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Analysis of Several Complex Variables Takeo Ohsawa
American Mathematical Society
Titles in This Series 211 Takeo Ohsawa, Analysis of several complex variables, 2002 210 Toshitake Kohno, Conformal field theory and topology. 2002 209 Yasumasa Nishlura, Far-from-equilibrium dynamics, 2002 208 Yukio Matsumoto, An introduction to Morse theory. 2002 207 Ken'ichi Ohshika, Discrete groups. 2002 206 Yuji Shimizu and Kenji Ueno, Advances in moduli theory, 2002 205 Seiki Nishikawa, Variational problems in geometry, 2001 204 A. M. Vinogradov, Cohomological analysis of partial differential equations and Secondary Calculus, 2001
203 Te Sun Han and Kingo Kobayashi, Mathematics of information and coding, 2002
202 V. P. Maslov and G. A. Omel'yanov, Geometric asymptotics for nonlinear PDE. I. 2001
201 Shigeyuki Morita, Geometry of differential forms, 2001
200 V. V. Prasolov and V. M. Tikhomirov, Geometry, 2001 199 Shigeyuki Morita, Geometry of characteristic classes, 2001 198 V. A. Smirnov, Simplicial and operad methods in algebraic topology, 2001
197 Kenji Ueno, Algebraic geometry 2: Sheaves and cohomology, 2001 196 Yu. N. Lin'kov, Asymptotic statistical methods for stochastic processes, 2001
195 Minoru Wakimoto, Infinite-dimensional Lie algebras, 2001 194 Valery B. Nevzorov, Records: Mathematical theory. 2001 193 Toshio Nishino, Function theory in several complex variables, 2001
192 Yu. P. Solovyov and E. V. Troitsky, C'-algebras and elliptic operators in differential topology. 2001
191 Shun-Ichi Amari and Hiroshi Nagaoka, Methods of information geometry, 2000
190 Alexander N. Starkov, Dynamical systems on homogeneous spaces, 2000
Library of Congress Cataloging-in-Publication Data Ohsawa, T. (Takeo) [Tahensu fukuso kaiseki. English] Analysis of several complex variables / Takeo Ohsawa ; translated by Shu Gilbert Nakamura.
p. cm. - (Translations of mathematical monographs, ISSN 0065-9282 v. 211) (Iwanami series in modern mathematics) Includes bibliographical references and index. ISBN 0-8218-2098-2 (soft cover : acid-free paper) 1. Functions of several complex variables. 2. Mathematical analysis. I. Title. II. Series. III. Series: Iwanami series in modern mathematics. QA331.7.03713 2002
® The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Information on copying and reprinting can be found in the back of this volume. Visit the AMS home page at URL: http://vwv.ams.org/
10987654321
070605040302
Contents Preface
ix
Preface to the English Edition
xi
Summary and Prospects of the Theory Chapter 1. Holomorphic Functions 1.1. Definitions and Elementary Properties 1.2. Cauchy-Riemann Equations 1.3. Reinhardt Domains
1
1
8 18
Chapter 2. Rings of Holomorphic Functions and 8 Cohomology 23 2.1. Spectra and the 8 Equation _ 23 2.2. Extension Problems and the 8 Equation 25 2.3. c? Cohomology and Serre's Condition 27 Chapter 3. Pseudoconvexity and Plurisubharmonic Functions 3.1. Pseudoconvexity of Domains of Holomorphy 3.2. Regularization of Plurisubharmonic Functions 3.3. Levi Pseudoconvexity
35 35 41 47
Chapter 4. L2 Estimates and Existence Theorems 4.1. L2 Estimates and Vanishing of 8 Cohomology 4.2. Three Fundamental Theorems
55 55 75
Chapter 5. Solutions of the Extension and Division Problems 5.1. Solutions of the Extension Problems 5.2. Solutions of Division Problems 5.3. Extension Theorem with Growth Rate Condition 5.4. Applications of the L2 Extension Theorem VII
83 83 87 93 100
viii
CONTENTS
Chapter 6. Bergman Kernels 6.1. Defi nitions and Examples 6.2. Transformation Law and an Application to H olomorphic Mappings 6.3. Bou ndary Behavior of Bergman Kernels
105 105
Bibliography
115
Index
119
107 110
Preface This book does not intend to explain the whole theory of complex analysis in several variables as it stands today. The goal of the book is to introduce methods of real analysis and see these methods produce a variety of global existence theorems in the theory of functions based on the characterization of holomorphic functions as weak solutions of the Cauchy-Riemann equations.
Chapter 1 starts with the definition and elementary properties of holomorphic functions, and in Chapter 2 the problem of extension of functions and the division problem are converted to the problem of solving the Cauchy-Riemann equations of inhomogeneous form. These are called a equations. The theme to observe up to Chapter 3 is that the solvability of the a equation on an open set 11 in C" imposes on 11 a geometric restriction called pseudoconvexity. Chapter 4 shows, to the contrary. the solvability of the a equation on a pseudoconvex open set; and, as an application, we generalize to several variables the Mittag-Leffler theorem, Weierstrass' product theorem, and the Runge approximation theorem, which are included in many textbooks for complex analysis in one variable. This approach is called the method of L2 estimates. By virtue of this method, in Chapter 5, we solve the extension and division problems. The point of this argument is that the solutions are evaluated by the estimates, and thus the application immediately becomes wider. The content stated so far is like the view down from a high place, while Chapter 6 invites the reader to climb the untrodden mountains, which is, so to speak, the view of the great mountains gazed upon from the base camp. The reader will see the author break down from exhaustion there. It is left to the reader to decide whether he has fallen down forward or backward. Several people offered help with the present publication. In particular, Professor Kazuhiko Aomoto, a great pioneer in analysis, rec-
ommended that I write the book and provided useful advice. The editorial board of Iwanami Shoten, Publisher, paid careful attention ix
x
PREFACE
to the appearance of the book. Mr. Tetsuo Ueda and Mr. Haruo Yokoyama pointed out many mistakes. I am grateful to these people.
I regret that just before the approval of the book, my teacher, Professor Shigeo Nakano, who had saved me from dropping out and introduced me to this field, passed away. I would like to offer this book
on the altar with my respect. Further, criticism on the book from readers will be considered as my teacher's reprimands from heaven, which I look forward to hearing. Takeo Ohsawa May 1998
Preface to the English Edition Voluminous textbooks have already appeared in several complex variables. In this concise booklet the author assumes a basic knowledge of analysis at the undergraduate level, and gives an account for the L2 theory of the 6 equation. Emphasis is put on recent results which have brought us a deeper understanding of pseudoconvexity and plurisubharmonic functions, and opened a major new way of developing complex analysis.
Xi
Summary and Prospects of the Theory The concept of analytic function was introduced by L. Euler, J. L. Lagrange and others during the 18th century, and it was A. L. Cauchy,
C. F. Gauss, G. F. B. Riemann, K. T. W. Weierstrass and others of the 19th century who made the theory of complex functions of one variable as complete and elegant as it is today. Entering the 20th century, breakthroughs to the world of complex functions of several variables were made by F. Hartogs, E. E. Levi, P. Cousin and others. The problems which they proposed in the field were extremely difficult at the time, but it did not take even a half century to settle all these problems and establish the foundations for the theory of analytic functions of several variables. As a matter of common knowledge, all the core problems in this area were solved by one mathematician. His name is Kiyoshi Oka (1901-1978). He grasped all the central problems as a system in the course of so-
lution and gave the last polish to this system by solving affirmatively the so-called Levi problem, which asserts the crucial proposition that if a domain satisfies a geometric condition called pseudoconvexity, one can construct a holomorphic function such that every boundary point of the domain is an essential singularity of this function. The methods created by Oka were of striking originality. (One of the methods was neatly named "the hovering principle.") On the other hand, some of the methods contained expressions that were difficult to understand and became obstacles which were hindrances to the succeeding development. However, it is fortunate that Oka's theorems have been widely
accepted today as a lucid fundamental theory, due to H. Cartan's formulation by virtue of cohomology with coefficients in sheaves, and H. Grauert and L. Hormander's application of methods of functional analysis. X iii
xiv
SUMMARY AND PROSPECTS OF THE THEORY
analysis
Theory of analytic functions of several variables
Based on Oka's work, the theory of analytic functions of several variables has developed in a variety of directions. Oka himself suggested how groups of problems should be located by using the schema above.
Borrowing this schema of his, the subjects raised in this book mostly belong to the intersection of the theory of analytic functions of several variables with mathematical analysis. The book hardly touches the other four parts, as noted in the Preface. As for mathematical analysis, what comes to mind first as an object is those functions dependent on space and time variables that satisfy some sort of differential equations. In order to describe properties of these functions and determine their exact formulae, we need to analyze the distribution of values taken on by these functions and their behavior at points at infinity or singular points, as is often experienced in solving even elementary problems. This methodology forms a complete theory by restricting the range of functions under
consideration to analytic ones. This is exactly the theory that was founded by Weierstrass and others within the theory of complex functions of one variable, and the basis for this methodology consists of the
Mittag-Lefer theorem, the Weierstrass product theorem, the Runge
SUMMARY AND PROSPECTS OF THE THEORY
xv
approximation theorem, and so forth. The Oka theory has had this basis transplanted into the soil of the theory of analytic functions of several variables. On this account, the purpose of this book is to tell the detailed story about how the basis of one variable took root in the soil of several variables, while deferring the sight of what trees grew and blossomed on it. In what follows, the precise contents of the book are explained. Chapter 1 gives the definition and fundamental properties of holomorphic functions. P. Montel's theorem and Weierstrass' double se-
ries theorem guarantee that the space of holomorphic functions is closed with respect to the topology induced by uniform convergence on compact sets. This is similar to the completeness of the real numbers, and is quite fundamental in deriving existence theorems for holomorphic functions. From thorough investigation on the structure of the space of holomorphic functions by methods of real analysis, one can derive the fundamental existence theorem. This is grounded on the characterization of holomorphic functions as weak solutions of the Cauchy-Riemann equations in the sense of distributions. Namely, this allows the space of holomorphic functions to be identified with a closed subspace in the space of locally square integrable functions, where functional analytic methods are applicable. As to individual problems, I. M. Gel'fand, A. Grothendieck and others pointed out that geometric problems of specific spaces can be interpreted as algebraic problems of the function rings on these spaces,
and the latter formulation offers an unbroken vista to approach the problems. This relies on the general tendency that the ring structure can be studied well through its ring extensions. From this point of view, the fundamental problem of the ring of holomorphic functions concerns the relation between the set of all the maximal closed ideals of the ring and the domain of definition for the functions. More precisely, the question remains whether 1 belongs to a closed ideal generated by a system of holomorphic functions which have no common zero point, and also whether one can construct a holomorphic function which takes values determined beforehand over a given discrete set. In Chapter 2, these problems are converted into the Cauchy-Riemann
equations of inhomogeneous form, or the so-called a equation. It is too early to treat the a equation to the fullest, but we introduce the concept of '5 cohomology group and make an elementary observation on conditions to solve the above problems. This observation results
xvi
SUMMARY AND PROSPECTS OF THE THEORY
in a necessary condition that the function ring must contain a function which cannot holomorphically extend beyond a given boundary point of the domain of definition. An open set that has this property is called a domain of holomorphy, and Chapter 3 connects this to a certain geometric concept called pseudoconvexity. The definition and fundamental properties of plurisubharmonic functions, a generalization of subharmonic functions to several variables, are described. This is done by Oka in a 1942 paper based on Hartogs' results. In addition, we prove basic results on the regularization of plurisubharmonic functions, as differentiable plurisubharmonic functions become important later. The approximation theorem due to J.-P. Demailly belongs to the same family of results and hence is introduced here. However, this is rather a deep result, and the proof of the theorem is not elementary and thus is postponed until Chapter 5. We also introduce the concept of Hartogs function, due to Bochner and Martin. This enables us to feel as if we were observing the development of the Levi problem around 1948. In case an open set in Cn has a smooth boundary, the pseudoconvexity implies some property of the boundary as a real hypersurface. This is what is called the condition of Levi pseudoconvexity. Open sets of this sort enable one to argue minutely about the boundary behavior of holomorphic functions and mappings. This is a subject of Chapter 6, but Chapter 3 also prepares for this subject. _ Chapter 4 explains a new methodology for L' estimates of the a operator. We begin with basics of closed operators on a Hilbert space, establish estimates involving the a operator and its adjoint operator on the completion of the space of differential forms with respect to the L2 norm with some weight function, and from these estimates derive the existence theorem on solutions of the a equation in Theorem 4.11. Further, we apply the theorem and generalize the M Iittag-Leffier theorem, the Weierstrass product theorem, and the Runge approximation theorem to several variables. The thread of our argument itself is not different at all from that of Hormander's book [27), the standard textbook in the theory of analytic functions of several variables. But it is worth emphasizing that what is different between our approach and his is the vehicle in which we are traveling, in spite of the same path. That is to say', the calculations that provide the foundation of the arguments in this book are a re-formation of S. Nakano's formulae used for the proof of the vanishing theorem of cohomology in
SUMMARY AND PROSPECTS OF THE THEORY
xvii
the theory of complex manifolds, and these calculations contain new estimates. Differing from Hormander's, in the background of our estimates, there is W. V. D. Hodge's book [24] that discusses the theory of harmonic integrals on projective algebraic manifolds, grounded on the symmetry of complex Laplacians with respect to the complex conjugate. It would be of no use if the new vehicle were cheap, but in Chapter 5 by virtue of this new method we show in Theorem 5.10 that holomorphic functions defined on a closed subspace can be extended under an estimate given by the L2 norm with some weight function. This estimate is what Hormander's method could not reach, and is
the main theme of this book. The calculations of Chapter 4 were, in fact, designed by the author together with Kensho Takegoshi in order to prove this extension theorem. (Refer to [37].) The proof of Demailly's approximation theorem is an application of the extension theorem. Although we cannot treat this topic in this book, the extension theorem has recently shown to have some applicability to some subtle problems in algebraic geometry and the theory of complex manifolds. (Refer to [44], [46], and [47].) Chapter 5 touches on H. Skoda's division theorem. With this, all the problems posed in Chapter 2 have been solved. We do not give full details of the proof of Skoda's theorem, but only introduce the essential part of the argument. The author would put an emphasis also on this point as a special feature of this book which is not found in any other books.
We throw in Chapter 6 as an extra, "just for fun". It contains a lighthearted approach to the difficult open problem of determining the singularity of the Bergman kernel, which is a reproducing kernel of the space of L2 holomorphic functions. After describing C. Fefferman and Bell-Ligocka's example of applying the Bergman kernel to holomorphic mappings, we show our own recent results about the Bergman kernel on a general Levi pseudoconvex domain. This might seem too scanty and miserable, but may rouse those who want to do some research in this field from this point forward. That is why the author did not shrink from "cutting a sorry figure".
CHAPTER 1
Holomorphic Functions To begin with, we define holomorphic functions as convergent power series, describe elementary properties of them, and achieve the main goal of characterizing them as weak solutions of Cauchy-Riemann equations. Our proof restricts itself to locally square integrable functions, while the concept of holomorphy of weak solutions is known to be extendible as far as hyperfunctions. This choice is made so as to keep our argument as simple as possible. At the end, we mention the Reinhardt domain, finding it important to study some properties possessed by domains of convergence for power series.
