[Grundlehren der mathematischen Wissenschaften] Entire Functions of Several Complex Variables Volume...

Grundlehren der mathematischen Wissenschaften 282 A Series o/Comprehensive Studies in Mathematics

Editors

M.Artin S.S.Chem 1.M.Frohlich E.Heinz H. Hironaka F. Hirzebruch L. Hormander S.MacLane W.Magnus C.C.Moore 1.K.Moser M.Nagata W.Schmidt D.S.Scott Ya.G.Sinai 1. Tits B.L. van der Waerden M. Waldschmidt S.Watanabe

Managing Editors

M. Berger B. Eckmann S.R.S. Varadhan

Pierre Lelong Lawrence Gruman

Entire Functions of Several Complex Variables

Springer-Verlag Berlin Heidelberg New York Tokyo

Professor Dr. Pierre Lelong

Universite Paris VI 4, Place Jussieu, Tour 45-46 75230 Paris Cedex 05 France

Dr. Lawrence Gruman

UER de mathematiques Universite de Provence 3 place Victor Hugo 13331 Marseille France

Mathematics Subject Classification (1980): 32A15

ISBN-13: 978-3-642-70346-1 e-ISBN-13: 978-3-642-70344-7 DOl: 10.1007/978-3-642-70344-7

Library of Congress Cataloging in Publication Data Lelong, Pierre. Entire Functions of several complex variables. (Grundlehren der mathematischen WissenschaFten; 282) Bibliography: p. Includes index. \. Functions, Entire. 2. Functions of several complex variables. I. Gruman. Lawrence, 1942-. II. Title. III. Series. QA353.E5L44 1986 515.914 85-25028

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© Springer-Verlag Berlin Heidelberg 1986 Softcover reprint of the hardcover I 5t edition 1986

2141/3140-543210

Introduction

I - Entire functions of several complex variables constitute an important and original chapter in complex analysis. The study is often motivated by certain applications to specific problems in other areas of mathematics: partial differential equations via the Fourier-Laplace transformation and convolution operators, analytic number theory and problems of transcen-dence, or approximation theory, just to name a few.

What is important for these applications is to find solutions which satisfy certain growth conditions. The specific problem defines inherently a growth scale, and one seeks a solution of the problem which satisfies certain growth conditions on this scale, and sometimes solutions of minimal asymp-totic growth or optimal solutions in some sense.

For one complex variable the study of solutions with growth conditions forms the core of the classical theory of entire functions and, historically, the relationship between the number of zeros of an entire function f(z) of one complex variable and the growth of If I (or equivalently log If I) was the first example of a systematic study of growth conditions in a general setting.

Problems with growth conditions on the solutions demand much more precise information than existence theorems. The correspondence between two scales of growth can be interpreted often as a correspondence between families of bounded sets in certain Frechet spaces. However, for applications it is of utmost importance to develop precise and explicit representations of the solutions.

If we pass from

VI Introduction

were introduced in 1942 by K. Oka and P. Lelong. They occur in a natural way from the beginning of this book. Indicators of growth for a class of entire functions f are obtained as upper bounds for loglfl; for log If I. To solve Cousin's Second Problem, i.e. to find (with growth conditions) an entire function f with given zeros X in

Introduction VII

bibliography, without pretending to be exhaustive, gives an overview of those areas of current interest. Each chapter has a short historical note which is an attempt to explain the origin of the given results.

III - Chapter 1 gives the basic definitions of the growth scales in (Cn, the notion of order and type, the indicator of growth and proximate orders. These classical notions extend trivially to plurisubharmonic functions and to entire functions in (Cn. In Chapter 2, we introduce the reader to the funda-mental properties of positive differential forms and of positive and closed currents. Chapter 3 studies the solution with growth conditions of the equa-tion iaaV=e for e a positive closed current of type (1,1) in (Cn, from which we deduce for V=loglfl the solution with growth conditions in (Cn of Cousin's Second Problem and the representation of entire functions with a given zero set. The result for an entire function of finite order in (Cn gives an extension of classical results of J. Hadamard and E. Lindelof for n= 1. Chap-ter 4 studies the class of entire functions f of regular growth. Certain results are given here for the first time. The importance of this study, which is based on the preceeding chapters, is in the numerous applications (Fourier transforms, differential systems) and the possibility of associating the regular growth of log If I with the regular distribution of the zero set of f

Chapter 5 studies the problems of entire maps F: (Cn --* (Cm. The first portion is devoted to the development of a representation of an analytic subvariety Y of (Cn as the zero set of an entire map F: (Cn --* (Cn+ \ that is Y =F- 1 (0), F=(fl' ···,jn+l)' with control of the growth of the function IIPII. The second part studies the growth of the fibers F-l(a)nB(O, r), where B(O,r)={z: Ilzll

VIn Introduction

distributions defined in a bounded domain Q of

Table of Contents

Chapter 1. Measures of Growth 1

§ 1. Preliminaries .............. 1 § 2. Subharmonic and Plurisubharmonic Functions 2 § 3. Norms on (Cn and Order of Growth .... 5 § 4. Minimal Growth: Liouville's Theorem and Generalizations 6 § 5. Entire Functions of Finite Order 8 § 6. Proximate Orders . . . . . . 14 § 7. Regularizations ....... 18 § 8. Indicator of Growth Functions 20 § 9. Exceptional Sets for Growth Conditions 24 Historical Notes . . . . . . . . . . . . . 28

Chapter 2. Local Metric Properties of Zero Sets and Positive Closed Currents 30

§ 1. Positive Currents 30 § 2. Exterior Product 35 § 3. Positive Closed Currents 37 § 4. Positive Closed Currents of Degree 1 40 § 5. Analytic Varieties and Currents of Integration 48 Historical Notes . . . . . . . . . . . . . . . 58

Chapter 3. The Relationship Between the Growth of an Entire Function and the Growth of its Zero Set 59

§ 1. Positive Closed Currents of Degree 1 Associated with a Positive Divisor . . . . . . . . . 60

§ 2. Indicators of Growth of Cousin Data in (Cn . . 63 § 3. Canonical Potentials in 1R m • • • • • • • • • 64 § 4. The Canonical Representation of Entire Functions of Finite Order 67 § 5. Solution of the aa Equation . . . . . . . . . 73 § 6. The Case of a Cousin Data . . . . . . . . . 77 §7. Slowly Increasing Cousin Data: the Genus q=O;

the Algebraic Case . . . . . . . . . . . . . 79 § 8. The Case of Integral Order: Extension of a Theorem of LindelOf 82

X Table of Contents

§ 9. Trace of a Cousin Data on Complex Lines 86 § 10. The Case of a Cousin Data of Infinite Order 89 Historical Notes . . . . . . . . . . . 94

Chapter 4. Functions of Regular Growth 95

§ 1. General Properties of Functions of Regular growth 96 § 2. Distribution of the Zeros of Functions of Regular Growth 106 Historical Notes . . . . . . . . . . . . . . . 114

Chapter 5. Holomorphic Mappings from ([:" to ([:m 116

§ 1. Representation of an Analytic Variety Yin ([:" as F- 1(0) 117 § 2. Local Potentials and the Defect of Plurisubharmonicity 118 § 3. Global Potentials ............. 123 § 4. Construction of a System F of Entire Functions

such that Y=F- 1(0) 126 § 5. The Case of Slow Growth 130 § 6. The Algebraic Case 133 § 7. The Pseudo Algebraic Case 136 § 8. Counterexamples to Uniform Upper Bounds 136 § 9. An Upper Bound for the Area of F-l(a) for a Holomorphic Map 138 § 10. Upper and Lower Bounds for the Trace of an Analytic Variety

on Complex Planes 143 Historical Notes . . . . . . . . . . . . . . . . . . . . 154

Chapter 6. Application of Entire Functions in Number Theory 155

§ 1. Preliminaries from Number Theory 155 § 2. A Schwarz Lemma . . . . . . . . . . . 160 § 3. Statement and Proof of the Main Theorem 162 Historical Notes . . . . . . . . . . . . . 165

Chapter 7. The Indicator of Growth Theorem 167

Historical Notes . . . . . . . 176

Chapter 8. Analytic Functionals 177

§ 1. Convex Sets and the Fourier-Borel Transform 178 § 2. The Projective Indicator ......... 17'13 § 3. The Projective Laplace Transform . . . . . 183 § 4. The Case of M a Complex Submanifold of ([:" 185 § 5. The Generalized Laplace Transform and Indicator Function 186 § 6. Support for Analytic Functionals 188 § 7. Unique Supports for Domains in ([:" 191 § 8. Unique Convex Supports 195 Historical Notes . . . . . . . . . . . 200

Table of Contents XI

Chapter 9. Convolution Operators on Linear Spaces of Entire Functions ............. 201

§ 1. Linear Topological Spaces of Entire Functions 201 § 2. Theorems of Division .......... 207 § 3. Applications of Convolution Operators in the Spaces E~(r) and EO 210 § 4. Supplementary Results for Proximate Orders with p> 1 212 § 5. The Case p = 1 ............ 217 § 6. More on Functions of Order Less than One 225 § 7. Convolution Operators in (Cn 227 Historical Notes . . . . . . . . . . . . . . 229

Appendix I. Subharmonic and Plurisubharmonic Functions 230

Appendix II. The Existence of Proximate Orders . . . . . 242

Appendix Ill. Solution of the E-Equation with Growth Conditions 245

Bibliography 254

Index . . . 269

Chapter 1. Measures of Growth

§ 1. Preliminaries

We will let (C represent the field of complex numbers and lR the subfield of real numbers. Let Z=(ZI' ... , z,,) be an element of (Cn and lR 2", the underlying space of real coordinates. The transformations from the complex to the

I d · . b d Zk + Zk rea coor mates are gIVen y zk=xk+iYk' zk=xk-iYk an xk=--, - 2 Yk = Zk ~ Zk. Unless specified to the contrary, we equip (C" with the Euclidian

metric of lR 211:

" " (1,1) ds 2 = L (dx~+dYD= L dzk·dzk

k~l k~1

and we choose for (C" the volume form

" '[ = /\ (dXk /\ dh) = (iI2)" dZ I /\ dZI /\ ... /\ dz" /\ dz". k~ 1

By a domain Q, we shall always mean an open connected set. We let dQ(z), the distance to the boundary, be defined for ZEQ by dQ(z) = inf Ilz-z'll

z'¢Q

(where II II represents the Euclidean norm) and set dQ(z) = + 00 if Q = (C". Let 1X=(1X1 , ... ,IX,,) be a multi-indice of non-negative integers. We then define IIXI

n al'l by IIXI = L lXi' the differential operator D' by D' , and z' by

i=l az~l ... az~n za = Z~l ... z~n.

We let ~k(Q) be the set of functions defined on Q all of whose deriva-tives up to order 11X1;£k are continuous and ~(.{)(Q) the set of functions whose derivatives of all orders are continuous. By ~~(Q) (resp. ~;; (Q)) we will mean the subset of ~~(Q) (resp. ~;;(Q)) composed of those functions whose support in Q is compact. We let a and a be the exterior differential operators defined by

n a a= L -a'7 dzk ,

k~ 1 ~k

and set

2 1. Measures of Growth

A function f: 0 c

§2. Subharmonic and Plurisubharmonic Functions 3

families included in a larger class of functions, the plurisubharmonic func-tions introduced by K. Oka and P. Lelong. This family is closed under the operation of taking the smallest upper semi-continuous majorant of a fil-tered family uniformly bounded above on compact subsets (in fact, one can show that the functions Clog If I, f E x( 1 complex variables that the subharmonic function play in complex analysis of one complex variable. Moreover, in


v) if cpE PSH(Q) and D(z, w, r) c Q, then the two functions r ---+ l(z, w, r, cp) and r ---+ sup cp(z) are increasing convex functions of log r, and as a con-

ZED(z, lV, r)

sequence },(z, r, cp) and M (z, r, cp) = sup cp(z') are increasing convex func-tions of log r (Proposition I.17); liz' -zll ;;r

vi) if F: Q ---+ Q' is a holomorphic homeomorphism (analytic isomor-phism) of Q onto Q', then the map T:CPEPSH(Q)---+cpoF- I EPSH(Q') is a bijection (i.e. the class of plurisubharmonic functions is invariant with re-spect to holomorphic homeomorphisms and is thus an object of the analytic structure; this is false for the larger class SeQ)).

