Pierre Lelong
Lawrence Gruman
Entire Functions of Several Complex Variables
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Pierre Lelong
Lawrence Gruman
Entire Functions of Several Complex Variables
Springer-Verlag Berlin Heidelberg NewYork 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 3-540-15296-2 Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-15296-2 Springer-Verlag New York Heidelberg Berlin Tokyo
Library of Congress Cataloging in Publication Data Lelong, Pierre. Entire functions of several complex variables. (Grundlehren der mathematischen Wissenscharten; 282) Bibliography: p. Includes index. I. Functions, Entire. 2. Functions of several complex variables. I. Gruman, Lawrence, 1942-.
II. Title. III. Series. QA353.E5L44
1986
515.914
85-25028
ISBN 0-387-15296-2 (U.S.) This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, spefifically those of translation, reprinting, re-use of illustrations, broadcasting reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use a fee is payable to "Verwertungsgesellschaft Wort", Munich.
© Springer-Verlag Berlin Heidelberg 1986 Printed in Germany Typesetting, printing and bookbinding: Universitlitsdruckerei H. Stiirtz AG, Wiirzburg 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 complex variable and the growth of
III
(or equivalently log
III)
I(z)
of one
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 Fn:chet spaces. However, for applications
it is of utmost importance to develop precise and explicit representations of the solutions.
If we pass from (C to (Cn, new problems such as problems of value (Cn to Cm arise. On the other hand, new techniques are often needed for classical problems to obtain solu
distribution for holomorphic mappings from
tions and representations of the solutions. Zeros of entire functions I are no longer isolated points; a measure of the zero set is obtained by the repre
sentation of the divisor XI of I (and more generally of analytic subvarieties)
by closed and positive currents, a class of generalized differential forms. .
Paradoxally, it is the non-holomorphic objects, the "soft" objects (ob [C]) of complex analysis, principally plurisub harmonic functions and positive closed currents, which are adapted to problems with growth Conditions, giving global representations in (Cn. Very
Jets souples in French, see
often properties of the classical (i.e. holomorphic) objects will be derived from properties obtained for the soft objects. Plurisubharmonic functions
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 I are obtained as upper bounds for logl/l; for logi/i. To solve Cousin's Second Problem, i.e. to find (with growth conditions) an entire function I with given zeros X in (Cn, we solve first the general equation j aJ v = () for a closed and positive current (); if () = [X], the current of integration on X, we then obtain I by V =log III. Properties of plurisubharmonic functions appear again in a remarkable (and unexpected) result of H. Skoda (1972): there exists a representation for the analytic subvarieties Yin (Cn of dimension p(O~p~n-l) as the zero set Y=F-1(0) of an entire mapping F=(fl' ... ,jn+l) such that IIFII is controlled by the growth of the area of Y. Plurisubharmonic functions obtained from potentials seem well adapted to the construction of global representations in (Cn; the method avoids the delicate study of ideals of holomorphic functions vanishing on Y and satisfying growth conditions. The same methods using the soft object's properties of the current (iaaV)p and the Monge-Ampere equation for plurisubharmonic functions V are employed for recent results obtained in value distribution theory of holomorphic mappings (Cn -+ (Crn or X -+ Y, two analytic subvarieties in (Cn. II - Before summarizing the content of this book, we would like to make some remarks. a) We have not sought to give an exhaustive treatment of the subject (problems for n> 1 are too numerous for a single book). We have tried to introduce the reader to the central problems of current research in this area, essentially that which had led to general methods or new technics. Applications appear only in Chapter 6 (to analytic number theory) and in Chapters 8 and 9 (to functional analysis). b) On the other hand, we have tried to make the book self-contained. Some knowledge in the theory for one complex variable is required of the reader, as well as on integration, the calculus of differential forms and the theory of distributions. A list of books where the reader can find general results not developped here is given before the bibliography (such references are given by a capital roman letter). The proofs of complementary results appear in three appendices: Appendix I for general properties of plurisubharmonic functions, Appendix II for the technic of proximate orders Appendix III for the J resolution for (0, 1) forms with L2 -estimates by Hormander's method. c) The importance of analytic representations, particulary for some applications, has made it necessary to give certain calculations in extenso. The authors are aware of the technical aspect of some developments given in the book. We recommand that the reader first read over the proof in order to assimilate the general idea before immersing himself in the details of the calculations. d) The literature on the subject of entire functions is enormous. The
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 fundamental properties of positive differential forms and of positive and closed currents. Chapter 3 studies the solution with growth conditions of the equation iaaV=() for () a positive closed current of type (1,1) in ern, 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 ern gives an extension of classical results of J. Hadamard and E. Lindelof for n = 1. Chapter 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: ern ~ er m • The first portion is devoted to the development of a representation of an analytic subvariety Y of ern as the zero set of an entire map F: ern ~ ern+ \ that is Y =F- 1(0), F=(fl' "''/n+l)' with control of the growth of the function \IF\I. The second part studies the growth of the fibers F- 1 (a)nB(0, r), where B(O,r)={z: \lz\l 0, I ~ k ~ n] is called the polydisc of center z', of radii rk • For A (z', r)~ Q, and f holomorphic in Q, the iteration of the Cauchy Integral Formula for one complex variable gives for zEA(z', r) the integral representation: 2"
f(z)=(2n)-n
(1,2)
As for
f
o tl
= I, we deduce from (1,2) a Taylor series expansion: (a)
which converges uniformly for Iz~ - zkl ~ r~ < rk. Then we obtain a Taylor series expansion on each compact polydisc of Q. We designate by .n"(Q) the family of functions holomorphic in Q. By an entire function, we shall mean an element of .n"(CCn ). Thus, an entire function f(z) has a Taylor series expansion f(z' + z) = L L Pa(z')za which, for every point z', converges unim lal=m
formly in z on compact subsets of CC n• We say that
L
p"(z')za is the ho-
lal=m
mogeneous polynomial of degree m in the Taylor series expansion of f(z) at the point z'.
§ 2. Subharmonic and Plurisubharmonic Functions In our study of entire functions f of several complex variables, we shall be interested in the asymptotic growth of If I, fE.n"(CC n), or equivalently by the asymptotic growth of log If I· Suppose for instance that q>(t) is an increasing function of t for t ~ 0 such that lim sup q>«tu» < 00, for u ~ 0 and lim q>(t) = 1-00 q> t t-oo and consider in .n"(CC n) the subclass Mtp defined by the condition
00,
log If(z)1 ~q>(llzll)+ C(f). Then the function . 10glf(tz)1 Xf(z) = lim sup () t-oc q> t
measures an asymptotic growth with respect to the weight factor q>(t) on the real lines through the origin. Thus we are led to consider expressions of the form lim sup Ci log 1.1;1, .l;E.M"(CCn ), CiElR +. This leads us to study filtered iel
§2. Subharmonic and Plurisubharmonic Functions
3
families included in a larger class of functions, the plurisubharmonic functions introduced by K. Oka and P. Lelong. This family is closed under the operation of taking the smallest upper semi-continuous majorant of a filtered family uniformly bounded above on compact subsets (in fact, one can show that the functions Clogl!l, !E£( 1 complex variables that the subharmonic function play in complex analysis of one complex variable. Moreover, in ( - 00 ~ q>(x) < + 00) is said to be subharmonic in Q if a) q> is upper semi-continuous and q>(x) $ - 00 in Q, b) q>(X)~A(x,r,q»=w;;;1 q>(x+rOt)dwm(Ot) for r on sm-I relative to the Haar measure dwm(Ot).
J
Definition 1.2 Let Qc(-oo~q>(z) < + <Xl) is said to be plurisubharmonic in Q if it has property (a) above and in addition b 2 ) q>(z) ~ I(z, w, r, q» =_1_ z+uwcQ for UE(z + wre
i9 )d8
for all w, r such that
0
In the sequel we denote by D(z, w, r) the compact disc {Z'EES(Q), then q>(x)~,;;;lr-m J q>(x+x')d,(x')=A(x,r,q» for r(z+uz') is identically - 00 or an increasing convex lui ~r
function of log r ; b) if p(z) is a complex norm, then Mq>,p(r) is an increasing convex function of log r. Proof For a): q>(z+uz')= -00 for all ueCC or q>(z+uz') is a subharmonic function of the variable u = IX + i f3 in CC = 1R 2 (cf. Remark 2 after Definition I.2). For b): Consider Ma,p(r)= sup [sup q>(uz)] and remark that zep-'(l)
lul~r
sup q>(uz) is an increasing convex function of log r or identically -
00,
but is
lui ~r
not identically -
00
for all z.
0
§ 4. Minimal Growth: Liouville's Theorem and Generalizations The existence of a minimal growth for a non-constant function q>ePSH(CCn ) is just a consequence of the convexity properties of Proposition 1.4 and formula (1,5). Theorem 1.5. i) Let p(z) be a norm and q>(z) a plurisubharmonic function in . M (r) . M (z, z', r) .. . ccn. Then C=hm q>.p and C(z,z')=hm ~ eXIst, eIther fimte or r-x logr r-x ogr infinite, with the following properties: a) C~O; moreover C(z,z')~O with the possible exception C(z,z')=-oo in which case q>(z+uz')= -00 for ueCC. b) C(z,uz')=C(z,z') for every ueCC, u=l=O.
§4, Minimal Growth: Liouville's Theorem and Generalizations
if p(z) is a norm on
then _O_Mtp p(r) ologr ' and ~I mtp(z, z', r) exist except perhaps for a countable set of rand u ogr , a ()r = I'1m Mtp ' p(r) . I Im--M r-oo alogr tp,p r-oo logr ' ii)
a
(C"
and 0 such that M tp, p(r0) > - 00, and since it is an increasing convex function of logr, Mtp,p(r) > -00 for r~ro, which proves that C~O, If O, k>O. This choice is motivated by the fact that the most familiar (and useful) transcendental entire functions faIl into this class as weB as the Fourier transforms of measures and distributions with compact support. Thus, the theory has an enormous range of applications from number theory to partial differential equations.
Definition 1.8. The order p of a positive real valued function a(z) with respect to a norm p(z) is given by . log Ma.p(r) p= I1m sup I . r-oc ogr If p < + 00, a(z) is said to be of maximal, normal, or minimal type according to whether the positive number .
(J'=hm sup r-oc>
M
a.:r (r)
§5. Entire Functions of Finite Order
9
is + 00, finite, or zero, and (1 is said to be the type of a with respect to p(z). For cpePSH(CC n) we define the order p of cp by using a(z)=sup[cp(z),O] =cp+(z). If f is an entire function, by an abuse of language, we shall say that f is of order p if log If I is of order p, and we shall denote Mloglfljr) by Mf,p(r) for simplicity. An entire function of order at most 1 and finite type is said to be of exponential type. Remark. The order and the nature of the type are not changed by a change in norms or a translation (cf. (1,5» and thus depend only on the topology of the space CC". Examples. i) If Jl. is a measure with compact support in CC n, then f(z)=
J exp (t
zk·ek) dJl.(e) is of order 1 and of normal or minimal type;
ii) if P(z) is a polynomial, exp P(z) is of order deg P; iii) if f1 (z 1)' ... ,fn(zn) are entire functions of finite order, and P(z) a polynomial, then poF is an entire function of finite order. iv) cos ~ in CC 2 is an entire function of order p = 1 and type 1
(1=
0. 00
L ~(z)
Theorem 1.9. If f (z) =
q=O
is the expansion of an entire function in
q
homogeneous polynomials and C = sup
1~(z)l,
then the order p and for p>O
p(z)~1
the type (1 of f with respect to p(z) are given by
a) -.!..=limsup log C q =lim sU P - I 1_ [sup log 1~(z)l]' p q~oo qlogq q~oo q ogq p(z)~1 b) log (1= lim sup [IOlg Cq +.!..] p logq -log p-1. q-oc q ogq P
p
h') log (1e = lim sup c)
q~GC
(~log Cq + log q). q
(1=li~}~p [r (~) cqfq.
d) For p=l, (1=limsup[q!Cq
r/
q•
q~oc
Proof Let Zq be a point on the unit p-ball for which IP (z )1 = C . By a . q q q J rotatIon, we can assume that Zq=(Xq, 0, ... ,0). If f(u)=f(u, 0, ... ,0) and (u) 00
=
L
... =0
•
a",um is the Taylor series expansion of
J
at the origin, then la xql
=Cq, and by the Cauchy Integral Formula,
q
Cq~r-qexpMf,p(r).
q
If
10
1. Measures of Growth
MI.p(r)~Ark for r>R(A, k), then Cq~r-q expArk, Since
d dr (r- q exp Ark) =r- q-
1(
-q + Akrk) exp Ark,
q )llk the minimum of this expression occurs when r= ( kA and is equal to eAk)q/k (-q- ,We then have for q sufficiently large log Cq ~~ [1 + log Ak -log q].
(1,6)
Thus, given e>O, for q sufficiently large,
k~ ~I::~q -e,
so if
f is of finite
, 1ent1y, -p- 1 ~ I'1m sup 10gC , ' sup qlogq or eqUiva q ord er p, p ~ I 1m -, If f IS q~oo -log Cq q~oo q logq of finite type (J and A>(J, (1,6) gives (with k=p): Aep~qC~/q for q sufficiently large (depending on A), so that (Je p ~ lim sup q C~/q, q~oo
Suppose now that
Cq~ (e:kr
k
for
q~qo(A, k),
If q>m r, where mr is
the largest integer smaller than or equal to 2keAkrk and r is sufficiently large, then for p(z) ~ r,
m,
Thus
L
If(z)1
O and
•
L 1';:1 = 1. k=l
Proof We let rk=r in (1,9). Then
r~=r7k.
D
Corollary 1.13. Let fe.Jf(CC"). If f has order Pk with respect to the variable Zk' k = 1, ... , n, then f is of finite order P and
sup Pk;£ p;£ k= 1. ...••
•
I Pj' k= 1
§5. Entire Functions of Finite Order
13
Proof The first inequality is a result of the inequality Mjkl(r)~Mf(r) for r ~ 1. To prove the second, we note that if Pk is finite, then for every P~ > Pk' n
L
there exists Ak such that Mjk\r)~AkrP~, 1 ~k~n, so if we set A= p~ and Yk =). Pi: I, then from Theorem 1.12 we obtain k= I Mf(r) = Mf(r, ... ,
r)~).-I
Ctl
PkAk)r".
o
We shall now apply (1,8) in another context by letting the numbers u l ' ... , Un_I remain fixed as Un goes to infinity and choosing the C k to be variable functions of un'
Theorem 1.14. Let fEJ'l'([n) and let 'I' ... , '.-1 be fixed pusitive numbers. Then there exists a positive function E(r) and ro> 1 such that E(r) goes to zero when r tends to infinity and for r > r 0' Mf('I' ... ".-1' r) ~ M f (l, ... , 1, rl
(1,11)
The function E(r) and ro depend on the
'j
and
H(r») =
Mj·l(r '
H(r l )
f
Proof From (1,9), we obtain
n-I ( ) R. d
Proof dr (r P(r))=p(r)r P(r)-l +rP(r)p'(r)logr. By (i) of Definition 1.15, for r>R p p(r»p/2 and by (ii) of Definition 1.15, for r>R2' Ip'(r)rlogrlsup(RpR2): . d
-(rP(r))>p/4 dr
rP(r)-l>o.
o
16
1. Measures of Growth
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 r > o.
Proposition 1.20. Given 6>0, there exists an R(6) such that (l-6)k PrP(r)«kr)p(kr) p(p2-1) rP(r)-1
for r sufficiently large.
D
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 Mr.p(r) for fEJf'(CC"). In Theorem 1.9, we obtained a formula for the type of an entire function of finite order p in terms of its Taylor series expansion in homogeneous polynomials. A similar formula exists for proximate orders. Since by Proposition 1.19, rP(r) is an increasing function for r>O, if p>O the equation t=rP(r) admits a unique solution for t>O. We will denote by r=cp(t) this solution; cp(t) is just the inverse function of rP(r). Of course, cp(t) depends on p(r), but the proximate order in question will be clear from the context, so we will not note this dependence. Theorem 1.23. Let f (z) = L ~(z) be the Taylor series expansion of the entire q
function f(z) of finite order p>O and of proximate order p(r), and let Cq = sup 1~(z)l. Then the type (J of f(z) with respect to the norm p(z) and to p(z)~l
the proximate order p(r) is given by
1 [1-log C + log cp(q) ] -1 - log -log (J = lim sup - -P, q p q~CX) q p p
p>O.
The function r=cp(t) is the inverse function of t=rp(r). . cp(kt) Proof 1) We first show that lim -(-) =kllP, OP. Then for q>2PO'''P1rP1, we have the bound Cqrq(Z)E PSH(CC n) c S(JRzn), we call ht(z, z', qJ) its circled indicator of growth function with respect to center z'.
Remark 1. Our principal interest will be the case when qJ=loglfl for fEJff(CC n ) an entire function of order p. In this case, we will say that h'~(z, z', qJ) and h~(z, z', qJ) are the radial and circled indicator functions of f Remark 2. The dependence on the function qJ will usually be clear from the context, and so will not always be noted. Proposition 1.30. For X'EJR m fixed, the functions hr(x, x', qJ) and h~(x, x', qJ) are positively homogeneous of order p. For Z'ECC n, the functions hc(z, z', qJ) and ht(z, z', qJ) are complex homogeneous of order p (i.e. hr(tx, x', u) =tPhr(x, x', u), t ~ 0 and h~(uz, z', qJ) = lul Ph(z, z', qJ)UECC).
Proof We shall prove only the case of the radial indicator, as the proof for the case of the circled indicator is practically identical (cf. Proposition 1.34). From Theorem 1.12, if L(r)=rP(r)-p, then for t fixed, t>O, lim L(tr)=1. Thus r .... oc L(r)
, . qJ(rtx+x'). qJ(rtx+x') (rt)P(rr) hr(tx,x,qJ)=hmsup () =hmsup ) (t) ._(-)pr r ..... oc r r .... oc (rt pr r pr _ . qJ(rtx+x') (rt)p(rt)-p P_ P , -hmsup (t)p(rt) p(r)_p·t - t hr(x,x,qJ) rt .... oo
r
r
22
I. Measures of Growth
and lim sup h,(Y, y', q»=t P lim sup h, (y,Y')-('X,x')
(y,y')-('x,x')
=
tP
(~, y', q» t
lim sup h,(Y, y', q»
= t P h~ (x, x', q».
(y,y')-(x,X')
o
Theorem 1.31 (Hartog's Lemma). Let v, (x), t>O, be a family of subharmonic functions uniformly bounded above on compact subsets in the domain DcJRm• Suppose that for a compact set K in D there exists a constant C such that w(x) = [lim sup v,(x)]* ~ C on K. Then for every 6> 0, there exists T" such that v,(x)~
C +6 for
t~
T" and XEK.
Proof We replace D by an open neighborhood Q of K relatively compact in
D such that w(x) ~ C +
i
in D. Since v,(x) is bounded above in D, by subtract-
ing a constant, we may assume that v,(x) ~. Since v, 0 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 Theorem 1.31. Since K is compact, we can choose a finite number of balls B(xi, b) which cover K. Then for t>sup TXi ' v,(x')0, there exists 8>0 such that hr(x", x o, cp):~ h(x)+ £/2 for IIx" - xii < 8 by the upper semi-continuity of h, and so by Theorem 1.31, there exists Re such that for r > Re and I x" - x I < 8, cp(rx"+xo) . . . p(r) ::::h(x)+£. Let Xl be an arbitrary pomt m 1R." and suppose r
Ily-xlll < 1. Then if Ilx" -xii ~ 8/2 and r is sufficiently large (depending on xo), X= (y-xo) +x" satisfies r
Ilx-xll ~8. Thus we see that cp(rx+xo) - ) p(r) ::::h(x +£ r
and hence hr(x',y,cp)~h(x)+£ for Ilx'-xll and Cq> such that cp(x)~Aq>rP(r) + Cq> for Ilxll ~r and XEr. The indicator function is then defined and subharmonic in r; it is positively homogeneous of order p; if r is convex,
XEr, and (j>*(x', x)= [lim sup cp(r:::x)], then (j>*(x)=cp*(x). r- 00
r
Theorem 1.34. Let cp(z) be a plurisubharmonic function of finite order p and normal type with respect to the proximate order p(r). Then h~(z, cp) = sup h~(zei6, cpl. 0~8~2"
Proof
It
follows
h~(z,cp)~h~(zei8,cp)
=h~(zo, cp) for
from the definition of the two functions that for all e. Suppose that sup h~(zoei8,cp)=bO such that limsupv,(z)~O
p(z)~p-e.
p~sup
p(z). Suppose that there 10glf(rz)1 If p =p-e/2 and v,(z)= P' ,then /
z
r
and so for Ilzll=l, by Theorem 1.31, there exists Ro such
, ... 00
that for r>Ro' v,(z)~ 1. Thus, Mf.p(r)~ Cfr P' and p~p/, which is a contradiction. 0
§ 9. Exceptional Sets for Growth Conditions Our purpose here is to classify those complex lines in £" on which the growth of an entire function differs from its global growth. A natural way of describing these exceptional sets is in terms of the pluripolar sets. We recall the definition:
Definition 1.36. Let Q c £" be a domain. A set E c Q is said to be pluripolar in Q if there exists q>EPSH(Q) such that E c {z: q>(z) = - oo}. Proposition 1.37. Let Q be a domain in £". Then a countable union of pluripolar sets in Q is pluripolar in Q. 00
Proof Let A~cAq={ZEQ: q>q(z) =
-00,
q>qEPSH(Q)} and let E= 00
be an exhaustion of Q, that is Qq~Qq+l and
U Aq.
q=l
UQq=Q. Since measure (Aq) =0, q= 1
00
there exists ~¢
UA~. Let Qq
m
Let Mq=supq>q and set Sm(z)= L Cq[q>q(z)-Mq], q=l Q.:x; q=l where the Cq>O are chosen so that LCqlq>qR)-Mql m-+oo
- 00.
