l have 8. singularity of order :::;; 2n - 1 at = z. Therefore, for every integer v ~ n~ and every bounded differential form Jon D, we can define
e
RD/(z) : =
(~~)'
(/(!;)
(C.I)E
~ x [0. 1)
A
rp"(z,!;) ro..c.l(i*(z, !;, A))
A
"'C(B(Z,!;)) , zED. (4.10.13)
Then it is easy to see that RDJ is continuous in D and RDJ is a (0, q - I)-form if JiB a (0, q) form (RDJ = 0 if q = 0). 4.10.4. Theorem (Koppelman-Leray formula). We use the notation8 trom SUb8ecti0f&8 4.10.1 and 4.10.3, and we 8uppose that in Definitions (4.10.9) and (4.10.13) an integer 11 ~ max (2n~, n) i8 used (the 8ame 11 Jor both deJiniti on8). Then (i) For every continuous bounded Junction J on D 8uch that -8J is alBo continuou8 and b&u7lded on D, we have (4.10.14) (il) Lee f be a coneinuouB bounded (0, q)-form on D Buch that of i8 alBo continuous and bOttnded on D, 1 ~ q :::;; n. Then
J
=
-
8R DJ
+ RD -8f
in
D.
(4.10.15)
In particular, if 8f = 0 in D, then u := RDf i8 a continuous bolution of
(4.10.16)
au = f
in D.
Proof. Repetition of that of Theorem 3.1.3• • 4.10.5. The Hormander diameter. We use the notations of Subsection 4.10.1. For every E E 8D and d 0 we define
>
Ht({) := {z
E
jj: I~(z, E)I
< {)I }.
(4.10.17)
HT({) is called the Hormander ball of radius {) centered at E. If W cD, then the number diamR W := inf {d
> 0:
there exists E E 8D such that W C HT({)}
will be called the H5rmander diameter of W. Remark. For X = on this definition is equivalent to that of Subsection 3.2.1, that is, 0 and 0 < 00 such that, for all ~ E aD and {) 0, there are constants c Ht(c{)) c H,(6) ~ Ot(06) , where He({)) is as in Subsection 3.2.1. This follows from (3.2.2) and the fact that, for some cl > 0, I~(z, e)l ~ c< Ie - zp' for all zED, E 8D.
>
>
e
4.10.8. Theorem. We U8e the notation8 of Sub8ection8 4.10.1 and 4.10.3. Let n~ and 1 ~ q ~ 11, be integers. We denote by L(8,Il)(D) the Banach space of bounded (0, q) forms f on D, with the norm 11·llo.D := BupzEDllf(z)11 (cp. Subsection 2.11.2). (i) For every bounded (0, q)-form f on D and each point ZED, the integral 11
>
nt RD/(z) := (2m)"
f
f(e) /\ lP"(z, e)
-
Wz.C.A(t*(Z,
e, A)) /\ mc(s(z, e))
(C.A)ED x [0,1]
converges, and the so def1:ned (0, q - 1)-Jo'rm R Dfi8 continuous on ii.
(4.10.18)
196
4.11. Bounded extension of holomorphio funotions
(ii) There is a COO8tant 0
<
00
suck that, for every! E L(8,,)(D),
IIRDfllo,D ~ 0 dia.mu (supp!) !I!llo,D, and, moreover, iJ ~ E aD,"
(4.10.19)
> 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<
00, the solution from b)
a sup If(Z)1 • ZeD,.
d) Prove that the solution from lOb) does not admit uniform estimates if n > 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.
<<< <
9. Open problem (BlIIDFORD/FoRNA1II8S [I]). Let X be 8 Stein manifold, and let Dec X be a domain of holomorphy with smooth real-analytic boundary. a) Let f be an arbitrary oontinuous function on if that is holomorphic in D. Can f be approximated uniformly on 15 by holomorphic functions in a neighbourhood of jj! The case D CC Os is considered in FORNAESSJNAGEL [1] and FOBNAESS/0VRELID [I].
EXfll"ciBf'R,
I'f'mal'kR
201
a.nd pl'oblAms
b) Do therf.'l Axist unum.'1ll estimates for the
6-equ~tion
in D't
10 (BISHOP [1]). Let X be 8. connected Stein manifold and pee X an open Bet of the form P = {z E X:IJj{Z)1 J; j = 1, ... , n}, whereA , ... ,i", are holomorphic functions on X and n = dime X. (Such sets arc called lipeeial analytic polyhedra.) SetJ :--= (/1' ... ,J",). Let XI be the set of points Z E X such tha.t J is biholomorphic in a neighbourhood of z. Observe that by Bard's theorem the Lebesgue measure of f(X" XJ) vanishes. a) Prove that thore is an integor k ~ 1 (the multiplioity of J) such that. for each W E f(X" XI) tho sot J-1 (w) consists of k points. b) For each continuous function rp on X and each W E f(X " XI)' W~ define P(-IL') : = ~(Zl) + ... ~(zk), where {Zl, ... , z1t} = i- l (w). Prove that 'qJ admits a continuous exteIlBion to f(X), which will be dt'noted also by i;. If tp is holomorphic on X, thon " is holomorphic on J(X). c) Let S := {z E X=lil(z)1 = ... = lin(z)1 = I}. Denote by dll " ... " din the measure on S which is defined by the diffel'untial form dfl " ... 1\ di", in the smooth points of S and by zero in the other points. Prove that this measure i~ finite. d) Prove that for each holomorphic function tp on X and for each Z E P
<
+
fj;{j(z))
=
_1_ (2~i}1~
J(it -
s
din . Jl(Z}) ... (fn - J",(z))
dfl/\ ... A
and S ~ Xl this formula was obtainod by MARTINELLI [3] (seo also AJSENBERG/JU~AKoV [1]). e) Let, 'P be a. holomorphic function on X. ~'or Z, v E XI' wo put iP(z, to.') =: (tp(Z2) - ",(v») ... (",(zL) - 'P(v»). where {z, Z2, ••• , Zk} = i- 1 (f(z»). Prove that 'P admits a. holomorphic oxtension toJ(X) X f(X). P.l"OVO that fOl' each holomorphic function q> 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
\x - W'
Ix -
~0
I' - 8\ lIn It - 811 •
Proof
;; J
I'"
_11'-- dq. + 0 II - 81 1 -
0
J
rut. ;;;; 0' It - 81
(1)
Is-tl ~t'-'1/2
Is-'I :oi;J,-.1/2
and
JI
Is-II :iil,-.I/I
Xl-Btl
Xj-~ -- daft ~ 0 It - 8\. \x - 'I" Ix - 8\" -
(2)
Appendix 1. Estimation of some integrals
Further, Xl - tl Xl S1 ---- ---
IIx-tin
I
Ix-sl" ft-l
i
~
203
i (Zt - t1) (Ix - al -Ix - tl) !: Ix - all' Ix - tl ft - . - l - (tl - ~) Ix - WI
I
---- -- -___~.--=o______ .__ .-... -....
n-ll x - al"lx - tI'~-I' It - sl :::; \Ix - al -Ix - til ~ ----. - 1 I - - n + ,,-0 Ix - al Ix - tl Ix - al
11
~ n It _ -
sl (max { Itl~
Consequently, since
lsi
~
J
Is-'I.III:-.1 ii:U -.1/2, Ixl < R
~ 0 It - al
,I} +
1
Ix - sl" Ix - tl"
J
1 . ).
Ix - al"
R,
I~=t~-~ -~~Ida. dO' IYI: ~ 0' It - 81 Jdr -;:- ~ 0" It 28
ailln It - all· •
\&-.\12
,,-.1/2;:5;1111 ;:5;28
Proposition 2. Let n ~ 1 and let D be a bounded domain in B" with piecewiae amooth boundary (for the definition of a piecewise smooth boundary see Seotion 1.8). Then
there is a comtant 0 < 00 with the following properly: If f i8 a (J1-function in D B'UCh tkat for some K < 00 and 0 < IX < 1, Ildf(x)1I ~ K[dist (x, aD)]-CI Jor x € D,l) then
If(x) - f(y) I s OK Ix - yll-CI for x, Y ED. Proof. Since aD is piecewise smooth, we can find a number e > 0 so small that, for all x, y E D with dist (x, aD) ~ e and dist (y, aD) :::; e, there exists a Ol-funotion r~: [0, 31x -111] -+D such that
Ix - y\) = y , dist (r_(l), aD) ~ A for A E [0, Ix - yl] ,
r~(O)
=
x ,
,,~(3
dist (rSJ((l), aD) ~ Ix. -!II
for A E [Ix - yl, 2 Ix - yl] ,
dist (r~(l), aD) ~ 3 Ix - yl -1 for A E [21x -
l~l''''(A) \ ~ a
yl, 3 Ix -
yl] ,
for A E [0,31'" -1111.