The preceeding remarks shall playa crucial role, since for fE£'(

~ 3. Norms on


Given a function a(z): O}, we consider

(1,4) Ma,p(r)= sup a(z), p(z) :;;;r

It then follows from (1,3) that there exist constants C and C, 0< C < C < 00, depending only on p(z) and q(z), such that for every real valued function a(z)

(1,5)

The functions we shall consider will often be plurisubharmonic, and in this case we have:

Proposition 1.4. If cp(z) is plurisubharmonic in

§4. Minimal Growth: Liouville's Theorem and Generalizations 7

ii) if p(z) is a norm on ° such that M "', per 0) > - 00, and since it is an increasing convex function of logr, M""p(r) > -00 for 1';:;:1'0' which proves that C;:;:O. If cp(z+uz')= -00, then for 1'>1, (logr)-lm",(z,z',r)= -00, hence C(z, z') = - 00. Otherwise, cp(z + uz') is an lR2 -subharmonic function of u and hence by the above reasoning. C(z, z');:;: 0. From the definition, by an obvious calculation we obtain C(z, z') = C(z, uz') for u =l= 0. Part (ii) follows directly from Proposition 1.4. 0

Theorem 1.6. Suppose fEJ'l'(p. If fez) is a polynomial of degree m, then If(rz')I~!pm(z')rml+O(rm-l), from which the second part follows. 0


Since M

§5. Entire Functions of Finite Order 9

is + 00, finite, or zero, and a is said to be the type of a with respect to p(z). For 0 p(z)~l

the type a of f with respect to p(z) are given by

a) -~= lim sup IOlg Cq = lim sup -11- [sup log 1~(z)l]' p q~oo q ogq q~oo q ogq p(z)~l

b) log a = lim sup [IOlg Cq +~] P log q -log p-1. q~oo q ogq p

b') log ae p = lim sup (~log Cq + log q). q~(jJ q

c) a=li~}~p [r (~) cqfq. d) For p=l, a=limsup[q!Cqr/q.

q~oo

Proof Let Zq be a point on the unit p-ball for which 1~(zq)1 = Cq. By a rotation, we can assume that Zq=(Xq, 0, ... ,0). If J(u)=f(u, 0, ... ,0) and /(u)

00

= L amum is the Taylor series expansion of J at the origin, then laqx~1 m=O

= Cq, and by the Cauchy Integral Formula, Cq;£r- q exp Mf,p(r). If


MI,p(r)~Ark for r>R(A, k), then Cq~r-q expArk. Since

d dr (r- q exp Ark) =r-q- l ( -q + Akrk) exp Ark,

( q )l/k the minimum of this expression occurs when r = kA and

(e~krk. We then have for q sufficiently large

(1,6) log Cq~f [1 +logAk-Iogq].

IS equal to

Thus, given 8>0, for q sufficiently large, k~ qllogq -8, so if f is of finite - og Cq

. q log q. . log C . order p, p~hmsup or eqUlvalently, -p-l~hmsup--q. If f IS

q~oo -logCq q~if) qlogq of finite type a and A>a, (1,6) gives (with k=p): Aep~qC~/q for q suf-ficiently large (depending on A), so that ae p ~ lim sup q C~/q.

q~if)

(eAk)q/k Suppose now that Cq~ -q- for q~qo(A, k). If q>mr, where mr is

the largest integer smaller than or equal to 2keAkrk and r is sufficiently large, then for p(z)~r,

(eAk)q/k (1 )q/k 1~(z)l~ -q- rq~ 2krk rq~2-q Thus If(z)1 < L Cqrq +2 for p(z)~r. Let f1(r)=sup Cqrq. Then

q=O q

expMI,p(r)~(1 +2keAkrk)f1(r)+2. By our assumption on Cq

f1(r)~s~p (e~krk rq~s~p (e~krk rX.

This latter function attains its maximum for x=Akr\ and so f1(r)~expArk and MI , p ~ A rk + (k + 1) log r for r sufficiently large. This shows that

By Stirling's Formula, we have

logF(x)=x(logx -1) -~ logx+~ log2n+O(x-l).

Thus, we obtain . [r(qjp)Jl/q

hm l/p (ep)-l/p. q~if) q o

§ 5. Entire Functions of Finite Order 11

Corollary 1.10. If f is an entire function of order p and type (J with respect to a norm p(z), then its restriction flL to any linear subspace L is of order at most p and type at most (J (with respect to the restriction pIL)'

Remark. If f (z) = I aaza is the Taylor series expansion at the origin of an a

entire function, then the order p can be calculated and the type (J (with respect to a given norm) estimated by the Taylor series coefficients aa (as was previously shown, the order p is in dependant of the norm, but not the type (J). We consider the classical situations p(z)= Ilzll and p(z)= sup Izkl. We compare mq = sup laal and Cq = sup 1~(z)1 l;;;k;;;n

lal~q p(z)';l

a) p(z) = 114 The unit ball contains-the disc IZkl ~ 1 ~, k = 1, ... , n. By the Cauchy Integral Formula, we have V n

laal~Cq(Vn)q for lal=q so mq~nq/2Cq.

Conversely, Cq ~s(n, q)mq, where s(n, q) is the number of monomials in ~, and s(n,q)~(q+l)". Then

q log mq -"2. log n ~log Cq~log mq + n log (q + 1).

From Theorem 1.9 (i), we deduce that -~=lim sup 10lgmq and from (ii) that p q~oo q ogq

n- P/ 2 (ep)-1 B'~(J~(ep)-l B' with B' = lim sup (q sup laaI P/q).

b) p(z)= sup IZkl. In the same way, we obtain 1 ;;;k;;;n

q~oo lal~q

logmq~log Cq~logmq+n log(q+ 1).

Therefore, p and (J are given by (i), (ii), and (iii) of Theorem 1.9 where we replace Cq by mq = sup laJ

lal~q

The order p defined above is sometimes called the total order of f(Zl' ... , Zn)E£,((Cn). We are sometimes led to study the growth of an entire function f E £' ((Cn X (CP) where the variables ZE (Cn and ~ E (CP play different roles. This is the case, for instance, in the study of Fourier transforms of linear differential and pseudo-differential operators. This leads us to define the growth and the order with respect to each variable Z and ~ separately.

Let f E £' ((Cn) where (Cn = (C X .•• x (C. We consider the .. function M f. P (I') = sup log If(zl' ... , zn)1 with respect to the norm p(z)=sup IZkl and write

IZkl ;;;r k

Mf(r) instead of Mf.p(r). In addition, we let Mf(rl' ... , I'll) = sup log If (Zl' ... , zn)l·

IZkl ;;;rk

Definition 1.11. Given fE£,((Cn), we define the indicator of growth of f with respect to the variable Zk' Mik)(r), by

(k) _-(1,7) Mf (r)-Mf (1, ... , 1,r, 1, ... , 1)


(/] is replaced by 1 in Mj (r1 , ""rn) if j*k and rk=r), We say that f is of order Pk in Zk if Mik) (I') is of order Pk'

Remark, If f is of finite order P, then Pk:'£P, for we have Mik)(r):,£Mj,p(r) for r~ 1.

Reciprocally, we shall see that there exists an upper bound for the total order P in terms of the Pk' We note that for I Zk 1= rk (1,8) Mj (r1 , "', rn)=sup log If(1X 1 Z1 , "', IXnzn)1

lakl ~1

is a convex function of each Uk = log IZkl with finite values for - 00 < ak:,£ uk:'£ bk < + 00, It follows from the continuity of log If I in a neigh-borhood of every point Z for which f(zHO that ljJ(u 1 , "', Un) = Mj (r1 , "', I;') is continuous for U = (u 1 , "', Un)ElR n, By (1,8), it is a plurisubharmonic func-tion of Z=(ZI' ""zn) which depends only on Izkl, so ljJ(u) is a convex function of U (cf, Proposition 1.25), Hence Mj (r1"", I;,) is a continuous convex function of the variables log rk = Uk which is increasing in each Ik,

Let (u 1 , "', Un)E(lR+t The convexity of ljJ(u) then gives n

ljJ( C1 U1 , "', Cnun):,£ I CkljJ(O"", 0, Uk' 0, ",,0) k~1

n

for every vector C=(CI"",Cn) such that O:,£Ck:,£l and I Ck=1. We rewrite this k~ 1

n

(1,9) M- (r r) < " C M(k)(r') wl'th vk' = l"kl/Ck , j I"'" n = L., k j k ,; k~1

From formula (1.9), we can immediately make two observations:

i) if the growth in Zk is of finite order for every k, then f is of finite order;

ii) the order in Zk could as well be defined using any polydisc Izjl:'£ rjO, Iznl:,£r,rjO fixed, in place of the polydisc IZjl:,£l, Izkl:,£r,

Theorem 1.12. We have n

(1,10) M.r(r) = Mj(r, ",,1'):,£ I Yk 1 Mik) (r1'k) k~1

n

for every system of numbers (Yl' "', Yn) with Yk>O and I Yk 1 = 1. k~1

Proof We let rk=r in (1,9), Then r~=rYk, D

Corollary 1.13. Let f EYl'(

§5. Entire Functions of Finite Order 13

Proof The first inequality is a result of the inequality Mt)(r);£Mf(r) for r ~ 1. To prove the second, we note that if Pk is finite, then for every P~ > Pk'

n

there exists Ak such that Mjk)(r);£AkrP~, 1 ;£k;£n, so if we set A= L p~ and "h=APi:\ then from Theorem 1.12 we obtain k=1

Mf(r) = Mf(r, ... , r);£A- 1 (± PkAk)ra • k=l o

We shall now apply (1,8) in another context by letting the numbers U 1 , ... , un - 1 remain fixed as Un goes to infinity and choosing the Ck to be variable functions of Un'

Theorem 1.14. Let fEJ'f(l such that 8(r) goes to zero when r tends to infinity and for r > 1'0'

(1,11) Mf (, l' ... "n-l' r);£ Mf (l, ... ,1, r1 +e(r» = Mj")(r1 +e(r»

The function 8(r) and ro depend on the 'j and f

Proof From (1,9), we obtain n-1

M- ( )ro' where ro is large enough so that M(n) (ro) > sup Mt)('j ln-l). For r>ro, O((r) is a continuous increasing function

k

[ 0((1') J- 1 of r and lim O((r)= + 00. Let 8(r)= ---1 . Then 8(r) decreases to zero d r~w n-1 an

C((r);£ k= /~~ n-l Ck(r)-I;£ (n -1) [t~ Ck(r) r 1 hence 8(1') ~ [Cn(r)] -1 -1. o


Remark. Theorems 1.12 and 1.14 can be applied in the same way to an entire function f E JIf (E), E = E 1 X ... X E" with Ek =

§6. Proximate Orders 15

Definition 1.17. A function L(r) defined on 1'>0 is said to be slowly increas-ing if for every compact interval I of (0, + (0), for every 8> 0, there exists 1'0

IL(kr) I such that for 1'>1'0' --)--1 0 be given. By definition

(L(kr)) log -- =[p(kr)-pJ logk+[p(kr)-p(r)] logr. L(r)

We first assume that 0< a ~ k < 1. By the Mean Value Theorem,

Ip(r) - p(kr) 1= Ip' (~)I r-kr for some ~, kr ~ ~ ~ r so by (ii) of Definition 1.15, there exists R1 such that

8 Ip'(~)1:S3(a-1-1)~log~ for ~>aR1' Thus, for I' sufficiently large, we have

the following bounds: 8(lOg 1') (r - kr)

(logr)lp(kl')-p(r)l< -1 3(a -l)krlogkr

8 (a- 1 -1) ~3( -1 1) 1 logr


Note. Since in the study of the asymptotic properties of entire functions we are only interested in their properties for r sufficiently large, we can always change p(r) on a bounded set without affecting the asymptotic properties we study. Thus, for p > 0, we can always assume that rp(r) is everywhere strictly increasing on the set I' > 0.

