Notions of Convexity

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Modern Birkhauser Classics Many of the original research and survey monographs In pure and applied mathematics published by Birkhauser in recent decades have been groundbreaking and have come to be regarded as foundational to the subject. Through the MBC Series, a select number of these modern classics, entirely uncorrected, are being re-released in paperback (and as eBooks) to ensure that these treasures remain accessible to new generations of students, scholars, and researchers.

Notions of Convexity

Lars Hormander

Reprint of the 1994 Edition Birkhauser Boston • Basel • Berlin

Lars Hormander Lund University Center for Mathematical Sciences SE-22100Lund Sweden

Originally published as Volume 127 in the series Progress in Mathematics

Cover design by Alex Gerasev. Mathematics Subject Classification (2000): 00A05, 01A60, 03-02, 26A51, 26B25, 31-02, 31B05, 31C10, 32-02, 32F05, 32F15, 32T99, 32U05, 32W05, 35A27, 52A40 Library of Congress Control Number: 2006937427 ISBN-10: 0-8176-4584-5 ISBN-13: 978-0-8176-4584-7

e-ISBN-10: 0-8176-4585-3 e-ISBN-13: 978-0-8176-4585-4

Printed on acid-free paper. \®

©2007 Birkhauser Boston BirkMuSCr i^ All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Birkhauser Boston, c/o Springer Science-fBusiness Media LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. 9 8 7 6 5 4 3 2 1 www.birkhauser.com

(IBT)

Lars Hormander

Notions of Convexity

Birkhauser Boston • Basel • Berlin

Lars Hormander Department of Mathematics University of Lund Box 118,8-221 00 Lund Sweden

Library of Congress Cataloging In-Publication Data

Hormander, Lars, 1931Notions of convexity / Lars Hormander. p. cm. - (Progress in mathematics ; v. 127) Includes bibhographical references and indexes. ISBN 0-8176-3799-0 (acid free). 1. Convex domains. I. Title. II. Progress in mathematics (Boston, Mass.) ; vol. 127 QA639.5.H67 1994 94-32572 515.'94-dc20 CIP

Printed on acid-free paper © Birkhauser Boston 1994

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PREFACE

The term convexity used to describe these lectures given at the University of Lund in 1991-92 should be understood in a wide sense. Only Chapters I and II are devoted to convex sets and functions in the traditional sense of convexity. The following chapters study other kinds of convexity which occur in analysis. Most prominent is the pseudo-convexity (plurisubharmonicity) in the theory of functions of several complex variables discussed in Chapter IV. It relies on the theory of subharmonic functions in R^, so Chapter III is devoted to subharmonic functions in R"^ for any n. Existence theorems for constant coefficient partial differential operators in R'^ are related to various kinds of convexity conditions, depending on the operator. Chapter VI gives a survey of the rather incomplete results which are known on their geometrical meaning. There are also natural classes of "convex" functions related to subgroups of the linear group, which specialize to several of the notions already mentioned. They are discussed in Chapter V. The last chapter. Chapter VII, is devoted to the conditions for solvability of microdifferential equations, which can also be considered as a branch of convexity theory. The whole chapter is an exposition of a part of the thesis of J.-M. Trepreau. Thus the main purpose is to discuss notions of convexity — for functions and for sets — which occur in the theory of partial differential equations and complex analysis. However, it is impossible to resist the temptation to present a number of beautiful related topics, such as basic inequalities in analysis and isoperimetric inequalities. In fact, this gives an opportunity to show how conversely the theory of partial differential equations contributes to convexity theory. Originally I also planned to discuss the role of convexity in linear and non-linear functional analysis, but that turned out to be impossible in the time available. Another topic which is conspicuously missing is the theory of the real and the complex Monge-Ampere equations, which should have been presented in Chapters II and IV. Convexity theory has contacts with many areas of mathematics. However, only applications in complex analysis and the theory of linear partial differential equations are discussed here, without aiming for completeness. I hope that in spite of that the book will prove useful for readers with main interest in other directions, and that it does justice to the beauty of the subject. To minimize the number of references relied on I have often referred to

my books denoted by ALPDO and CASV (see the bibliography at the end) instead of original works. Further references can be found in these books. At the beginning of the notes no prerequisites are assumed beyond calculus and linear algebra. Measure and integration theory are required in Section 1.7 and from Chapter III on. Distribution theory has been used systematically from Chapter III when it simplifies or clarifies the presentation, even where it could be avoided. However, only the most elementary part of the first seven chapters in ALPDO are required. Some background in differential geometry is assumed in Section 2.3, and the proof of the FenchelAlexandrov inequality there requires some knowledge of elliptic differential operators. At the end basic Riemannian geometry is also required, and Section 6.2 assumes familiarity with pseudodifferential operators. The last section. Section 7.4, assumes some background in analytic microlocal analysis, and some knowledge of symplectic geometry is needed in Section 7.3. Only the simplest facts from functional analysis are needed except in Section 6.3 where deeper results on duality theory are used. However, these are exceptions which can be bypassed with no loss of continuity. Apart from these points the notes should be accessible to any graduate student with an interest in analysis. As already mentioned Chapter VII is based on J.-M. Trepreau's thesis. The presentation here owes much to the patience with which he has corrected and improved earlier versions; any remaining mistakes are of course my own. I wish to thank him for all this help and for informing me about improvements that he made in a recent unpublished manuscript. In the final version they have been partially replaced by still more recent unpublished results due to A. Ancona presented in Section 1.7 and at the end of Sections 3.2 and 4.1. I am grateful for his permission to include them here. I would also like to thank Anders Melin for his critical reading of a large part of the manuscript, and M. Andersson, M. Passare and R. Sigurdsson who agreed to the inclusion in Chapter IV of some material from an unpublished manuscript of theirs. Thanks are also due to the publishers and their referees. Lund in June 1994 Lars Hormander

CONTENTS

Preface

iii

Contents

v

C h a p t e r I. C o n v e x functions of one variable 1.1. Definitions and basic facts 1.2. Some basic inequalities 1.3. Conjugate convex functions (Legendre transforms) 1.4. The r function and a difference equation 1.5. Integral representation of convex functions 1.6. Semi-convex and quasi-convex functions 1.7. Convexity of the minimum of a one parameter family of functions C h a p t e r II. Convexity in a vector space 2.1. Definitions and basic facts 2.2. The Legendre transformation 2.3. Geometric inequalities 2.4. Smoothness of convex sets 2.5. Projective convexity 2.6. Convexity in Fourier analysis

1 1 9 16 20 23 26 28

finite-dimensional 36 36 66 75 94 98 111

Chapter III. Subharmonic functions 3.1. Harmonic functions 3.2. Basic facts on subharmonic functions 3.3. Harmonic majorants and the Riesz representation formula 3.4. Exceptional sets

116 116 141

C h a p t e r IV. Plurisubharmonic functions 4.1. Basic facts 4.2. Existence theorems in L^ spaces with weights 4.3. Lelong numbers of plurisubharmonic functions 4.4. Closed positive currents 4.5. Exceptional sets

225 225 248 265 271 285

171 203

4.6. Other convexity conditions 4.7. Analytic functionals

290 300

Chapter V . Convexity w i t h respect t o a linear group 5.1. Smooth functions in the whole space 5.2. General G subharmonic functions

315 315 324

Chapter V I . Convexity w i t h respect t o differential operators 6.1. P-convexity 6.2. An existence theorem in pseudoconvex domains 6.3. Analytic differential equations

328 328 332 344

C h a p t e r V I I . Convexity and condition (*) 7.1. Local analytic solvability for d/dzi 7.2. Generalities on projections and distance functions, and a theorem of Trepreau 7.3. The symplectic point of view 7.4. The microlocal transformation theory

353 353

Appendix. A. Polynomials and mult linear forms B. Commutator identities

391 391 396

Notes

403

References

407

I n d e x of notation

411

Index

413

372 375 382

CHAPTER I

CONVEX FUNCTIONS OF ONE VARIABLE S u m m a r y . Section 1.1 just recalls well-known elementary facts which are essential for all t h e following chapters. Section 1.2 is devoted t o proofs of basic inequalities in analysis by convexity arguments. T h e Legendre transform (conjugate convex functions) is discussed in Section 1.3 in a spirit which prepares for t h e case of several variables in Chapter II. Section 1.4 is an interlude presenting an interesting characterization of t h e F function by t h e functional equation and logarithmic convexity, due to Bohr and MoUerup. We introduce representation of convex functions by means of Green's function in Section 1.5, as a preparation for t h e representation formulas for subharmonic functions. In Section 1.6 we discuss some weaker notions of convexity which occur in microlocal analysis. Section 1.4 and most of Section 1.6 can be bypassed with no loss of continuity. T h e last section. Section 1.7, studies when the minimum of a family of (convex) functions is convex. T h e extension t o (pluri-) subharmonic functions in Chapters III and IV will be essential in C h a p t e r VII.

1.1. Definitions and basic facts. Let / be an interval on the real line R, which may be open or closed, finite or infinite at either end, and let / be a real valued function defined in / . Definition 1.1.1. / i s called convex if the graph lies below the chord between any two points, that is, for every compact interval J C / , with boundary dJ, and every linear function L we have (1.1.1)

sup(/-L)=sup(/-L). J

dJ

One calls / concave if — / is convex. Let dJ = {3:i,X2}. An arbitrary point in J can then be written Airî A2a:2 where Xj > 0 and Ai + A2 = 1. Since L{XiXi + A2X2) = XiL{xi) A2i(x2), and we can choose L and a constant a with L -\- a = f on 9 J , follows that (1.1.1) is equivalent to (1.1.1)' f{XiXi^X2X2) < Xif{xi)-^X2f{x2), if Ai,A2 > 0, A1 + A2 = 1, Xi,X2 E

+ + it

/.

If / is both convex and concave, then there must be equality in (1.1.1)', that is, / = L + a where L is linear and a is a constant. Such a function

2

I. CONVEX FUNCTIONS OF ONE VARIABLE

is called affine] it can of course be uniquely extended to all of R. More generally, a map / between two vector spaces is called afEne if it is of the form f = L -\- a with L linear and a constant. This is equivalent to (1.1.2)

/ ( A i x i + A2:r2) - Xif{xi)

+ A2/(x2),

when Ai + A2 = 1.

Indeed, ii L = f — /(O) we obtain L{Xx) = XL{x) when X2 = 0, hence L{xi + X2) = L{xi) + L{x2) follows if Ai = A2 = | . This means that L is linear. Conversely, if / = L + a with L linear we obtain not only (1.1.2) but more generally

(1.1.2)'

/ ( ^ X^x,) = 5 ^ X,f{xj),

if X ; A,- = 1.

The following statements are immediate consequences of (1.1.1) or (1.1.1)': T h e o r e m 1.1.2. If fj are convex functions in I and Cj G R are > 0, J = 1 , . . . , n, then f — Y^ Cjfj is a convex function in I. T h e o r e m 1.1.3. Let fa, a E A, be a family of convex functions in I, and let J be the set of points x e I such that f{x) = sup^,^^/^(rc) is < -l-cxD. Then J is an interval (which may be empty) and f is a convex function in J. If fj, j = 1 , 2 , . . . , is a sequence of convex functions and J is the set of points x ^ I where F{x) — limj_,oo /j(^) < +00, then J is an interval and F is a convex function in J unless F = —00 in the interior of J or J consists of a single point. To prove the second statement one just has to write F{x) = lim.Fp^(x) where FN{X) = sup^^^y fj{x) and use the obvious first part. Exercise 1.1.1. Prove that one cannot replace sup by inf or lim by lim in Theorem 1.1.3. Exercise 1.1.2. Let / and J be two compact intervals with J C I and lengths |/|, I J|, and let / be a convex function in / . Prove that if m and M are constants such that / < M in / and f > m in J then f>M-{M-

m)\I\/{\J\

+ d(J, dl))

in / ,

where d{J, dl) is the shortest distance from J to dl and the denominator is assumed 7^ 0.

DEFINITIONS AND BASIC FACTS

3

T h e o r e m 1.1.4. Let f be a real-valued function deGned in an interval I, and let ip be a function defined in another interval J with values in I. Then f o cp is convex for every convex f if and only ifcp is afEne; and f o cp is convex for every convex (p if and only if f is convex and increasing. Proof. li f ocp is convex for f{x) = x and for f{x) = —x^ then cp is both convex and concave, hence afRne. Conversely, if ^ is affine it is obvious that f o (p inherits convexity from / . Now assume that f o cp is convex for every convex cp. Taking (p{x) = x we conclude that / must be convex. If yi < y2 are points in / , then (p{x) = yi -h (2/2 — 2/i)|^| is convex in [—1,1] , and ii f ocp is convex it follows since fo(p{±l) = f{y2) and f o(p(0) — f{yi) that f{yi) < f{y2), so / must be increasing. Conversely, assume that / is increasing and convex, and let xi,X2 G / , Ai, A2 > 0, Ai + A2 == 1. Then f{(p{XiXi + A2X2)) < f{Xi(p{xi)

+ X2(p{x2)) < Xif{(p{xi))

+ A2/((^(x2)),

where the first inequality holds since cp is convex and / is increasing, the second since / is convex. This completes the proof. If xi < X < X2 then x = AiXi + A2X2 for Ai = (x2 — x)/{x2 X2 = {x — rci)/(j:2 — rci), so (1.1.1)' means that {x2 - xi)f{x) (1.1.1)''

< {x2 - x)f{xi)

{fix) - f{x,))/{x

-\-{x-

xi)f{x2),

— xi)^

that is,

- X,) < {f{x2) - f{x))/{x2 - X).

Hence we have: T h e o r e m 1.1.5. f is convex if and only if for every x E I the difference quotient {f(x •}- h) — f{x))/h is an increasing function ofh when x -\-h E I and h ^ 0. Corollary 1.1.6. If f is convex then the left derivative f[{x) and the right derivative f!^{x) exist at every interior point off. They are increasing functions. If xi < X2 are in the interior of I we have (1.1.3)

//(:ri) < / ; ( x i ) < (/(X2) - fix,))/{x2

- x^) < //(X2) < fl-ix^).

In particular, f is Lipschitz continuous in every compact interval in the interior of I.

contained

There is no need for / to be continuous at the end points of / , but f{x) has a finite limit when x converges to a finite end point of / belonging to / , again by Theorem 1.1.5. Changing the definition at the end points if necessary we can therefore assume that / is continuous also there. The right (left) derivative exists then at the left (right) end point but may be

4


—oo (+00). If we allow / to take the value +00 we can always make the interval / closed. Using the continuity of / we obtain from (1.1.3) if xi < X2 are points in / (1.1.3)'

lim fl{xi

•\-e)< U{X2) - f{x,))/{x2

- x,) < lim fl{x2 - e).

If we let X2 i xi or xi t ^2? we obtain T h e o r e m 1.1.7. If f is convex in I and x is an interior point, (1.1.4)

/ ; ( a ; ) = lim / ; ( x + £ ) = limfUx

(1.1.5)

//(x)= l i m / ; ( x - £ ) =

We shall therefore write f'{x-{-0) conditions are equivalent

= f!,{x),

then

+ e),

\imflix-e). f'{x - 0) = //(a:). The following

(1) / / is continuous at x; (2) /^ is continuous at x; (3) frix) — fl{x), that is, f is differentiable at x. These conditions are fulfilled except at countably many Proof. The last statement follows from the fact that iixi in / , then E

points. < X2 are points

(/;(a;)-//(^)) 0 one calls f strictly convex. E x a m p l e 1.1.11. f{x) = e^^ is a convex function on R for every a G R. If 7* > 1, then fr{x) = a:'^ is a convex function when x > 0, if r < 0 then fr is convex when x > 0, but if 0 < r < 1 then x'^ is concave when x > 0. The functions g{x) = xlogx and h{x) = — logx are convex when x > 0. Another immediate consequence of Theorem 1.1.9 and Corollary 1.1.6 is: Corollary 1.1.12. Convexity is a local property: If f is defined in an interval I and every point in I is contained in an open interval J C I such that the restriction of f to J is convex, then f is convex. We have defined convexity in terms of affine major ants, but there is also an equivalent definition in terms of affine minorants: T h e o r e m 1.1.13. A reai-vaiued function f defined in an interval I is convex if and only if for every x in the interior of I there is an affine linear function g with g < f and g{x) = f{x). Proof. Assume that / is convex. Choose k E [fi{x), fri^)] and let g{y) = f{x) + k{y — x). Since g{x) — f{x) and (1.1.3) gives f{y) > fix) + iy-

x)r,{x)

> g{x),

iiy>x;

f{y) > fix) + iy-

x)f[ix)

> giy),

iiy<x,


7

the necessity is proved. Now assume that / satisfies the condition in the theorem. We must prove that (1.1.1)' holds. In doing so we may assume that xi 7^ X2 and that A1A2 > 0, which impHes that x = AiXi -f- A2X2 is an interior point of / . If g is an affine minorant of / with f{x) = g{x) then 2

2

2

1

1

1

which completes the proof. In view of (1.1.2)' the second part of the proof gives a much more general result with no change other than extension of the summation from 1 to n: T h e o r e m 1.1.14. Let f be convex in the interval I, and let xi,... / . Then we have n

(i.i.ir

n

^x^ G

n

/(E^^-^^)Ê^^/(^^)' ifAi,...,A„>o, Y.^, = i. 1

1

1

If Aj > 0 for every j , then there is equality in (1.1.1)'" if and only if f is affine in the interval [minxj,maxxj]. E x e r c i s e 1.1.8. Prove (1.1.1)'" directly from (1.1.1)' by induction with respect to n. (1.1.1)'" is usually called Jensen^s inequality, and so is the following more general version involving integrals instead of sums: E x e r c i s e 1.1.9. Let / be a convex function in the interval / , let T be a compact space with a positive measure dji such that Jrpd/jL(t) = 1, and let x(t) be a // integrable function on T with values in I. Prove that

/ ( / X{t)dti{t)) < I f{x{t))dtL{t). JT

JT

E x e r c i s e 1.1.10. Let 11 be an orthogonal projection in a finite dimensional Euclidean vector space E. Show that if yl is a symmetric linear operator in E then

Tv(n/(nyin)n) < Tr(n/(^)n) for every convex function / (Berezin's inequality). (Recall that if B is a linear transformation in E then Tr B = Y^{Bej, Cj) if Cj is any orthonormal basis in E and (•, •) denotes the scalar product. If B is symmetric then f{B) has the same eigenvectors as B with every eigenvalue A replaced by /(A). — Hint: Express both sides in terms of the eigenvectors of IIAII in HE and of A in E.)

8


E x e r c i s e 1.1.11. Wîth the notation in Exercise 1.1.10 show that for any / G C'{I) Mminf

< Tr(n/(A)n) - T r ( n / ( n A n ) n ) < Mmax/",

where M — | Tr ( n A ( I d - n ) A n ) and / is the interval bounded by the largest and smallest eigenvalues of A] Id is the identity operator. A number of applications of Jensen's inequality will be discussed in Section 1.2. We shall first end this section by discussing some seemingly weaker definitions of convexity which are sometimes useful. T h e o r e m 1.1.15. If f is continuous but not convex in the open interval I, then one can find ?/ G / , c G R and e > 0 such that (1.1.7)

f{y -i-h) < f{y) + ch - eh^,

when \h\ is small.

Proof. Let J = [a,b] C I be an interval such that for some affine g we have f < g on dJ but s u p j ( / — g) > Q. Then fs{x) = f{x) - g{x) + e{x - a){x - b) is < 0 in dJ but sup j /^ > 0 if e is small enough. The maximum is then taken at an interior point 7/ G J , so f{x)-g{x)-{-e{x-a){x-b)

= fe{x) < fs{y) = f{y)-9{y)

+

e{y-a){y-b),

when X ^ J. With x = y -\- h it follows from Taylor's formula that f{y -^h)
0 if / is convex.

SOME BASIC INEQUALITIES

9

Exercise 1.1.12. Let / be a continuous real-valued function in the interval / . Prove that / is convex if for arbitrary e^6 > 0 and x in the interior of / there is a positive^measure with support in [0,6] such that

/ {f{x-hh)-\-f{x-h)-2f{x))dfi{h)>-e

h'^dfi{h)^0.

Jo

Jo

We shall end the section with some more esoteric conditions for convexity. Exercise 1.1.13. Let / be a real-valued function in an open interval / such that / is bounded above on some open non-empty subinterval and (1.1.9)

/(i(a; + 2 / ) ) < i ( / ( x ) + /(y)),

x,y € I.

Prove that / is convex. (Hint: Prove in order the following statements: (1) ( L l . l ) ' is valid if Ai, A2 are rational numbers with a power of 2 as denominator. (2) / is bounded above on every compact subinterval of / . (3) / is bounded below on every such interval. (4) / is continuous. (5) (1.1.1)' is valid in general.) On the other hand there are unbounded functions satisfying (1.1.9) such as all functions / satisfying the functional equation f{x^y)

= f{x) + f{y),

x,2/GR.

It is well known that there are such functions which are not linear. However, they are not measurable, which is confirmed by the following: Exercise 1.1.14. Prove that if / is measurable and satisfies (1.1.9), then / is a convex function. (Hint: The set Ea = {x;x E I^fi^) < a} is measurable, and it has positive measure for some a. We have x E Ea ii X -\- y E Ea and x — y e Ea ioi some y. Show that this implies that Ea contains an interval.) 1.2. S o m e basic inequalities. When combined with Jensen's inequality the convex functions listed in Example 1.1.11 yield some of the most important inequalities in analysis. T h e o r e m 1.2.1 (Inequality b e t w e e n geometric and arithmetic m e a n s ) . Ifaj > 0, A^ > 0, j = 1 , . . . , n, and ^ ^ A^ = 1, then n

(1.2.1)

n

n^'Êv.'

10


with strict inequality unless all aj are equal. Proof. With aj = e^^ the inequahty becomes n

n

exp ( 5 ^ XjXj) < Y^ Xj expXj, 1

1

so (1.2.1) follows from Jensen's inequality since x i-^ expx is convex and not afBne in any interval. T h e o r e m 1.2.2. Ifaj

> 0, Xj > 0, j = 1,... ,n, and Xli A^ = 1, then

n

(1.2.2)

n

^ I

"' ^•^P>1'

EV.^(EM) 1

1

with equality only if all aj are equal. Proof. If we raise both sides to the power p this follows from the fact that X 1-^ x^ is convex when x > 0 iip > 1, and is not affine in any interval. The right-hand side of (1.2.2) is called the P mean of a = ( a i , . . . ,an) with weights A = ( A i , . . . , A^). More generally we define

(1.2.3)

A^p(a;A) = ( E A , < )

\

P ^ 0-

1

When p -^ 0 we have, since Yl^ Aj = 1, n

n

p-Mog(;Â,a^)=p-Mog(Â,(l+ploga,+0(/))) 1

1 n

= X]Ajlogaj+0(^), 1

so M.p{a] A) becomes a continuous function of p for all p eR

if we define

n

(1.2.3)'

Aô(tt5 A) ~ TT^j^

{^^^ geometric mean).

1

M.i{a; A) is the arithmetic mean of a and A^_i(a; A) is called the harmonic mean of a, with weights A. When p > 0 we have maxttjA^- ^ < Mp{a]X)

< maxa^,

and when ;? < 0 we have mina^ < Mp{a;X)

< minaÂ^

,

so we get Mp{a] A) -^ ^4^00(0-; A) as p -^ ±00 if we define (1.2.3)''

A^_oo(o^;A) — minttj;

A^+oo(tt;A) = m a x a j .

Theorems 1.2.1 and 1.2.2 are now special cases of the following:


11

T h e o r e m 1.2.3. If aj > 0, Xj > 0, j = 1,... ,n, and J^l Xj = 1, then Mp{a] X) is a strictly increasing function ofp G [—oo, +CXD] unless all aj are equal; in that case Mp{a^ X) is this common value, for all p and X. Proof. Assume that all aj are not equal. By Theorem 1.2.2 applied with aj replaced by a j , g > 0, we conclude that Mq{a] X) < Mpq{a] A),

p > 1,

which proves the statement for p > 0. Since M-.p{a;X)

=

{Mp{a-';X))-\

the statement follows for p < 0 also. If we drop the condition ^ ^ Xj — 1 and apply (1.2.2) with Aj replaced by Xjl Y^^ ^k: we obtain for arbitrary Aj > 0

1

1

1

Here p' is called the exponent conjugate to p; note that p + p' = pp'. This inequality gets a more familiar form if we replace Aj by &^ and aj by ajh^

^ , noting that p(\ — p') + p' == 0, which gives

T h e o r e m 1.2.4 (Holder's inequality). For arbitrary positive aj and bj we have ifp > 1, p^ > 1, 1/p + 1/p' = 1 n

(1.2.4)

n

-, /

n

i / '

E«A-(E«0 (E 0, bj > 0 for j = 1 , . . . , n . Then

(B«.+M')""s(E4)""+(E'^)"". ^>i. 1

with strict inequality early dependent.