Clearly, we only have to prove the proposition for points x, 11 ED with dist (x, aD) and dist (y, aD) ~ e. For such points we obtain
J:/(y..
Slz-JlI
1/("') -
1(11)1 = \
(A») dA\
o Is-rl
J
21a1-1I1
+J
a'x
~
o )s-rl 30'K Ix - yll-IX. •
1) IId/(x)Il:=
" (l'::
;-1
J
I
IIdf(l'..()'»)II\ ~ l'..().) d.l
0
:0;;
j.-o d)'
~a
8ls-rl
18/(x)/8x111)1/2.
Ix -111-
Blz-Ifl
0
dA + J
(31'" -111 - ).)-.
21:1-,,1
dist (xt aD) := inf {Ix - Y!: Y E 8D}.
dA
~e
204
Appendix 1. Estimation of some integrals 11. ~
Proposition 3. Let 0 that for all e
>
I and 0
< R < 00.
Then there exists a constant 0
< 00 such
(i)
( ii)
Proof. (i)
J- +- - - - - s J J + R
dtYn
(e
Ixl!) Ixl n -
Ixl:5 n
(ii). :For
have
(e
'1'2) r n - 1
sJ
ux
Ixl;;;;R
For 11,
~
'1'2
=
soJ-?X+
+ X)2
(e
.
2 O"N~.
R
2x dx
+ Ixl + IxI2)2Ixl-l -
(e
-dr - < 0 ,/,r:: ye
e
0
R
J
=
Ixl-r
0
= 1 we
11.
dr
1 -
J+ R
- -dtYn_l ---<0
e2
-
0
dx S O'ln e S x2 -
0
2 we obtain
J ,-1 J ~fn--2 ~ R
J
dtYn
+ IXII + Ix12)2
(e
1:l:I;;;;R
S 0 -
(e
Ix'i ~R
J(e
dtY1
+ I~I- +Ix'12) Ix'ln=-2 dX 1
0
[dT J + R
dtYn-l
--- < C
dtY,,_2
+ Ix'12) Ix'l n - 2 = .
Ix'i ~ R
0
(e
'1'2) r n - 2
1:1:'1 =T
R
-$:
o'J~ ~ CliNe. _ e+r
o Proposition 4. Let n, d :;;::::: 1 be integers. Then there exists a constant 0 = 0(11., d) 00 such that the following statement is true: Let PI' ... , p, (1 ~ 8 -$: 11.) be real-valued polynomials of degree ~ d defined on /R,", let e, /-" 11,~,,, > 0, and let D c tB,n. Then (for the definition of 11·11 see Subsection 1.6.3)
<
i J
+
IIdpl(x) A ... A dp,(x) II dXl A ... A dx" Ip1(x)\
:&eD
=
[e t€T(D)
+ ,,(Ixl + ~ Ipl(X)I)fJ]" [Ixl + ~ Ipl(x)IY ;=1
i-I
;=1
dttA ... Adtn _ _
+;=1 i:. Itll +" Itl
ll
]"
(3)
IW' '
where T(D) := {(lpl(X)I, ... , Ip,(x)l) : xED} T(D) :=
U
l~il< ...
A ... A
8
= 11.
{(lpl(X)I, ... , Ip,(x)l, IXi I, ... ,IXi J
Proof. From the definition of IIdpl IIdpl(x)
if
dp,(x)1I -$: 0
A ... A
and
n-'
I): xED}
dp.1I it follows that
max l:ijl< ... <j.~n
Idet [(8p,(x)/8xj,Jtl_lJI .
.A.ppendk 1. Eetbnatlon of BODle iaUepate
< ... <
On the other hand, if 1 s: 31 j. ~ • and 1 {iI' ..• ,j,} u {it, i._.} = {I, ... ,ft}, then 0"
-
i 1-.
:a it < '" < i._. =' • noltitMt
,
\dP1«(l;) A ••• A dp.(:I:) This implies that
<Wi,
A
d2:,-_.1
A ... A
dp.(:I:)11
IldtJt(~) A ••• A
~0
• :.: ; .• :~, t •
,-I·J.
=
Idet [(~(:I)/e.JJ'. '-1] d:rt. A ... A ck.1
0
~ A ••• Adz.
max Id.f'1(~) ... < .._.~.
1\ ... 1\ d.f'.(~)
" cb'l
A '0' 1\
1~"<
d.'a_,I".
Consequently, the integral on the'left.hand side of (8) can be estimated by the mul\mum over all oollections 1 ~" "'-~ of " . IdP1(z) 1\ ••• "dp.(z) A~, A "-~_.I
=' .,.
< <
"f
Ot'
Ot.
i l.PI(~)1 + ,,(J-1D:if'f
D
0
1-1
i-I
i-1
~ tI,
pull-back of the integrand of the integral on the riBht.hand side of (3) with reepeot to the map "~'''''._.: D -+ P(D) defined by "ilo •••• i._.(Z) := (1P1(~)I, ... , Ip,(z)\, I~I, •• I~~I) • It is a corollary of Bezout's theorem that for every 1/ E • there are no more than",. points :r: E r 8uch that (1l1(:r:), '0' , .P.(~), ~fa' '0' ,:t':,-_.) = 1/. Consequently, for every 11 E 8ft there are no more than 2"4' points ~ E r such that «'u.... '-_.(.) = til .Tbia completes the proof of the proposition. • . 0
Proposition I. Let n
<5
> 0 aM (i)
J
00
~
;
1. Tken there e:NI8 a r..onsIa1&l 0
< < 00 the lollowi"f/ utimalu Aoltl:
< 00 811M ,., lor
(I
lor ~ =
du.. S; lM·
IZ/ft-IC -
J
1, 2
J
IIlEBA,leI<'
C') 11
du. + Ixl') tzl
Uztl
do'.. ~ 06" Jor + ta:al + 1:q·)·I~I·-·-· -
Ill."'. IGII <'.
J SE"'.
~ 06· J.or ,,== 1, 2 ,
(I~I
.
ft -
1- .
II<~<.+"
(iii)
I~
J
,
, dO',. =Jdr JdO'ft-l ~ OJ"'-l dr:iit 0'6". Ixl"-IIC r"-N -
1_1<'
I~ -r
0
0
(ii)
1.
(1~1
s;'J&+I'w..J'dr
dan
+ \xll) Ixl,,-I-. -
I <'t s
= 1 2.
.
.
a
Proof. (i)
H
•
ta+"
,
06"-IJ ~J II
0
+ ,.. ='
cit
IZtI
0
_+"
1
(1-.1
I-r
Ol."-I! )iiiJ
dtt._1
+ tI) ....-1-.
cIA;.:it O"ll- •
•
aU
206
Appendix 1. Estima.tion of Borne integra.ls
(iii)
dO'"
(1~1+·lx21
+ IxI Z)2Ixl"-2-'=-;
Proposition 6. Let n ~ 2. Then there exists a constant C
E
> 0 the following estimates hold: (i)
f
I~I-.
(ii)
(iii)
Proof. (i)
(ii)
(iii)
dO',,_l
(Ixll
~ 08"-4
+ sa)1 -
,
<
00
8uck that
lor every
207
Appendix 1. Estimation of Borne integrals
< <
Proposition 7. Let n ~ 2 and 0 R 00. Tken there exi8t8 a wnatant 0 0 the following e8timatea hold : that for all e
>
I
(i)
-
(e 2
dan
--
+' "'" ,. .
II --I --
< 00 8uck
-::;. Oe _1
I ,
-
:l:EBfl.\Z\
(iii)
dO'.
",,,•• 1<1<"("
+ 1";\ + 1"'1 2)'T",r- 2 ~ 01· ,
(e l ""•• 1<1<1<
TIT
"'''''.1<1<1< (.
+ l"'I'I"iZi;;:":'
(ii)
(iv)
Ix 1-
I -
(v)
I
I
dan
+---l"'Ii)-'
"'.1
dan
.- .:. 01· ·
~ 0/. ,
dO'•
+ 1.,,1 +-lxl')31"'1"-3 ~ 01£·
.,,,,,.1<1<1«£
Proof. Parts (i) and (ii) follow from Proposition 3 if we replace e bye 2 • Proof of (iii) for n = 2.
off---+ BIt
I
Ixl
+ IXII +
(e 2
da2 IX21
~
+
\zl
3. dan
+ IZtI + IXil + Ix12)3 \XI"-3
(e 2
\xl
R
R
I I dx Id rI ~ G'Idrff ---- ~+ ~
-
+
0 0
Proof of (iii) for n
I
dzt dxll :::; O'/e IZtI IXs1)6/2 .