Proposition 1.20. Given B > 0, there exists an R(B) such that

(1 -B)kPrP(r) < (kr)p(kr) < (1 + B)kPrP(r)

uniformly for 0< a;£ k;£ b < + 00 for I' > R(B).

Proof By Theorem 1.18, for I' large enough

(kr)p(kr)rP (l-B)< () ::;;(1 + B). - (kr)prP r - o

Definition 1.21. A proximate order p(r) is a strong proximate order if p(r) is twice continuously differentiable for 1'>0 and lim p"(r)r210gr=0.

r~oo

Since we are interested in convexity properties, the following is essential:

Proposition 1.22. If p(r) is a strong proximate order, p > 0, then rP(r) is a convex function of log I' for r large. If p> 1, then rP(r) is a convex function of I' for r large.

Proof By Proposition 1.19, rP(r) is an increasing function of r. A simple calculation shows that

d ~---,- rP(r) = p(r)rP(r) +rP(r) p'(r)r log r, d(log r)

which is positive for r sufficiently large by (i) and (ii) of Definition 1.15. Furthermore,

d2 ~~--;o-rp(r) = I' {p' (r)rP(r) + p(r)2 rP(r)-l d(logrf

+ p(r) rP(r) p'(r) log r+ p"(r) rP(r) log r + p'(r) rP(r)-l + p(r) p'(r) rP(r)-l log r + [p'(r) r log r] 2 rP(r)}

2

>L rP(r) for I' sufficiently large 2

by (i) and (ii) of Definition 1.15 and Definition 1.21. Similarly, a simple calculation shows that

§6. Proximate Orders 17

d2 r P(r) --= p'(r)rp(J-)-1 + p(r)(p(r) _1)rP(r)-2

dr 2

+ per) rP(r)-1 p'(r) log I' + p"(r) rP(r) log I' + p'(r) rP(r)-1

+ per) pi (r)(log r)rP(r)-1 + [pi (r) log r J2 rP(")

> pcp 2- 1) rP(r) -1 for r sufficiently large. o

A fundamental result that we shall need (for the proof see Appendix II) is that for any positive continuous increasing function a(r) of finite order p there exists a (strong) proximate order with respect to which a(r) is of normal type. We apply this result to Mj,p(r) for fEYl'( 0, if p > 0 the equation t=rP(r) admits a unique solution for t>O. We will denote by r=


If we integrate from t to kt, we obtain

(1) (CP(kt)) (1 ) p-8 logk (J. Then, as in the proof of Theorem 1.9, it follows from

the Cauchy Integral Formula that log Cq < (j rP(r) -q log r for r large, and if rq is the solution of the equation q = (j p rP(r), then for q large we have

10gCq p. Then for q>2P(J"PlrPl, we have the bound Cqrq

§7. Regularizations 19

plurisubharmonic). However, if the family is infinite, even when it is uni-formly bounded on compact subsets, the supremum is not in general upper semi-continuous and so is not subharmonic (resp. plurisubharmonic). We seek to remedy this situation by finding the smallest subharmonic (resp. plurisubharmonic) majorant of a family {qJJ iEl of subharmonic (resp. pluri-subharmonic) functions defined in a domain Q.

Definition 1.24. Let qJ(x) be a function defined in a domain QclRm and locally bounded from above. Then cp*(x), the regularization of cp(x), is defined by cp*(x)=lim sup cp(x').

x'cQ--+x

Obviously, qJ*(x) is the smallest upper semi-continuous majorant of qJ(x).

Lemma 1.25. Let Q be a domain in (Cn. Let qJ be an upper semi-continuous function locally bounded from above in Q. Let WE(Cn, W =1=0, be fixed and suppose that for some zoEQ, D(zo, w, l)={zo+uwEQ: lul~l}cQ. Then

1 21t . g(z)=- S qJ(z+we'O)dB is an upper semi-continuous function of Z for

2n ° zEQ'={ZEQ: D(z, w, l)cQ}.

Proof Q' is open and non-empty, since zoEQ', so g is defined in a neigh-borhood of zEQ'. Let zm -+ Z in Q'. Then the functions cp(zm + weiO ) are uniformly bounded in m and B. Fatou's Lemma then gives

1 2n lim sup g(zm)=lim sup - S qJ(zm+weiO)dB

Zm.--+Z Zrn--+Z 2n 0 2n

~ 1/2n S lim sup qJ(zm + weiO ) dB o Zm--+ Z 2n

~1/2n S qJ(z+weiO)dB=g(z). o o

Theorem 1.26. Let {CPJieI be a family of plurisubharmonic functions defined on a domain Qc (Cn and locally bounded from above and tf;(Z) = [sup CPi(Z)]. Then tf;*EPSH(Q). i

Proof Let D(z, w, 1) c Q. Then

1 2n 1 2n qJi(z)~-2 S qJi(z+weiO)dB~- S tf;*(z+wei8)dB,

n 0 2n 0 1 2n

since CPi~SUP CPi~tf;*· Thus CP(z)~-2 S tf;*(z+weiO)dB. We take the regula-i n 0

rization of both sides and apply Lemma 1.25. 0

Theorem 1.27. Let {cp J ieI be a family of plurisubharmonic functions defined in Q c (Cn and locally bounded above uniformly in I. Suppose that the family I is


an ordered filter with the property that the filter of sections Si=ijEI;j~i} has a countable basis Sm' Then if t/I(z) = lim sup CPi(Z) = lim [sup CPi(Z)], t/I*(z)EPSH(Q) or t/I*= - 00. ieI m-oo ieSm

2n

Proof Let t/lm(z)=SUPCPi(z) and let J represent the lower Lebesgue integral ie8m .0

on the boundary of the disc. Then 1 2n .

t/I(z)= lim t/lm(z)~ lim -2 J t/lm(Z + we,O)d(}. m-+ 00 m-+ 00 n *0

We apply Fatou's Lemma to the lower Lebesgue integral to obtain

1 2n . t/I(z)~-2 J lim sup t/lm(Z + we'O)d(}

n.o m-+oo 1 2n 1 2n

~- J t/I(z+weiO)d(}~-2 J t/I*(z+weiO)de. 211: .0 11: 0

We now take the regularization of both sides and apply Lemma 1.25 to

obtain t/I*(z) ~f- Y t/I*(z + weiO)d(}. Thus, if t/I*(z) =1= - 00, it is plurisubhar-monic. 11: 0 D

In exactly the same way, we have the following result:

Theorem 1.28. Let {CPiheI be a family of subharmonic functions defined in Qe]Rm and locally bounded above uniformly in I. Suppose that the family I is an ordered filter with the property that the filter of sections Si=ijEI, j~i} has a countable basis Sn' Then if cp(x) = lim sup CPi(X) = lim [sup CPi(X)], CP*(X)ES(Q)orcp*(x)=-oo. ieI n-oo xeSn

Remark. If the family {CPiLeI' CPiEPSH(Q), is locally bounded above uniform-ly, then t/I(z)=sup CPi(Z) is in fact integrable in (} over the set {z+weiO :

ieI O~e~211:} for every disc D(z, w, l)eQ which is not contained in n {ZEQ: CPi(Z) = -oo}. The set {(}: t/I(z+weiO)O t P t

t-+ + 00

§8. Indicator of Growth Functions 21

cp(Z) is in addition a plurisubharmonic function in


and lim sup hr(y, y', cp) = tP lim sup hr (~, y', cp)

(y,y')~('x,x') (y,y')~(tx,x') t

=tP limsup hr(Y,y',CP)=tPh~(x,x',cp). D (ji,y')~(x,x')

Theorem 1.31 (Hartog's Lemma). Let v,(x), t > 0, be a family of subharmonic functions uniformly bounded above on compact subsets in the domain QcIR'". Suppose that for a compact set K in Q there exists a constant C such that w(x) = [lim sup v,(x)J* ~ C on K. 711en for every 8> 0, there exists T" such that

,~oo

v,(x) ~ C + 8 for t~ T" and XEK.

Proof We replace Q by an open neighborhood Q of K relatively compact in

Q such that w(x) ~ C + ~ in Q. Since v,(x) is bounded above in Q, by subtract-

ing a constant, we may assume that v,(x) < 0. Choose I' so small that B(x, 3r)cQ for XEK. Then v,(x)~A(x, r, v,), and

by Fatou's Lemma, lim sup A(x, 1', v,) ~ C + 8/3. Thus, for xEK, there exists ,~oo

Tx such that A(x, r, v,) ~ C + 8/2 for t > Tx' Since v, ° be given. Since K is compact, g is uniformly continuous on K, so there exists b such that Ig(x')-g(x)I Tx and lx' - xl < b by Theo-rem 1.31. Since K is compact, we can choose a finite number of balls B(x;, b) which cover K. Then for t>sup Tx" v,(x')

§8. Indicator of Growth Functions 23

Let xoElRm and define hex) = lim sup hr(x", Xo' cp), which is a subharmonic x"--+x

function of x. Suppose x =1= O. Given 8> 0, there exists 6> 0 such that hr(x",xO' cp);£h(x) + 8/2 for Ilx"-xllR£ and Ilx"-xll


Proposition 1.35. Let f E£(

§9. Exceptional Sets for Growth Conditions 25

cpEPSH(QxO, M(z,r)cp(z, 0). Then cp(z,m)= -log 6(z, m) is plurisubharmonic on every connected component of Qq for m>q, where Qq={ZEQ: M(z, l)q.

Proposition 1.39. Let {cp q} be a sequence of plurisubharmonic functions uniform-I y bounded above on compact subsets in a domain Q c en, with lim sup CPq ~ °

q~ 00

and suppose that there exists ';EQ such that lim sup CPq(';) =0. Then A={zEQ:limsuPCPq(z)


Then for g(z')=lim sup t/J(z', m), and g*(z')=lim sup g(z"), there exist two pos-sibilities: m~ 00 z" ~z'

1) g*(z')=O in Q"; then there exists z~EQ' such that g(z'o)=g*(z~)=O (App. I. 27). By Proposition 1.39, the set

E={Z'EQ": g(z')Mp and Ilz'll ~p, supt/J(z',mp)~ -iXpMp we define:

t/J l' (z') = sup [t/J(Z', mp), log II ~ "] for liz' II ~ p

t/J (z')=logJ!.:1 for Ilz'll ~p. l' P

Ilz'll . Because t/J(z',mp) is bounded by -iXp

§9. Exceptional Sets for Growth Conditions 27

Z'=(ZI' ... ,Zn_l)E0, M",(r z) = sup cp(uz) mo= sup cp(z). Then liz II ~ 1

-logb(uz,m)= -logb(z,m)+loglul

and for m > mo > 1, the functions

I/Im(z) = (log m)-1 [ -log b(z, m)] EPSH(


Then S(z)=~ [t/tmq(Z)-Co- ;2] EPSH(

Historical Notes 29

by the Russian school (cf. Gold'berg [lJ). Relations between the total order and the orders relative to each variable were first given by Borel [IJ; the first comparison with respect to the growth on complex lines was made by Sire [lJ at the beginning of the century. The modern treatment of the indicator function as given here is primarily due to Lelong [2]. This ge-neralizes the classical Phragmen-LindelOf indicator function and was first considered by Lelong [2J and by Deny and Lelong [IJ and [2J for sub-harmonic functions. In particular Lelong developped in his early works Hartog's Theorem in ([2 in the context of subharmonic functions and potential theory. After the introduction due to Oka and Lelong [5, 6J of the class of plurisubharmonic functions (1942), the properties of the indicator function were obtained from the general properties of locally bounded families of plurisubharmonic functions; the characterization of the indicator functions for entire functions of finite order in terms of plurisubharmonicity was given by Kiselman [2J and Martineau [4, 5J and will be presented in Chapter 7. The proof given here that h*(x, x', cp) is independent of the center has the advantage of working in the class of sub harmonic indicators defined in cones. The results of § 9 and the Inverse Function Theorem for pI urisub-harmonic functions (see Appendix I) were given by Lelong [15J for complex topological vector spaces.