Thus, S(z) = lim Sm(Z)E PSH (Q q ) for every q (Proposition 1.3). Hence m-+ ce·
S(z)EPSH(Q) (Corollary 1.20) and EC{ZEQ: S(z)= -oo}.
o
Proposition 1.38. Let q> E PSH (£") be bounded. Then q> == q> (0) is a constant. Proof By Proposition 1.17, Mtp(r)= sup supM(z, 1) r- 00 og r zeD' and m>sup(Mo, 1). Choose rm so that M(z',rm)=m. Then ,I'
.,
.
ProoJ. By defimtlon, p(z')=hmsup
p(z') = lim sup(logm) (log Mp= sup cp(z',u)~2. Now we replace I/I(z',m) by a sequence Ilz'll :;i,p,lul:;i, I
I/Ip(z')EPSH(CC n). We remark that for mp>Mp and Ilz'll ~p, supl/l(z',mp)~ -~pMp we define: 1/1 p(z') = sup [1/1 (z', mp), log II ~ "] 1/1
p
(z')=log~ p
for II z' II
~p
for Ilz'll ~p,
and log ~ is a p continuous function vanishing for Ilz'll = p, the function 1/1 p is well defined and 1/1 pE PSH (CC n- I). Now there exists z~, Ilz~ II < 1 such that Because 1/1 (z', mp) is bounded by -lXp F(z', zn) is finite for all z/. From Theorem 1.41 it is so if and only if p(z/) has finite values on a non pluripolar set. Note that F can be of finite order with respect to zn even if its total order is infinite.
Corollary 1.43. Let cpE PSH (CC n) and let p(z) be the order of cp=(u): u -> cp(uz). Then p(z) is a constant Po (finite or infinite) except on a pluripolar cone A with vertex at the origin where p(z) 1, lim mq = +
(log mq)-l
mo > 1 (see Proposition 1.39) such that
(1)
1
I/Im q(z)-C O - 2 ro and zEQI'
Proof Let ma = sup M (z, 1). The equation 1/1 (log r) = m has as solution log r ZEQI
= log lJ(m). The equation M(z, r) = m has as solution r = 15(z, m). Let I/I(z, m) -log b(z, m) I () for m>ml=sup(ma,I/I(O)). Then I/I(z,m)EPSH(QI) and
oglJ m I/I(z,m)A(Z)=SUP {r(z)} rEPSH(QI) r~O r~
-Ion A.
Since A is non-pluripolar, we have q>1(z)$O and q>1EPSH(QI)' Thus, for zEQI by the Maximum Principle. Let
q>~(z)~(z) log lJ(m) b(z,m)(z) = log If(z)l, f an entire function in ccn. As remarked before, the indicator functions for the growth of If I are plurisubharmonic functions (not necessarily continuous); later we shall apply the same technics to the indicator functions of the zeros of f
Historical Notes The idea of using intermediate functions in the definition of type is due to Lindel6f, but the use of proximate orders is due to Val iron [1]. The calculation of the order and type in terms of the Taylor series coefficients is classic for n = 1. For n ~ 2, this as well as variants has been studied in detail
Historical Notes
29
by the Russian school (cf. Gold'berg [1]). Relations between the total order and the orders relative to each variable were first given by Borel [1]; the first comparison with respect to the growth on complex lines was made by Sire [1] at the beginning of the century. The modern treatment of the indicator function as given here is primarily due to Lelong [2]. This generalizes the classical Phragmen-Lindelof indicator function and was first considered by Lelong [2] and by Deny and Lelong [1] and [2] for subharmonic functions. In particular Lelong developped in his early works Hartog's Theorem in (C2 in the context of subharmonic functions and potential theory. After the introduction due to Oka and Lelong [5, 6] 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 [2] and Martineau [4, 5] and will be presented in Chapter 7. The proof given here that h*(x, x', lfJ) is independent of the center has the advantage of working in the class of subharmonic indicators defined in cones. The results of § 9 and the Inverse Function Theorem for plurisubharmonic functions (see Appendix I) were given by Lelong [15] for complex topological vector spaces.
Chapter 2. Local Metric Properties of Zero Sets and Positive Closed Currents
§ 1. Positive Currents A biholomorphic mapping F: : (D)
onto
([>:(0').
Proposition 2.2 permits the definition of positive forms on a complex submanifold Me D: it is those forms which are positive for every choice of local coordinates.
§ 1. Positive Currents
For p = 0,
tPt (D)
31
is just the set of positive continuous functions on D. n
For p=l, qJEtPi(D) if and only if qJ=i L
qJjkdz/,dzk, where the matrix
j,k= ,
[qJjk(Z)] is positive semi-definite for every (2,1)
ZED.
If for any p we have
qJ=iA,I\I,1\ ... l\iApl\J..p,
where the ;.p are complex linear in dZ j with coefficients in CCo(Q), then qJEtP;(D). Those qJ 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 = ... =un=O. We define the form r(11')EtP;(O and in Q, QclR2~ the conditions .1=1=0 and Al =A[.(E's-P)] =1=0 are equivalent; we shall say that the system A={E'.-P} is regular. We conclude that in each open set ofR. 2Nn we can find points such that AI =1=0. If Gn_p(CC") is the p(n-p) dimensional complex Grassmannian manifold of (n -p) dimensional linear subspaces of ccn (cf. [H]) then in each open set of Gn_p(CC n ), we can find a regular system A = {L~-P} which allows us to calculate lfJI.J(Z} as a linear combination of Crp,s(z), If lfJ is a positive form, the Crp .• (z) are positive functions (later we use the same algebraic process on positive currents, and Crp,s. 13n will be a positive measure).
§ I. Positive Currents
33
Proposition 2.6. Let M be a complex submanifold of Qc (Cn of dimension p (cf. Definition 2,33) and CPEtP; (Q) with compact support in Q. Then
J cP = [M](cp)~O.
(2,4)
M
If cP =dt/l for a form t/I with f6'1 coefficients, then
Jdt/l =
[M] (dt/l)=O.
M
Proof Let {Uk} be a locally finite covering of M by relatively compact local coordinate patches, and let {Q(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 Y,., an open neighborhood of the origin in (CP, and
J Q(kCP= J Ft(Q(kCP)= J Q(~CP~· M
Vk
Vk
Since Q(~=Q(koF-l>O and cP~=cpoF-lEtP;(Vk)' we have which (2,4) follows. If cP = dt/l then [M](dt/l)=L
J Q(~CP~~O,
from
Vk
J Q(~dt/l~=L J d(Q(~t/I~)-L J dQ(~"t/I~.
k Vk
k Vk
k Vk
It follows from Stokes' Theorem that each summand in the first sum is zero,
since supp Q(~ is compact in Y,.. On the other hand L J(LdQ(k) "t/I=O, since LQ(k=1. k
JdQ(~" t/I~ =
Vk
0
M
The area of a manifold is a positive measure
(J
defined for fEf6';'(Q) by
(2,5) i
n
where P="2 L dzk"dzk and Pp=(p!)-l(3P. k= 1
This leads us to:
Proposition 2.7. If M is a complex submanifold of Qc (Cn, then the area of M defined by (2,5) is the sum of its projections on the coordinate spaces
.)P
pIp-I)
Proof We have Pp=(p!)-I (3P= (~ (-1)-2- ~ dz l from (2,5) we obtain
(J(f)
"
dzl = ~ PI so that
= L [M] (f PI) = L (J I (f), where (J I' given by the I
I
integration of PIon M, is the projection with multiplicity of M on (CP(I)c(C", where (cP(I) is defined by equations Zj=O for j¢I. 0
34
2. Local Metric Properties of Zero Sets and Positive Closed Currents
Remark. The positive measure [M]
1\
r(lJ') is the projection of the area of M
on the subspace lJ'. We recall that re~.(p,q)(.o) is the space of differential forms qJ of degree (p, q) with coefficients in re~(.o). We let Cg'(.o) = reO~(p.q)(.o). Let .oj be
U
p,q
an exhaustion of .0 by compacts sets, .ojq;;;.oj+lq;;;.o and .0=
00
U .oj. j=
We
1
equip re;:(p,q)(.oj) with the topology of uniform convergence of the coefficients qJI,] and all of their derivatives. With this topology, reg:(p,q)(.oj) is a Frechet space. Finally, we equip reg:(p,q)(.o) with the strict inductive limit topology, re~(p,q)(.o) = lim re~(p,q)(.oJ The dual space of reg:(p,q)(.o) is the set
-
of linear functionals t(qJ) for which lim t(qJm)=O for every sequence {qJm} which tends to zero in reg: (p, q) (.0); a sequence {qJ m} tends to zero in reg:(p,q)(.o) if and only if
i) there exists .oj such that, supp qJm c.oj for every m, ii) for every multi-index (x, lim [sup sup IDaqJm,I,](z)l] =0, where Da is a m-+oc,
zeD 1 J 9
differential operator with respect to the underlying real coordinates in For further details, we refer the reader to [E, F].
(C ••
Definition 28. The elements of the dual space of reg:(p,q)(.o) are the currents of type (p',q') with p'=n-q and q'=n-q. If in addition, we have lim t(qJm) =0 whenever {qJm} satisfies (i) and (ii'):
lim [sup sup IqJm I Az)l] =0, we say that
m-oo
zIt}
t
is continuous of order zero; t
• ,
then extends as a linear functional to the space reg.(p,q)(.o) of differential forms of order (p, q) with continuous coefficients having compact support. Currents of type (n, n) apply to functions in reg'(.o) and are thus distributions; if they are continuous of order zero, they are measures. Currents of type (0,0) apply to forms of maximum type and play the role of densitydistributions. We will call them generalized functions; the product of such a function with P., the volume element, is a distribution.
Definition 2.9. A current t will be said to be positive of degree (n - p) if i) t is zero on the subspaces re~(r.s)(.Q) if (r,s)=t=(p,p) (i.e. tE[reg:(p,p)(.o)],), ii) for every system (Xl' ... , (Xp of complex linear forms in dZ j with constant coefficients and every qJEreg'(.o) with qJ ~O, (2,6)
In particular, a form r/lEcP:_p(.o) defines a positive current via integration qJEre;{:(p,p)(.o)- Jr/I I\(p. We will let T,,~p(.o) be the space of positive Q
currents of degree n -
p
in .0. 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 I! of dimension p, t 1\ r(l!) is a positive distribution (hence a positive measure). We see that an element tETn~p(Q) can be identified with a measure in Q (depending on l!). A current tET,,~p(Q) is represented by a differential form homogeneous of type (n - p, n - pl. We can express it in a canonical form
(2,7)
L tl,Jdzll\dzJ
t=k~
I=(i1 < ... *i U k
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:/;(Z)=O, i=l, ... ,p, /;EJf'(Q)} is defined by p holomorphic functions, then dim: Yk"?,(n -p) for all z. In particular, for a holomorphic function f$O, fEJf'(Q), if Yf = {ZEQ: f(z)=O}, then the dimension of the regular points of Yf is exactly (n -1). Definition 2.38. Let Qc:CC n be a domain. Then a Weierstrass pseudo-polyk-l
nomial P(u; Z)E£(CC x Q) is a function P(u; z)=u k + L ai(z)u i, aiEJf'(Q), ai(O) =0. i=O Now, we recall a classical result. Proposition 2.39 (Weierstrass Preparation Theorem, cf. [A, BJ). Let f be holomorphic in a neighborhood Q of 0 in en and assume that z;;Pf(O, zn) is holo-
48
2. Local Metric Properties of Zero Sets and Positive Closed Currents
morphic and does not vanish at 0. Then we can write f ill one and only one way in the form f = hPp, where hand Pare holomorphic in a neighborhood of p-l 0, h(O)=!= 0, and Pp is a Weierstrass polynomial, that is: P(z) = z~ + I a/z')z~, where the aj are holomorphic functions in a neighborhood of vanish when z' =(ZI'"'' Zn_l)=O.
°
o
in CC n-
1
and
The following proposition is a first step in the proof of the existence of a precise notion of "area" for analytic varieties: first we have to prove the bounded ness of the area of ~ in a neighborhood of the singular points ZE Y~. We use the property that Y,. can be locally imbedded in a complete intersection.
Proposition 2.40. Let Q c cc n be a domain and Y c Q an anal ytic variety such that OEY and Y = {ZEQ: fj(z) =0, 1 ~j ~ t, fjEYf(Q)} dimo Y = p. Then 1) for every system of axes (not necessarily orthogonal) such that isolated point of cc n - P(z p+ l ' ... , Zn) n Y there exists a domain
°
is an
D= {ZEQ: IZil 0 such that the area U y of the manifold Y c Y in K satisfies: (2,21)
J
Uy~ C(K, Y)r 2p
for B(z, r)cK.
B(z,r)
Proof It is enough to prove (2,21) for a compact neighborhood D of a point zEYnD. For simplicity, we assume z=O. Let r o- p be an (n-p)-dimensional subspace such that r o- p n Y contains 0 as an isolated point. By Proposition 2.42, we can find a neighborhood w of Lo in Gn_p«Cn) such that for Ln-PEW,
r-Pn Y contains 0 as an isolated point. For N
n!
'( _ )" we choose a
p. n p. system A={L"I-p, ... ,rNP}cwNcG~_p«CO) such that A'={~, ... ,I!'N} forms is the subspace orthogonal to E:-p. a regular system, where Let us first consider the form T(~) and [Y] (T(I!'I»= JT(~). By Proposi-
n:
y
tion 2.40, we can find a neighborhood DI of the origin and a system of axes for which we can c~oose (C0-P(zp+l> ... , Zn) = r l- p and (CP(ZI' ... , Zp)=I!'1 such that (Y nDl)c(Y nD) where YnD={z: lj+p(Zj+p;
Zl'
""zp)=",=Pn(zn;
Zl' ... ,zp)=O}
and deglj+p=vj+p' Thus, for z'=(zl> ... ,zp) fixed, there are at most
n Yj+p points in n-1(z')n Y in
n-p
Y=
j=1
DI
;
the same is then true for any ball
52
2. Local Metric Properties of Zero Sets and Positive Closed Currents
B(z, r)cD 1 • It follows from Corollary 1.12 and the fact that subvariety of Y that
J
J
T(li1)=
It- 1 (W)n
Y is a
T(~).
YnB(z,r)
YnB(z,r)-,,-'(W)
Thus
where z' = n(z) is the projection on CC P (zl' ... , zpl. Thus
J
T(li1)~YT2pr2P=C1r2p
for B(z,r)cD1'
Y nB(z,r)
We proceed in the same way for li2, ... , liN and obtain similar bounds J T(ll.)~ Cs r 2P for every ball B(z, r)cDs' where Cs is independent of YnB(z, r)
B(z, r).
n N
Let A be a connected open neighborhood of 0 in Ds' Then for B(z,r)cA we obtain, T(ll.)~( sup Cs )r 2P . s=1
J
YnB(z,r)
1 ~s~N
By Theorem 2.16, there exists a constant C" depending only on the system A={ll.} such that for every positive :urrent t, Ut~C" Applying this to the current of integration [Y], we obtain (2,22)
uy[B(z, r)] ~ C(A, Y)r2p
(f t/\T(ll.»). =1
for B(z, r)c A.
Since K is compact, we can cover it by a finite number of domains Ai for which (2,22) is true for a constant C(A;, Y). If we let C=sup C(A;, Y), we obtain (2.21). D 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{~} is a locally finite covering of a domain Dc en by subdomains and P.iECC;(~) such that L~(x)=l on D, we write for cp with coefficients in rc;(D); t(cp) = Lt(Pjcp)= Lt(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 Dc Rm to obtain a continuation of t as a closed current to Rm. The problem is local, therefore we suppose D relatively compact. l1leorem 2.44. Let t be a current continuous of order zero defined in a bounded domain Dc 1R.m. i) in order that t extends across the boundary of D to a current t' continuous of order zero, it is necessary and sufficient that t be bounded in D, that is that the measure coefficients of t be of finite total mass in D. In this case, the simple extension i of t, which has no mass on is obtained by
CD,
(2,23)
i(cp)= lim t[lXqcp] q-oo
§5. Analytic Varieties and Currents of Integration
53
where ocq(x) is a family of functions in fC;'(Q) such that O~ocq(x)~I, ocq+ dx)~ ocq(x) and lim OCq(X) = Xn, the characteristic function of Q; m-oc,
ii)
if
and only
t is closed, the simple extensioll
i of t,
defined by (2,23) is closed
if
if
for one sequence ocq(x) with the properties stated in i). Proof. If t extends to a current t' defined in Q'::::> Q and continuous of order zero in Q', obviously the mass lit'll G of t' in G~ Q' is an upper bound of the mass of t in GnQ, and for G=Q, the mass Iltll n must be bounded. Conversely, if t is of finite total mass in Q, (2,23) by convergence defines a current i in R m of bounded mass and so i is seen to be continuous of order zero, and the definition of t does not depend on the particular sequence ocq(x) with the above properties. Furthermore, for a form cp with coefficients in fC;'(Rm) t(dcp) = lim t(ocqdcp)= lim [t(d(ocqcp»-t(doc q A cp)]. q-oo
q-oo
The first term vanishes since t is closed and supp(ocqcp) is compact in Q. Then t(dcp)= - lim t(docq A cp) for each sequence {oc q} with the given properq-oo
0
ties, and ii) is proved.
Corollary 2.45. Let Q be a domain in Rm=RPxRm- p, O~p<m, y=(x 1 , ••• ,xp), y'=(x P+ 1 ' ••• ,xm) and Q 1 =[XEQ: 11y'11 =FO]. Then if t is a closed current continuous of order zero in the open set Q 1 (it is defined on the forms cp with coefficients in fC;'(Ql»' a sufficient condition for the simple extension t of t to be a closed current in Q is that for each domain G~Q (2,24)
lim r-11ItIIG=0 r-O
where IltliG is the mass of t in G nQ 1 n [11y'11 0. But we also have VI + V2EL(Q). To see this, we note that if V EL(Q), n
then vlexp
10gV+Re
ktl
L
AkzkEPSH(Q)
for
A=(AI, ... ,An)ECCn.
every
Thus
k=1
AkZkIEPSH(Q) for every A ECC n. A simple calculation shows that
this condition is necessary and sufficient to have (d) (cf. Lemma 3.46). But if it is verified for VI and V2, it is verified for J.'t + V2. In particular, if we let n
(2,28)
ifi .(z) = I/Ik(z) + t: 1, since X cannot be supposed to be a complete intersection. The positive currents and closed positive currents were introduced by Lelong [10] 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 holomorphic function of one complex variable with a given zero set was solved by Weierstrass in the middle of the nineteenth century. He showed that if D is a domain in the complex plane, if {a.} is a sequence of points without limit point in D, and if {m.} is a sequence of positive integers, then there exists a function f(z)EJft'(D) 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 D is a domain in (Cn, then for every zero set defined locally in D, does there exist a global holomorphic function which defines the same zero set? More specifically, if Ui is an open covering of D and J;EJft'(Ui) are such that J;Jj-1 EJft'(Ui nUj ) and I j J;-I EJft'(Ui nUj ) for all pairs i,j, does there exist IEJft'(D) such that IJ;-I EJft'(Ui) and I - 1J;EJft'(Ui) for all i? The answer is in general negative, even when D is a domain of holomorphy, and depends upon the topological as well as the complex analytic properties of D (cf. [A, B]); however, when D is a simply connected domain of holomorphy (as in the case of (Cn), the answer is always affirmative. We shall be interested in studying a quantitative version of Cousin's Second Problem. The Cousin data X = (Ui , J;) defines a divisor in (Cn composed of an analytic variety Y(X) of dimension (n -1), which is just the zero set of J; in Ui' and a set of non-negative integers mk , the multiplicity of Yk(X) in the Cousin data, where Yk(X) is an irreducible branch of Y(X); mk is the order of J; on Yk(X), the regular points of Yk(X) (see below). It follows from the comp~tibility conditions JjJ;-1 holomorphic in Uir. Uj and the connectivity of Yk(X) that the mk are well defined. This permits us to define an "area with multiplicity" for the Cousin data X in the ball B(O, r) by a(r) =L mk area (Yk(X)nB(O, r)). The problem then is to find an entire function k
[ such that IJ;-1 and J;[-1 are holomorphic in Ui and 10gMf(r) has minimal asymptotic growth. For entire functions of finite order, this is equivalent to constructing a solution of minimal order of growth. The solution [ for Cousin data of finite order and the properties of 1 generalize to several complex variables the well known results of E. Borel, J. Hadamard, and E. LindelOf for one complex variable (cf. [D]).