1.1. Definitions and Elementary Properties Let C be the complex plane and consider the n-dimensional comn
x C. Let z = (zl, , zn) be the plex number space Cn := C x coordinate system of Cn. Essentially z is a vector valued function on Cn, but we also denote a point of Cn by the same symbol z as long as there is no fear of confusion. Write x2j _ 1 and x2j for the real part Re zj and the imaginary part Im zj of zj, respectively. Let R be the real line, and identify Cn with R2n by the correspondence (z1,... ,zn) ~'
(x1,x2,... ax2n)
For z E Cn, set and IzI = (1z112 + + Jzn12)' i These are norms on C' that are topologically equivalent to each other. Let Z4. be the set of all nonnegative integers, and for an element a= (al,- an) in Z n' set IzImax = max Izj1
n
a!
f aj! ,
n
(a)
j=1
z°
z1°1...zn°^.
> aj, j=1
1. HOLOMORPHIC FUNCTIONS
2
Furthermore, we use the following notation: 1( a a
Va
2
az2
1(
a
azj
Also, f o r $ = (Q1.
2
, )32n)
(9X- 2i _ 1
ax2i
a + 8x2j-1
a ax2j
E Z+1, let ;31
(aan)$2"
From now on, let (1 denote a nonempty open set in C. DEFINITION 1.1. A complex valued function f on (1 is holomorphic if for each point a of P. there exists a power series
P. (z) := E c. (z - a)° with c" E C aEZ+
that converges to f on some neighborhood of a, where the convergence
is regarded with respect to some linear order of Z+, or a bijection a(k) E Z+. f is called antiholomorphic if the complex Z+ E) k conjugate of f, z'-- f(z), is holomorphic. Let A(R) denote the set of all holomorphic functions on R. First, let us describe briefly the most elementary properties of holomorphic functions: 1. A(Q) is a C algebra with the usual four rules of arithmetic. (This is obvious from the definition.) 2. Holomorphic functions are of class C'°. In fact, (2) is included in the following proposition: PROPOSITION 1.2. If a power series
E b"(z - a)" with 6" E C "EZ+ , a point a' = (a...... aft) around a point a = (a1, with a' # aj (1 < Vj 2, clearly U(n) is not equal to the semidirect product of U(1)1
and Can. From this, it is seen that in general On and 3n are not holomorphically equivalent to each other'. Aut 13n acts on l3n transitively. To see this, it is sufficient to note that l3n is holomorphically equivalent to the open set n
Dn
c=( 1,...,(n)ECn Im(I>EICi12 j=2
under the Cayley transformation: z1
_ (I - -,/--1
2(2
2(n
(1+V-122
1In general, the group of holomorphic automorphisms on a bounded domain is known to be a Lie group with respect to the compact open topology (see 1331).
I.I. DEFINITIONS AND ELEMENTARY PROPERTIES
5
In fact, the transitivity of Aut B' is obvious from the facts that U(n) C Aut B" and Aut Dn contains the transformations:
fort>0, ((1 +t,(2i ,(n) fort ER. w
One of the holomorphic automorphisms on Bn that map a point (0, , 0) to the origin is given by the formula
1 --I w (z - (iwl2) w) - w +
(Iwl2) w
1 - (z, w)
'
n
where (z, w)
E zjwj.
j=1
It is an interesting task to calculate the groups of holomorphic automorphisms on various bounded domains, but we do not study more than these examples in this book. Let us return to the exposition of elementary properties of holomorphic functions. We wish to describe topological properties of the ring A(.R) of holomorphic functions. For this purpose, it is necessary to clarify the range in which the Taylor expansion of a holomorphic function on !l is convergent. An immediate consequence from Proposition 1.2 is that the sum of a power series that converges on 0(a, r) is a holomorphic function on A (a, r). The converse statement to this does not hold for real analytic functions in general, and this difference between holomorphic and real analytic functions is fundamental.
PROPOSITION 1.4. A holomorphic function on A(a, r) is equal to the sum of a power series whose convergence holds on the same domain 0(a, r).
PROOF. It is enough to show the proposition for 0(a, r) = An. Let
f (z) = E caza with c,, =
f
(0)
a! be the power series expansion of an element f in A(A") at the origin. n
Since there exists a positive number a such that the power series converges for IzImax < c, cQ is expressed by Zw
(1.1)
cQ =
2a
I f (re`B' , ... , re'8^) r-(a)
e-i(a,e)
d9,
1. HOLOMORPHIC FUNCTIONS
6
where r is an arbitrary real number with 0 < r < e, and we set 0:= (01, - ,B") and However, since f is holomorphic on 0", it follows that the right hand side of (1.1) takes on finite determinate values for all r with 0 < r < 1, and is of class C" with respect to r. Hence, (1.1) holds for all r with 0 < r < 1. From this we obtain the estimate of cQ: Ic0I u
/
u in
l a9 J
The definition of I 7
I
u is a natural generalization of the con-
cept of the usual derivative, since t
I
h=
ax 19 ) element h in C°°(Il), by integration by parts.
(ax) 3t(h) for an
The following theorem provides a characterization of holomorphic functions. THEOREM 1.10 (Theorem of L2 Holomorphy).
A(.R) = f E L1 °(fl)
a = 0 for j = 1, a j
n
.
us review, below, complex differential forms in order to prepare some calculations that are used in the proof of this theorem. A differential form of degree r (or r-form) on !l (C C) can be written as
E uIJ dzj Adz J, where ujj = sgn
IJ
I1
I sgn ( j1 ) uj,j,
for multi-indices I = (i1, ... , ik) and J = (ii,... , j!) with k + I = r whose components are taken from the natural numbers from 1 to n, and dzl = dz;, A ... A dz;k ,
dzJ=dzj,A...Adzj,. A differential form is often written as
u = F,'uIJdzj AdzJ,
I,J in which the multi-indices I and J are only allowed to have strictly increasing components. In this case, u is said to be a differential form of type (p, q), or simply a (p, q) form, if the lengths k and 1 of I and J are equal to the given constants p and q, respectively. Let C(")(fl) be the set of all r-forms of class C°° on fl, and let Cp,9(fl) be the set of all (p, q)-forms of class C°° on 12. In general,
1.2. CAUCHY-RIEMANN EQUATIONS
11
let C°(K) be the set of all complex valued functions of class C'° on a subset K of C". When K = !l, we define CM(72) C CM(Q) and C(r) (.R) C C(') (,f1) in a similar fashion. Let us give an example of a calculation in which differential forms are effectively used.
Given a holomorphic mapping F = (f1,
,
fn) :.fl --> C", we
have
F' (dz1 A ... A dzn A dz 1 A ... A dzn ) 2
ZIn...AdznAdz1n...ndzn det(af') \azkjk From this, the Jacobian of F with respect to the real coordinates 2
(x1. x2, ... . X2n) is equal to det af3 \ a`k
7,k
The complex exterior derivative operators
a : Cp,q(Q) _ Cp+1,q(l?) of type (1, 0) and a : CP,q(Q) .4 Cp.q+I(Q)
of type (0,(El 1) are defined respectively by
AdzJl = F_': aulJdzk Adzj Ad4J, aulJdzl Id k azk Id \/
(>'u1dzJ A dJ I Id /
>' ° ! iiA dzl n dj. I,J k
k
From this definition, it is obvious that the ordinary exterior derivative d is equal to a + D. Let L ,q(17) denote the set of all differential forms of type (p, q) whose coefficients ujj are elements in L2 C(0), and let LP?9(j?) be the set of those whose coefficients are in L2(fl). L1oC(!2) and L(r)(Q) are defined similarly. Also, Cop, q(12) expresses the subset of CP.q(.R) whose elements have compact supports, and C0(') (f2) the subset of C(r) (,f1) whose elements have compact supports. Using this notation, through the involution L(r)(f2) X
C012n-r)(n)
-+
C W
W
(u, v)
Jo
uAv,
1. HOLOMORPHIC FUNCTIONS
12
Co(2n-T) (fl )', and similarly L joy (fl) is identified with a subspace of L o'C(fl) with a subspace of Co -p'n-Q(fl)'. Accordingly, the domains of definition for the exterior derivative operators d, 0, and a extend
to &)(fl) or L oC(fl). The proof of Theorem 1.10 needs the mean-value property for differentiable solutions of the Cauchy-Riemann equations.
PROPOSITION 1.11. Assume that f E C1(fl) and 8f = 0. 1. For an arbitrary n-open ball B(a, R) C= 12, it follows that 1
(1.4)
Vol(OB(a, R))
J
f dS = f (a), B(Q, R)
where dS denotes the volume element of 8B(a, R) induced by the Euclidean metric, and we set Vol(818(a, R))
LB0. R)
dS = (n - 1)!
2. f E A(fl).
PROOF OF (1). Set g(z) := f (Rz+a) -f (a); then (1.4) is equivalent to
r
JOB" gdS=0.
(1.5)
Now that the restriction of a (2n -1)-form Slog I z I A
(
n-1 A 08 log IzI
to 8En is not equal to 0 but unitarily invariant, (1.5) is equivalent to Jean g(z) 8log Izl A
(
A109
log IzI I = 0.
In general, for a function u of class C1 onStokes' formula implies n-1
r
(1.6)
/eBn u(z) 8log Izl A
J +
-
A aalog Izl
d{u(z)alogIzIA ("A'oThogIzI)} ^ \B^ (0, e )
f
as^(o,e>
n-1
u(z) 8log IzI A
A 070 log IzI
,
1.2. CAUCHY-RIEMANN EQUATIONS
13
for 0 < e < 1. In the case u = g, since the first term of the right hand side is 0 by the condition ag = 0, it follows that n1 alogIzl n A aalogIzI s n Jag(z)
f
8B" (0, e)
/n 1 g(z)alogIzI A ( A Ca logIzl
\
.
Therefore, we obtain (1.5) by letting e \ 0, since g(0) = 0.
REMARK. When n = 1, in the above argument, take g(z) = 1, and replace z by an element f in C1 (a) fl Ker a that has no zero point on aA. A similar calculation provides Argument Principle:
JUlogIf(z)l
=o
where n f denotes the sum of the orders of zeros of f in A.
Before proceeding to the proof of (2), we need to prepare the following proposition:
PROPOSITION 1.12. Under the same assumption as in Proposition 1.11, (1.7)
1
Vol(BB(a, R)) JB(a, R)
where
f dV = f(a), 7nR2n
Vol(B(a, R)) _ n!
PROOF. This is due to (1.4) and Fubini's theorem.
Apply the Cauchy-Schwarz inequality to the left hand side of (1.7); then we obtain (1.8)
If(a)12 1.
PROOF. Take a neighborhood U of On that provides a repre-
sentative v E C°. (U) fl Ker a of an arbitrary element in H°() (q > 1). Set v
=
L,' Vrdzn A dzI + >' vjdz j
IT,
Jon Ill=9-1 IJI=9 with VI', v'j E CO°(U),
2. FUNCTION RINGS AND 8 COHOMMOLOGY
30
where III denotes the length of the multi-index I, and I that I does not contain n.
n means
From Lemma 2.8 and the succeeding remark, there is an element u11i in C°° (Cn) satisfying
NO)
(2.9)
r
a -z"
= vi
on some neighborhood of On. By setting UM
In
u1lidz1,
(II=q-1
from (2.9) v is transformed into the following form that does not contain dzn any more: v - Du(1)
w'1dzn-1 A dz1 +
>'
1 on,n-1
Jj1n,n-1
11I=q-1
IJI=q
ZUJdwJ,
where w'1 and w; on the right hand side are holomorphic with respect to Zn.
Therefore, we can take, this time, a C°° function u (2) that satisfies
a-Zn -1
tv'1 and is holomorphic with respect to zn. Set u(2)
then v-auili -aui2i contains neither dzn nor dzn_1. f3Jn.n-1 I1I=q-1
In this way, if we keep producing new forms starting with v, then eventually the form reaches the (0, q)-form that does not contain any of dzn,
,
(w).
dzl, or the form 0 at which v = d E
j=1
COROLLARY 2.10. f E C°°(0) if both f E Li C(fl) and of E
C0.11Q). PROOF.
Since the equation au = of for an unknown function u has locally a C°° solution, f is holomorphic modulo C° functions. Hence, in particular, f E C'(17). Let us show one consequence of Serre's criterion. PROPOSITION 2.11. If HO,q(Q) = {0} for 1 < q
n - 1 and if
an is a real hypersurface of class C', then for every boundary point
2.3. 8 COHOMOLOGY AND SERRE'S CONDITION
31
zo of 12, there exists an element f in A(f2) that satisfies
lim If(z)I =oo. zU
PROOF. Let L be a complex line that intersects transversely with
8.2 at zo. Take a holomorphic function on L \ {zo} that has a pole at zo, and extend it to .f2. Then this holomorphic extension f will satisfy the above condition. 0 This proposition naturally poses the problem of relating the vanishing of a cohomology groups on .fl with such a geometric condition as the logarithmic convexity of complete Reinhardt domains.3 For brevity, we call the condition
H°'9 (Q)=0 for all 1 C°° (.R) by it
a(vi ... v,,) _ E zjvj. j=1
Let. {e1,
,
en} be the standard basis of V. In terms of a. define
k k-i contractions ak : AV -i AV pointwise by k
1.7
ak(eil A ... A eik) = L (-1)3a(ei, )ei1 A ... Aeik j=1
and by linearity. Then the following exact sequence is obtained:
0-,nV an, n1Vck- ...
AV
V
3This will be described in detail starting in the next chapter.
2. FUNCTION RINGS AND $ COHOMOLOGY
32
If we extend the range of definition for the operator 8 to vector-valued differential forms, noting that ak commutes with the operation of 8,
then 8 cohomology groups HM(Ker ak) with coefficients in Kerak are defined by HP,q(Kerak) := (CM(0) ® Ker ak) fl Ker B/8 (Cp,q-1(12) (9 Ker ak) .
4
Then the same argument as for (2.2) induces that if H°,k+I (Ker ak+1) _ {0}, then the homomorphism derived from ak+I, HO,k(H)®(k+1)
_, H°A(Kerak) ,
becomes surjective. Therefore, it follows that
Im(aI A(.fl)®")=A(,fl) H°"1(Ker a) = {0} HO-1 (0) = {0} and
{0}
H°, I (f2) = H°,2 (f2) =
= HO,n-1(.12) = {0}
and H°""(Keran) = {0}. Since Ker a" = {0}, the proof is complete.
0
PROOF OF COROLLARY 2.13. If a sequence of points {z(µ)}°O µ=1
C 11 is a convergent sequence with respect to the weak topology, then the z(µ) are clearly a bounded sequence and must converge to some point inside .fl. Otherwise, as there is a subsequence z(µk) that converges to some point of 8,11, by choosing this point as a and taking gi , , g" produced in Theorem 2.12, it would follow that
lim F,
k-oc 1=1
Ig;(z(µk))I
= 00,
which contradicts the fact that the z(µ) form a weakly convergent sequence.