Chapter 2. Local Metric Properties of Zero Sets and Positive Closed Currents

§ 1. Positive Currents

A biholomorphic mapping F: ce" -> ce" induces a mapping of the underlying real coordinates lR211 whose determinant is just IJ(FW, where J(F) is the Jacobian of F, and this determinant is positive. Thus, if one chooses once and for all a volume form an ce", this choice determines an orientation in cen which is invariant with respect to holomorphic isomorphisms, and its re-striction to subspaces IJ', p < n, or to complex submanifolds, determines a volume element and hence an orientation. This led to the introduction of a positive differential form in the exterior differential algebra E211 (dz) with in-volution (given by dz->dz) and its generalization, the positive closed current.

We write ,8=~aallzI12=~ I dZ j 1\ dZj and set ,8p=(p!)-I,8P, which is just 2 j~ I

the p-dimensional Euclidean volume measure in cen•

Definition 2.1. A differential form q>(dz) with complex-valued coefficients will be said to be positive of degree p in E 2n (dz) if

i) it is homogeneous of type (p, p), ° ~p ~ n; ii) for every system of forms lXI' ... , IXn_ p linear in dZ j (that is such that

n

IX; = L a;jdzj , aijEce, is a (1,0) form), the product q> 1\ ilX l 1\ IXl 1\ ... j~ I

1\ i 1X 1I _ p/\ IXII _ p = tjJ-,811 is such that ljJ ~ 0. For 0 a domain in ce", we let cP; (0) be the positive forms of degree p in

o with continuous coefficients. As a consequence of Definition 2.1, we obtain:

Proposition 2.2. A C-linear change of wordinates in E 211 (dz): dzj= L Cj.kdzk, 11 k~ I

dZj= L Cjkdzk transforms a positive form into a positive form. As a con-k~ I

sequence a biholomorphic map F: 0 -> 0' induces a map of CP: (0) onto CP: (0').

Proposition 2.2 permits the definition of positive forms on a complex submanifold Me 0: it is those forms which are positive for every choice of local coordinates.

§ 1. Positive Currents 31

For p = 0, cPt (Q) is just the set of positive continuous functions on Q. n

For p=1, cpEcPi(Q) if and only if cp=i L CPjkdzjl\dzk, where the matrix j.k= 1

[CPjk(Z)] is positive semi-definite for every zEQ. If for any p we have

(2,1)

where the Ap are complex linear in dZ j with coefficients in ceO (Q), then CPEcP;(Q). Those cP which can be represented as in (2,1) will be said to be decomposable.

If 11' is a complex subspace of dimension p, there exists a rotation gE U (n) given by u = g(z), such that g(11') is defined by the_equations up + 1 = ... =un=O. We define the form ,(lJ')EcP;(

32 2. Local Metric Properties of Zero Sets and Positive Closed Currents

if p is odd. Then C",(z) is equal to the coefficient of T(IJ') in (2,3). We ob-tain finally:

Proposition 2.5. A necessary and sufficient condition for a form

O, l~s~N. Then LI=LI(ws) =BLI [T(E's-P)] for B=b l ... bN>O and in Q, Qc1R.2"; the conditions LI =1=0 and LlI=LI[T(E's-P)]=I=O are equivalent; we shall'say that the system A = {E's- P} is regular. We conclude that in each open set of 1R. 2Nn we can find points such that LlI =1=0. If Gn_p(

§ 1. Positive Currents 33

Proposition 2.6. Let M be a complex submanifold of Q c (Cn of dimension p (cf. Definition 2,33) and CPEP; (Q) with compact support in Q. Then

(2,4) S cp=[M](cp)~O. M

If cP = dt/! for a form t/! with ~l coefficients, then S dt/! = [M] (dt/!) = O. M

Proof Let {Uk} be a locally finite covering of M by relatively compact local coordinate patches, and let {(J(k} be a partition of unity subordinate to {Uk}' Then

(where the sum is finite since cP has compact support). For each Uk' there exists a holomorphic homeomorphism Fk of Uk onto

Vk , an open neighborhood of the origin in (CP, and

S (J(kCP= S Fk*((J(kCP)= S (J(~CP~· M Vk V k

Since (J(~=(J(koF-l>O and cP~=cpoF-1EP;(V,J, we have S (J(~CP~~O, from which (2,4) follows. If cP = dt/! then Vk

[M] (dt/!) = L S (J(~dt/!~ = L S d((J(~ t/!~) - L S d(J(~ /\ t/!~.

It follows from Stokes' Theorem that each summand in the first sum is zero, since supp (J(~ is compact in ~. On the other hand L S d(J(~ /\ t/!~ = S (L d(J(k) /\ t/! =0, since l:>k == 1. k Vk 0 M

The area of a manifold is a positive measure 0' defined for f E~;{' (Q) by

(2,5) O'(f) = [M] (f fJ p)

Proposition 2.7. If M is a complex submanifold of Q c (Cn, then the area of M defined by (2,5) is the sum of its projections on the coordinate spaces

(i)P p(p-l) Proof We have fJ p =(p!)-lf3P= 2: (-1) 2 ~dZI/\dZI=~fJI so that from (2,5) we obtain O'(f)=L[M](ffJI)=LO'I(f), where O'I' given by the

I I

integration of fJI on M, is the projection with multiplicity of M on (Cp(I)c(Cn, where (CP(I) is defined by equations Zj=O for UI. 0


Remark. The positive measure [M] /\ ,ell) is the projection of the area of M on the subspace ll.

We recall that ~~.(P,q)(Q) is the space of differential forms qJ of degree (p,q) with coefficients in ~~(Q). We let t&'g'(Q)=U ~O~(p,q)(Q). Let Q; be

p,q 00 an exhaustion of Q by compacts sets, Qi~Qi+l~Q and Q=UQi' We

i= 1

equip ~!f:(p,q)(Qi) with the topology of uniform convergence of the coef-ficients qJ J,J and all of their derivatives. With this topology, ~!f:(p,q) (Q i) is a Frechet space. Finally, we equip ~~(p,q/Q) with the strict inductive limit topology, ~~(P,q)(Q)=li~ ~~(p,q)(QJ The dual space of ~~(p,q)(Q) is the set

of linear functionals t(qJ) for which lim t(qJm)=O for every sequence {qJm}

which tends to zero in ~!f:(P.q)(Q); a sequence {qJm} tends to zero in ~~(P,q)(Q) if and only if

i) there exists Qi such that, supp qJm C Qi for every m,

ii) for every multi-index rY., lim [sup sup IDaqJm J J(z)l] =0, where Da is a m-+oo ZEQ [,J ' ,

differential operator with respect to the underlying real coordinates in S tjJ /\ qJ. We will let T,.~p(Q) be the space of positive

Q

currents of degree n - p in Q. As in Proposition 2.4, we have:

§2. Exterior Product 35

Proposition 2.10. A current t defined in Q belongs to T..~ p(Q) if and only if for every linear subspace II of dimension p, t /\ T(ll) is a positive distribution (hence a positive measure).

We see that an element tE Tn~ p(Q) can be identified with a measure in Q (depending on ll). A current tE T..~ p(Q) is represented by a differential form homogeneous of type (n - p, n - p). We can express it in a canonical form

(2,7) t=k~ItI,JdzI/\dzJ I=(i 1


For a measure fl defined in a domain Q, we define IflIQ=suplfl(f)I, for fE~o(Q) and IflQ=sup If(z)l;£ 1.

ZEQ

Definition 2.14. Let t be a current defined in a domain Q of C" and continuous of order zero. We shall say that the nates the current t if there exists a constant

positive measure fl domi-Cil such that for every

§3. Positive Closed Currents 37

Proof If tEqJ;;_p(Q), then this is true, and according to Proposition 2.11 and Theorem 2.16, it remains true for tE~~p(Q) by passing to the limit. 0

Remarks. From the definition and from the fact that positive currents are continuous of order zero, we see that

i) T/ (Q) and qJ: (Q) are cones over the set of positive continu-ous functions (i.e. t 1,t2ET/(Q) and Cll,Cl2E~O(Q), Cl1~0, Cl2~0, then Cl 1 t1 +Cl2 t2ET/(Q));

ii) Since the degree of a positive current is even, cp /\ ljJ = ljJ /\ cp if CPEqJ;(Q) and ljJEqJ:(Q), but in general cp /\ ljJ¢:qJ;+q(Q).

iii) A current t = i i)p,qdzp /\ dZq is positive if and only if the distribution 8(A)=C2:.t p,qApIq)f3n = I Tp,qApIq is a positive measure for AE


=t(a

§3. Positive Closed Currents 39

Theorem2.22. Let tEf,,-:'p(Q), OEQ and B(O,R)~Q. Let r1


lim V,(B, r) = 1:Zpl r- 2p (JJa, r) - v,(a) = S V~ .~o 0< Ilz-all:o;r

or equivalently

o

The existence of the number v,(a), called the Lelong number of the current t, is an essential property of positive closed currents.

Definition 2.24. Let tET,,::'p«Cn). We call the function

(2,15)

the indicator of growth function of t; it is the mass of the measure v, carried by the ball B(O, r).

§ 4. Positive Closed Currents of Degree 1

Proposition 2.25. Let V E PSH (Q) for Q a domain in

§4. Positive Closed Currents of Degree 1 41

We shall now show that the converse of this is true, at least locally, that is, for every tE 11+ (Q) and every ZE Q, there exists a neighborhood Uz of every ZEQ such that t=i8aV for VEPSH(UJ For the proof, we shall use an integral operator which gives solutions of the a-equation with special regu-larity properties.

Let k(Z)E~2(


Proof. Let z be fixed. Applying Stokes' Theorem, we obtain

S o-h(~)!\K(z,~)=lim S o-h(~)!\K(z,~) Q ,-+OQ-B(z,,)

= lim S d(h(~)K(z, m ,-+0 Q-B(z,,)

= S h(~)K(z,~)+lim S h(~)K(z,~) bdQ £-+0 bdB(z,~)

= S h(~)K(z, ~)+h(z) bdQ

since

lim S ,-+0 bdB(z,')

-en -2) lia 2n"' ~llz-~112-2"!\[3"_1=1. D

Corollary 2.27. Let [3 be a (0,1) form in a neighborhood of B(O, 1) with rc oo coefficients such that 0-[3 = ° and let

a(z)= S [3(~)!\K(z,~). B(O,l)

Then arx = [3 in B(O, 1).

Proof Choose k(z)=llzI12-1. Let a be any rc oo function such that oa(z) = [3(z) in a neighborhood of B(O, 1) (cf. Appendix III). Then

a(z) = - S am K(z, ~) + S [3(z) !\ K(z, ~). bdB B

But K(z, ~) is holomorphic in ZEB for ~EbdB. Thus aa(z) = [3(z). D

Theorem 2.28. Let 0 be a positive closed current of degree 1 defined in a

neighborhood of B(O, 1). Then there exists a plurisubharmonic function Y defined in B(O, 1) such that iaay=O as a current.

n

Proof Let 0 = i L 0kjdzk !\ dZj and set OL = Okj* p" which, for s small enough, k,j= 1

is rc oo in a neighborhood of B(0,1). Let O'=O*p,. Since 0 is a positive closed current, dO'=O, and we find v~ and v~, a (0,1) and (1,0) form respectively, such that d(v~ - v1)= 0'. We can calculate .v~ and v~ explicitly in terms of the coefficients of 0':

It follows from degree considerations that av~ = av~ =0, and v~ = i!f. Let V. = [J v~(~)!\ K(z, ~) + S v~ (~) !\ K(z, ~)] = 2lR. e S v~ m !\ K(z, ~).

Then iaav. = i[av~ -av~] = id(v~ -vD = 0'.