1

1

unless a = ( a i , . . . , a^) and b = (&i, • • •, &n) ^^^ lin-

Proof. The theorem is trivial if a = 0 or & = 0. Otherwise we can choose a > 0 and /? > 0 so that n

1/

n

1

1/

1

Then the convex function of A G [0,1] defined by

fiX) = Y.iaXaj + P{l-X)bjr 1

is equal to 1 when A = 0 or A = 1, and it is not afline in [0,1] unless aa — /3b. Otherwise we conclude that /(A) < 1 when aA = /3(1 — A), hence A -r p/(a + p) and aX = ap/{a + /?). Thus n

-. ,

(5^(a,+6,r) ' < l / a + lM 1

which completes the proof.


13

Exercise 1.2.2. Derive Minkowski's inequality from Holder's inequality. We give two more exercises involving Jensen's inequality, which are related to the notion of entropy: Exercise 1.2.3. Prove that — X^^^jloga;^ < logn if 0 < Xj, j = 1 , . . . , n, and Yî ^j — 1- (We define rrloga: = 0 when x = 0.) E x e r c i s e 1.2.4. Let 0 < Xjk^ j = 1 , . . . , J , k = l , . . . , i i r , and let E i = i Efc=i ^jk = 1. Prove that J

K

J

K

- X^ X! ^^^ ^^s ôk 0, Xj > 0, j = 1,... ,n, and ^ ^ Xj = 1, then p I—> plog A^p(a; A) is convex on R and not affine in any interval unless all aj are equal. Note that the difference quotients at 0 are logjVtp(a; A), so Theorem 1.2.7 contains Theorem 1.2.3 in view of Theorem 1.1.5. For the proof of Theorem 1.2.7 we need a lemma to which we shall return several times in related contexts: L e m m a 1.2.8. If g is a positive function defined in an interval I, then logg is convex (resp. affine) in I if and only ift H^ e^^g{t) is convex (resp. constant) in I for every (resp. some) c G R. Proof. If log^ is convex, then t ^-^ ct -i- log^(t) is convex. Since u y-> e^ is increasing and convex it follows from Theorem 1.1.4 that t i-> e^^g(t) is convex. Conversely assume that t y-^ c^^g{t) is convex for every c. For every compact interval J C I the maximum of e^^g{t) when t G J is then assumed when t G 5 J , which means that the maximum of ct -\- log g{t) in J is taken when t G dJ. Hence log^ is convex. The condition for log^ to be affine is trivial. Proof of Theorem 1.2.7. The convexity of the exponential function implies that p ^ e^^Mpia; Xy = ^

Xjieâjf

14


is a convex function, and it cannot be constant unless all aj are equal. Hence Theorem 1.2.7 is a consequence of Lemma 1.2.8. Corollary 1.2.9. Ifaj > 0, Xj > 0, j ^ 1,... ,n, and Y^'l Xj = 1, then X I—> log Aî/a;(a; A) is a convex function for x > 0 and a concave function for X < 0, and not affine in any subinterval unless all aj are equal. Proof. Let (p{p) = PôgMp{f), which is a convex function by Theorem 1.2.7. The claim is that x H-> tlj{x) = x(p{l/x) is convex for x > 0 and concave for x < 0, and this is clear since

^\x)

= ^{1/x) - ip\l/x)/x,

i^"{x) =

^"{\lx)lx'^.

Exercise 1.2.5. Prove that Corollary 1.2.9 follows from Holder's inequality Exercise 1.2.6. Prove that '0(/ o ip) is convex for every convex / if and only if -0 and ijjip are afRne and V^ > 0, unless ip is constant. (This explains the proof of Corollary 1.2.9.) Exercise 1.2.7. Prove that liui and U2 are positive functions such that log Til and log'U2 are convex, then \og(ui + U2) is convex. The mean values Mp{a\X) studied above can be generalized further. Let (/p be a strictly monotonic continuous function defined in an interval / , and let ip"^ be the inverse function defined in the range which is also an interval. If a i , . . . , a^ G / and A i , . . . , A^ > 0, J^^ Aj = 1, then ^ ^ Xj(p{aj) belongs to the range of (^, which makes

M c + k{y — x)

for every y G R.

Proof. This is a consequence of Theorem 1.1.13 and (1.3.1) if x is in the interior / of the interval where / < oo. If rr is, say, the right end point of this interval we choose ^ G / and note that since f{y)>f{0

+ {y-Of'i^

+ o)

VyGR

the statement follows unless f{() -h{x — 0 / ' ( ^ + 0) < c for all ^ G / . Since fiO "^ / ( ^ ) > ^ when ^ T ^5 tliis would imply that / ' ( ^ -f 0) -^ —oo when ^ T X, which is absurd since / ' ( ^ + 0) is increasing. This proves the statement except when x ^ I. Then we just have to take k sufficiently large positive or negative. Definition 1.3.2. The Legendre transform (also called the conjugate function) / of / is defined by (1.3.2)

fiO = supix^ -

fix)).

X

The term Legendre transform may be preferable because of the ambiguity of the term "conjugate function". Note that only the interval where / is finite matters in the definition (1.3.2). By Theorem 1.1.3 it is clear that / is convex, for it is the supremum of a family of functions which are affine, hence convex; / is lower semi-continuous since the supremum of any family of lower semi-continuous functions inherits this property. Since X i-> x^ — f{x) is increasing for x < XQ \i / ' ( X Q — 0) < ^, the restriction of / to [xo,oo) determines / on ( / ' ( X Q — 0),oo), if / < oo in a neighborhood of XQ. Thus / ( ^ ) is determined for large ^ if / ( x ) is known for large x. T h e o r e m 1.3.3. For every convex lower semi-continuous have the inversion formula f = fj that is, (1.3.3)

Proof

fix)

= sup(xe - / ( O ) .

Prom the definition (1.3.2) we obtain at once

Hence fix)

> sup(xe - fiO)

=

fix).

function f we

18


So far we have not used the hypotheses on / . By Lemma 1.3.1 they imply that for every c < f{x) we have for somefcE R ky — f{y) < kx — c^

V?/ E R,

hence f{k) < kx — c.

Thus c < kx — f{k) < f{x)^ which proves that f{x) < f{x). The proof is complete. If / is differentiable we can determine / ( ^ ) by differential calculus in the interior of the interval where / < oo; this leads to elimination of x from the equations (1.3.4)

fix) + m=î,

/'(^) = e

If / € C^ and / " > 0 the second equation (1.3.4) determines x locally as a C^ function of ^, and differentiation of the first equation (1.3.4) with respect to ^ then gives / ' ( O = oo, so f e C^ (locally) and we obtain the symmetric formulas (1.3.5)

fix)+fiO=^^^,

f'{x)=C,

f'iO=x-

Exercise 1.3.1. Find the Legendre transform of the following functions: a) fix) = \x\^, p > 1; b) fix) = e^; c) fix) = x l o g x ii x > 0, /(O) = 0, fix) = +00 if X < 0. Conjugate functions are often presented in a different way in the literature. Let / be a strictly increasing continuous function on [0, CXD) with /(O) = 0, and denote the inverse function by g. Set

J^fit)dt,

ifrr;>0;

^^^ ~ \ J^ gis) ds,

if ^ > 0.

Then F and G are conjugate convex functions. In fact, if J: > 0, ^ > 0 then x^ — Fix) is the area of the rectangle with vertices (0,0), (0, x), (a;, ^), (0,^) minus the area of {(/;,?/); 0 < y < fit),0 < t < x}. For given ( it is clear from a picture that it grows until fix) = (, and decreases afterwards, so the maximum value is equal to Gi^). Hence F(^) = G(^) when ^ > 0, and Fi^) — 0 when ^ < 0, with the maximum taken for x = 0. E x a m p l e 1.3.4. With fix) = x ^ - \ p > 1, we obtain ^ ( ^ = C^'~^ where 1/p -f- 1/p' = 1, for this is equivalent to (p — l ) ( y — 1) = 1. Hence Fix) = x^/p for X > 0 and G(<J) = (^'^ /p' for (^ > 0 are conjugate functions. Similarly Fix) - e^ - 1 - x for x > 0 and G ( 0 = ( 1 + 0 log(l + 0 - ^ for ^ > 0 are conjugate functions. (Compare with Exercise 1.3.1.)

CONJUGATE CONVEX FUNCTIONS

19

If / and / are conjugate functions, we have for arbitrary a^, bj^ j = l,...,n, n

(1.3.6)

n

âjb, I

n

< ^ / ( a , ) + J2f{b,). l

l

The first case in Example 1.3.4 gives, if aj > 0, bj > 0

1

1

1

Hence n

Yâjbj

< 1,

if A = l,

B^l,

1

which imphes Holder's inequality. The inequality (1.3.6) becomes a substitute when other means are known. Some minimum problems can be solved by means of Legendre transforms. We give an example: P r o p o s i t i o n 1.3.5. Let f be an everywhere Unite convex function on R such that f{x) -^ H-oo as \x\ -^ oo. Let 5 i , . . . , 5 ^ be given positive numbers, let ti^... ^tn be given numbers ^ 0, and set n

(1.3.7)

n

Fs,t{A) = inf { 5 ^ s J ( a O ; a, G R, J ^ t.Siai = A}. 1

1

Then Fs^t is a continuous convex function with Legendre

transform

n

FsA^) = Y^Sif{ati). 1

Proof. Since / -^ H-oo at cx) it is clear that the infimum in (1.3.7) is attained in a fixed compact set when a bound for A is given. Hence the continuity follows. If ^^ tiSidi ~ A and J^^ tiSibi = B, then n

Y^tiSiiXa^

+ (1 - X)bi) = A^ + (1 - A)S,

1 n

n

n

Y^ SifiXa, + (1 - X)bi) <XJ2 Si/(«^) + (1 - A) ^ 1

1

Sifibi),

1

which proves the convexity. By definition we have n

= snp{aA - Fs^t{A)) = sup ^ ^ 1

which proves the proposition.

n

atiSiai - ^ 1

n

sj(ai)j = ^

Sif{ati), 1

20


1.4. T h e r function and a difference equation. is defined by the Eulerian integral

The F function

/*CX)

(1.4.1)

T{x) = /

e-H''-^ dt,

x>0.

The convergence is obvious, and integration by parts yields the functional equation

(1.4.2)

r{x + i)=^xr{x),

x>o.

Since (1.4.3)

r ( i ) = 1,

we have T{n) ~ {n — 1)\ for every positive integer, so the F function interpolates the factorial to non-integer arguments. There are many functions satisfying (1.4.2) and (1.4.3), for these properties are preserved if we multiply T{x) with any function of period 1 which is equal to 1 when x = 1. However, one can characterize the F function uniquely by a convexity property: T h e o r e m 1.4.1. log F is a convex function on the positive real axis, and there is no other positive solution of (1.4.2), (1.4.3) having this property. Proof. To prove that logF is convex it suffices by Lemma 1.2.8 to prove that X — f > e^^F(x) is convex for every c G R. This follows at once since J2

poo

—^(e"^F(x))= / rfx^ Jo

e - ' ( c + logt)2e^^t^-irf^>0.

(Instead of differentiating we could interpret the integral defining e^^F(a;) as a limit of sums of exponential functions and use Theorem 1.1.2, which is also true for "continuous sums".) To prove the uniqueness we first note that (1.4.2) can be written (1.4.2)'

logF(x - h i ) - logF(x) = logrc,

where the right-hand side is a concave function which is o{x) as x -^ cx). The uniqueness is therefore a special case of the following: T h e o r e m 1.4.2. Let h{x) be a concave function on {x E Ii]x such that h{x)/x —> 0 as x -^ oo. Then the difference equation (1.4.4)

g{x + l)-g{x)

= h{x),

x > 0,

> 0}

THE r FUNCTION AND A DIFFERENCE EQUATION

21

has one and only one convex solution g with g{l) = 0, and it is given by n-l

(1.4.5)

g{x) = -h{x)

+ lim (xh{n) + Yîhij)

-

h(x-\-j))).

Proof. The function h is increasing, for {h{x-^y) — h{x))/y and -^ 0 as 7/ -^ oo for fixed x. Moreover, if ^ > 0 then (1.4.6)

{h{x + 2/) - Hx))/y

< {h{x) - h{x/2))/{x/2)

-^ 0

is decreasing

as x -^ cx).

In particular, the sequence a^ — /i(n + 1 ) — h{n) is decreasing and converges to 0 as n -^ oo. Now (1.4.4) and the condition ^(1) = 0 give n-l

(1.4.7) g{x)-\-h{x)-xh{n)-^{h(j)-h{x-\-j))

=

g{x-\-n)-g{n)-xh{n).

If g is convex and k is an integer with x < k, then h{n - 1) == g{n) - g{n - 1) < {g{x + n) < {g{k + n) - g{n))/k

g{n))/x

= {h{n) -\-• • •-j-h(n +k -

l))/k.

Thus -xan-i

< g{x + n) - g{n) - xh{n) < x{{k - l)a^ H

f- 0^+^-2)/^,

which proves that g{x -hn) — g{n) — xh{n) —> 0 as n -^ 00. Hence it follows from (1.4.7) that g must be of the form (1.4.5). This uniqueness suffices to complete the proof of Theorem 1.4.1, but to prove Theorem 1.4.2 we must also show that (1.4.5) converges to a convex function with ^(1) — 0 satisfying (1.4.4). Only the convergence needs some motivation, for it is clear that ^(1) = 0, and (1.4.4) will follow since h{n) — h{x + n) -^ 0 as n —> 00, by (1.4.6). We write the limit (1.4.5) as Yl^ '^n(^)> where uo{x) = xh{l) — h{x) and y^n{x) — x{h{n + 1) — h{n)) -{- h{n) — h{x + n),

n > 0.

It is clear that Un is convex. Using the concavity of h we also obtain when k >X h(n — 1) — h{n) < {h{n) — h{x + n))/x x{an

- an~l)

< Un{x)

< X (ttn ~ {an -{

< {h{n) — h{k + n))/k^ h an+fc-l)/^),

n > 0.

hence

22


Thus Un > 0, and since oo ^

k (ttn -

( t t n H- • • • +

fln-f-fc-l)/^)

= ^

1

% ( 1 " J/k)

< OC,

1

the convergence of ^ ^ Un{x) follows, and the sum is convex since the terms are. This completes the proof. If we apply (1.4.5) to the F function, we obtain the product formula (1.4.8) ^ ^

r ( x ) = lim — ^, , ^ ^ n->oox(x-f l ) - - - ( x + n)

Using the concavity of h we can also get estimates for g. First note that for every x > 0 px + l

^{h{x) + h{x + 1)) < /

h{t) dt < h{x + I ) ,

for h has an af&ne majorant equal to /i at :r + | and an affine minorant equal to ft at a; and x + 1. Hence n-l

xh{n)^Y.{h{j)-h{x^j)) 1 nn — l

< xh{n) + /

px-\-n—\

h(t) dt -f | ( / i ( l ) -f /i(n - 1)) - /

/•cc+l

/i(t) dt

px-\-n—^

= / ft(^) rfi^ + | / i ( l ) + | / i ( n - 1) - /

/i(t) 6/^ + xft(n).

It follows from (1.4.6) that the sum of the last three terms —> 0 as n —> oo. Similarly, n-l

xh{n) +

Y.{h{j)-h{x^j)) 1 rn—\

> xh{n) + / J\

= /

nx+n—l

h{t) dt-

h{t) dt - \(h{x -f-1) -h /i(x + n - 1)) Jx-\-l

h{t) dt - \h{x + 1) -

/

h{t) dt + xh{n) - \h{x -h n - 1).

Again the sum of the last three terms -^ 0 as n -^ oo, so we obtain /

h{t) dt - ^h{x + 1) < g{x) + h{x) < /

h{t) dt + | / i ( l ) .

INTEGRAL REPRESENTATION OF CONVEX FUNCTIONS

23

We estimate the left-hand side from below by noting that \h{x^l)-

/

h{t)dt=

/

{h{x-^l)-h{x-\- t))dt

is decreasing, so we have proved the following estimate: T h e o r e m 1.4.3. For the convex solution of the difference (1.4.4) with g{l) = 0 given by Theorem 1.4.2 we have

(1.4.9) \g{x) + h{x)- I

h(t)dt\ < \h{l)-

equation

I h(t)dt < |(/i(l)-/i(|)).

Applied to the F function this estimate gives 24 < r ( x ) x e ^ ( x + | ) " ' ^ ~ 2 < 2 4 . Stirling's formula gives a much more precise result for large x: __

r{x)e''x

. 1

>

—

^"^2 -> V27r,

as rr -^ oo.

It can be further improved by Stirling's series. However, this has little to do with the topic of convexity. Exercise 1.4.1. Prove under the hypotheses of Theorem 1.4.2 the existence of the limit rx-\-

lim {g{x) -^h{x) -

a:;—>+oo

I

h /i

2

h{t)dt).

•^2

1.5. Integral representation of convex functions. Let I = (a, 6) be a bounded open interval and let / be a convex function which is bounded in / . We can define / ( a ) and f{b) so that / is continuous in / = [a, b]. Set ., ^ , . (1.5.1)

^ . . ( ix~b){y-a)/{b-a), when a < y < x < b, Gi{x,y) = < / / /A t (x — a){y — b)/[b — a), when a < x < y < b.

Note that Gj is continuous and < 0 in the square I x I^ and that Gj = 0 on the boundary. Furthermore, Gi{x,y) is symmetric in x and y and affine in each variable outside the diagonal where the derivative has a jump equal to 1. In particular, Gj{x,y) is therefore a convex function in each variable when the other is fixed.

24


T h e o r e m 1.5.1. There is a uniquely determined positive measure dfi in I such that (1.5.2)

fix) -

/ Gjix,y)dfx{y)

+ ^f{b)

JJ

+ ^ / ( a ) ,

0— a

x

el.

a— 0

In particular, (1.5.1) implies that (1.5.3)

{x- a){b - x)d[i{x) < oo.

Conversely, for every positive measure satisfying (1.5.3) the integral in (1.5.2) defines a continuous convex function f in I which vanishes on dl, and f" = dpi in the sense of distribution theory. Proof. Assume at first that / is convex in a neighborhood of / . Then ^(x) = fr{oo) is an increasing function in / . Since Gj = 0 on the boundary, an integration by parts gives / Gi{x,y)dfi{y)

= -

dGi{x,y)/dyfi{y)dy

= -F~; nr{y)dy-'^ b-a

J^

fK{y)dy

b-a J^ 0—a

0— a

(cf. Theorem 1.1.9) which proves (1.5.2). If we choose x = {a-\- b)/2 and let d{y) — {b — a) 12 — \y — {a-{- &)/2| be the distance to the complement of / , we obtain from (1.5.2) (1.5.4)

^ d{y)d^l{y) = / ( a ) + /(&) - 2 / ( i ( a + b)).

Dropping the assumption that / is convex in a neighborhood of / we can apply (1.5.4) to the interval {a-\- £,b — e) for small e > 0, and when e -> 0 we conclude that (1.5.4) holds for / , which implies (1.5.3). When e -^^ 0 we also conclude that (1.5.2) is valid. li (f e CQ{I) we have found that ip{x) = /

Gi{x,y)ip"{y)dy

without any convexity assumption on if. If / is defined by (1.5.2) with d/jL >0 satisfying (1.5.3), it follows from the convexity of x \-^ Gi{x^y) that / is convex, and / f{x)(p"{x)dx Ji

=

Gi{x,y)(p'\x)dxdfx{y) JJixi

= / ip{y)dii{y). Ji

INTEGRAL REPRESENTATION OF CONVEX FUNCTIONS

25

Here we have used the symmetry of Gj. Thus d/x = f" in the sense of the theory of distributions which proves the uniqueness. Until now we have only used differentiation in a classical sense, and this would still suffice here. Thus we have not underlined the fact that for a convex function / in an open interval / , the derivatives /^ and / / both define f in the sense of the theory of distributions. This follows at once since for a non-negative test function if G CQ{I) we have by monotone convergence, for example, [ f^{x)ip{x)

dx = lim

l{f{x-\-h)-f{x))h-îp{x)dx

= lim / f{x){(p{x — h) — ip(x))h~^ dx = —

f{x)(p\x)

dx.

In analogous discussions of subharmonic functions later on, the language of distribution theory will be much more essential. Theorem 1.5.1 means that every convex function in a finite interval is a superposition of a linear function and functions of the form x \-^ G{x^y)^ or equivalently, x \-^ (x — y)^ oi x y-^ \x — y\^ where t_|_ = max(t, 0) when t G R. Jensen's inequality N

N

N

is trivial i f / ( x ) = 1 or f{x) ~ ±x^ and it follows from the triangle inequality when f{x) = \x — y\. Hence it is true in general. Similarly, Berezin's inequality (Exercise 1.1.10) follows if we prove that

Tr(n|nAn|H) < Tr(n|^|n). li Si,... jSi^ G HE is an orthonormal basis of H ^ consisting of eigenvectors for HAH, then

1V(H|HAH|H) = 5 ] ( | H A H | . „ . , ) = 1

Y^\{UAUe,,ej)\ 1

= Y, \iAei,e,)\ 1

< Y,{\A\e^,e^)

= Tr {U\A\U).

1

We give as an exercise to prove another inequality due to Berezin:

26


Exercise 1.5.1. Prove that if £" is a finite-dimensional Euclidean vector space and A i , . . . , AN are positive symmetric maps in E with ^-|_ Aj = Id, the identity, then N

TV f{Y,AjXj) 1

N

< TV 5 ] A,/(x,),

xj eJ, j =

h...,N,

1

provided that / is convex in the interval J C R. (Hint: Prove first that if A is symmetric then TV \A\ = max||5|| f in dJ, then L{t) > f{t) when a > t E J, since / — L is decreasing. Hence / ( a ) = / ( a -h 0) < L{a) by condition (iii), if a G J , and the convexity gives that f < L when a < t £ J. If / is monotonic the necessity is also obvious. To prove the necessity otherwise we note that if ti < ^2 and f{ti) > 7(^2), then f{t) > / ( t i ) for t < ti since f{t) < f{ti) > /(^2) would contradict the definition even with L = 0. If we denote by a the supremum of all ti e I with f{ti) > f{t2) for some ^2 > î, ^2 ^ It it follows that / is increasing to the right of a and decreasing to the left of a. The lower semi-continuity gives /(a)<min(/(a + 0),/(a-0)). If / ( a ) < f{a-\- 0) we get a contradiction, for if a < t G / and f(t) > f{a), then / ( ( I - e)a + et) < (1 - e)f{a) + ef{t) -> / ( a )

SEMI-CONVEX AND QUASI-CONVEX FUNCTIONS

27

when e -^ +0. This completes the proof. Semiconvexity is of course invariant under composition with increasing affine functions. It does occur naturally in some analytic contexts; see ALPDO [IV, pp. 145-147]. In the beginning of the proof we actually encountered an even weaker concept, which does not depend at all on the affine structure of R but only on the notion of intermediate point, so that it is invariant under composition with monotonic functions. Definition 1.6.3. A function / defined on an interval / C R with values in R will be called quasi-convex if / is lower semi-continuous and for every compact interval J C I the equality (1.1.1) holds when L is a constant, say 0. There is a description analogous to Theorem 1.6.2: T h e o r e m 1.6.4. / is quasi-convex in the interval / C R if and only if either (i) / is decreasing and continuous to the right, or (ii) / is increasing and continuous to the left, or (iii) there is a point a e I such that f satisfies (i) in I fl {—oo,a) and (ii) in I n (a, +oc), and f{a) < m i n ( / ( a + 0), / ( a - 0)). Proof. The sufficiency is obvious and the proof of necessity was a part of the proof of Theorem 1.6.2. In spite of the fact that the notion of quasi-convex function seems quite trivial, it has a prominent role in the theory of linear partial differential operators, although in a somewhat different guise. In fact, if/ is differentiable at every point then / is quasi-concave if and only if (1.6.1)

/ ' ( ^ ) l^^s no sign change from -h to — for increasing x.