The above argument tells us that the smoothness assumption on 817 in Proposition 2.11 is redundant. Let us emphasize this fact because of its importance: 4The tensor products on the right hand side are regarded as COO (g)-modules.
2.3. 5 COHOMOLOGY AND SERRE'S CONDITION
33
THEOREM 2.14. If Serre's condition holds for fl. then for every zo E OR and every sequence {pµ} of points in fl that converges to zo, there exists f E A(f2) such that limoI f(p,,)I = 00.
In general, an open set fl of C' is said to be a domain of holomorphy5 if there is no connected open set U that possesses the following property:
!2, and there is a nonempty open subset V of f2fU such that for any element f in A(Q), there is an element g in A(U) that satisfies f I V = g I V. Let us leave to the reader the verification that logarithmically convex Reinhardt domains, convex domains, and open sets in the complex plane are all domains of holomorphy. By virtue of this terminology, it follows from Theorem 2.14 that (2.10)
U
every open set satisfying Serre's condition is a domain of holomorphy. The converse statement is also true, but the proof of it needs further preparation, and we postpone the details until the next chapter. Note that the following fact is derived from this converse statement, which we accept for the present. PROPOSITION 2.15. A necessary and sufficient condition for f2 to be a domain of holomorphy is that for any zo E 8f2 and any sequence {pµ} of points in f2 that converges to zo, there exists f E A(f2) such that slim If(p,.)I = oo. -CC
What we wanted to make clear in this chapter is that a region of definition f2 of holomorphic functions must be a domain of holomorphy in order to solve affirmatively the extension and division problems. Later on, we will indeed investigate whether these problems can be solved on domains of holomorphy. In deference to Kiyoshi Oka's methods, we will first characterize domains of holomorphy by the concept of pseudoconvexity, and then treat solutions of all the problems on domains of holomorphy as consequences of pseudoconvexity. Our lines are in imitation of what has been done repeatedly in mathematics, such as the reconstruction of Euclid's theory by means of Descartes' methods. 5Connectedness is not imposed on the definition of "domain" of holomorphy.
CHAPTER 3
Pseudoconvexity and Plurisubharmonic Functions As seen in the preceding chapter, f2 must be a domain of holomorphy
in order that the propositions corresponding to the Euclidean algorithm on the ring of integers and to Lagrange's interpolation on the ring of polynomials of one variable hold on the ring A(f2) of holomorphic functions. On the other hand, these propositions are closely related to Serre's condition, the vanishing of a cohomology on f2, and this condition provides a unified grasp of various phenomena on a domain of holomorphy. a cohomology is essentially under control of the pseudoconvexity of an open set. Pseudoconvexity is a concept similar to geometric convexity, but is much weaker as a condition. This chapter begins with the definition of Hartogs pseudoconvexity and verifies that a domain of holomorphy is Hartogs pseudoconvex. Secondly, we show that some canonical function on a Hartogs pseudoconvex open set expressed by a distance function is plurisubharmonic. In consequence of this, it is derived that a domain of holomorphy is pseudoconvex. Hartogs and Oka's discovery of this relevancy gives a unique vitality to the theory of analytic functions of several variables. In Chapter 4, in order to show that a pseudoconvex open set is a domain of holomorphy, some kind of differentiable plurisubharmonic functions will be needed. For this purpose. we detail the regularization of plurisubharmonic functions in the present chapter. Finally, we mention the Levi pseudoconvexity and introduce basic
facts and important examples of pseudoconvex open sets that have smooth boundaries. This also serves as an introduction to Chapter 6.
3.1. Pseudoconvexity of Domains of Holomorphy A handhold is a domain
T={ (z l, Z2) E A 2
I
I z l I< E or 1- c.< I Z21 < 1}. 35
3. PSEUDOCONVEXITY AND PLURISUBHARMONIC FUNCTIONS
36
which is called a Hartogs figure.
DEFINITION 3.1. fl is said to be pseudoconvex in the sense of Hartogs, or, for short, Hartogs pseudoconvex, if any holomorphic map-
ping from a Hartogs figure TE with a arbitrary to !l always extends to a holomorphic mapping from 02 to P. THEOREM 3.2. C is Hartogs pseudoconvex.
PROOF. It suffices to show that the restriction mapping A(02) A(TE) is surjective. Let f E A(TE). From Proposition 1.4, f(z) expands into the power series k t
Ck,lz 1 z 2 k.1
which is convergent on 02(0, (e,1)). After changing the order of the summation, let us observe the range on which the following equation holds:
f (z) _ >
(Eckzz)
Z
.
By setting Ck(Z2) := E ck,tz2 for Iz21 < 1, we regard the right hand
side as a series with terms of holomorphic functions ck(z2)zi. Since the left hand side is holomorphic on A x (Z2 I 1 - e < I Z21 < 11, from the same argument as in (1.2) it follows that when 1 - e < Iz2I < 1, (3.1)
Ick(z2)I < sup If (zI, z2)Ir-k for 0 < r < 1. Izi I=r
Hence, from the maximum principle, when Iz21 < r < 1, (3.2)
ICk(z2)I :-5
SUP
If (Z1, Z2)Ir-k
121 I=Iz21=r
a
Therefore, the series E ck (z2) zi converges uniformly on compact sets k=1
in 02, and Theorem 1.6 says that this series is a holomorphic exten-
sion off to 02
0
COROLLARY 3.3. C" is Hartogs pseudoconvex.
THEOREM 3.4. A domain of holomorphy is Hartogs pseudoconvex.
PROOF. Let f be a holomorphic mapping from TE to P. From Corollary 3.3, f has a holomorphic extension f : 02 _ C. If it were
3.1. PSEUDOCONVEXITY OF DOMAINS OF HOLOMORPHY
37
true that f (A2)
fl, then, letting T. be the maximum among connected open sets U that contains T. and satisfies f (U) C fl, it would follow that 8TE n A2 54 0. The maximum property of TE implies f (p) E 89 for a given point p E 8T£ fl A2. _Hence, from Proposition 2.15, choosing a sequence of points p in TE fl A2 that converges to p, there exists an element g in A(fl) such that lim 1g(f(pµ))J = oo.
(3.3)
This contradicts the fact that g o f has a holomorphic extension to O
2
COROLLARY 3.5. The following are all Hartogs pseudoconvex open sets: 1. Logarithmically convex Reinhardt domains. 2. Convex domains. 3. Open sets in the complex plane.
on 1fl defined by Next, observe arfunction 6'(z) 12
6n(z):=inf{iS z+II ¢fl}, where vEC'1\{0}, 1
in order to relate Hartogs pseudoconvexity to a metric character. From the definition, 6n(z) is lower semicontinuous as a function from f2 to (0, oo]. In addition to this, the following remarkable property of 6a1' (z) emerges if 12 is Hartogs pseudoconvex.
THEOREM 3.6 (F. Hartogs, K. Oka). If fl is Hartogs pseudocon-
vex, then for z E fl, v E Cn \ {0}, w E C", and r > 0 that satisfy
{z+tesew10
- r'Ia0(z, r) then
,
3.1. PSEUDOCONVEXITY OF DOMAINS OF HOLOMORPHY
39
for z' E A(z, r).
Re h(z') >_
When these conditions are met. tr} is called a subharmonic function on
V. When 0(z, r) C .R, given a subharmonic function i on !1, define a function AI(v, t) by 1
AI(t t) :=
2
f
2n
t (z + te`B)dO.
Then AI (i', t) is monotone increasing on t. This is obtained from (3.9) and the mean-value property of harmonic functions. We can take this condition on AI( , t) as the definition of subharmonic function. The next four properties follow immediately from the definition of subharmonic function: (3.10) If Y and 2' are subharmonic, so are w + and aY (a >- 0). (3.11) For a family {V.,,\}A of subharmonic functions that are bounded from above uniformly on compact sets in Q. set th := sup v'a A
Then tb' is subharmonic on
and r/,*(z) := lieu sup e-.0
z'ESC(z._)
Si.
(3.12) A real-valued function 0 of class C2 on Si is subharmonic if and only if the following differential inequality holds everywhere: a2 i.
a:ati
'-°
(3.13) If a sequence {j} I of subharmonic functions on I? is monotone decreasing, then lim bj is subharmonic. y-.00
A fact that is immediate from (3.12) and often applied is that if V is subharmonic for any is subharmonic and of class C2. then increasing convex function A of class C2 on R. Combining this with (3.13), for instance, it is easily seen that log E I f j 12 and Ul I fj I2j=1
(a ? 0) are subharmonic on ft for f1.... f,,, E A(Q). Let us return to the topics on C". The formula (3.5) indicates the subharmonicity of - log bv(z + cup) with respect to (. Namely, - logJv is subharmonic on L fl Si for any complex line L. DEFINITION 3.7. An upper semicontinuous function
!':.R - (-oc, oc)
40
3. PSEUDOCONVEXITY AND PLURISUBHARMONIC FUNCTIONS
is said to be plurisubharmonic if O(z + (w) is subharmonic as a function of ( for a given (z, w) E f1 x C".
Denote by PSH(fl) the set of all plurisubharmonic functions on fl. For simplicity, f1 may be omitted from this notation. The basics of plurisubharmonicity are totally the same as in the case of subharmonic functions, but we list them for convenience:
(3.14) If pand0areinPSH(Q),soarecp+0 andacp(a>0). (3.15) If 0 E PSH(fl), then, given z E fl, 111(z), t) ==
Vol 8I8(z, t) J8B(z,t)
dS
is monotone increasing on t with B(z, t) C 17. (3.16) If {z'A}AEA C PSH(.fl) and if t?', are bounded above uniformly
on compact sets in fl, then V)`(z) = lim
sup
e-0 z'EB(z,e)
sup 7/iA(z') E PSH(Q). A
(3.17) Given a real-valued function , of class C2 on fl, 0 E PSH(12) if and only if the n x n matrix 8zj8zk Ca20) is semipositive definite everywhere on Q.
(3.18) If a sequence of functions {Oj}'?O I C PSH(fl) is monotone decreasing, then lim 1' E PSH(Q). From Theorem 3.6 and (3.16), it is seen that fl is Hartogs pseudoconvex
- log bn E PSH(Q),
where bn(z) := inf Iz - wI. Unlike bn, bn is a continuous function wE8l1
that has some finite determinate values on .fl as long as fl 0 C". DEFINITION 3.8. Q is said to be pseudoconvex if there exists a continuous plurisubharmonic function Eli: fl - R such that the set fly,,c :_ {z E fl 1 '(z) < c}
is relatively compact inside 9 for each c E R. It is clear from the definition and either (3.14) or (3.16) that if fll and f22 are pseudoconvex, then so is S11 fl f12. Pseudoconvexity and Hartogs pseudoconvexity are equivalent, but before giving the proof,
3.2. REGULARIZATION OF PLURISUBHARMONIC FUNCTIONS
41
we will describe the regularization of plurisubharmonic functions in the next section.
REMARK. From the proof of Theorem 3.6, it turns out that if Il is not pseudoconvex, then there exists a biholomorphic mapping t from 0" onto an open set U C C" such that
t(TE x 0"-2) C fl and U 0 .R. This means, by Theorem 3.2, that all elements in A(V) extend to U as holomorphic functions. But the value of the analytic continuation of a holomorphic function does not necessarily coincide with the value of the original function at a point in 12nU except c(TE x An-2). Therefore, in order to discuss, in general, the global theory of holomorphic functions, it would be insufficient to restrict the domains of functions to open sets in C. However, as long as we stay only within the theory explained in this book, the replacement of .fl with a more general space such as a complex manifold will not affect the frame-
work of the theory (although this replacement would put things in much wider perspective). Hence, in order to give priority to brevity, we will be restricting ourselves to open subsets of C" from now on, too.
3.2. Regularization of Plurisubharmonic Functions The existence of differentiable plurisubharmonic functions is important, as we will need to differentiate formulae that contain plurisubharmonic functions later on to solve the 8 equation on pseudoconvex open sets. As preparation for this, we describe below the regularization of plurisubharmonic functions. Let .RE and µE be as defined after (1.9) in § 1.2. Let be a plurisubharmonic function that is locally integrable on fl, and set
E(z) := f 'Y(z + ()µe(()dV( for z E
f?
.
^
Clearly, & E C°°(QE). Furthermore, PROPOSITION 3.9. WE E
'NE
has the following property:
PSH(!lE), and pE \ 0 as e \ 0.
PROOF. The monotonicity of the family of functions ,0E follows Y'E is shown as
from that of Ill (i , t). The plurisubharmonicity of
42
3. PSEUDOCONVEXITY AND PLURISUBHARMONIC FUNCTIONS
follows: 1
2.
27r 1
J0
=
te(z + re'Bw)dO
r ;ii (()dV0(z + re0w - e()dO J f27T e()dV(
where r is chosen to be sufficiently small. COROLLARY 3.10. Let V) E PSH(f1). If A is an increasing convex
function (in the broad sense) defined on an interval that contains the range of V;, then A(0) E PSH(fl). PROOF. There is a family of increasing convex functions Au of class C''° with A. \ A. For this family. (3.17) implies a,1(O5) E PSH(fQE) by direct differentiation, while since \,,(V,) \ A(v), (3.18) yields A(t,1') E PSH(fl).
COROLLARY 3.11. Let f11 C Cm and f12 C C" be open sets, and
let F: f11 - f12 be a holomorphic mapping.
Then, for any t' E
PSH(f22). one has that v o F E PSH(f11). PROOF. If ,IL E C2 (02) n PSH, then tai o F E PSH(f11) follows from taking its derivatives. For the general case, use an approximate family as in Corollary 3.10. THEOREM 3.12. Hartogs pseudoconvexity and pseudoconvexity are equivalent to each other.
Pseudoconvexity: As PROOF. Hartogs pseudoconvexity take I z 12 for the case f1 = C', and 1z12 - log 5n otherwise. Hartogs pseudoconvexity: From the propPsuedoconvexity erty (3.15) of plurisubharmonic functions and Corollary 3.11, it suffices to repeat an argument similar to the proof of Theorem 3.4. COROLLARY 3.13. For an increasing sequence {flk}k I of pseu-
x
doconvex open sets, U flk is pseudoconvex. k=1
If plurisubharmonic functions are continuous, a stronger approximation theorem holds. To describe this, let us introduce the concept of strictly plurisubharmonic function.
3.2. REGULARIZATION OF PLURISUBHARMMONIC FUNCTIONS
43
In general, given a locally integrable function cp on 12 and z° E Q. let L[yp](zo) (E [-oo. oo]) be the supremum of real numbers E such that V(z) -EIzl2 is plurisubharmonic on some neighborhood of z°. Clearly, L[V + 01 > L[4o] + L[O] and
L[max(ip,11b)] >-
L[V;])
It is also obvious that L[V] is lower semicontinuous.
If L[ i J(zo) > 0, 0 is said to be strictly plurisubharmonic at zo, and if L[y,] > 0 on fl, 4' is called a strictly plurisubharmonic function on f2.' Denote by PSH*(f2) the set of all strictly plurisubharmonic functions on fl. THEOREM 3.14 (Richberg's theorem). If 4 E PSH*(f1) n C1 (fl)? then for any positive-valued continuous function E on fl, there exists a function V E PSH*(fl) n C' (0) that satisfies the inequalities
0
2 3L['U]
1
'In the literature, the condition "of class C2" is also frequently included.
