Complex Convexity and Analytic Functionals

171 Pages·2004·5.62 MB·English

by  Mats Andersson

Progress in Mathematics
Volume 225
SeriesEditors
H. Bass
1. Oesterle
A. Weinstein
Mats Andersson
Mikael Passare
Ragnar Sigurdsson
Complex Convexity
and
Analytic Functionals
Springer Basei AG
Authors:
Mats Andersson Prof. Mikael Passare
Department of Mathematics Department of Mathematics
Chalmers University of Technology Stockholm University
41296 Goteborg 10691 Stockholm
Sweden Sweden
e-mail: [email protected] e-mail: [email protected]
Ragnar Sigurdsson
Science Institute
University of Iceland
Dunhaga 3
107 Reykjaviik
Iceland
e-mail: [email protected]
2000 Mathematics Subject Classification 32F17, 32A26, 46Fl5
A CIP catalogue record for this book is available from the Library of Congress,
Washington D.C., USA
Bibliographic information published by Die Deutsche Bibliothek
Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie;
detailed bibliographic data is available in the Internet at <http://dnb.ddb.de>.
ISBN 978-3-0348-9605-4 ISBN 978-3-0348-7871-5 (eBook)
DOI 10.1007/978-3-0348-7871-5
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Originally published by Birkhauser Verlag in 2004
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Contents
Introduction .... . . . . . . . . . . . vii
1 Convexity in Real Projective Space
1.1 Convexity in real affine space . 1
1.2 Real projective space . 2
1.3 Convexity in real projective space. 5
2 Complex Convexity
2.1 Linearly convex sets . 15
2.2 {>convexity: Definition and examples. 25
2.3 C-convexity: Duality and invariance . 33
2.4 Open C-convex sets . 39
2.5 Boundary properties ofC-convex sets 45
2.6 Spirally connected sets . 63
3 Analytic Functionals and the Fantappie Transformation
3.1 The basic pairing in affine space .. 74
3.2 The basic pairing in projective space .. 83
3.3 Analytic functionals in affine space . . . 93
3.4 Analytic functionals in projective space 101
3.5 The Fantappie transformation . . . . 107
3.6 Decomposition into partial fractions 115
3.7 Complex Kergin interpolation . . . . 120
4 Analytic Solutions to Partial Differential Equations
4.1 Solvability in C-convexsets . . . . . . . 129
4.2 Solvability and P-convexity for carriers. 140
References 151
Index ... 159
Introduction
In this monograph we are going to study certain classes ofdomains and compact
subsets in <en which can be described in terms ofconcepts that are analogous to
those ofusualconvexity theory in realaffinespace. Recall that there are (at least)
twonaturaldefinitionsofwhataconvexset E inlRn shouldbe. Eitheronedemands
that the intersection ofE with each line be connected, or else that through every
point in the complement ofE there passes a hyperplane which does not intersect
E. Under the mild assumption that E is connected these two definitions are in
fact equivalent.
We shall consider analogues of these two definitions in <en. Sets whose in
tersection with any complex line is both connected and simply connected will be
called <e-convex, whereas those sets whose complement is a union ofcomplex hy
perplanes are said to be linearly convex. Any open or compact <e-convex set is
linearly convex, and for domains with C1 boundary the two notions are in fact
equivalent.
Much ofthe motivation for studying complex convexity notions stems from
a desire to represent holomorphic functions in terms of particularly simple ones.
Let us therefore briefly recapitulate some facts regarding representation formulas
for holomorphicfunctions. Iff is holomorphic in a neighborhoodofsomecompact
E c C, then the classical Cauchy formula provides the representation
~1
f(z) = f(()d(, z E E,
2m 'Y (- z
where "f is a smoothcurve encircling E on which f is holomorphic. This represen
tation formula is useful in several respects: It expresses the values of f on E in
terms of its values on "f, and the formula looks the same for any set E. Further
more, it provides a decomposition of f as a (infinite) linear combination of the
simple fractions z f--+ 1/((- z), which after the substitution a = -1/( is the same
as a combination ofthe functions
1
z
f--+ -1+-a-z.
Now suppose we want to similarly write a holomorphic function defined in a
neighborhood ofa compact set E c <en as a superposition ofthe simple fractions
1
z
f--+ 1+(z,a) ,
viii Introduction
Let us see what is required for this to be possible. First of all it is clear that
for such a fraction to be holomorphic on E it is necessary that the hyperplane
1+(z,0) = 0 does not intersect E. Let us denote the set ofall such hyperplanes
by E*. Then we wish to write 1as
r
I(z) = iE- dp,(o) , z E E,
1+(z,o)
wherep,isameasure,ormorepreciselyananalyticfunctional, onE*. Ourfunction
1is now called the Fantappie transform of the functional p, and we write this as
I=Fp,.
