Analyse Complexe: Proceedings of the Journées Fermat - Journées SMF, held at Toulouse, May 24–27, 1983

192 Pages·1984·3.765 MB·English-French

by  E. Bedford, S. Bell (auth.), Eric Amar, Roger Gay, Nguyen Van Thanh (eds.)

Lecture Notes ni
Mathematics
Edited by .A Dold and .B Eckmann
4901
esylanA exelpmoC
Proceedings of the Journ6es Fermat - Journees ,FMS
held at Toulouse, yaM 24-27, 1983
Edit6 par .E Amar, .R Gay et Nguyen Thanh naV
III
galreV-regnirpS
nilreB Heidelberg New York oykoT 1984
Editors
Eric Amar
Roger Gay
Unite associee CNRS 226, Universite de Bordeaux I
35t Cours de la Liberation, 33405 France Talence,
Thanh Nguyen naV
Laboratoire Complexe d'Analyse Analyse et Fonctionnelle
Universite Paul Sabatier
811 Route de Narbonne, 31062 Cedex, Toulouse France
AMS Subject Classification (1980): 32 22, A 32 99, D 32 25, E 32 30, E
32F99, 32H99
ISBN 3-540-13886-2 Berlin Heidelberg Springer-Verlag Tokyo New York
ISBN 0-38743886-2 Heidelberg New York Tokyo Berlin Springer-Vertag
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© by Springer-Verlag Berlin Heidelberg 1984
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PREFACE
Le Colloque de Toulouse en Analyse Complexe a eu lieu
l'Universit@ Paul Sabatier du 24 au 27 Mai 1983, organis4 autour des
deux axes suivants :
)I Th4ories du Potentiel Complexe : capacit4s, 4quations
de Monge-Amp6re Complexe, probl6me de Dirichlet,...
2) Etude des fonctions holomorphes : noyaux reprodui-
sants, r4solvant l'@quation ~ , extension et division,...
L'analyse complexe en plusieurs variables est relati-
vement jeune en ce sens qu'elle a pris un nouveau d4part avee l'in-
troduction de noyaux "presque" explicites reproduisant et r4solvant
l'4quation ~u = f, en 1969-1970, par Ramirez de Arellano, G. Henkin,
I. Lieb, H. Skoda et d'autres depuis.
Au cours du Colloque, on a pu constater les progr6s ef-
fectu4s depuis : presque toutes les questions "naturelles" sont r4so-
lues pour les domaines strictement pseudo-convexes born4s de {n ; en
voici quelques exemples :
- solutions de l'4quation u~" = f avec estimations cri-
tiques ;
- division et extension de fonctions holomorphes avec
contr61e de croissance.
- existence de fonctions int@rieures.
- "bonne" d4finition d'une capacit4 complexe.
- solutions ~ C de l'4quation de Monge-Amp6re complexe
quand le second membre est a C et non nul et bien d'autres r4sultats
encore, ce qui fait que d4j~ les domaines faiblement pseudo-convexes
et les domaines non born4s, sont 4tudi4s efficacement par de nombreux
chercheurs.
Nous exprimons notre gratitude envers les 4tablissements
et organismes qui ont bien voulu financer le Colloque :
Universit4 Paul Sabatier de Toulouse
Universit4 de Bordeaux I
Soci4t~ Math4matique de France
Direction de la coop4ration et des relations
internationales
IV
Nous ne saurions oublier l'aide d4vou~de Pierre Bonneau,
Anne Cumenge et Ahmed Z4riahi. Qu'ils en soient remerci4s ici.
Enfin nous remercions la Maison Springer qui a bien vou-
lu accepter de publier les Actes du Colloque dans sa s@rie
"Lecture Notes in Mathematics", favorisant ainsi une diffusion large
et rapide.
E.A., R.G. et N.T.V.
LISTE DES CONFERENCES
(par ordre chronologique)
IDRAM 42 IAM
I. LIEB (en collabo, avec )EGNAR : Formules int~grales dans la th~orie du pro-
em~lb ~ -Neumann.