This is closely related to the so-called condition ( ^ ) . (See Definition 7.3.3.) Remark. In the applications referred to, the natural continuity condition is upper semi-continuity and not lower semi-continuity. It has been changed here to agree with the standard condition for convex functions. However, upper semi-continuity has some obvious advantages such as fixing / ( a ) as m a x ( / ( a -h 0), / ( a - 0)) in Theorem 1.6.4. T h e o r e m 1.6.5. Semi-convexity is a local property: If f is defined in an open interval / C R and for every point x e I there is an open interval J with x e J C I where f is semi-convex, then this is true in I. Proof. Let J be a maximal open subinterval of / where / is strictly increasing and convex. If J is not empty then the right end point XQ is

28


equal to that of / . In fact, if XQ E / then / is by hypothesis semi-convex in a neighborhood of XQ, and Theorem 1.6.2 proves that the convexity extends to a larger interval. Hence there can only be one such maximal interval J , and / must be decreasing to the left of it, again by Theorem 1.6.2. Quasi-convexity is not a local property, for a locally quasi-convex function can be monotonic in a number of intervals separated by intervals where it is constant. However, if / is locally quasi-convex and not constant in any open interval, then / is quasi-convex. We leave the verification to the reader. 1.7. Convexity of t h e m i n i m u m of a one parameter family of functions. The maximum of a family of convex functions is convex by Theorem 1.1.3, but the minimum is usually not. However, the following theorem shows that convexity of the minimum of a family of functions, convex or not, is decided by local conditions. T h e o r e m 1.7.1. If X C R is an open interval, / = [a, 6] C R is a compact interval, and u ^ C'^{X x I), then U{x) = mint^/ u{x,t) is convex in X if and only if the following three conditions are fulfilled: (i) Ifx G X I;u{x,t) (ii) Ifx eX (iii) Ifx GX (1.7.1)

then u'^{x,t) does not depend on t when t G J{x) = {t E = U{x)}. andtG J{x) then u'^^{x,t) > 0. andte J{x) \ dl then < , ( x , t) + 2 < , ( x , t)X + < ( x , t)X^ > 0,

A G R.

Then U G C^'^(X), that is, U G C^(X) and U' is locally Lipschitz uous. IfY is the open subset of X defined by (1.7.2)

Y = {x e X;t ^ dl and u'^ti^,t) > 0 when t G J{x)},

then U G CîY) (1.7.3)

contin-

and

U"{x) = min K J x , t ) - < , ( x , t ) V < ( ^ , ^ ) ) .

For almost all x e X (1.7.4) U"{x) = u'^,{x,t) - u'^,{x,tflu'l,{x,t)

^^Y-

ift G J{x) \ dl and < ( x , t ) > 0,

U"{x) = u'^^{x,t) ift G J{x) and t G dl or < ( x , t ) = 0. Proof. First we prove that the conditions (i)-(iii) are necessary. If U is convex then for every x G X there is some c G R such that U{x -\- h) > U{x) + ch when x-\-h e X, hence u{x -\-h,t~[- s) > u{x, t) + ch,

if t G J{x),

x + h E X, t-h s e I.

CONVEXITY OF THE MINIMUM OF A FAMILY

29

With 5 = 0 this imphes u'^{x,t) — c and u'^^{x^t) > 0 since X is open, which proves (i) and (ii) (and also the uniqueness of c, so that U ^ C^). li t ^ dl we also obtain u'f.{x^t) = 0 and that the second differential of u is non-negative at (x,t), which proves (iii). (The condition (1.7.1) is also necessary at points t G J{x) fl dl where u'f.{x^t) = 0.) To prepare the proof of sufficiency we prove a lemma which also clarifies the role of the condition (i) in Theorem 1.7.1: L e m m a 1.7.2. Let X cH be an open interval and I a compact set. If u E C{X X / ) then U(x) = min^^/ u{x, t) is continuous. If X 3 x H^ U{X^ t) is in C^ fortel and u'^ is continuous in X x I^ then U is locally Lipschitz continuous in X. We have U G C^{X) if and only ifu'^{x, t) is independent oft G J{x) = {t E I;u{x,t) = U{x)}, and then we have U'{x) = u'^{x^t)

when t G J{x)^ x G X.

If in addition I is finite and x H^ u{x^t) is in C^ then U G C^(X) and

U'\x)=

min < , ( x , t ) ,

xeX.

If I = / i U /2 where Ij are disjoint, and X 3 x \-^ u{x, s) — u{x, t) is convex when 5 G / i and t G /2, then either U{x) = miute/i u{x, t) for all x E X or U{x) — mintG/2 u{x,t) for all x ^ X. Proof. Ii K C I is a compact subinterval then the uniform continuity of u in K X I implies that U is continuous, and ii \u'^\ < M in K x I then M is a Lipschitz constant for U in K. If C/ G C^ then the derivative of the non-negative function u{x,t) — U{x) with respect to x must vanish at every zero, that is, when t G J{x), so u'^{x^t) — U'{x) for every t G J{x). Conversely, if u'^{x,t) is continuous and independent of t when t G J{x) and V{x) is defined as this common value, then V is continuous since {{x,t) e X X I;t e J{x)} is closed. By Taylor's formula U{x + h) < min u{x -\-h,t) < U{x) + V{x)h -f- o{h). t£J{x)

For every ^ > 0 we can find a neighborhood u of J{x) in / where \u'^{x, t) — V{x)\ < 6. For sufficiently small h we obtain U{x -\- h) = m.inu{x 4- ft, t) > U{x) 4- m.inu'^{x, t)h -f o{h) >U{x)

+

Vix)h-6\h\+o{h),

which proves that U is difFerentiable at x with derivative V{x).

30


Now assume that / is finite and that u{x, t) is a C^ function of x. To prove the last statement, assume that the set F = {x G X; U[x) = minu{x,

t)}

is not empty. It is obvious that F is a closed subset of X . li x E F and U{x) — u{x^s) for some 5 G / i , we choose t £ I2 such that u{x^t) = U{x) = u ( x , s ) , hence u'^{x^t) — u'^{x,s). Then the convex function X 3 X 1-^ u{x, s) — u{x, t) is non-negative in X , so s can be dropped from / in the definition of U. But this implies that x is an interior point of F , so F is open and closed, hence equal to X. Let xo G X and let /Q be the set of all t G J{XQ) such that u'^^{x{)^t) has the minimum value a. From the result just proved it follows that there is a neighborhood XQ of XQ in X such that U{x) = mint^/o ^(^5^)5 hence IQ n J{x) 7^ 0, when x G XQ. Since 7i^(x, f) — u\xo, t) — a{x — XQ) = o{x — xo),

t G /Q,

it follows that U'{x) — U'{XQ) — a{x — XQ) = o(x — XQ), hence U"{XQ) — a. Thus lJ"{x) = u'^^{x^t) for some t £ IQ when x G XQ, so U" is continuous at Xo, which completes the proof. End of proof of Theorem 1.7.1. From condition (i) and Lemma 1.7.2 it follows that U G C^{X) and that U'{x) = u'^{x,t) for every t G J ( x ) . If Xo G y then J(xo) is a finite set { t j , . . . , t ^ } . By the implicit function theorem the equation 'uj(x, t) = 0 has for A: = 1 , . . . , A/' a unique C^ solution t — tk{x) in a neighborhood of XQ such that t^(xo) = t^, and J ( x ) = { t i ( x ) , . . . ,tiv(3:)} then. Thus Y is open, and in some neighborhood of Xo we have U{x) = in.miemY, (b) {x E E'^ J{x) n dl{x) 7^ 0} is discrete, hence countable. In fact, if x^y E E and a G J{x) fl J{y) then

(C/'(2/)-C/'(x))+ 0 when t = v?(x), X £ LJ. Then U{x) < u{x, ^{x)) = v{x), x e uj, and v G C^(a;). In F = {x e u;]v{x) = U{x)} we have v'{x) = C/'(x), hence v"{x) — U"{x) iix is not an isolated point in F and U' is differentiable at x. Isolated points in F are countable, and we can account for all (x, t) £ Xx (a, 6) with u^{x^ t) = 0, u'l^{x^t) > 0 by countably many functions ip as just considered. Since v'[x) = u'^{x,t) and v"{x) = u'^^{x,t) - \y-

x\/N,

if \y - x\ < 1/N.

If y is also in E^ then \U'{y) - U'ix) - u'Ux, t{x))iy - x)\ < o{\x -y\

+ \tix) -

t{y)\)

by Taylor's formula, for U'{y) = Uy{y^t{y)), U'{x) = n^(x,t(x)), and IAJ'^(X, t{x)) = u'^ti^^ ti^)) — 0 by definition oiE^ and condition (iii). Hence we can for fixed N choose (^ > 0 so that for x^y E EN \y - x\/N

< il/2N){\x

~y\ + \t{x) - t{y)\),

if \x-y\
0 so that x i-^ U'{x) + Ax is increasing in a neighborhood of K^ which means that U{x) + \Ax'^ is

34


convex there. If we apply Theorem 1.7.1 to v{x^t) = u{x,t) -f- Âx"^ then (i), (ii) follow from (i), (iii) there, for (1.7.1) means that v^^ — u'l^ > 0 and that < ? = Q in X, and that ul^l^

- u'^^^ > 0

inX

when u[ = 0.

Let I C H be a compact interval and set ujix) — inf u(x, t), when I HTx ^ ^. teinT^ If the interval Jx = {t E I CiTx] u{x^ t) = ui{x)} is compact and non-empty when I nTx 7^ 0, then uj is a convex function in every component of the open subset of R where it is defined. Proof. Let XQ be a point such that iHTx^ ^ 0. We can choose a compact interval K such that

Jxo CK CinTx,,

dKnJx,

Cdl,

CONVEXITY OF THE MINIMUM OF A FAMILY

35

for TxQ is open. Then we have u{xo^ t) > uj{xo)^

when t G dK \ 9 / ,

and we can choose an open neighborhood u of XQ such that u x K C X and u{x, i) > u{x, s), ii X e (jj, t e dK \ (9/, s E JXQBy the quasi-convexity of u it follows that UT(X) = inf u(x,t), ^ teK ^ ' ''

X E u,

hence uj is convex in u by Corollary 1.7.4. Remark. If/ is any interval such that the hypotheses in Corollary 1.7.5 are fulfilled with / replaced by any compact subinterval, then it follows that ui is convex in the intervals where it is defined. In fact, if Ij are compact subintervals increasing to / , then uj. j uj^ and since uj. are convex it follows that uj is convex.

CHAPTER II CONVEXITY IN

A

FINITE-DIMENSIONAL VECTOR

SPACE

Summary. Section 2.1 presents basic facts on convex sets — such as convex hulls and extreme points, intersection and separation properties — and on convex functions. It ends with some convexity properties of hyperbolic polynomials which will be important in Section 2.3. The Legendre transformation is extended to several variables in Section 2.2, where we also give some applications to game theory and linear programming. The role of the Legendre transformation in Fourier analysis is discussed in Section 2.6. The main topic in Section 2.3 is inequalities between mixed volumes, in particular the Brunn-Minkowski and Fenchel-Alexandrov inequalities. A related result of H. Weyl on the volume of tube domains in a Euclidean space is also given. Section 2.4 is a brief discussion of the smoothness properties of projections of convex sets, and in Section 2.5 we study convexity in a projective rather than an affine space.

2 . 1 . D e f i n i t i o n s a n d basic facts. In Chapter I convex functions were always assumed to be defined in intervals. When we pass to functions of several variables we must first introduce convex sets which will replace the intervals. Definition 2.1.1. Let F be a vector space over R. A subset X of F is called convex if every line intersects X in an interval, that is, (2.1.1) Aiî + A2a;2 G X

when Ai, A2 > 0, Ai + A2 = 1, and a:i,a;2 G X.

A function / : X —> ( - 0 0 , +00] is then called convex if (2.1.2)

/ ( A i x i + A2X2) < A i / ( x i ) + A2/(x2) when Ai, A2 > 0, Ai 4- A2 = 1, and 0:1,3:2 G X.

The condition (2.1.2) means precisely that the epigraph of / , {(x, t) eVeR;x

eX,t>

f{x)},

is convex. We can therefore concentrate our discussion on convex sets and obtain results on convex functions as corollaries.


37

Exercise 2.1.1. A subset X of F is called starshaped with respect to XQ if XXQ + (1 — X)x E X for every x E X and 0 < A < 1. Show that for every X the set of XQ such that X is starshaped with respect to XQ forms a convex set (which may be empty). The condition (2.1.1) implies a seemingly stronger one, N

(2.1.1)'

N

YJ ^3^3 ^ ^

if >^3 > 0^ XI \ = 1^ ^3 ^ ^•

1

1

This follows by induction with respect to N. For let iV > 2, Ajv 7^ 1 and assume that (2.1.1)' has been proved with N replaced by iV — 1. Set fjij = Xj/{1 — Aiv)j 3 < N. Then ^^ ~ IJLJ = I so Jî " /^j^j G X by the inductive hypothesis, hence N-l

X = {1 — Ajv) 2Z ^3^3 "^ ^N^N G X.

It follows at once that we can make a corresponding extension of (2.1.2). In Section 1.1 we touched briefly on notions of affine geometry. It will be useful to make some additional remarks here before proceeding with the main topic. A subset W of the vector space V is called an affine subspace if Wo = {x-xo;x

G W}

is a linear subspace of V for some XQ eV. This requires that XQ eW^ if xi is any other element of W it follows that Wo = {x -xi,x

and

G W},

for a; — a: 1 = x — xo — {xi — XQ). The definition as well as Wo is therefore independent of the choice oi xo E W. In view of the definition of a linear subspace of V it follows that W is an afiine subspace if and only if for arbitrary Ai, A2 G R and xi^X2 £W we have Ai(xi - xo) + A2(x2 - xo) + xo G VF, which implies that (2.1.3)

Aioî 4- A2a;2 eW

if Ai + A2 - 1, x i , 0:2 G W.

38

II. CONVEXITY IN A FINITE-DIMENSIONAL VECTOR SPACE

The proof that (2.1.1) imphes (2.1.1)' also shows that (2.1.3) is equivalent to the seemingly stronger condition N

(2.1.3)'

N

Yl ^J^J ^ ^

if Z l î " ^' ^^ ^ ^ •

1

1

When N = 3 this means that Aiî + A2X2 + (1 - Ai - A2)x3 = Ai(xi - xs) -f A2(a;2 - X3) -\-xs

eW,

which shows that (2.1.3)', hence (2.1.3), is sufficient for W to be an affine subspace. The dimension of W is by definition equal to the dimension of WQ. If xo,xi,... ,Xn G W, then xi — XQ, ... ^Xn — XQ are linearly independent if and only if Y^ Xj{xj - xo) = 0 =^

Xj = 0, j =

1,..., n.

This means precisely that XQ, ... ,Xn G W are affinely independent in the sense that n

n

The dimension of VF is thus one less than the supremum of the number of affinely independent elements. If XQ, •.. ,Xn G W are affinely independent and n = dimVF, then every element x e W can be written uniquely in the form X

= J2^^^^

where ^ A ^ - 1 ,

0

for this is equivalent to x — XQ = Yî ^ji^j ~ ^0) and AQ = 1 — ^ ^ Xj. We shall therefore call XQ,. .. ,Xn an affine basis. If Vi and V2 are vector spaces then a map / : Vi —> V2 is called affine if the graph {(x, / ( x ) ) ; a: G Fi} is an affine subspace of Vi 0 V2, that is, n

(2.1.4)

n

/ ( X ; A,^,) = 5 ] A,/(a;,), 1

1

n

if x, G Fi, j ; A,- = 1. 1

For / to be affine it suffices that this condition is fulfilled with n = 2. (See also Section 1.1.) Note that (2.1.4) also defines the notion of affine map


39

from an affine subspace of Vi to an affine subspace of V2; such a map can of course be extended to an affine map defined in all of Vi. The intersection of any family of affine subspaces is an affine subspace in view of (2.1.3). For every subset EofV the intersection ah(J5^) of all affine subspaces containing E is therefore an affine subspace, the affine hull of E^ and N

(2.1.5)

N

8ih(E) = { ^ XjXj; Y, ^j = 1. Xj eE,

N = l,2,...

}.

Equivalently, if XQ E £", then ah(£') is the sum of XQ and the linear hull oi {x — xo,x e E}. If the affine hull has finite dimension n it is therefore sufficient to take iV = n -I- 1 in (2.1.5) and fix n H- 1 affinely independent elements x i , . . . , Xn-î E E, one of which can be chosen arbitrarily. The following proposition is trivial but important: P r o p o s i t i o n 2.1.2. IfT is an affine map V\ -^ V2 where Vj are vector spaces, and Xj is a convex subset ofVj, then TXi = {Tx;x E Xi} and T~^X2 — {x e Vi\Tx G X2] are convex subsets ofV2 and V\. By definition a subset X of F is convex if and only \iT~^X is an interval for every affine map T \^-^V. Thus the definition of convex sets is forced by the second part in Proposition 2.1.2 if convex subsets of R are to be intervals. P r o p o s i t i o n 2.1.3. The intersection of any family X^, a E A, of convex subsets ofV is convex. For every subset EofV the intersection ch.{E) of all convex sets containing E is therefore a convex set, called the convex hull of E. We have N

(2.1.6)

ch(J5) = {YXjXj]\j

N

> 0 , ^Xj

= 1, Xj e E, i V - 1 , 2 , . . . } .

Proof. Since {(AQ, . . . , XN) ^ R^"^^; Xj > 0, ^ Q XJ = 1} is a convex set, the set on the right in (2.1.6) is convex for fixed N and XQ^ . . . ^x^. All points obtained from the points XQ, ... ,X]\^ or 2/0? •• • ^VM can also be obtained using the points XQ,. .., xjsf^yo,..., yu together, which proves that the set on the right of (2.1.6) is convex. By (2.1.1)' it is contained in every convex set containing £", which proves the proposition. Note the analogy of (2.1.6) with (2.1.5), and recall that in (2.1.5) it suffices to take N — 1 equal to the dimension of F , or even the dimension of ah(J5). To prove an analogue for convex sets we shall first make some

40


geometric observations concerning simplices, which we actually encountered already in the proof of Proposition 2.1.3. The convex hull E oi k -}- 1 afRnely independent points XQ, .. • ,Xfc G V is called a k simplex. All k simplices are afRnely equivalent. One representation is obtained by taking rro = 0 and Xj = ( 0 , . . . , 1 , . . . , 0) G R^ with 1 at the j t h place only when j = 1 , . . . , A:. Then ( A i , . . . , A^) G E ii and only if Aj > 0, j = 1 , . . . , fc, and Xî Aj < 1. Another more symmetric representation is obtained by taking for Xj the point (AQ, . . . , A^) G R^"^-*^ with Aj = 1 and the other coordinates 0. Then E is defined by

A , > 0 , j = 0,...,A:,

5Â,-l.

Indirectly we have already encountered these representations several times. In fact, given a standard simplex E in some vector space W, with vertices ^^"05 • • • 5 ^fc5 and points Xj G V there is a unique affine map (f from ah(£') to V such that ^{(TJ) = Xj, jf = 0 , . . . , fc, and

'P(E) = { J ] XjXyAj

> 0, ^

A^ = 1}.

Thus (2.1.1)' means that the affine image of a simplex is contained in X if this is true for the vertices. L e m m a 2.1.4. IfT is an afRne map from the affine hull of the k simplex S with vertices (JQ, . . . ,cr^ ô an affine space W of dimension < k, then TS = UiTSj where Sj is the k — 1 simplex obtained by omitting the vertex Proof. It is obvious that yJ^TSj C TS. li x e S then L = T'^Tx is an affine space of dimension > 1 which contains a;, and TL = Tx. In the realisation above of S as a compact subset of R^, the boundary dS is equal to UQSJ. If y G L , 2/ 7^ X, then / = {t G R; (1 — t)x -\- ty E S} is a compact interval, and if t is in the boundary then (1 — t)x -^ ty E L D dS, which proves that Tx G TdS = UQTSJ. We can take j 7^ 0 here unless the affine line spanned by x and y intersects S only in So- But then it would be contained in the affine subspace spanned by So and intersect the boundary dSo of So there, by the (fc — l)-dimensional version of the result already proved. Since dSo C U^dSj^ this completes the proof. Combining Lemma 2.1.4 with Proposition 2.1.3 we obtain the following result:


41

T h e o r e m 2.1.5 ( C a r a t h e o d o r y ) . If E is a subset of a vector space V and the afRne hull W of E has finite dimension n, then n n ch{E) = (^XjXj-.^Xj 0

= l,Xj > 0 , Xj G-E, i = 0 , . . . , n } .

0

Here XQ can be fixed arbitrarily in E. If E is compact, then ch{E) is also compact. Proof. We have to show that if x is of the form ^ Q XJXJ as in (2.1.6) with N > n^ then x is also of this form with N replaced hy N — 1 and one of the points x i , . . . , x^ omitted. But that is just Lemma 2.1.4 applied to the affine map sending the vertices d o , . . . , crjv of the N simplex to XQ, .. •, xj^. Prom the part of the theorem already proved it follows that ch(£^) is the range of the continuous map n (Ao, . . . , Xn^Xo, . . . , Xn) ^-^ / ^ XjXj, 0

defined in the subset of R'^"*"^ x E'^'^^ where Xj > 0, j = 0,... ,n and Xô ^j ~ ^- ^^ ^^ compact HE is compact, so ch{E) is then compact, which proves the theorem. If the intersection of a family of aSine subspaces of a vector space of dimension n is empty, then it is clear that one can find n-\-l subspaces with an empty intersection, for the dimension must decrease if the intersection decreases. There is an analogue of this too for convex sets; as we shall see later it is closely related to Caratheodory's theorem: T h e o r e m 2.1.6 (Helly). If Xj, j = l , . . . , i V , are convex subsets of a vector space V of dimension n, and if Xi^ fl • • • fl Xi^ ^ 0 for arbitrary z'o,..., 2n ^ {I5 • • • 5 -^}j then H^Xj ^ 0. This remains true for Xj, j E J, when J has arbitrary cardinality, if all Xj are closed and some finite intersection is compact. Proof. By a standard definition of compactness the second part of the statement follows from the first, which will be proved by induction with respect to N. Thus assume that iV > n 4- 1 and that the statement has already been proved with N replaced by iV — 1. Then we can for i = 1,... ,N find Xi e r\i<j n + 1 these points are aSinely dependent, so we can find ( A i , . . . , A^v) 7^ 0 so that N

Y^ XiXi = 0,

N

^Xi

= 0.

42


Write Xi = Xf - A" where Xf = max(±Ai,0).

Then A = E f \ ^

=

E f A- > 0, and N

N

N

N

^ = J2iK/^)î = E(VA)^^, E(^^Â) = E(v/A) = 1We have x G H^Xj. In fact, if A^ > 0 then A~ = 0, and since Xi E Xj when i ^ 3^ the second representation of x proves that x £ Xj, in view of the convexity of Xj. Similarly, using the first representation, we find that X E Xj if A^ < 0, and all together this proves that x E HJ^Xj, so the intersection is not empty. E x e r c i s e 2.1.2. Write down explicitly what Kelly's theorem states when n = 1, and give a direct proof. Prove that if Xj C R"^ are intervals, that is, products of intervals in the different coordinates, then d^Xj ^ 0 if Xj n Xfc 7^ 0 for arbitrary j , k < N. We can give Theorem 2.1.6 a more precise form if we make the following definition: D e f i n i t i o n 2.1.7. By the dimension of a convex subset £" of a (finitedimensional) vector space V we shall mean the dimension of the affine hull Sih.{E), that is, the supremum of all n such that there exists an affinely independent (n -f- l)-tuple XQ,. .. ,Xn E E. Note that the simplex with vertices XQ,. .. ,Xn has interior points as a subset of the affine hull W of E/ii XQ,. .. ,Xn is an affine basis for W. The interior (boundary) of E considered as a subset of W will be called the relative interior (boundary) of E. The relative interior is dense in E, for it contains some point y, and if x is any point in E, then Xx -\- {1 — X)y is in the relative interior when 0 < A < 1. E x e r c i s e 2.1.3. Show that the convex set E is relatively open if and only if every affine line intersects E in a. point or an open interval. E x e r c i s e 2.1.4. Show that Theorem 2.1.6 holds if n is replaced by 1 + m i n d i m X j or by dimch(UXj). We have seen that every point x in a fe simplex has a unique representation of the form x = ^ Q Xjaj where aj are the vertices and Xj > 0, 5^0 A, - 1. We shall now discuss to what extent a similar representation holds for an arbitrary convex compact set K in a. finite-dimensional vector space V. It is clear that such representations cannot avoid using points which are extreme in the sense of the following definition:


43

Definition 2.1.8. A point x in the convex set K is called extreme if 2

X = A i ^ i + A2a:2, Xj G K , Xj > 0, j = 1, 2, Y"] Aj = 1 1

An apparently stronger condition is that N

x = ' ^ XjXj,

N

Xj e K,

Xj >0,

j = 1,...,N,

^

1

Xj = 1 1

—^

X J ^= x^ J = i , . . . , i V.