3. PSEUDOCONVEXITY AND PLURISUBHARMONIC FUNCTIONS
44
on
IR
(pi, R1). Hence, given a positive number 5, set 1z - R2112
1P a := 06 - 771,\ I
then, for a sufficiently small 6,
L[ib] > L[ei] ands < -0 + E
(3.23)
3
on B(pj, R1). In addition, we have
b
k+1
and V)(k+1) < V)+ s j=1
on B(pk+1,Rk+1), and furthermore a
(k) < P(k)
(3.29)
on some neighborhood of 8B(pk+1, Rk+1). In this case, (k+1) coincides with 'tl k) on some neighborhood of .fl \ 18(pk+1, Rk+1). Also, since
>on B
t
Rk+1 1 J , it follows that z/i(k+1) coincides 2
with (1-Xk)O(k)+Xklk(k) and is of class C°° on this open ball. Hence, all the requirements are met. k
From the construction, we have
Therefore, cp :=
0(k) on U
m 1 (k) exists, satisfying i&
k 00 cp E PSH* n COO (17).
j=1
lE$
rpj,
l
2
l
< cp < 0 + c and 0
In general, a real-valued continuous function cp on a topological space X is said to be an exhaustion function on X if the subset {x I 'p(x) < c} of X is relatively compact for every real number c that is less than the supremum of the values of 'p. By virtue of R.ichberg's theorem, the definition of pseudoconvexity can be strengthened as follows:
THEOREM 3.15. A pseudoconvex open set has an unbounded strictly plurisubharmonic exhaustion function of class C°°. In recent years a remarkable result on the approximation of plurisubharmonic functions has been obtained. For the present, let us introduce the statement of this result while putting off its proof. Let V) be a plurisubharmonic function on Q. The Lelong number v(0, x0) of i at a point x0 of .R is defined by sup V) = lim v(7/,, xo) := liminf r\0 logr z-.x0 log Iz - xol
Recall that log If I E PSH(Q) for f E A(Q). It is easy to see that v(log If I, xo) = sup{k E Z+ I (a) < k implies f (a) (xo) = 0}.
46
3. PSEUDOCONVEXITY AND PLURISUBHARMONIC FUNCTIONS
For a locally integrable plurisubharmonic function Hilbert space
l
((
{ f E A(n) f
A2my (n)
n
ll
define a
IfI2e-2mL4dV < oo } JJJ
Take an orthonormal basis {Q1 }i ° I of A2,.,,t (.R), and set 2m.
log X00 L
IQ1I2.
1=1
THEOREM 3.16 (J.-P. Demailly, 1992). There exist constants CI and C2, independent of m, which satisfy the following conditions:
a. ?'(z) - CI fin
v,,,(z)
and r < bn(z).
b. v(,, z) -
m
Sup
K-zI_ c}
is an analytic subset of Q. PROOF. From Theorem 3.16 (b),
Ec(') = n Ec-n/m(Wr) m> 1
However, since
Ec-n/m(1m)
_{zI (a) <me-nimplies a")(z)=0forl=1, 2,
}, it follows that, the Ec_n/m(t&m) are analytic subsets, and so is the intersection EE(') of these sets. REMARK. Ec(ip) is clearly monotone decreasing with respect to c, but not much more than this is known. From a property of analytic subsets, it has been known that Ec( r/)) is left continuous on c and has only countably many discontinuous points. At a discontinuous point c', there occurs the phenomenon that a family of analytic subsets of
3.3. LEVI PSEUDOCONVEXITY
47
EE(?P) as c \ c' is absorbed by a higher-dimensional analytic subset in EE' (0).
REMARK. The first conjecture on the approximation of plurisub-
harmonic functions was made by Bochner and Martin [4, p.145] in relation to the Levi problem in the following form: DEFINITION 3.18. A Hartogs function on fl is by definition an element of the smallest among the families _F(fl) of functions on P_ with values in IR U {-oo} that satisfy the following conditions:
1. f E A(fl) implies log If E F(fl). 2. fl, f2 E .F(.R) and c E [0, oo) imply fI + f2, cf1 E .F(fl). 3. If { fA}.EA C F(fl), and if the fA's are uniformly bounded from above on compact subsets in fl, then sup fa E .F(fl).
4. Both {fj}j- I C.F(fl) and fj
fj+I imply slim f? E F(fl). -00 5. f E F (S?) implies lim f (z') E .F(fl). Z' z 6. Letting 1-1(fl) be the smallest of the .T(fl)'s that satisfy (1) (5)1 if f satisfies f I fl* E f(fl*) for any relatively compact open set fl* of fl, then f E .F(12). Bochner-Martin conjecture: Plurisubharmonic functions should be Hartogs functions. If fl is pseudoconvex, we see that the B-M conjecture is correct
by using the solution of the Levi problem (Bremermann [6]). On the other hand, clearly the B-M conjecture would affirmatively solve the Levi problem, but there is a domain that gives a counterexample against the B-M conjecture ([ibid.]). Roughly speaking, Bremermann's result corresponds to the polyhedral approximation of figures, while Demailly's theorem corresponds to the approximation by surfaces.
3.3. Levi Pseudoconvexity Open sets that have boundaries of class C2 are mainly described. A boundary point p of fl is said to be of class Ck if there exist a neighborhood U of p in C' and a real-valued function ru of class Ck on U that satisfy the following two conditions: (3.30)
U n P = {z I rj(z) < 0}.
(3.31)
{zEUIdru(z)=0}=0.
3. PSEUDOCONVEXITY AND PLURISUBHARMONIC FUNCTIONS
48
In this case, ru is called a defining function of .R on U or simply around p. When every point of 8.R is of class Ck, .R is said to have boundary of class Ck, and we write al? E Ck. By means of a partition of unity, we can construct a real-valued function r of class Ck defined on a neighborhood of Si that satisfies (3.32)
.R = {z I r(z) < 0},
(3.33)
{zE0(11dr(z)=0}=0.
In general, we call such a function r a defining function of .R.
Let p be a C2 boundary point of .R. Since the tangent space to 8f2 at p is of real dimension 2n - 1, this space contains a complex hyperplane. This is called a complex tangent space of 8(l at p, and is denoted by Tp. Given a defining function r of (1 around p, the quadratic form z
(3.34)
_
j,k 8 8jazk (P)CjZk
on the vector space {l; E Cn I l; + p E Tp} (= KerOr(p)) is said to be the Levi form of r at p.
DEFINITION 3.19. A C2 boundary point p of .R is said to be strongly pseudoconvex if either n = 1 or the Levi form of r at p is positive definite. .R is called a strongly pseudoconvex open set if f1 is bounded and its boundary points are all strongly pseudoconvex. The positive definiteness of the Levi form does not depend on the choice of a defining function r, since another defining function is written as a multiple ur by a positive-valued function u of class C', and, given p E 012, we have (3.35)
8(ur)(p) = (u8r)(p),
and
(3.36)
08(ur)(p) = (u8ar + 8u A Or + Or A 5u) (p).
In particular, take u = eBr; then the right hand side of (3.36) becomes (88r + 2B8r A 8r) (p). Hence, if the Levi form of r is positive definite at p, then ur is strictly plurisubharmonic near p for a sufficiently large B. From this the next proposition follows.
3.3. LEVI PSEUDOCONVEXITY
49
PROPOSITION 3.20. A strongly pseudoconvex open set is pseudoconvex.
PROOF. For a strongly pseudoconvex open set 12, from the above argument, there is a defining function r of (2 that is strictly plurisubharmonic on a neighborhood of 8.(1. Since 12 is bounded, there is a sufficiently large positive number C such that - log(-r) + Cjzl2 is exhaustive and strictly plurisubharmonic on .R.
From (3.36) it follows that the signature of the Levi form does not depend on the choice of a defining function. (2 is said to be Levi pseudoconvex at p if the Levi form of a defining
function is semipositive definite at p. An open set whose boundary points are all Levi pseudoconvex is called a Levi pseudoconvex open set.
THEOREM 3.21. A Levi pseudoconvex open set is pseudoconvex.
PROOF. For p E an, set 12p,e := {z I r(z)+EIz-p12 0. From the assumption, there are a neighborhood U of p and a positive number co that satisfy the following two conditions: 1. Given z E U and e < co, there is a point w in anp,E fl U such that 5Q, (z) = I z - w1. 2. Given e < e'o and q E a12p,e fl U, the Levi form of r + elz - p12 is positive definite at q. In this case, since every point of a12p,e fl U is a strongly pseudoconvex boundary point of f2p,e i it follows that - log 8n, is plurisubharmonic on ,(2p,e fl U. Moreover, since - log 5np.r \ - log 5n as e \ 0, (3.18) implies that - log bn is plurisubharmonic on .(? fl U. As p was chosen arbitrarily, there is some neighborhood W of a12 such that - log bn E PSH(f2 fl W). Finally, compose - log 5n with an appropriate increasing convex function; then we obtain a plurisubharmonic exhaustion function on 12. (See Corollary 3.11.)
That the Levi pseudoconvexity is a fundamental concept is also understood from the following:
PROPOSITION 3.22. A pseudoconvex open set that has a C2 boundary is Levi pseudoconvex.
3. PSEUDOCONVEXITY AND PLURISUBHARAIONIC FUNCTIONS
50
PROOF. Let .0 be a plurisubharmonic exhaustion function on fl. If there were a boundary point p at which f1 is not Levi pseudoconvex, then there would be some v E Tp such that 82r (p) (v - p)iv - P)k < 0. uZj0Tk ,.k
(3.37)
Let vp be the unit inward normal vector WOO at p, and consider a family of holomorphic mappings: 7i :
C -p
C'
W
W
--1 p + tvp + ( (v - p)
fort>0.
Then from (3.37) there is a positive number E such that
(3.38)
J70(0(0, E) \ {0}) C fl, 7rt(A(0,E))Cf2 for00.
2. A(t)=0when t50,and)(t)> 1 when t - 1. 3..1"(t) ? 100A'(t) > 0 when t > 0. 4. A'(t) > 100 for t with A(t) > 2 The following domain is called a worm domain, due to Diederich and Fornaess: .fQDF,r
{z
I'
E C2I z2
+A (Iz112
2 1
- 1 ) + A(Iz1I2 - r2) < OT
where r > 1. i?DF,r is a bounded pseudoconvex domain with boundary of class C°°. If r >_ e', then
1DF,r D {(zl,0)I15IzlI<e"}U{(zl,z2)I Izll=1ore',Iz2+1151}. Hence, by an argument similar to the proof of Proposition 3.22, any pseudoconvex domain containing S2DF,r must contain {(z1, z2) 11 < Izi I < e" and 1z2 + 11 < 1}. In order to construct .fQDF,r, first take a disk on the z2 plane whose boundary contains the origin, and then let this disk revolve around the origin, changing the center and radius appropriately at the same time the disk travels along the zI plane. The point of deriving the pseudoconvexity of l1DF,r is that the angle of rotation is a harmonic function of zI. (For the details, refer to the original article or [25].)
REMARK. Recently, a remarkable result on the a equation on a worm domain has been obtained. Namely, due to M. Christ [9], it is known that P(C0 (.fQDF,r)) 0 C'(QDF,r) for the orthogonal projection P: L2(.RDF,r)
A2(I1DF,r)
3.3. LEVI PSEUDOCONVEXITY
53
This property of worm domains contrasts finely with the next results.
THEOREM 3.28. The closure of a pseudoconvex domain whose boundary is of class C" possesses a system of pseudoconvex neighborhoods.
For the proof, refer to [14]. THEOREM 3.29 (Kohn's theorem). If (l is a strongly pseudoconvex domain with boundary of class C°°, then P(C°D (f1)) C C'(77), where P: L2(37) - A2(fl) denotes the orthogonal projection.
For the proof, refer to [18]. As it will be seen in Chapter 6, Theorem 3.29 is useful in studying the boundary behavior of holomorphic mappings.
CHAPTER 4
L2 Estimates and Existence Theorems We have already observed that the division and extension problems on A(.(1) are reduced to the problem of solving the a equation under appropriate constraints, and have shown that in order for the a equation to be solvable, Q must be pseudoconvex. In this chapter, let us go ahead and solve the a equation on a pseudoconvex open set once without any constraint. The argument stated here follows basically
the approaches by J. J. Kohn, L. Hormander, and others, but the core method of deriving the L2 estimates is due to the article [371. This modification has practically no effect on the existing results included in this chapter, but will make an essential difference in the next chapter. _ In § 4. 1, we derive the vanishing of the a cohomologyon a pseudoconvex open set by means of the L2 estimates. Here the 0 equation is solved under estimates with respect to the L2 norm with weight function. Fundamental formulae and inequalities are shown in this section. From the vanishing of the a cohomology, it follows that Serre's condition and pseudoconvexity are equivalent. In § 4.2, we apply the existence theorem proved in § 4. 1 to generalize the classical results of function theory in one variable to several variables. These results were all established by Kiyoshi Oka, but we follow Hormander's methods here.
4.1. L2 Estimates and Vanishing ofd Cohomology The goal of this section is to show the solvability of the a equation on a pseudoconvex open set by means of the so-called L2 estimate. This method is one of the fundamental ideas in functional analysis which originates with the Fredholm alternative, and uses an infinitedimensional system that keeps the equivalence between the existence of solutions for linear equations and the uniqueness of solutions of their adjoint equations. 55
4. L2 ESTIMATES AND EXISTENCE THEOREMS
56
Let us review closed operators briefly before starting the details.
Let HI and H2 be Hilbert spaces. A closed operator from HI to H2 is by definition a linear mapping T that is defined on a dense linear subset D in H1 and has values in H2 such that the graph
GT:={(u,Tu)IuEV} is a closed set of the direct sum H1 ® H2. V is called the domain of T and denoted by Dom T. Also, T (V) is the image of T and denoted by Im T. The denseness of Dom T determines a closed operator f from H2 to H, whose graph is the orthogonal complement of GT in H1 ® H2. -T is called the adjoint operator of T and denoted by T. The definition of T* may be rephrased in terms of the inner product (, ); of H; as follows:1 (4.1)
T'v = u :'
(Tw, v)2 = (w, u)I for any element w in Dom T.
From (GT) l = UT = GT, we obtain (T')* = T. In what follows, let II Ili denote the norm of Hi. THEOREM 4.1. The following two statements on an element v in H2 and a positive number C are equivalent: 1. There is an element u in Dom T such that
Tu=v, 1 Ilulll -< -C-
2. For any element w in DomT', I(V, W)21:5 C
PROOF. 1
:
1.
2: Let w E Dom T'. From (4.1) it follows that
I(V,w)21= I(Tu,w)21= I(u,T'w)II IIull1 IIT*wII I 2
C IIT`wIII
1: Define an antilinear mapping over Im T' by 1:
Im T'
C
w
w
y = T'w
(v, w)2 .
From the assumption, Il(y)I < C IlyIII for any y E Im T*. Therefore, from the Hahn-Banach theorem, we obtain an extension T of I to H1 I A :tea B reads 'A is defined by B.' This is a common usage nowadays.