Notice that1is thenautomaticallyholomorphicin E**,thesetofall zwhich
do not lieinanyofthehyperplanesfrom E*. HenceifE isadomainofholomorphy
and such a decomposition is possible, we must have that E = E**. Sets with this
property are said to be linearly convex.
We can thus rephrase the problem of representing each 1on E as a super
position of simple fractions 1/(1 + (z,0)) into a question of surjectivity of the
Fantappie transformation F: O'(E*) --+ O(E). The necessary condition that E
be linearly convex is not sufficient in general. This was observed already by Mar
tineau, and he called a linearly convex set E strongly linearly convex if F is an
isomorphism. He showed that convex sets always have this property, but it was
only later that a geometric criterion for strong linear convexity was found. The
correct notion turned out to be what we have chosen to call «::>convexity.
cn
A subset E of is said to be C-convex if for every complex line £ the
intersection E n£ is both connected and simply connected. Comparing with the
usual definition ofconvexity, namely that E c IRn is convex if for any two points
a and b in E the line segment between them is also contained in E, and noting
that this is equivalent to saying that the intersection of E with any real line in
IRn should be connected, we see that the notion ofC-convexity is a very natural
complex analogue to ordinary convexity.
All convex sets are C-convex and all C-convex sets are linearly convex,
though this latter fact is far from obvious. Roughly speaking, representability
in terms of simple fractions 1/(1 + (z,o)) is equivalent to the domain E be
ing C-convex, whereas representability in terms of the more involved fractions
1/I17=1(1+(z,Oj)) amounts to E being linearly convex.
The class of C-convex sets is not closed under intersections, so it is not
a convexity concept in the usual sense, but it still has plenty of nice properties
analogousto thoseofconvexsets. Inordertofully exploit this analogyit is best to
placeoneselfin projectivespace, and to this endwe have included a short chapter
explaining how ordinary convexity can be defined in real projective space. It is
important to keep in mind that the projective viewpoint does not create any new
convex sets, but rather puts the usual ones in a new perspective, and in a more
elegantsetting. Forinstance,thedualcomplementE*,whichistheanalogueofthe
polarofa convexset, will be viewed asa subset ofthedual projectivespace. Some
Introduction ix
of the basic properties of C-convex sets, in parallel with the real case, can then
be formulated as follows: IfE is C-convex then so is E* as well as all images and
preimages of E under projective mappings. Such mappings include for example
affine projections and projections from a point. Notice that this does not hold in
general for linear convexity.
Let us give a hint as to how the surjectivity of :F can be proved using the
Cauchy-Fantappie-Lerayformula. Take a compact C-convexset E and choose an
openneighborhoodw:J E withsmooth boundaryow. The ideais tofind for every
( on ow a hyperplane s(() not intersecting E, and depending in a smooth way on
(. Ifthis can be done then the Cauchy-Fantappie-Lerayformula
r
f(z) = _1_ f(() s1\ (8s)n-l
Jaw z E E,
(21fi)n (s,( - z)n
holds, and since the kernel is homogeneous in s, one can replace s by the section
a = -sf(s,() and then get
r
1 f(()al\(8a)n-l
f(z) = (21fi)n Jaw (1 +(a,z))n z E E.
In order to remove the exponent in the denominator we makeanexplicit inversion
ofthe kernel function ¢(z) = 1/(1+(z,Q))nand obtain a standard functional flo:,
a so-called simplex functional, with :Fflo: = ¢. Combining this with the Cauchy
Fantappie-Leray formula above, we then obtain a functional fl with :Ffl = f, and
the surjectivity is proved.
Even though all C-convexsets can be regarded as subsets ofcn, their study
is deeply pervaded by the concept ofduality, and in order to have a satisfactory
duality theory one is led to adopt a projective point of view. We will therefore
often consider our sets as subsets ofprojective space. The open C-convex subsets
of the projective line C U {oo} are the simply connected domains. The open C
convexsubsets ofClP'n arealso topologically trivial, for they are homeomorphic to
the unit ball B in cn. In fact, if we assume that E does not contain any affine
n
line and we choose the coordinates so that 0 E E, then as a consequence of the
Riemann mapping theorem weget anexplicit homeomorphismc.p: Bn --+ E, whose
restriction to any line through the origin is a Riemann mapping. The compact C
convexsets are also topologically simple in the sense that their Cech cohomology
groups vanish.