The equivalence follows as usual by induction for increasing N. It is obvious that the extreme points of ch(j5') must belong to E \i E is an arbitrary set. E x e r c i s e 2.1.5. Determine ch(£'i U E2) and the extreme points of ch(£;i [JE2) where Ei - {(xi,X2,0);x? + x | = 1} and E2 = {(1,0, ± 1 } . (Note that the set of extreme points is not closed!) T h e o r e m 2.1.9 (Minkowski). IfK is a compact convex set of dimension n in a Gnite-dimensional vector space, then every point x E K can be written in the form n

(2.1.7)

n

X = Y^XjXj,

Xj > 0, j = 0 , . . . , n , ^ A ^ - = 1,

0

0

where all Xj are extreme points of K. The representation all X in the relative interior of K unless K is a simplex.

is not unique for

Proof. The existence of a representation (2.1.7) is obvious if n = 1, for K is then an interval and the extreme points are the end points of the interval. For n > 1 the proof will be given later by induction with respect to n, after we have proved the required separation theorem. However, the lack of uniqueness is easily proved at this time. If K is not a simplex then there are at least n-\- 2 different extreme points XQ,. .. ^x^-^-i- These must be afBnely dependent, so we have n+l

n+1

0

0

where not all JJLJ are equal to 0. Then x = ^ Q ^ J / ( ^ + 2) can also be written a: = ^ Q Xj(H-e/Xj)/(n + 2), and all the weights (1-f £/Xj)/(n + 2)

44

IL CONVEXITY IN A FINITE-DIMENSIONAL VECTOR SPACE

are positive if e is small enough, but not for all £ > 0 nor for all £ < 0. Hence we can find two representations of x as center of gravity of only n -f 1 points among XQ, . . . , Xn-\-iRemark. Theorem 2.1.9 has an infinite-dimensional analogue called the Krein-Milman theorem. It states that a compact convex set is the closed convex hull of its extreme points. Refinements due to Choquet [2] state that it is enough to take averages of extreme points weighted by positive measures, and he has also proved an analogue of the last part of Theorem 2.1.9 with an appropriate definition of simplices. We shall now discuss separation theorems. We shall refer to them as the Hahn-Banach theorem although that is only appropriate in the infinitedimensional case. T h e o r e m 2.1.10. Let X be a finite-dimensional vector space hyperplane W such that XQ ^ W function f in V with f{xo) = 0
0, then tiXi + t2X2 = (ti + t2){{ti/{ti + t2))Xi + (t2/(ti + t2))x2) G X. If X G X then —x ^ X. Since V \ {0} is connected we can find a point y ^ 0 in the boundary of X , and y ^ X because X is open. In every neighborhood of y we can find x G X , hence —x ^ X , which proves that —y ^ X. Hence X f) Ky = 0, which proves the theorem when d i m F = 2. (b) If dim V > 2 we let W he a. subspace of maximal dimension with W^ n X = 0. We have to prove that W is a hyperplane, that is, that dim{V/W) = 1. Let T : V —^ V/W be the canonical map, assigning to an element in V its residue class mod W. Then T X is convex (by Proposition 2.1.2) and relatively open (see Exercise 2.1.3). If diin{V/W) > 1 we take a two-dimensional subspace H and note that H fl T X is also convex and relatively open. By part (a) of the proof there is a line Hi C H with HiHTX -= 0, which means that ( T - ^ i f i ) H X = 0. Since T-^Hi D W this contradicts the maximality of W and completes the proof.


45

Corollary 2.1.11. Let X be a convex and closed subset of a finitedimensional vector space V. If XQ ^ X there is an afHne hyperplane containing XQ which does not intersect X, that is, there is an affine function f with f{xo) < 0 < f{x), xeX. Proof. Let U be an open convex set such that XQ e U and [/ D X = 0. Then Y = {x-y;xeX,yeU}= \J{x-y]yeU} xex is open and convex, and 0 ^ Y. By Theorem 2.1.10 we can find a linear form L such that L > 0 in y , that is, L{x) > L{y) ii x e X and y ^ U. Now sup L{y) = L{xo) + sup L{y - XQ) = L{xo) + 6 yeu yeu where ^ > 0, so f{x) = L{x) ~ L{XQ) — 6 has the required properties. The following statement is closer to Theorem 2.1.10: Corollary 2.1.12. IfX is a closed convex subset of a Gnite-dimensional vector space V, and ify is on the boundary of X, then one can find a nonconstant affine f such that f{y) = 0 < f{x), x E X. The affine hyperplane {x E V\ f(x) = 0} is called a supporting plane of X. Proof. The statement is trivial if d i m X < d i m F , so we may assume that the interior X° of X is not empty. By Theorem 2.1.10 we can choose / so that f{y) = 0 < f{x), x G X ° , which implies that 0 < f{x), x eX. Corollary 2.1.13. An open (closed) convex set K in a finite-dimensional vector space is the intersection of the open (closed) half spaces containing it. Proof. This is an immediate consequence of Theorem 2.1.10 and Corollary 2.1.11. Remark. The infinite-dimensional analogue of Theorem 2.1.10 and its corollaries is the Hahn-Banach theorem which can be found in every text on functional analysis. E x e r c i s e 2.1.6. Prove, with the notation in Corollary 2.1.12, that if V = BJ^ with the Euclidean norm | • | and scalar product (•,•), and if X is compact, then {x-yj')t\f'\ > 0.

Conclude that X is the closed convex hull of the set of all x G dX such that there is a closed ball B D X with x G dB; such points are extreme.

46


(Hint: Assume that this is not true and derive a contradiction by taking t large in the preceding statement.) End of proof of Theorem 2.1.9. It is no restriction to assume that K has interior points. If x G dK^ it follows from Corollary 2.1.12 that we can find an affine function / with / > 0 in if and f{x) = 0. The intersection Ki = {y E K; f{y) = 0} is convex and compact, x G iî, and dim if i < n. It is clear that an extreme point of if i must be an extreme point of K. If the theorem has been proved for lower dimensions we can therefore write n —1

(2.1.8)

n —1

X = Y ^ AjXj,

where Aj > 0, Xj G i f i , 0 < j < n, Y ^ A^ = 1,

0

0

and Xj are extreme points of i f i , hence of if, which is better than the claim for a general point. In particular, we conclude that extreme points exist. If X is in the interior of if we choose an extreme point Xn and note that the intersection of if with the line through x and Xn is an interval with end points Xn and rr' G OK. Hence x' is of the form (2.1.8), and since X = fiXn + (1 — f^)x^ for some fi G (0,1), we obtain n-l

X = y . ^ji^ ~ f^)^j + M^n5 0

which completes the proof. We shall now discuss an example. An nxn matrix {ajk) is called doubly stochastic if the elements are non-negative and the row and column sums are equal to 1, djk > 0, j,fc = l , . . . , n ; ^

a^-fc = 1, fc = 1,. • •, n; ^

j=i

a^-fc = 1, j = 1 , . . . , n.

k=i

Important examples are matrices where all the elements are equal to 0 or 1, which means that each row and each column contains precisely one element equal to 1. Thus there is a permutation a of { 1 , . . . , n} such that ajk = 1 if k = aj and ajk = 0 ii k ^ aj. One calls (ajk) a permutation matrix then, fo^ Yl'k = l^jk^k = âjT h e o r e m 2.1.14 (G. BirkhofF). The nxn doubly stochastic matrices form a compact convex set Dn of dimension (n — 1)^, and the extreme points are the permutation matrices. Thus every doubly stochastic matrix (ajk) is of the form ^jk = X ^ ^^^ k=aj

where A^ > 0, ^

A^ = 1,


47

and a runs over the group 6 ^ of permutations of { 1 , . . . , n } . One can take A^ = 0 except for at most (n — 1)^ + 1 permutations. Proof. (2.1.9) defines Dn as the intersection of closed half spaces and 2

affine hyperplanes in R'^ , so it is clear that Dn is closed and convex. Since 0 < ajk < 1 it is also clear that D^ is bounded, hence compact. The equations in (2.1.9) mean that (Ink = l - ^ J ^ - ^ ' ^ ' ^ < '^5 o.jn = l - X / ^ - ^ ' ^ ' ^ ^ '^' ^^^ ^ 2 - n + 22 j Q \i Xj > 0, j — 1 , . . . , n, and we shall write ax = ( x ^ i , . . . , x^^) if cr G Sn- If 0 < a G R"^ we shall write x^ — Y[^ ^V • T h e o r e m 2.1.15 (Muirhead). Ifa,/3e the following conditions are equivalent:

IC" and a > 0,/3 > 0, then

(i) There is a constant C such that

(2.1.10)

Z l ^""^ - ^ 5 ] ^'''''

0 < X G R^;

48


(ii) the inequality (2.1.10) holds with C = 1; (iii) P is contained in the convex hull of {crof; a G S n } ; (iv) ifa"" and ^* are the decreasing rearrangements of a and (3, then

^/3;<x:«;, i an when a E A. E x e r c i s e 2.1.8. Determine the extreme points in the set of n x m matrices (ajk) such that (2.1.9)''

ajk>0,

i =

l,...,n,fc-l,...,m;

n

m

âjk

4-Onj sin6>,

0{6)n,j =

-Ok-\-i,jsin6-\-On,jCos6,

the other elements being the same as in O. Then

o{e)^^\ai,. . . , < ) = (A,.. .,Pk,aie),pk+i, •. .,pn-2,bie)) where a(0) = a and a(|7r) = b. Hence we can choose 0 so that a{6) = 13^-1 and 0 ( ^ ) ^ ( a i , . . . , a * ) is then a permutation of /?, for h{6) = /J^ since a{0) -f- b{0) — Pn-i + f^n for every 6. This completes the proof. Exercise 2.1.9. Prove that the diagonals /? of the Hermitian symmetric n X n matrices H = (Hjk) with prescribed eigenvalues ( a i , . . . , a^) G R^ form a convex set described by Theorem 2.1.15. Conclude using Exercise 2.1.7 that if / is a convex symmetric function in R^ then H H-> / ( a i , . . . , o^n) is convex in the n^-dimensional space of Hermitian symmetric matrices. (Hint: Prove that n

j,k

= l

n

j,k

= l

where [/ = (C/j/^) is a unitary matrix.) Our next example of convex sets occurring in analysis concerns the numerical range of an operator. T h e o r e m 2.1.18 (HausdorfF). Let V be a (Rnite-dimensional) vector space over R with a Euclidean scalar product denoted by E{-, -), and let Ti, T2 be two linear transformations V -^ V. Then the numerical range O = {{E{Tix,x),E{T2X,x))]x

G V,E{x,x)

= 1} C R^

is convex if dim V > 2, and if dim V = 2 it is an ellipse, possibly to an interval or a point. Proof. Let 61,62 be two orthonormal vectors, that is, E ( e i , e i ) = ^(62,62) = 1, £"(61,62) = 0,

degenerated

52


and write Qj{x,y) = \{E{TjX,y) •Ê{x,Tjy)), which is a symmetric bilinear form. Then e{9) = cosêi + sin0e2 is a unit vector for every ^ G R, so f7 3 ( Q i ( e ( e ) , e ( ^ ) ) , Q 2 ( e W , e ( ^ ) ) ) . Since 2cos2e = 1 + cos 20, 2sin^0 = 1 - cos26>, 2 sin l9 cos 0 = sin2 0 (if D < 0), when 6 goes from 0 to TT. When d i m F = 2 the proof is now complete. When d i m F = 3 we can choose a third unit vector 63, orthogonal to ei and 62, and continuously deform 61,62,63 to 62,61,-63 by changing 61,62,63 to - 6 3 , 62, 61 to 62,63,61 to 62, 61, - 6 3 by rotations around one of the vectors. Let the corresponding vectors be 6^, 0 < t < 3. Then the range i?(6*,e2) must for some t G [0,3] pass through an arbitrary point y in the interior of ^(61,62) — i?(6i,62) = i?(6i,62), for the winding number of the oriented ellipse around y changes sign as t goes from 0 to 3, and it is a continuous function of t when y ^ i?(6i, 62). Hence Q contains the interior of i?(6i, 62) also. Since two arbitrary points in V belong to some two-dimensional plane, the convexity of Q follows. C o r o l l a r y 2.1.19. Let V be a Gnite-dimensional complex vector space with positive definite Hermitian scalar product denoted by E, and let T be a complex linear transformation in V. Then the numerical range n = {E{Tx,xy,x is a convex subset

e V,E{x,x)

= 1}

ofC

Proof. If V has complex dimension one then T is just multiplication by a constant, and the statement is true. If the complex dimension of ^ is > 1, we can regard I^ as a real vector space of dimension > 4, with Euclidean form Re E. If we write T = Ti -\- iT2 where Tj are Hermitian symmetric, then E{Tx, x) = E{Tix, x) + iE{T2X, x) = Re E{Tix, x) + i Re E{T2X, x), and the statement follows from Theorem 2.1.18. It is the corollary which is usually referred to as the theorem on the numerical range.


53

Exercise 2.1.10. Prove that with the notation in Corollary 2.1.19, the numerical range of T contains the convex hull of the spectrum of T, that is, the set of 2: E C such that T — z Id is not invertible if Id is the identity operator, and that there is equality if T is normal. Exercise 2.1.11. Prove that the numerical range of a doubly stochastic nxn matrix acting on C^ with the standard Hermitian metric is contained in the convex hull of the roots of unity e^^*^/^ where 0 < z/ < /x < n. Theorem 2.1.14 studied a special convex polyhedron: Definition 2.1.20. The convex hull of a finite set E C V is called a convex polyhedron. It is clear from the definition that a polyhedron is compact and that the extreme points form a subset of E\ hence it is finite. T h e o r e m 2.1.21. A convex polyhedron X CV is the intersection of a, finite number of closed half spaces H.IfX has interior points, they can be chosen so that X fl dH has interior points relative to dH. Conversely, any bounded intersection of a finite number of closed half spaces is a convex polyhedron. Proof. Let ^ be a finite subset of V with ah(£J) = V^ so that the polyhedron ch(jE') has interior points. If a: G dch{E) then there is a half space H D ch{E) such that (dH) H ch{E) contains x and dim{{dH) D ch{E)) = n — 1 where n — d i m F . This is obvious when n = 2. A general proof can be made quite parallel to that of Theorem 2.1.10. Let x = 0^ and choose H D E containing 0 so that i^ = dim(W fl ch{E)), W = 9 i ? , is as large as possible. If z/ < n — 1 we let W be the linear hull oiWr]ch{E) and observe that the image K of ch{E) in V/W, which has dimension n — i/ > 2, is also a convex polyhedron with 0 on the boundary. From the (n — z^)-dimensional case it follows that there is a half space Hi in V/W containing K with 0 E dHi and diin{K fl dHi) = n — u — 1 > 1. Then the inverse image H2 of Hi in V contains ch(£^), and (dH2) fl ch(E) contains W f) ch.{E) as well as points not in W, so the dimension is > z/. This contradiction proves our claim. Now an afline hyperplane is uniquely determined by n afiinely independent points, so it follows that ch{E) is the intersection of at most (dimv) half spaces, where \E\ is the number of points in E. Conversely, let X be the intersection of a finite number of closed half spaces i J j , j = 1 , . . . , J , and assume that X is bounded. If x is an extreme point of X , then it follows (as in the proof of Theorem 2.1.14) that X must be in n = dim V aSine hyperplanes dHj intersecting only at x, which proves that there are at most {^^^y) extreme points. By Theorem 2.1.9 X is the convex hull of this finite set.

54

11. CONVEXITY IN A FINITE-DIMENSIONAL VECTOR SPACE

We shall now prove some simple facts on convex functions which are analogues of results proved above for convex sets. We label them to suggest the connection. T h e o r e m 2.1.2'. Let T be an affine map V\ —> V2 where Vj are vector spaces, let Xj be convex subsets ofVj, and let fj be a convex function in Xj with finite values. Then Fi{x) = f2iTx),

X G T-'X2,

F2{y) = inf h{x), y G TXx, Tx=y are convex functions if F2{y) > —00 for some y in the relative interior of TXi. Proof. The statement on Fi is an immediate consequence of Proposition 2.1.2 or (2.1.2). To prove the one on F2 let yi,y2 G TXi and Ai,A2 > 0, Ai 4- A2 = 1. For arbitrary Xj G Xi with Txj = yj we have F2{Xiyi + A2y2) < /i(AiXi + A2X2) < Ai/i(a:i) + A2/i(x2). Taking the infimum over all permissible Xj we obtain F2{Xiyi + X2y2) ^ XiF2{yi)-hX2F2{y2)- Ifi^2(2/i) = - o c it follows that F2(Ai?/i + A22/2) = - 0 0 for all ?/2 G T X i , so F2 would be equal to —00 in the relative interior of T X i , which is against the hypothesis. The proof is complete. T h e o r e m 2.1.3'. If X is a convex set and fa, a E A, is any family of convex functions in X with values in (—00, H-oo], then f{x) — sup^,^^ / a ( ^ ) is also convex; it is the smallest convex majorant of all f^- If info, ^ A / a ( ^ ) < 00 for every x E X, then the largest convex minorant is g{x) = i n f { ^ Xafa{Xa)]Xa

>0,XaeX,

^

Ac, = 1, ^

XaXa = x},

provided that g{x) > — 00 for some x in the relative interior of X. In the definition of g it is assumed that only a finite number of X^ are ^ 0, and it does not change if we require that at most 1 + d i m X of them are / 0. Proof. The statement about / is a very easy exercise. The convex hull of the epigraphs of the f^ is defined by t > g{x) with equality excluded if the infimum in the definition of g is not attained. We may assume that X has interior points. Hence it follows from Caratheodory's theorem 2.1.5 that it sufiices to take 2 + d i m X of the AQ, non-zero in the definition of g. In taking the convex hull over the point x we may let one of the points be (/a(^) + 1?^) fo^ some a such that fa{x) < 00. But a positive weight for this point would increase the infimum, which completes the proof. (That g{x) > —00 everywhere follows as in the proof of Theorem 2.1.2'.) The results on diff'erentiability of convex functions proved in Section 1.1 can easily be extended to several variables:


55

T h e o r e m 2.1.22. Let f be convex and finite in an open convex subset X of a Gnite-dimensional vector space V. Then f is locally Lipschitz continuous, f'{x;y)= Ihn (fix+ hy)-fix))/h exists for every x £ X and y E V, and the limit is uniform in y when y is bounded. The Gateau differential (or subdifferential) f'{x]y) is convex and positively homogeneous, f{x',ty) (2.1.12)

= tf{x-y),

t>Q,yeV,

f ( x ; y i + y 2 ) < f ( : r ; ^ i ) + /'(x;7/2), f'{x-y) 0; x/t e B} = 1/ sup{t; tx G B}

is a convex positiirely homogeneous the origin, that is, F{x)>0 (2.1.14)

function,

ifO^xeV,

which is positive except at

F{0)=0]

F{tx) = tF{x),

t > 0,

xeV-

F{x + y) 0 such that dx^B is locally concave in the subset of XR = {rr G X ; |x| < R} where dx,B < 2^^. Then f{x) = max{-dx,B{x),

6R{\X\

- R)/R),

x G

XR,

is locally convex in XR, the supremum is 0 in every component, and when t < 0 then {x G XR\f{x) < t} is compact since it stays away from the boundary of X and from {rr; |x| = R}. Hence it follows from Theorem 2.1.25 that every component of XR is convex, and when i? —> 00 it follows that X is convex. Thus the first three conditions are equivalent, and by

60


Theorem 2.1.25 we also have (v) => (i). On the other hand, if X is convex then is convex in X (see Exercise 2.1.13) and tends to +oo at the boundary and infinity, so (iv) is fulfilled. The proof is complete since (iv) = > (v). Note that (iii) is a local condition on X at dX: If X is locally convex in the sense that every point in dX has a (convex) neighborhood U such that C / n X is convex, then it follows that X is convex (if connected). This local property is also evident from the following: T h e o r e m 2.1.27. Let X he an open connected set in R"^ which is not convex. Then there is a point z G dX and a quadratic polynomial q with q{z) — 0, ^'(/2^) 7^ 0, such that q{x) < 0 implies xEXifx is close to z, and {t,d)q{z) = 0,

{t,dfq{z) eh^^

\h\ < 6,

so when x = ZQ the minimum in (2.1.18) is a non-degenerate minimum achieved Sit h = 0. By the implicit function theorem it follows for x near ZQ that the minimum is still a non-degenerate minimum taken at a point h which is a C^ function of x. Hence (/? is a C^ function in a neighborhood of ZQ, ^'{ZQ) = i^, and since (p{zo + ht) < —eh'^ it follows that {t^d)(p = 0 and that {t^d)^cp < —2e at ZQ. For the quadratic polynomial

q{x) = Yl ^ X ^ o ) ( ^ - zor/a\

+ rj\x - zo\'

|a| cp{x) if rj > 0 and |x — ZQI is small, so q{x) < 0 implies ip{x) < 0, hence x ^ X. If r/ is sufRciently small we have {t,d)^q{zo) < 0. The proof is complete. Corollary 2.1.28. An open connected set X C R'^ with C^ is convex if and only if the principal curvatures are > 0 at every point.

boundary boundary

We shall finally discuss another characterization of convex sets in terms of a distance function, now in the exterior of the set. We begin with a general fact about distance functions, for the sake of simplicity stated only for the Euclidean distance in R^. L e m m a 2.1.29. Let F be a closed set in IV^ and set f{x) = min|x — z\'^ where \'\ is the Euclidean norm. Then we have f{x -\-y) = f{x) + f'{x] y) + o(|y|), f'{x\y)

7/ -> 0,

= min{(2y,x - z);z e F,\x - z\^ =

where f{x)}.

This is Lemma 8.5.12 in ALPDO, and we refer to the proof given there. Note that the Gateau differential f'{x]y) is a concave, positively homogeneous function of y just as it would have been if / had been a concave

62


function. If dpix) = \/f{x) is the distance from x to F , it follows from Lemma 2.1.29 that dp is Gateau differentiable at every point x ^ F , with differential y H^ min{(y, x - z)j\x - z\\ z^F,\x-

z\ =

dpix)},

and that dp is differentiable at x if and only if there is a unique point in F at minimal distance from x. This is used in the statement of the next theorem: T h e o r e m 2.1.30 ( M o t z k i n ) . Let F be a closed set in R^. Then F is convex if and only if the Euclidean distance function dp is differentiable at every point in ZF, or equivalently for every x ^ F there is a unique point in F at minimal distance from x. Proof. If F is convex and z ^ F has minimal distance from x G C F , then {y — z,x — z) < 0 for every y E F, for

\x-zf

< \x-z-e{y-z)\'^

= \x-z\^-2e{x-z,y-z)+e^\y-z\^,

0 < 6 < 1.

(Note that this gives another proof of Corollary 2.1.11.) Taking e = 1 we conclude that |x — >2^| < |x — ?/| if F 3 y ^ z. The condition in the theorem is therefore necessary. Assume now that F is not convex. We must then prove that there is an open ball B with B H F = 9 such that B H F contains more than one point. That F is not convex means that we can find an interval [xi, 0:2] with xij^2 ^ F , xi 7^ X2, such that the interior is in C F . We place the origin at the midpoint so that X2 = —xi, and write B{w^r) = {x; |x — it;| < r } , w eIC',r>0. Choose ^ > 0 such that 5 ( 0 , ^ ) D F := 0. If B{w, r) D 5 ( 0 ,

Q),

Xj i B{w, r), j = 1,2,

then r > |t(;| + g,

\w ± x i p > r^,

hence \w\'^ + | x i p > r^ > {\w\ + ^)^, so (2.1.19)

\W\ 0, so x G T^ \ip{x) 7^ 0. Hence VQ is open and closed in {x G V]p{x) ^ 0}. Now TQ is connected, for if X G r^, then x^t0 eVe when t > 0, hence Ax + /x0 G Tg for all A > 0 and // > 0. This proves that TQ is even starshaped with respect to 0, and

that r^ = r. If 2/ G r and ^ > 0 is fixed, then Eyê = [x eV] p{x + iE0 + isy) 7^ 0, Re s > 0} is open, and 0 G Ey^^ since p{ie0 + isy) = {is)^p{£0/s + y) = 0 implies 5 < 0. If X G Eyê, then p{x + ie0 + isy) 7^ 0 by Hurwitz' theorem if

64


Re5 > 0, and this remains true when R e s = 0 since x + isy is real then. Hence Eyê is both open and closed, so Eyê — V- Thus p{x + i{ee + y)) ^ 0 ,

if X G R^, y G r , £ > 0.