(e l
IxI2)S -
0
dx
1
o
2
0
+ ~ + X 2 + r2)3 r,,-3
1:1:"\ =r
0
R
R R
(8 2
-
o
----
dO'n-3
(ell
-I-
R
2
dx
Xl
X2
+
r2 )3
I
S 0" -
0 0
e2
d_~ + r2-
SOli' Ie .
-
0
(iv)
I I ~ + ,.2)2 R
I
da'"
(;-+-';;1 2)2I xl"-2
\ xl < R
=-
0
dr
j' (;-+--r R
dO'n_l
r"-2
<
0
Ixl = r
dt" r 2)2
=
0
(v)
I re-+ -I~I +-1~12f3-1~ln-3 ~ I I I ~ G' Idx I---+ + OIlI ___+ R
dan
0
Ix\
S -
dx1
0
B
o
0
£11"
0
R
1
R
--t-
da,,_2 + r ll )3,-"-3
Xl
Ix'i =r
R
(e
_r dr 2 S Zt r )3 -
(e
0
dXI __ S O"'/e . • X1 )2 -
C
.
/e .
PropoaitiOD 8. ~,. ~AoW:
(I)
I
ctm81tJ", .0
I
1/" ~ 3, ,116"
data.,. 8 :;; Os • + 1%.1) Iztta-
Uztl
..... I..
< cc nM tlKJl lor all 8 > () lhe folotJJi1l4
ore.
11...1!'i"'1l1it ;;;
••• 1111<.
(ii)
. . ,.
Proof. (i)
I
(Iztl
I~<'
•
a
0[."Jr ~ d:rs ~ 0'1 ~ s: O,,·t;r
+daBlz.n'/l ~
I
(~+ at.)'/!
0
(li)
I ~ o'fl
(lZtI
.... l.t<}';:
~.
~ -
~
+ I:r.O"'lzl"-8 -
•• r~
r.
Of"J dztj dr j 0 0
0
(Zl !:D"j-r
I~I.I-.l<.
• •
o
f
Vi
cb:,. eb.
(lI:t
+ Ss)'"
.
0
•
dr
e •
0
~()"Ys/~ (:i;. ~0"'8 . • 0
data_a
+ .zs)8/2 ,.a- 8
,"":"
. • •
• .(.
i'
'-'
~~
Appeaclix 2. On Ba."~1 OpeD . .app• .lIle8.... ,
. '
'.
f'··.1
",
.'
':
'.-
If .A: B ~ F is a continuous linear map between Frichet spaoea Band', then we Bet KerA !II::I' {a: E 8:.At: = O} and IaA·;= .A(~4 ' \\ Proposition 1. Let
~E.
El
!An HI
An
, HI
1....
.(1)
~JiJ.
be fJ commutaliw diagram 0/ FrkAeJ BpGCU 14, _ lhe /ollotciftg coMtIiou are /td/iUed: (1) (Ii)
coMita~
li,..,. fMJ'f A,. lUi Mol
.
1m All ~ 1m All .
(I)
P1I,ere eNt. a oontmuou Im P u
p.z
~
"fHM operolor p.: Bt
~ 11. __ ...,
KerAa ,
= A I1t1:
(8)
lor all
Z E Ker
.Aodu .
(')
Phen 'Mre ~ fJ comin16016l lifteQ.r OJ)ertHor Au IUCA ,AcIt 1M tI"ram . "\
ID•
~E.
E.
..... B.
....1
lA"
"(1)
'S commufGfiw. Proof. Set At. := All - Pal Then, by (3), the commutative ~ (1) the commutative diagram
.IC. !-t". ..... .. B,
~1
EJ. II.
Au
~"tl
.IC.
It follows from (4) that Ker that JleBldnJLelterer
(6)
AL ~ Ker AaA... 8ince(l) II oommutative, this impllee
Ker AL ~ Ker Au . 16
ImPliet
(7)
210
Appendix 2. Ou Ba.nach's open ma.pping theorem
'Ve denote by K the canonical map from El onto the factor space E1/Ker Au; let A3J be the operator which is defined by the condition that the diagram - - - E8 E1 - -AU
KLE IKerA 1
81
Jon
(8)
Is rommutative. Further, by (7), there is a unique operator A)l such that the diagram E1
..t~l
E
-
K \
S
.... E1/Ker ASI -
r1S1
(9)
is commutative. From the commutative diagrams (6), (8) and (9) we obtain the
following commutative diagram
~'l
Ani
I -----_ E 181
l..4 8
.Au -----::..:c-_E~
n.
(10)
..
l.A1O
E6 Since, by (2), 1m AS6 ~ 1m Au Ker A81 such that the diagram
= 1m A3l , there is a unique operator AIS: E"
-+ Ell
(11) '"
is commutative. Since (11) is commutative, Au is closed and therefore, by Banach's open mapping theorem, continuous. Define Asa := Ag1 A16 • Then the commutative diagrams (10) and (II) imply that (5) is eommutative. •
Proposition 2. Condition (ii) in Proposition I is fulfilled if there exist a Hilbert space Hand continu0U8linear operators A": H -> 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·
Bibliography
AOBANOVSKIJ,
M. L., and
n. E.
VAL'SKIJ
[1] Maxi.mality of invaria.nt algebras of functions (Russian). Sibirsk. Mat. ~. 12 (1971), 3-12; Engl. transl. in Siberian Math. ,T. 12 (1971), no. 1,1-7.
P., amI R. SOHNEIDER [I] Holomorphio Lipschitz fWICtioIl8 in pBeudoconvax domains. Amet'. J. Math. 101J (1079), no. 3 543 - 566.
AHERN,
R. A. [1] Boundary properties of an integral of Cauchy-Leray type in piecewise smooth domains in DR (Russian). Dokl. Akad. Nauk 8SSR 246 (1979); no. 4, 779-782; Engl. transl. in Soviet Ma.th. DokI. 20 (1979), no. 3, 619-522.
AJBAPBTJAN,
AJBAPlCTJAN, R. A., and G. M. HENKIN [I] IntegraJ representations of differentia.l forms on OR-manifolds and the theory of CRfunotions (Russian). Uspehi Mat. Nauk. In press. AJSENBERO, L. A .• and A. P. JU!AKOV [1] Integral representa.tio:qs and residues in multidimensional complex a.nalysis (Russia.n). Nauka, Sibirsk. Otdel., Novosibirsk 1979.
ALE][SANDBOV,
A. B.
[1] The existence ofinnel' functions in the baJl (Russia.n). Mat. Sb. 118 (1982), 147-163. AHAB,
E.
[1] Extension de fonotions holomorphes et courllnts. Proprint, Universite de Paris-Sud, Departement de Mathematiquc, December 1980; Bull. Sci. Math. (2) 107 (1983), 25-48. [2] Generalisation d,un th~ol'~me de Wolff a Is. houle de On. Preprint, Universite de
Paris-Sud, Departement de Mathematique, December )980. ANDJlB80N, M .• and B. BERNDTSSON [I] Henkin-Ramirez formulas with weight factors. Ann. Inst. Fourier. To appear. ANDBEIAN CAZACU. C.
[1] Theorie dar Funktionen mehrerer komplexer Veranderlicher. VEB Deutscher Verlag der Wissensohaften. Berlin 1975. ABmN8oN, E. A. [1] Gleason valleys and the Choquet boundary of the algebra of functions on a convex compa.ctum. Zap. Nauon. Sem. Leningra.d. Otdel. Mat. Inst. Steklov. (LOMI) 188 (1981),
204-207. BASBNEB,
R.
[1] Peak points, barriers and pBcudoconvex boundary points. Froc. Amer. Math. Soc. 86 (1977), 89- 92. BEDJ'ORD, K., and J. E. FORNAESS [1] Approximation on pseudoconvex doma.ins . .Pre print, Prinoeton University 1979. BDGMAN,
S.
[1] ttber eine Integraldarstellung von Funktionen zweier komplexerVerfmderlioher. Mat. Sh. 1 (1936),861-861.
.Bibliography
213
[2] tiber eine in gewissen Bereichen mit Maximumflachc gUltige Integraldarstellung der Funktionen zweier komplexer Variabler, 1. Math. Z. 39 (1934),76-94; II. Math. Z. 89 (1934),605-608. BERNDTSSON,
B.