.E DROFDEB : Boundary behaviour of proper holomorphic correspondances. (*)
S. LLEB : Boundary behavior of holomorphic mappings dna B-N problem. (*)
oG YKSSUOLD : Surfaces ed Kato.
J. BRUNA : Id~aux de type fini dans les alg~bres Am(D). (*)
.P ed SIEMOLOTRAB : Fibr6s positifs et applications harmoniques.
IDERCREM 52 IAM
emM .M HAKIM (en collabo, avec .N )YNOBIS : Valeurs au bord des modules ed fonc-
tions holomorphes.
Chr. NAMLESIK : Monge-Amp6re ne dimension finie. (*)
J.J. NHOK : ehT Dirichlet Problem for the complexe Monge-Amp~re Equation.
L. TREPMEL : Solving the degenerate Monge-Amp~re Eauations with one concentra-
ted singularity.
L. NAMURG : Prolongement d'ideaux. Applications. (*)
LE RIM : Prolongement des courants positifs ferm~s.
hT MOOLB : Interpolation polynominale des fonctions analytiques. (*)
IDUEJ 62 MAI, matin
B.A. ROLYAT : Comparaison of emos capacities in C n. (*)
J. SICIAK : Highly non continuable functions on polynomial convex sets. (*)
.U LLERGEC (avec remarques par P. )GNOLEL : Discontinuities of the complexe
er~pmA-egnoM Operator, (*)
IDERDNEV 72 IAM
K. HCIREDEID : Proper holomorphic mappings with fibers of positive dimension.
J.P. YLLIAMED : Propagation des singularit6s des courants positifs ferm~s. (*)
.hP REITNEPRAHC : Z~ros sed fonctions ed la classe ed Nevanlinna ne plusieurs
variables. (*)
J. AREDREV : Quelques r~sultats d'approximation par des modules holomorphes. (*)
emM .H TNERUAL : Th~or~me ed Plemelj et th~or6me ed Bochner sur les vari~t~s
ed Stein. (*)
J. ELRETSE : Continuit~ des caract6res d'une alg6bre de Fr~chet et applications
de Bieberbach, (*)
seL conferences marquees d'un ast~risque ont donn~ lieu aux textes publi~s
dans ce volume.
DES LISTE PARTICIPANTS
RAMA Eric (Bordeaux, France)
ED SIEMOLOTRAB Paolo (Florence, Italie)
DROFDEB Eric (Indiana, Etats-Unis)
LLEB Steve (Princeton, Etats-Unis)
UOTTAYNEB demahoM daassaL (Tunis, Tunisie)
NOSSTDNREB oB (G~teborg, )edeuS
MOOLB samohT (Toronto, )adanaC
IMANOB Aline (OrlEans, France)
UAENNOB Pierre (Toulouse, France)
ANURB Joaquin (Barcelone, )engapsE
SEUGRUB Josep (Barcelone, )engapsE
OLLITSAC naoJ (Barcelone, )engapsE
LLERGEC nabrU (Uppsala, )ed6uS
ITNEPRAHC RE Philippe (Orsay, France)
TAMUAHC seuqcaJ (Orsay, France)
TELLOHC Anne-Marie (Orsay, France)
TEPUOC Bernard (Marseille, France)
IFUC Julia (Barcel one, )engapsE
EGNEMUC ennA (Toulouse, France)
NONEUGED Charles (Toulouse, France)
EHGNALED Richard ,dnaG( Belgique)
YLLIAMED Jean-Pierre (Grenoble, France)
JDIRRED lhkaM ouf ,neuoR( France)
ZARTED Jacquel ine (Marseille, France)
HCIREDEID Klas (Wuppertal, lA )engamel
DINI Gilberto (Florence, Italie)
YKSSUOLD segroeG (Marseil le, France)
TLUAEBLOD Pierre (Paris, France)
LE RIM Hassine (Nancy, France)
ENIMRE J.L. (Bordeaux, France)
ELRETSE J. (Bordeaux, France)
REIRREF J.P. (Nancy, France)
IKSNEZULAG .G (Bordeaux, France)
YAG .R (Bordeaux, France)
TECRAG Andre (Liege, leB gique)
REGNARG .M (Bordeaux, France)
NAMURG .M (Marseil le, France)
llV
EDNEYOKOMI A. (Toulouse, France)
IHCIAHCAH demahoM Salah (Alger, Alg~rie)
MIKAH Monique (Orsay, France)
TLUANEH .hC (Bordeaux, France)
NAIRAHCAKREK (Nancy, France)