Since F is open, this remains true when ^ == 0, so the equation p(x + ^y) = 0 has only real roots, for ii t = ti -^ it2 is a root with ^2 7^ 0 we would get p{{x + tiy)/t2 4- iy) = 0. Hence y can play the role of ^, so 7 is starshaped with respect to every point in F, hence convex. Finally we shall prove that if 7/ G F and x G V, then 5 1—> {p{sx + y)Y^'^ is concave when sx -\-y ^V. With t^ G R we have m

p{x+ty)=p{y)Y[{t-ti),

m

thus p{sx+y) = s'^pix-hy/s) =

p{y)Y[{l-sti),

and sx -i- y ET means that 1 — st^ > 0 for every i. If f{s) = logp{sx + y) then

hence

by Cauchy-Schwarz' inequality, which proves that s f-^ p{sx + y)^^'^ concave. The proof is complete.

is

E x a m p l e . The simplest example, and the one motivating the name, is the hyperbolic quadratic form p{x) ^=^ x\ — x\ — - • • — x^ in R^. The convexity of F just means the convexity of the forward light cone, defined by xi > \/x\ + • • • 4- x^, and the concavity of -yjp means that we have the reversed triangle inequality \/p[x + y) > \/p{x) + \/p{y) for all x, ?/ in the forward light cone. Other important examples are the space of n x n real symmetric matrices (or hermitian symmetric matrices), with p{x) — d e t x and 6 equal to the identity matrix. The cone F consists of the positive definite matrices then. If X G F and y EV^ then the fact that t \-^ p{x + ty) has only real zeros implies that this is also true for the derivative with respect to t; if x G F then all the zeros are negative. The derivative is equal to mq{x-{-ty) where q[x) = p{y^x^... ,x) with p denoting the polarization of p. (See Appendix


65

A.) Hence q is also hyperbolic with respect to every vector in F. If ^(^) > 0 then p > 0 and g > 0 in F. By the concavity of p^/"^ it follows according to Proposition A.l, condition (iv), that

Since q satisfies the same hypotheses as p^ with F replaced by a cone containing F, we can prove by induction: Corollary 2.1.32. If the hypotheses and p{6) > 0, then (2.1.21)

p{xu ...,xm)>

of Proposition

V^XiY'"^ • "V{xmf''^,

2.1.31 are fulhlled

X i , . . . , x ^ E F.

Ifp is complete in the sense that p{x + ty) = p{x) for all x^ t impUes y = 0, then X H^ p{x, ...,x, yn+i, • • •, Um) is complete ifn > 2 and 2/n+i, "-.ym ^ F. In particular, this is a non-degenerate quadratic form if n = 2, and p ( x i , . . . , Xm) > 0 ifxi G F \ 0 and Xj G F when j = 2 , . . . , m. Proof. Taking y = xi and assuming that (2.1.21) is already proved for hyperbolic polynomials of degree m —1, we obtain with q{x) = p{y^ x , . . . , x)

which proves (2.1.21). Alternatively we could use that X^p{x,X,X^,...,Xm)

is a hyperbolic quadratic form with forward light cone containing F, if 3^3,... ,Xm G F. This implies the condition (A.7) with f = p (see Remark 2 after Proposition A.2), and hence the inequalities (A.8)-(A.10) follow. To prove the last statement it suffices to show that q is complete iim, > 3 and q is defined as above. Suppose that q{x + tz) = q{x) for all x and t. In particular, q{y-^tz) = q{y)^ that is, q{ty-\-z) — q(ty)^ sop{ty-\-z)—p{ty) = a is independent of t. Since the zeros of p{ty) + a = t^p{y) + d must all be real, it follows that a = 0. Thus p{y -h sz) — p{y) ^ 0 for all 5, so it follows that y + 52; G F for every s. Hence (sx 4- y + sz)/{s + 1) G F,

if x G F, 5 > 0,

and letting s —^ oo we conclude that j:-f >2: G F for all x G F, hence x-hz ET then. We can replace z by tz for any t, so x -\- tz E V for all t and x G F. Thus p{z 4- sx) cannot have any zeros ^ 0, so p{z + sx) = s'^p{x)^ that is, p{x + tz) — p{x) for all t and all x G F. But this implies 2; = 0 since p is complete.

66


Exercise 2.1.14. Prove that ii p{z) = Yl^ ^j^^ i^ a polynomial in z G C, then the zeros o{p'{z) are either zeros of ^(z) or else contained in the relative interior of the convex hull of the zeros oi p{z). (Hint: Study p'{z)lp{z).) Exercise 2.1.15. Let if be a convex subset of C, and let ^{z) = Y^^ djZ^ be a polynomial in z G C. Prove that the set of all w E C such that all the zeros of p{z) = w are contained in K is a convex set. (Hint: Apply the preceding exercise to a product {p{z) — wi)'^^{p{z) — W2)'^'^-) 2.2. T h e Legendre transformation. In Definition 2.1.1 we introduced the notion of convex function, with values in (—oc, cx)], defined in a convex subset X of the vector space V. It is clear from the definition that the function remains convex if we extend it to V by defining / = -hoo in V \X. For the sake of convenience we shall always assume in this section that the convex functions considered are defined in all of V. P r o p o s i t i o n 2.2.1. If f is a convex function in V then X = {xeV-f{x)

0, for it is clearly convex on a line where x is constant, and d? f{x^ ax + b)/dx'^ = 2b'^/x^ > 0. The limit of f{x,y) as (x^y) -^ 0 along a ray is 0, but the limit along the parabola x — ay'^ is 1/a. The best way out of the problem is to make / lower semi-continuous: P r o p o s i t i o n 2.2.2. Let f be a convex function in V, and set fi{x)=

lim/(y),

xeV.

Then / i is convex and fi{x) < f{x) for all x, with equality if x is in the interior of X = {x e V]f{x) < oo} in ah(X) or interior in V \X. The function / i is lower semi-continuous and is called the lower semi-continuous regularization of f. Proof. We just have to verify that fi{x) > —oo for every x, and we may then assume that X has an interior point XQ. But then f{x) — / ( X Q ) is

THE LEGENDRE TRANSFORMATION

67

everywhere bounded from below by the Gateau differential f'{xo,x (cf. Theorem 2.1.22).

— XQ)

In the example above we can make / convex in V by defining f{x, y) = -j-oo when a; < 0, but the lower semi-continuous regularization is equal to 0 at the origin. The reason for the importance of lower semi-continuity is that while the epigraph

{{x,t)

evîi-xev,t>f{x)]

is convex if and only if / is convex, it is dosed if and only if / is lower semi-continuous. This will be a decisive point when we extend the notion of Legendre transformation. Denote by V the dual space of V^ and let V xV' 3 ( ^ , 0 ^~^ (^^0 be the bilinear form defining the duality. Definition 2.2.3. Let / be a convex lower semi-continuous function in V^ not identically -j-oo. Then the Legendre transform ( = conjugate function = Fenchel transform) / of / is defined by (2.2.1)

/(0-sup((a;,0-/(^)),

i ^ V .

xev

As the supremum of a family of affine, hence continuous, functions on V it is clear that / is convex and lower semi-continuous (cf. Theorem 2.1.3'), and we have an inversion formula: T h e o r e m 2.2.4. If f is a convex lower semi-continuous function in V, not identically -j-oo, then the Legendre transform f has the same properties in V , and (2.2.2)

f{x)=snp{{x,0-f{0)-

Proof. From (2.2.1) we know that f{x) + f{() > (x,^), hence f{x) > snp^^Y,{{x,() — / ( O ) . To prove the opposite inequality we choose any c G R with c < f{x). Since (x, c) is not in the closed convex epigraph of / , it follows from Corollary 2.1.11 that we can find ^ G V^', & G R and ^ > 0 such that (2.2.3)

( x , 0 ^ c b > {y,0+tb^£.

iiyeV,t>

f{y).

Assume at first that f{x) < oc. When y = x it follows from (2.2.3) that 6 < 0, and dividing by \b\ we obtain with rj = ^/\b\ {x,rj)-c>{y,r])-t

ifyeV,

t>f{y).

68


Hence /(r/) < (x,?/) - c, that is, c < {x.rj) - f{rj) < f{x), or f{x) which proves (2.2.2) when f{x) < oo. In particular, we conclude not identically H-oo. Now assume that f{x) = +oo. Since f{y) some y eV/it follows from (2.2.3) that 6 < 0. If 6 < 0 we obtain

> /(x), that / is < oo for as before

that f{x) > c, so it remains to examine the case where b = 0. Then it follows from (2.2.3) that (x,^) > ( y , 0 + ^ if f{y) < oo, so it follows from (2.2.1) that f{v + to < fiv) + t{{x,0

-e),

iit>0,rje

V.

Hence / > ) > {x, 7/ + to - t{{x, 0~e)-

hv)

= {x, v)+t£-

m ,

t>0,7,€V'.

If we choose rj so that f{rj) < oo and let t —> +oo, it follows that f{x) +00, which completes the proof.

=

Exercise 2.2.1. Show that the Legendre transform (i) of / -f C is / — C, if C is a constant; (ii) o f / ( . - a ) i s / + ( a , . ) , i f a E y ; (iii) o f / ( ^ ) i s / ( . / ^ ) , i f t > 0 ; (iv) o f t / is tf{-/t) iit>0. Show that if / = +00 in the complement of a linear subspace Vi oiV, then / is the pullback of the Legendre transform of the restriction of / to Vi by the natural map V -^ V^, with kernel equal to the annihilator of Vi in V. T h e o r e m 2.2.5. Let / i , . . . , /iv be lower semi-continuous convex functions in V such that f = ^^ fj ^ +oo. Then f is the lower semicontinuous regularization of the convex function (infimal convolution)

(2.2.4)

S{0 = jni

Ê/.-fe).

Proof. Let f{x) < oo. Then ^( —oo for every ^, for fj{x) j = 1 , . . . ,iV, hence

< oo,

It is clear that g is convex, and it follows from (2.2.4) and the inversion formula (2.2.2) for fj that N

sup((a;,0 - 9(0) = s u p E ( { x , e , ) - fjiQ)

N

= 5;/,(a;) =

f{x).


69

If ^1 is the lower semi-continuous regularization of ^, it follows that gi > f, hence gi < f hy Theorem 2.2.4, and since gi > f we obtain gi = f. The preceding result simplifies if miiij fj{^) < oo for every ^, for g is then finite everywhere, hence continuous, so f = g. In particular, we obtain when iV = 2 by taking the value at the origin: Corollary 2.2.6. Let / i and /2 be convex lower semi-continuous functions in V with / i + /2 ^ +oc, and assume that /2 < -f oo everywhere. Then we have (2.2.5)

inf (/i(x) + ^(a;)) + inf ( / i ( 0 + / 2 ( - 0 ) = 0-

We shall give an application of (2.2.5) later on, but we continue now with an analogue of Theorem 2.2.5: T h e o r e m 2.2.7. Let fc^, a ^ A, be convex and lower semi-continuous. Then f = sup^,^^ fa has the same property. Iff ^ -hoo then f is the lower semi-continuous regularization of the largest convex minorant g of{fa},

where X^ > 0,a ^ A, only Gnitely many AQ, are ^ 0, and YlêA -^Q; = 1The proof is so close to that of Theorem 2.2.5 that we leave it as an exercise for the reader. If iT is a convex subset of V then (2.2.6)

^K{X) = \ ' 1^ H-oo,

.^ , ^ It X f K

is convex, and cpx is lower semi-continuous if and only if K is closed. We have (pK = H where (2.2.7)

if(0 = sup(x,0, xeK

i^V,

is called the supporting function of K. That (fK just takes the values 0 and +00 is equivalent to t^K = ^K for every t > 0, and therefore equivalent to positive homogeneity of H, HitO = mo,

t>0,^£V'.

Hence the following theorem is a consequence of Theorem 2.2.4:

70


T h e o r e m 2.2.8. If K C V is closed and convex, then the supporting function H defined by (2.2.7) is lower semi-continuous, convex and positively homogeneous, that is, (2.2.8) ^ ( A i ^ + A26) < AiF(î) + A 2 F ( 6 ) , if Ai > 0, A2 > 0, ^ 1 , 6 e r . Conversely, every such function H in V is the supporting function of one and only one closed convex set K CV, and it is defined by (2.2.9)

K = {xe

V; {x, 0 < H{0,

V^ G V'}.

Note that UK = V then H{0) = 0 but i f ( ^ = +00 when

CÔ.

T h e o r e m 2.2.9. Let Ki,...,K^ be closed convex subsets of V with supporting functions i i f i , . . . , H^. If K = Ki H • " H Kjsf ^ 0 then the supporting function of K is the lower semi-continuous regularization of N

If K = ^ then hm^_Q h{^) = - 0 0 . If Ki is compact, then either h = —00 or else h is the supporting function of a convex compact set. Proof. The first statement follows from Theorem 2.2.5. In any case h is positively homogeneous. If lim^_^Q /i( —oc then h is bounded from below in a neighborhood of the origin, hence > — 00 everywhere, so the lower semi-continuous regularization of h is the supporting function of a set k C K, hence K ^ 0. Since HjOj + 6>) < Hj{^j) + Hj{e) we have If Ki is compact then Hi is continuous, and it follows that h is continuous or = —00, which proves the last statement. Remark. Theorem 2.2.9 gives another proof of Helly's theorem (Theorem 2.1.6) for closed sets one of which is compact. In fact, if h{0) < 0 then Caratheodory's theorem applied to the epigraphs of Hj (see the proof of Theorem 2.1.3') proves that this remains true if we just keep n + 1 of the sets Kj suitably chosen. E x e r c i s e 2.2.2. With the notation in Theorem 2.2.9, describe a set with the supporting function if 1 H h H^ and a set with the supporting function maxj=:i^...^jv HjLet K now be a closed convex set containing the origin, and denote the supporting function of K by H. That 0 G if is equivalent to if > 0. Set (2.2.10)

K° = U; ^ G V, {x, 0 < 1 Va; G i^} = U ; ^ G V, H{0

< 1}.

It is clear that K° is closed, convex and contains the origin. One calls K° the polar of K.


71

T h e o r e m 2.2.10. The polar K"" defined by (2.2.10) of a closed convex set K CV containing the origin is a closed convex set in V containing the origin, and we have (2.2.11)

K = {x; xeV,

{x, ^ < 1 V^ G X ° } .

Proof. It is obvious that {K°y D K. If x G (i^°)°, then ( x , 0 < 1 when ^ G K"", that is, when i J ( 0 < 1, and it follows that {x,^) < H{0 for all (. This is obvious if H{(,) = oo and otherwise we obtain {x^t^) < 1 when tH{() = H{t^) < 1, t > 0, and that means precisely that {x,(,) < H{(,). Hence x G if by (2.2.9), which proves (2.2.11). The other parts of the statement have already been verified. Exercise 2.2.3. Show that if K is a closed convex cone then K° is a closed convex cone, if°-U;êF',(x,0.) If iiT is a compact convex set with the origin as interior point then H{() < cxD for every (^, and H is the distance function associated with the polar K° of K, which is also convex and compact with the origin as interior point. Hence the supporting function iif° of K° is the distance function associated with K^ and we have (2.2.12)

H{0

= snp{x,0/H''{x);

H%x) =

sup{x,0/H{0-

If K is symmetric with respect to the origin, then K° is also symmetric, and (2.2.12) is the well-known relation between dual norms. Exercise 2.2.4. Let (p be an even non-constant convex function on R with (p{0) = 0, and denote the Legendre transform by ip. Show that n

if = {:Z;GR^; J](^(xj) < 1} 1

is convex and closed, and that n

2_]xjyj 1

n

< max(l, yîp{xj)),

if x G R'^, y G K°.

1

Deduce that X^^ ^(T/^) < 1 if 2/ G if °. From Theorem 2.1.22 we know that a convex function is Gateau differentiable in the relative interior of the set where it is finite. The differential is convex and positively homogeneous, so it is the supporting function of a set in the dual space. It has a natural interpretation in terms of the Legendre transform:

72


T h e o r e m 2.2.11. Let f be a convex lower semi-continuous function in V which is Rnite in a neighborhood ofxeV. Then the Gateau differential y j-> f'{x; y) at x is the supporting function of K = {^;êV',{x,0

= fix) +

f{0}-

Proof. If ^ is in the subset Ki of V with supporting function f^{x; •), then f{x + y)-f{x)>r{x;y)>{y,0. V^V, and we obtain 0 > sup((y, 0 - fi^ + 2/) + fix)) yev

= fix) - {x, 0 +

no-

Since f{x) + / ( ^ ) > (x,^) for 3\\ x e V and ^ E V, it follows that f{x) + / ( O — (^5^5 SO ( e K. On the other hand, if ^ G ii' we have by definition fix + y) > {x + y,0 so f'{x\y) claimed.

- ho

= (y,0 + / W ,

y^V,

> (y," "> 2/71)5 which are arbitrary points in the simplices X and y , m

X = {x 6 R'";a;i > 0 , . . . , a ; ^ > 0 , ^ 0 ; ^ = 1}, (2.2.13)

/ y - { j / G R " ; ? / i > 0 , . . . , y „ > 0 , 5 3 y f c = l}.

Then A expects to pay the amount m

(2.2.14)

A{x,y)

n

= ^ây^x^j/fc.


73

For a given strategy x oi A the player B can choose his strategy y so that A must pay raaxy^y A{x^y). If A chooses x to minimize his cost, he can count on paying at most

no matter how B bets. On the other hand B can choose his strategy so that A must pay at least\ maxmin A(x,y) yeY xex no matter how he bets. The following theorem shows that these two values are equal, so two rational players may as well pay this amount and quit! T h e o r e m 2.2.12. Let A be the real bilinear form (2.2.14), and define X,Y by (2.2.13). Then (2.2.15)

maxminA(x,2/) = minmax A(x,?/). yeY xex xex yeY

If s is this common value one can therefore choose x^ £ X, y^ E Y such that A(x^7/) <s, ye y ; A(a;,7/^) >s, xeX. Proof. The right-hand side of (2.2.15) is min/i(x), xex

fi{x) = maxA{x,y) yeY

= max (:r,afc), ajt == (aijt, • • • ,ttmfc)l 0, {x, ak) > Ck, k = 1,..., L = {yeR'^;y>0,

,a^ €

{y,a;) < bj, j =

m},

l,...,n};

here x > 0 and y > 0 means that all coordinates are non-negative, a* e R - is defined by YJ^{^.cik)yk - E i (2/,^*)^^'

and

Proof. The closed convex set K is not empty since ( t , . . . , t) G K for large positive t. Since inf {x,b) = — sup (a;, —b) = —HK{—b), the problem is to give another expression for the Legendre transform of the function / which is 0 in iiT and +00 in CK. Let g{x) = 0 when 0 < X G R^ and let g{x) = +00 otherwise, and let hk{x) = 0 when x G R'^ and {x,ak) > Ck^ hk{x) = +00 otherwise. Then g{(,) = 0 when ^ < 0 and g{(,) = +00 otherwise (cf. Exercise 2.2.3), and hk{0 — ^^k if ^ = ^Cbk^ A < 0, and hk{() = +00 otherwise. Since f = g -{-^hk it follows from Theorem 2.2.5 that / ( ^ ) is the lower semi-continuous regularization of m

m

F{0 = i n f i j ; XkCk] A - ( A i , . . . , A^) < 0, ] ^ Aâ^ > 0 If all coordinates ^j are negative, then F{^) < 00, so F is then continuous, hence equal to / . In particular, replacing A by —y we obtain m

-f{-b)

= -F{-b)

^ snp{Y^ykCk]y

m

> 0 , ^ ^ ^ ^ ^ < b}.

GEOMETRIC INEQUALITIES

75

The condition YlVk^^k < & can be written m

{b,x) > ^yk{x,ak) 1

n

= ^Xj{y,a)),

x > 0,

1

so it means that {yâ^) < bj, j = 1,... ^n^ which completes the proof. The advantage of results like (2.2.16) is that they allow one to find both upper and lower bounds for the quantity in question. 2.3. G e o m e t r i c inequalities. Let F be a vector space of finite dimension n, and denote by IC{V) or /C for short the set of convex compact subsets of V. In /C we have a natural partial vector space structure: If Ki,K2 ^ IC and Ai, A2 > 0, we can define (2.3.1)

A1K14-A2K2 = {AiXiH-A2X2;xi eKi,X2

e K2}.

In terms of the corresponding supporting functions this is equivalent to (2.3.1)'

Hx^Ki-hX2K2 =

ÎHKI

+

^2HK2'

Thus the map K \-^ HK from /C to the vector space Ci{V') of continuous functions on the dual space F ' , which are homogeneous of degree 1, identifies /C with a convex cone in Ci{V'). The identification carries the operation defined by (2.3.1) to the standard operation on functions, so we conclude that all the usual rules of computation are valid for (2.3.1). We shall write W = {HK\K G /C}. The linear space W — W is much smaller than Ci{V'). We can regard /C as a convex cone in this vector space. Fix a Lebesgue measure in F , for example, by choosing a basis for F or a Euclidean metric. Then the volume Q3(if) is well defined for every K e IC. (We could avoid fixing a measure by regarding 2T(if) as an element in the space of translation invariant densities on V.) The following basic theorem is due to Minkowski (see Appendix A for the definition of a polynomial): T h e o r e m 2.3.1. The map IC 3 K \-^ V{K) mial of degree d i m F .

is a homogeneous

polyno-

In the proof of the theorem as well as later results in this section it is convenient to approximate general elements in /C either by polyhedra or smoothly bounded ones: L e m m a 2.3.2. If K e IC and ft is a neighborhood find Ki^K2 E )C such that K cKiCn,

K

cK2Cn,

of K, then one can

76


Ki is a convex polyhedron with interior points, and K2 has C^ with strictly positive Gaussian curvature.

boundary

Proof. We may assume that J7 is convex. Every point in K is in the interior of some simplex with vertices in Q. We can cover K by finitely many such simplices. The convex hull of all of them has the properties required of Ki. When constructing K2 we may assume that the origin is an interior point oi K. Denote the corresponding distance function by p{x)^ so that K = {xe V]p{x) < 1}, and let 0 < x ^ C^{V)^ Jxdx = l. Then the C ^ function Pe{x) =

p{x-

ey)x{y) dy + £\xf,

5 > 0,

is strictly convex, since p is convex, and for small e Pe{x) < p{x) -h Ci£ < 1 + CiS, X eK]

ps{x) > C2 > 1, X

^n.

For small e the unique minimum point of p^ belongs to K, so K2 = {x;p^{x) < 1 + 2Cie} has C^ boundary with strictly positive principal curvatures since pe is strictly convex, and if C if2 C Jl if 2Cie < C2 — 1. The proof is complete. Proof of Theorem 2.3.1. Let i f i , K2 be polyhedra with the origin as interior point, and denote the supporting functions by ii"i and H2. If Ai, A2 > 0 then the supporting function of Aiifi + A2if2 is îHi + X2H2. I f ^ G F ' \ { 0 } , then (2.3.2) {x e Aiifi + A2if2; {xÔ = A i i i i ( 0 + X2H2{0} = Aiifi(e) 4- X2K2{0.