It is C-convexity that is the main object of study in this book, but there
are other notions of convexity that we are also interested in. We have already
mentioned the concept oflinear convexity, and a related versionofthis is what we
call weak linear convexity. An open subset E ofClP'n is said to be weakly linearly
convex ifthrough every boundary point ofE there passes a hyperplane that does
not intersect E. A compact set E in ClP'n is called weakly linearly convex ifthere
exists a basis ofopen weakly linearly convex neighborhoods ofE. Aset is weakly
x Introduction
linearly convex if it is a connected component of its linearly convex hull E**, so
every linearly convex set is also weakly linearly convex. For sets with a smooth
boundaryhowever, weaklinearconvexitycoincideswith linear convexityand with
«:::-convexity.
There is more to be said about the boundary properties of«:::-convexsets. In
lRn everyopen or closed convexset E is the intersectionofall the half-spacescon
taining it. The affine hyperplanes separatethe spacelRn but an affine hyperplane
,
in cn is of real dimension 2n - 2 and its complement is a connected set. Hence
in C1P>n and cn we have nothing comparable with the separation property ofreal
convexity theory.
Given an open set E and a boundary point a, we let f(a) denote the set
of all ( E E* so that the hyperplane defined by ( in C1P>n passes through a. We
can also describe f(a) as the intersection ofthe hyperplane defined by a in C1P>m
with E*. Then E is C-convex if and only if f(a) is non-empty and connected
for every a E 8E. In particular, if E is an open weakly linearly convex set with
smooth boundary, then each f(a) consists of a single point, namely the point
corresponding to the complex tangent plane at a.
When the smoothness assumption is strengthened, so that
isgiven bya defining function pofclassC2, thenoneconsidersthequadratic form
Hp(a;w) = 2Re(LPjk(a)WjWk) +2LPjk(a)WjWk'
j,k j,k
where a E 8E, W E cn, Pjk =82p/8zj8zk,and P{k =82p/8zj8zk. This quadratic
form is called the Hessian ofPat a, whereas its hermitian part
Lp(a;w) = LPjk(a)WjWk
j,k
is called the Levi form. A smoothly bounded domain E is convex if and only
if the Hessian of its defining function is positive semi-definite when restricted
to the real tangent plane at any a E 8E. Similarly, a domain E is pseudoconvex
preciselyiftherestrictionofthe Levi form to thecomplextangent planeis positive
semi-definite. The notion ofC-convexity lies in between: A domain E c 1P>n with
boundary of class C2 is C-convex if and only if for any a E 8E the Hessian is
positive semi-definite on the complextangent plane at a.
Complexconvexityturns up inmanydifferentcontexts,andwehavenot been
able to coverthem all. Interestingtopics that we haveomitted include for instance
the papersofLemperton invariant metrics and pluricomplexGreen functions, the
connections to Radon transforms discovered by Henkin and his collaborators, as
well as the study ofa-problems by Diederich, Conrad and others.
Introduction xi
The plan of the book is as follows: In Chapter 1 we discuss real convexity
from the projective point of view. This chapter is quite independent and not a
prerequisite for the following chapters. In Chapter 2we prove the main theorems
on the structureoflinearly convexand «>convexsets. We consider their topology,
their duality properties, and their invariance with respect to projective mappings.
InChapter3westudyanalyticfunctions andfunctionals definedon«>convexsets.
We relate them via the Fantappie transformation. Finally, in Chapter 4 we char
acterize C-convexity in terms ofsolvability oflinear partial differential equations.
Bibliographical references and historical comments are included in the "Notes" at
the end ofeach chapter.
The present text is the result of a joint effort protracted over several years.
We were lead into this area via the study of complex interpolation problems.
We found an intriguing subject with several nice results, and many natural open
questions, but unfortunatelyalsoagreat uncertaintyabout what wasreallyknown
and what was not. Severalofthe papers wecame acrosscontained sketchyproofs,
vague references etc., and the results seemed to be very scattered in the existing
literature. That is why we took on the task of providing a unified treatment of
the subject, mainly presenting known theorems in a coherent fashion, but also
contributing a few new results and giving simplified proofs. This work has taken
much more time than we expected, since the job proved to be a painstaking one.
Duringtheearlystagesofourworkwe benefittedmuchfrom theappearanceofthe
book of Hormander [3], which contains a chapter on open C-convex and linearly
convex sets. Among other people whose work has inspired us we wish to mention
ChristerKiselman,SergeiZnamenskil,andofcourseAndreMartineauwhowasthe
pioneer of the subject. Many colleagues and friends have helped us scientifically
through valuable discussions, and also by kindly reading earlier versions of our
manuscript. Particularly we want to thank Lev Alzenberg, Bo Berndtsson, Lars
Filipsson, LarsHormander, Christer Kiselman, Hans Rullgard, Jan Alve Svensson
and Sergei Znamenskil for pointing out errors and suggesting improvements. It
goes without saying that any remaining mistakes are all ours.