4.1. L2 ESTIMATES AND VANISHING OF & COHOMOLOGY
57
such that
Il(x)I0) in order to show that there exists a solution u in L °' -1(Il) under an appropriate condition. For this purpose, we consider the Hilbert space
L° (n)
{f EL i
feIfI2 dV < oo }
for a real-valued function p of class C2 over fl. Also, let Lo , q(Q) be the set of all (0, q)-forms with coefficients in L' (.fl), which inherits the structure of Hilbert space as the direct sum of L2,(Q)'s. Let u:= E' uI dd1 and v E' v, dz1 be elements in L°, q(,fl). We define the weighted L2 norm f fl on L°W q(Q) with weight function W by (4.5)
IIutI , := E' IIuiV ,, where 11U111,
J e-(Iuj12 dV.
(This definition of norms of differential forms is not, in general, compatible with the patching of coordinate neighborhoods in a manifold, but the advantage of fixing a local coordinate system is an enormous simplification of arguments.) Similarly, the weighted inner product (, ),y on L°,q(Q) with the weight function cp is defined by (4.6)
(u, v),1, :=v) dV, where (u, v) :_ E' ujUj . I
Also, for a multi-index I:= (i1, ... , ip), we set (4.7)
I :_ {il,...
,
ip}
4. L2 ESTIMATES AND EXISTENCE THEOREMS
58
and
Ik" :_ (i1, ... , k, ... , ip), where iµ is replaced by k.
(4.8)
µ
For u E Co'q-1(Q) and v E C°,q(0), if-`,' (au, v) dV
E, E' sgn (J) auJ vi dV
n
_
I iu{k} = J eE-'
-J
E' I
n
kI azk
sgn (:)
Iavj ur
1 aW vr/ dV Ozk 994
iu{k} = J
_- f
ur
In
sgn
/.Il(av, kIJ azk
iu{k} = J
ac &k
vi)
dV.
Therefore, if we define a differential operator `pt9 of the first order by (4.9)
acp
"19v :_
sgn iu{k} =J
it follows that
(a u, v)= Je
(au, v) dV =
vj) (,) U (avJ &k - azk f
Jn
dzr.
e(u, W19v) dV = (u, t9v).
Similarly to the case of a, we extend the domain of `pT9 to L (Q). °C Define an operator a from Lw, Q-1(.f2) to L°p, q (.R) by Dom`65:= {u E Lo,q-1(.fl) I DU E
LOP, q(.R)}
w
U -- `eau := au E L°q(f2).
Then it turns out that a is a closed operator2. In fact, Doma is dense in L Q-1(Q), since clearly Co' q-1(Q) C Dom "a. In order to show that the graph of 'a is closed, it is sufficient to show that i.e. there is a sequence (u, v) E G,,3 provided that (u, v) E (u,,, 4'c9-u,) E Gj9 such that
u
u and
v
(v -+ co).
2Closed operators of this kind are, in general, called 5 operators.
4.1. L2 ESTIMATES AND VANISHING OF t'3 COHOMOLOGY
59
From (u, `°19w),, = lim (u,,, "'19w)w V
00
= Vlim0C (au,,. W)"' = (v. W)" . it follows that au = v. Therefore, (u, v) = (u, 8u) = (u, `58u), which
tells us that Gj = C. In a similar fashion The adjoint operator of `08 is written by to `18, by restricting '10 to the following subset of L°,-4(f2): {v E L° 4(12) I "19v E L0, Q-1(12)}
we obtain another closed operator, which we denote by the same notation I19 for convenience.
In this section, we show that some differential inequality holds by applying integration by parts to an element in Co'1(1l), and that the range for this inequality to hold, in fact, extends to Dom va * fl Dom "o n L°,9(0) by means of a sort of approximation theorem. First of all, let us prepare this approximation theorem.
Let p : min {
bn
l 2
11
(0, 1] be a C°° function such that. 5n > p
and sup I dp(z) I < oo.
,1 .
ZeR
THEOREM 4.2. For any element v in Dom `'8f1Dom ''19fL°,q(.R).
there exists a sequence {v,}µ 1 in C0'Q(.fl) such that (4.10)
in the sense of convergence with respect to the norm In fact, by calculation, we obtain (9(XR v) = aXR A V ± XR COV.
4. L2 ESTIMATES AND EXISTENCE THEOREMS
60
For the second term of the right-hand side, we have
IIP (v - XROV) III; Ii5v-XR3VIIy, - 0 as R
oo. As for the first term OXR A v, we have
8XR = - Rx' (-RP(z)) X (IRI
- 3) aP
+aRIX(-RP(z))x'(IR 3). Since p < R on supp x'(-Rp(z)), from the boundedness of I apl, it follows that IPOXR A vI is bounded when R -' oo. On the other hand, because of the choice of p, for any compact set K of fl, there is a sufficiently large R such that aXR I K = 0. Therefore, since IIP'XR A vli, -' 0 as R --+ oo, the combination of these results yields
P0(XR V) - P&The other two cases in (4.11) follow similarly. In order to deduce the desired conclusion from (4.11), it suffices to use the following three kinds of convergence with respect to II Ilp
a(xR v)e 0 a(XR v), (4.12)
`°'9(XR V)E
I '19(XR v),
(XR V)E -' XR v as e --+ 0, where (XR v)E is the e-regularization of XR v. This conver-
gence is nothing but a general property of regularizations. (Refer to [281.)
For the time being, we exhibit several formulae in order to arrange efficiently those terms produced by integration by parts. For a continuous (a, b)-form won fl, an operator e(w) : L oq (fl) LI a,4+b(!l) is defined by e(w)(u) := w A u. Let t(w) denote the adjoint operator of e(w), that is, t(w) is an operator from L oq(fl) to Lp-a,Q-6(p) such that
(w A u, v) = (u, t(w)(v)) for u E Llp-"9-b(fl) and v E L oq(fl). c(w)(v) is simply written as w _j v.
4.1. L2 ESTIMATES AND VANISHING OF 5 COHOMOLOGY
61
>2Badza E C°"1(17) and u
PROPOSITION 4.3. For B
C1
uldzl E
C°'q(n),
1
(4.13)
BJu=E'EE'sgn(
J)BauldTi.
PROOF. For v = E'vjdzj E C°'q-1(9), (B Av, u)
_ E' E E' sgn ( I BavJut
)
EvJ (F- "Baut sgn
(aj
j)dz.1) _ (v,E'EE'Bautsgn( \ Ja I
.
Set V :_ °z9, and define t9 by :u := (T9u). Then from (4.9) it is easy to see that ai9 +,9a = at9 + We,
8t9+t9a=o,
(4.14)
a,9+t9a=0. PROPOSITION 4.4. For a C2 junction?? on (l and an element u in Cn,q(fl), (4.15)
au + a
an n Wu +
8277
_ Z' E
I j 6 j k OZjazk
u) ujdz1 A ... A dzn A dz1k
(9277
= E' E OZjazk urkdzl A ... A dzn A d'21, j I j,k
where we set ulf = 0 for j and k when I is not defined. PROOF. From (4.9), (4.16)
N A :au
a aut dz1 A ... A dzn A dz j.
1
j azj azj
4. L` ESTIMATES AND EXISTENCE THEOREMS
62
From (4.13),
aul dz1 A (4.17) art, au = F_' E az I jgi j j a
A dzn A dzl
I dzI A ... A dz A dztk .
j¢IkElazk azj Similarly, (4.18) a(art _j u)
/
.I
j
I
azj
a277
I j.k aZja zk
ulsgn` I )dzl
jJ
uldzl A ... A dzn A dzljk
art au j _ +E'E. I jElak azj azk 19277
E'I j.k E azja.Zk uldzl A ... A dzn A d7-IAA aul
'Iq
+E'
dzl n ... A dz,, A dzIkj
I jEIkOI az7 a z k 1: ±1 a ut dzI A ... A dzn A d'51.
+EI AEI azJ
J
By adding these formulae, the desired (4.15) is obtained. From (4.14) and (4.15) it is easy to derive the following: PROPOSITION 4.5. If 77 is a positive-valued C2 function on 11,
then (4.19)
II
1auII2 + IIv'
9uII2 2
a aaAUjkfjdV
2Re (t9u,art_j u) I 1j. k
j
for u E Co "(Q).
In what follows, (4.19) will be proven again in a more general form for the L2 norm 1111,, with weight function o. For this purpose, define an operator Pa by (4.20)
''au := au -
A u.
4.1. L2 ESTIMATES AND VANISHING OF 5 COHOMOLOGY
63
Then, for u E C"'(Q) and v E COP}1*9(Q). (4.21)
_
(Wau, v)s,
(e-1' 'au, v)o v)o = (e-yu,19v)o (u,
Namely. the adjoint operator of a with respect to II II, is equal to 0 on Cop" (0).
On the other hand, for the complex conjugate TO-:= a - e(0 ) of 'P,Y, we have (VC9u, v)0
=
(e Y(a - e(acp))u. v)-Y
_
(a(e
'u), Z')_ , =
-'Ov)_
(u. -w19v)o
Therefore, the adjoint of 'a with respect to II IIo is equal to -Pi9 on Cop,
q(Q). Direct calculation shows that
(4.22)
z9 pa + "'a19 =19a + a19 - z9e(8(p) - e(acp)19
.
Now take the complex conjugate of both sides of (4.22), consider their adjoint operators, and change the sign of cp so as to obtain (4.23)
&(a0.
V19a + v x'19 =19a + a19 +
For simplicity, given a C2 function 5= on n and a differential form u = E'uldzl A A dzn A dz1 or u = E'u jdz1, we define 1
1
2
(4.24)
Ly,u := E' E as a4k ulkdzl A ... A dzn A dz1 I
3,k
'
a2
or F_' F,
I ;,k az;azk
ulkdzl,
respectively.
PROPOSITION 4.6. Let V and p be C2 real-valued functions on !1.
Then, for any element u in Co,4(.R), (4.25) IIPauI12 +
= (p2u, Pt98u),, - (8p2 A U, 7u),, + (u, p28'Pt9u),,
+(u, 8p2 A °jqu)W - (u, P2'°8t9u),P - (u, 8p2 A t9u) 2 A 9u),, au + e(eW _j u) + = (P U, + 2 Re (u, aP2 A `'ft),, - (Lp2u, u)W = 4 Re (8p J u, p `Pt9u),, + (p2Lwu, u)ti, - (Lp2 u, u),p,
where we have used both (4.22) and (4.23) for the third equality and (4.16) for the last equality. COROLLARY 4.7. Under the same condition as above, for every positive number C, (4.26)
IlPeull + (1 + C)IIPpt9uiI ,
,
> (p2L ,u, u),, - (Lp2u, u),
-2
U
112
The calculation passed through (n, q)-forms, but by looking into (4.26) we see that the same result also holds for (0, q)-forms. Since we previously set up the 8 equation for (0, q)-forms, though overlapping
a little, Corollary 4.7 can be restated in this form: PROPOSITION 4.8 (Fundamental inequality). If cp and p are C2 real-valued functions on (1, then, for any element u in C0'q(.R), (4.27)
Ilpeu1l,2p + (1 + C)IlppVuII , >_ (P2
u, u)v - (Lp2u, u),p
-
112)P
u llw
,
where C is an arbitrary positive number. In practice, we have to estimate the right-hand side of (4.27) from below in order to derive the existence theorem from this inequality. A (1, 1)-form w = Ew,kdzj A dzk on fl is said to be nonnegative j'k (or positive) at a point xo E Q if the matrix (wjk(xo)) is a semipositive (or positive) definite Hermitian matrix, respectively. We write w > 0 (or w > 0) when w is nonnegative (or positive), respectively.3 In 3The condition wl - w2 > 0 is written as wi > w2. (wi > w2 is understood similarly.)
4.1. L2 ESTIMATES AND VANISHING OF a COHOMOLOGY
65
terms of this expression, from the definition of L,p it is clear that (L,Du, u) >_ 0 for any u E CO.9(12) if and only if 809W ? 0 on Q.
Also, set
wVu:=wjkuI
(4.28)
I
k
dzJ
j,k
for u = >'ujdzl. Then from this definition and (4.13) it is easy to see that (4.29)
IN-j ul12 = ((ap A p) V u, u),
Therefore, the fundamental inequality is written as follows: (4.30)
IIp au11
, + (1 + C) Ilp p9ull2
((p2o_oP2......aPAP)VU,U).
>
From now on, let (1 be pseudoconvex, and fix an unbounded strictly plurisubharmonic exhaustion function 0 of class C°° on 11. LEMMA 4.9. Let p be a strictly plurisubharmonic function of class C2 on fl with L[W] >_ 1 everywhere. Then, given any c E R and any continuous function T : R -+ R, there exists a function A : R -+ R of class C2 that satisfies the following four conditions: 1. A(t) = 0 when t < c.
2. A(t) > r(t) when t > c + 1.
3. A'?0 andA"?0. 4. Letting W,\ := cp + A(' b), if u E Dom `Pa a n Dom wa t9 n Lwq (.fl),
then
IIauliv, + Il'' 9uIIW > PROOF. For a natural number v, we can take a C°° function
p :.fl -+ (0, 1] that satisfies both (4.31)
min { 26,?,1}
min{v6,o,1}
and (4.32)
ap 1, we conclude that aa,;, aaI-I2. Therefore, for any element u in Co'9(f1), (4.30) yields (4.34)
II Pv UII A + 1+
1)
Ilp.v
U112, > qjI P"uII2
Since, by Theorem 4.2, (4.34) holds for any element in Dom''aa n Dom P'A V n L°4 (f1), we obtain (4) by letting v - oc. Let us write down what follows immediately from Lemnia 4.9 and Theorem 4.1.
PROPOSITION 4.10. Let fl and 4;a be as above. If u' E Kera n
L°Q(f1) (q > 0). then there exists an element v in
n
such that (435)
at' = ur, gIlz'll a
2 i--1
IIWIL
Likewise. if w E KerY'd n L°,4(R) (q > 0), then there exists an element v in Dom s'' 0 n L°' (fl) such that val3v=w, (1.36) Ilu'll
gIIL'I12
PROOF. We prove only the first statement, since the second can be (lone by the same argument. It suffices to show that (4.37)
1I'aa*u1il
I(u',u')wa12 < g1lu'II2
n
for any element u' in Dom
a
Decompose u' as
u' = u1 + u2, where ul E Ker a and u21Ker a. Since (4.38)
E Ker a, it follows that (uw, u')
a=
(w. u1)ti,A.
4.1. L2 ESTIMATES AND VANISHING OF 8 COHOMOLOGY
67
On the other hand, since u21Ker a, (4.39)
v' E
0 = (c?v', u2),,A = (v','°''a*u2)a,,
.
Therefore, U2 E Ker `ea 8 * , which in particular implies that u l E Dom a * and u' = w) 8 * ul. Hence, Lemma 4.9 can be applied
to ul to yield (4.40)
II`Pxa*u'IIA
Combining this with (4.38), the Cauchy-Schwarz inequality implies (4.37), as we wish. The following existence theorem is a fundamental theorem with a wide variety of applications. THEOREM 4.11 (Hormander's theorem). Assume that f2 is pseudoconvex, and a C2 function ep :.R , ]R satisfies L[,p] > 1.