NAMLESIK Christer .O (Uppsala, Suede)
NHOK Joseph J. (Princeton, Etats-Unis)
FEIRK Pierre-Henry (Marseille, France)
ICCUDNAL Mario (Florence, Italie)
TNERUAL Christine (Paris, France)
GNOLEL Pierre (Paris, France)
TREPMEL Laszlo (Budapest, Hongrie)
LIEB Ingo (Bonn, Allemagne)
RAFFOG-TEBMOL Josette (Li6ge, Belgique)
OTSENUOL Pertti (Espoo, Finlande)
LIREM A. (Bordeaux, France)
NEYUGN Thanh naV (Toulouse, France)
ZARREVON Philippe (Nancy, France)
AGETRO Joaquin (Barcelone, Espagne)
TEUQAP J.C. (Mons, Belgique)
NOSSRETTEP Mikael (Stockholm, Su6de)
GULFP Peter (OsnabrUck, Allemagne)
ZEIMMOP Michel (Toulouse, France)
NIOBAR P. (Nancy, France)
VONADAMAR Ivan-Pierre (Sofia, Bulgarie)
SOOR yuG (Tunis, Tunisie)
YASOR Jean-Pierre (Marseille, France)
IGGAVLES OIRECEMIRP Angela (Florence, Italie)
YNOBIS Nessim (Orsay, France)
SICIAK Josef (Cracovie, Pologne)
ADOKS Henri (Paris, France)
RABBES .A (Bordeaux, France)
NEMMOS Franciscus (Gand, Belgique)
IUOAHLAT Abdellah (Marseille, France)
ROLYAT B.A. (Michigan, Etats-Unis)
SAMOHT Pascal (Paris, France)
AREDREV Joan (Barcelone, Espagne)
ONIHSOY OINUK (Gand, Belgique)
IHAIREZ demhA (Toulouse, France)
ZCIWONYRWA£ J. (~odz, Pologne)
ELBAT SED SEREITAM
Eric Bedford et Steve Bell
Holomorphic correspondences of Bounded Domains in n C ...................... I- 41
samohT Bloom
nO the convergence of Interpolating Polynomials for Entire Functions ...... 15- 91
Joaquim Bruna et Joaquim .M Ortega
Id~aux ferm~s ed type fini dans les alg#bres Am(D) ........................ 2o- 82
Urban Cegrell
nO the Discontinuity of the Complex er~pmA-egnoM Operator ................. 29- 13
Philippe Charpentier
Sur les z~ros sed fonctions de type Nevanlinna dans le bidisque ............ 32- 34
Philippe Charpentier et Aline imanoB
Estimations sed (1-1) courants positifs ferm~s dans les domaines ed 2 C .... 44- 25
Jean Pierre Demailly
Sur la propagation sed singularit~s sed courants positifs ferm~s .......... 53- 46
Jean Esterle
em@lborP ed Michael et fonctions enti@res ed plusieurs variables complexes 65- 38
Lawrence namurG
Solutions of Difference Equations with Non-constant Coefficients .......... 84-138
Christer .O Kiselman
Sur la d~finition ed l'op~rateur ed er~pmA-egnoM complexe ................. 139-15o
Christine Laurent-Thiebaut
em~ro~hT de Bochner sur enu vari~t~ de Stein .............................. 151-161
namroN Levenberg et B.A. Tayl~r
Comparison of Capacities in C ............................................ 162-172
Josef Siciak
Highly Noncontinuable Functions in Polynomial Convex Sets ................. 173-178
Joan Verdera
Uniform approximation yb solutions of higher order Cauchy-Riemann equations 179-184
Holomorphic Correspondences of Bounded Domains in ~n
Eric Bedford and Steve Bell
1. Introduction
Let ~ , Dcc {n be open sets, and let ~T : 9 × D ÷ ~ ,
~D : ~ × D + D denote projections. If V c ~i × D is a complex sub-
variety, then we consider the associated multiple-valued function
f ---o D given by f(z) : ~YD~l(z) This is a natural generaliza-
tion of the usual definition of a single-valued function and V will
be referred to as the graph of f , and we write V = f'l