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 =(J;, Vi) be a set of Cousin data in D, Y(X) the analytic variety given by Y(X)nVi={zEVi: J;(z)=O}, and Y(X) the (n-1) dimensional complex manifold of regular points of Y(X). Theorem 3.1. A Cousin data X =(J;, Vi) in a domain D defines in a canonical way a positive closed current ()x, which is called the current associated with i the Cousin data. The value of()x is given by ()x=-iJiJlogl.fj1 in V j . Moreover
n
if Z=(J;', U/)
is a Cousin data and
i
if Z
is equivalent to X, then ez=()x.
-
Proof Let tj=-;oiJlogl.fj1 in Vj and define ex in D by ()x=t j in Vj'Since .fjfk- 1 and J;..fj-l are holomorphic in VjnVk' logl.fjl-loglfkl is plurihar-
monic in VknVj . Thus
tj-tk=~iJt3[logl.fjl-loglfkl]=O,
so ()x is well den fined and is positive and closed, since each tj is positive and closed. By the same method, in a covering V. finer then {Vi n Vj}, we prove iJ t3[log If; I -loglJ;I]=O and 8z =()x· 0
In the classical case where n = 1, if f (z) is hoi om orphic in D and a is a zero of f, then there exists a neighborhood Va of a such that f (z) =(z-a)qg(z), where g(z) =1=0 in Va' Then n- 1 iiJt3log If(z)1 =!I iiJt3log Iz -al =2q A log Iz -al ,/l=qb(a)
n
n
where b(a) is the Dirac measure of the point a. Thus for n = 1, the current associated with a Cousin data {ak,m k} is ()x= Lmkb(a k). For n> 1, we show: k
Theorem 3.2. Let X =(J;, Vi) be a Cousin data and ()x the associated positive closed current. Then (3,1)
()x= L mk[Yk(X)], k
where the Y,.(X) are the irreducible branches of Y(X), [Y,.(X)] is the positive closed current of integration over the connected submanifold Yk(X) of regular points of Y,.(X), and mk = vx(z) is a positive integer, the multiplicity of Yk(X) in the Cousin data X. For every form qJECC;r:(1I_1,1I_1)(D), we have (3,2)
where the sum is taken over those Yk(X) which intersect the support of
qJ.
§ I. 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 v(x, [Y]) has the value 1. By using an analytic isomorphism, we can suppose x=o and Yk(X) defined by zn=O. Then
For lpE~O,(n-1,n-1)
and a, the trace measure a(r) in the ball Ilzll ~r of the characteristic function X, is just [Yk](Pn_1X)='t'2n_2r2n-2. Thus v(O, [1';.])= 1. Now we consider the Lelong number vx{z) for the current Ox associated with the Cousin data X =(J;, Ui) and prove: Proposition 3.3. The number vx(z) for the current Ox associated with the Cousin data X = (ti' Ui) is a positive integer mk which is constant for ZE Yk(X). Proof Since Yk(X) is connected, it is enough to show that vx(z) is locally constant on Yk(X). Let zoEYk(X). There exists a holomorphic map w=H(z) which is a homeomorphism of a neighborhood Uzo of Zo onto a neighborhood V of the origin such that H(zo)=O and wn(z)=O if and only if zeYk(X)n Uzo ; if fj defines ~(X) in U'o' then fl(w) = fjoH-1(w) is zero in V if and only if wn=O. By the Weierstrass Preparation Theorem (Proposition 2.39) we can find a neighborhood V' of the origin with V' c V and fl(w) = [w:+ ~I.1
aj(w')w~]gj(W)'
where gj=t=O in V'. Since fl(w)=O if and only if
)=1
wn=O, it follows that i
fj'(w)=w~gj(w).
Thus the current Ox has the value
-
Ox=q-oologlwnl and has q for Lelong number on the set Ul(w)=O} in a 1t
neighborhood of the origin. Since it is an invariant (with respect to the complex analytic isomorphisms), we have vx(z) =q in a neighborhood of Zo on ~(X). Proof of Theorem 3.2. For
lpe~~n-1,n-1)
with support in Uzo ' we obtain
Ox(lp)=~1t Joaloglfjl A lp=~1t Jaaloglfll A(lp0H- 1) =q
J Wn=O
lpoH- 1=q
J
lp,
Yk(Xj
and q is a positive integer which is associated by Proposition 3.3 with the Y,.. Then (3,2) is proved for lp with support in U'o' To end the proof, we proceed by a partition of unity subordinate to the open covering Uz of Y(X) - UY"(X),, where y"(X)' is the set of singular points of Y(X) on y"(X). k
62
3. The Relationship Between the Growth of an Entire Function
It follows from Theorems 2.46 and 2.47 that if Y'=U Yk(XY, 0x=Oxlu_y, == mk[Yk(X)J by Proposition 3.3. k 0
L "
Defmition 3.4. For X =(/;, Ui ) a Cousin data, the positive integers m k 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 Ux = Ox A Pn- l' trace of the current Ox, will be called the area of Y(X) with multiplicities. This definition is justified by the fact that f Pn-l is just the (2n - 2) dimensional area of the complex y.(x) manifold Yk(X). Remark 1. The majoration of II Ox I by CnU X 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(CC n) of the cone of the complex lines through the origin which intersect X nB(O, r); vx(O) is the degree of the cone of the directions of the complex tangents to X at O. Thus, vx(O) can be interpreted as a mean value relative to the Haar measure dW 2n on the unit sphere. More precisely, we state:
Proposition 3.6. Suppose that X is defined in a neighborhood of zero by f(z)=O. Then vx(r)=uX(r)[t2n_2r2n-2]-1 has the properties: 0 2(0, r, log Ifl), i) vx(r) =-01 ogr ii) vx(r) = W 2n1
f 11.11 =
n(lX, r)dw 2n (IX), 1
where n(lX, r) is the number of zeros of f(ulX) in the complex line Z=IXU of modulus at most r (with multiplicities). Proof Since Theorem,
u x(r) = i/lt0810g IflA Pn-l = (2lt)-1 A (log Ifl)Pn'
u x (r)=(2lt)-1
f
Ilzll ~r
by
Gauss'
r 2n - 1 0 AloglfIPn=-2--0 2(0,r,loglfl)w 2n 1t
r
= [t 2n- 2r2n - 2] -;;----1° i.(O, r, log Ifl). e ogr On the other hand, on the complex line Z=IXU, for lul=r,
c 2n . n(lX, r) = - - (2lt)-1 flog If(lXre,9)ldfJ, iJ log r 0
§2. Indicators of Growth of Cousin Data in
and hence averaging over all
cr"
63
IX such that IIIXII = 1, we have
win1 Jn(lX, r)dw 2n (IX)=-a la ),,(0, r, log Ifl).
o
ogr
§ 2. Indicators of Growth of Cousin Data in <en The Euclidean area O'x(r) and projective area vx(r) give indicators of growth for the Cousin data X. They are related by the formula (2,15) vx(r) = (r 2n- 2 r 2n -
2)-1 0' x(r).
It is rather the projective indicator that we shall use. If X is defined globally by a polynomial P(z) of degree m, it follows from
Proposition 3.6 that vx(t)=-a la },(O, t, log IPI) and hence lim vx(t)=m ogt
r-<Xl
= degree P. As in the study of entire functions, we shall be interested in the scale of finite order.
Definition 3.7. The indicator vx(r) will be said to be of finite order p if lim sup 10~ vx(r) r-oo ogr
p
lent
°
and n ~ 1, the following conditions are equiva-
<Xl
i)
J t-Sdvx(t) < +00,
a 00
ii)
J t-
s - 1 vx(t)dt
< + 00,
a 00
iii)
J t-s+ 2 - 2n dO'x(t)< + 00,
a ex:
iv)
J t-s+ 1 - 2n dO'x(t)< + 00,
a
and anyone of these conditions implies that lim vx(r)r S=0. Proof. Integrating by parts, we obtain
(3,3)
r
r
a
a
J t-Sdvx(t)=[t-'vx(t)]~+s J vx (t)t-
S -
1 dt.
64
3. The Relationship Between the Growth of an Entire Function
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- 5 vx(t) = C. Then C =0 follows r~
00
2r
from ii). Conversely, ii) implies that lim S t- 5 - 1 vx(t)dt =0 or, since vx(t) is r - 00
increasing, lim vx(r) r-oc,
Ytr
5 -
1 dt
=
lim C5 r-oo
r
vx~r) =0,
so ii) implies i) by (3,3).
r
The equivalence of ii) and iv) as well as i) and iii) follows from (2,15).
D
Defmition 3.10. The number, = inf {s} such that i) holds will be called the convergence exponent of the Cousin data X. 00
Definition3.11. The smallest integer q for which St-q-1dvx(t} and let () be the angle between the vectors (0, a) and (0, x) in Rm. Then
°
Iia - xl12 = IIal1 2[1 - 2t cos () + t 2] = Ila11 2(1 - te i8 )(1 _te- i8 ),
and hence (3,5) is majorized term by term by the series 00
(3,6)
(l-t)-P=
L
with b p,s=(s!)-I p(p+l) ... (p+s-l).
bpjS
s=o
The case p =0 corresponds to the classical case studied by Weierstrass for CC = R 2 of the potential related to the kernel -log II a - x II. The series is 00
then dominated by -log(l-u)=
L
uS/s, and the estimates obtained in the s= 1 classical case are valid for the kernel eo (a, x, q) for all R m: m~2,
Proposition 3.13. For p=O and every following estimates for all
i)
R (where a*,O and u = ::: :: > 0) : m
if q~l, leo(a,x,q)l~uq+1
u < e -I .
the kernel eo(a, x, q) satisfies the
for
u~-q-. For q=O, leo(a,x,q)l~eu for
.
q+ 1
ii) if q~l, eo(a,x,q)~euq(2+logq) eo(a, x, 0) ~ log (1 + u) for all u > O.
Proof The first part of i) stems for the fact that cc US uq + 1 1
L
-~--'
s=q+IS
__ ~uq+1
l-uq+l
q u~+t'
for
and
if
q=O
q
1'f
q URc l for r chosen as in Proposition 3.14. Then for (3,12), we obtain
k"2.1_ 2
J
le 2._ 2(a, z, q)ldCTe(a)
II all >R' 1 R'
IlaI12-2.-q-ldCTe(a),
§4. The Canonical Representation of Entire Functions of Finite Order
and the right hand side converges by Proposition 3.9, since ve(t) is of q. This establishes the uniform convergence. On the other e 2._ 2 (a,z,q) differs from h 2 ._ 2 (a,z) by a finite sum of harmonic nomials, from which it follows that Ae 2._ 2(a, z, q) = k 2._ 22nt5(a), and
69
genus hand, poly(3,13)
D
holds.
Theorem3.17. The canonical potential Iq(z) defined by (3,12) with respect to a positive closed (1,1) current 0 of genus q and such that B(0,ro)nsuppO=0 satisfies the inequality
Iq(Z)~A(n,q)rqLS t- q- 1ve(t)dt+r
(3,14)
o
for
Ilzll =r.
j t- q- 2ve(t)dt] 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
sup Iq(z)=M(r)~k2._2C(2n-2,q)r
q+1
Ilzll=r
OOI
ro
due(t) ( ) q+2. 2· t+r t
We integrate by parts in order to express the right hand side in terms of ue(t) and hence ve(t): ooI
ro
due(t)
(t+r)~+2.-2
[ =
ue(t) (t+r)t q+2.- 2
]00 ro
(at+br)
ooI
( )d
+ ro (t+r)2 t q+2.-1 Ue t
t
with a=q+2n-l, b=q+2n-2. The first term is zero, since ue(ro)=O and lim t- q- 2.-1 Ue(t) = lim t- q- 1ve(t) =0, and ve(t) is of genus q. Thus ooI
ro
due(t) (t+r)t q+2.- 2
~
(2
1) n+q-
ooI"
ro
ue(t)dt (t+r)t q+2• 00
=(2n+q-1)'2._2
1
ve(t)dt
I ( ) q+1· ro t + r t
It then follows that -1
M(r)~k2._2'2._2(2n+q-1)C(2n-2,q)r4 ~A(n, q)rq
+1
OOI
ro
+1
OOI
ro
(
ve(t)dt ) q+1
t +r t
ve(t)dt ( ) q+1' t +r t
and OCI
vg(t)dt
-"---...,.< r
ro (t+r)~+l-
-1 Ir
ro
ve(t)dt "'I" ve(t)dt -+ ~+l r ~+2·
D
Remark. A better estimate can be obtained by distinguishing between the intervals ro p+ 1, ex;
I~(r)=.r
tP(O-Adt=(),_p_1)-l rP(r)+I-.l.+ o (rP(r)+I-A).
Proof After an integration by parts, we obtain r
J tp-.l.tP(O-Pdt=(p+ 1_),)-1
[tP(')+I-.l.]~
R
-(p + 1_),)-1
r
J[tP(t)-.l.(p(t) -
p)
R
+tP(,)+I-.l.p'(t)logt]dt=II +1 2 , It follows from Definition 1.15 that given
£
> 0, there exists T. such that
r
II21~£
J tP(t)-.l.dt
for
r>T. (we
recall
that
p-).> -I
implies
that
R r
lim
J tp(t)-.l.dt= + 00). Hence
r-+ooR
11 tP(t)-.l.dt -(p + 1 _),)-1 rP(r)+ 1-.1.1 ~£(1 +£)-1 [rP(r)+ 1-.1. + C] where C =(p + 1 -A)-I RP(R)+ 1-.1.. For ), > p + I, we obtain OCJ
J tp-.l.tP(t)-Pdt=(), -
p _1)-1 rP(r)+I-A oc
+().-p-l)
J tP(O-A[(p(t)_p)
+p'(t)-t logt]dt=i l +i2 • It follows from Definition 1.15 that given
£
> 0, there exists
T;
such that
C1j
li21 ~£
J tP(t)-.l.dt and thus
I!
tP(t)-.l.dt -(I, - p _1)-1 rp(r)+I-.l.1 ~£(I _£)-1 rP(r)+I-.l..
0
§S. Solution of the
Proof of Theorem3.J9. From (3,16), we have
vo(t)~M(et)
iJa Equation
73
and hence
vo(t)t-p(1) ~ M(et)t-P(/) ~ [M(et)(et)-p(el)] [et]p(el)-p(/)·eP(I). It follows from Definition 1.15 that lim eP(/)=e P and from Theorem 1.18 that
lim [et]p(e/)-p(/) = 1. Thus
1-+00
limsupvo(t)t-P(/)~ePlimsupM(t)t-P(I).
t-tooo
t-oo
In the
t-oo
other direction, with qro, we obtain via (3,17)
{1
M(r) ~ A(n, q)rq
vo(t)-q-l dt + (C +e) [lq+ 1 (r)+ l~+ 2 (r)r]}
~A(l!,q)(C+e) [_1_+ p-q
1
q+1-p
]rP(rl+o(rPlrl)
o
by Lemma 3.20.
Remark. Starting with bounds for the growth of M(r)= sup Iq(z), it is easy Ilzll =r
to obtain a control of the mean values A(O, r, Iq) on
A (0, r, Iq)=('r2nr2n)-1
J
Ilzll =r and of
II q(z)ldT 2n ·
II zII ;ar
We set I:=sup(Iq,O), I;=sup(-Iq,O). From the subharmonicity of Iq we obtain O~A(O,r,Iq)=},(O,r,I:)-A(O,r,I;) from which it follows that
O=A(O, r, Iq) ~A(O, r, I:) ~ M(r) and hence A(0,r,IIql)~2M(r)
(3,18)
§ 5. Solution of the
and
A(0,r,IIql)~2M(r).
aa Equation
We have already seen that the canonical potential Iq(z) associated with a positive closed (1,1) current () of genus q and such that O¢supp () solves the equation 21n Alq=(Jo. In this paragraph, we shall show that it solves in fact the more restrictive condition of equation (3,10). Let i
i
()=-; L ()p,qdz p /\ dzq, p,q
-
i
()' = - oOI q -()=-
n
where the ()p,q are complex measures. Then
L ()~,qdzp /\ dZq has the following properties:
n p,q
L
i) its trace ()~p is the zero measure; ii) ()' is 0 and aclosed.
74
3. The Relationship Between the Growth of an Entire Function
Proposition 3.21. If a current 0' 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 coefficients O~,q are twice continuously differentiable. Then, since dO' = 00' + 80' = 0, we obtain for m=Fp, q. Thus
4L10' p,q
2 _"0' =" oz020~,q =_0_ OZ OZ OZ ~ m
m
m
p
q
~
=0.
m,m
To treat the general case, we take aEli&'oXJ(B(O, 1)) such that S a(z)d'2n= 1 and
a~O
and set ae(z)=a
(~)C2n,
where a depends only on Ilzll. Then
0~,q*ae=HO~,qd'2n(u)][ae(z-u)] is a li&'OO function, and the current (O~,q *ae)dzp1\ dZq satisfies the hypotheses of the Proposition. Hence the
I
p,q coefficients O~,q*ae are harmonic functions. Using the mean value property for harmonic functions we obtain [O~,q*ae']*ae=O~,q*ae' so [O~,q*ae' -O~,q]*ae=O for every s, s'. Hence when s--+O, we obtain O~,q=O~,q*ae" which shows that as a current O~,q is equivalent to a form with harmonic coefficients. 0
Lemma 3.22. Let h(x) be a harmonic function for Ilxll,a;>,o is bounded in modulus independently of t (resp. goes to zero uniformly when-t goes to infinity) for IIXII~I; iii) the coefficients a". t of P, are bounded in absolute value (resp. go to zero) when t goes to irifinity. Proof i)=ii). w contains a cube Llr={x: IXk-xio'lO. Let IPt(x)=P'(x-x(O,). Then IPt(xk)~M, for IXklR,p(z)=k 2L2
(3,31)
lIall
+k2L2
J
e2n _ 2(a,z,q-l)da x (a)
;:i;R
e 2n _ 2(a, z, q)dax(a) = II +1 2,
lIall >R
Then DR(z) is a subharmonic function in JR. 2n. We set
(3,32)
Mn(R)= sup DR(z). IIzll
;:i;R
We use (3,9) to estimate successively the two integrals I I and 12 ; I 0 and prove first the sufficiency for CPm' A simple calculation shows that 0 2 (P,."Jz) GZ/Jzk
13 2 CP.,m(Z) GZjaZk
Since CPaEPSH(Q) for all IX, CPa,mEPSH(Q) for all IX, which shows that the 13 2 log cP (z) . form L ~ m Wj"\~O for all WECC n• Hence 10gCPmEPSH(Q). Smce j,k
azjoz k
log CPm decreases to log cP and CP:$ 0, it follows from Proposition 1.3 that 10gcpEPSH(Q). To treat the general case, we choose IX(Z)Erc~(B(O, 1)) such that
SlX(z)dr(z) = 1 and IX depends only on Z and set IX,(Z) = IX (~ ) e- 2n. Then cP,=CP*IX,EPSH(Q,), where Q,={z:dQ(z»e}, cP,ErcOC(Q.) and CPt decreases to cpo Then exp Re
(JI IXjZj) cp,(z) = Scp,(z + z')lXt(z') exp [ - Re (t IXjZ') ] dr(z')
is in PSH(Q,) by Proposition 1.14, so 10gCPtEPSH(Q.) by the above, and since cP, decreases to cP and cP $0, log cpE PSH (Q) by Proposition 1.3. To see the necessity, we note that if 10gcpEPSH(Q), then for every IX, 10gcp(z)+Re
(tl IXjZj) EPSH(Q).
Since cp(t)=e' is an increasing convex
function, the result now follows from Proposition 1.24.