1. For any w E Kerafl L°,4(0) (q > 0), there exists an element V in L°-9-1(Q) such that av = w and gllvll;
2. If w E Ker B fl L a9 (.R) (q < n) and the support of w is compact, then there exists an element v in L o9-1(Tl) such that the support of v is compact, 8v = w, and (n - q)IIvII? < IIwI1?,,
PROOF. (1) For c E R and T - 0, let ac denote, expressing the choice of c, a function \ that satisfies the condition of Lemma 4.9. Since Ac is nonnegative, IIwII.A,
IIwIIw and w E L°Q (!l).
Therefore, Proposition 4.10 implies that there is some v, E LO,? 1(Q) such that Javc = w,
4.41 (
)
(- IIwII2)
gIIvcll;aC < IIwI12
is bounded with respect to c, {v} has a subsequence that converges weakly on compact sets. It is sufficient to choose the limit of this subsequence as v. Since l l vc l l v,,,
(2) For u = E'uldzl, set *
u :=
,
sgn (
12
I
n
J
ujdzj.
Then the defining equation (4.9) of 'PO is written as `°19u = -e`°(ae-`Pu*)* .
4. L2 ESTIMATES AND EXISTENCE THEOREMS
68
Therefore, if w E Ker 8, then &A- w* E Ker VA-d. Also, since the support of w is compact, IleW.%C W*
Ilsv.,C = II eS'w II, (= IIwII-w)
for a sufficiently large c.
Now that the latter half of Proposition 4.10 is applicable to &ac w*, there is vc such that W.%,Ovc = &acw*,
1(n -
IIwII',
for a sufficiently large c. Apply the above result to co,a, (s > 0) instead of spa,, and take a
sequence of numbers sµ -> oo and a sequence Iv,,}-, that converges with the LZ norm on compact sets such that f `R' 6v,, = e`° '>w*,
(n where we set
IIwII' , cp,µa, If we choose, in advance, ac(t) > 0 when
t > c, then, for v,,,, := lim vµ,
µo0
I supp voo C 11 ,,c ,
(4.42)
'*Vv,,. = ePw*,
(n - 9)IIv00II
IIwII'
0
Hence, it is enough to put v := a-`°v;. Supplement. It is readily seen from the above proof that if supp w C 11 y,c ,
then we can take v with supp v C .fl,,,c as solutions of the 8 equation. As an application of Theorem 4.11, we can derive fundamental results on the representation of 8 cohomology groups. Let us begin by setting up our notation. Define (4.43)
W (.fl)
u E L o'" (.fl)
I
E L o'q+I (!2)
Then from the complex
woe(')
...
the cohomology groups Hpo, (.fl) are determined by (4.44)
H('c(Q) := Kerafl Woc (.fl)/{8u I u E Woc-'(fl)} .
4.1. L2 ESTIMATES AND VANISHING OF 8 COHOMOLOGY
69
If Si is pseudoconvex, then, given u E L ,q([2), there is some C2 function o with L[p] >_ 1 such that IIuJI,p < oo. Therefore, from Theorem 4.11, we obtain, in particular, a vanishing theorem of cohomology.
THEOREM 4.12. If Si is pseudoconvex, then Ho,9(12) = {0} for
q>0. From the theorem of L2 holomorphy, we get Hoo(Q) = Hp,o(fl) . (4.45) In effect, this correspondence holds in general. THEOREM 4.13. For any open set .fl C C", the homomorphisms
a:
Hp,q(0)
induced by the injections Cp,q(f1)
- Hoc (Q) L oq(!1) are bijections.
PROOF. 4 We can assume that q > 0 from the above observation. Also, clearly it is sufficient to prove only the case p = 0. Proof of surjectivity of a. Let v E Ker 8 n L°oq (.R). It is enough to show that there is an element u in L °C-1(.R) such that v - au E CO,q(,R). For this purpose, fix a locally finite covering {Ui}901 of fl with U, C Si, where the Ui are open balls; then construct inductively via...;, E L oq-t-1(U. n n U,) (0 < 1 < q) that satisfy the next two conditions:
vUi=&i,
(4.46)
Uio n ... n ui, = 5vio...it
(4.47) V=0
where Uio n n U, # 0, and i means the exclusion of the index i,,. This process is possible since
((_1YV10......11)
(4.48)
0=o
I
V_p
V 0 from the pseudoconvexity
and H° " (Uio n Set
1
(4.49)
L
uio...ii
I Uio n ... n Ui,
.
V=0
For l = q, (4.48) implies that (4.50)
uio...iq E Ker a n LI c(Uio n ... n Uiq) = A(Uio n ... n Uiq) .
Also, (4.49) yields q+1
E
(4.51)
(-1)Vuio...;V
v=0
...i +i = 0. q
Use the partition of unity {pi} subordinate to {Ui} to set Piuiio...iq-i
uio...iv-i
then, first, by (4.50) we have uio...iq_i E COO(Uion q
nUiq). Secondly,
q
(4.52) V=0 i
V=0
(-1)VPiuiio...i,,...iq
rPiZ (-1)Vuiio...!2,...iq piuio...iq
=
by (4.51)
uio...iq .
Similarly, for 1 with 0 1 q - 1, we can construct u;o,,,it_1 E n . . . n Ui,) such that 1+1
1 (-1)Vu1
(4.53)
_a
= Vui0...i1
V_0
From (4.49) and (4.52), it follows that q
(4.54) V=0
(-1)V(vio...i,,...iq - uio...i....iq) = 0.
When q = 1, this means that (4.55)
vi-u;=V3-u1
.
4.1. L2 ESTIMATES AND VANISHING OF 5 COHO`MOLOGY
71
When q > 1. from (4.54) in the same way that we produced u;,..;,_, we can take vio...;q_2 E W10C (U;0 n - - . n U{ q_2) such that
q-1
u.
'Uio ...t 4-1
(4.56)
=
(_)Vv,
. , 10...1"...44-1
V=0
By applying a to both sides of this equation, we obtain
_=
_ avi... 0 t9
(4.57)
-
q-1
1
au'... ip 2q_2 _E(-1)"av:i,,...i,...t4_1 V=°
Rewrite this in terms of (4.47); then q-1
(4.58)
E
V_°
q-1
u:4p...i"...i4_1
- 51?:io...
0.
Repeat the process of producing (4.58) from (4.54) until we eventually reach the formula
vi - ui - t ' i = Z'; - u - av;
(4.59)
Therefore, finally we define an element u in L °C-1((1) by
u:=
(4.60)
vi-ui ifq=1, Vi - u= - av;
ifq > 1;
then this u satisfies
v-au(=ai)EC°"q(fl).
(4.61)
Hence, the surjectivity of a is proven.
Proof of the injectivity of a. From Corollary 2.10, this is true when q = 1. Let us assume that the assertion holds for all k with 1 < k < q - 1. Choose any w E Kera n C°,q(Q) with w = ag for some g E Recalling that H°-I(0") = {0} for l > 0, select a similar covering {Ui} as above so that for each i there is si E C°'q-1(Ui) satisfying asi = w I U2. Hence, using Theorem 4.12 and the induction hypothesis at the same time. we can inductively construct elements s10...i, in n ... n Ui,) such that t (-1)"si
(4.62) V=°
5q-1
f o r 0: 51 :
.
I U10 n ... n U1, = 5si....i1
4. L2 ESTIMATES AND EXISTENCE THEOREMS
72
Note that w = 8g yields 8(si-g) = 0. Hence, from Theorem 4.12, there are tio...i, E L oc-i-1(Uio fl . fl Ui,) (1 < 1:5 q - 1) such that
{ ti = Si - 9, (4.63)
8tio...if = sio...it
- v=0 (-1)Ytio...
In addition, for the case l = q - 1, (4.64)
8 (io...iq_ 1
q-1
- ` (%v...iv-1}
= 0'
and so the L2 holomorphy theorem implies that q-1(
-1)Ytio... is of class C°°.
Therefore, applying the gluing in terms of the partition of unity used for the proof of surjectivity, first take tio...iv-2 E C°°(Uio fl ... n Uiv-2)
such that q-1
q--1
E (-1)
Y=0
Y
v
-1 - Y=0 -1) v
and consecutively choose tio...ii E C°'q-1-2 (Uio fl
.
. fl Ui,) inductively
in the descending order of 0 5 1 < q - 2 such that sia ...i1
Y=0
1
tio...t,,...tt
Then eventually we obtain si - R, = sj - &I'. This represents an element of C°,q-'(fl), say, v, which results in 8v = 8(si - 8ti) =
0
w.
From Theorems 4.12 and 4.13, the following vanishing theorem for 8 cohomology groups is obtained: THEOREM 4.14. If 17 is pseudoconvex, then
HP,q(.fl) = {0} for q > 0. COROLLARY 4.15. Pseudoconvex open sets satisfy Serre's condition.
Therefore, the condition in Theorem 2.14, for instance, may be replaced by pseudoconvexity. That is to say, pseudoconvex open sets are domains of holomorphy.
This assertion, named the Levi problem (or the inverse problem of Hartogs), had been a central conjecture of long standing in the theory of analytic functions of several variables, but was solved by Kiyoshi Oka for the case n = 2 in 1942, and by K. Oka [39], H. J. Bremermann [6], and F. Norguet [35] independently for general n. The first half of Theorem 4.11 has played an active part, but also
from the second half we can derive an explicit consequence on the analytic continuation of holomorphic functions. THEOREM 4.16. Let K be a bounded closed subset of C", and fl an open set that includes K. If n >- 2 and if fl \ K is connected, then the restriction mapping A(Q) --i A(Q \ K) is a surjection.
PROOF. Fix a neighborhood U of K that is relatively compact
in Sl, and let X be a real-valued function of class C° on fl with X I K= 1 and supp X C U. Then, given a holomorphic function f on fl \ K, the trivial extension of 8((1- X) f) to C' belongs to C0011(C" ) Take an open ball 3(0, R) for which supp 8((1 - X) f) C 3(0, R) and 83(0, R) fl 17 0. From the latter half of Theorem 4.11, there is a solution u of the equation 8u = 8((1 - X) f) such that (4.65)
u E L2(3(0, R)) and supp u C= 3(0, R).
In this case, the theorem of L2 holomorphy implies (1 - X) f - u E A(Q), while, by the condition 83(0, R) fl 17 0 0 and the theorem of identity, (1 - X) f - u coincides with f on fl \ K. REMARK. Intuitively, K may seem to be extinguished thoroughly by repeating the process of embedding the biholomorphic image of a
Hartogs figure into 17 \ K and extending this image to the biholomorphic image of 02. However, it is uncertain whether the function extended by this method is single-valued. In fact, there is an example in which this process cannot be continued without allowing the image of A2 to stick out from fl in the course. (Refer to [19].)
4. L2 ESTIMATES AND EXISTENCE THEOREMS
74
Essentially, the same content as Theorem 4.16 can be described as the extension theorem for functions on a real hypersurface. (The assumption that n > 2 is kept valid successively.) DEFINITION 4.17. Let n be a bounded domain whose boundary is of class C'. A complex-valued function f of class CI on an is said to satisfy the tangential Cauchy-Riemann equation if there exists an element F in C'(77) such that
F I 8f2= f andaFABrI aft=0, where r is a defining function of n.
THEOREM 4.18 (Bochner-Harvey). Let an E C'. If an element f in C' (an) satisfies the tangential Cauchy-Riemannequation, then there exists an element f in CI(fl) fl A(fl) such that f 180 = f. PROOF. We prove this only in the case that an E COO and f E COC(an). For the general case, refer to [22] and [27]. Let F E COO (fl) with F I an = f. From the assumption,
aF = afar+13,r for some a, E C°°(n) and Q1 E Hence, setting FI := F - alr, we see that
F, I an = f and aF, = 0'r, where Ql := 01 -
Dal
.
From a(,3ir) = 8(aF,) = 0,
01AOr Ian=o, and so, Ql = agar +,32r for some a2 E C'(17) and 32 E Co" 1(f2) Setting
F2:=F-a1r- 2 r22
,
we see that
F2 I an = f and OF2 =,32r2, where , 2 = Q2 - 1 aa2 2 As the same operation can be repeated, there is a sequence of functions {ak}k , C C°°(n) such that, given a natural number N,
a F - k k, rk k=1
= QNrN for some ON' E C°" (n)
4.2. THREE FUNDAMENTAL THEOREMS
75
Therefore, there is an element F in C" (.R) such that k 1 00 = f and the derivatives of OF of all orders are equal to 0 on X2. If we set
w:=
8F on D. 10
onCf2
then w E CO, I (CI) fl Ker a and supp w C fl. Hence, the rest of the proof is similar to that of Theorem 4.16. 0
REMARK. As to Theorem 4.13, it appears that the proof connects the world of C°° functions with that of locally square integrable functions in terms of holomorphic functions. The generality that has developed from arguments of this kind is the so-called theory of cohomology with coefficients in sheaves; and, further, the unobstructed view that has grown by applying this theory to analyzing the singularities of solutions for linear partial differential equations is nothing but the microlocal analysis of M. Sato, T. Kawai. and M. Kashiwara [40].
4.2. Three Fundamental Theorems In the classical general theory there are results that display the perfection of the world of complex functions; here we find the Mittag-Lefer theorem, the Weierstrass theorem and the Runge theorem, and their generalizations to several variables are derived from the established existence theorems, Theorems 4.11-4.14.
4.2.1. Distribution of Poles and Zeros. When a function f is defined on an open set (l in C' except a null set E, f is said to be a meromorphic function on .fl if each point xo E fl has some neighborhood U (= U(f , x0)) such that on U \ E. f can be expressed as the quotient of two holomorphic functions defined on U. Let M(Q) denote the set of all meromorphic functions on 11. Given f E M(.R), we call
f.,c:={pE.Rl limIf (--)I=x} the pole of f. From the definition, it is obvious that f, is a closed set that is included in E. From Theorem 1.13, f extends over !l \ f«, as a holomorphic function. We identify this extension with f. DEFINITION 4.19. For an open set U in .fl, the subset
(f I U\ f.)+A(U)
4. L2 ESTIMATES AND EXISTENCE THEOREMS
76
of M(U) is called the principal part of f on U and is denoted by
P(f,U) In the case of one variable, the sum of terms of negative power in the Laurent expansion of a meromorphic function at a pole was called the principal part of the function. The above definition is a generalization of this.' The Mittag-Leffier theorem can be generalized to the case of several variables as follows:
THEOREM 4.20. Let .R be a pseudoconvex open set, U an open subset of .R, and g E M(U). If go. is a closed subset of 12, then there exists an element f in M(.R) such that g E P(f, U). PROOF. From the condition, there is a C°° function p on .R whose
value is 1 on some neighborhood of go. and such that supp p C U. Set (4.66)
V:=
Igap on U, 0
onQ\U.
(.R). From the pseudoconvexity of .R, there is Then v E Ker 8 f1 a solution u E C°° (.R) of the equation au = v.