[1] Integral formulas foI' the aa-equation and zeros of bound('d holomorphio functions in the unit ball. 'Math. Ann. 249 (1980), 163-176 . . [2] A formula for interpolation and division in (In. Preprint 1981. DIsnor, E. [1] 'MappingA of partially analytic spaces. Amer. J. Math. 83 (19tH), 209-242. f21 Some global problems in the theory of fUIletions of several complex variables. Amer . •T. Math. 83 (1961), 479-4!lR. [:~] Partially analytic spaces. Amer. J. Mat,h. ~3 (HI61), 669-692.
S. [1] Analytie and meroJlJorphic continuation by means of Green's formula. Ann. Math. 44: (1943),652-67:\. [21 Group invariance of Cauchy's formula in several variables. Ann. Math. 45 (1944). 686-707.
DOCHNER.
H. [1] 'Obf'r die AquivaiC'llz der ps('udokonvexen Gebiete und der Holomorphiegebiete im Raum von n komplf'xell Verandprlichen. Math. Ann. 128 (1954). 63- 91. f21 On the conjecture of the equivalence of the plurisubharmonic funetiolls and the Hartogs functions. Math. Ann. 131 (1956), 76- 86.
13REMERMANN,
B'C'RNS, D., and E. L. STOUT [1] Extcnding functions from suhmanifoldR of the boundary. Duke Math .•r. 4:3 (1976), :l!H-404. RVCKOV, R. N. [1] Geornet,ric propE:'rtieH of thf' boundary of u domain of holomorphy (Russian). Izv. Akad. Nank SSSR Ser. Mat. 44 (1980), no. 1, 46-62; Eng1. trans1. in Math. USSR Jzv. 16 (1981), no. 1,41-54.
L. [1] Sele('tf'd prohlmnH Oil pxpptiollltl sots. Van Nostrand, Princ('ton -Toronto-LondonMel bourne 1967. [2] Interpolations by bounded analytic functions and the corona prohlelll. Ann. Math. 78 (1962),547-559. [3] The corona theorem. Ll: Lecture Notes in Math., vol. 118. Springer. Verlag, BerlinHeidelberg-New York 1970, 121-132.
CARLESO:S,
CARTAN,
H.
fl] Sur les matrices holomorphes de n variahles eornpl('xes. J. Math. Pures Appl. 19 (1940), 1-26. [2] :Espacps fibres analydques. S:vmp. Internal. Topol. Algehraica, Agosto 1956, Mexico, 195R, 97-121. l3] Seminaires E. K. ~., 1951/52. CART AN, II., and 1'. 'J'HULLEN [1] Zur Theorie del' Ringularitaten del' Funktionen mehrerer komplexer Veranderlicher. Regularitats- und Konvprgenzuerpiche. Math. Anu. 106 (1932), 617 - 647.
J., and A. ~i. CnoLLET ll] Ensembles pics pour AOO(D). Ann. lnst. Fouriflr 29 (1979), no. 3, 171-200.
CHAUMAT,
CII~KA, E. M., and G. M. HENKIN [1] Bounda,ry properties of holomorphic functions of several complex variables (Russian). Current problems in mathematics, vol. 4, Moscow] 975, 12-142.
M., and P. GRIFFITHS []] Analytic cyeleR and vector bundles on non-compact. algebraic varieties. Invent. mat.h. 28 (1975),1-106.
CORNALBA.
15 Henkinl Leiterer
214
Bibliography
CUMENGE, A. [1] Extension dans lee classes de Hardy de fonctions holomorphes definies sur des sousvarietes en position generique. Ann. Inst. Fourier. To appear. DAUTOV, S. A., and A. M. J{YTMANOV [I] Boundary properties of an integral of Martinelli-Bochner type (Russian). l'ropcrtics of holomorphic functions of several complex variables, !nst. Fiz. Sibirsk. Otdel. Akad. Nauk SSSR, Krasnoyarsk 1973, 49-54. DAUTOV, ~. A., and G. M. HENKIN [1] The zeroes of holomorphic functions of finite order and weight estimates for solutions of the a-equation (Russian). Mat. Sb. 107 (1978), 163-174. DEBIART, A., and B. GAVEAU [1] Frontiere de Shilov des dOlllaincs faiblement pscudoconvexes de (1976),17-:n.
(Cn.
Bull. Sci. Math.lOO
DIEDE1UOH, K., and J. E. FOBNAESS [1] A strange bounded smooth domain of holomorphy. Bull. Amer. Math. Soc. 82 (1976), 74-76. [2] Pseudooonvex domains with real analytio boundary. Ann. Math. 107 (1978), 371- 384. DOOQUIBR, F., and H. GBA UEBT [1] Levisohes Problem und Rungesoher Satz fUr Teilgebiete Steinschf'r Mannigfaltigkeiten. Math. Ann. 140 (1960), 94-123. DOLBEAULT, P. [1] Formes differentielles et oohomologie sur une variete analytique complexe, 1. Ann. Math. 64 (1956),83-130; II. Ann. Math. 66 (1957), 282-330. [2] Sur la cohomologie des varietes analytiques complexes. C. R. Aoad. Sci. Paris 286 (1953),175-177. DYN'XIN, E. M. [I] Smoothness of Cauchy-type integrals (Russian, English summary). Investigations on linear operators and the theory of functions, IX. Zap. Nauen. Sem. Leningra.d. Otdel. Mat. Inst. Steklov. (LOMI) 92 (1979), 115-133,320-321. EHRENPBEI8, L. [I] A new proof and an extension of Hartog'S theorolll. Bull. ArneI'. Math. Soc. 67 (1961), 607-509. F ANTAPPIiJ, L. [1] L'indioatrioe proietti va dei funzionali lineari e i prorlotti funzionali proiettivi. Annali di Mat. 22, Serie 4 (1943),181-289. FEi'I'EBMAN, C. [1] The Bergman kemel and biholomorphie mappings of pseudoconvex domains. Invent. Math. 28 (1974),1-65. FEI'I'EBMAN, C., and E. M. STEIN [I] H1' spaoes of several variables. Acta Math. 129 (1972),137-193. FOLLAND, G. B., and E. M. STEIN [I] Estimates for the all-complex and analysis on the Heisenberg group. Comm. Pure Appl. Math. 27 (1974), 429- 522. FOBNAESS, J. E. [1] Embedding striotly pseurlooonvex domains in convex domains. Amer. J. Math. 98 (1976), 529-569. FOBNAES8, J., and A. NAGEL [1] The Mergelyan property for weakly pseudooonvex domains. Manuscripta Math. 22 (1977),199-208. FOBNA1Il88, J. E., and N. 0VBELID [1] Finitely generated ideals in A(D). Preprint 1982.
215
Bibliography
FORSTER, O. [I] Plongements des variates de Stein. Comment. Math. Helv. 46 (1970), no. 2, 170-184. FORSTER, 0., and K. J. RAMSPOTT [I] Ober die Anzahl der Erzeugenden von projektiven Steinsohen Moduln. Arch. Math. 19 (1968), 417 - 422. FUKS, B. A. [1] The theory of analytic functions of several oomplex variables. To appear. FRUMIN.1. L. [1] Estimates for solutions of the a-equation in analytic polyhedra in general position (Russian). Uspehi Mat. Nauk 37 (1982), no. 4, 189-190. GAMELIN, T. W. [1] Wolff's proof of the corona theorem. Israel J. Math. 37 (1980), no. 1- 2,113-119.
GINDIKIN, S. G., and G. M. HENKIN [1] Integral geometry for the a-oohomology in q·Hnearly ooncaN domains in' sian). Funkcional. Anal. i Prilo!en. 12 (1978). no. 4, 6-23.
orm
(Rus-
GLEASON, A. PJ The Cauchy-Weil theorem ..J. Math. and Mech. 12 (1963), 429 -444. [2] Finitely generated ideals in Banach algebras. J. Math. and Mach. 18 (1964). 125-132. GoHBERO, 1., and .J. LEITERER [1] Holomorphic vector-valued functions of one variable. 1. Functions on 0. compa.ct set (Russian). Mat. rssled. 7 (1972), no. 4, 60-84. [2] Holomorphic vector-valued fWlctions of one variable. II. Functions in a domain (Russian). Mat. Tssled. 8 (1973), no. 1, 37-58. [3] Cocycles, operator-valued functions and families of subspa.ces (Russian). Mat. Tssled. 8 (1973), no. 2, 23-56. GORIN, E. A., S. V. Hnu§(5Ev, and S. A. VINOGRADOV r1] Free interpolation according to Joneli (Russian). Zap. Nauen. Rem. Leningrad. Otdel.