0
§ 10. The Case of a Cousin Data of Infinite Order
91
Lemma 3.47. Suppose that V(z)EPSH(CP) and cp(z) is a real valued function such that JIV (zW exp - 2cp(z)dt(z) < C. Then there exists a constant ten. e) such that V(z)~C(n,e)·Cexp[ sup cp(z')]' Ilz'-zll ~, Proof By the Inequality of the Mean for subharmonic functions, we have V(z)~ C(n, e) J V(z+z')dt(z'). We now obtain by the Schwarz Inequality Ilz'lI (z) is a plurisubharmonic function and C>O. Then for every 0(>0, there exists a plurisubharmonic function V solution of (3,10) in cen such that (3,43)
J[V+(zW C(l + IlzI1 2 )-n-3-a exp -2t/t(z)dt(z) < C(n) C
and (3,44) 1
where V+ =sup (V, 0), t/t(z) = log Jt exp I"/>(t z)dt and o
x(z) =t log
J
exp 2t/t(z')dt(z').
Ilz'-zll ~,
Proof Let ()=iL()jkdzjAdzk. Then ()jk=()jk*CX,. Theorem2.16 gives an estimate of the coefficients of ()' in terms of the trace (10 of (): (3,45)
I()jk(z)1 ~ I() jkl *CX, ~t Jcx,(z -a)d u(a) ~t M,u,(z)~ C' C(e) expl"/>(z)
where M£=supcx,=c 2n supcx(z). Suppose now that V is any solution of (3,10) (cf. Corollary 2.30) and let w.= V*CX .. Then it follows from the Mean Value Property that V~ W. and what is more i/noaw.=i/noa(V*cx,)=()*cx,=()'. Thus every solution of the equation (3,10) is majorized by a solution W. of the equation i/ncBW'=()', whose coefficients satisfy (3,45).
92
3. The Relationship Between the Growth of an Entire Function
First we solve the equation idw=8'_ As in Theorem 2.28 the solution is given explicitely by the formula (3,46)
V=
L [S t8jk(tz)dt] zjdzk
j,k
0
-L j,k
I
d
t8jk(tz)dt] Zk dz j=v 2 -VI'
0
n
S
Let Ajk= t8jk(tz)dt, which are o
rtY.. functions, The forms C/>k=
j= I
n
o-closed and hence the forms
L
L 8jkdzj are
Ajkdz j are also o-closed, so
VI
is o-closed,
L=I
and in a similar manner,
V2
is o-closed, Then
and
, " (OA'k OA.k ) I 8\(tz) Smce L ~zs+ :0': Zs = S t 2 o-1-:o-' we see that s
uZ s
uZ s
dv =ov 2 -8vI = [2
ut
0
S 8jk(tz)dt + S t 2 O~jk dt] dZ j /\ dZk = -ie',
o
0
vt
which shows that we indeed have idv=8'. From (3,46) we obtain the estimates
1vj(zW ~ C- C(a) IIzI12 exp [2 ",(z)]
j = 1, 2
and hence for every (X> 0 we have (3,47) S Iv/z)12(1 + IlzI12)-"-I-~ exp -2",(z)dr(z)< C- C(a, (X),
j= 1, 2,
CCn
The function C/>(z) is plurisubharmonic. To see this, it is enough to show I
by Lemma 3.44 that the positive function h(z)= St exp C/>(tz)dt generates a o
plurisubharmonic function h~(z)=h(z) exp I(z) is plurisubharmonic, so is h(z) for every z, and I
h(x) exp I(t z) + Re o
is plurisubharmonic.
R 1 • Let ,,=llx'-x"lI. Then for r>Rl' by Theorem 1.18, u(ry)r-p(r)~Cl
)< bm Ir("~) (b+,,)m_b m(2 )PC I r( , i:)sup(R 1 , R 2),
-I~(y, b)~ - (!~r C~+b-m
[e:r
-bm] (2y)PC 1
and so we obtain
(4,1)
I~(x', b) - I~(x", b) ~ [(b !m,,)m -1 ] I~(x", b) (b + ,,)m - bm(2 )p C +
(b + ,,)m
y
no instead of
0
r>sup(R 1 , R2)'
Theorem 4.3. Suppose that u, vESHP(r)(r). If u is of regular growth for the ray rx o, then h!+v(X O) =h!(X o)+ h!(X o)' Proof Clearly h:+v(xo)~h!(xo)+h~(xo), so we prove the converse. If x~EEv' we are through, so suppose xo¢Ev' Let e>O be given. Since h~(xo) =limsuphv(x/), for every ,,>0, there exists x' with Ilx ' -x o llR(e,fJ) 3e (4,11) -r,;;I IX -m J II~(x', fJ) -h:(x')ld-r(x') 0 and b > 0 and a sequence ", increasing to infinity such that I~n(xo,b)~h~(xo)-e. By choosing a subsequence, if necessary, we can assume that rn+ 1 ~ 2rn. By Lemma 4.2,
there exists Nl and rf>O such that l~n(x',c5)~h:(xo)-~ for n~Nl and < rf l' and since l~n(x', c5) ~ u (rn x') . rn- p(rnJ, 2
!lx' - Xo I
(4,12) Let '" be a Cfjoo function of the variable tE1R with support in the interval ( -1, 1), O~ '" ~ 1 and", 1 on the interval ( -t, +t) and let
=
where rf2~rfl will be fixed later. By Proposition 1.22, for I x I sufficiently large, A I x I p( II x II ) =
P(lIx l ) (0 log Ilxll )2 8 1IxII----,,-:--;;I --:-:,---' 2
m
i~
1
o(log Ilxll)2
m
+ i~l
OX i
0 IlxIIP(llx l ) 02 log Ilxll o(log Ilxll) ox?
2
~P2 IlxII P(llx l )-2 and so for Ao sufficiently large, and ~ sufficiently small v(x)ESHP(r)(T) by Theorem 1.18. Furthermore, h~(xo)=(llxollp+O. We now choose rf2 suffi-
.
.
(l+rf) (1-rf2)
clently small, so that settmg y= _ _2_ (4,13)
2yp(h~(xo)-~) ~h:(xo)-~.
Suppose that Ilx'-xollR" and for 1/2~lIz-rmzoll~l, there exists a constant C such that q>(z)+log(1 + IlzlI» c. Proof Let z be such that
ei- 1 ~ Ilzll ~ei.
If m~/, then
log IIz- rmzo ll C-, "eL... = 00
> C- I =
m=1
I'
Furthermore, for rm log
Ilz-r z II m 0 ::::logt-logrm~logt-log(1 + Ilzl!). rm
o
Proof of Theorem 4.9. As in the proof of Theorem4.8, we find ZoEr, 8>0, '11 >0 and an increasing sequence rm tending to infinity such that h;(zo)=t, - 00 and 10glf(rmz')I2, then by the Mean Value Theorem, IIA(z')-A(rmzo)II~Collz'IIP' for p>p'>inf(0,p-1) by Proposition 1.19. Let V2(z) = VI (z)+ 2ncp(z) + 3n log (1 + Ilzll)+ Co IlzIIP'.
§l. General Properties of Functions of Regular Growth
105
Then for p=aa., suppPcU{z:t~llz-rmzoll~l}, so by Lemma 4.10, we m have 00
J IPI2exp-2V2(Z)d-r(z)~C L
r
00
exp-Iog(1+rm_l)~C
m= 1
L
el-2m~C'.
m= 1
Now, we apply the resolution of the a-equation (see Appendix III). There exists I' such that ay = p and
J lyI2 exp -2V (z)d-r(z)ECC;;'(r), then lim t~
Sq>dJlt
ro
= Sq>L1h~. Let Q be a bounded open set such that Qcr. If {q>n} is a sequence in CC;;'(r) such that q>n ixQ' the characteristic function of Q, then lim inf Jlt(Q) ~ lim Sq>n dJl t = Sq>n L1 h~ for all n, and hence lim inf Jl,(Q) t-oo
~L1h~(Q).
t-oo
(-00
In exactly the same way, if I/InECC;'(r) is such that I/Inhn, lim sup Jlt(Q)~L1h~(Q).
We have Jlt(Q) = t:'(!?;(t) (where tQ={X:
~EQ}).
Hence, if L1h~(aQ)=o, then I.
(4,14)
1m
Jl(t Q) -tm--"""z:-:-+-p(=t)
t-ro
By homogeneity, we have L1
h~(f' nB(O, 1))= !~~ L1 h~ (r' nB(O, l)n CB (0, ~) ),
and from (4,14) Jlr(t) > Jlt(r' nB(O, t)nCB(O, t/r)) >L1h*(r'nB(O 1))tm-Z+p(t) =". ,e
tm- Z+p(t) -
for r sufficiently small and is arbitrary,
t
sufficiently large (depending on r and e). Since e
liminftm~r;~~(t)~L1h~(f'nB(O, 1)). t~oo
§ 2. Distribution of the Zeros of Functions of Regular Growth
107
Now, given e > 0, there exists O"(e) such that J.l(r' nB(O, O"r)nC B(O,
1»~er2-m+p(r)
for r large enough. To see this, we note that if Q=r'nB(0,2)nCB(0, I), . J.l(tQ) ~hen by (4,14), there eXists to such that tm-2+p(t)~Ah~(Q)+1 for t~to, and If2qto~rO, there exists s(e) such that for 3(2n-2)
s~s(e),
Iiall>
IlzlI,
Proof We have 1P,,(a, z)1 ~
IlzliS . IlaI1 2n - 2+ s bn,s WIth
1 bn s=-(2n-2)(2n-l) ... (2n+s-3)
,
s!
by Proposition 3.14, and so le2n_2(a,z,q+s)l~
oc;
L
IIzllk
1
2n-2+k k' (2n-2) ... (2n+k-3)
k=s+q+lliall . Ilzllq+l (l)k 1 < 2 1 L - -k--(2n-2) ... (2n+k+q-3) -ilall n- +q k=s t (+q)! 00
for lIall >rllzll·
§2. Distribution of the Zeros of Functions of Regular Growth
109
If r = 3(2n - 3) and s > (2n - 2) is sufficiently large, then
Ilzllq+1 (2)k EIIzllq+1 z, q+s)l:S IlaI1 2n -1+ qk~S "3 ~ IlaI1 2n -1+ q' 00
le 2n _ 2(a,
o
Lemma 4.14. Suppose X={z:f(z)=O} OrJ:X and O"x(r) 0, we obtain I~(w, b) ~ y w - e/2 for r sufficiently large. We now turn our attention to Q(z)=kinl_2 J e 2n _ 2(a,z,q)dCT x (a). It r!o
follows from the estimates of Theorem 3.19 that there exists a constant C
§2. Distribution of the Zeros of Functions of Regular Growth
113
such that Q(z)= CA(cP o) IlzII P (li zll) and since Q(O)=O, O~
J
B(O, r(l + b»
Q(z)dr(z)=
J
B(rw,rb)
Q(z)dr(z)+
J
B(O, (1 + b) r) - B(rw, rb)
Q(z)dr(z)
or
J
B(O,r(l +b)-B(rw,rb)
and O~
J
B(O,(l+b)r)
Q(z)dr(z)~
Q-(z)dr(z)~
J
B(rw,rb)
J
B(O,(l+b)r)
Q(z)dr(z)
Q+(z)dr(z).
Using the increasing nature of rP(r) we obtain by Theorem 1.18, for r large
J
B(, r( 1 + b))- B(rw, rb)
Q(z) dr(z) ~ - CA( cP o)((1 + W r)p(1 + b)r ~
Thus IQ(w, (j) ~
- 2 CA(cP o)(1 + (5)P rP(r).
CA(cPo)(1+(5)P . suff"IClentI y smal I (d ependmg . (jn ,and'f I cPo IS
on (j) IQ(w,(j)~-e/2 for r large enough. But 10glfl=IReS(z)+I(z)+Cb, where S(z) is a polynomial of order q 1, there exists C a such that vx(r)~ Ca
sup (loglfl)+ Cf'
Ilzll
~/Zr
Can one estimate the size of F- 1 (a) by the growth of IIFII for a holomorphic map? ii) if Y is an analytic variety of co-dimension 1 in ([n, then we can define Y as f -1 (0) for an entire function whose asymptotic growth is related to v[Yl(r), and it follows from Jensen's Theorem for one complex variable that for any complex line L such that L¢X, the asymptotic growth of LnX nB(O, r) cannot be greater than the asymptotic growth of If I, hence not much greater than the asymptotic growth of v[Yl(r). If Y is of codimension superior to 1 and L is a linear subspace equal to the codimension of Y, can one estimate the intersection L n X n B(O, r) in terms of v[Y](r)? iii) if X is a Cousin data and f is an entire function such that X = (f, ccn) and log If I grows like vx(r), then for any complex line L, r
1
1
J-card(LnXnB(0,t»dt=-2 J
ot
log If(rei6 1X)ldO-log If(O)1
11:11"11=1
(where the points in the intersection are counted with multiplicity). Since the right hand side is a plurisubharmonic function whose average over
§ I. Representation of an Analytic Variety Yin
«::n as
F- 1(O)
117
s2n-l is equal to vx(r), we see that the set of L such that L"X"B(O,r) "grows much more slowly" than vx(r) is locally pluripolar in IP(CCn) (cf. Corollary 1.43). If Y is of arbitrary co-dimension p and If is a linear subspace of dimension p, for how small a set of I! can I! " y" B(O, r) grow "more slowly" than v[y)(r)? We shall see that the answer to the first two questions is negative; however, we shall show that the set of values for which such estimates are not possible is quite small. What is more, we shall show that the set of If such that If" Y grows more slowly than v[y)(r) is also quite small.
§ 1. Representation of an Analytic Variety Yin
(Cn
as P-l(O)
As in Chapter 3, for Y an arbitrary analytic variety in ccn, we are interested in expressing Y as F-1(0), where F is an entire mapping whose growth is related to the growth of vy(r). There are, however, two fundamental differences: i) if X is a Cousin data, X = (Uk.fk), then X determines an analytic variety Y(X) in ccn and in addition, for every irreducible branch ~ of Y(X), a non-negative integer mk , the multiplicity of X on Yk , and the solution f has a zero of order m k on ~; ii) the Cousin data gives an expression for the current of integration, Ox= Lmk[Xk] in ccn. In the general case, we define Y as a set in each Uk of an open covering of ccn; Y"Uk={z:fk/z)=O,j=l, ... ,jk}' Then we require only that Y= F- ' (0) as a set. The construction of the current of integration over [Y] as a positive closed current is carried out in Chapter 2. The basic plan is as follows: a) using the positive closed current [Y], we construct local potentials with density on Y; b) using a partition of unity, we construct a global potential with densityon Y; c) by adding a function whose Levi form is strictly positive and whose growth is related to v[y)(r), we construct a plurisubharmonic function V whose growth is related to v[y)(r) and such that, if t' =~ aav, then V,.(z) 1t
= v[Y](z) ~ 1, where
vt,(z) and v[y)(z) are respectively the Lelong numbers at x
of t' and [Y] the current of integration on the analytic set [Y] (cf. Theorem 2.23); d) using the existence theorems for the a-operator with growth conditions (cf. Appendix III), we construct an entire mapping F such that Y= F - I (0) with estimates for the growth.
118
5. Holomorphic Mappings from
cr" to cr
m
The construction, with modifications, is also valid in any pseudo-convex domain and for any positive closed current t, but we shall not treat it in its full generality. These technics are due to H. Skoda.
§ 2. Local Potentials and the Defect of Plurisubharmonicity Let cp=W"i.;, where w 2p is the area of the unit sphere in 1R2P. We define the kernels gp(a,z)=-cplla-zll-2P for l~p~n-l and go(a,z)=loglla-zll for p = 0; these functions are JR 2n subharmonic but plurisubharmonic only for p=O. Furthermore, the Laplacian
Llzgp(a, z)=2p(2n -2 -2p)c p Iia _zll-2 p-2 ~O is (for a fixed) locally integrable if p ~ n - 2. For p = n -1, we have Ll z =2n 0. We can choose a partition of unity Pj and the associated "Ij such that for U given by (5,10), L(U, Je)~ - C(e, n, p)(l +r)-2 v,((1 +e)(1 +r» IIJeI1 2. In order to obtain a plurisubharmonic function V = U + W, it is enough to construct W such that
L(W, Je)~ C(e, n, p)(1 +r)-2 v,[(1 +e)(1 +r)] IIJeI1 2. We shall obtain Was a continuous function of r. Let q(r)=log(1 +r2)+! log2(1 +r2). Then L(q, Je)~(1 +r2)-IIIJeI1 2. We choose W=hoq, where h(r) is an increasing convex function of r such that (5,12)
h' oq(llzll) ~ C(e, n, p)v,((1 + e) (1 + r».
Then we shall have L(W,Je)~(I+r2tlh'oq(r)IIA.112. Let q-l(r) be the inverse function of q(r). Condition (5,12) will hold if we have h'(r)~
C(e, n, p)v,((1 + e)(1 + q-l (r))).
Since v,(r) is increasing, we can take h(r) to be r
ho(r) = C(e, n, p)
J v,((1 + e)(1 +q-l (~)))d~.
o
Then Wo(z) ~ C(e, n, p)q(r)v,((1 + e)(1 + r» and
(5,13)
Wo(z) ~ C(e, n, p) log2 rv, [(1 + e)(1 + r)].
r
If d>O, if in place of q(r), we take the function
G(r)=(1 +r)d
1
V'((;d~~W d~,
then L(G, i.)~d/4v,((1 +e)(1 +r»(1 +r)-2, which permits us to choose 1 +r
(5,14)
Wo(z) = C(e,d)(1 +r)d
J 1
~-(d+l)v,((l +eWd~;
§3. Global Potentials
125
this gives a better estimate when Vt is of finite order. When vt(r) is of infinite order, we obtain a better growth estimate by using the following partition of unity: we let X(t)ECC;, O~X(t)~ 1 for t~e, X(t)=O for t~2e and set xiz) =x(llzll-je+e) for j a positive integer and PI =XI' Pj=Xj-Xj_I' j~2. Then SUPPPj is contained in the annulus U-1)e~llzll~U+1)e. We then set 'liz) =x(IIzll-je-e)=Xj+2(z). If we define U(z) as in (5,10), we obtain the estimate L( U, A) ~ - C(e, p, n) 1IJ.11 2(jt(r+ 4e). We choose Wo(z)=h(llzI12), where h is an increasing convex function, so that r2
L(w, A)~h'(llzI12) IIAI12, and thus Wo(z) = C(e, p, n) to the estimate
J(jJ-vt +4e)dt, which leads 0
(5,15) Now we can replace W(z) by Wo(Z)ECC OO with a similar bound.
Theorem 5.7. Let Y be an analytic variety of pure dimension p in ([n and t the current of integration on Y or in general a positive closed current of degree p (i.e. type (n-p,n-p». Let (jt=t/\!3p be the trace of t and vt(r) =(r 2pr2p)-1 (jt(r). Then there exists a plurisubharmonic function V in ([n such that i) for every compact set K c ([n and ill an open bounded neighborhood of K, V +c p J Ilz-all- 2P d(jt(a) is CC oo on K; w
ii) if M y(r) = sup V (z), we have one of the following liz II
~r
My(r) ~ C(e, n, p) log2 rv t ((l + e)r) (5,16)
1
My(r)~C(e,d)(l+r)d
for r> ro,
I+r
J
Vt((l+e)~)~-d-Id~,
I
M y(r) ~ C(e, n, p)r2 (jt(r + e).
iii) Let v~(z) be the Lelong number of t'=i/naov. Then v~(z)=Vt(z).
Proof Parts i) and ii) follow from the above construction and (5,13), (5,14) and (5,15), since V = U + Wand U ~O. Part iii) follows from Proposition 5.1 and Corollary 5.2, since W is CCOC and hence its density is identically zero.
o Theorem 5.S. Let Y be an analytic variety of pure dimension p in ([n. Then there exists a plurisubharmonic function V which can be chosen to verify any one of the three conditions (5,16) and such that Vy(z)=v[y)(z) and Vy(z)=O if z¢ Y, where v is the (2n - 2) dimensional density of (2n) -1 Ll V.
126
5. Holomorphic Mappings from
cr" to cr
rn
§ 4. Construction of a System F of Entire Functions such that Y=F- 1 (O) Given a plurisubharmonic function V in CC", we consider the analytic set E(c, V)= {ZECC": vv(z)~c} for c>O. We are interested in constructing a representation E(c, V) = F- 1 (0) for an entire mapping, where we obtain an estimate of the growth of IIFII in terms of Mv(r). We have already obtained for an analytic set Y a function V such that Y=E(I, V) and an estimate of M vCr) in terms of v[y](r), the indicator of Y. The solution will then give a solution Y=F-1(0) with estimates for the growth of IIFII in terms of v[Yj(r). In fact, this problem is more general. We shall see that every analytic set Y in CC" can be represented as E(l, V) for V a plurisubharmonic function, "-I
for if is true for a pure dimensional analytic variety, then Y =
U Y"
where
.=0
Y, is of pure dimension sand Y,=E(I, v,,), vvJz)=O for z¢y'. Then if "-I
V=
L v" , Y=U Y,=E(I, V) .