§4. Positive Closed Currents of Degree 1 43

n n

Let (J = L {)ii dr, (JE = L {)~i dr. Choose (j so small that 0 is defined m i= 1 i= 1

. - S (n - 2)! 2 2 ~ a neighborhood of B(O, 1 + (j) and set V(z)= - ~ Ilz- ~II - nd(J(s). _ _ _ B(O.I) _

Then V (z) and v,,(z) = v* Pe are subharmonic functions such that v,,(z) de-creases to 11 (z) as e goes to zero.

(n- 2)' By Stokes' Theorem, we see that, setting C~=---',

2 nil

S -iC;,llz-~112-2nv~ /\f3n-l bdB(O, 1)

=iC~ S av~/\(-llz-~112-2n)/\f3n_l B(O,I)

+iC~ S v~/\allz-~112-2n/\f3n_l B(O,I)

= C~ S -llz_~112-2nd(Je+iC~ S V~ /\allz_~112-2n /\f3n-l' B(O,1) B(O,I)

Hence

(2,16) v,,(z)=2IRe S v~(~) /\ [K(z,~) B(O,I)

+ic~allz-~112-2n/\f3n_lJ+2C~ S _llz_~112-2nd(Je B(O,1)

+2IRe S iC~llz-~112-2nv~/\f3n_l' bdB(O,I)

Since S d(JE is bounded independently of e, the coefficients of v~ have B(O,I)

bounded Ll norms on bdB(O, 1) independently of 8, so we can find a sequence 8m 1 0 such that each coefficient of v~m converges weakly to a measure. Since K(z,O= -C~allz-~112-2n/\f3ll_1 in a neighborhood of z and since C~ S -II z - ~ 112 - 2n d(JE ~ v,,(z) - C for some constant C < + 00, we can find

B(O,I)

a subsequence e~ of 8m such that v";,, -+ V pointwise. Furthermore, we have (2,17) v,,(z)~ CK + V.(z)~ CK + V(z) for zEK~B(O, 1).

Thus, if q>E~~(n_l,n_l/Q), it follows from (2,17) and the Lebesgue Dominat-ed Convergence Theorem that

i a a V (q» = S V i8 a q> = lim S v" i a a q> = lim S i a a v" /\ q> Q f.~OQ f.~OQ

= lim S {)E /\ q> = S () /\ q>. o f.~0 Q Q

Remark. We sketch here a second proof of Theorem 2.28 which is of histori-cal interest and provides motivation for much of the technics to be de-velopped in Chapter 3.


1) Suppose that 0' is a (1,1) form whose coefficients O~,q are polynomials in the 2n real variables x, y;

(a) 0' = ~ P (z) Q (z) p,q '-" p,q,A p,q,Jl A,Jl

where the Pp, q, A and Q p, q, Jl are homogeneous polynomials of degree }, and fl respectively. If there exists a solution V of the equations

a2 v (b) --=0'

azpazq p,q

then we set W(z, Z, t, t')= V(zt, zt') for t, t'E

§4, Positive Closed Currents of Degree 1 45

Proposition 2.29. Let h be a pluriharmonic function defined in a neighborhood of the ball B(O, 1), Then there exists a function f holomorphic in B(O, 1) such that h=lR.ef

Proof If h is pluriharmonic, then oah=O and so d(oh-ah)=O, Thus, by Poincare's Lemma, there exists a function g such that dg=oh-ah, Since h is real valued, ah=oh, Hence idg=ioh-ioh, which implies that iag= -iah =(ioh)=(iog), so that ig is also real valued, If f=h+g, then af=ah+ag =ah-ah=O, so f is holomorphic and h=lR.ef 0

Corollary 2.30. Let e be a positive closed current of degree 1 in


Proposition 2.32. Let FS (Zl' ... , Zn)EYl' (Q), S = 1, ... , m and let

V=1log (t1 IFs(SW)EPSH(Q), t=iaav. Then

(2,18) vt(a)=min vS ' where vs=multiplicity of the zero of Fs at a. In particu-s

lar, if V = log IF(z)l, vt is the multiplicity of the zero at a, that is the degree q(a) of the first homogeneous polynomial in the Taylor series expansion of F at a which is not identically zero (i.e. F(z+a)= I Pm (z)).

m i;q(a)

Proof We suppose that a=O. Then Fs(uz1, ... , uzn)=uv F;(u, Zl' ... , zn) where F; is holomorphic in the (n + 1) variables (u, z). Thus

V(u, z)= v log lui +1log (t1 IF; 12 ) = v log lui + W(u, z),

where W is a plurisubharmonic function of (u, z). For u = 0, m

t/J(Z) = IIF,'(O,zW$o.

Thus, when r goes to zero ,1.(0, r, V) = v log r +1 A + e(r) where A=},(O, 1,logt/J» -00 and s(r) goes to zero with r, from which (2,18) follows. 0

§ 5. Analytic Varieties and Currents of Integration

Since we are interested in studying the properties of the common zero set of several holomorphic functions, we recall here some of the complex analytic structure of such sets.

Definition 2.33. Let Q c (Cn be a domain. A closed subset Me Q is said to be a complex submanifold of dimension p if for every zEM, there exists a neighborhood Uz c Q of Z and holomorphic functions J;EYl'(Uz ), i = 1, ... , n such that

M n Uz = {z': f1 (z') = ... = fn_p(z') = O} and rank [,aJ; (Z)]n = n - p. aZj i,j~ 1

Definition 2.34. Let Q c (Cn be a domain. A subset Y c Q is an analytic variety in Q if for every zEQ, there exists a neighborhood Uz and functions J;EYl'(Uz ), i = 1, ... , tz such that Y n Uz = {z' E Uz : f1 (z') = ... = frz(z') = O}.

If Uz n Uz =t= 0, the set Y n Uz n Uz is defined by two different systems, fj(z') =0 in Uz and gj(z') =0 in Uz, and the number of the equations can be different.

§S. Analytic Varieties and Currents of Integration 47

Definition 2.35. Let Q be a domain in (Cn and Y c Q an analytic variety. Then for ZE Y, the complex dimension of Y at z, dimz Y, is the minimal

n

number of linear equations L ais(z; - Zi) = 0 which, when added to the i=1

y=O, j= 1, .~., t z ' give Z as an isolated solution of the system of equations

l.!j(Z') =0, i~laiS(Z;-ZJ=O}. We say that Y is of pure dimension p if dimz Y = p for all ZE Y.

Remark. If Y is of pure dimension 0 in Q, then Y is just a discrete set of points.

Definition 2.36. If Q is a domain in (Cn and Y is an analytic variety in Q, then Y is said to be a complete intersection if Y is of pure dimension p and Y = {ZEQ: fl (z)= ... = fn_p=O, hEJIl'(Q)}.

Definition 2.37. An analytic variety Y c Q is said to be irreducible in Q if for every pair of analytic varieties Y1 and Yz such that Y = Y1 U Yz, either Y = Y1 of Y= Yz.

The complex structure of analytic vanehes can now be resumed as follows (cf. [A, B]): suppose that Y is an analytic variety in Qc(Cn; then Y is a union (finite in fach Q' ~ Q) of irreducible analytic varieties Yk such that

i) there exists a proper subvariety Y~ c Yk called the set of the singular points of Yk such that Yk = Y" - Y~ is a connected analytic complex sub-manifold of (Q - Y~) of dimension Pk; ZE Yk is called a regular point of Y,,;

ii) for Q' relatively compact in Q, Yk (\ Q' = 0 for k ~ kQ ,. iii) dim Y = sup dim Yk , and the closed set Y' = U Y~ U (Yo (\ 1j) is an

k b*j

analytic subvariety of dimension at most (dim Y -1); Y - Y' is a union of disjoint complex manifolds in (Q _ Y'),

We note that if Y={zEQ:h(z)=O, i=l, ... ,p, hEJIl'(Q)} is defined by p holomorphic functions, then dimz Yk ~ (n - p) for all z. In particular, for a holomorphic function f$O, fEJIl'(Q), if Yf = {ZEQ: f(z)=O}, then the di-mension of the regular points of Yf is exactly (n -1).

Definition 2.38. Let Q c (Cn be a domain, Then a Weierstrass pseudo-poly-k-l

nomial P(u; Z)EJIl'(c x Q) is a function P(u; z)=uk + L ai(z)ui, aiEJIl'(Q), ai(O)=O. i=O

Now, we recall a classical result.

Proposition 2.39 (Weierstrass Preparation Theorem, cf. [A, B]). Let f be holo-morphic in a neighborhood Q of 0 in C" and assume that z;/ f (0, zn) is holo-


morphic and does not vanish at O. Then we can write f in one and only one way in the form f = hPp, where hand Pare holomorphic in a neighborhood of

p-1 0, h(O)=l= 0, and Pp is a Weierstrass polynomial, that is: P(z) = z~ + I a/z')z~,

o where the aj are holomorphic functions in a neighborhood of 0 in

§5. Analytic Varieties and Currents of Integration 49

Set Z'=(Zl, ... ,Z,,_l) and D1 ={(z',zlI): Ilz'112


By the same argument as above, we see that Fn continues as a holomorphic function to D 2' Since Fn is a pseudopolynomial in Zn' we can replace it by P" which has the same zeros as Fn but no multiple factors. Then Ye ¥ = {zED 2 : Pp+! = ... =P,,-! =Fn=O} and n(Y nD)=n(¥ nD)=n(Dz), which establishes (1).

We now show (2). Let zo=(zZ, ... ,Z~)EY such that zoiW. Then each of the equations Pp+j(zp+j; zl' ... , zp)=O has in a connected neighborhood Vof (zZ, ... ,z~) in (CP a unique root zp+j=~p+j(z!"",zp) which is holomorphic with value ~p+j(z~, ... , z~)=z~+j' Thus, there exists a neighborhood U={z:IZk-z~l


as defining an analytic variety X in Gn_p(


B(z, r) c Dl . It follows from Corollary 1.12 and the fact that n- l (W) II Y is a subvariety of Y that

_ S r(ill ) = _ S r(ill)' YnB(z.r)-1r-'(W) YnB(z.r)

Thus

where z' = n(z) is the projection on (CP(ZI' ... , zp). Thus

J r(ill ):;;; yr 2pr2P = C1 r 2p for B(z, r) c D 1 • Y nB(z.r)

We proceed in the same way for il2, ... , ilN and obtain similar bounds S r(ll,):;;; Csr 2p for every ball B(z, r)cDs' where Cs is independent of

Y nB(z.r) B(z, r). N

Let .1 be a connected open neighborhood of 0 in n Ds' Then for B(z,r)cL1 we obtain, S r(ll,):;;;( sup Cs)r2P. 5=1

YnB(z,r) I ~s~N

By Theorem 2.16, there exists a constant Cft depending only on the

system A={ll,} such that for every positive ~urrent t, O't:;;;C" (f tAr(ll,)). Applying this to the current of integration [Y], we obtain = 1

(2,22) O'y [B(z, r)]:;;; C(L1, Y)r 2p for B(z, r) c .1.

Since K is compact, we can cover it by a finite number of domains .1; for which (2,22) is true for a constant C(L1;, Y). If we let C=sup C(L1;, Y), we obtain (2.21). 0

Let us remark that the definition of a current t is local and the same is true for the current dt, the closure of t defined by dt(cp)=t(dcp): if{Lj} is a locally finite covering of a domain Qc en by subdomains and PjEC€;{'(U) such that L~(x)=l on Q, we write for cp with coefficients in "t';{'(Q): t(cp) = It(pjcp)= L t(cp) for Cpj= PjCP. We prove now a generalization of Stokes' Theorem and use assumptions on the mass of a closed positive current t defined in a domain Q c Rm to obtain a continuation of t as a closed current to Rm. The problem is local, therefore we suppose Q relatively compact.