Mat. lnst. Stoklov. (LOMI) 118 (1981). GRAUERT, H. [1] Approximatiollssiitze fUr holomorphe :Funktionen mit WertRn in komplexen Riumen. Math. Ann. 188 (1957),139-159. [2] Rolomorphe Funktionen mit Worten in komplexen Lieschen Gruppen. Math. Ann. 138 (1957),450-472. [3] Analytische Fasel'ungen libel' holomorph-vollstandigen Raumen. Math. Ann. 186 (1958),263-273. [4] On Levi's problem and the imbedding of real-analytic manifolds. Ann. Math. 68 (1958). 460-472. GRAUERT, H., und K. FRITzSOHE [1] Einfiihrung in die Funktionentheorie mehrerer Vera.nderlicher. Springer-Verlag, Bedin-Heidelberg-New York 1974. GRAUERT, H., und 1. LIED [1] Das Ramirezsche Integral und die Losung der Gleichung 81 = '" im Bereioh der lJ('Bchrankten Formen. Rice Univ. Studies 56 (1970), no. 2, 29-50. ORAUERT, H., und R. REMMERT [1] Zur Spaltung lokal-freier Garben liber Riemallnschen Floohen. Math. Z. 144 (1975), 35-43. [2] Kohii.rellte o.nalytische Garben. To appear. GREINER, P. C., and E. STEIN [I] Estimates for thp a-Newllann problem. Princeton University Press, Princeton 1977. GRIFFITHS, P., and J. HARRIS. [I] Principles of algebraio geometry .•fohn Wiley, New York 197R. I.')·
216
Bibliography
GROMOV, M. L., and ,TA. M. ELIABBERG [1] Nonsingular mappings of Stein manifolds (Russian). Funkcional. Anal. i Prilo!en. 5 (1971), no. 2, 82-83; Bngl. transl. in Func. Anal. Appl. 5 (1971), 156-157. GUNNING, R. C., and H. ROSSI [1] Analytic funotions of several complex variables. Prentice-Hall, Englewood Cliffs (N.J.) 1965. HAKIM, M., ann N. SIBONY [1] Frontiere do Shiloy et spectra de A(iJ) pour des domaines fuihlement pseudoconvexes. C.R. Aoad. ~ci. Paris 281 (1975), A 959-A 962. r2] Spectre d~ A (D) pur des domaincs bornes faiulement psoudocon YPXOS regulipfS. ,J. :F'unot. Anal. 87 (1980), no. 2, 127 - 135. HARTOGS, F. [1] Einige Folgernngen aus der Cauchyschen IntegralfornlPl bei Funktionen mphr{'rpr Veranderlioher. MUnchener Sitz. Bel'. 36 (1906). 223-242. HARVEY, n. [1] Holomorphie chains anu thpir houndaries. Proe. Symp. l'ure Math. 30 (J977), no. I,309-:lS2. [2] Integral formulas connected by Dolueault's isomorphism. Rice University Studies 56 (1970),77-97.
n., and H. B. LAWSON [1] On boundaries of complex analytic varieties, I. Ann. Math. 102 (I ~)75), 233 - 290.
HARVEY,
H., and J. POLKING [1] Fundamental solutions in complex analysis. Duke Math. J. 46 (I97f), no. 2, 253 - 340.
HARVEY,
HARVEY, F. R., and R. O. WELLS. ,Jr. [I] Zero sets of non-negative strictly plurisuhharmonic funclions. Math. Ann. 201 (1 !l73), 165-170. [2] Holomorphic approximation and hyperfunction theory on a Cl totally 1'('0.1 suulIJo.nifold of a. complf'x manifolcl. Math. Ann. 1U. (1972),287-318. HAVIN, V. P., and R. A. VINOORADOV [I] li'ree interpolation in Hoo and in cortain other classes of functions (Rnssian). Investigations on linear operators and the theory of functions. Zap. Nauen. Sem. Leningrad. Otdcl. Mat. lnst. Steklov. (LOl\II) 47 (1974), 15-54, 56 (1 !l76), 12-58, and 92 (1979), 317. H. [J] ZUI' Funktioll£'ntheorif" mehrf"rer V£'Iranderlicher. Math. Ann. 122 (1950), 276-27R.
HElI'ER,
G. M. [J] Integral representation of functions in strongly pseudoconvex domains and applications to the a.problem (Russian). Mat. Sb. 82 (1970), 300--~~m,; Eng!. trans!. in Mat.h. USSH-Rb. 11 (1970), 27~~-281. [2] ~olutionfo\ with bounds for th£' <'qllations of H. Lewy and PoinciHl>-Lelong. Construction of functions of Ncyanlinna class with given zeroes in a st.rongly pseudoconvex domain (Russian). Dokl. Akad. Nauk S8SR 224 (1975), no. 4, 771-774; Engl. tralll~1. in Roviet. Math. Dokl. 16 (1975), no. 5, I:HO-1314. [3] H. Lewy's equation and analysis on pseudoconvex manifolds, I (Russian). Uspehi, Mat. Nnuk. 32 (1977), no. 3, 57-ll8; Engl. trans!. in Russian Math. Rurveys 82 (1977). no. 3, 59-130; and n, Mat. Sb. 102 (1977), no. 1, 71-108; Engl. transl. in Math. USSR-Sb. 102 (1977)., no. 1, 63-64.J [4] Integrall'opres('ntation of functions holomorphic in strictly pseudoconvex domains and some applications (Russian). :\'Iat. Sb. 7S (1969), 611-6:~2; "Engl. transl. in Math. USSH-Sb. "2 (1969), 597 -616. [5] Continuation of bounded holomorphic functions from submanifolds in general position in a. strictly psendoconvex domain (Russian). Izv. Akad. Nauk SRSH. SOl'. Mat. 36 (1972),540-567; Eng!. transl. ill Math. USSR Izv. 4) (1972), no. 3,536-563.
HENKIN,
217
Bihliography
[6] A uniform estimate for the solution of the ()-problem in a Weil domain (Russian). Uspehi Mat. Nauk 26 (1971), no. 3, 211-212. [7] Approximation of functions in pseudo convex domains and a theorem of Z. L. Lelbenzon (Russian). Bull. Acad. Polon. Sci. Serf Sci. Math. Astronom. Phys. 19 (1971),37-42. HENKIN, G. M., and .J. LEITERER [I] Global integral formulas for solving the a-equation on Stein manifolds. Preprint, Berlin and Wnrszawa H179; Ann. Poloniei Math. 39 (1981). 93-116. [2] Theory of functions on strictly pseudoconvex sets with non-smooth boundary. Report R-MATH-~~/81 IMath del' AdW del' DDR, Berlin 1981. HENKIN, G. M., and A. 1. PETROSJAN [I] Rolntion with a uniform estimato of the a-equation in a real non-degenerate Weil polyhedron (Russian). Tzv. Akael. Nauk Armjan. SSSR Serf Mat. 18 (1978), no. 5-6, 428-441. HE~KIN, G. 1\1., aud A. V. l{OMANOV [1] Exact Holder estimates of solutions of the a-equation (Russian). Izv. Akad. Nauk SSSR Serf Mat. 3:) (1971), 117l-IIR3; Engl. transl. in Math. USSR Tzv. I) (1971), 1180-1192.
HENKIN, G. M., and A. G. SERGEEV [I] Uniform estimat{'f; for 801ution8 of tho e.oquation in pseudo convex polyhdera (Russian). )lnt. Sb. 112 (1980), no. 4, 522-567; Engl. transl. in Math. USSR-Sb. 40 (1981), 4fHl-n07. G. M., and A. E. TUMANOY [1] Interpolation submanifolds of pseudoconvex manifolds (Russian). Mathematical programming and related questions (Proc. Seventh 'Vinter School, Drogobych, 1974). Theory of functions and functional analysis, Central. Ekonom.-l\1at. lnst. Akad. Nauk SSSR, Moscow H)76, 74- 86.
HENKIN,
HE~RIKSEN, S. [1] A peak S(!t of ITuusdorff dimeusion 1 for the algebra A(D) in the bOlmdary of a domain D "lit.h Goo-boundary in C!".. Proprint, Univ. of Oslo 1981.
HEUNE::M.ANN, D. [I] Estimates for t.hp. a-equation on Stein manifolds. Math. Nachr. To appoar. HOFF~fAN, K. [I] Banach spaces of analytic functions. Prentice-Hall, Englewood Cliffs (N.J.) 1962.