• =0
•
Definition 5.9. We say that co> 0 is a number of complete left stability for VEPSH(CC") if E(c, V)=E(c o' V) for O 0 and all rx> 0, there exists C(n, e, rx) independent
§4. Construction of a System F of Entire Functions such that Y=F- 1 (O)
129
of rand (n+ 1) entire functions F=(fl' ... , f"+I) such that E(I, V)=F-1(0) and (5,17)
sup log IIFII Ilzll
~
~nMy(r+ (X) +(n +e)
log (1 +r)+ C(n, e, (X) + CF'
r
a-
Proof We shall use the L2-estimates with weight for the solution of the equation (cf. Appendix III). Let e>O and cpEPSH(CC"), Zo a point such that e-'" is integrable in a neighborhood of Zo (Corollary 5.11). Then, by Theorem 5.12, there exists fE.tt'(CC") such that f(zo)=l and (5,18)
IlfII;=
J If(zWe-"'(Z)(l + IlzI1 2 )-n-£dt(z)< + 00. C'
Wr; let H", be the Hilbert space of all fEJt'(CC") such that iifli",<w. The closed set 1'/ C (C" of points z for which e-'" is non integrable in a neigh-
borhood of z is an analytic variety, the set of common zeros of the elements in H",. We recall that
E(2n, cp)cI'/.
(5,19)
On H"" a point ZoECC" defines a linear functional Zo given by zo(f)=f(zo), which is zero for ZoEI'/. To see that Zo is continuous, we use the CauchySchwarz Inequality and set t/I(z)=cp(z)+(n+e) 10g(1 + Ilz112) for e>O: If(zo)1 ~(t2"r2n)-1 J If(z)1 dt(z) by subharmonicity and 8(zo,r)
8(zo,r)
8(zo,r)
so Izo(f)I=lf(zo)I~Cllfll", with C=(r2n)-1/2exptM",(1+llzoll)=C(zo) and C(z) is bounded on every compact subsets of (Cn independantly of f EH",. The linear form Zo thus belongs to the dual space H~ and A",: z-+zEH~ is a mapping of CC n into H~. Furthermore, I'/cI'/'={z: cp(z)= -oo}, so 1'/ is of measure zero in CC". Let zortl'/' and set cp = 2n V. The function e-'" is integrable in a neighborhood of Zo by Corollary 5.11. Thus, we can find fl EH", such that fdzo) = 1, and from (5,18) and Lemma 3.47, we obtain the estimate sup log IfI (z)1 ~nMv(r+(X) +(n+e) 10g(1 + r) + C(n, e, (X)+ C I Ilzll
~r
with C(n, (x, e) =(n + e) log (I + (X) -n log (X -1/210g t 2n' What is more, E(l, V)cfl-I(O), since if vy(z);;;I, for cp=2nV, we have v",(z);;;2n and zEE(2n, cp),
(5,20)
E(l, V)=E(2n, cp)
with r'" non integrable for every point of E(l, V). This shows that fl(z)=O for zEE(l, V). Let Xj be the irreducible branches of /1-1 (0) which are not contained in E(I, V). For every j, we choose a point ZjEXjnC E(I, V). Since co = 1 is a
130
5. Holomorphic Mappings from
cr"
to
crm
number of complete stability for V, we have vy(Zj)=O and v",(Zj)=O with e-'" integrable in a neighborhood of Zj. We can thus find fEH", such that f(Zj) = 1. Then Zj(f)=O defines a proper closed subspace of H",. Since a countable union of closed subspaces is of first category in H "" there exists f2EH", such that f2(Z):f:0 for every j. Then E(1, V)cfl- I (0)nf2- 1 (0)=X 2. We continue in this way by considering the countable family Xj2) of the irreducible branches of X 2 not contained in E(1, V). We choose zjEXy), zj¢E(1, V). As before, there exists f3EH", such that f3(zj):f: 0 and E(1, V)cX 3 = {z: /,.=0, n= 1, 2, 3}. By iteration, we obtain fk' k= 1, ... , n such that
E(1, V)cZ = {z: /,.(Z) =0, k= 1, ... , n} and the set of points in Z n CE(1, V) is discrete. Thus, as before, we find
!,.+ 1 EH", such that fn+ 1 (z):f:O for ZEZ n CE(1, V). Then
n/,.-1(0)
n+1
E(1, V)=
k= 1
and IIFII satisfies (5,17), since each fk does.
D
Theorem 5.14. Let Y be an analytic variety in (Cn of pure dimension p with indicator v(r). Then Y={Z:fk(Z)=O, k= 1, ... , n+ 1}
where the /,. satisfy one of the following estimates:
I
Mk(r) ~ C(t;) log2 rv(r+u),
(5,21)
1+,
Mk(r)~
C(t;, cx)(1 +r)d [ v(t+M)t-d-1dt,
Mk(r)~
C(t;)r 2 u(r + t;).
Remark 1. Theorems 5.13 and 5.14 show that if tEf,,~p(Cn), Acsuppt, and v/(z)~ 1 for ZEA and v/(z)=O for zr/:A, then A is an analytic subset of (CR, and we can obtain A as F-1(0) where log IIFII satisfies one of the estimates in (5,21). Remark 2. The entire functions fj are zero on Y and have no common zero outside Y, but the theorem does not give at ZE Y the value of the integer n+ 1
)
vw(z)~ 1, where W=t log ( j~1 Ifjl2 .
§ 5. The Case of Slow Growth The use of a partition of unity in the construction of U(z) means that there is a certain degree of arbitrariness in the behavior of U(z). It is perhaps worth the effort to try to extend the method of canonical potentials, which
§5. The Case of Slow Growth
131
permit a constructive solution for a Cousin data of finite order, to the case of general analytic varieties. Even in the case of co-dimension 1, the use of canonical potentials loses much of its precision when one treats Cousin data of infinite order. Thus, it is more reasonable to treat only the case of finite order, and we give below an extension of the canonical potential to analytic varieties Y of dimension p such that v1y1(r) is of finite order. We shall use kernels g~(a, z)= -cplla-zll-2P, 1~p~n-1 g~(a,
z)=log Ila-zll,
p=O
and construct as in Chapter 3 the kernels ep(a, z, q), for q an integer,
q~O.
Theorem 5.15. Let t be a positive closed current of degree n - p and U t = t /\ f3 p the trace of t. Suppose that the indicator vt(r) satisjies 00
J vt(r)r-
(5,22)
3 dr
O such that Mllzll-l~lax) and Mllzll-2~laaxJ Since aXj=o for Ilzll~j and for Ilzll ~2j, there exists a constant C(p, n, X) such that, if Ij(z)= -cpJ Ila-zll-2Pxia)dut(a), we have by Proposition 5.5: (5,23) with Icpj(z)l~
J Iiall
[lIz-all-llaXj(a)I+laaxia)IJllz-all-2Pdut(a), by (5,11).
>j
Let Ilzll ~R andj>2R Then for lIall =r andj>2R, ac
(5,24)
Icpiz)1 ~ C'
J vt(r)r- 3 dr+ C", j
where C' and C" depend only on t, Rand n but not on zEB(O, R). Thus, cP j(z) tends uniformly to zero on every compact subset of ([n. Furthermore, by (5,23), if q=O or q= 1, ep(a,z,q)= -cplla-zll-2p+ Iq(a,z)
132
5. Holomorphic Mappings from
crn to cr m
where lq(q, z) is pluriharmonic (by Lemma 5.16). Thus L(1i' }.) = L(1j, ;,).
Since L(1, }.)= lim L(1i' }.);;;;O by (5,22) and (5,23), the theorem is proved.
0
j-cc OC;
Remark 1. The hypothesis
J v,(r)r- 3 dr < + 00
implies that the terms J I' J 2
I
and J 3 disappear in (5,7), the representation for ioaI. We thus have i) ioaI(z)=q~[t*(nIXP+ I) A q*t], O~p ~n -2, ii) iaaI(z)=nt, p=n-l. Remark 2. Theorem 5.15 was proved under the hypothesis that O~supp t. If in fact OESUPP t, we subtract from I(z) a pluriharmonic function
II (z)= -c p J [llz-all- 2p + 2(1 + IlaI1 2P )-I]dO",(a) (5,25)
(;"
J [llz-all-2p+2(1+llaI12P)-1
II(z)= -cp
(;"
+ 2p(1 + lial1 2p + 2)-I1Re(a, z) ] dO", (a), oc
if
J v,(r)r-
3 dr
O. If p=O, we use -loglla-zll and write t instead (2p)-I. The method of the canonical representation allows us, under the hypothesis (5,22), to improve the estimates given in (5,21) for F- I (0) if [Y] is the current of integration on an analytic subvariety of dimension p. First of
§6. The Algebraic Case
133
all, we obtain the equality V(z)=Iq(z) and hence, from (3,14)
(5,26)
Mv(r)=M/(r)~A(p,q)r'I[i vr(s)s-q- 1 ds+r
I
V r (S)S-q-2 dS].
Thus, by (5,21), we obtain Y=F- 1 (0), with oc>O, e>O and M F(r) ~ nM v(r + oc) + e log (1 + r) + C(n, e, 0). We resume this result in the following theorem: Theorem 5.17. If Y is an analytic subvariety of pure dimension p with in00
Jv[Yl(t)t- 3 dt< + 00, then for every e>O, there exists
dicator v[Yl(t) such that
I
a representation Y = F- 1 (0) with MF(r) = sup log 11£11 ~nMv«1 +e)r)+e 10g(1 +r)+ C(n,p, e) Ilzll
~r
and Mv(r) satisfies (5,26). Thus for q=O or q=1 (5,27)
sup log IIFII IlzllO and a biholomorphic map ~z: Uz--+B(O, l)ccr p • For W fixed, set
n-I
{
}
Yw= uEB(O, 1): ;~I z;(u)w;+zn=O
(n-I
)
}
~ {UEB(O,I):-. a .I Z;(U)W;+Zn =O,j=I, ... ,p, Yw= au) 1 1=
and A={WEcr n - l : YwcYw}' Then A is an Fa-set (and hence measurable), since Fv={w: Yw nB(O,I-I/v)cYw nB(O,I--=-i/v)} is closed and A=UFv' Let - I Z;(U)W; - Zn(U) f(u)
n-I
in B(O, 1). Suppose for w'=(w 2 ,
... ,
wn) fixed, w=(w I , w')EA. Then for uEYw
af(u) -I ( n~1 az;w; OZn) --=ZI(U) - L. - - - CUj ;=2 aU j OUj +ZI(U)
-2 (n-I az; .I Z;(U)W;+Zn(U) ) -a .=0; 1=
But we have seen that
2
j=I, ... ,p.
Z)
f == constant on each connected component of
{ UEB(O,I): of =O,j=I, ... ,P}, and thus for every w', there are at most a oU j countable number of WI = f(u) such that (WI' w')EA, so A is of measure zero in cr n - I . We now choose a countable dense set Z;E Y-0 and neighborhoods 00
UZi and sets A; as defined above. Then A 3 =
UA; is of measure zero. ;=1
Suppose that w¢AI uA 2 uA 3 . Let Zw= yn{z:
~t>;W;+Zn=O} with Zw as
regular points and Z~ as singular points. Then, since w¢A z , It follows from above discussion that since w¢A3'
dim(Z~)~p-2.
1 _ In_1 1 -icc log I Z;W;+Zn /\8, 2n ;= 1
is the closed positive current of integration over Zw in cr n - Y' - Z~. Its simple extension t is a closed positive current of degree (p - 1) in cr n - Y', since dim(Zw)~p-2, and the simple extension of t to cr n _ Y' is again a closed positive current of degree (P-l), since dim(Zwn Y')~p-2 for w¢A 2 (cf. Chapter 2). Lemma 5.31. Let Y c cr n be an analytic variety of dimension p ~ 1 and let elY] be the closed current of integration on Y. Let Kccr n - I be a compact set and
148
5. Holomorphic Mappings from q:. to q:m
IP.E~; (CC n - l ) a sequence of functions such that IP. decreases to XK' the characteristic function of K. Then for r> I, if VIP. is defined as in Lemma 5.27, lim J__ ioaVIP'(z)"OIYl"PP_1 .-OOB(O,r)-B(O,I)
2n [uIYl(w; r)-ulyl(w, 1)]XK(w)dr(w), m(K) 1["-1
J
=--
where Ow is the current of integration over the set yn{z:
~r.l ZiWi+Zn=O} 1=
and
uIyl(w;r)=
1
J
0w"Pp-l'
B(O,r)
Proof Let Y' be the singular points of Y and let 1/1 "E~; (B(O, r)) be a sequence such that the 1/1" increase to X(B(O,r)-Y'-B(O,I))" An integration by parts gives us
J1/1"ioaVIP. "Oly] " Pp-l = JVIP'ioal/l" " 0IYl" Pp-l' and it follows from Fubini's Theorem that
JVIP'ioal/l""olY]"P P_1 =C;;}
I["t
[JiOgl:t: ZiWi+Znlioal/l."OIY]"PP_l]IPV(W)dr(W)
l
A second integration by parts gives us
C;;.1
I:t:
)_1 [J 1/1"ioalog =C;;}
Zi Wi + Znl " 0ly]" Pp-l ] IP.(w)dr(w)
+
)_1 [JiOgl~t: Zi Wi+ z
oal/l""OIYl"PP_l]IP.(W)dr(W).
It follows from Lemma 5.30 and the Monotone Convergence Theorem that lim
,,-00
J1/1"ioaVIP." 0ly]" Pp-l = C;;.1 )_
1U~~ J1/1" ioalog
I:t:
Zi Wi + Znl "OIYl" Pp-l ] IP.(w)dr(w).
or
J__ ioavlP." 0IYl" Pp-l B(O,r)-B(O,I)
=2n C;;.1
I["J- [UIyl(W; r) -Uly](W'; I)] IP.(w)dr(w). 1
Finally, it follows from the Lebesgue Dominated Convergence Theorem that lim Jic2VIP'"O[Yl"Pp_1
v-x
=2nm(K)-1
I["J- [u[Yj(w;r)-u[Yj(w; l)]XK(w)dr(w). 1
0
§ 10. Upper and Lower Bounds for the Trace of an Analytic Variety on Complex Planes
149
Lemma 5.32. Let YcCC n be an analytic variety of pure dimension p ~ 1 and let K c CC n - 1 be a compact set and q>.E~.f (CC n - l ) a sequence of functions such that q>. decreases to XK' Then if k> 1 is a constant, there exist constants 1'1,1'2,1'3 and 1'4 which depend only on K and k such that for r> 1
J
r- 21'1 0"[Y](Y2r) ~ 1'3 O"[yp)+
iaaV0).
Then we can find a compact set K c E c U0 of strictly positive measure, where
Let rt=(1 +e)t/2p, k=(1 +e)1/2p. By Lemma 5.32, we obtain
J [O"[Yl(w; r)- u[Y](w; I)JxK(W) d r(w);;:;; C•. Ku[Y]((1 + ::)1 /2Pr)· r- 2.
a:n - ,
By reasoning as in Theorem 5.26, we conclude that . CT[Yl(W; r)-CT[Yl(W; I) } { WEK: hmsup -2(1 )fl «I )l lp )=1=0 '-00 r ogr CT[YI +e r is of Lebesgue measure zero, which is a contradiction. Thus measure (E)=O. We now assume the induction hypothesis for t;;:;;q -1. Suppose now that L¢E as defined above. By the induction hypothesis, the set
E = {i(L p ... , L q_ 1)E [G n- 1 (CC)]q-l: lim sup '-+00
CT[YnL,n ... nLq_d(r) q-l r- 2(q-l)(logrf(q-1l CT[Yl«1 + e)2P r)
=l=0}
is of measure zero. This establishes the induction hypothesis for all q. To terminate the proof, we show that if '1: [G n_ 1 (CC)]q is given by
f/(L 1 ,
...
,Lq)=
n
L i , then for EcGn_q(CC) such that the 2q(n-q) Lebesgue
i=l
measure of E is positive, the product measure of '1-1 (E) is positive in [G n_ 1 (CC)]q. Let ccs(q) = A cc n as a linear space. For 1= (i\ < ... < iq), we let -
q -
I=U\< ... <jn-q:ji¢I} and sign (I, I) the sign of the permutation (I, ... ,q)-+(l,l). If e\, ... ,en is the standard basis in CC n, we set el =ei , /\ ... /\ eiq , and we ~efine the map ~q: CCs(q)-+ccs(n-q) to be the linear map such that eq(e l ) = sign (I, l)el' Let 1tq: CCS(q) - to} -+ IP(CCS(q). Then the map ~q=1tn_qeq1t;;\: Gq(CC)-+Gn_q(CC)
is holomorphic and ~q(L)=H, the space orthogonal to L. To see this, we calculate in local coordinates. Let UI be the coordinate patch defined at the
152
5. Holomorphic Mappings from
cr"
to
cr
m
beginning of Section 9. For simplicity, we assume that 1= (1, ... , q). Then to
we associa te
(-CI(n_q)' ... , -cq(n_q)' 0, ... , 1) which is clearly a holomorphic homeomorphism of VI onto VI' We define the holomorphic map WI: [Gn _ 1 (O}
is of Lebesgue measure zero. The conclusion now follows in the same manner as that of Theorem 5.33. 0 Of course, for the purpose of applications, one wants to choose the sequence rm so that (f[y](rm) has "maximal growth". As an example of an illustration, we present the following corollary:
Corollary 5.35. Let Y c ccn be an analytic variety of pure dimension p such that r- 2p (fy(r) is of finite order p. Then the set of LEGn _ q, O O. Then there exists a set E c Gn _ q of Lebesgue measure zero such that for L¢E, we can find a subsequence rm , for which lim (r ,)-2(p-q) (f[YnLl(rm,) >0. m (rm,)p(r m ,)
m'-x,
It then follows from Theorem 1.18 that
lim sup r-oo
(f
(r)'r- 2(p-q) [YnLJ
rP(r)
>0
.
D
154
5. Holomorphic Mappings from (C" to (Cm
Historical Notes The representation with growth conditions of an analytic subvariety X of (Cn of co-dimension greater than one poses certain delicate problems which are only partially resolved. First of all, X is not in general a complete intersection, and secondly even when it is, the Lelong- Poincare equation Ox=iaaloglFI (studied in Chapter 3) for X a subvariety of co-dimension one (which reduces the problem to a linear equation) is replaced by a nonlinear equation of complex Monge-Ampere type: 8x =(iaalog 11F11)q for X a complete intersection of pure dimension q. Two results which were published the same year (1972) have enriched the theory of holomorphic mappings. One is the counter-example of Cornalba and Shiffman [lJ, which shows that there is no control (even asymptotically) as a function of I F I for the growth of the analytic subvariety F- 1 (a) of co-dimension two for F an entire holomorphic mapping of (C2 into (C2. An average estimate has been obtained by Carlson [2J and Gruman [8, 12]. The latter shows that the set of a for which no such estimate is possible is pI uri polar. Another result, due to Skoda [2, 3J was thus surprising. He showed that for X an analytic subvariety of CC n of pure dimension q, 0 ~ q ~ n -1, one can express X as X = F- 1 (0) for F an entire holomorphic mapping F: (Cn->cc n + 1 where IIFII has an asymptotic growth controlled by that of vx(r), the indicator of X. A comparaison of the results shows the "statistical" nature of the control (which is the point of view adopted by Stoll [8J in his study of the transcendental Bezout problem). The study of the trace of an analytic subvariety X on linear subspaces (undertaken in § 9) was also motivated by the counter-example of Cornalba and Shiffman [1]. An upper bound outside exceptional sets was first given by Carlson [2J using the Crofton Formula and general properties of positive monotonic functions. Much more surprising was the fact, first shown by Gruman [8, 11, 12J, that one could obtain a lower estimate of the trace of X on linear subspaces outside a small exceptional set. In fact, one can obtain estimates outside sets much smaller than those of Lebesgue mesure zero. For results in this direction, we refer the reader to the articles of Molzon, Shiffman and Sibony [IJ, Alexander [2J, Gruman [8, 11, 12J, and Molzon [4].
Chapter 6. Application of Entire Functions in N urn ber Theory
As has already been noted, one of the basic motivations for studying entire functions of finite order is that all of the familiar transcendental functions fall into this category (and what is more, for meromorphic functions of finite order, one arrives at the familiar elliptic functions). Thus, by studying the algebraic or arithmetic properties of entire functions of finite order, one implicity studies the algebraic or arithmetic properties of the fundamental transcentental functions. This in turn has a wide variety of applications in transcendental number theory. We give here an illustration where the methods of holomorphic functions of several complex variables were used to solve a problem which contains several classical problems of number theory. The following method uses an idea which goes back to c.L. Siegel and consists of constructing an entire fonction with "many zeros" in the class of those whose values have given algebraic properties. We present here the solution of E. Bombieri in CC· given in connection with his joint paper with S. Lang. The solution relies heavily on the technics of positive closed currents and the resolution of the operator aa.