Theorem 2.44. Let t be a current continuous of order zero defined in a bounded domain Q c IR m.

i) in order that t extends across the boundary of Q to a current t' continuous of order zero, it is necessary and sufficient that t be bounded in Q, that is that the measure coefficients of t be of finite total mass in Q. In this case, the simple extension t of t, which has no mass on C Q, is obtained by

(2,23) t(cp) = lim t[lXqcp] q~GO


where (Xq(x) is a family of functions in ~;'(Q) such that O~(Xq(x)~l, (Xq+l(X)~(Xq(x) and lim (Xq(x)=Xo, the characteristic function of Q;

m-oo

ii) if t is closed, the simple extension i of t, defined by (2,23) is closed if and only if

for one sequence (Xq(x) with the properties stated in i).

Proof If t extends to a current t' defined in Q'::::l Q and continuous of order zero in Q', obviously the mass II till G of t' in G~ Q' is an upper bound of the mass of t in GnQ, and for G=Q, the mass Iltllo must be bounded. Con-versely, if t is of finite total mass in Q, (2,23) by convergence defines a current i in IR m of bounded mass and so i is seen to be continuous of order zero, and the definition of i does not depend on the particular sequence aq(x) with the above properties. Furthermore, for a form qJ with coefficients in ~;' (IR m)

i(dqJ)= lim t(aqdqJ) = lim [t(d(aqqJ))-t(daq!\qJ)]. q-oo q-+oo

The first term vanishes since t is closed and supp(aqqJ) is compact in Q. Then i(dqJ)= - lim t(daq!\ qJ) for each sequence {aq} with the given proper-

q-oo

ties, and ii) is proved. 0

Corollary 2.45. Let Q be a domain in IRm = IRPx IRm- p , O~p


Theorem 2.46. Let Q be a domain in lRm=lRPxlRm- p , O~p


Remark 2. The essential point of the proof of Theorem 2.47 is Theorem 2.43 which corresponds to a property of the area of analytic sets. This also leads to the following:

Proposition 2.48. The area of an analytic variety Y c Q of pure dimension p exists in the real dimension 2p and is given by

a= J !3p= J [Y] /\ !3p= [Y]!3p' y

is finite on every compact subset of Q, and has the property a= L aI' where aI is the projection of a on the subspace (CP(ZI)' I

Thus we see that for t = [Y], at' the trace of t, is just the area of Y.

Proposition 2.49. If tEf;,:"p(Q), vt(a) is upper semi-continuous.

Proof Since at (a, r) is the mass of at carried by the closed ball of radius r and center a, at(a, r) is upper-semi continuous for r fixed, and since vt(a) = inf '2; r- 2p at(a, r), it is also upper semi-continuous. D

r-O

Theorem 2.50. Let t'=F*t be the image of a positive closed current tETn:"p by an application z' =F(z) which is biholomorphic from a neighborhood of Zo onto a neighborhood of z~ = F(zo). Then the Lelong numbers Vt'(z~) of t' at z~ and vt(zo) of t at Zo are equal.

Proof We shall divide the proof into several steps. i) For simplicity, we assume that zo=O, z~=O and that F is biholomor-

phic between the two open neighborhoods Q and Q' of Zo and z~ respective~ ly in (Cn, n ~ 2. The current t' = F * t on a form q> with coefficients in c'6'; (Q) is given by

(2,26) t'(q» = F*t(q» = t(F* q»

where F* q> is obtained by replacing in q> the variables z' as functions of z. We set t/t(z)=F*(llz'11 2), that is, for F=(Fk)

n

(2,27) t/t(z) = L Fi(z)· Fi(Z).

We then obtain from the definition of vt,(O) in Q' that

1 (i - )P vt,(O)=lim-( 2)P J t/\ -2 aa t/t . r-O nr I/I(z)(z) = Ilz112.


Let L(Q) c: PSH (Q) be those functions V (z) such that a) V(z)~O b) V(Z)E'6'2(Q) n PSH(Q)

c) V(z)~r is compact in Q for O


As in Theorem 2.22, if we set t=d8, by Stokes' Theorem, we have

which proves (2,31). In order to prove that V""t(O)=v""t(O) (i.e. vt(O)=vt,(O)), we shall prove

that V"',t(O)~v""t(O). The inverse inequality then follows from applying the same reasoning to F- \ from which we deduce the equality. From (2,31), we s~e that it is enough to show that for k>l~2, we have V",k,t(O)~v""jO). Let ljJe(z) be the function defined by (2,28). For every 8> 0, there exist re> 0,

C1 (8) and C 2 (8) positive constants such that for Ilzll


Corollary 2.51. If X is an analytic variety of pure dimension p, for every point ZoEX which is regular, v[XI(zO)' the Lelong number with respect to the current of integration on X, is equal to 1.

Proof We can find neighborhoods U of Zo and V of OE(Cn and a biholomor-phic map F(U)-+V such that F(zo)=O and F(UnX)=Vn(CP(zl'''''Zp) = Y. By Theorem 2.50,

V[Xj(Zo) = v[Y](O) = lim ''2; r- Zp O'[Y](r) = 1. D r-O

Remark 3. The area 0' of an analytic variety in Q is the trace of t = [Y]. By (2,13), O',(r), the area of Y in the ball B(a, r) c Q has the property that the quotient ('zprZp)-lO't(r) is an increasing function of r. Then Proposition 2.50 gives a lower bound for O',(r)

O',(r);:;;, zprZPv,(a);:;;, zprz p .

Historical Notes

The first attempt to study the "area" of an analytic set X goes back to Poincare [2J, who showed that if f is holomorphic, then log If I is locally the sum of an Rzn-harmonic function and a potential - Cn J II a _zllz- zndO'(a), where 0' is the "area" of the divisor f =0. In 1938, Kneser [lJ, in an attempt to generalize the Weierstrass product to (Cn, used the projective area and constructed a (locally convergent) representation of the holomorphic function logf, where f defines a given divisor X, and in 1952 Stoll [2J by this method succeeded in giving a bound for a solution of the Cousin problem in (Cn for X of finite order. In 1950 Rtitishauser [1J showed that for X an analytic manifold in (Cz, 0' x(r);:;; nrz. In 1950, Lelong [8J gave a convergent representation for log IFI, P a polynomial in (Cn, by a potential. Using the technics of distributions [FJ and currents [EJ, in 1954 Lelong gave a modern formulation for integration over a divisor X by [XJ(CP)

i -= J - aa log If I /\ cp and gave the current associated to a Cousin data of

n zeros.

The general problem of proving the existence and closure of the current of integration [XJ for X an analytic variety was different for co-dimension X> 1, since X cannot be supposed to be a complete intersection. The positive currents and closed positive currents were introduced by Lelong [1OJ in 1957, and the closure was obtained as a consequence of bounds for measures. Now the positive closed currents are a classical notion in complex analysis, as will be illustrated in the following chapters.

Chapter 3. The Relationship Between the Growth of an Entire Function and the Growth of its Zero Set

The problem of constructing a hoI om orphic function of one complex vari-able with a given zero set was solved by Weierstrass in the middle of the nineteenth century. He showed that if Q is a domain in the complex plane, if {aJ is a sequence of points without limit point in Q, and if {mJ is a sequence of positive integers, then there exists a function j(Z)E£'(Q) which has a zero of order exactly mv at every point avo The equivalent problem for several complex variables is Cousin's Second Problem, which we state as follows: if Q is a domain in

60 3. The Relationship Between the Growth of an Entire Function

§ 1. Positive Closed Currents of Degree 1 Associated with a Positive Divisor

Let X = (/;, Vi) be a set of Cousin data in Q, Y (X) the analytic variety given by y(X)"Vi={ZEVi:/;(Z)=O}, and Y(X) the (n-1) dimensional complex manifold of regular points of Y (X).

Theorem 3.1. A Cousin data X =(/;, Vi) in a domain Q defines in a canonical way a positive closed current ex, which is called the current associated with

i -the Cousin data. The value of ex is given by ex=-aaloglfjl in Vj . Moreover

n if Z=(J;', Vi) is a Cousin data and if Z is equivalent to X, then ez=ex .

Proof Let tj=~aalOglf) in Vj and define ex in Q by ex=tj in Vj. Since f j fk- 1 and fJj- 1 are holomorphic in Vj"Vk, loglfjl-loglfkl is plurihar-

monic in Vk"Vj. Thus tj-tk=~aa[lOglf)-loglfkl]=O, so ex is well de-fined and is positive and closed, since each tj is positive and closed. By the same method, in a covering V. finer then {Vi" Vj}, we prove aa[log If;1 -log 1/;1] =0 and ez=ex · 0

In the classical case where n = 1, if f (z) is holomorphic in Q and a is a zero of f, then there exists a neighborhood Va of a such that f (z) = (z -a)q g(z), where g(z)=j= 0 in Va' Then

n- 1 iaalog If(z)1 =5i iaalog Iz -al =!L Lllog Iz -al' f3 = qi5(a) n 2n

where i5(a) is the Dirac measure of the point a. Thus for n = 1, the current associated with a Cousin data {ak,mk} is eX=Lmkb(ak). For n>l, we show:

k

Theorem 3.2. Let X =(/;, Vi) be a Cousin data and ex the associated positive closed current. Then

(3,1) ex = I mk[Yk(X)], k

where the Yk(X) are the irreducible branches of Y (X), [Yk(X)] is the positive closed current of integration over the connected submanifold Yk(X) of regular points of Yk(X), and mk=vX(z) is a positive integer, the multiplicity of Yk(X) in the Cousin data X. For every form CPE~~(n-1,n-1)(Q), we have

(3,2)

where the sum is taken over those Yk(X) which intersect the support of cpo

§1. Positive Closed Currents of Degree 1 Associated with a Positive Divisor 61

First we remark that as a consequence of Proposition 2.31 and 2.32, at each point XE Yk(X), the Lelong number vex, [Y]) has the value 1. By using an analytic isomorphism, we can suppose x=o and Yk(X) defined by zn=O. Then

For


It follows from Theorems 2.46 and 2.47 that if y'=U Yk(XY, Ox=OxIQ_Y' = L mk [Yk(X)J by Proposition 3.3. k D

k

Definition 3.4. For X = (1;, Ui) a Cousin data, the positive integers mk which appear in the expression (3,1) for Ox are called the multiplicities of the Yk(X) in X, and the current Ox associated with the Cousin data is the current of integration with multiplicities.

Definition 3.5. The measure C5X =OxAf3n_l' trace of the current Ox, will be called the area of Y (X) with multiplicities. This definition is justified by the fact that S f3n-l is just the (2n - 2) dimensional area of the complex

Yk(Xj

manifold Yk(X).

Remark 1. The majoration of IIOxl1 by CnC5X can be interpreted as the majoration of the current of integration over the analytic variety Y (X) by the area of the analytic variety Y (X).

Remark 2. In the same way, Vx can be interpreted as the projective area, and vx(r) is the measure relative to the metric of IP(

§2. Indicators of Growth of Cousin Data in


Since the non-constant terms on the right hand side are positive and vx(t) is positive, i) implies ii) and the existence of lim t-Svx(t) = C. Then C =0 follows

from ii). Conversely, ii) implies that lim S t- s - i vx(t)dt =0 or, since vx(t) is r-+- CX) r

increasing, lim vx(r) Y t- s - i dt = lim Cs vx~r) =0, so ii) implies i) by (3,3). r-IX! r r-oo r

The equivalence of ii) and iv) as well as i) and iii) follows from (2,15). 0

Definition 3.10. The number r =inf {s} such that i) holds will be called the convergence exponent of the Cousin data X.

00

Definition 3.11. The smallest integer q for which S t-q- i dvx(t) < + 00 IS 1

called the genus of the Cousin data. We have, from Proposition 3.9, that 00

S t- q- 2 vx(t)dt< + 00. 1

Proposition 3.12. The order p of vx(t) is equal to the convergence exponent of vx(r). If p is not an integer, the genus q of vx(t) is the largest integer less than p; if p is an integer, we have q -1 ~ p ~ q. If p = q -1, then X is of minimal type with respect to the order p.