Hence, it suffices to define f := pg - u, where pg I .R \ U = 0. O
The pole of a meromorphic function is an analytic subset, although we do not prove this fact in the present book. DEFINITION 4.21. For a holomorphic function f on .R and an open set U C .0, f A(U) is called the divisor class of f on U and is denoted by D(f,U). A generalization of the Weierstrass product theorem to several variables is made possible on a pseudoconvex open set whose second Betti number is 0. THEOREM 4.22. Let .R be a pseudoconvex open set, and H2(12, Z)
_ {0}. If V(g) is a closed set in .R for an open set U in .R and g E A(U), then there exists an element f in A(.R) such that g E D(f, U). 51n the above situation, unlike the case of one variable, there does not exist anything that corresponds to the Laurent series. Hence, we are obliged to define this concept as an equivalence class, as in Definition 4.19.
4.2. THREE FUNDAMENTAL THEOREMS
PROOF. Take a locally finite family {B}
77
of open balls Bj
B(pj, Rj) in fl such that 00
fl= UBj
(4.67)
j=1
and Bj n V (g) # 0
(4.68)
implies
Bj C U.
Define gj E A(Bj) by gj
(4.69)
g I Bj
if Bj n V(9) 34 0 ,
1
ifBjnV(g)=0,
and define gjk E A(Bj n Bk) by 9jk := 9jlgk, where Bj n Bk # 0. Note that Bj nBk is simply connected since it is convex, and that from the definition, gjk does not have any zero point. Hence, we can have some branch ujk of log gjk be in one-to-one correspondence to on Bi n Bj n Bk. (j, k). Then uijk := uij + ujk + uki E Adjusting the Ujk'S in advance so that ujk = -ukj, we can assume Uijk + 'ujk, + Ukli + ulij = 0. Therefore, from the assumption on fl, there is a set { m consisting of elements of Z such that
-1(mij + mjk + mki) = uij + ujk + Uki
(4.70)
on each Bi n Bj n Bk. If we set uij :=uij - 21rmij, then
uij+ujk+uki=0.
(4.71)
Let {pj} be a partition of unity subordinate to the open covering {Bj }, and define (4.72)
ui
pjuij , where pjuij I Bi \ Bj
0.
Then it follows that (4.73)
ui - uj = E Pkuik - F, Pkujk k
k
_ >Pk(uik - ujk) _ k
k
Pkii:j
Hence, since uij E A(Bi n Bj), we have c3ui = 0u, on Bi n Bj, and so this determines an element in Ker c? n C°-'(fl). Therefore, from Theorem 4.14, there is an element u in C°°(fl) such that &u = aui.
4. L2 ESTIMATES AND EXISTENCE THEOREMS
78
If we set h= := e"--", then h; has no zero point and h= E A(131), while from (4.74)
(u; - u) - (uj - u)
log(gi /gj)
mod
27r
Z
it follows that he/h, = g;/gj. Consequently, if we define f := g;/h=, then f E A(R) and g E f A(U), which completes the proof. O
4.2.2. Approximation Theorem. According to the Runge approximation theorem in the theory of functions of one variable, a necessary and sufficient condition for the polynomial ring C[z] to be dense in A(Q) for a given open set .fl C C is that C \ .f1 be connected. This topological condition is related to the theory of functions by the following proposition: LEMMA 4.23. A necessary and sufficient condition for C \ .R to
be connected is that for every compact set K of .R. there exists a continuous subharmonic exhaustion function cp : C -+ R such that
KC{zE.Rlcp(z) 0), there exists an element u in L°p4-1(0) such that 0u = v and IIuII,, 5 CIIvll,,, where C is a constant that depends only on the diameter of .fl (:= sup Iz - z'I). z.z'E1
The interpolation problem raised in Chapter 2 can be solved perfectly as an application of Theorem 5.1. THEOREM 5.3.1 The following are equivalent: 1. !1 is pseudoconvex.
2. For any discrete set t c (1, the restriction mapping A(Q) Cr is a surjection. PROOF. (1)
(2): Take h E C°°(!1) such that ah = 0 on
some neighborhood of F. Then it suffices to show that there is an element g in L C(Q) such that 0g = 0h and g I I' = 0. Let p and UU (l; E r) be as defined in § 2. 1, and decompose p as (5.1)
p = E pt with supp p C UU. FEr
1 Onecan see also from this theorem that pseudoconvex open sets are domains of holomorphy.
5.1. SOLUTIONS OF THE EXTENSION PROBLEMS
85
Then for a function c defined by 4 (z) := 2n> Pa(z) log Iz
(5.2)
CEr
- I,
there is a continuous function -r : 11- R such that (5.3)
O
(z) > T(z)aa1z12 for z E .R\r.
Hence, if we choose an appropriate exhaustion function '4' E C°°(Q) fl PSH`, then (5.4)
on . ? with
L[(D + ] ? 1
_ _ and Ilahll,+ , < 1. From Theorem 5.1, there is g E Lb+,y(fl) such that ag = 8h Also, from Theorem 2.7, g is of class C°°. and II9Il4,+' < II Noting that a-'-'0 is not integrable around r, it follows that g I r = 0, which is what we wished to show. (2) (1): This follows from Theorems 3.4 and 3.12.
Next, coming to the extension of holomorphic functions on an analytic subset X of dl, two new problems arise if we apply the same argument as in the case of discrete sets: 1. Can holomorphic functions on X be extended to holomorphic functions on some neighborhood of X? 2. Does there exist a function that corresponds to (D + 0 in the proof of Theorem 5.3? As to (1), in general there does not exist any holomorphic mapping from a neighborhood of X to X that coincides with the identity mapping when restricted to X. This problem has already been as difficult as the extension of functions to the whole .fl. Now let us think
in a more adaptable way: Given a holomorphic function f on X, construct an extension f of class C°° by patching local holomorphic extensions of f in terms of the partition of unity, and apply Theorem 5.1 to 5f. Then in order to ensure the finiteness of the norm of a f by adjusting 0, a-1 15f I2 must be locally integrable in the first place. Since a-D is also required not to be locally integrable along X, it turns out, in turn, that (2) is quite a subtle problem. Precise argument for this point calls for two fundamental theorems on analytic subsets. THEOREM 5.4 (for the proof, see [34] or [25]). Let .fl be an open
set in C", and X an analytic subset of (l. Then there exists a family
5. EXTENSION AND DIVISION PROBLEMS
86
{XQ}QEi of analytic subsets of .R that satisfies the following conditions: (5.5) X = UXQ, each XQ is non-empty, and Q
#{XQIXQnK#0}<x for any compact set K of Q. (5.6) Every XQ contains a connected differentiable manifold Xn as a dense open set in it, and XQ fl X3 = 0 for a : 3. An Xa that appears in Theorem 5.4 is called an irreducible component of X. Also, the maximum open differentiable manifold contained in XQ is called the regular part of XQ and is denoted by Reg Xa. Reg XQ is a locally closed complex submanifold of Q. Namely, for any
point x E Reg X. there are a neighborhood U of x in I? and a biholomorphic mapping F to On such that F(U fl RegXa) = {z E An I Z'.+1 = = Zn = 01, where rn.Q is an integer that is independent of the choice of x and is called the dimension of X. The dimension of XQ is denoted by dim Xa. In addition, U Reg XQ is called the regular part of X and denoted by Reg X.
THEOREM 5.5 (for the proof, see [34]). Let Si and X be as defined above. Then for any x E X. there exist a neighborhood U x in Si and a system {wQ}a=1 (1 # oc) of local defining functions of X on U that possess the following property: (5.7) For any y E U fl X and any system {h,3}- 1 (1 m IpjwQ12 (in:=dimX) j,a
by means of a partition of unity { pj } associated with {U}. Then (5.9) and (5.10) clearly hold. We leave it to the reader to verify (5.8). (Recall the Gauss-Codazzi formula.)
5.2. Solutions of Division Problems Given a vector f = (f1, , f,,,) E A(f1)3)"' of functions that have no common zero point on .fl, as described in § 2.1, a necessary and sufficient condition for there to exist holomophic functions gj that satisfy the equation m
>2 fjgj = 1
(5.12)
j=1
is that the vector-valued a equation (5.13)
au = u :_
\a
CIf121
1
a 0-0)
S. EXTENSION AND DIVISION PROBLEMS
88
fjuj = 0.
has a solution u E L'10C(Q)e- with j=1
Let us proceed with the calculation under the assumption that cp is a C°° function with 8(f j/I f I2) E L°W,1(Q). Define (5.14)
S,°'q :=
a E L°q(Q)®"`
fjaj = 0
I
j=1 , Da"' ) Then, by the holomorphy of Ii, the D operator a '- (Da I, becomes a closed operator from S,°p,q to SO q+', which is denoted by
DS for distinction. In order to solve the division problem, it is enough to choose an appropriate w so that the ratio of I (w, v),, I to 1IDs w1j,' is bounded on Ker Ds (l Dom Ds. We will try to express Ds in terms of '"D * (which operates componentwise) and f. If the problem is solvable, this calculation should
naturally produce an L2 estimate. Let us first introduce the following notation: For elements w1 and w2 in Lwq(f2)®m, set m
(5.15)
(WI, W2)
(5.16)
(WI, w2),p
(5.17)
IIwl IILP
E (w1j,w2j),
j=1
Jn (
e-4'(wl, w2) dV, wl )lp
The orthogonal complement of the set SO,'0 in LO,°(f2)®m is denoted
by (4'0)1. Set (5.18)
(S°p'0)0 -L := {w I w = (cf1,
cfm) for c E Co (.R)}.
, fm do not have Then (S °)o is dense in (S°p'0)1. In fact, since f1, any common zero point, any element in L°Y,O(Q)®m that is orthogonal , f1). to S0.'0 must be a function-multiple of (f 1, Go back to the definition of adjoint operator; then for an element w in Dom Ds and an element h in Dom Ds,
(5.19)
(as w, h),p = (w, ah),P.
5.2. SOLUTIONS OF DIVISION PROBLEMS
89
From this, we see that as satisfies Dom'B * fl SO.,I C Dom Bs, and
that
BSw=P`°Xw
(5.20)
by using the orthogonal projection P : Loo(n)®m -. So'° This formula can be expressed as
BSw=vB*w--(`°Xw,ej),pej
(5.21)
j=I in terms of an orthonormal basis {ej} I of (S°'0)1. We may assume ej E in advance. Using the equation ej) , _ (w, Bej )r the self-duality of Hilbert space allows us to identify Bej with an element in (LO,I(.f1)Om)`, which results in the expression
BSw =' ''B
(5.22)
- f ej ®BejI w. j=1
Since ej = (c171/If, ... , cjfm/Ifl), cj E Co (.fl), and IIcjII'v =
1, it follows that (5.23) D ej
= (f15
L(-)
Ifl
l Ifl+ cj
,
(I,
cj
...
,
l
Ifl afm)
If W E Dom'B * fl S-1, the inner product of w with the first term of the right hand side of the above formula is equal to 0. Hence, (5.24)
0a w- [.. L. (wk' j=1k=1
BS w
_
oo m j=1k=1
(
'°BOw(k'=
IfIa
ej
fk
Bfk Jwk,Cjl,p
/
If I lafk-Jwk
ej
ff.
Set for simplicity
_ af (w)
fm
if
k=
12157k -J Wk-
Then from the above formula, for p E C°°(Q) and X E Co (.R), IIpa;(XW)II ,
Combine this with the fundamental inequality (4.27); then, when p > 0, we obtain IIPas(Xw)II; + IIPa(X011, i + E IIP °a*(xw)II , + IIPa(xw)II2 - EIIPQf(Xw)III 6
> 1+
E
1+C
Ilp ((P2 L, - LP2) XW, XW ),P
2 C I I aP j X W 112
-6IIPQf(Xw)II(C > 0). Hence, from this point on, as in deriving Theorem 4.11, we obtain the following existence theorem by running p and X by means of the auxiliary weight function cpa.
THEOREM 5.7. Let fl be a pseudoconvex open set. Assume that , f,,, in A(fl) (m < oo) have no common zero point elements fl, and that a Cx plurisubharmonic function o on .f1 satisfies both
Fl
fne-°Ie( f 12)I'dV (l+E){Ifl_2> 18fk12+1}forsomeE>0. k=1
m
JJJ
Then the re exist elements gk in A(fl) (k = 1,
,
m) such that
fkgk = 1, and k=1
fe_lg2dV
ftE) (I+
2 mEl
la (di;)
dV.
COROLLARY 5.8. If .(1 is pseudoconvex, then for any system { fk}"0 1 of holomorphic functions on (1 that has no common zero
point, there exists a system {gk}k 1 of holomorphic functions on f1
such that Eoc fkgk = 1. In particular, Spec,,, A(f1) = Q. k=1
5.2. SOLUTIONS OF DIVISION PROBLEMS
91
ao
PROOF. We may assume that 0 < E IM2 < oo. Then from k=1 a Cauchy's estimate it follows that E Iefki2 E C°(f1). Hence, there k=1
is a plurisubharmonic exhaustion function cp of class CO° on fl such
that r
"0
\fI2)l dV> 1. Afterwards, accurate evaluation of the quadratic form implies the following theorem:
THEOREM (Skoda's Theorem [45]). Let Q be a pseudoconvex open set, and cp a plurisubharmonic function on Q. Suppose that we are given p holomorphic functions gl, , gP (or a sequence {gj }?_-1 of holomorphic functions) on (1. Let a > 1, and q := inf{n, p - 1} (or q := n). If a holomorphic function f on !l satisfies If121gl-2«q-2e-wdV
fn
gjhj
or f = >00 gjhj
j=1
j=1 Ih121gl-gage-1,dV 1,
( I nl ),set forzElll,5, forzEIll\J7,6.
V6
10
Then it follows that v6 E Ker a n C"(121) and
suppv6C{z2
>
(881og(Izn12 + 52) V w, w)w, b2
J
e-w' (IznI2 + 52)2IwnI2 dV
{ 1. Then for arbitrary strongly pseudoconvex open set !2' C f2 and e > 0, there exists a positive number 60 such that for 5 with 0 < 6 < 60 and an element v of L,+,,,,2(fl') that satisfy
.36) (5.36)
l -0 (9'=1 ,n-1) ot,
8-2j
(fl ') such that
there exists an element u of
8u=vdz (5.37)
IIu II,o+El:l2 < Cl IwIIs+e z 2 , Zn
E LIOC(n') ,
where Cl is a constant that depends only on no and h1*.
PROOF. Let V + G, s + KoG E PSH(fl), and G E go. V may be taken to be locally integrable. If we define µa by putting n = 1 and c = 5 in (1.9), then since 8a-(log IznI2)a
= 2/25(zn)dzn A din,
from the assumption, for any e > 0 we can take a sufficiently small 5 such that for any n, with 1 < n < no,
5.3. EXTENSION THEOREM WITH GROWTH RATE CONDITION
99
aa( s + hG6 + EIzl2)
(5.38)
> 1p5(z,,)dzn
(z E f2').