HORl\fANDER, L. [I] An introduction to complex analYBis in several variablcs. D. Van Nostrand, PrincetonToronto-London 1966. [2] L2 €lstirnates and existence theorems for the a-operator. Acta Math. 118 (1960), 89-] 52. [3] LP estimates for (pillri)-subharrnonic functiolls. Math. Scand. 20 (1967), no. 1, 65 - 78. HOR)1ANJ)RR, L., and J. \\-'RRMER [I] Uniform approximation on compact sets ill
(fJn.
Math. Scand. 2:1 (1968), no. 1, 5- 21.
M. LI] Globale holomorphe Kerne 7.ur LORung uer Cauchy.Riemannschen Differentialgleichungen. Preprint 1979; In: FORNARSS, J. (ed.): Recent dev€llopments in several complex variables. Princeton University Press, Princeton 1981.
HORTMANN,
JORICKE, B. [1] The multidimensional analogue of Privalov's theorem (Russian). Math. Nachr. 107 (1982), 221-233. [2] The analogue of Tamrasov's theorem and continuity of tho Cauchy projection in HOlder norms for certain multidimensional domains (Russian). Math. Nachr. 112 (1983). KERZMAN, N. [I] Hold~r and LP estima.tes for solutions of Bull. Amer. Math. Soc. 76 (1970), 860-864.
au = f in strongly pseudoconvex domains.
218
. Bibliography
au
[2] Holder aud [;P estimates for solutions of = Comm. Pure Appl. Math. 24 (1971), 301-379.
I in strongly pscudoconvex domains.
KlIlRZMAN, N., and E. M. STEIN [1] The 8zego kernel in terms of Cauchy-Fantappie kernels. Duke Math. J.45 (1978), 197 - 224. KNESER, H. [1] Die singularen Kanten bei analytischen Funktionen mehrerer komplexer Vcranderlieher. Mat.h. Ann. 106 (1932), 656-660. KOHN, J. J. [1] Harmonio integrals on strongly pseudo-convex manifolds 1. Ann. Mat.h. 78 (1963), 112-148; IT. Ann. Math. 79 (1964), 450-472. [2] Boundaries of complex manifolds. In: Proo. Conf. Complex Analysis, Minneapolis 1964. Springer-Verlag, Berlin-Heidelberg-New York 1965, 81-94. [3] Global regularity for -8 on weakly pseudoconvex manifolds. Trans. Amer. Math. Soc. 181 (19n),273-292. [4] SubeJlipt.ioity of the:' a-Neumann problem on pseudo-convex domains: suffioientconditions. Acta Math. 142 (1979), 79-) 22. KOHN, J. J., and L. NIRENBERG [1] A psE>udooonvex domain not admitting a holomorphio support function. Math. Ann. 201 (l97:i), 265-268. K0081S, P. [I] Introduction to HP-spac~s. London Math. Soo. Lecture Note series, vol. 40, Cambridge University Pl'~Ss, CambridgE> 1980.
KOPPELMAN, 'V. [1] The Ctuwhy integral for differential formA. Bull. Amer. Math. Soc. 73 (1967),554-556. KRANTZ, S. G. [1] Charaoterizations of various domains of holomorphy via a-estimates and applications to a prohlem of Kohn. Illinois J. Math. 23 (1979), 267-285. [2] Function theory of several complex variables. John Wiley, New York 1982. LBITBRER. J. [1] Ober analytische Operatorfunktionen einer komplexen Veranderliohen mit stetigen Randwerten. Math. Nachr. 72 (1976), 247 - 274. [2] Theorem B filr analytische It'unktionen mit. stetigen Randwel'ten. In: Beitrage zur Analysis, Band 8. VEB Deutsoher Verlag der Wissensohaften, Berlin 1976, 96-102. [3] Equivalenoe of Steinness and validity of Oka's prinoiple for Subdomains of Stein manifolds. Math. Naohr. 89 (1979), 181-183. [4] tJber das Okasche Prinzip fUr Schnitte in holomorphen Vektorraumbiindeln. Preprint, ZIMM der AdW der DDR, Berlin 1977. LlIlLONG, P. [1] Les fonot-iolls plurisousharmoniques. Ann. Ecole Norm. Sup. 62 (1945), 301-338. [2] Sur quelqucs problemes de 10. theorie des fonotions de deux variables complexes. Ann. Eoole Norm. Sup. 68 (1941),83-177. [3] ProprieteB metriques des variet.es analytiques oomplexes de£inies pal' une equation. Ann. Ecole Norm. Sup. 67 (1950), 393-419. LERAY, J. [1] Fonction de variables complexes: so. representation comme somme de puissances negatives de fonctionslineaires. Rend. Accad. Naz. Lineei. serie 8, 20 (1956), 589-590. [2] Le oaloul differentiel et integral sur une variete analytique complexe. Bull. Soo. Math. France 87 (1959), 81- 180.
LEVI, B. [1] Studii sui punti singolari essenziali delle funzioni analitiche di due complessc. Ann. mat. pura ed appl. 17 (UHO), no. 3, 61-87.
0
phI variabili
Bibliography
219
LEWY, H. [I] On the local character of the solutions of an atypical linear differential equation in three vw:iables and a related theorem for regular functions of two complex variables. Ann. Math. 84 (1966). 614-622. [2] An example of a smooth linear partial differential equation without solution. Ann. Math. 88 (1957), 166-158. LIEB,
I.
[I] Ein Appl'oximationssatz auf streng pseudokonvexen Gebieten. Math. Ann. 184 (1969), 66-60. [2] Die Cauchy-Riemallnschen Differentialgleichungen auf streng pseudokonvexen Gebieten. Math. Ann. 190 (1970;71), 6-44; 199 (1972), 241-266. MALGRANGE, B. [1] Lectures on the theory of functions of several complex variables. Bombay 1968. [2] Ideals of differentiablo functions. Oxford University Press. Oxford 1966. MARTINELLI,
E.
[1] AIcuni teoremi integrali per Ie funzioni analitiche die piu variabili complesse. Mem. r. Acad. Ita1. 9 (1938), 269-2ts3. [2] Sopra una dimostraziono di H. Fueter per un teorema di Hartogs. Comment. Math. Helv. 16 (1943), 340-349_ [3] Contributi alIa teoria dei residui per Ie funsioni di due variabili cornplessc. Ann. mat. purs cd appl. 89 (1966), 4, 335-343. MILNOR,
J.
[1] Lectures on the h-cobordisltl theorem. Princeton University Press, Princeton 1965. C. B. [I] The analytic embedding of abstract real analytic manifolds. Ann. Math. 68 (1958), 169-201.
MORREY,
N. I. [1] Singular integral equations (Russian). Nauka, Moscow 1968. Engl. trans1. ed. J. R. M. RADOK, Noordhoff, Leyden 1977. MUBKHELISVILJ,
NAGEL, A. [1] Smooth 7.ero sets amI interpolation sets for some algebras of holomorphic funotions on strictly pseudoconvex domains. Duke Math. J. 48 (1976). 323-348. NAGEL, A., and W. RUDIN [1] Local boundary behavior of bounded hololllorphic functions. Canad. J. Math. 80 (1978), 683-592. NARASIMHAN, R. [1] Compact ana.lytic varieties. L'Enseignoment Math. 14, (1968), 75-98. [2] Ana.lysis on real and complex manifolds. North Holland, Amsterdam 1968. [3] Imbedding of bolomorphically complete complex spaces. Amer. J. Math. 82 (1960), 917-934. R., and R. O. WELLS, Jr. [I] Approximation theorems on differentiable submanifolds of a. complex manifold. Trans. Amer. Math. Soc. 142 (1969), 15-35.
NIRENBERG,
NORGUET, F. [I] SlIr les domaines d 'holomorphie des fOllctions ulliformes de plusieurs variables complexes. Bull. Soc. Math ..France 82 (1954), 137- 169. [2] Problemes sur les formes differentielles et lee couranta. Ann. lnst. Fourier 11 (1961), 1-82.
OXA,K. [1] Sur les fonctions analytiques de plusieurs variables complexes. IV. Domaines d'holomorphie et domaines rationellement convexes. Jap. J. Math. 17 (1941). 617-621. [2] Sur les fonctions analytiques de plusieurs variables oomplexes. II. Dornaines d'holomorphie. J. Sci. Hiroshima Univ. 7 (1937), 115-130.