Proof The proof follows immediately from the definitions and Proposition 3.9. 0

§ 3, Canonical Potentials in 1R. m

For XE1R.m, we let

hp(a,x)=lla-xll-P 1~p~m-2

ho(a, x)= -log Ila-xll for p=O. For p=m-2, -hm _ 2 (a,x) is the Newtonian kernel in 1R.m and

L1 xhm_ 2 (a, x) = 2nrm _ 2 b(a).

For q a non-negative integer, we define: 1 aqh

ep(a, x, q)= -hp(a, x)+h/a, 0)+ ... +qy atqq (a, tx)lt~o,

which we call the canonical kernel of genus q and dimension p in 1R. m, Then

~ aqhp(a, tX)\ = Pq(a, x, p) q! atq t~O IlalI P '

§3. Canonical Potentials in IRm 65

where Pq(a, x, p) is a homogeneous polynomial of x in lRm of degree q. For p = m - 2, the functions Pq(a, x, m - 2) are harmonic polynomials in lR m, and for p=m-2s, 2:£2s:£m, they verify LI~Pia,x,p)=O, where LIS is the Lap-lacian iterated s times. We then have, for a =1= 0, 0:£ p :£ m - 2

00

(3,5) = -ilall- P L Ij(a, x, p) j=q+l

where the latter expression converges uniformly on every compact subset of the ball Ilxll < Iiali.

Let Ilxll =tllall for t>O and let e be the angle between the vectors (O,a) and (0, x) in lRm. Then

Iia _x11 2 = IIal1 2 [1-2t cos e + t2 ] = IlaI1 2 (1-te i6 )(1 - te- iO), and hence (3,5) is majorized term by term by the series

00

(3,6) (l-t)-P= L bp.stS with bp.s=(S!)-lp(p+ 1) ... (p+s-l). s= 0

The case p = ° corresponds to the classical case studied by Weierstrass for ° ) : i) if q~l, leo(a,x,q)l:£uq+ 1 for u:£-q-. For q=O, leo(a,x,q)l:£eu for

< -I q+1, u e . q ii) if q~1, eo(a, x, q):£euq (2+logq) for u~+I' and if q=o

eo(a,x,O):£log(1+u) for all u>o. q

Proof The first part of i) stems for the fact that 00 US uq + 1 1 L _:£ __ ' __ :£uq+ 1

s=q+lS 1-uq+l 'f q I u


Proposition3.14. For p~l, we let r(p,q)= ( p+q )P. Then for a=l=O and Ilxll p+q+ 1

U=-Iiall' i) (3,7)

ii) (3,8) where

and

if u~r(p, q);

if u~r(p, q)

C1 (p, q)= [(p -1)!J- 1(p+q)p-1(p+q+ 1)

C 2 (p,q)=[(p-l)!J- 1 (q+1)(q+l) ... (q+p-l) exp (-.!!.!L) p+q < eP(p + q -1)P [(p -1)!J-1

for p~ 1 and q>O. For p~ 1 and q=O,

( p+q )P Proof From (3,5), (3,6) and 0 < U < r = < 1, we have p+q+l

00

lep(a,x,q)I~llall-P I bp.suS q+1

a) For p=l, (3,6) gives bp.s =1 and

lep(a, x, q)1 ~ Ilall- puq + 1 (1 + r + ... )= Iiall- Puq + 1 (l-r)-1 = Iia 11- Puq + 1(p + q + 1)

and (i) is proved with the value C(1, q).

b) Forp~2,wewrite

b =(p+s-l)! (s+I) ... (s+p-l)

§4. The Canonical Representation of Entire Functions of Finite Order 67

Thus (J'r:= p+q and (l-eJ'l:)-1=p+q+1, which proves (3,7) with the p+q+l

value C1 (p, q). In order to calculate C 2 (p, q), we use the first equality of (3,4). Since

-hp(a, x) is negative, we obtain

ep(a, x, q);£ Iiall- P[l +bp,l u + ... +bp,quq] ;£ Iia 11- Pr-quq [1 + bp, 1 r + ... + b p,q rq].

This gives immediately C2 (p, 0) = 1. In the general case, since r < 1 and there are (q + 1) terms in the brackets and since b p, s ;£ b p, q' we have:

(p+q+ l)pq ep(a, x, q);£uq(q+ l)bp,q p+q

(P+q-1)! ( pq ) ;£uq(q+1) ( -1)' , exp -.' '. p .q. p+q o

Proposition 3.15. Let a, x EIR n with a =1= 0 and m ~ 2 and let p, q be positive integers. Then

(3,9)

eP a) for p~l, C(p,q)=(p_1)! (p+q+i)P;

b) for p=O and q~l, C(0,q)=3e(2+logq);

c) for p=q=O, C(O,O)=l.

Proof We choose C(p,q)=sup[(l+r)C1(P,q), (1+r)-lC2 (P,q)J and use

Proposition 3.l4 to obtain the estimate. If we replace r by ( P + q )P, we p+q+1

obtain the value for p ~ 1. The case p = 0 follows easily from Proposition 3.13. 0

Remark. The bounds for the kernel e p(a, x, q) do not depend on m in IR/n. In the sequel, we use these in (Cn = IR 2n.

§ 4. The Canonical Representation of Entire Functions of Finite Order

If X =(1), U) is a Cousin data in (Cn and vx(r) is of finite order p, we are interested in finding an entire function F(z) whose zero set is exactly X such that MF(r)=sup log IF(z)1 is minimal. In fact, we shall treat this problem as

II z II ;ir


a special case of a larger problem. For V = log IFI, we have

(3,10) i - i - aa V=- aa loglFI=8x n n

where 8x is the current associated with the Cousin data X. Then we have to ,

determine a plurisubharmonic function V which is a solution of ~aaV=8, n

where 8 is a given (1,1) positive closed current of degree 1. From (3,10) we deduce, with 8=8x :

(3,11)

ae is a positive distribution, so it is a positive measure, the area of the Cousin data X with multiplicity (see Theorem 3.2). We first construct Vas a potential in IR. Zn =


and the right hand side converges by Proposition 3.9, since ve(t) is of genus q. This establishes the uniform convergence. On the other hand, e2n- 2 (a, z, q) differs from h2n - 2 (a, z) by a finite sum of harmonic poly-nomials, from which it follows that L"le2n_2(a,z,q)=k2n_22nc'5(a), and (3,13) holds. D

Theorem 3.17. The canonical potential I q(z) defined by (3,12) with respect to a positive closed (1,1) current 8 of genus q and such that B(0,ro)nsupp8=0 satisfies the inequality

(3,14) Iq(z) ~ A(n, q)rq [S t- q- 1 ve(t)dt + r S t- q- 2 ve(t)dt ] ~ r

for Ilzll =r. We can choose A(n, q) = (2n _2)-1 C(2n -2, q)(q + 2n -1).

Proof From (3,9), we obtain that

_ < -1 _ q+1 COs d(Je(t) sup Iq(z)-M(r)=k2n_2 C(2n 2, q)r q+2n-2.

Ilzll =r ro (t+r)t

We integrate by parts in order to express the right hand side in terms of (Je(t) and hence ve(t):

00 d(Je(t) 1 (t+r)tq +2n - 2 with a=q+2n-1, b=q+2n-2. The first term is zero, since (Je(ro)=O and lim t- q- 2n-1 (J e(t) = lim t- q- 1 Ve(t) = 0, and ve(t) is of genus q.

Thus

00 d(Je(t) 00 (Je(t)dt 1 (t+r)tq+2n - 2 ::::(2n+q-l) 1 (t+r)tq+2n - 1 W vo(t)dt

= (2 n + q - 1), 2 n - 2 S ( ) q + 1 • ro t + r t

It then follows that

-1 +1 Ws ve(t)dt M(r)~k2n_2'2n_2(2n+q -1) C(2n -2, q)rq q+1

ro (t+r)t

+1 WS· vo(t)dt


rt -1

+Cz(2n-2,q)rq S Ve(t)t-q-1dt] ro

ve(rr- 1 ) +rq (2n-2) [C z(2n-2, q) -C1(2n-2, q) r].

The canonical potential is lR2n subharmonic. Thus, using (3,12) and Gauss' Theorem, we have

where, since .,l(O, r, Iq) is a convex function of _rZ-2n, it has a derivative except perhaps for a countable set of values of r. Thus

(3,15) ( a/,(O, r, Iq)

v r)= . o a log r

Theorem 3.18. The canonical potential Iq(z) with respect to the positive closed (1,1) current e whose support does not contain the origin has the following properties:

i) M(r) and veer) are of the same order p,

ii) if p is not an integer and if vo(r) is of minimal, normal, or maximal 00

type with respect to rP, then so is M(r) and the integrals S M(t)t-P-1dt and 00 ~

S ve(t)CP-1dt converge or diverge together; ro

iii) if p is an integer, M(r) and veer) are not necessarily of the same type, but if S ve(t)t- p-l dt < + 00, that is, if the genus of e is q = p -1, then M (r) is of minimal type with respect to rP.

Proof i) From (3,15), we see that J.(O, r, I q) is a convex increasing function of logr. Thus, since .,l(O, r, Iq)~Iq(O)=O, we obtain

(3,16) ve(r)~J,(O, er, Iq)-l(O, r, Iq)~.,l(O, er, Iq)~M(er),

and hence p'=order Iq~p=order veer). In the other direction, we use (3,14). If ve(t)~ C(e)tPH for e>O, then (3,14) gives M(r)~ C(e)A(n, q)rp+e, so that p'~p.

ii) If p is not an integer, the genus q of e satisfies q < p < q + 1. If y is the type of ve(t) then ve(t)~(y+e)tP for t~R, and we obtain from (3,14), letting


R

0(= S t-q-1vo(t)dt, ro

(3,17) M(r)~A(n,q)[o(+(Y+B)] (~+ rP ), p-q q+1-p

which shows that the type y' of M(r) is at most C1 y. On the other hand, from (3,16), we see that y~ey'. In the same way, (3,16) shows that the

00 00

convergence of S M(t)t-P-1dt implies that of S vo(t)t-P-1dt. Conversely, using (3,14) ro ro

By changing the order of integration and observing that q - p < 0 < q + 1 - p, we obtain

R r R R

S rq+p-1dr S t-q-1vo(t)dt= S t-q-1vo(t)dt S rq-p-1dr o 0 0 t

R 00 ~(p_q)-l S t-P-1vo(t)dt«p_q)-1 S t-P-1vo(t)dt

o 0 and

00 00 00 t

S rq-Pdr S t- q- 2 vo(t)dt= S t-q- 2 vo(t)dt S rq-Pdr R r R R

00

~(q_p+1)-1 S t-P-1vo(t)dtro be such that for B>O, Vo(t)R and S vo(t)t-P-1dtR, we obtain from (3,17) that R

~A(n, q) [cr- 1 + B(r~R) +B],

which shows that M(r) is of minimal type of order p.

We now develop an analogue of Theorem 3.18 for proximate orders.

o

Theorem 3.19. Let 8 be a closed positive (1,1) current such that O¢supp 8 and such that its indicator vo(t) is of finite order p which is not an integer and


normal type with respect to the proximate order p(r). Then Iq(z), its canonical potential, is also of normal type with respect to the proximate order p(r).

We shall need the following Lemma:

Lemma 3.20. If p(r) is a proximate order, then for llR>ro,

r

l(r)= S tP(t)-Adt=(p+ 1_1l)-l rP(rl+1-J·+ 0 (rP(rl+1-J.) R

and for },> p + 1, co

l~(r)= S tp(t)-Adt=(Il_p_1)-lrP(rl+1-A+o(rP(rl+1-A).

Prool After an integration by parts, we obtain

r

S tP-Atp(t)-Pdt =(p+ 1_},)-1 [tP(tl+1-A]~ R

r

-(p+ 1-1l)-1 S [tP(t)-A(p(t)_p) R

+tP(t)+1-)'p'(t)logt]dt=I 1 +12 ,

I t follows from Definition 1.15 that given 8> 0, there exists ~ such that r

IIzl;£8StP(t)-Adt for r>~ (we recall that p-},>-1 implies that R

r

lim S tP(t)-Adt= + co). Hence r-+(JJ R

11 tp(t)- A dt -(p + 1 - },)-1 rP(r)+ 1- AI;£ 8(1 + 8)-1 [rP(rl+ 1-A + C]

where C=(p+ 1-).)-1 RP(R)+l-A. For },>p+1, we obtain

co S tP- AtP(t)- P dt = (), - P _1)-1 rP(r)+ 1-A

00

+(),-p-1) S tP(t)-A[(p(t)_p)

+p'(t)·tlogt]dt=ll +lz.