Set T1A :_ -G6+A (A > 0), and define a quadratic form Qp,.D(w) to be the one for p:=
On the one hand. from the condition +KG E PSH(Q) (1 < rc widi) From this, in order to assert the existence of u that satisfies (5.37) for v that satisfies (5.36), it is sufficient to solve the a equation in L 9+GT+£I=12 (f2 *) and let T --+ 0. (Note that C1 does depend on If
as well, because of the appearance of Gr in the right hand side of (5.40).)
We use Lemma 5.11 and execute limit operations similar to those in the proof of Theorem 5.10 in order to deduce the following extension theorem: THEOREM 5.12. For a pseudoconvex open set fl and cp E II,..(fl)
(ho > 1), there exists an interpolation operator I.: A ,(f?) --+ A ,(fl) if lK # -oo. The norm of I, depends only on ho and lK .
If sup Iznl < oc, then PSH(fl) C II,,(fl) for any r, > 0, and n
pp ? -tc sup log lzn 12 . Therefore, Theorem 5.12 is a generalization of
n
Theorem 5.10. A similar method can prove the following extension theorem. (For the proof, see [36].) THEOREM 5.13. Let Q be a pseudoconvex open set, and let both cp
and ip be plurisubharmonic functions on fl. If there exists an element G of Cn such that +G is plurisubharmonic and bounded on fl, then there exists an interpolation operator from A2 +v (fl') to A2v (fl).
This allows us to evaluate the Bergman kernel (Chapter 6) in terms of a geometric invariance of 8f1.
5.4. Applications of the L2 Extension Theorem 5.4.1. Locally Pluripolar Sets. A set of boundary points of a pseudoconvex open set may happen to be removable for functions with low growth rate, such as L2 holomorphic functions. The problem of characterizing such a function-theoretically small set has been deeply
studied in the case of one variable, and in particular, the characterization of removable singularities of bounded holomorphic functions
5.4. APPLICATIONS OF THE L2 EXTENSION THEOREM
101
in terms of analytic capacity and that of L2 holomorphic functions in terms of logarithmic capacity are well-known. In what follows the latter will be generalized to several variables. DEFINITION 5.14. We say that a subset E of the complex plane is locally polar, or the logarithmic capacity of E is 0, if for each point x of E, there exist a connected neighborhood U of x and a subharmonic function cp $ -oo on U such that E n U C {y E U I cp(y) = -oo}.
THEOREM 5.15. For an open set f2 in the complex plane and a closed subset E of f1, we have A2(f2) = A2(f1\E) if and only if E is locally polar.
For the proof, see (7] and [42].
The concept of local polarity naturally extends to several variables.
DEFINITION 5.16. A subset E of Cn is said to be locally pluripo-
lar if each point x of E has a connected neighborhood U and 0 E PSH(U)\{-oo} such that E n U c {x E U 1,0(x) = -oo}. An analytic subset X of a domain f2 is locally pluripolar. In fact,
for a system {f} of local defining functions of X, it will do to set := log E If. I2 . a
The next statement can easily be proved by applying Theorem 5.15 to a function with parameter (the details are omitted). THEOREM 5.17 (J. Siciak). If a closed subset E of f2 is locally pluripolar, then A2(f2\E) = A2(fl). As an application of this theorem, we show that a bijective holomorphic mapping F : f21 - f12 between domains is biholomorphic. In fact, if f denotes the Jacobian of F, then F-1 is holomorphic on i72\F(V (f )). However, setting (z)
floglf(F'-1(z))I for z ¢ F(V(f)). for z E F(V(f)), 1-oo
since Sard's theorem implies -0 # -oo, F(V(f)) turns out to be locally pluripolar, and from Theorem 5.17, it follows that F- I is holomorphic on $22.
A generalization of Theorem 5.15 in a rigorous sense is as follows:
THEOREM 5.18. For two bounded pseudoconvex open sets f11 D f12 # 0, we have A2(f11) = A2(f12) if and only if for any point zo of
5. EXTENSION AND DIVISION PROBLEMS
102
fl2 and any complex line l through zo, l n (nl\n2) is locally polar in 1.
PROOF. Sufficiency is a direct consequence of Theorems 5.15 and 5.10, and necessity is obvious from the same theorems with p = 0.
REMARK. Due to Josefson's theorem, given a pluripolar set E. there exists an element i of PSH(C)\{-oo} such that E C {z E Cn I z':(z) = -oo}. From this, in particular. it follows that a countable union of locally pluripolar sets is locally pluripolar.
5.4.2. Proof of Demailly's Theorem. Siu's theorem has shown a similarity between locally polar sets and analytic subsets. Demailly's theorem, used for the proof of that theorem, is a beautiful application of the L2 extension theorem as stated below. PROOF OF DEMAILLY'S THEOREM. Let the notation be that of
§3.2. First, regarding the sum of the series E Ia,(z)12, since {al} is the orthonormal basis, E Iai(z)12 coincides with the square of the norm of the following linear function on A2mtP(fl):
am: A2 W
W
f
f(z).
From this by Cauchy's estimate it follows that E 1at12 converges uni-
formly on compact sets in P and is of class C° on fl, and that the following equation holdds: (5.41)
zitm(z) = sup j
log If(z)I i f E A2mv(n), 1If II2mC' = 1 } .
For 0 < E < an(z) and f E A2.1,..,(17), since If 12 E PSH(fl), nl .nE2n
If(()12 dl' J It-=I 0, it follows that. 881ogKo(z, z) is a (1, 1)-form that is invariant under the action of Aut S2. In other words, as it is clear that 88 log Kn(z, z) > 0, the action of Aut .f2 is isometric with respect to the Hermitian metric: n
E 1,k=1
82 log Kn(z, z) dxj o d---k azjazk
.
This metric is called the Bergman metric of V. THEOREM (Bremermann's Theorem). If fl is complete as a metric space with respect to the Bergman metric, then fl is obviously a connected domain of holomorphy. Hence, a bounded homogeneous domain in en turns out to be pseudoconvex.
108
6. BERGMAN KERNELS
We will describe an application of the Bergman kernel to holomorphic mappings.
Caratheodory's theorem in the theory of conformal mappings states that if there exists a biholomorphic mapping F between domains 111 and 112 of the complex plane, and if each of 0111 and 0172
consists of a finite number of simple closed curves, then F can be extended to a homeomorphism from 111 to 112.1 A generalization of this to several variables is the following theorem:
THEOREM 6.3 (Fefferman's theorem). Assume that there exists a biholomorphic mapping F between strongly pseudoconvex domains 1l1 and 02 in Cn , and 0111 E C°°. Then F can be extended to a diffeomorphism of class C°° from j71- to 112. PROOF. Let Pi : L2(112) -p A2(11i) be the orthogonal projection. If we set
u := det
0Fj
84
then from the transformation formula (6.3) we see that
dgEL2(f12). By the method of indeterminate coefficients, for an arbitrary h E there exists an element v of CO.1(?12)f1Dom0* such that any derivative of h - e* v is equal to 0 on 0112, where we apply the method of indeterminate coefficients to the coefficients of the formal power series of the defining functions of 112. (See the proof of Theorem 4.18.) Therefore, for any h E C°°(172), there exists an element ho of Co (C) such that suppho C 722 and P2(ho 1112) = P2 h. Regarding an element g of C°°(111), we write, for simplicity, g E C'(771-) when g can be extended to .71 as a C°° function. Since the transformation formula implies in order to show that Fj E C°D (111) (1
n), it suffices to prove
that (6.5)
u (ho o F) E C°°(f11) .
'See Einar Hille, Analytic Flunction Theory, vol. II, Theorem 17.5.3, Ginn and Co., Boston, Mass., 1962.
6.2. TRANSFORMATION LAW AND AN APPLICATION
109
In fact, in this case, since Kohn's theorem (see Theorem 3.29) implies P, (u (ho o F)) E C' (71), we have (6.6)
u (h o F) E C°° (511) .
Hence, by putting h = 1, we obtain u E C°°(521).
The same argument is applicable to F-1, and in particular, it
_
turns out that u does not have any zero point on 511. Therefore, from (6.6), it follows that h o F E C°°(711). If we use this for h = zj (1 < j < n), then F3 E C°O(Q1), and F can be extended to .f11 as a C°° function.
As a similar argument applies to F'1, it follows that F can be extended to a diffeomorphism of class C°° from n1 to 512. PROOF THAT u (ho o F) E C°° (5l2). Since F3 is a bounded holomorphic function, by Cauchy's estimate we have IFIfa)(z)1 < cabnI
(z)-(a),
z E 111,
_
where ca is a constant that does not depend on z. Since from ho E Co (C') we see that supp ho C .R2, in order to say that every derivative of u (ho o F) is bounded, it is sufficient to show that there exists a constant C such that (6.7)
JR2 (F(z)) < C5n, (z),
z E 121.
However, since for a strictly plurisubharmonic defining function r of l1,, r o F-1 is both negative-valued and subharmonic on 512, from Hopf's lemma we get (6.8)
r o F-1(w) < -C1bn2(w), WE 172,
for some constant C1 > 0. (For Hopf's lemma and its proof, see Proposition 6.4 below.) From (6.8) and the self-evident inequality
-r(z) < C26n, (z) for some constant C2, if we set C = CIC2, then (6.7) holds. Hopf's lemma used in the above proof is the following proposition: PROPOSITION 6.4 (E. Hopf). Let f2 be a bounded domain in 1R''
whose boundary is of class C2, let a be a boundary point of 12, and let v(a) be the inward unit normal line to 80 at a. Then for an arbitrary negative-valued subharmonic function u(x) on .R, there exists
6. BERGMAN KERNELS
110
a positive number c such that lim u(x) < -c,
(6.9)
Ix - al -
where the superior limit on the left hand side is taken when x E v(a)
and x - a. PROOF. If we take a point x0 on v(a) sufficiently close to a, there exists an open ball B(xo, R) with center x0 of radius R within .R whose
boundary is tangent to OR at a. For a fixed R' with 0 < R' < R. we can take a sufficiently large positive number A such that a function v(x) := e-ajX_b0 2
- e-AR2
is subharmonic2 on B(xo. R) \ B(xo, R'), and satisfies v 18B(xo, R) _ 0. If we choose a positive number e such that
sup{u + ev I Ix - xol = R'} < 0, the maximum principle for subharmonic functions implies u < -Ev on B(xo, R) \ B(xo, R'). From this, (6.9) is evident.
6.3. Boundary Behavior of Bergman Kernels The proof of Fefferman's theorem contained in the original article was
based on a rigorous analysis of the boundary behavior of Bergman kernels. This would be the main road in the sense of tackling the singular points of reproducing kernels directly, but this much analysis of Bergman kernels requires some treatment of the so-called degenerate elliptic boundary value problems that involve slightly more precise tools than merely the L2 estimates. In fact, Kohn's theorem used in the above proof is one of those precise tools, and we have no space for
them in the present book. However, as the method of L2 estimates is able to deduce interesting general properties on the singularity of Bergman kernels, we will describe these below. The next characterization of the value of Kn(z, z) is often used: (6.10)
Kn(z. z) = suP{if(z)I2 I f E A2(Q) 11f 11 = 1}.
2This means Av ? 0 only for this place.
6.3. BOUNDARY BEHAVIOR OF BERGMAN KERNELS
111
In fact, Kn(z, z) is nothing but the square of the norm of the following linear mapping: A2 (.fl) w
f
C w
'' f (z)
Also, since Kn is a reproducing kernel, it turns out that the function that realizes the right hand side of (6.10) is the following one:
eie Kn(.. z)
(0 E R).
V K0(z, z)
It is clear from (6.10) that for an open subset (1* of (1,
(zE.fl`).
(6.11)
Concerning the boundary behavior of Bergman kernels, the following fact is the most fundamental: THEOREM 6.5. If rt E C2(C') satisfies li I z1 -2 n(z) = 0, and if m an open set
fl1 = zECnImzn+1zj1 2+n(z) 1. Kno(z, z) _
0 COROLLARY 6.6 (L. Hormander, K. Diederich). Let fl be a pseu-
doconvex domain in Cn, and zo a strongly pseudoconvex boundary point of f1. Let r be a defining function of fl around zo, and k(zo) the Jacobian of the Levi form of r at zo. If Igradr(zo)I = 1, then (6.13)
lim Kn(z, z)bn(z)"+1 =
47r" k(zo).
For general domains, no clear relation, as seen above, between the Levi forms of defining functions and the boundary behavior of Kn is known. But the next statement is fundamental in a different sense from the above.
THEOREM 6.7. If fl is a bounded pseudoconvex domain with boundary of class C2, then lim Kg (z, z)bn(z)2 > 0. (6.14)
z-8n
6.3. BOUNDARY BEHAVIOR OF BERGMAN KERNELS
113
PROOF. In the case n = 1, from the condition we conclude that for each point zo of Oft there exists a circle of constant radius that is contained in C \ 12 and is tangent to 01? at zo. Since the Bergman kernel for the outside of the closed disk satisfies (6.14), from (6.11), (6.14) holds even for .f1. In the case n ? 2, it suffices to make use of the L2 extension theorem, Theorem 5.10 (we leave the details to the 0 reader). Concerning the Bergman metric, the following is basic:
THEOREM 6.8 (K. Diederich). If f1 is a pseudoconvex domain, and zo is a strongly pseudoconvex boundary point of f2. then there exist a neighborhood U of zo and a positive number C such that for any z E f1 fl U. 85 5 as e6 2 00 log Ks? (z, z) >
-
± COOT z I
(the double symbols read in the same order), where b := Sn. For the proof, see [12]. A natural question arises on the estimate of the distance function d(z, w) with respect to the Bergman metric: CONJECTURE. If 1? is a bounded pseudoconvex domain with
boundary of class C', then for any zo E f1,, there exists a positive number C such that d(zo. z) >
1 1 logbn(z)I - C.
At present the following is known in this direction: THEOREM 6.9. If f1 is a bounded pseudoconvex domain with
boundary of class C2, then for any zo E f1. there exists a positive number C such that d(zo, z) >
log (I log bn(z)I) - C.
For the proof, see [15]. C For strongly pseudoconvex domains with boundary of class C", the following decisive result is obtained, and there are several studies modeled on this: THEOREM 6.10 (C. Fefferman [17]). If ft is strongly pseudoconvex, and if OS? E CO°, then there exist functions cy, W E Cx(f1) and a number c > 0 such that KQ(z. z) = y7(z)8n"-1(z) + i'(z) logbn(z). z E fl\f1E .
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Index dim Xa. 86 Dom T, 56 log D, 19 (log D) ", 10
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180 A. A. Milyutin and N. P. Osmolovskii, Calculus of variations and optimal control, 1998 179 V. E. VoskresenskiT, Algebraic groups and their birational invariants, 1998
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170 Viktor Prasolov and Yuri Solovyev, Elliptic functions and elliptic integrals, 1997
169 S. K. Godunov, Ordinary differential equations with constant coefficient, 1997
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163 A. Ya. Dorogovtsev, D. S. Silvestrov, A. V. Skorokhod, and M. I. Yadrenko, Probability theory: Collection of problems, 1997
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One of the approaches to the study of functions of several complex
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Emphasis is on recent results, including an L' extension theorem for holomorphic functions, that have brought a deeper understanding of pseudoconvexity and plurisubharmonic functions.
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