K,{0 =

{êKj;{x,0=Hj{0]-

Let ^^, u — 1 , . . . , A/", be the directions such that the set on the left has dimension n — 1; it is clear that this condition is independent of the choice of the positive numbers Ai, A2. By Theorem 2.1.21 the boundary of Aiifi + A2if2 is the union of the sets (2.3.2) with i — iv, and these can only have sets of dimension n — 2 in common. Hence N

5J(Aiifi + A2if2) - 5^2J(ch(0, Aiifi(^,) + A2if2(^.))) 1 ^ = - 5 ] ( A i i f i ( ^ , ) + A 2 i i 2 ( e . ) m . ( A i i f i ( e . ) + A2if2(^.)). ^

1


77

Here 03^ denotes the measure ^ ( ( - , 0 ) ^^ ^he hyperplane {x G V] (x,^) = 0}, transported to the parallel supporting plane. We may assume that Theorem 2.3.1 has already been proved for lower dimensions, for it is obvious when n = 1. Then we know that 53^(AiKi(^) + X2K2{0) is a polynomial in Ai,A2 > 0, of degree n — 1, and the theorem follows for polyhedra in dimension n. In the general case we can use Lemma 2.3.2 to choose sequences of polyhedra K^ IKj, /i-ôo. Then ^{XiKi

+ X2K2) = lim 2J(Aiiff +

X2K^),

fJL—KX)

and since the limit of a polynomial of degree n is a polynomial of degree n, the theorem follows in general. As shown in the appendix it follows from Theorem 2.3.1 that we can polarize the volume function: D e f i n i t i o n 2.3.3. The symmetric n-linear form 93(iCi,..., Kn) on IC^ with Q3(if) = 53(jFf,... ,i oo of the same expression with f± replaced by the defining functions / ^ of K^. Here ip is any continuous function on the unit sphere. But this follows at once since / ^ -^ f± uniformly in A;, and Theorem 2.1.22 shows that for x' E k minus a null set the functions f± and / ^ are all diff"erentiable and df*^{x') —> df±{x') as /x -^ oo. (Note that we have a uniform bound for these derivatives.) Thus we have defined the surface measure dS and defined the exterior unit normal almost everywhere with respect to dS. In view of Lemma 2.3.2 we have at the same time proved: P r o p o s i t i o n 2.3.4. IfV points, then (2.3.4) is vaUd.

is Euclidean, B,K

e IC and K has interior

Remark. With V Euclidean and B equal to the unit ball we have in particular that n^{B^K^... ^K) is the Euclidean area of dK. Also for more general, non-convex sets K, one can adopt the definition ]h^{^{K

+ eB) -

V{K))/e

as the definition of the area (outer Minkowski area) of dK. Prom the proof given above it is easy to see that this agrees with the usual definition whenever dK is smooth. Taking for B some other convex compact set with interior points we get a definition of area with respect to a general distance function as in Theorem 2.1.23. Another interesting case of (2.3.4) occurs when we take for B an interval By = ch{0,y} or equivalently ch{ —|^, | ^ } , where y EV. Then

nQJ(B„if,...,i^) = | /

\{y,a^))\dS{x)

JdK

is equal to the Euclidean length of y times the area of the projection of K in the plane perpendicular to y. Note that y »—> ^{By, K,... ,K) is convex,

80


positively homogeneous and symmetric, so a symmetric convex compact set in V and case where K is the unit ball, n^{By, K,... (n — l)-dimensional unit ball if \y\ — 1, so Cn-i = \ l

it is the supporting function of a norm in V. In the particular ,K) is the volume Cn-i of the we have

\{y.i)\du^{i).

M-1,

where du is the surface measure on the unit sphere. Hence we obtain for any K E IC nf

V{By,K,...,K)du{y)

= \ j

J\y\ = l

dS{x) [

JdK

|(y,^(a;))| da;(y)

J\y\ = l

= Cn-1 [

dS{x).

JdK

This means that Cn-i times the area of dK is the integral over y E S"^'^ of the area of the projection in the orthogonal plane of y. We shall now return to the mixed volume and prove another basic result, essentially due to Minkowski: T h e o r e m 2.3.5. IfK'-, Kj G /C and Kj C K'j for j = 1 , . . . , n, then (2.3.6)

0 < 2 J ( i ^ i , . . . , Kn) < « ( i ^ I , . . •,

K).

Proof. The statement is obvious when n = 1, so we assume that n > 1 and that the theorem has already been proved in dimension n — 1. It suffices to prove that ^(Ki^... ^Kn-i^Kn) < 9 J ( i f i , . . . ,iir^_i,if^), for iteration of this result gives the second inequality (2.3.6) in view of the symmetry, and if Kj consists of a single point in Kj, then it follows that ^{Ki,..., Kn) > » ( i f { ' , . . . , i^n) = 0- In view of Lemma 2.3.2 we may assume in the proof that Ki,..., Kn-i are polyhedra with interior points, and we assume that V is Euclidean. For any polyhedron K with interior points it follows from (2.3.4) that N

(2.3.7)

n 2 J ( K , . . . , K, K J - 5 ] F x ^

{ûMK{^,))

where d denotes (n — l)-dimensional volume in a hyperplane, K{0

=

{X£K;{X,0=HK{0},


81

and the sum is taken over the exterior unit normals of all the (n — 1)dimensional faces of K. li K = XiKi + • • • + X^-iKn-i and A i , . . . , A^-i are positive, then (2.3.2)'

K{0

= XiKiiO

+ ••• +

Xn-iKn-dO-

The vectors ^^ in (2.3.7) are then those for which this is a set of dimension n — 1, so they do not depend on A. The sets Kj{^) lie in parallel planes, but we can shift them by parallel translation to the plane [x] (x^^) = 0}, so D ( A I K I ( ^ ) H h Xn-iKji-iiO) is equal to ( n - l ) ! A i - . - A n _ i t ) ( i f i ( 0 , . . . , i f n - i ( 0 ) + --where the dots indicate terms not divisible by all Aj, j = 1 , . . . , n — 1, and the (n — l)-dimensional volumes are defined using the parallel translation just mentioned. (Recall that (mixed) volumes are invariant for separate translation of each argument.) Hence

V

By the inductive hypothesis the coefiicient oiHKyîîv) here is non-negative. If Kn C K'^ then HK^ < HK^, SO it follows that ^{Ki,.. .,Kn-i,Kn) < QJ(iî,..., Kn-i->K'^), which completes the proof. We shall now make the left inequality (2.3.6) more precise; for the sake of convenience only we assume that 0 belongs to all the bodies: T h e o r e m 2.3.6. If Ki,... conditions are equivalent:

,Kn e IC and 0 G n'^Kj,

then the following

(i) ViKu...,K^)>0. (ii) One can choose Cj G Kj such that e i , . . . , e„ is a basis for V. (iii) Ifl n'^{R -

rf


83

since the roots are real and differ at least by R — r. This is a strong form of the isoperimetric inequality. Thus the polynomial ^{K) is strictly hyperbolic in the direction 5 , considered in the vector space generated by convex bodies with non-empty interior, and then it is strictly hyperbolic in all such directions. This will prove Theorem 2.3.7 when n = 2. Proof of Theorem 2.3.8. (See Burago and Zalgaller [1, Section 1.3].) It is sufficient to prove the theorem when if is a convex polygon and r < X < R. Consider a circle of radius A with center which slides along dK. When it slides along a side we see that each half circle bounded by points with parallel tangent will slide over a set of area 2AL where L is the length of the side (Cavalieri's principle). Counted with multiplicities the area covered by the circle when the center travels once around dK is equal to ^XL{dK). The multiplicity can only be in doubt at points which have distance exactly A to a vertex or a side, and these are of measure 0. Otherwise the multiplicity with which a point x is covered is decided by counting the number of times the circle with center at x and radius A intersects dL. The multiplicity must always be even, for a circle starting outside K will finish outside K. If the multiplicity is 0 at a point x^ then the circle of radius A with center X is either entirely in the interior of K or entirely in the exterior, which cannot happen if r < A < i? and the distance from x to if is < A. Thus K -f \B is entirely covered at least twice, apart from a null set, so we have 5J(if + \B) < |(4AL(aif)) - 4AQJ(if, 5 ) , which proves the theorem. Proof of Theorem 2.3.7. Throughout the proof we shall assume that n > 2, and we shall no longer work with convex polyhedra but shift to convex sets with C^ boundary of positive Gaussian curvature with the origin as interior point. We shall denote this subset of /C by /Creg- When K G /Creg we can rewrite (2.3.4) by introducing ^{x) E S''^"^ as a new variable. Recall that dK 3 x f-> ^(x) G S'^~^ is a diffeomorphism. We extend the inverse gn-i 3 ^ )_^ x(^) G dK to R'^ \ 0 as a positively homogeneous function of degree 0. Then H{(^) = (x(^),(^), and since (dx,^) = 0 on Ta,(^)5M, it follows that H'{(^) = x{^). The surface area duj{^) on S'^~^ is equal to hidS{x) where K is the Gaussian curvature at x G dK. The principal curvatures and the radii of curvature are defined by the equations n

dxj{i) = Yl ^jkiOd^k == Rd^j, i = 1,...,n, k=i

n

Yijd^^j = 0. 1

Here HjkiO — d'^HiOl^^jd^kSince H'{l^) is homogeneous of degree 0, the radial direction is an eigenvector with eigenvalue 0 while the other

84


eigenvalues corresponding to the principal curvatures are solutions of the equation -R-^ det{Hjk - R6jk) = 0. The leading term is (—i?)"^"^, so the product K~^ of the roots is the constant term, that is, with Id denoting the unit matrix, « - ' - ^ d e t ( i ? " ( 0 + i?Id)|fl=o. Denote by D{A) the determinant of a symmetric nxn matrix A^ and let D be the polarization of D. (We recall from the example after Corollary 2.1.32 that D is hyperbolic with respect to the cone of positive definite matrices.) Thus we obtain

^-'^nD{H"{0,...,H"{OM), and we can rewrite (2.3.4) when B ^ K and K G /Creg in the form

By polarization of this polynomial in K we obtain for if i , . . . , Kn G /Creg (2.3.4)' Q J ( K i , K 2 , . . . , i f n ) = /

H,{OD{H!^{0,---.H':{OM)du;{0.

where Hj = HKJ is the supporting function of Kj. (Note that (2.3.4)' gives another proof of Theorem 2.3.5 since D{H2^... ,H!^,ld) > 0 by Corollary 2.1.32.) The left-hand side is symmetric in Ki,... ^Kn^ so this must also be the case for the right-hand side. Thus we may exchange the index 1 with any one of the indices 2 , . . . , n in the right-hand side. Our goal is to prove (2.3.8), which means that for the extension of the mixed volume function to (/C — /C)^ (2.3.8)'

2J(fc,fc,i f 3 , . . . , ifn) < 0,

if Q3(fc, K 2 , . . . , i^n) = 0,

when k is in the plane spanned by Ki and K2. (Note that if 93(fc, 7^2? • • • 5 Kn) is equal to 0 then Q3(AiA; + X2K2, Xik + AziiTz, i^s, • • •) = X'Mk,

A:, i f a , . . . ) + ^MK2,

i^s, i^3, • • •)

is not positive definite if and only if the coefficient of Af is < 0.) That k is in the plane spanned by Ki and K2 is no significant restriction on


85

the smooth homogeneous "supporting function" Z of A:, for Z + IH^ will for large positive i be strictly convex non-radially, hence the supporting function of some K\ E /Creg- Rewriting (2.3.8)' using (2.3.4)', we conclude that what must be proved is that (2.3.10)

/

Z{i)D{Z"{i\E'i{i\...,K{i)M)i^{i) 1. Now the side condition in (2.3.10) means precisely that Z is orthogonal to the harmonics of degree 0, the constants, so we conclude that (2.3.10)

86


is valid for iir2 = * • • = Hn = H with strict inequality except when Z is linear. (The vanishing for linear Z is no surprise for it expresses precisely the translation invariance of the mixed volumes.) The extension of (2.3.10) to general K2,... ^K^ G /Creg will now be carried out by a continuity argument, adapted by Alexandrov from Hilbert's proof of the Brunn-Minkowski inequality when n == 3. The symmetric bilinear form corresponding to the quadratic form in (2.3.10) is (W, Z)^

j

W{OD{Z"{il

H'^{i\

. . . , H';[i\

Id) du;{i\

for the symmetry of the right-hand side inW^Ze /Creg extends by linearity to W^Z e /Creg — /Creg- The differential operator

z^D{z"{0,HUO,---,K(OM) is elliptic, with negative principal symbol. To verify this we make it explicit at (^ = ( 1 , 0 , . . . , 0) again. As above the coefficient

n\D{Z^'{0:H!;{0.-'-,K{OM) of Ai •.. A, in D{XiZ'\0 + ^2HîO + ' ' • + K-iK(0 + A, Id) is equal to the coefficient of Ai • • • A^-i in the determinant of AiZ''(^) + X2H'^{^) -f • • • -+• Xn-iH!ll{^) with the first row and column removed. The principal symbol at the (co-)tangent vector 0 = {0i,..., On) with î = 0 is obtained when Z"{^) is replaced by —6 (8) ^, so it is equal to

(2.3.13)

e^

-Did®e,H'îo,...,Kio,id).

Apart from a factor 1/n the right-hand side is not changed if we remove the first row and column. Since D is hyperbolic with respect to the cone of positive definite matrices and complete, and since 6 0 will be chosen in a moment. The eigenfunctions Z and eigenvalues A are solutions of the elliptic equation (2.3.15)

D{Z",H'^,...,H':,ld)=^XQZ.


87

The side condition in (2.3.10) can be written

0=/ = /

zb{H'{,...,H';M)MO H2D{Z",Hl...,H';,ld)dw{i).

We choose (2.3.16)

Q = b{Hl...,H':,M)lH2

to make H2 a solution of (2.3.15) with A = 1. Note that both numerator and denominator are strictly positive. Then the side condition in (2.3.10) becomes orthogonality in the sense of (2.3.14) to this eigenfunction. Thus proving (2.3.10) means to show that all other eigenvalues are < 0. The next step is to determine the eigenspace with eigenvalue 0, that is, the solutions of the equation (2.3.15)'

b{Z",Hl...,H';M)

= Q.

By the ellipticity all eigenfunctions are in C^. In view of the hyperbolicity oiD and Corollary 2.1.32 it follows from (2.3.15)' that

b{z"{i), z"{0, H'l{ii..., K{OM) < 0, with strict inequality except when Z"{^) = 0. Since

0=/ = f

zb{z",Hi...,H';M)duj{o HsD{Z'\ Z\ R'l..., i/;', Id) rfa;(0

and jEfa > 0, we conclude that Z" = 0 so Z is a linear function. Conversely, every linear function satisfies (2.3.15)'. Thus the multiplicity of the eigenvalue zero is always equal to n, and the eigenfunctions are always the same. Now we deform our eigenvalue problem to the case of the ball studied above by introducing mj

= fiHj-\-{l-fi)H,

j = - 2 , . . . , n , 0 < / x < 1.

Here H is the supporting function of the unit ball. When /i = 0 we have only the simple eigenvalue 1, the eigenvalue 0 of multiplicity n, and otherwise negative eigenvalues. For any /x G [0,1] we know that 1 remains an

88


eigenvalue, with eigenfunction ^H2, and that 0 remains an eigenvalue with the same eigenspace. Hence standard ellipticity theory shows that no new eigenvalues > 0 can appear and that 1 remains a simple eigenvalue, which proves (2.3.10). In fact, for large enough M the inverse of the operator

z^Mz-

^Q-"^b{z",^H'^,... ,^^;',id) v ^

is compact in L^, uniformly continuous with respect to /x, has only a simple eigenvalue ( M — 1)~^ in (M~^, oo) when // = 0, and has M"^ as an eigenvalue of fixed multiplicity when /z G [0,1]. The eigenvector corresponding to the eigenvalue (M - 1)~^ depends continuously on /x G [0,1]. By continuity it follows that the largest eigenvalue in the orthogonal space of this eigenvector and the eigenvectors with eigenvalue M~^ must remain < M~^ as is known when /x = 0. Remark. By Theorem 2.3.7 and Proposition A.2 the mixed volume has much of the properties we established in Section 2.1 for hyperbolic polynomials. It is therefore natural to ask if it might actually be hyperbolic. The answer is no for n > 2 by an example already given by Minkowski. Let K be the convex hull of a ball of radius R and a point XQ outside the ball, and let B be the unit ball. Then it is geometrically clear from a picture that K 4- hB only differs from (1 + h/R)K at a distance 0{h) from XQ. Hence V{K + hB) = (1 + hlRYV{K) 4- 0(/i^), and since the coefficient of /i^ is QJ(5), it follows that V{K + hB) - (1 4- h/R)''V{K)

+ /i^(5J(B) - Q3(ii:)/i?^).

If we choose K so that V{K) — ^(-B), then i? < 1, and any number R G (0,1) can occur. Thus the equation 5J(iir + hB) = 0 has the form (l + / i / i ? r = / i ^ ( i ? - ^ - l ) which implies 1 + h/R = LJ{R-'' - 1)^/^/^ where CJJ is an n t h root of unity. Hence there are at most two real roots, so there is a non-real root if n > 2. A simple consequence of the Fenchel-Alexandrov inequalities is the Brunn-Minkowski inequality (2.3.17)

^{Ki-^K2)^

>QJ(iCi)n+9J(i^2)*,

iî,K2e/C,

expressing the concavity of K H^ QJ(i^) ^, and its corollary (2.3.18)

V{Ku . . . , K u K 2 ) > Q 3 ( K i ) ' ^ 0 3 ( ^ 2 ) ^ .

(See Propositions A.l and A.2.) When K2 is the unit ball, then (2.3.18) states that the Euclidean area nV{Ki,..., K i , ^^2) of dKi divided by the


89

volume of Ki raised to the power (n — l ) / n is at least as large as the value for the unit ball, since there is equality in (2.3.18) when Ki = K2. This is the isoperimetric inequality which has a very old history. Also (2.3.17) has been known since the turn of the century for convex sets. Since the 1930's it is also known for non-convex sets. There is such an elementary proof that it cannot be resisted although it falls outside the main topic. We use the notation m for the measure rather than 93 to indicate that in this generality Lebesgue measure must be used. T h e o r e m 2.3.9. If A and B are compact subsets ofR^, {x -{-y]X ^ A^y E B} then (2.3.17)'

and A-\- B =

m{A + 5 ) ^ / ^ > m ( ^ ) ^ / ^ -f m ( 5 ) i / ^ .

Proof. It suffices to prove this when A and B are unions of finitely many disjoint (products of) intervals, and that can be done by induction over a-\-b if a and b are the number of intervals which constitute A and B. In fact, if A and B are both intervals, with side lengths a^, 6^, 2 = 1 , . . . , n, then

by the inequality between geometric and arithmetic means, if aj -\- bj = 1 for every j . For homogeneity reasons this gives (2.3.17)' in general if A and B are both intervals. Now assume that a > 1 and that (3.2.17)' is proved already for smaller values of a H- 6. Then there is a plane Xj — constant separating two of the intervals defining A, In view of the translation invariance it is no restriction to assume that it is the plane Xj = 0 , and that m(A+)/m(A_) = m(B+)/m(5_), if A± and B± are the intersections of A and B with the half spaces defined by Xj > 0 and Xj < 0 respectively. These are constituted by at most a — 1 and at most b intervals, so by the inductive hypothesis m{A + S ) > m{A^ + 5 + ) + m(A_ -h BJ) > (m(A+)^/^ + m ( 5 + ) ^ / ^ ) ^ + (m(A_)^/^ + m ( S _ ) i / ^ ) ^ -(m(A)^/^+m(5)^/^)^, for iim{A^) = Xm{A) then m(A_) = (1 - A)m(A), m{B^) = Xm[B) and m{B-) = (1 - X)m{B). This completes the proof.

90


Remark. It is known that there is strict inequahty in (2.3.17)' unless m{A + 5 ) = 0, or A or B consists of a single point, or A and B are convex sets which differ only by a translation and a homothety. For a proof we refer to Burago and Zalgaller [1, §8.2]. Chapter 4 of the same book contains an extensive discussion of applications of the Fenchel-Alexandrov inequalities and also an algebraic proof. Prom Theorem 2.3.7 and Proposition A.2 one can obtain numerous inequalities between the mixed volumes, but we shall not write them down here. Instead we shall end our geometric discussion by deriving some formulas for mixed volumes with the unit ball B in a Euclidean space. We shall write h{^) = \^\ for the supporting function of B. Let K G /Creg have supporting function iJ, and let (2.3.19)

W^ = 5 J ( K , . . . , i^, S , . . . , B).

If 0 < z/ < n it follows from (2.3.4)' that

JS^-^

'

V

' ^

V

'

Assuming as in similar calculations above that ^ == ( 1 , 0 , . . . , 0) we find as there that the coefficient of Ai • • • A^ in D{{\i

+ • • • + X.)H'\0

-f (A,+i + .. • + \n-i)h"{0

+ A, Id)

is equal to the coefficient of Ai • • • A^-i in the determinant of aH"{C> + &Id,

a = Ai + • • • + A^, h = Kî

+ • • • + A^-i,

with the first row and column removed. We may assume that H"{£^) is diagonal, with the principal radii of curvature i ? i , . . . , i i ^ _ i as diagonal elements. The determinant is then equal to Y\!x~ (O'Rj-^b)^ so the coefficient of a^6^~^~^ is the z/th elementary symmetric function { i ? i , . . . , Rj^}. Thus the coefficient of Ai • • • A^_i is equal to { i ? i , . . . , R^,} times u\{n — u — 1)!, and after dividing by n! we obtain (2.3.20)

^ n - v ^ — i ^ l {Ri,...,R^}du, n( ^ j J 5 - 1

v

0, then

92


is for small r > 0 the volume of the set of points at distance < r from dK. Following Weyl [1] we shall now study this volume not only for convex hypersurfaces in R'^ but for arbitrary smooth compact submanifolds M of a Euclidean space. It is convenient to change notation now, so we shall denote the dimension of M by n and assume M embedded in R*^"^^ for some p > I. Let N{M) be the normal bundle of M . For sufficiently small r > 0 the map N{M) 3 (x,n) n-y x + n G R^"^^ is a diffeomorphism of {(x,n) G iV(M); ||n|| < r} on the set of points in R ' ^ + P at distance < r from M. To calculate the volume we parametrize a piece of M by uj 3 X i-^ f{x) e M , where cu is an open subset of R"^. We assume the coordinate patch u chosen so small that there is an orthonormal basis ni(x)^... ^Xip{x) for Nf(^x^(M) depending smoothly on x. The considered map is then p

LjxRP

3 (x, t) ^ F{x, t) = f{x) + Y^

t^nj,{x).

1

We have dF/dtj^ l,...,n

— n^, and modulo the normal plane we have for j = p

p

dF/dxj = fj + jÛdjn,

n

= /i - E

1^=1

E

t-(hij,n,)g'''fk

u=li,k=l

where hij G N are the coefficients of the second fundamental form and fj^ = dkf- (See e.g. Klingenberg [1] for the Riemannian geometry used here and below.) The volume spanned by the partial derivatives of F is therefore equal to the volume y/g spanned by / i , . . . , / n multiplied by the determinant of {6jk - YZ=i^^(^h^''^')lk=i^ where h^ = Yîî îjQ^''Hence the volume of the tube over f{uj) is

(2.3.21)

/ ,/^dx I JLJ

det((5,-,

J\t\0 (2.3.22)

/

(t,a)^dt = r^+^|arCp

T]

^ ^ '

a G R^,

where Cp is the volume of the unit ball in R^ which makes (2.3.22) obvious when a = 0. For reasons of homogeneity and rotational invariance it suffices


93

to prove (2.3.22) when a = ( 1 , 0 , . . . ,0) and r = 1. Then the integral is equal to

= Cp_iS(|(a+i), i(p+i)) = c,.ir(|(a+i))r(|(^+i))/r(|(a4-p)+i). When a is replaced by a + 2 the right-hand side is multiplied by the factor {a + l)/(c»" -hp + 2), so (2.3.22) follows by induction. If we polarize the two sides it follows more generally that for a i , . . . , a^j G R^ (2.3.22)'

/

{t,ai)---{t,a,)dt

= r''+PECp

TT

- i - ,

where and the sum is extended over all a\ permutations i / i , . . . , z/^^ of 1 , . . . , a. (We have used that a\ = a\\{a - 1)!!.) The integral in (2.3.22)' vanishes when (7 is odd. If we expand the determinant in (2.3.21) it follows that the inner integral of the terms of even degree 2a with respect to t is equal to

Here the summation runs over all a-tuples of unequal integers a and P chosen among the integers 1 , . . . , n which are permutations of each other with signature sgn (^). If we let /3i and P2 change places we can replace the first factor by

by the Gauss equations. Hence we obtain T h e o r e m 2.3.10. For a manifold M of dimension n embedded in R^+^, the volume of the set of points at distance < r from M is for small r > 0 equal to

(2.3.23)

c,

Y: 0 0 it is positive if X2 is small enough. Since d^f > 0 when X2 7^ 0 we conclude that there is a unique solution xs = X{x2) which is a C^ increasing function of \x2\ for small 0:2 7^ 0 and —> 0 as X2 ^^ 0. By (2.4.3) we have F"{x2) = dlf(x2,X)

-

{d2dsf{x2,X))ydlf{x2,X).