If ~ : V + ~ is proper, then f : ~[ -o D is upper semicontin-
uous, i.e. for 0 z c 9 and 6 > 0 , there exists s > 0 so that
f(z) c {w c D : dist(w,f(z0) ) < 6}
for all z E {z c ~ : Iz-z01 < E) (see [15]). The inverse correspon-
dence f-i : D -~ ~ is defined as the correspondence whose graph is
F-i : { (W,z) < D × q~ : (z,w) { Ff}
f
A correspondence is proper_ if and only if both ~ : V Q and
~D : V ÷ D are proper. In this case both f and f-i are upper
semicontinuous.
A correspondence f : ~ -o D is irreducible if Ff is an irre-
ducible variety. If f is proper and if ~ and D are connected,
then there are finitely many irreducible, proper correspondences
3'f : ~ ~-- D , j = 1,2,...,k such that if : I~fl u...u ['fk Corre-
spondences have the following properties, which are valid if ~. is a
3
complex space (see [16]).
Proposition i.i. Let f : ~i o-- q2 ' g : 2~f o-- ~3 ee__b proper corre-
spondences. Then the set-valued mapping gf(z) = g(f(z)) defines a
Rroper, holomorphic correspondence gf : i]~ -~ Q3 " Further, the iden-
tit[ map i~2 is an irreducible component of both g -i g and ff-i
Holomorphic correspondences were studied in a rather general con-
text by K. Stein [16]. For instance, correspondences give a useful
generalization of meromorphic mappings. Correspondences also arise
naturally in some algebraic-geometric constructions (see [9]). Here
we consider the special case of correspondences between bounded
domains, which rules out many kinds of singular behavior. In partic-
ular, dim f(z) is both constant and equal to zero, i.e. these are
"Uberlagerungskorrespondenzen" (cf. Rischel [13]).
Here we give some elementary facts and examples of correspon-
dences. These are given in Sections 2 and 3. In Section 4 we discuss
briefly some results on the boundary regularity of correspondences,
obtained in [4,5]. One tool used for boundary regularity is the
Bergman kernel function, and we show in Section 4 how the kernel func-
tion transforms under proper correspondences.
Our principal motivation for studying correspondences has been
in the factorization of a proper mapping h : ~i ~ D (see [2]). That
is, correspondences are used to prove the following.
Theorem 1.2. If 9 is simply connected and has C , strongly pseudo-
convex boundary, then there is a subgroup I ::c Aut([!) such that
)i( hg = h for all g ~ ! , and
(ii) h-lh )z( : u g(z)
(This result is proved in [5]; and an easier proof is given in [4]
under the assumption that ~.:i; is also real analytic.)
In case ~ fails either to be simply connected or strongly
pseudoconvex, the group i need not exist. In this case, however,
we have a set of correspondences
Aut~(f~) : {proper correspondences T : ~ o-- i;i such that hT : h}
It follows that Aut~(~) satisfies (i) and (ii) above. Furtherr
Aut~(~) yields some understanding of the mapping h : ~ * D as a
branched covering space, since it plays in some sense the role of the
group of covering transformations.
In Section 5 we discuss the set Aut~([~) under the operation of
composition. In general, Aut~(~) does not form a group but has the
structure of a graph on the finite set h-lh(z )
0