§ 1. Preliminaries from Number Theory A complex number (X is said to be algebraic of degree p if there exists an irreducible polynomial P(x) = a o x P + ... + a p with ak a real integer and ao =1= 0 such that P(oc)=O. If we suppose that the ak have no common divisor, then P is unique and is called the minimal polynomial for (X. If a o = 1, we say that IX is an algebraic integer. A complex number which is not algebraic is said to be transcendental. We denote by Q the field of rational numbers. If K is a subfield of CC, we say that K is an extension of finite type if there exist Yl, ... ,YmEK such that K=Q(Yl, ... ,Ym). If K is a vector space of finite dimension over Q, we denote this dimension by [K: Q]. If K, an extension of finite type, is a field of algebraic numbers, then it follows from the Theorem of the Primitive Element that K is then simple, that is K = Q(oc) for some algebraic oc, and in addition we then have Q(oc) = Q [oc], that is the field generated by adding (X to the rationals is the same as the ring
156
6. Application of Entire Functions in Number Theory
generated by adding IX to the rationals. We then say that K is a number
field. Proposition 6.1. Let IX be an algebraic number. Then the following two
properties are equivalent: i) there exists a non-trivial unitary polynomial Q(X)EZ [x] such that Q(IX)=O; ii) there exists a non-trivial Z-module M generated by a finite number of algebraic elements such that IXM c M.
Proof If p is the degree of Q in i), then M=Z[IX, ... ,IX P] is a non trivial Z-module of finite dimension such that IX M c M, so i) implies ii). To see m
that ii) implies i), we let v l '
..• ,
vm be a basis for M. Then IX Vi =
L aij Vj
j= 1 with aijEZ. Let A=[aij]{:;::::::::' Then det(A-IXI) annihilates the module M, so this determinant must be zero. Then P(x)=det(A-xl) is a polynomial with integral coefficients such that P(IX)=O. 0
It follows from ii) above that the set of algebraic integers is a ring, since (IX + f3)MN =IXMN +f3MN cMN and IXf3MN cMN. It also follows from ii) that AIX is an algebraic integer for IX an algebraic integer and },EZ, since AIXM=IX(},M)=IXMcM. For IX an algebraic number, we let Da be the ideal in Z defined by Da={A.EZ: AIX is an algebraic integer}. n
This ideal is non-zero, since if Q(x) = L CiX i is the minimal polynomial of IX, then CiIXED a, for i=O n
Q(x)=xn+c n_ l x n- l +c n_ 2Cnxn-2 + ... + COC:- 2 =
L CjC:- j -
l Xj
j=O
satisfies Q(CnlX)=O, and hence (ii) of Proposition 6.1 is fulfilled. A positive element of Da is called a denominator for IX, and the positive generator d(lX) of Dais called the denominator of IX. Let IX be an algebraic number and P(x) its irreducible polynomial. Denote by 1X1' ••• , IXm the complex roots of P (which are all different, since P
n (X-IX). The IXj are j= n
is an irreducible polynomial in Q[x]). Thus P(x)=
1
said to be the conjugates of IX, and we set fi1 = max IlXjl. We then define the size of IX, S(IX), by S(IX) = max (log r;]. log d(IX». 1 ~j~m Proposition 6.2. Let K be a number field. Then for IX EK, IX =F 0, -2[K: Q]S(IX);;;; - [K: Q] 10gd(IX)-([K: Q] -l)1og~ ;;;;log IIXI.
§ 1. Preliminaries from Number Theory
157
Proof Let m be the degree of the irreducible polynomial for tx, so that Q]. Notice that the irreducible polynomial for d(tx)·tx is
m~[K:
?(x)= xm +a m_ 1d(tx)x m- ' +a m_ 2d(tx)2 xm- 2 + ... + aod(txt
(where xm + am_ 1x m- I + am_ 2 xm- 2 + ... + ao is the irreducible polynomial for m
tx). Hence ?(x) = m
we have
TI j~
(x - d(tx)tx). Since the coefficients of ?(x) are integral,
1
TI d(tx)'ltxjl~l, from which the conclusion follows. j~
0
1
We shall need the following technical lemma from linear algebra. Lemma 6.3. Let A be a IIOIl-zero subgroup oflRn. Assume that in any bounded region of lR" there exists only a finite number of elements of A. Let m be the maximal number of elements of A which are linearly independent over lR. Then we can select m elements of A which are linearly independent over lR
which form a 7l-basis for A. Proof Let {WI' ... , wm} be a maximal set of elements of A linearly independent over lR. We proceed by induction on m~n. Suppose that m= 1. Let w=tw l , where ItI is the smallest possible non-zero value. Suppose that wEA, and let t be such that w=tw i . Choose qE71 such that qt~t«q+ l)t. Then w-qw=(t-qt)w1EA and O~t-qt«q+l)t--qt-=~ which contradicts the choice of t unless t = q ~ in which case w is a 7l-basis for A. Suppose now that m> 1. Let V be the vector space over lR generated by {WI' ... , wm } and let Vm _ 1 be the space generated by {WI' ... , wm _ I }. Set Am_I = An Vm_ l . Then in any bounded region of Vm_ l , there exists only a finite number of elements of Am - I ' By the induction hypothesis, we can find {w~, ... , w~_ I} which form a 7l-basis for Am_I' Let S be the set of elements of A which can be written in the form m
L tiw; with O ms and suppose that the aij are algebraic integers in
K
*I. Preliminaries from Number Theory
159
n
fG;jl < A. Then there exists XiEZ not all zero such that
with
m
j= 1, ... ,m and IXil< 1 +(cnA)n-ms, where c depends only on K.
l:aijx = 0, i
i=1
Proo}: Let WI"'" Ws be a basis for I Kover Z, which exists by Lemma 6.4. Let 0'1' ... , as be the s embeddings of K into ce. For aEI K' let rea) =(0' I (a), ... , 0'. (a», which embeds I K as an additive subgroup of ce s =1R 2s. Then sk=r(w k ), k= 1, ... ,s, are linearly independent over the real numbers. Since s
a=
L
•
lX(k)W k ,
L
r(a)=
k= I
lX(k)Sk
and so there exists a constant c depending
k= I
only on K (or rather on the basis (WI' ... , ws» such that
Ia1 =
sup lUi(a)1 ~ c- I sup lac( i1 I· 1 ~i~s
1
~i~s
s
In particular, we can write aij=
L
1X1~)Wk with IXI~)EZ and IlXijl 0 for t a positive closed current were analytic varieties (as obtained later by Siu [1] in 1974 using the L2 -estimates of Hormander). For more details, see Lelong [12b]. The estimate for the degree of the algebraic hypersurface S was improved by Skoda [6] and Demailly [1]; Bertrand and Masser used the n dimensional result to obtain results on transcendence and algebraic independence. For a very thorough treatment of the above as well as many related topics, we refer the reader to the book by Waldschmidt [1].
Chapter 7: The Indicator of Growth Theorem
We have seen in Chapter 1 that the radial indicator h1 of an entire function f(z) of normal type with respect to a proximate order p(r) satisfies i) h1(z)=tPh1(z) for t~O; ii) h1(z) is plurisubharmonic in CC n• We now show the converse for strong proximate orders; that is, suppose that h(z) is a function which satisfies i) and ii) above. Then for every strong proximate order p(r) there exists an entire function f(z) such that h1(z) = h(z). Let k be an integer, (1 ~ k ~ n), and", a function plurisubharmonic in CC n • Suppose that o(E~~(B(O, 1», 0(~1, O(z) dr(z)=l, 0( a function of Ilzll. We define by induction a sequence {"'~} of plurisubharmonic functions as follows:
J
i) "'~ = ... = "'~ = '" ii) "'~=O(*"'jkforj>kwith
"'t=[ sup "'~_l(Zl,· .. ,Zj_l' Zj+~j' Zj+l,· .. ,Zn)]*' I~jl< 2
For 1 ~j~n, we identify CCj with the subspace of cc n defined by {ZECC n: zp=O for p>j} and z(j) will be the projection of z on CC j , that is if z = (Zl' ... , zn), z(j) = (Zl' ... , Zj' 0, ... ,0). We will let drj be the Lebesgue measure on CC j . We shall use this to prove the following result on the extension of entire functions with growth conditions.
Theorem 7.1. Let ",EPSH(CCn) and f holomorphic on CCk (k g when e-->O in L2(B(0,r)) for every r so g,=g almost everywhere, and hence g is (equivalent to) a holomorphic function. Finally, from (7,1) and (7,2), since Ilzj+1112~1+llz(j+llf, it follows that
S
(:j+l
Ig(Z(j+l»)1 2 exp-I/Ik (ZU+II) . (1 + Ilz(j+ 1'112)3(/+\ - kl dr j+ d zU + 1»
_
~ (1 + 4 C 2 )MJ .
o
Let PSHp(t)(CC n ) be the set of plurisubharmonic functions of order p and normal type with respect to the proximate order p(t).
7. The Indicator of Growth Theorem
169
Proposition 7.2. Let I/IEPSHp(I)(CCn ) and Zo *0. Then given e>O, there exists R(e, zo) and J(e, zo) such that for Ilz-zoll R
I/I(tz)~ (h:(Zo) +e) tP(I). -
if. Let Proo.
'
Zo Zo =~.
IlzollP
Since h:(z) is upper semi-continuous, there exists a
neighborhood 2z~...of -z~ such that h:(z) ~h:(z~)+e/2 for ZE Uz~' We apply Hartog's Lemma (Theorem 1.31) to the family V,(z)=t-P(I)I/I(tz) on Uz ' (which we suppose compact by passing to a smaller set if necessary). D Corollary 7.3. Let I/IEPSHp(I)(CC n) and let A be a continuous positively homogeneous function of degree p with A ~ h:. Then given e > 0, there exists r > such that l/I(z)~(A(z)+ellzIIP)llzIIP(lIzll)-p for IIzll ~r.
°
Proof Since the sphere of radius 1 is compact, for t> t I' we have by D Corollary 1.32: l/I(tz)~(A(tz)+etP)tP(')-P for IIzll=l and t>r l ·
Theorem 7.4. Let 1/1 EPSHp(,)(CC n). Then the functions t/I~ introduced in Theorem 7.1 all have the same radial indicator function h:. Proof There exists a>O such that I/I~I/I~~I/I' for I/I'(z) = [ sup I/I(z+m*. For zo*O and e>O, there exist 15 and r such that II~II ~a
I/I(tz)~ (h:(Zo) +e) tP(I) - IlzollP and
for
t~r
for
t~r'.
Ilz-zoll ~J, so that I/I'(tz )< (h:(zo) +e) tP(I)
o = IlzollP
But this implies that
h",,(z)~h:(z)
for all z and hence
h:,(z)~h:(z).
Theorem 7.5. Let I/IEPSHp(,)(CC n ) and f an entire function such that
JIfl2 exp-I/Idr< 00. 4:n
Then f is of normal type with respect to pet) and
h!(z)~
1/2h:(z).
Proof By the Cauchy Integral Formula, we have f(z)2= (2 ~)n Jf (rl ei61 + zl ' ... , rnei6n + zn)2 dlJ I so
...
dlJ.,
170
7. The Indicator of Growth Theorem
Thus If(z)12~ Cn[sup exp I/I(z+ I~jl ~
2
m
J
If(z+ ¢)12
1~I~jl~2
·exp-( sup I/I(z+mdr I~JI ~ 2
~Cn[sup expl/l(z+m l~jl~2
J
If(z+¢Wexp-l/I(z+¢)dr
J
If(z+~)2Iexp-l/I(z+~)dr)
1~I~jl~2
~C~[supexpl/l(z+m ( l~jl~2
1~I~jl~2
so (7,3) loglf(z)I~C~+1/2 sup I/I(z+~) and hence f(z) is of finite order II~II ~2n
with respect to p(t) and h1(z)~!h:(z) by Theorem 7.4.
D
Theorem 7.6. Let I/IEPSH(Cn) be positively homogeneous of order p. Then we can find a decreasing sequence of plurisubharmonic functions {I/I q} each positively homogeneous of order p and ~oc on (Cn - {O} such that lim 1/1 q(z) = 1/1 (z).
q~oc
J
Proof Let IX(Z)E~oX (B(O, 1)), 0 ~ IX(Z) ~ 1, IX(Z) dr(z) = 1 and IX depending only 1 on Ilzll. Let IX,(Z) = e2n IX(Z/e). Then I/I,(z) = Jl/I(z')IX,(z-z')dr(z') is ~c>C, plurisubharmonic, and decreases to 1/1 (z), but it is not in general positively homogeneous of order p, so we must change the construction slightly. Let
~,(z)= Ilzll- 2nJ1/1 (Z')IX, el~ln dr(z')
for Ilzll =FO or equivalently
~,(z)
= JI/I(z -II zll w)IX,(w)dr(w) for all z. Then ~ ,(z) is ct x on (Cn - {O}, is positively homogeneous of order p (since 1/1 is), and since ~,=I/I, for Ilzll=l, ~, decreases to 1/1 when e tends to zero. It remains to show that 1/1, is plurisubharmonic in (Cn. Since IX,(W) depends only on Ilwll, there exists a positive continuous function A(r) such that
,
I/I'(z) =
JA(r)T,.(z)
o
with
1 T,.(z)=w2n
J
Ilwll ~r
l/I(z-llzII w)dwzn(w).
Thus, it is sufficient to prove T,.(z) plurisubharmonic in (Cn. Let i be the unitary group on (Cn, which is compact, and dy the normalized Haar measure on 1. Let Zo be a fixed point of (Cn of norm rand l/I(y)=I/I(z-llzlly(zo))· Then T,.(z) = I/I(}')d~'. Furthermore, there exists '1Ei
J
such that r'1(z)= Ilzllzo. so
T,.(z) =
r 1/I(~')=I/I(z-r~''1(z)),
and if ¢(";')=I/I(z-r~'(z)),
J¢(i''1)d"'/= J¢(r')d,'= JI/I(z-r,'(z))d)'.
r
r
r
Since for every YEi, z-r·),(z) is a holomorphic function of z, I/I(z-ry(z)) is plurisubharmonic in z and hence so is T,.(z). D
7. The Indicator of Growth Theorem
171
By a Lipschitz continuous function, we will mean II/I(z)-I/I(z')1 ~ C Ilz-z'll for z, Z' ES 2n - I.
Proposition 7.7. Let p(r) be a strong proximate order and I/I(z) a Lipschitz continuous plurisubharmonic function positively homogeneous of order p. Then there exists a plurisubharmonic function t/i(z) such that 1/1 (z)t P(liz 11)- P~ t/i(z) and h~(z) ~ I/I(z) (where the indicator is taken with respect to p(r)). Proof Suppose that e(r) is a continuous decreasing function of r such that lim e(r)=O. Then there exists an increasing convex function ~ such that r- oc
i)
~'(logr)~e(r)rp(r)
.. ) l'
Wogr)
and
We
r~ro,
s
~(s)
define
for
0
Im~=. r-oo r
II
e"(logr)~e(r)rP(r)
by
I
~(s)= S ~'(t)dt,
~'(t) =
CpS e(er)erp(er) dr
with
o
o I
C p =2sup(p, 1). Then ~'(t)~e(el)Cp S erp(e'")dr~e(el)e'P(et) for t large enough, 0 1 and since e(s)~- for s~sn' we have n C I 1 ~'(t)~
Cn+---.£. S erp(er)dr~ Cn+- C~elp(et). n s n Thus ~(t) ~ Cnt + C~ e'p(et), whic~ shows ii). By adding a multiple of n log(l + r2), we may assume that i) holds for all r. We note that if a = (aI' ... , an) is a complex vector, then ,,2 ,,2 02!,() [111121L..ajrjl] lL.,ajr) (7,4) L ~ ajiik=e'(logr) -;- j 4 +~"(logr) j 4 j.k OZjOZk r r r ~!e(r)rp(r)-21IaI12.
Since I/I(z) is Lipschitz continuous and positively homogeneous of order
p,
10~(Z)I~CIIZIIP-I;
that is, as a distribution °ol/l(z) is equivalent to a
c~
~
function with the above bound. Let a be a complex vector. Then as a distribution, setting Ilzll =r, we have
L j.k
02(I/I(z)rP(r)-p) _ _ p(r)-p 021/1(Z) _ ~ 0ajak- L r ~;:)- ajak cZj Zk j,k CZj{/Zk crP(r)- pol/l(z) _
+L-;:)--~ajak
j,k
(/Zj cZ k orP(r)- p ol/l(z)
_ + L ---;=--~-ajak j,k L Zk cZj c2 r P(rl- p +1/I(z)I ~ ~ aiik j,k czjczk ~
-e(r)rP(rl-21IaI1 2
172
7. The Indicator of Growth Theorem
for some e(r) such that lim e(r)=O. Thus we can find Wogr) plurisubhar. ~(logr) monic with 11m ----;>ir)=0, such that rfr(z)=Wogr)+t/J(z)rP(r) is plurisubharmonic. r~ x r 0
Theorem 7.8. Let t/J be a subharmonic function in CC positively homogeneous of order p. Then for 0 ~ e~ 2n, there exists an entire function f (z) of order p(depending perhaps on 0) such that . log If(te i6 ) 11m sup p I~'X: t
. t/J(e I6 ) and
If (z)1 ~ C exp rfr(z)
where rfr(z)= [sup t/J(z+ ~)]* + C 1 10g(1 + IzI2)+ C 2(log(1 + IzW. I~I ~3
n (l-2- i z), which defines an entire function. 00
Proof. Let h(z)=
i= 1
Suppose that for somej, 1/4~lz-2il~t, so that
and
k
independent of j. Let cpErt';-(B(O, 1/2)) such that O~cp~1 and cp=1 for Izl~i, and set g(z) x
=
L
cp(z-2iei6)expt/J(2iei6) (for every z, there is at most one summand).
i= 1
We shall write f(z)=g(z)-h(ze- i6 )v(z) where v(z) is chosen so that f(z) is holomorphic. We must have 2f =0 or 8g=h8v. Since 8g=0 for
Iz-2iei61~i if p=
8:,
then
IPI~cl:~1 and
JIpl2 exp-2~(z)dr(z)< + ex; C[
where ~(z)=[ sup t/J(z+m*+log(1+lzI 2). I~I ~1:2
By Appendix III, there exists r such that h = p and
J Irl2 exp-(2~(z)+ 10g(1 + IzI2))dr(z)< + "x. C[
Then g - h r defines a holomorphic function.
7. The Indicator of Growth Theorem
173
Suppose that 2j~lzl~2i+l. Then, since log (I +x)~x for x~O, x (IZI) i+ I 1 loglh(z)l~k~,log 1+2k ~k~,log(I+2i+'-k)\~,2k 'Yo
i+ I
~k~1 U+2-k)log2+1=
U 1)U 2) + 2 +
log2+1
~ C 1 (log(lzl + 1))2 + 1.
Let l/i(z)=I/i(z)+log(I+lzI 2)+C, (log(I+lzI))2, which is subharmonic. Then h,(eiO)=IjJ(eiO ) and by Theorem 7.5,
!fl ~ C exp ([ sup I/I(Z + ~)J* + 2log (I + !Z!2) + C~ (log (1 + Iz!2))2.
0
I~I~P
Theorem 7.9. Let ljJ(z) be a plurisubharmonic function positively homogeneous of order p in ccn. Then for ZoECC n, there exists an entire function g(z) (depending perhaps on zo) such that h:(z)~IjJ(z) and h:(zo)=IjJ(zo) (where the indicator is with respect to rP). Proof By a rotation, we may assume that Zo = (zp 0, ... ,0). Let ljJ(u) = ljJ(u, 0, ... ,0). Then by Theorem 7.8, we can find an entire function in CC feu) such that h;(zo)= ljJ(zo) and log If (u)1 ~ [sup ljJ(u + I~I ~3
m* + C
1
log (1 + lu1 2)2
+ C 2 log(1 +luI 2)+log c. By Theorem 7.1, there exists an entire function g(z) such that g(u,O, ... ,0) = feu) and Slgl2 exp-~(z)dT(z)< + rx; where
~(z)=2[ sup ljJ(z+ m* + Cn [log(1 + Ilz112)]2 + C~ log(1 + Ilz112) II~II ~a
for some a>O. Then h:(z)~h$(z) by Theorem 7.5 and h$(z)=h~(z)=IjJ(z) by Proposition 7.2. 0
Corollary 7.10. Let ljJ(z) be a Lipschitz continuous plurisubharmonic function positively homogeneous of order p in ccn and p(t) a strong proximate order. Then for ZoECC n, there exists an entire function g(z) (depending perhaps on zo) such that h:(z) ~ ljJ(z) and h: (:0) = ljJ(zo) (where the indicator is with respect to r PI ,)).