Hence ]i^F"{x2) 2v{0), solim^2_+oi^"(^2) > 2^7(0) also. On the other hand, id2d,f{x2,X))'/dlf{x2,X)

= ixlv'{Xy/{xlv"{X)

+

< with equality iiu"{X)

u"{X))

W{Xf/v"{X),

= 0. Hence

lim F"{x2) = 2(^(0) - 2t;'(0)V^"(0)) iiu" has zeros arbitrarily close to the origin. Such functions exist, although they cannot be analytic, and for them we have lim F'\x2) X2-+0

= 2(i;(0) - 2^'(0)V^"(0)) < 2^(0) -

Ih^

F'\x2),

a:2->H-0

which proves that F ^ C'^. It is easy to find a convex set with C^ which is defined by (2.4.4) near 0.

boundary

In the example the second derivative of F was bounded although not continuous. This suggests that one should consider the regularity condition in the following definition:

SMOOTHNESS OF CONVEX SETS

97

Definition 2.4.2. A closed set K C R^ is said to have C^'^ boundary dK if for every point XQ G dK one can choose coordinates with the origin at XQ and a neighborhood U of XQ such that (2.4.5)

UnK

= {xeK-xi>

f{x')},

where x' — {x2^ •.. ,x^) and / E C^'^, that is, / G C^, and df is Lipschitz continuous. For convex sets this condition takes a very simple form: P r o p o s i t i o n 2.4.3. Let K G /C(R^). Then dK G C^'^ if and only if there exists some R> 0 such that K is the union of balls with radius R. Proof. Assume that (2.4.5) holds with U = {x;\x\ /'(O) - 0 and \nx^)-ny')\ f{x') if \x\'^/2R > C\x'\^{2^ which is true if CR < 1. This gives the desired ball containing the origin. To handle points {f{x'),x') near 0 we recall that \f'{x')\ < C\x'\ and that \f{x')\ < C\x'\^/2. There is a coordinate change differing from the identity by 0(|:z;'|) which shifts the origin to this point and the direction (1, —f'{x')) to the direction of the new rî-axis, so we get the same conclusion at this point with C replaced by C ( l + 0(|:r'|)) < 2C if \x'\ is small enough. If R is sufficiently small it follows that there is a ball with radius R contained in K and with any given point in dK on its boundary. Every point x £ K can be written X = Xxi + (1 — X)x2 where 0 < A < 1 and Xj G dK. If Bj C if is a ball of radius R containing Xj, then XBi + (1 — A)52 is a ball with radius R contained in K and containing x. Now assume conversely that the interior ball condition in the proposition is fulfilled by K. Choose the origin at an arbitrary point in dK so that K is contained in the half plane where xi > 0. Then the ball in the hypothesis must be defined by |xp < 2i?xi, so it follows that if fl C/ is defined by î > / ( ^ ' ) with / convex, iiU = {x; |:z;| < R}. Since 0 < f{x')

X2 sinxa}, then the projection ivK in the X1X2 plane is defined by xi > —\x2\- Thus the boundary is C^ except at the origin where it is only Lipschitz continuous. Exercise 2.4.1. Show that if iiTi, ^ 2 ^ /C(R'^) have boundaries in C^'-^, then the boundary of Ki -f K2 is also in C^^^. The regularity of projections and sums of convex sets has been studied in great detail by J. Boman and C. Kiselman. We refer to Kiselman [1] for a survey of the results. 2.5. P r o j e c t i v e convexity. If F is a vector space over R of dimension n + 1 , then the corresponding projective space P{V) is defined by identifying points in F \ {0} which are multiples of each other; thus P{V) is the space of one-dimensional subspaces of V. One defines d i m P ( y ) = n. li W is a linear subspace of the vector space V, then P{W) is called a subspace of P{V). It is a projective line if it has dimension 1 and a projective hyperplane if the dimension is n — 1. The identity dim 1^1 + dim 1^2 = dim(Iî + 1^2) + dim(Iî fl W2) valid for linear subspaces of V implies that d i m P ( t ^ i ) + dimP(I^2) = dimP(Tî + W2) + dimP(Tî H 1^2). Here P{Wi -f 1^2) is the smallest subspace of P{V) containing P ( W i ) and P(1^2)5 cind one should interpret P({0}) as the empty set with dimension — 1. For any coordinate system XQ,...,Xn in V we can take

PROJECTIVE CONVEXITY

99

x i / x o , . . . ^Xn/x{) as coordinates in P{V) outside the projective hyperplane defined by XQ = 0, so P{V) is identified with R"^ extended by a "plane at infinity", a projective space of dimension n — 1. The coordinates XQ, . . . , x^ are called homogeneous coordinates, and we shall often use the notation {XQ : xi : •' • : Xn) for a point in P^ = P(R'^"'"^). More generally, for any vector space W the projective space P ( R 0 W) can be identified with the union of W and the plane P{W) at infinity. If Vi —> V2 is an injective linear map between vector spaces, then the induced map P{Vi) —> ^ ( ^ 2 ) is called projective. In a projective space P{V) a projective hyperplane is the image of a hyperplane W CV, defined by an equation L == 0 where L is a linear form on V. We can then identify P{V) \ P{W) with {x G V-L{x) = 1}. This is an affine space which becomes a vector space if we choose a point in it as origin. This affine structure is independent of the choice of L. Thus P{V) \ P{W) has an affine structure which makes the notion of convex subset in the usual affine sense meaningful. The projective line is topologically a circle. Two different points in it determine two intervals bounded by them. If we have four different points A = {ao : a i ) , B = {bo : 61), C = {CQ : ci), D = {do : di) in the projective line P ( R ^ ) , then the cross ratio is defined by rA r? ry T-i\

CLQCI OQCI

- aiCo /aodi-aido - bico I bodi - bido

note that it is homogeneous of degree 0 in each variable in R^ and therefore defined on the projective line P^. lia^ = bo = CQ = do = 1 it is the quotient between the ratio in which C divides AB and that in which D divides AB. If we make a linear transformation in R^, then each factor is multiplied by the determinant and the cross ratio is unchanged. Hence it is invariantly defined on any projective line. Note that C and D lie in the same open interval bounded by A and B if and only if {A, B] C, D) > 0. Definition 2.5.1. A set K in a projective space P{V) is called projectively convex if every straight line in P{V) intersects K in an interval (which may be open, half open or closed, empty, a point, the whole line, the whole line except one point, or an interval with two end points). The definition is clearly symmetric under passage to the complement, that is, K is projectively convex if and only HCK = P{V)\K is projectively convex. We shall say that a pair of sets Ki^K2 C P{V) is convex if K2 = ZKI and K i , hence 7^2? is projectively convex. Both Ki and K2 are then connected. In terms of the cross ratio the definition can also be stated as follows: lixi^yi G Ki and X2^y2 ^ ^ 2 lie on the same line, then (^1,2/1; ^2,2/2) > 0.

100


If if is a convex subset of R"^ in the usual (affine) sense, then iiT is a projectively convex subset of P^;, for it has empty intersection with Hues contained in the plane at infinity and intersect other lines in an interval. Also the complement is projectively convex. Another example is obtained if we consider a quadratic form Q{x) = aoxl + aixl H

h aâ:^

and define K = {{XQ : • • • : Xn); Q(^) > 0}. Since a quadratic form in R^ is positive either in R^ \ 0 or in a symmetric set bounded by one or two lines, it is clear that K is projectively convex. If there are at least two coefficients of each sign in Q then every hyperplane intersects both K and the complement of K, so it is not possible to choose coordinates to make K convex in the affine sense. This example is important; we shall see that essentially there are no projectively convex sets beyond those we have listed now. L e m m a 2.5.2. If K is a projectively convex subset of P{V) which is not contained in any subspace, then the interior K° is not empty, and it is dense in K. Proof. We prove the lemma by induction with respect to the dimension n -\- 1 oi V. It is trivial when n = 1 so we assume that n > 1. For an arbitrary X G iiT we can choose X i , . . . , X^ G K such that X , X i , . . . , X^ are independent (that is, corresponding elements in V are linearly independent). We can choose coordinates in V so that X is the origin (1 : 0 : • • • : 0) and Xj is the point at infinity on the j t h coordinate axis. By the inductive hypothesis the intersection Kj of K with the coordinate plane 11^ where Xj = 0 for every j 7^ 0 has points arbitrarily close to the origin which are interior with respect to 11^. Let Uj C Kj be an open set in 11^ close to the origin with all coordinates except Xj diff'erent from 0. If yj G /7j, j = 1, 2 then there is a unique line between them and one of the intervals with end points yi,7/2 isin K. If the whole lines are in K then K° D U1UU2. On the other hand, if there is a point z ^ K on the line determined by y 1,2/2 5 then all lines nearby through z cut both Ui and U2, and the interval determined by these intersections not containing z must be in K. Thus 2/1 and 2/2 are in the closure of iC°, and the proof is complete. If Ki^K2 (2.5.1)

is a convex pair in P{V), we shall write V = ~K[f\'K~2 = dKi = dK2,

thus P{V) = KlUTl}

K^.

This boundary never has any interior points. In fact, assume that U C T is open. Since U H Kj = 0, it follows from Lemma 2.5.2 that U H Kj = 0 unless Kj is contained in a hyperplane 11. Then C/ C F C 11, so C/ is empty.


101

If K is a projectively convex set in P{V) which is contained in a subspace P ( W ) , then K is projectively convex as a subset of P{W) and conversely, so we have a trivial reduction to lower dimensions unless K has interior points, by Lemma 2.5.2. From now on we shall therefore always assume that both Ki and K2 have interior points. Note that Ki^K2 and K°^K2 are also convex pairs then. In fact, if Ki 3 Xj —> x, Ki 3 yj -^ y then an interval bounded by Xj and yj is contained in iiTi, and after passage to a subsequence it converges to an interval bounded by x and y contained in Ki. Hence Ki is projectively convex, and so is K2. We shall primarily study convex pairs where one of the sets is open and the other closed, with interior points. L e m m a 2.5.3. If L is a line then we have either (i) L n r - 0, and L C KI or L C K^; (ii) L n r is an interval, possibly reduced to a point, with complement in L contained either in K^ or in K2; (iii) L n r consists of two points and LHCT consists of one interval in KI and one in K^. Proof. Since Kj fl L is an interval, the intersection F fl L is empty or consists of one or two intervals, possibly reduced to points, and the complement in L consists of one or two open intervals / then. Since / is the union of the disjoint open sets / fl KI and / fl X | , the connectedness of / implies that one is empty. If there is only one interval I we have case (i) or case (ii). Now suppose that we have two such intervals, and that both are contained in K ° , say. Choose Xi,yi in the two intervals. They are separated by points ^2,2/2 ^ F n L. We can choose 2:2,2/2 ^ ^2 so close to X2,2/2 that the line between them intersects neighborhoods of Xi and yi contained in Ki, which contradicts the definition of convexity. This contradiction proves that we have one interval in if ° and one in K | . If a separating interval contains two points ^2,2/2 ^ F we get a contradiction in the same way, so F fl L must consist of precisely two points separating an interval C K^ and an interval C K2- The proof is complete. L e m m a 2.5.4. IfP{W) is a subspace ofP{V) such_that Tw = PjW)^^ has interior points relative to P{W), then P{W) C Ki or P{W) C ^ 2 , and Fv^^ is a projectively convex subset of P{W). Proof. First we prove that Fv^^ is the closure of its interior u with respect to P{W). By hypothesis a; is not empty. Ii x ^ Tw \uJ we consider the lines connecting a; to a point y £ UJ. If they are all contained in Vw^ then X is the vertex of an open cone C u, so the assertion is true. Assume now that on one such line there is a point z ^ Tw^ say z ^ K2, that is, z E K^. A line through x and a point ( G Ki fl P{W) close to z must still intersect

102


u) in an open non-empty set, so it follows from Lemma 2.5.3 (ii) that the interval between x and uj not containing (^ must be a subset of Vw Hence r^^ contains an open set in P(W) with x on its boundary. An immediate consequence is that Vw is projectively convex in P{W). In fact, if x,?/ G Tw^ then we can find Xj G u with Xj -^ x. By Lemma 2.5.3 one of the intervals bounded by Xj and y must be contained in Tw-, for the intersection of the line through these points and Tw has a nonempty interior. Since Tw is closed it follows when j -^ oo that an interval bounded by x and y is contained in Fw If we join two points x,y e P{W) \ Tw by a line L, then the fact that L n r^^ is a closed interval implies that the complement, which contains x and y, is an open interval J. Hence we have one of the cases (i) or (ii) in Lemma 2.5.3, so x and y are both in K^ or both in K2, which proves that P{W) C K T or P{W) C ^ . The proof is complete. L e m m a 2.5.5. IfP{W) is a subspace ofP{V) such that P{W)nK] + 0, j = 1,2, then Fvi/ = P(VF) fl F has no mteiioi point, and the closure of

p(w) n K ; isP{W)nlK~j^ {P{w) n K ; ) U r ^ , J -1,2. Proof. That Tw has no interior point is a consequence of Lemma 2.5.4. We must prove that Tw is contained in the closure of Uj = P{W) 0 K^. It suffices to discuss the case j = 1. Let x G Vw and suppose that a neighborhood ?7 of x in P(W) does not meet uji. If the line L through x and a point y E uJi contains a point in a;2, then we have the case (iii) in Lemma 2.5.3 and uji contains an interval on L bounded by x since x G Tw This is a contradiction. Hence we must have the case (ii), so L C Ki, since y ^ K2, hence LnU C Tw But if this is true for all y ^ uJi, then Tw has interior points which is again a contradiction proving the lemma. L e m m a 2.5.6. IfxGT or L C 1^2-

then there is a Une L with x E L and L C Ki

Proof. A point in T cannot be isolated, for then either Ki or K2 has an isolated point and must consist of that single point. Hence we can choose Xj ET\ {X} with Xj -^ X SiS j -^ 00. By Lemma 2.5.3 the line Lj through X and Xj is entirely contained in one of the sets Kk except in case (iii); in that case the whole line except the interval bounded by x and Xj close to x is contained in one K^. When j —> 00 we conclude that a limit of the lines Lj is contained in one of the sets K^. We can now study the two-dimensional case. (In the one-dimensional case a projectively convex set is by definition just an interval.) If Ki,K2 is a projectively convex pair in P^, then one of the closures, say K2, contains a line, which we can put at (X). Then K^ C R^ is an open convex set in the


103

affine sense, and the complement has interior points. We have three cases: (a) if ° is contained in two half planes with linearly independent normals. By a suitable choice of coordinates we may assume that xi > 0w and that X2 > 0w in jFf°. Thus ^^ ^ ^ ^ x v ^ %.xx^w ^ ^ ^ J.XX -*^*-l • -^""'-' ^ O,xi/a:o > 0,a:2/xo > 0 or XQ = 0,xiX2 > 0}.

iî C {(xo : x\ : x^\x^

The line defined by XQ + xi + ^2 = 0 has no point in Ki. If we put this line instead at infinity, then Ki is a bounded, hence compact, convex subset of R^, which means that Ki is a bounded convex subset of R^. (b) KI is a half plane, say KI

= {(xo

: xi

: X2)\XQ

> 0 , X I > 0}.

Taking X2 = 0 as the line at infinity we see that if° consists of two opposite angles, including the interior of the limits at infinity. Hence Ki consists of one of the two open components of the complement of two intersecting lines, together with an interval on each of the lines. (c) KI is bounded by two parallel lines, say KI = {(xo : xi : X2); 0 < Xi
3 and that the theorem has been proved for lower dimensions. Assume at first that d i m L i > 0. Take a point XQ G L I and a plane P{W) C P{V) of codimension one with XQ ^ P ( l ^ ) . The projection K2 C P{W) defined in Lemma 2.5.8 has interior points and is compact and projectively convex. Let L[ = LiH P{W), and let L2 be the projection of L2 in P{W) from XQIt is clear that L[ is a maximal subspace of P{W) contained in the open set P{W) \ K2, and that L2 C K2' By the inductive hypothesis there is a subspace M D L'2 oiP(W) such that M C K2 and d i m X i + d i m M = n-2. If N is the subspace of P{V) spanned by XQ and M , then AT D L2? so L2 is maximal in ^2fliV. In KiON the point XQ is maximal since every line in TV passing through XQ contains a point in K2. By the inductive hypothesis it follows that dimZ/2 = dimiV — 1 = d i m M , and since d i m L i = 1 -h dimL^, we obtain d i m L i + dimL2 — n — 1. We must also consider the case where d i m i i = 0. Then we have dimL2 > ^ — 2 > 1, for otherwise we could choose a plane of dimension < n — 1 through L i , L2 and an interior point of K2 and conclude by the inductive hypothesis that L2 is not maximal even in that plane. Now we choose XQ G Z/2 instead and form with P ( l ^ ) ^ XQ the projection K i , which is an open set. It is clear that L2 = L2 H P{W) is maximal in P{W) \ Ki. If P ( l ^ ) \ Ki has interior points and dim 1^2 = n — 2, it follows from the inductive assumption that there is a line M C Ki containing the projection L'l of L i . In the two-dimensional plane N spanned by XQ and M , we have an interior point Li of A/" fl iiTi, and XQ is a maximal point in N r\K2 since


105

every line in N through XQ intersects Ki. li N H K2 has an interior point we conclude that N f) Ki contains a line. If AT fi if2 has no interior point, it is a proper subset of a line, and again there is a line contained in the complement NdKi^ so Li was not maximal. It remains to discuss the case where P{W) \ Ki is contained in a subspace H of codimension 1 in P{W) and L'2 = i>2 n P(W) has codimension 2 in P{W). Since L'2 is maximal we know that P{W)\Ki is not equal to ff, so we can choose a line M C P{W) intersecting H only at a point in Ki. Then it is contained in i f i , and we can argue as before to show that Li was not maximal. The theorem just proved is of course also valid with KI replaced by Ki and K2 replaced by K2. Note that it follows from the theorem that if (2.5.2)

n = dimF(y),

n^ = max dimL,

n° = max dimL,

LcKi

LCK°

then every maximal subspace of Ki has dimension n^, and every maximal subspace of if ° has dimension n^. Moreover, (2.5.3)

0 < n° < rij, 2 = 1,2;

ni-\-n^=n\-\-n2

— n — 1;

hence n i — n° = n2 — ^2 — z^, where we shall call v the defect If we extend a maximal plane 11^ C K^ to a maximal plane 11^ C Kj, then (2.5.4)

dim(ninn2) = v - l .

(Recall that we have defined the dimension of the empty set to be —1.) In fact, n 4- d i m ( n i n 112) > dim Hi + dim 112 = n i + 77,2 = n — 1 + z/, hence d i m ( n i Pi 112) > z^ — 1. On the other hand, since IIi fl 112 — 0? we have d i m ( n i n 02) + d i m n 2 < d i m n 2 - 1, for Hi n 112 and II2 are disjoint and contained in 112, which means that d i m ( n i n 112) < i/ - 1 and proves (2.5.4). All values of the numbers Ui^n^ satisfying (2.5.3) can occur. For let A;,/ be non-negative integers with A: + / < n — 1, and let k

g(x) = 5 ^ a ; 2 _ 0

k+l+l

^ ^ 2 ^ fc+1

x = (a;o,...,a;„)€R"+\

106


The image Kj in P £ of the set where { — lyQ of the set where (-1)^(3 < 0. It is clear that

< 0 has as closure the image

nl > k, n 2 > / , n i > A : - h n —fc— / — l = n — 1— /, n2 > n — 1 — k, and by (2.5.3) equality must hold in all these inequalities. We shall prove later that if min(nj,n2) > 0, that is, max(ni,n2) < n — 1, then this is not only an example but equivalent to the general case. However, we shall first discuss the case where ni = n — 1 or n2 = n — 1, which is close to affine convexity and can be studied by a small modification of the proof of Theorem 2.5.7. Assume that n2 = n—1. If we choose the plane at infinity as a subset of K2 in P{V) — P ^ , then K\ is a convex subset of R'^, not equal to R'^. Let k be the maximum number of linearly independent normals of supporting planes. By a suitable affine coordinate change in R'^ we can arrange that Kl C {(a;i,...,Xn);î > 0, ...,a:ifc > 0}, and K\ is then the intersection of half spaces of the form k

{a;GR^;â;,Ci e^^''^\x'^Dîp{x)\ < oc,

Va,^.

This implies that e-^(^) < C ( l +

\x\)-''-^J^'^\

where / is the Legendre transform of / . Hence the Fourier-Laplace form of (^, (2.6.2)'

^ ( 0 -

/e-^f^'^VW^^,

is defined in {( G C^; /(ImC) < oo}. Since {idldiY{i the Fourier-Laplace transform of x^Dîp, we have (2.6.4)

sup\e-f^'^\^

trans-

+ ir]f{d/dO'"^{^'hiv)\

+ ir]Y(p{i + iv) is < oo,

Va,/?.

Here e~^^'^\^ + ^v)^{9/d^)'^(f{^ + irj) should be interpreted as 0 if f{rj) = -f oo, and the absolute value is then an upper semi-continuous function in C"^. The convex set M = {rj ^ R'^;/(r/) < oo} may not have interior points, so in general we cannot differentiate with respect to rj. An example is the case / = 0 where Sf is the standard space S, and the Fourier-Laplace transform is the Fourier transform, defined only in R'^. The aflSne space spanned by M is of the form {a} + MQ, where MQ is a linear subspace of R'^. The fact that / = H-oo outside {a} + MQ means that {x,r]) = {x,a)^ a x e MQ- and /(?/) < oo, so f{x) — {x,a) is then constant along MQ- by Theorem 2.2.4, that is, it is a function in the dual space IV^/MQ- of M Q . Conversely, this implies that / = 4-oo outside {a} + MQ. NOW we have the Cauchy-Riemann equation (2.6.5) _ (t, d/dOip{^ + iv) = W, d/di + id/d7])ip{( + zry) = 0, te MQ, V ^ M\ where Af° is the relative interior of M . This follows at once from the uniform convergence of (2.6.2)' in any compact subset of R ' ^ + i M ^ . We can then differentiate with respect to r/ also in (2.6.4) provided that derivatives are only taken in directions in MQ.

CONVEXITY IN FOURIER ANALYSIS

113

T h e o r e m 2.6.1. Let f be a convex lower semi-continuous function in R'^ which is finite in some open set. Then the Fourier-Laplace transformation (2.6.2)'is an isomorphism of the space Sf, defined by (2.6.1)', on the space of functions (p in R"^ + zM, M = {rj £ R'*; /(r/) < oo} satisfying the following conditions: (i) {d/d(,)^(p{^ -f- IT]) is continuous for all a when r] is in the relative interior M° of M. (ii) IfL is any complex line then (fii+irj) is continuous in Lfl (R^ -j-iM) and analytic in the interior. (iii) The absolute value

is upper semi-continuous

and bounded in C^ if defined as 0 when

f{7]) = -fCX).

The inverse is given by (2.6.3)'

ifix) = (27r)-^ /e^(^(^ + i77)de is defined and e^*'^^(^^ G S. In view of (iii) we have (2.6.6) e('^^'i)-fM\x''DîPr,{x)\

= e-^'W|x"(jD + 277)ê,(^)| 2 throughout this chapter.) If X G R'^ \ {0} ^hen the reflection x' of x in the sphere OBR is defined by x' — i? x/|rr| . It has the same direction as x, and \x\\x' I = i?2. Note that the inversion x \-^ x' is the identity on the sphere OBR, and that it is an involution, that is, {x')' = x for every x ^ 0. If x,x' and y,y' are two pairs of corresponding points, then the equation \x\\x'\ = \y\\y'\ shows that the triangles O^x^y and O^y'^x' are similar; we may interchange x and x' in this conclusion. Hence

(3.1.4)

\y\/W\^\x\/\y'\

=

\x-y\/\x'-y'\,

(3.1.5)

\y\/\x\^\x'\/\y'\

=

\x'-y\/\x-y'\.

HARMONIC FUNCTIONS

119

If \y\ = R then y' = y and (3.1.4) gives \x - y\/\x' -y\ = y/\x\J\x^\ = \x\/R, which means that the sphere is harmonic with respect to x and x'. Now we define the Green's function (cf. (1.5.1)) (3.1.6) GR{x,y) = E{x -y)E{{x' - y)\x\/R) = E{x - y) - E{{x y')\y\/R), when x^y E Bn and x ^ y. Here the second equaUty follows from (3.1.5). The first expression shows that GR is a harmonic function of y for fixed X ^ y^ and the second expression that it is a harmonic function of x for fixed y ^ X. In fact, GR{x^y) = GR{y^x). We have (3.1.7)

GR{X, y) = Oii \x\ =Roi

\y\ = R; and GR{X, y) < 0,

since for fixed x with |x| < R the inequality |x — ?/| < \x\\x' — y\/R is satisfied for all 7/ in a ball with OBR on its boundary, by (3.1.4), so it is equal to BR. Uue C^CBR) satisfies (3.1.3), then (3.1.8) u{x) =

GR{x,y)f{y)dy-\JBR

dG{x,y)/dnycp{y)

dS{y),

x e BR,

JdBn

where Uy is the exterior unit normal y/R, and dS is the Euclidean surface measure. If n G CI{BR), then (3.1.8) follows from (3.1.2)', for

/

E{{x'-y)\x\/R)/lu{y)dy

=Q

because BR 3 y ^-^ E{{x' — y)\x\lR) is harmonic. On the other hand, if u vanishes in a neighborhood of the point x, then GR is a harmonic function of 2/ in a neighborhood of the support of -u, and

/

{GR{x,y)Au{y)

- {AyGR{x,y))uiy))

dy

JBR

L

dGR{x, y)/dnyu{y)

dS{y)

dBR

by the Gauss-Green formula and the fact that GR{X, V) = ^ when y G OBR. By means of a partition of unity we can write t6 as a sum of two functions satisfying one of the preceding conditions, and this proves that (3.1.8) holds iiu E C'^(BR). Conversely, given / G C{BR) it is clear that the first term ui in (3.1.8) is continuous in BR and vanishes on OBR, and it follows from (3.1.2)'' that Aui = f in BR in the sense of distribution theory. (It is not always true that u G C^{BR), but we ignore this point for the moment.