220
Bibliography
[3] Sur lOB fonctions analytiques de plusieurs variables complexes. III. Deuxieme probh~me de Cousin. J. Sci. Hiroshima Univ. 9 (1939), 7-19. [4] Sur les fonctions analytiques de plusieurs variablos complexes. VI. Domaines pseudoconvexos. Tohoku Math. J. 49 (1942), 15-22. [6] Sur les fonctions analytiques de plusieurs variables complexes. VII. Sur quelques notiolls arithmetiques. Bull. Soc. :Math. France 78 (1950), 1- 27. [6] Sur les fonctions analytiques de plusieurs variables complexcs. IX. Domainos finis sans point critiquo intcrieur. J ap .•T. Math. 2S (1953), 97 -155. [7] Sur les fonctions analytiques de plusieurs variables complexes. T. Domaines convexes par rapport ttllX fonctions rationelles. J. Sci. Hiroshima Univ. 6 (1936), 245-255. [8] Sur leR fonctions analytiqueH de plusieurs variables complexes. V. L'integrale de Cauchy. Jap. J. Math. 17 (194,1), 525-531.
N. [1] Integral repr~Rpntation formulas and LP-estimates for the a-equation. Math. Scand. 29 (1971), no. 1,137-160. _ [2] Generators of the maximal ideals of A(D). Pac. J. Math. 39 (1971) 1,219-223. [3] Integral representation formulas for differential forms and solutions of the a-equation. Colloque International du C.N.R.S., N 208, Agora. Mathematica, Gauthier-Villars, Paris 1974,180-198.
0VRELIU,
PALM,
A.
[I] Integral represontation formulas on strictly pseudoconvex domains in Stein manifolds. Duke Math. J. 43 (1976), 41-62. PETROSJAN, A. 1. [I] Uniform approximation of functions by polynomials on Weil polyhedra (Russian). Izv. Akad. Nauk SSSl-t Ser. Mat. 84 (1970),1241-1261; Engl. trans!. in Math. USSR Izv. 84 (1970), no. 6,1250-1271.
IJnoNo, D. H., and E. M. STEIN [I] EstimatC'ls for the Bergman and Szego projections on strongly pseudoconvex domains. Duke Math. J. 44 (1977), 695-704.
S. 1. [1] A boundary uniqueness theorem for holomorphic functions of several complex variables (Russian). Mat. Zametki 15 (1974), 205-212; Engl. transl. in Math. Notes 15 (1974), no. 2,116-120.
PINCUK,
POLJAKOV, P. L. [I] The Cauchy-Wcil formula for differential forms (Russian). Mat. Sb. 86 (1971), 388-402; Engl. transl. in Math. USSR-Sb. 14 (1971), no. 3, 383-398.
[2] Extension of bounded holomorphic functions from an analytic curve in general position in a polydisc. Ji""unkcional. Anal. i Prilozen. In press. [3] Banach cohomology on piecewise strictly pseudoconvex domains (Russian). Mat. Sb. 88 (1972),238-255; Eng!. transl. in Math. USSR-Sb. 17 (1972), no. 2,237-256. POMPEIJU,
D.
[I] Sur les siugularites des fonctions analytiques uniformes. C.H. Acad. Sei. Paris 1Sn (1904), 914 - 915.
RANGE,R. M. LI] Holder estimates for on convex domains in Symp. Pure Math. 80 (1977), no. 2, 31-33.
a
([J2
with real analytic boundary. Proc.
HANGE, R. M., and Y. T. SIU ll] Uniform estimates for the a-equation on domains with piecewise smooth strictly pseudoconvex boundaries. Math. Ann. 206 (1973), 325-364. RAMIREZ DE AREI.LANO,
B.
[1] Ein Divisionsproblem und Randintegraldarstellungen in der komplexen Analysis. Math. Ann. 184 (1970), 172-187.
ROHBL, H. [1] Das Riemann-Hilbertsohe Problem der Theorie der linearen Differentialgleichungen. Math. Ann. ISS (1957), 1- 25.
221
Bibliography H.ONKI~,
L. I.
[I] Introduction to the theory of entire functions of severa.l varia.bles (H.ussian). Moscow 1971. Engl. transl., Amer. Math. Soc., Providence 1974. ROSSI,
IT.
[1] Holomorphieally convex sets in sevE'ral complex variables. Ann. Math. 74 (1961), 470-493. RUDIN, \"1. [1] Peak interpolation 8Pts of class 0 1 • Pacific J. Math. 76 (1978), no. 1, 267-279. [2] Projections on invariant subspaces. Proc. Amer. Math. Soc. 18 (1962). 429-432. [3] Function theory in the unit ball of CT•• Springer-Verlag, New York 1980.
SABAT.
n. V.
[I] Introduction to complex analysis (Russian). Nauka, Moscow 1976. RAKAI,
A.
LI] Uniforlll approximation in several complex variables. Osaka
J. Math. 16 (1978),
689-611.
V. V. [1] Direct and inverse estimates for a singular Ca.uchy integral along a closed curve (H.ussian). Mat. Zametki 19 (H176), no. 3, 365-380; Engl. transl. in Ma.th. Notes 19 (1976), no. 3, 221-2:11.
SALAEV,
SOli NEIDER, M. [1] Vollstalllligo Dllrchsehnitte in Steinscheu Mannigfaltigkeiten. Math. Anu. 186 (1970), 191-200. SERGEEV,
A. O.
_
[1] Uniform estimates for the a-problem in a Levi-Wail domain (Hussian). Dokl. Akad. Nauk SSSR 286 (1977). 1080-1082; Engl. transl. in Soviet Math. Ook!. 18 (1977), no. 6, 1335-1338. SIBONY. N. [1] lJn eXAmple de flomaine pscudoconvexe regulior ou I'equation au - / ll'u,dlllflj, pas de solution bornee pour f bOnlee. Invent. math. 62 (1980). 235 - 242. [2] Prolong€lmant des fonctions holomorphes born~es et metrique de Carathco(lol'Y. Invent. math. 29 (I !l75), 205 - 230. [3] Valeurs au bord de fonctions holomorphcs et ensembles polynomialement convexes. In: Lecture Notes in Math., vol. 578, Springpr-Vcdagllerlin-Heidelberg-New York 1977, 300 - 31:1. SKODA,
H.
[I] Valfmrs au bord pour los solutions de l'operat(,llr d" clans les ouvcrts strictement pseudocollVCXCS. C.H. Acad. Sci. Paris 2S0 (1975), ti:l:~-636. [2] Valeurs au hord pour les solution de I'operateur d" et charaetcrisutioll dos zeros des fOllctions de 10. clu.sse de Nevanlinnu.. Bull. Soc. Math. Frunce 104 (I !)7IJ). 225 - 299. SOMMER,
F.
[1] Dber die Integralformeln in del' FunktionenthcOI'io mehrerer komplexer VeraudeI'licher. Math. Aml. 125 (1952), 172-182. STOUT,
E. L.
[I] An integral forlllull1 for holomol'phie functions on strictly pscudoconvex hypersurfaces. Duke Math. ;J. 42 (1975), 347-356. [2] The boundary values of holomorphic functions of several complex variables. Duke ::vrath .•1. 44 (1977), 105-108. D., and Y. L. TONG [1] A parametrix for 0- and Riemann-Hoeh in Cech-t.heory. Topology 16 (1976), 27:i-:iOI.
TOLEDO,
A. E. [1] On the boundary values of holomorphic fWlCtions of several complex variables (Russian). Uspohi Mat. Nauk 29 (1974), no. 4, 158- I 59. [2] Peak sets of metric dimension 2.5 for algebras of holomorphic functions on the 3-dimensional sphere in 0 2 (Russian). lzv. Akad. Nauk SSSR Sere Mat. 41 (1977), 115-13~.
TUMANov,
222
Bibliography
V AL'SKIJ, R. E. [1] On measures orthogonal to analytic functions in en (Russian). Dokl. Akad. Nauk SSSR 198 (1971), 502-505; Engl. transl. in Soviet Math. Dokl. 12 (1971), no. 3, ROS-SI2. V AROPOULOS, N. TH. [1] B.M.O. funotions and the a.equation. Pacific J. Math. 71 (1977), no. 1, 221 - 273. VITU8KIN, A. G. [I] Analytic capacity of sots in problems of approximation theory (Hussian). Uspchi Mat. Nauk 22 (1967), no. 6, 141-199; Engl. transl. in Russian Math. Surveys 22 (1967), no. 6,139-1200. [2] On a problem of W. Budin (Russian). Dokl. Akad. Nauk SSSR 218 (1973), no. I, 14-15; Engl. transl. in Soviet Math. Dokl. 14 (197;{), no. 6, 16IS-161!).