Proof Let I/i(z) be the plurisubharmonic majorant of ljJ(z)tPlt)-P constructed in Proposition 7.7. Then, as in Theorem 7.8, we construct an entire function f(uz o) of the variable u such that f(2 i z o)=1/i(2i z o) and If(z)1 ~ [sup I/i(z+ ()]* + C 1 log(l + Ilz112)+ C 2(log(1 + Ilzll))2 I~I ~3
174
7. The Indicator of Growth Theorem
(we choose g(UZO)=jtl
7.1, to extend
f
cp(UZo-2jzo)exp~(2jzo)).
We then use Theorem
to (Cn, as in the proof of Theorem 7.9.
0
If cp is a continuous positively homogeneous function of degree p in (Cn, we let B~(t) be the Banach space of entire functions f such that
lim If (z) exp - cp(z) IlzlIP( liz 11)- PI = 0 Ilzll~'"
with supremum norm. Let E~(I)=
n
B:~lllzIIP' which is a Frechet space. If
q
q
I/IEPSHp(t)(Cn), let m(l/I) be the set of continuous plurisubharmonic functions q; positively homogeneous of order p such that h~(z);£qJ(z). Theorem 7.6 shows that h~ (z) = inf cp and that m(l/I) is an ordered filtered set with a ",Em(I/!)
countable basis. We set E~(t) =
n E:(I), which is also a Frechet space.
",Em(I/!)
Theorem 7.11. For I/IEPSHp(t)(Cn), let f be an entire function of normal type with respect to p(t). Then fEE~(t) if and only if h1(1/I);£h~(z). Proof If fEEl/!' then fEE", for every cpEm(l/I) and hence 1
If (z) exp ( - cp(z) -- Ilzll P) IlzlIP( liz I )-PI q
is bounded for every q. Thus
JIf(z)1
2
exp - (2cp(z) + e liz liP) liz IIP(lIzll)- P dr(z) < +
e for every e>O. By Theorem 7.5, h1(z);£h:(z)+21IzIIP, so
OC>
h1(z);£h~(z).
On the other hand, for qJEm(I/I), 10glf(z)I;£(cP(z)+ellzIIP)llzIIP(llzll)-p for Ilzll ~rt by Corollary 7.3, so fEE", for every cp and hence fEEl/!' 0
Theorem 7.12. Let 1/1 be a plurisubharmonic function positively homogeneous of order p. There exists an entire function f(z) in (Cn whose indicator function h1(z) with respect to rP is 1/1 (z). Proof Suppose that 8EPSH p(0
Since II
.Q(z,~(z))=L(-ly+l~j(z) /\ a~k(Z) /\ dZ I
k*j
is of degree
+n
in
- 2
~,
0
-
1
"' "11-1
(t/I(~))
0\;0
-;;-
1= 1
-
Q(z, ~(z)) is of degree zero. In a neigh-
'>0
borhood of infinity (~o=O), we can choose local coordinates in which, for • 1 an d hence -. t/I(~). I h · ./,. . f· . . Instance, \;j= - IS hoomorp IC· SInce 'I' IS zero at In Inlty. 20 - 1 (t/I(~)) \;0 Thus O~~-I ~ is everywhere defined and holomorphic and the integration is well defined. We show now that the value is independent of the choice of K. It suffices to show that for K 2 ~ K 1 ~ w the value remains unchanged, for if K and K' are two compact subsets of w with nonempty intersection, we need only choose K 2 c K n K' and if the integral has the same value for K and K 2 and for K' and K 2' then it has the same value for K and K'. Thus, we verify
t f(z)~.o_1 ('11-1
A=
bdK,
8 - 1 ~ -11-1 ( ~o 0
__
-.- Q(z,~(z))
C\;o
- J f(z) bdK,
(t/I(~)) \;0
(t/I(~)) -.~o
- ""
Q(z, ~(z))=O.
Suppose that Pj defines K j , that is Kj=[z'Pj(z)~O, Pj strictly convex]. Let X=([,ll x IP((['I1) and let L j be the manifold II
C
L -Zk (fj,
~~)=
(z,~(j)(z)),
where
~V)=~~j, (
':'k
which we identify in a natural way with bdK j • Choose a
cZk (to remain fixed). For zEK I -K2' we let miz) be the point on bdK j where the half line from Zo to infinity passing through z intersects k=1
point
ZOEKl
§2. The Projective Indicator
181
bdKj' If tE[O, 1] set
Since for zEK I -K 2, we can write z in a unique way as z=tm 1(z)+ (l-t)m 2 (z), we have (1_t)(~2)(Z),Z>
(~I)() >=t(~I)(Z),Z> z , z (m 2, ~1)(Z»
(mp ~2)(Z» since (mi , ~j)(z» =0. We show that Y,={z': (Z',~(I)(Z»=O} does not intersect K for zEK I -K2 so that l/IWI)(z» is well defined. Let
. { I
Y.= z': t
1Re(7' ~1(7» 1Re(z', ~(2)(Z» ,-, -(I-t) 1Re(m 2, ~1 (z» 1Re(m 1, ~(2)(Z»
} 0 .
Then Y,c ~ (we note that since we can multiply ~Ul(z) by any complex number, we may assume without loss of generality that (m2' ~1)(z» and (ml' ~2)(Z» are both real, so the inclusion is trivial). For zEK, 1Re(z, ~(1» • ~ ~ ~1) 1Re(z, ~1)(Z» z
1Re(ml,~(2)(z»>0, so -(I-t)1Re(Z,~(2)(z»>0 1Re(m1, ~(2)(Z»
.
The manifold 1'12=(Z,~I(Z» has boundary 1'1-1'2' We apply Stokes' Theorem on this manifold: A=
an-l J d ( f(z) a~n-l 1'12
'>0
(l/I(~») ) -~- Q(z,~) '>0
an- 1 (l/I(~») _ = J ozf(z) o~n-l -~- Q(z, ~)+ 1'12
'>0
'>0
n
1 (l/I(~») _ -~- Q(z, ~).
a J f(z)d~ o~n-l -
'>0
1'12
'>0
The first term is zero since f(z) is hoi om orphic. n
On 1'12' (z,~>= -1 and
L
(dZk~k+Zkd~k)=O so, since
k=1
o
this last factor is also zero and the Lemma is proved.
Proof of Theorem 8.4. Let C(2)= J.l(z") for J.lEJt'(K') and IX a multi-index of positive numbers. Then there exists C(K) such that for sup l~i~O 11 < C(K) j
the series converges uniformly on K and if (~Z)=(~1 ZI' ... , ~nzn):
~o
L (_1)121 ~ (~Z)2
~O+~IZ1+"'+~nZn"
IX!
~O
•
182
8. Analytic Functionals
of the origin
In
.
Let t/lEPo(K). In order to calculate the ClaP we consider the closed ball BR with center at the origin and radius R > Ro so that K c: Bw For all ~EjjR' the Taylor series at the origin of t/I converges uniformly on BR . Let t/I(u) = L a(a)u a. Then (a)
(l)n-1 on-I (t/I(~») ""(I r:x 1+ 1) ... (I r:x 1+n- l)a(a)~a o):n-I -):- =L... ):a+n'
(8,2)
0, and hK(uvt)=hG(vt). Hence If(uvt)exp-vtl~exp-'1ltl for some '1>0, which implies that the integral converges absolutely. Suppose If(z)I~C£exp(hK(z)+£llzll) for £>0. By the Cauchy Integral Formula, if (i = (0, ... , (, 0, ... ,0) (( in the /h place), then
so
l I~ aaf Zj
~
sup If(u 1, ... , uj +(, ... , un)1
1~1=1
c£ exp( sup hK(())(exp £) exp(hK(z)+ £ Ilzll) 1,1 :5 1
since
hdz+U~hK(Z)+hK(()'
Thus we also have the absolute convergence
of each of the integrals Saf (- u vt) exp - (v t)(v dt), and we can differentiate aZ j _ under the integral sign, so £.If(~) is holomorphic. 0
Theorem 8.9. Let feu) be an entire function of exponential type. We slIppose for some compact convex set K and for every £>0, If(u)1 ~ C£ exp (hK(u) +£ilull). Then f is the Fourier-Borel transform of an element JiEK(K)' and if £.I f(~) is the projective Laplace transform of f then n(n+ 1) -(-1)-2-
f(u)= where
(hon
a
n- 1
JK exp O, there exists UE~<X)(Q) with ou=f and supu~C",suplfl+e. K
'"
Proof Let WI be an open neighborhood of K, WI ~ Q and WI holomorphically convex in Q. Then -log du(z) is plurisubharmonic in Q, and so we can find a plurisubharmonic function cp (depending only on WI' C and f) such that cp == 0 on WI and
[f Ifl2 exp-cpd,r /2 =[J lil2 exp-cpd,+ u
J U-WI
(01
~ Co(W I) sup Iii +e/C "'l
lil2 exp-cpd,r/ 2
192
8. Analytic Functionals
where
C is
a constant to be fixed later (if c=sup( -Iogda(z», we choose y(t)
=
to be a sufficiently rapidly increasing function"" of t such that y 0 for t;£ c and set cp(z)=y(sup( -Iogda(z), c)). Then there exists a solution uEICOO(Q) of the equation = f such that
au
JluI Z [exp-cp](1 + IIzII Z )-3'dr;£ JIfl z exp-cpdr
a
(cf. Appendix III).
a
(n-2) , Let t/J(iICOO(Q) such that 1/1= 1 on K. Then if C,=~, for zEK, -1 i u(z)= C.lllz_allz,-z LJ (I/Iu)dr ="2 Cn
J
-1
-
Ilz-allz,-z aa(l/Iu) "
13.-1
i [ ( 1 )="2 C, lua Ilz-aII Z' - Z "al/l"p._1
+ l I/Ia (1Iz-a\z.-z) "f "13.-1]. Since for zEK, aEsuppal/l, liz-ail >15>0, it follows from the Schwarz Inequality that for zEK, lu(z)l;£ C(w I)[
J
lulzr 'z + C' sup If I,
l.L)l-K
Cl)t
where C(w I) depends only on WI
where C'(w l )
lu(z)l;£ C'(wl)[J Ifl z exp- cpdrr 1z + C' sup Ifl, a "" depends only on WI' since cp=O on WI' Thus G lu(z)l;£ C'(w l ) [Co(W I ) sup If I+"c ]
+ C' sup If I,
WI
Wt
D Theorem8.21. Let Ko and KI be compact sets in a domain of holomorphy Qc{:' and let L be the holomorphically convex hull of Ko u K I . Suppose that K
is such that L- K is a disjoint union of two sets M 0 and M I closed in L - K such that Kj-KcM j , j=O, 1. Then every analytic functional J1.EJff(Q)' carried by Ko and KI is carried by K. Proof Let W be any open neighborhood of K. We begin by constructing a function I/IEctt(Q), such that i) 0;£1/1;£1; ii) I/I=j on wj-w for some open neighborhoods
Wj
of K j
;
§7. Unique Supports for Domains in
cr n
193
iii) IjJ is constant on every component of U - ro for some open neighborhood U of the holomorphically convex hull of Wo u W1. Let mj=Mj-w. Then L-w=mouM 1 and mOnm1 =0. Furthermore, the mj are closed in L, hence compact. Let
m1= {ZEQ:
inf Ilz-wll
Then for some I> > 0, the sets ~. are disjoint and contained in Q. Let cxE~;(B(O, 1») be such that Jcx(z) dr(z) = 1 and set IjJ = Xm, *cx (Xm, is the CI:"
characteristic function of md. Then IjJ = j in mj and 0 ~ IjJ ~ 1. Furthermore, m~ u m~ u w is a neighborhood of L, so we can find two open neighborhoods U and V of L such that V' c U c(m~ um~ uw), where V' is the holomorphically convex hull of V. Set wj=(mjuw)n V so that WOuw 1= V; hence the holomorphically convex hull of Wo u W1 is contained in U. Since U -roc(m~um~), IjJ is constant on every component of this set. By Theorem 8.20, we can find a constant C' such that for every fEJff(Q) and every 1»0 there exists UE~;:'(Q) with au=faljJ and sup lui ~ C' sup If aljJl + I> ~ C' sup If aljJl + 1>, U
roOuCOl
co
since aljJ=o on U -ro. Now J1.(f)=J1.(ljJf-u)+J1.«l-ljJ)f+u), and since J1. is carried by Ko and K 1, we obtain 1J1.(f)1 ~ Co sup IljJf -ul + C1 sup 1(1-IjJ)f+ul "'I
"'0
~
Co sup IIjJI + Co sup lui + C1 sup 1(1-IjJ)fl + C1 sup lui co
CIJ
CJJ
Wl
because ljJ=j in wj-ro. Hence 1J1.(f)I~(Co+ C1)(sup Ifl+ C' sup IfaljJl+l»
'" and since
I>
'"
o
was arbitrary, the proof is complete.
Corollary 8.22. Let Q be a domain of holomorphy in ern and Ko and K1 carriers of J1.EJff(Q)'. Then J1. is carried by K=Kon«L-Ko)uK 1 ), where L is the holomorphically convex hull of Ko u K 1. If Ko U K1 is holomorphically convex, then J1. is carried by K 0 n K 1 . Proof Set S=(L-Ko)uK 1 ,
Mo=Ko-K=Ko-S=L-S,
M1 =(L-K)-Mo·
Then MonM1 =0, MouM1 =L-K and Mo is closed in L-K, for
(L-K)nMo=LnC (KonC Mo)nMo =Ln«C KonMo)uMo)=LnMo=Mo.
194
8. Analytic Functionals
On the other hand, Mo= Mo -K =(L-S)-K =(L- K)nC S is open since Cs is open in L-K. Finally Ko-KcMo and KI-KcCKocCM o, so KI -Kc(L-K)nC Mo=MI' We can then apply Theorem 8.21. 0 Theorem 8.23. Let Q be a domain of holomorphy in (Cn and J-LE.tt'(Q)'. If Ko is an .tt'(Q)-convex support of J-L whose boundary is twice continuously differentiable, then Ko is the unique .tt'(Q)-support of J-L. Proof We show that every convex carrier KI of J-L contains Ko. To show this, it suffices to construct for every Jf'(Q)-convex compact set KI with Ko - KI =1=0 two plurisubharmonic functions F and G continuous in Q such that i) supF;;:;O, supF>O; K,
Ko
ii) sup G;;:;O, hence supG;;:;O (where L is the holomorphically convex KouK,
L
hull of KouK I) and zrtK o, G(z);;:;O implies F(z);;:;O. If zEL -Ko then F(z);;:;O by (ii) and so by (i) sup F;;:;O. Thus sup F;;:;O, where (L-Ko)c:K,
K
K=KonI.; and I.; is the hull of(L-Ko)uKI' Hence K is a holomorphically convex proper subset of K o, since F>O somewhere in Ko. Then Corollary 8.22 shows that K carries J-L, which is a contradiction. Hence Ko is the unique holomorphically convex support of J-L. We now carry cut the construction of F and G. An essential ingredient in the proof will be the fact that the hull of a compact set K c Q with respect to the holomorphic functions, the plurisubharmonic functions, and the continuous plurisubharmonic functions in Q, is the same if Q is a domain of holomorphy. Since KI is supposed holomorphically convex, there exists a CC oo plurisubharmonic fonction G in Q which satisfies sup G 0 if and only if x is extremal. Proof Suppose that K is contained in some p dimensional subspace of 1R. 2n = CC n• If there exists eo> 0 such that the convex hull of K n C(B(x, contains x, there exist (p + 1) points of K at a distance at least eo from x such that x is in the convex hull of these points. Thus the point x is not extremal in this simplex and thus not extremal in K. On the other hand, if x is not extremal, x is contained in the interior of some line segment in K, say x=tx o +(1-t)x 1 O no' If this
This will be true if we can show that
c
last statement does not hold, since CK ~ CL n , for every n there exists a
. .
•
complex hyperplane i) ~n ( l
~n
y
of V such that
Ln =0,
ii) ~n(lK*0,
.
iii) ~n¢Wy(xO)'
.
.
If ~o is a limit point of ~n then from ii) ~o ( l K =1=0 and ~o is a supporting plane at Xo by i), which contradicts iii). Thus, Lno is a carrier of J1 which contains the origin. This is the final contradiction which proves the lemma. D
§8. Unique Convex Supports
199
Corollary 8.30. Suppose that K is strictly convex and that for each point of bdK there is only one complex tangent plane. Then if K is the support of an *
analytic functional /1, it is the unique convex support and CK is the domain of definition of CPl" Proof By translating K, we may assume without loss of generality that K contains the origin in its interior (a strictly convex set always has non*
empty interior). Then CPI' is holomorphic in CK and cannot be extended to a neighborhood of any boundary point by Lemma 8.29. Thus, if L is a convex *
*
carrier CLee K so K c L.
D
We say that a compact set K is linearly convex if its complement is a **
union of complex hyperplanes or equivalently if K = CCK. Let IP be the family of linearly convex sets. A IP-support will be said to be linearly *
convex. If /1 is carried by K, then CPI' extends to CK and if the hypotheses of Corollary 8.30 are fulfilled, K is the unique linearly convex support. Suppose that K is a convex set and V the smallest linear subspace of (Cn containing V. We shall say that K has the property (u) if ( ) {for every extremal point x of bd K, there exists at most u one complex hyperplane which has support xo' Lemma 8.31. If K is a convex carrier of /1 and /1 is carried by a complex subspace V, then /1 is carried by K n V. Proof We first show that K u V is holomorphically convex if V is a convex subset of V. Let V be defined by fp ... ,fn-q (i.e. V={z:fl= ... =fn_q=O}). Suppose KnV=I=K. For x¢KuV there exists CPx such that CPx(x)=l and cp(y) = 0 for YEV. Let M =sup ICPx(x)l. Since K is convex, there exists I/Ix such that 1/1 x(x) = 1 K
and
supll/lx(z)I: t wm(z)= { p(z)+-llzll we define m ( I,) I }PP(Z)+;;;!lZ'1 { w;"(z)= p(z)+; Ilzll '
we define an equivalent metric on E:(r).
§ I. Linear Topological Spaces of Entire Functions
203
~
Lemma 9.2. Suppose that for fEE~(r) (resp. EO), f(z)=
I
~(z) is the Taylor q= ° series expansion of f in homogeneous polynomials of degree q. Then A,.{z) \'
=
I
~(z) converges to f for the topology of E~(r) (resp. Eo)·
q=O
Pm(z)=p(z)+~ Ilzll,
Mm=sup Ifexp-p~(Pm)1 and g(z,J-)=f().z), m ~n By the Cauchy Integral Formula, we have
Proof We let ).E f for the topology on for EO is identical.
The proof D
we expand f at the ongm in homogeneous polynomials, x (CP(q)P )q/P f(z)= I ~(z). Let A~)= ~where r=cp(t) is the inverse function of q=O ep If
f
E~(r).
E E~(r),
7:
I
A~) ~(z). If we apply Theorem 1.23 to the function q= ° h().z) as a function of one complex variable, we see that the power series for h converges for II.I 1 and p(r) is an associated proximate order, we assume that in addition for all r ~ 0 i) p(r»
1
ii) :r (rP(r, -I) > o. Since both of these properties hold for r sufficiently large, there is no loss of generality. In this case, the equation r = t P(II-1 has a unique solution for all r~O.
§ I. Linear Topological Spaces of Entire Functions
205
Definition 9.3. Let p*(r)=
p(t) where t is the unique solution of the p(t) -1 equation r=t P(I)-I. We define p*(r) to be proximate order conjugate to p(r). Proposition 9.4. For p> 1, the conjugate proximate order is indeed a proximate order.
Proof We first note that lim p*(r)=-P- exists, so (i) of Definition 1.15 is r - 00 p-1 verified. Furthermore d * d p(t) (dr)-1 dr P (r)=dtp(t)-1' dt
=-
p'(t)(p(t) -1)- 2 t 2- p(r) [t p'(t) log t+ p(t) -1]- \
so
r I d *() r -t(logt)p'(t) r:~r ogr dr P r =,~~ (p(t)-1)2
0
by (ii) of Definition 1.15. Thus, the same property is verified for p*(r).