It follows from Definition 1.15 that given 8> 0, there exists ~' such that CD

Ilzl;£8 S tp(t)-)'dt and thus

§S. Solution of the aa Equation 73

Proof of Theorem3.19. From (3,16), we have vo(t)~M(et) and hence

vo(t)t-p(t) ~ M (et)t- pet) ~ [M(et)(et)- p(et)] [etJP(et)- p(t). ep(t).

It follows from Definition 1.15 that lim ep(t) = eP and from Theorem 1.18 that t~oo

lim [et]p(et)-p(t) = 1. Thus limsupvo(t)t-P(t)~ePlimsupM(t)t-P(t). In the t--+eo t-+oo t--+oo

other direction, with qro, we obtain via (3,17)

M (r) ~ A(n, q)rq{j vo(t)-q-l dt + (C + 8) [lq+ 1 (r) + l~+ 2 (r)r]}

~A(n, q)( C +8) [_1_+ 1 ] rP(r) +o(rP(r») p-q q+l-p

by Lemma 3.20. o

Remark. Starting with bounds for the growth of M (r) = sup I q(z), it is easy Ilzll =r

to obtain a control of the mean values ),(0, r, Iq) on Ilzll =r and of

A(O, r, Iq)=('2nr2n)-1 S IIq(z)ld'2n' II z II :;;;r

We set I:=sup(Iq,O), I.;-=sup(-Iq,O). From the subharmonicity of Iq we obtain O~),(O,r,Iq)=A(O,r,I:)-),(O,r,I';-) from which it follows that

O=A(O, r, Iq)~A(O, r, I:)~M(r) and hence

(3,18) A(O, r, IIql)~2M(r) and A (0, r, IIql)~2M(r).

§ 5. Solution of the a a Equation We have already seen that the canonical potential Iq(z) associated with a positive closed (1,1) current 8 of genus q and such that O¢supp 8 solves the

1 . equation 2 n AI q = (J o' In this paragraph, we shall show that it solves in fact

the more restrictive condition of equation (3,10). i

Let 8=; I 8p,qdzp/\dzq, where the 8p,q are complex measures. Then p,q

i - i 8'=-88Iq-8=- I 8~,qdzp/\dzq has the following properties:

n n p,q

i) its trace I 8~p is the zero measure; ii) 8' is 8 and a closed.


Proposition 3.21. If a current (J' of type (1,1) is closed and has zero trace, then it can be represented by a differential form with harmonic coefficients.

Proof Let us first suppose that the ~oefficients (J~,q are twice continuously differentia ble. Then, since d (J' = 8 (J' + 8 (J' = 0, we obtain

Thus

8(J~,q _ 8(J~,q. 8zm --az;:'

8(J~,q _ 8(J~,m 8zm - 8zq

for m*p,q.

4L1(J,=,,82(J~,q 82 "(J' ° p q L... 8 8 8 8 - L... m, In = . , m Zm Zm zp Zq

To treat the general case, we take IXE~~(B(O,l)) such that SIX(z)d'211=1

and IX ~ ° and set IX,(Z) = IX (f) 8- 211, where IX depends only on liz II. Then (J~,q*IX, = H(J~,qd, 2n(u)] [IX,(Z -u)] is a ~oo function, and the current I ((J~,q *IX,)dzp /\ dZq satisfies the hypotheses of the Proposition. Hence the p,q coefficients (J~, q * IX, are harmonic functions. Using the mean value property for harmonic functions we obtain [(J~,q*IX.]*IX, =(J~,q*IX" so [(J~, q * IX. - (J~, q]* IX, = ° for every 8,8'. Hence when 8-*0, we obtain (J~,q=(J~,q*IX., which shows that as a current (J~,q is equivalent to a form with harmonic coefficients. D

Lemma 3.22. Let hex) be a harmonic function for Ilxll

§5. Solution of the aa Equation 75

Corollary 3.23. i) If hex) is harmonic in 1R.P, then its complexification f(X) in


ii) From Theorem 2.16, we know that the coefficients 8p ,k satisfy 118p,kIIK~2110'1IK for every compact set K. Thus, if Mo=supI1X11,

IS 2 (z)1 = 18 p,k *lXel ~ 2 S lXe(z -u)dO'(u) ~ 211 0'11 B(z, elMO 8- 2n,

and if 8 = r, we have

o

Theorem 3.26. The canonical potential I q(z) with respect to a positive closed current 8 of degree 1 whose support does not contain the origin and which is of finite order p and genus q is plurisubharmonic in

§6. The Case of a Cousin Data 77

Proposition 3.27. If the positive closed (1,1) current e has an indicator ve(r) of finite order p ~ 0, then there exists a real finite dimensional vector space E p of all the solutions of minimal order p of the equation i88V=8:

i) if p is not an integer p -1 < q < P then

(3,21 ) V(Z)=V(0)+2Re{f ~88jj [V(tZ)]t=o}+Iq(Z), j= 13! t

where V (z) = ~(z) + Iq(z) and Pq(z) is determined by the value of V and its derivatives up to order q at the origin; we then have order Pq ~ q < P = order I q •

ii) if p is an integer and q = p, the solutions of order p are all given by (3,21); if q=p-1, then V(z)=~_l(z)+Iq(z)+Pp(z) where Pp is the real part of a homogeneous polynomial of degree p in z.

In conclusion, we resume:

Theorem 3.28. If e is a positive closed (1,1) current of finite order p in


= 2a Iq and thus z

(3,23) G=log Fo(z)=logF(0)+2 S alq(~), o

where the integral is taken over a polygonal path from 0 to Z compact in (Cn - Y (X).

Let us show that (3,23) defines the logarithm of a non-zero hoi om orphic function. If we replace the path y by the path y', we must verify that we obtain a multiple of 2ni over the closed path Yo=(Y, y'). The open set (Cn - Y(X) is locally arc connected. It is enough thus to prove the result when Yo is in U. and is the boundary of a manifold Yo' By Stokes' Theorem and J _ _ the equation aa I q = aa log lijl in Uj, we write successively

2 S aIq(Z) =2 S dalq(z)= -2 S a8Iq= -2 S a810glijl )'0 'Yo Yo Yo

2 S alq(z)=2 S da log Ii) =2 S a log Ifjl = S d logij=2niN. Yo Yo Yo '10

This shows that G=logFo is determined by (3,23) and Fo=eG is well defined in all (Cn.

To prove that Fo(z) is holomorphic in (Cn, we choose zoct Y(X). Then by the definition of G in (3,23), dG is a (1,0) form in a neighborhood of zo; dG=2alq implies 8G=0. Then G=logFo(z) is holomorphic in zo; moreover log IFol =Iq is locally bounded above on compect subsets, and hence !Fol is locally bounded and can be extended as a holomorphic function to all of (Cn by Riemann's Theorem (cf. Corollary I.23).

In Uj, we have ~a810gIFo(z)l=e=~a810glijl, which shows that Foij-l n n

and ijFol are never zero in Uj. Hence, we obtain Fo=ij'({Jj where ({Jj is holomorphic in Uj , ({J j =1= O. Furthermore, if F is an entire function vanishing on the Cousin's data X, then FFol =(F.fj-l)({Jj-r, and so g=F·Fo- 1 is holomorphic in every ~, hence entire, with F =gFo' If F is a solution of Cousin's Second Problem for the data X, then F=gFo, with g=eh=l=O in (Cn, hE£' (Cn). 0

Using Theorem 3.26, we obtain:

Theorem 3.30. Let X = (ij, U) be a Cousin data such that oct Y (X) and such that ex is of finite order p. Then there exists an entire iimction Fo(z) which has exactly X as its zero set (that is, which solves Cousin's Second Problem with data X) such that

i) log !Fo(z)1 =Iq(z)=k;,L2 S e2n _2(a, Z, q)dCIx(a)

and z z log Fo(z)=2k;,L2 S alq{O=2k;'nl_2 S dCIx(a) S ae2n_2(a,~, q),

o 0

§7. Slowly Increasing Cousin Data: the Genus q=O; the Algebraic Case 79

where (0, z) is any compact polygonal path in ([n - Y(X) and q is the genus of X;

ii) loglFo(z)1 ~A(n, q)rq [i t- q- 1 vx(t)dt+ J t- q- 2 V x (t)dt], ~ r

where vx(t) = [r 211- 2 t2 /1- 2J -1 (j x(t) is the projective indicator of X and B(O, ro)n Y(X)=0;

iii) Fo is of the same order as X and if vx(t) is of normal type with respect to a proximate order pet) and p is non-integral, then Fo is also of normal type with respect to the proximate order pet);

iv) Fo divides every entire function which is zero on X;

v) Among the set of entire functions which are zero on X and only on X (with the given multiplicities), Fo is the unique function which has order equal to the order of vx(t) and such that Fo(O) = 1 and all derivatives up to and including q = genus vx(t) are zero at the origin;

vi) any entire function which has properties i)-iv) can be written as Fo(z) exp P(z), where P(z) is any polynomial of degree at most p.

Corollary 3.31. An entire function F of finite order p is determined by the set X of its zeros and its value at a finite number of points in ([n equal to dim E p (cf Proposition 3.27).

Definition 3.32. The genus q' of an entire function f of finite order will be defined by q' = sup (q, p), where q is the genus of the zero set of 1, and p is the degree of the polynomial P such that f(z)=Fo(z) expP(z).

The genus of f is at most equal to its order p and is strictly smaller if p is not an integer.

§ 7. Slowly Increasing Cousin Data: the Genus q=O; the Algebraic Case

Estimates of the growth of Fo(z) for a Cousin data X of finite order in ([n depend on a constant C(2n - 2, q) (Proposition 3.15), which in its turn depends upon two constants C 1 (2 n - 2, q) and C 2 (2 n - 2, q) (Proposition 3.14). For q = 0, we have C 2 (2n - 2,0) = 1, which is in dependant of the dimension 11, but

(2n - 2)2n-2 C1(2n-2,0) depends on n. If r= 2n-1 ' for n;;;;2 then O


where (j = 'C 1. It is interesting to consider the case where the second integral on the right is negligible with respect to the first - in this case, we obtain an estimate independant of the dimension n of the space. This will be the case if vx(t) satisfies an estimate of the type

(3,24) vx(t)~ C(log+ t)'+ C C>O, C>O, s~O. 00

Let Is= S (logt)Sc 2 dt. An integration by parts shows that

Thus, we obtain

(3,25) Mo(r)= sup 10glFo(z)1 liz II ~r

~ C(S + 1)-1 (log+ r)'+l + An,,(log+ r)'(1 +8r )

where 8r=0(r). We resume these results as follows (cf. Theorem 1.6).

Theorem 3.33. If X is a Cousin data such that vx(t) satisfies (3,24), then

. M(r) . v (r) (3,26) hmsup +lS;(s+1)-lhmsup-x-

r~ro (logr)' - r~oo (logr)'

holds for the canonical solution of Cousin's Second Problem. If s =0, that is if o

§ 7. Slowly Increasing Cousin Data: the Genus q = 0; the Algebraic Case 81

Theorem 3.34. Let F(z) be an entire function and let lim (logr)-l M(r)=a

i -and lim "x(r)=b, where vx(r) is the indicator of -aologlFI and vx(r)

r-7oo n o

=-a 1- A(O, r, log IF i). Then ogr

i) if a=O, F(z)=F(O) is constant and b=O; ii) if O


§ 8. The Case of Integral Order: Extension of a Theorem of LindelOf

We have not yet given a complete treatment of the comparison between the growth of vx(

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