V. S. [I] Methods of the theory of many complex variables (Russian). Moscow 1964. Engl. trans1., MIT Press, Cambridge 1966.
VLADIMIROV,
WElL, A. [1] L'integralc de Cauchy ot 1es fonctiolls de plusieul's variables. ::\fHth. Ann. 111 (1935), 178-182. WEINSTOOK, B. M. [I] Continuous boundary values of analytic fuuctiouEI of several complex variables. Proc. Amer. Math. Soc. 21 (1969),463-466. WELLS,
Jr., R.O.
[1] Differential analysis ou complex manifolds. PI'putice Hall, Englewood Cliffs (I\ .J.) 1973.
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<:t:"""i8e Cl-hOlmdal'Y; 169
X()B X@N
8K
Differential operators d
l'xtcrior uiiferentilll operator holomorphi(, ('omporll'nt of d; 9,43,44 antiholomorphie l'omponent of d: 9, 43,44, 86
n
8 iJ,
==
i..l/ [;;j
<\
=
fJ/oz j :=
Dfl ... IJ!'"
Int~gral
operators
(ik
nD , Bnv L~D' LeD Nlw
f)
if
1t4 =
(k1 , ... ,kn ) is a mult i-order: 9
47,165,173 50,51 13i>, I 37, I fir, luu, l74
Itw
LDI
RZ
]j::lK'
RD
Ill, 112,
1~3
125 R,sK
171,180
Spac('!s of functions or differential forms r) BCO(J') Ck(y)
l'ontilllLOl1K ('ompkx-valued functions on .f; 9, 48 bound(·d fUJl(~tions in ()o( Y); 48
('O(
k t.imes l'ontinuollsly differentiable functions in OO( Y); 9, 75 fllJ1ctioJl8 f f Cfc(y) with l:mppj CC n; 9 holomorphic function!:) on Y; 11, 14, 78 := O( J") EB 4" EB ()( Y) (m tlme!3); 12, 14, 78 (.'unt.inllolls phlToisuuharulOnlc fllnction~ on D; 23 hOllnclC'd measuI'n.ble complex-valued functions on Y; 4R flllwt,iollH f t: OO( Y) with 11/11«, Y 00: 49 fllJwtion~ l' E (}O(.Y) with II/II~. K (Xl for each J( C C Y; 49 fl1nctions Ie Ck(.l') whos(' derivatives of order k belong to C~(Y); 4!-J difforent,lal form!:! of hidegroe (p, q) with coefficknts in S( Y), where S i~ OlW of the sym b()l~ L=, (,10, RGo, H~, Ok, Ck+ c¥; 43, 44, 49
C~(Y)
CV(Y) 0 111 ( .1') PO( Y)
LOO(y)
< <
llC¥(}' )
Ci1I(y) Ck+,",( Y) S(p,q)( Y)
forms
f E C(~', l)(D) with I ( C~o.q)(j)) \vit.h
formH
JE
form8
at
=
0; 75
at = 0; 72 Ill~~q)(lJ) wit,h aj ( C?o,q+l)(D); 72
: = ~)P~~~q-lij»); i2
Spaces defined for holomorphic vector bundles B holomol'phic SCCt.jOll~ of B over 1) sod,ions of B over]); 88
O(D, JJ) CO(D, n)
('(mtinIl0118
Ht!l(lJ, B)
r.fo,q)(D, 13)
Hl~(·tioml I E: 0°(1), lJ) with I!/II~J D oo-.i 88 {Oont.hl1lous }J-valued (0, q)-fornls on J); 88
Z?o.9)(D,
<
B)
form~ I
F~J7qijj, B)
forms I
EM:q)(D, B)
:= aFid~!1-1)(D, B); 88
al
OrO,g) (ii, B) with = 0; 88 ( Ilt/,7qiiJ, B) with af E CrO,IJ+l)(D, B); 88
f
Subject index
ana.lytic polyhedron 22, 182 analytic polyhedron, T£'al non-degenerate 182 analytic Shp.aveH, coherent 159 analytic subsheaf, coherent 160 analytin subsheaf of xOlJ 160
fWlction, function, function, function, function,
bidegrP8 45, 46 biholomorphic map J 4, 16 bihololllorphically equivalent manifolds 16
holomorphic atlas 16, 87 holomorphic continuation; uniqu('nesB 13 holomorphic coordinates 16 holomorphic Cousin data 88 holomorphic function 13, 16 holomorphic homomorphism betweon holomorphic vector bundles 88 holomorphic map 14, 16 holomorphic soction of a holomorphio w'ctor bllndh~ 8R holomorphic trivialization of a holomorphic vector bundle 88 holomorphic vector bundle 87 holomorphio vector bundle; holomorphic section 88 holomorphic vector bundle; holomorphic trivialization 88 holomorphic voct.or bundle, holomorphically trivial 88 holomorphic vector bundles; holomorphic homomorphism 88 holomorphically convex complex manifold 98 holomorphically trivial holomorphic v('('lor bundle 88 Holder space 51 Hormandcr ball 116 Hormander diametP.r 116, 194
canonical c:omplex coordinates II Ca.uchy's inequality 14 Cauchy-Green formula 9 Cauchy-Riemann equations 9, 17 centre of an algehra 48 closed complex submnnifold 16 COhf'rfmt ana.lytic sheaves 159 coherent analytic subsheaf 160 compact linear operator 210 complex cotangent bundle 162 complex line 30 complex manifold 16 complex manifold, holomorphically convex 98 complex manifold; orientation 163 complex plane 30 complex Bubmanifold 16 complex subrnanifold, closed 16 complex tangent bundle 162 complex tangent plane 36 cotangent bundle, complex 162 Cousin data, holomorphie 8S Cousin problem 88 crit.iC'al point 35 differential form 43 differential form with values in a. ho10morphic vector bundle 88 discrete subset 36 domain of holomorphy 19 expression of a with respect to (U1, h 1) form 43 Fredholm operator
210
163
holomorphic 1 :l, 16 plurisubharmonic 25, 26, 32 strictly plurisubharmonic 25 strictly subharmonio 24 subharmonic 23
implicit function theorem Jacobi matrix
15
14
I{oppelman fOI'mula 57, 174 Koppelman-Leray formula 59, 112, 178, 194 Koppelman-Loray-Norguct formula 180, 183, 191
226
Subject index strictly convex set 67 strictly plurisubharmonic cxhaU!~ting function 93 strictly plurisubharmonic fWlction 25 st.rictly pseudoconvex polyhedron 189 strictly pseudoconvex aet 38, 89 st,rictly subhatrnonic function 24 subharmonic function 23 support 43 systAm of holomorphic coordinates 16
lemma of Morsn 35,93 lemma of Oka-Hefer 81 Leray formula 56, 167 Leray map 52 Leray section 165 Leray-Norguet formula 171 Leray -N orguet section 170 Levi form 25 Levi polynomial 27 Levi problem 33 linear operator, compact 210 local trivialization 87 manifolds, biholomorphically equivalent map, holomorphic 14, 16 Martinelli-Bochner formula 54, 166 maximum principle 13 multi-order 11 multi-radius 12 orientation of a complex manifold orientation of q;tI 48 orientation of aD 48
It)
163
uniqueness of holomorphic continuation 13
piecewise Ol-boundary 48, 163 plurisubharmonic function 25, 26, 32 polyrlisc II polyhedron, analytic 22, 182 polyhedron, real non-degenerate analytic 182 polyhedron, real non-degenerate strictly pseudoconvex 189 polyhedron, strictly pseudoconvex 189 pseudoconvex set 32 real non-degenerate analyt.in polyhedron 182 real non-degenera.te strictly psoudoconvAx polyhedron 189 real plane 30 shea.f of germs of hoi om orphic sections Stein manifold 98
tangent bundle, complex 162 tangent plane, cornpl~x 36 Theorem A of Cartan 160 Theorem B of Cart an 160 thcorem of Dolbeault 88 theorem of Hartoge 18 theorem of Oka 160 theorem of Stieltjes-Vitali 14 totally real plane 30 totally real submanifolc..l 30
160
Weil formula
184
ex-Hendel' continuous function 51 ex-Holder norm 51 Ok-form 44 Ofal-form 44 otl1 , f)-form 45, 46 Ok+~-form 51 O~~~f)-form 51 O~~I¥-function 51 qJn; orientation 48 aD; orientation 48 a-equation 17 t'(D)-convex compact subset 19, 97 O(D)-hull 19,97 PO(D)-convex compact. subset 33 PO(D)-hull 33 (p, q)-form 45, 46 a-form 43