0
p-I
F or p> 1, se t A p (p - p1)-p- an d FP*(r) Ap' -
U EP*(r) Apm' m
Theorem 9.5. The mapping J1--+ ji(u) is a one-to-one linear mapping of (E:(r))" (resp. EO), onto . i) Fi:~! for p> 1 ii) the set Q:!r) of formal power series at the origin which satisfy (9,5) for some m for p < 1 (resp. the set Qo of formal power series at the origin which satisfy (9,5) for some p > 0 for (EO)'),
Proof Of course (ii) is just a restatement of facts already observed, so we must only verify (i). Since A 1/q= 0 be given and let 00
00
G(u)=
L
q=O
Rq(u),
H(u)=
L
q=O
00
Pq(u),
and
F(u)=
L
q=O
Tq(u)
with s the smallest integer such that T,.(u) $0. We choose mo so large that (9,5) holds for both H(u) and F(u) for m~mo' Thus, there exist constants C 1 and C 2 such that for m~mo+1 (since Pm(u)~Pmo(u)+'1llull for some '1>0) IPq(u)1
~ C [P~(u)Jq (qJ~~prp (~r
Tq(u)1
~ C [P~(u)]q (qJ~~prp (~r
1
Since ~+s(u) =
L
l+k=q
1
2
Rl(u) Tr.+s(u),
Rq(u) = T,.-I (u) [~+s(u) -
L
Rl(u) Tr.+s(u)].
l+k=q l*q We now show by induction that there exist constants Kq with K q_ 1 ~ Kq and Kq=K q_ 1 for q~q such that for c5>0
~ K [P' (u)]q(l +c5)q q (qJ(q)p)q,p (_e_) (qJ(q))s+ I. q q m ep q+s q For q=O, by Lemma 9.9, we have q+s IRq(u)1 ~ C 2 Kb[P~(UW(l + b)q (qJ(q +S)p)-p- (_e_)q+s ep q+s IR (u)1
§2. Theorems of Division
209
and if «r)=rl-p(r), we have [1P(q + S)]q+s = q s q+s = [1P(q)]q+s [(rq+s)]q+s q+s (r + ) q (rq)
~(1 +b)q+s [1P~q)r+S+1 for q sufficiently large. We now assume the conclusion for q ~ qo-1. Then by Lemma 9.9 IRqo(u)1 ~ I T.(U)I-I [1~o+s(u)1 +
L
I+k=qo I*qo
IRI(u) T,.+s(u)IJ
We assume that the function (r)=rl-p(r) increases. Since this holds eventually, we lose no generality. For simplicity, we let i=k+s, j=qo+s, IX
( 1 )-1 P -P Now sincej'=r!'(rj ) and (/IU)=r.
2'
J"1'
J'
p. = [(r}] -I [(r}] - i [ 1P(l)IIP~il] IPUP II i' (r ) (r;) l
Suppose for the moment that i ~ 3j/4. Then
Thus
if 1+ i = j.
210
For
9. Convolution Operators in Linear Spaces of Entire Functions i~241l,
we have
4
T//4]i>.' ['T' / ]'.' [ -((r}]i ~ [ 1 + ~ 1 + I r.~ (Jr,.)" + ... + K Tli' ~ r ((rJ ((rJ" I
/
where ,-' ~ 31l (since ((rJ = O(iI/2H) for G>0). For i ~ 24 !X + 1 and {3 = 2 max ((rJ, ( (r.)] i
((r.)
i
~ 242
we have [ ~ ~ -{3J ~ 1 for qo, and hence j, sufficiently large, since drJ p < I. By symmetry, similar inequalities hold when we replace i by I. We K (1 + i5)S . . . choose qo so large that b )3 «3q6)-I. Thus, SInce for qo suffIcIently r(qo+s large, either I or (k + s) is greater that 241l if 1+ k = qo, we obtain {
1+ I+tqo Kb(l I *qo
l and 1X,,=I1*v). a sequence in the image such that IX" -+a weakly. Then cx).(u)=,u(u)v;.(u) as entire functions. For ZE(Cn, let GC
00
L ~(z,u'),
L ~"(z, u'),
q=O
q=O
be the Taylor series expansions of jl(u), cx,,(u) and v;.(u) respectively at the point z. Suppose s (depending on z) is the smallest integer for which T.(z, u') $ O. Then as above, R;(z, u') approaches a limit Rq(z, u') for all i. and Rq(z, u') = [T.(z, u')] - I [~+s(z, u') -
L
R/(z, u') 7;.+s(z, u')].
I+k=q '*q
L
We now show that Rq(z, u') converges in a neighborhood of u' =0. Let LI (z, r) be a polydisc with center z and polyradius r such that on A={u: IUj-zjl~rj,j=l, ... ,n-I, lun-znl=rn}, T.(z, u') =1= 0 and set
~ = mln IT.(z, u')I. There exists a constant C ~ I such that
for uELI(z,r), 1~(z,u')I~C2-q, 1~(z,u')I~C2-q. We show by induction that on L1'=L1(z,r/2), IRq(z,u')I~(4CK)q. We have by the Cauchy Formula
f
IRq(z, u')I;;;;
I[~+s(z, u~, ... , u~_ l ' ~n) .
I~n-unl='n
- L
R,(z, u~, ... , U~_I' ~n)]
I+k=q I*k
X
T.(u~, ... ,U~_I'~n)-I(~n-u~)-ld~nl
212
9. Convolution Operators in Linear Spaces of Entire Functions
so for q = 0 the result is immediate, and once it is established for q - 1 we ~ili~
00
IRq(z,u')1~2K(2CK)q-l
L
(W~(4CK)q.
n=O
~(u) = F(u) is actually an entire function, and by Theorem 9.11 and Jl(u) Theorem 9.5, F(u)=v(u) for vE(E:(r), so the image is closed and the TheoThus
D
~~~~
§ 4. Supplementary Results for Proximate Orders with p> 1 We will show that for strong proximate orders that we can improve the precision of our results. This will stem from Proposition 1.22 which says that for a strong proximate order, rP(r) is a convex increasing function of r, so if we compose with a plurisubharmonic function, the result remains plurisubharmonic. In particular if p(z) ~ 0 is a support function (i.e. p(t z) =tp(z), t>O, P(ZI +Z2)~P(ZI)+P(Z2»' then we can write p(z)=sup1Re(z,u) ueK
for some convex compact set K, so p(z) is plurisubharmonic and p(z)P(P(Z)) is plurisubharmonic also. Note that for p(z) a complex norm, then logp(z) is plurisubharmonic and (p(z»P(P(z)) is also plurisubharmonic for every p (by Proposition 1.22). This goes a long way in explaining why one must take a complex norm for p < 1 but only a positive support function for p> 1. Let
Pm(z)=p(z)+~ Ilzll. Then Km={z:Pm(z)~l} m
is a compact convex set
and so p;"= sup 1Re(u, z) is also a positive support function and k~m.
n co
ueKm
We let E:(r)=
B!~
with
wm(z)=Pm(z)P(P~(Z))
p~~p;"
for
and F;tr) = UB!. with
m=l
m
w;" = p;::(P;"). We equip E:(r) with the projective limit topology, so that it becomes a Fn!chet space, and we equip F;(r) with the inductive limit topology. Then (F;(r)" the dual space of continuous linear functionals, is
n
just (F;(r) = '" (B!.), and if we equip m=l
(9,6)
(B!. )' with the dual
m
topology
m
Ilvllm=
sup
Iv(f)I,
/EB*w m
IIJIIBw~=
1
then we can give (F;(r), the projective limit topology, under which it becomes a Fn!chet space. Lemma 9.13. Every element (xE(E:!r), can be represented by a measure Jl such that Jexp w;"dlJlI < + 00 for every m.
Proof We recall that C!;.. is the space of continuous functions k(z) such that lim Ik(z) exp -w;"(z)1 =0. A Cauchy measure Vy is integration on a rectifiiizi!-:x.
§4. Supplementary Results for Proximate Orders with p> I
213
able curve), contained in some complex line. We note that the closure of the linear subspace spanned by the Cauchy measures is just (B!J\ since if f is continuous and vy(f) = 0 for every Cauchy measure, f is holomorphic in every complex line by Morera's Theorem and hence f is globally holomorphic by Hartog's Theorem (cf. [B]). Note that Ilvllm+1 ~ Ilvll m in general. Let J1.1 represent cx in (Bw)' and let J1.~ represent cx in (B wi )" Then the measure (J1.~ - J1.1) is orthogonal to B w \' so we can find a finite linear combination of Cauchy measures v 2 such that 11J1.~ -J1.1 -v 2 11 1 < 1/2. Set J1.2 = J1.~ -v 2· We choose by induction J1.1' ... , J1. m-1 such that IIJ1.m-1 -J1. m- 2 1I m- 2 < 1/2m- 2 • Then we can find J1.~ which represents cx in BW'm' and we can find a finite linear combination of Cauchy measures vm such that IIJ1.m-vm-J1.m-Illm-1 1) and p*(r) is its conjugate proximate order, then p*(r) is also a strong proximate order. We leave this simple calculation to the reader.
Theorem 9.16. The Fourier-Borel transform establishes an isomorphism between the spaces i) (E~(r)' and F:;,~(r) and between the spaces ii) (Fp~(r)' and E~p*(r) where r
p
A -I p'
(p -1)(P-l)/p
vE(E~(r)'. Then by Lemma 9.13, there exists an m such that Iv(f)1 ~ C msup If (z) exp - Pm(z)P(Pm(ZHI. Thus
Proof Let
z
Ifv(u)1 ~ C msup lexp (u, z) -Pm(z)p(Pm(ZHI z
~Cmexp(sup(
sup {1R.e(u,z)t-t P(t)})
t;:;O Pm(z)=t
~
C mexp sup (p~(u)t -tp(t). t;:;O
Now
:t(P~(U)t-tP(t)=P~(U)-(P'(t)IOgt+P~t»)tP(t)
and since p(t)-+p and
t p' (t) log t -+ 0, it follows that for large values of Ilu II, this function takes on an absolute maximum. For (j>0 and Ilull sufficiently large (depending on (j), the maximum occurs at
tP(tu)-1 u
~ {( P, (U)P(tul-1 m
=
p~~u)
p+~(u)
1
p+~(u)
for
1~(u)I I
215
which is less that or equal to [(r+l:)p~(u)]p*(k(u)p;"(U)) where 1:--+0 as 1 and p(z) a non-negative support function and suppose that f" has minimal order with respect to p*(r). Then the convolution equation P(x)= f has a solution gE(E:(r,)'. Proof The map v -+ Jl* v is one-to-one and has closed image as one sees easily by repeating the second half of Theorem 9.12. We need only apply Proposition 9.8. 0
§5. The Case p= 1 In what preceeds, we have considered proximate orders for p < 1 and p> 1. The case of proximate orders for p= 1 is extremely delicate since for p=l= 1, either rP (r)-1 increases or decreases but for p= 1 this becomes problematical, and the theory and calculations become impossible without making additional assumptions. In a certain sense, this also translates the central role that the exponential functions play in the theory. We shall not consider proximate orders but we shall assume that p == 1. Let p(z) be a support function and Ep the Frechet space of functions that we get by setting
wm=P+~ 114 m
If
JlEE~ we
define its Fourier-Borel trans-
form by f,,(u) = Jl(exp (z,u». Ifr:=v*Jl is the convolution of the measures Jl and v, then ft(u) = fv(u)f,,(u). Suppose that g is any function holomorphic in a neighborhood of K. Then g defines a continuous linear operator Sg from Ep into Ep by Theorem 8.9: "("+ 1)
_(_1)-2(2 ')" 1t1
J g(z)exp(z,u)~_n_l 0"-1 E(w,
(
C;o
(i!F(~»)
- "
--]: Q(z, c.),
Kcw
'00
via the projective Laplace transform.
Lemma 9.19. Let 1/1 Zo = i!exp(zo. u) for zoEK. Then the linear functional on Jf'(K) determined by I/Iz o is Tt/lz o =b(zo), the Dirac measure with support zoo
Proof Let f be a representative of jEJt'(K) defined in some strictly convex neighborhood w of K. Since w is a Runge domain, f can be uniformly approximated by polynomials in an open neighborhood of K, and since 'r 1y approximate . d by exponentials. · ~i).-1 . CC f can be unlIorm Zi= 11m --.-, I.E, P·I-O
....
Since
"("+1,
F(u)
-( -1)-2(21ti)"
J
E(w,
;;0-1
exp(z,u) 0]:0-1 '>0
(i!F(C;») Q(z,e), __ -]:-
'>0
218
9. Convolution Operators in Linear Spaces of Entire Functions
is just f(zo) for the exponentials. It now follows from the uniform 0 convergence in a neighborhood of K that T",zo = f(zo) for all fE.Yt'(K). T",=o
Lemma 9.20. Let vEE~. If fv is its Fourier-Borel transform, then the linear operator QJ.: Ep-+Ep is just the transpose of the convolution v* /1 (i.e. (QJJF), /1) = (F, v*/1».
Proof We can represent /1 by a measure such that /1 exp(p(u)+ellull) has bounded mass. Thus from Fubini's Theorem, n(n+1)
-( -1)-2/1(F(u» = (hit
on-l rt /1(exp ... , Yn) is a support function. We define the set B: (resp. B:*) to be the Banach space of all fEBg (resp. ~) which satisfy IlfII~=
J If (x)IPdx < 00
1 ~p
_ "/2 =a i;; . ,p () rm m
But this is a contradiction, so hj(z) ~ hJ(z). By noting that /(z) = Jf(z+w)drx(w),
§ 7. Convolution Operators in (["
we can reverse the roles of ht(z) = hj(z).
f
and
j in the above calculations. Thus
ii) Suppose now that f is of regular growth along the ray Let I] be so small that for r> Rq (9,9)
II!(z',b)-I!(w,b)I~~/8
227
t
w,
WES 2n - 1•
for Ilz'-wll R~, we can find w~ with Ilw~ -wll R~
and
bO, by the uniqueness of analytic continuation, P(f) is holomorphic in Q.
Theorem 9.35. Let
~Eyt(Cn)' be carried by the compact convex set K and let
~~ be the Fourier-Borel transform of ~. If h"} ~ (~)=hK(~) and JT(~) is of Il
regular growth in (Cn (with respect to p =- 1), then for g E JIf (Q), there exists a solution jEJIf(Q+K) of the equation {l(rx)=j. If Qc:(Cn is a bounded strictly convex domain with 2 boundary, then (l: JIf(Q+ K)->JIf(K) only if h"} =hKm and ~(~) is of regular growth in (Cn. ~
ee
m
228
9. Convolution Operators in Linear Spaces of Entire Functions
Proof For rt.EYi (D)" let ~(~) be its Fourier-Borel transform. Then if pt(rt.) is the transpose operator of Ji, ~I(.)(~)=~(~)~(~)' Suppose that pt(rt.J converges weakly to an element /3EYf(Q + K)'. Then 3'p(~)=~(~). G(~) for some entire function G(~), since the Taylor series of 3'pm at each point (E I:
~
1
where dW m is the Lebesgue measure on the unit sphere sm -I and Wm is the total mass of sm-I. We denote by S(Q) the family of subharmonic functions in Q. If cp and - cp are subharmonic in Q, we say that cp is harmonic in Q. Remark. If cp is subharmonic in Q and r < dg(x), then cp(X)~T';; 1 r- 2m
S Ilx
cp(X+X')dTm(X')=A(x,r, cp)
;iir
where dTm is the Lebesgue measure in IRm and Tm is the total mass of the unit ball R(O, 1).
Definition 1.2. Let Qc 2, of uz(r) =logr if m=2. If QcCC n and cpEPSH(Q)n~Z(Q), then i.(z, r, cp) and A (z, r, cp) are increasing with r and convex in log r. Proof It follows from (1,1) that (m-2)r
m-l
ci. ci.(x, r, cp) -;;-(x,r,cp)= ~ ( cr CUm r)
is increasing, which shows the first part for m>2. For m=2, we have
ci. r -;;- (x, r, cp) cr
ci.(x,r,cp) cUz(r)
-~-'-------
Appendix I. Subharmonic and Plurisubharmonic Functions
233
If QczEPSH(Q) (or S(Q)). the set {z: C!>!(z)= -00 or C!>z(z) = -oo} is of measure zero by Proposition 1.9. Thus, tCPt +(1-t)CP2 is not identically - 00, O~t ~ 1, and hence is in PSH(Q) or S(Q). 0 Definition 1.11. A subset E c Q, a domain in 1Rm (resp. CC n) is said to be polar (resp. pluripolar) if there exists cpES(Q) (resp. PSH(Q)) such that E c {x: cp(x) = - oo}. Corollary 1.12. A (pluri)polar set in a domain QcCC n is of Lebesgue measure
zero. PropositionI.13 (Maximum Principle). Let Qc1Rm be a domain and cpcS(Q). Let m=sup cpo If there exists xoEQ such that cp(xo)=m, then cP =m. Q
Proof If B(xo, r)cQ, then m=cp(xo)~A(x, r, cp)~m. Thus, cp(x)=m in B(x o, r), for otherwise, by the upper semi-continuity of cP there would exist e>O and an open subset U of B(xo, r) for which cp(x)<m-e on U and so A(x,r,cp)<m. Thus, the set M={XEQ: cp(x)~m} is open, and it is closed since cp is upper semi-continuous. Since M =1= 0, M = Q. 0 PropositionI.14. Let cp(z,t) be a real valued function of ZEQcCC n and tET, T a locally compact space. Let j1 be a positive measure on T. Suppose that i) z-+cp(z,t) is plurisubharmonic in Q and (O,t)-+cp(z+we i6,t) is dOxdj1 measurable. ii) For every compact subset K c Q, there exists a constant M (K) such that cp(z,t)~M(K) for every tET. Then I/I(z) = Jcp(z, t)dj1(t)EPSH(Q) or is identically - oc.
Proof We shall verify properties i) and ii) of Definition 1.2. Let I/I*(z) = lim sup I/I(z) be the upper regularization of I/I(z). z'-z
Then there exists a sequence Wq such that wq-+O and
I/I*(z) = lim I/I(z+ wq)= lim sup Jcp(z+ wq, t)dJ1(t). q-oc·
q-x
Appendix I. Subharmonic and Plurisubharmonic Functions
235
From Fatou's Lemma and the uniform bound for cp(z+wq , t), we obtain
J
J
1/1* (z);£ lim sup cp(z + Wq' t)dl1(t);£ cp(z, t)dl1(t) = 1/1 (z), q~
00
where the second inequality stems froms the semi-continuity of cp(z, t) for fixed t. Thus, I/I*(z)= 1/1 (z), and so I/I(z) is upper semi-continuous. To show ii) we observe that
J
I/I(z) = cp(z, t)dl1(t);£J dl1(t)
2"
J cp(z+we o
i6 ,
dO t)-2 1t
for every disc {z+uw: lul;£r} contained in D. We then conclude from the measurability in dl1xdO that dO 1 l/I(z);£J dl1(t)-2 cp(t+we i6, t)=-2
J
2"
J l/I(z+we
i6 )dO.
1t 0
1t
D
Remark. Proposition 1.14 remains valid for cp(x, t) subharmonic in xeD for te T, where we replace condition ii) by condition ii'): (a, t) -+ cp(x + ra, t) is dwmxdl1 measurable. Let a(x)efCcf(B(O,I» such that a(x)~O, a depends only on Ilxll and a(x)drm= 1. We consider the positive functions a.(x)=c ma(x/e), which
J
form, as e tends to zero, an approximation to the Dirac measure with point mass at the origin.
Proposition 1.15. Let cpeS(D) (resp. PSH(D» and set
J
CP.(x) = cp*a.(x)= cp(x + x')a.(x')dr(x'). Then
i) cp.(x)eS(DJ n fcx (D,) Crespo cp,(z)e PSH(D,) n fcOO (D,)] where D.= {x: dQ(x»e} ii) cp.(x) is an increasing function of e for e < dQ(x) and lim cp,(x) = cp(x). ,~o
Proof By ii) of Definitions l.l and 1.2 we obtain cp.(x)~CP(x) for e -;)->0. Since M",(z,ul)=m, we obtain l
u\
UI
=Iogb(z,m).
From the Implicit Function Theorem, we obtain
and hence
where
so in this case, -log b(z, m) is plurisubharmonic. For the general case, we let CPv(z) be a sequence of CC'" plurisubharmonic functions which decrease to cp and let b.(z,m) be the associated functions on a domain Q'~Q, for zoEQ'. Then -Iogb,,(z, m) decreases to -Iogb(z, m), which is plurisubharmonic in Q' by Proposition I.3. 0
Appendix II. The Existence of Proximate Orders
Theorem 11.1. Let M (r) be a continuous positive function for r > 0 such that . log M(r) . . hm sup --I -= P < + x. Then there eXists a strong proximate order p(r) r~ C/O og r such that M(r);£rP(r) for all r>O and M(rm)=r~(rm) for an increasing sequence of values rm tending to + 00.
log 1/1 (X) for which is possible by (iii). Then there exists a e\m)O such that IIYI12~CIIA*YIII for every YEFnDA*. Then for vEHln[A-I(O)F, there exists wEDA* such that A*w=v and Ilw112~ Cllvll l · Proof Since A is closed, if xnEDA' Xn -+XO' and Axn=O, then xoEDA and Axo=O, so A-I(O) is a closed subspace of HI. Now Ax=O is equivalent to 2 = J(J/j (::)) e-2E~1(Q) and n
e-3)' Proof Since Pm has compact support for all m and is in ~oc, aPm A f and fJmaf have coefficients in L~o, 2)(q>3) if f E...PB· Furthermore, B(Pmf)-Pm(Bf)=oPmAf, and from (III,7), we have IB(Pmf) - pm(BfWe-dr ~ I for t:vt:ry r, and so iui 2 e-