120

III. SUBHARMONIC FUNCTIONS

which is an advantage from using distribution theory.) The second term U2 in (3.1.8) is harmonic in BR since dG{x^y)/dy is a harmonic function of X G BR when y G OBR. We shall prove in Theorem 3.1.5 below that U2 is continuous with boundary values if for every tp G C{dBR), but first we shall calculate dGR{x^y)/dny explicitly. To calculate dGR{x,y)/dy we use the first expression in (3.1.6) which gives CndGR{x,y)/dy

= {y - x)\y - x | - " - {y - x')\y - x r ^ W I ^ I ) " " ' -

If we use that \y — x'\/\x — y'\ — \y\/\x\ — i?/|x|, x' — i?^x/|xp and y' = y then this formula can be simplified to CndyGRix, y) = \y- x\-^{y

- x - {\x\IR)\y

-

R^xl\x\^))

=

y\y-x\-^{\-\x\^lB?).

Hence we obtain dG{x,y)/dny

= {Rcn)-\R'

- \x\^)\y ~ x p " ,

|y| = R.

We introduce a notation for this Poisson kernel when i? = 1, (3.1.9)

P{x,y)

= c-Hl-\x\')\y-x\-",

\y\ =

l,\x\ 0, da2 > 0, so da,dai,da2 are multiples of the Dirac measure at y then. Remark. Corollary 3.1.9 means that the set T-L^{BR) of positive harmonic functions u in BR with u{0) = 1 is a convex subset of C{BR) with extreme points CnP{-/R,y)'-, these are called minimal positive harmonic functions. By Exercise 3.1.2 HJ^{BR) is compact. Thus Theorem 3.1.8 can be considered as an infinite-dimensional analogue of Theorem 2.1.9, with T-ij^{BR) forming an infinite-dimensional simplex. The Choquet theory referred to after Theorem 2.1.9 allows one to deduce results like Theorem 3.1.8 from a direct proof of results like Corollary 3.1.9. We shall now for harmonic functions in a half space prove analogues of the preceding results on harmonic functions in balls. As a preliminary we shall discuss some further properties of the inversion x H-> R^XI\X\^ that will also explain the formula (3.1.6) above for GR. Note that the second term there can be written {\X\IRY~'Ê{X' — y) if n > 2, E{x' — y)-\(27r)~^ log(|x|/i?), if n = 2. That it is a harmonic function of x was proved above by appealing to another version of (3.1.6), but it is also a special case of the following theorem. (For the sake of simplicity we take JR = 1 in what follows.) T h e o r e m 3.1.10. Let X C R*^ he an open set, and let X be the reflection {x elU^X {0};x' = x/lx]"^ G X}. Then (3.1.17)

\dx'\ =

\dx\/\x\\

so that X \-^ x' is conformal with magnification | x ' | / | j ; | . Ifu is a harmonic function in X and u{x) = \x\^-''u{x/\x\^),

factor l/\x\'^ — \x'^ —

xeX,

126


then u is a harmonic function in X. Proof, Since dXj = dxjl\x\^ — 2xj 'Yj^k dxkl\x\^^ we have |rz;pda:;' = rxdx where rx is the reflection in the plane orthogonal to x, hence orthogonal. This proves (3.1.17). Since |x| and Uj{x) = X j \X\ a r e harmonic functions in R^ \ {0}, it is clear that u is harmonic if ?/ is a first-order polynomial. A quadratic form q{x) — YTi,k^\ Qjk^j^k is harmonic if and only if Y^l Qjj = 0, and then we have n

q{x) = \x\~''~'^ ^

n

QjkXjXk = Yl Qjk{duj/dxk -

j,k=l

6jk\x\~'^)/{-n)

j,k=l n

= 51

Qjkduj/dxk,

j,k=i

so q is harmonic. Thus Au = 0 if li is a harmonic polynomial of second order. If we apply this to the Taylor expansion of ti at x', it follows that Au{x) = 0. The proof is complete. Remark. If n = 2 it is well known that for every conformal map X 3 X H^ (p{x) G X, that is, for every analytic or antianalytic function from X C C to X C C, the composition uocp is harmonic in X if i^ is harmonic in X. The conformality means that V ' is proportional to an orthogonal matrix, that is, dicpi = ±d2(p2,

^2^1 =

Tdi(p2,

hence Acpi = ±8182^2 T ^2^1 2 every conformal map (^ is a product of inversions and orthogonal transformations (see Berger [1, p. 223]), so it follows from Theorem 3.1.10 that X 3 X ^-^ \ip^\~^~{uoip) is harmonic in X if tt is harmonic in X. Exercise 3.1.5. Prove with the notation in Theorem 3.1.10 that {Au){x) = |x|-2-^(A^)(x/|xn, first when u E C^(X), then when u G V'{X). If X is the half space {x G R'^'.Xn > | } , then X = {x e IC';2xn > |xp} is the ball with radius 1 and center at ( 0 , . . . , 0,1). Thus Theorem 3.1.10 gives a linear one-to-one correspondence between harmonic functions in the

HARMONIC FUNCTIONS

127

half space and in the ball which can be used to carry our results from the ball to the half space. One should just keep in mind that the boundary point 0 of the ball X corresponds to a boundary point at infinity in the half space X , and this point can carry a positive measure in the analogue of (3.1.15). We have u{x) —> c as x —> 0 if and only if \x\'^~'û{x) -^ c as a; —> oo in X . Since \x\'^~'û{x) = u{x/\x\'^), re G X , we see that if u e C\X) then 9^(|x|^-2^(j:)) - 0(|x|-2|«l), |a| < 1. To relax these rather unnatural conditions we shall repeat some of the arguments for the ball in the case of the half space instead of using the inversion to just carry the results over. We define the Greenes function for the half space H = {x E R"^; Xn > 0} by

(3.i.6y

_

GH{x,y) = E{x-y)-E{x-y'')

= E{x-y)-E{x^-y),

x.yeH,

x^y,

where y* = (?/i,..., ^/n-i? —Vn) is the refiection of y in the boundary plane dH. It is clear that Gnix^y) = Gniv^x) < 0 is harmonic in x (in y) for fixed y (fixed x), x ^ y, and that G{x,y) = 0 if x E dH or y E dH. We define the Poisson kernel when x E H and y G dH by (3.1.9)' Pnioc^y) = -dGH{x,y)/dyn = -2dE{x - y)/dyn = 2xn\x - yl'^'/cn. (We shall often identify dH with R ^ - ^ ) Then the analogue of (3.1.8)' is that (3.1.8)" uix)=

[ GHix,y)Au{y)dy+

[

PH{x,y)u{y) dS{y),

X e H,

JdH

JH

provided that u G C'^{H) and that u{y) and (1 -\-yn)du{y)/dy are bounded in H. The integral over H is defined as the limit of the integral over {y G JH"; \y\ < ^} as ^ -^ oo. The only difference in the proof is that we have to apply the Gauss-Green formula to

/

{GH{x,y)Au{y) -

{AyGH{x,y))u{y)dy

integrated over {y G H; \y\ < g} and then let ^ -^ oo. We must show that the integral

/

{GH{x,y)du{y)/dny

-

dGH{x,y)/dnyu{y))dS{y)

taken over the spherical part of the boundary converges to 0. The area is CnQ^'^/'^ and dGnix^ y)/dy = 0{\y\'~'^) by the mean value theorem, so the

128


second part of the integral is 0(1/g). Since Gni^^y) — 2xn{xn - 2/n)|^ — y\~^ jCn + 0(1^1"^) the same is true of the first part, which gives (3.1.8)''. In particular, when u — \ê obtain (3.1.10)'

/

PH{x,y)dS{y)^l,

x e H.

We can now prove: T h e o r e m 3.1.5'. Let u be a bounded continuous function in H which is harmonic in H, and let ^{y') = u{y',0), y' G R'^"-^. Then we have (3.1.1iy

u{x)=

I PH{x,y')^{y')dy', JdH

x e H.

Conversely, if cp is a given continuous function in R^~-^ such that the integral J \ip{y')\{l H- \y'\)~'^ dy' is finite, then (3.1.11)' defines a harmonic function in H which is continuous in H, u{y'^0) = ^{y')Proof. Since u is harmonic we know that u G C^{H)^ and ii \u\ < M in H then yn\du{y)/dy\ < 2nM in i?, by (3.1.14)' applied to S i 9 a; H^ M =b u{y + ynx). Hence we may apply (3.1.8)" to u{x',Xn + 6:) if e > 0, which gives u{x',Xn-ê)=

/

PH{x,y')u{y',e)dy'.

JdH

In view of (3.1.10)' and the boundedness of u we obtain (3.1.11)' when e —» 0. The proof of the strong converse is essentially a repetition of the corresponding part of the proof of Theorem 3.1.5, so it is left for the reader. Next we extend the Riesz-Herglotz theorem. Note that x i-^ Pnix^y) for y G dH is clearly a minimal positive harmonic function for the boundary point y, for it vanishes at all finite boundary points ^ y and \x\'^~'^PH{x,y) -^ 0 as a; -^ oo. For the point at infinity we have the minimal positive harmonic function which vanishes at every finite point of the boundary. It is therefore clear that Theorem 3.1.8 must take the following form: T h e o r e m 3.1.8'. Let u be a harmonic function which is non-negative in H. Then there exists a unique positive measure da on dH = IV^~^ and a constant a > 0 such that (3.1.15)'

u{x) = axn+

/ PH{x,y')d(j{y'), JdH

x e H.

HARMONIC FUNCTIONS

129

We have / ( I -I- \y'\)~"^ da{y') < oc, da is the weak Umit of the measure u{x\ Xn)dx' as Xn -^ 0, and a is determined by the behavior ofu at infinity in the sense that u{tx)/t -^ axn in L\^^{H) as t -^ -f oo. Moreover, (3.1.18)

/ ( I + \x'\^)-û{x\xn)

dx'

< ( 1 - f Xn) / (1 + k T ) ~ ^ da{x')

and for every ^p G C{W^)

/

+ \CnaXn,

Xn > 0,

with (p{x') = 0 ( ( 1 + | x ' p ) - t ) we have

ip{x') da{x') = lim /

ip{x')u{x'^Xn)dx'.

Proof. As already indicated we could obtain Theorem 3.1.8' from Theorem 3.1.8 by means of an inversion, but we shall give another proof to avoid some computations. First we assume that u is continuous in i J , and we write down (3.1.11) for the ball {x G R"^; |xp < 2Rxn} with radius R. This gives for every x ^ H when R is large enough u{x) = {2xn - \x\yR)c-'

J \ x - y\-My)

dS{y)

with the integral taken over the boundary sphere. When R -^ oo we obtain by just keeping the integral where yn = \y^\'^/{R + ^/R^~-^W\^) ^^^ v' i^ in a compact set u{x) > /

PH{x,y)u{y)dS{y)

= v{x),

x e H.

JdH

Thus u{y)/{l + 1^1)"^ is integrable over dH^ so it follows from Theorem 3.1.5' that t; is a continuous function in H with boundary values equal to those of It, and that v is harmonic in H. Hence u — v is a non-negative harmonic function which is continuous in H and has boundary values 0. The inversion argument shows that all such functions are multiples of the minimal positive harmonic function x H-> x^ corresponding to the point at oo. Hence we obtain u{x) — v{x) = aXn for some a > 0, which proves (3.1.15)' when u is continuous in H. To extend (3.1.15)' to general positive harmonic functions in H we apply the result already obtained to u{x',Xn + s) which gives u{x)=

/ JdH

P{x\xn-e,y')u{y\e)dy'-\-a^{xn~e),

Xn>e.

130


For 0 < £ < 1 it follows that 0 < a^ < C, / ( I + Wiyuiy'.e) dy' < C. We can therefore choose Sj -^ 0 such that u{y'êj)dy' converges weakly to a limit d(j{y'), and we have / ( I + \y'\)~'^dG{y') < oo. When 0 < e < 1 and \x'\ < i? > 1 , we have P{x', Xn - £, y') u{y\ e) dy' < C'{xn - e), / \y'\>2R for (1 + \y'\)l\x' - 2/'| < 3 when \x'\ < R< Poisson integral v{x)^

f

||2/'|, so it follows that for the

P{x,y')da[y')

JdH

we have v{x) < u{x) and u{x) — v{x) < Cx^ in H. Hence u{x) — v{x) is a minimal positive harmonic function for the point at infinity, so u{x) = v{x) + aXn: which proves (3.1.15)'. Now we obtain u{x'^Xn)dx' —> da{x') as Xn -^ 0, for ii cp e (7o(R'^) then / u{x'^Xn)^{x')dx'

= axn I (p{x')dx' -\- I da{y')

/

P{y\xn')X')ip{x')dx'

where the inner integral converges uniformly to 0}, and Re/(2;) > 0 there, then

/(.) = -i(a. + 6) + i / ; j (-1^ + j l p ) Mi). where dcr > 0, a > 0, & G R, and / da{i)j{l

+ i^) < CXD.

E x e r c i s e 3.1.7. Show that if ?x is a positive harmonic function in if, then either t'^~û{tx) -^ oo as t -^ oo, uniformly on every compact subset of i7, or else a = 0 and ^Qjjda{y') < oo in the representation (3.1.15)' of 16, and then we have t'^~û{tx) -^ PH{XÔ) J^jjda uniformly on every compact subset of H. We shall now discuss the extension of Theorem 3.1.8' to functions which are not semi-bounded. To obtain representations of the form (3.1.15)' it is clear that one must impose restrictions both as xî —> 0 and as x -^ cx), where we have to rule out functions like nx^ — I^P which is bounded above when Xn is bounded but grows too fast with Xn- (We shift to upper instead of lower bounds now to conform with the analogous discussion in Section 3.3.)

132


T h e o r e m 3 . 1 . 1 1 . Let u be a harmonic function in H such that u~^ = max('u, 0)

(3.1.20)

lim

/ u'îx', Xn){l + \x'\y

(3.1.21)

lim

/

with

dx' < oo,

u-^{x)dxlBJ' + 1 < 0 0 .

Then there is a unique measure da with / ( I -j- |2/'|)~'^ \da{y')\ < oo on dH = R^~^ and a constant a such that (3.1.15)' is vaUd. da is the weak limit of the measure u(x',Xn)dx' as Xn —> 0, and a is characterized by the behavior of u at infinity in the sense that u{tx)/t -^ axn in L\Q^{H) as t ~> +00. Moreover, (3.1.180

j{l^\x'\^)-'^\u{x',xn)\dx' 0,

and for every ip G C ( R ^ - ^ ) with ip{x') = 0 ( ( 1 + l^r'p)-?) we have

/

ip{x')da{x')

— lim Xn^^

/

J

ip{x')u{x',Xn)dx'.

Proof. So far we have only used the Green's function and the Poisson kernel for the ball and for the half plane. However, for the half ball S ^ = H n BR we can also easily define a Green's function by G\{x,y)

= GR{X,y)-GR{X,y*)

= GR{X,y)-GR{x%y),

x,y e B^,x^

y,

where * denotes reflection in dH just as in (3.1.6)'. In fact, it is clear that G%{x^y) = 0 if X or y is in d B ^ , and G\{x,y) — E{x — y) is harmonic in X and in y. The Poisson kernel Pt{x,y)

= dG+{x,y)/dny,

x E B+, y € dB+

is positive, for G^{x,y) < 0 in S j x B j by the maximum principle since G^{x,y) —> - o o as a; -^ y, and Green's formula gives as before if v is harmonic near BR (3.1.11)"

v{x)=

f JdBt

P^{x,yHy)dS{y).

HARMONIC FUNCTIONS

133

If 7/ G B^ then the harmonic function G\{x^y) — Gnix^y) of x E B^ is > 0 when x G dB^^ so it follows that G j > GH in 5 ^ x 5 J , which proves that (3.1.22)

PR^X.V)

< PH{x,y),

iix e B^ and y G

dB^ndH.

When X G J5j and y G 5 5 J \ 9iJ we have by the mean value theorem (3.1.23)

P^{x, y) = R'-îP{x/R,

= R{1 - \x\yR')c-'{\x

y/R) - P[x^lR,

- y\-- -\x-

yTn

y/R))

< ^nXn{R -

\x\)-^c-\

By (3.1.20) we can take a sequence £j —> 0 such that the integral Ju'^{x',ej){l + Ix'l)"^ dx' has a finite limit as j —> oo and the sequence u'^{x'êj)dx' converges weakly to a measure dv > 0, necessarily with / ( I -h \x'\ydiy{x') < oo. If we apply (3.1.11)" to u{x',Xn + Sj) and let j -^ oo after integrating with respect to i? for ^ < i? < 2^, it follows that for large g gu{x) < f

dR [

JQ

P^{x,y)u^{y)dS{y)^Q

JHndBR

[

PH{x,y')dv{y').

JdH

Here we have used that u'^ G Ll^^{H). Hence we obtain using (3.1.23) (3.1.24)

u{x) o. Choosing T'(0) = —aid as we may, we conclude that (3.1.35) ^ ( ^ ) ( 0 ; e , e , y , y ) - 6au''\0;e,Y,Y) -\-Ga\'\0]Y,Y)

> 0,

Y G R^.

The left-hand side is a harmonic quadratic form, for u is harmonic, and since it takes its minimum at 0 it must vanish identically. Returning to (3.1.34) we can choose T so that T{0)Y = -aY-\-eZ for any Z with (Z,'u'(0)) = 0, and since there is equality in (3.1.34) when ^ = 0, the coefficient of £ must vanish, that is, 47i'"(0; e, Z, Y) - 8au''{0] Z,Y) = 0

HARMONIC FUNCTIONS

139

for all Y and Z with {Z,u'{0)) = 0, which implies (3.1.32). Hence the harmonic quadratic form u"'{0] e, F, Y) — 2au'\0] Y, Y) must be a constant times {Y,u'{0))'^, and the harmonicity implies that the constant is 0. This proves (3.1.29) with j = 2. The remaining condition in (3.1.34) is only that 2u"{0; Z,Z)>0 when (Z, u'(0)) = 0, which follows from the fact that u"{0; •, •) is positive semi-definite in the tangent plane of the level surface of u at 0. With T chosen so that T'{0) = —aid we know now that (3.1.31) must vanish of third order, hence of fourth order because of the positivity, and we can repeat a similar argument. However, it is now time to specify the inductive statement which will prove (3.1.29). We shall prove for z/ = 1,2,... that (3.1.29) is valid for 1 < j < z/ + 1 and that for 0 <j 1. Then we can choose the tangent vector field T so that for a given symmetric u-\-l linear form Z in R'^ with values in the orthogonal plane of u'{0) in R"^ (3.1.30)' T(0) = e,

T'(0) = - a i d ,

T^^\0) = 0, K

j < u,

T(^+I)(0)

= Z.

In fact, the j t h differential of (T(x),'a'(x)) at 0 in the direction Y is

(3.1.37)

Yl (^)^^"^^\0',T^'~'\0;Y^-'),Y').

If T has the Taylor expansion required in (3.1.30)' and j < ^' -I- 1, this reduces to ^(^-^^)(0;e,y^)-ia'a(^)(0;y^) = 0 by the inductive hypothesis. Thus R{x) = {T{x),u'{x)) = 0{\x\'''^'^), and replacing T{x) by T(x) - R{x)u'{x)/\u'{x)\'^ we obtain all the conditions (3.1.30)'.

140


When T satisfies (3.1.30)' and j < 2i/ + 1 the j t h differential of u''{x;T{x),T{x)) at 0 in the direction Y is 3-1

(3.1.38)

^^^•+2)(0;e,e,y^) + 2 ^

('•^\(^+2)(Q.g^yy-0(^

+ i(i - l)^«)(0;T'(0;y),r'(0;y),y^-') + 2 J^

-^^

^7x(^+')(0; T'(0; Y), T ( ^ - ^ - ^ ) ( 0 ; y^"^"^), F^)

if j < 2i/ + l, for terms where both "factors" T are differentiated more than once must vanish by (3.1.30)' since one dijfferentiation is of order < u. If j = 2i^ + 2 there is an additional term

Using (3.1.30)' we simplify (3.1.38) for j < 2z/ + 1 to u (^•+2)(0; e, e, Y^) - 2aju^^+^\0; e, Y^) J-^'-i

+2 E 2= 0

.,

7T^^:Tn^^'^'^(0;e,TO-)(0;y^-^),yO+i(i-l)aV^)(0;y^^^ ^^

*^**'

Since j — u 1. Thus the inductive hypothesis remains valid with u replaced by z/ + 1, which completes the proof. 3.2. Basic facts on subharmonic functions. Using harmonic functions instead of linear functions we can now copy Definition 1.1.1 for functions of several variables: Definition 3.2.1. A function u defined in an open subset X of R"^ with values in [—cx), oo) is called subharmonic if (a) u is upper semicontinuous; (b) for every compact subset K oi X and every continuous function h on K which is harmonic in the interior of if, the inequality u < h is valid in K if it holds in dK. A function u is called superharmonic

if —u is subharmonic.

Remark. It may seem inconsistent that we require upper semi-continuity here while we imposed lower semi-continuity on the convex functions in Chapter II. However, we only consider subharmonic functions in open sets, and a convex function in an open set is continuous. Since upper semicontinuity means that a strict upper bound valid at one point is also valid in a neighborhood, it is clear that (a) is the natural condition to go with (b). The function u = —oo is subharmonic according to Definition 1.3.2. This is sometimes convenient but some authors exclude this function in the definition, which is convenient at other occasions.

142


By Theorem 3.1.6 every harmonic function is of course subharmonic and superharmonic. The most important example of a subharmonic function in R'^ which is not harmonic is a fundamental solution x — t > Ey{x) = E{x — y)^ where E is defined as in Theorem 3.1.2, equal to —oo at 0. It is clear that Ey is then continuous with values in [-00,00). If y ^ K then it is also obvious by Theorem 3.1.6 that condition (b) in Definition 3.2.1 is fulfilled, and ii y e K then Ey < h in some open neighborhood V of y^ and since diK \ V) C (dK) U (dV) we have Ey < h in d{K \ V), hence inK\V and in K. In the same way we see that a finite sum a; i-> ^ ajE{x — yj) is subharmonic if a^ > 0. In particular, if / is an analytic function ^ 0 in an open connected set X C C, then log \f{z)\ is subharmonic, for if we write f{z) = g{z) H i {z — Zj) with g analytic and ^ 0 in K, then log |^| is harmonic near K and Yli log \^ ~ ^j\ is subharmonic. This example is the reason for the importance of subharmonic functions in analytic function theory. The case n = 2 often diff'ers from the case where n > 2 because the fundamental solution is not negative in the whole space when n = 2, but since many of the most important applications occur when n = 2 we cannot ignore these special features by assuming n > 3. Exercise 3.2.1. Show that if Xând X are open sets in R'^ and O is an orthogonal transformation, X = OX, then X 3 x — t > u{Ox) is subharmonic in X if lA is subharmonic in X . Also prove that X 3 x t-^ \x\'^~'û(x/\x\'^) is subharmonic in X if iz is subharmonic in X and X, X are related as in Theorem 3.1.10. Show that if X, X C C = R^ and cp is a. complex analytic bijection X —> X , then u o (p is subharmonic in X ii u is subharmonic in X. T h e o r e m 3.2.2. If u is subharmonic in X and 0 < c E R, then cu is subharmonic in X. If ui,... ,Uy are subharmonic in X , then u — m a x ( t i i , . . . ,Uj^) is also subharmonic in X. If Ui^, t E / , is a family of subharmonic functions in X and u{x) = sup^^j Ui^{x) is upper semi-continuous with values in [—00,00), then u is subharmonic in X. If ui,U2-,- • • is a decreasing sequence of subharmonic functions in X , then u = lim_^_,oo '^j is also subharmonic in X. Proof. The first three statements are obvious from the definition. To prove the last one we take a function h which is continuous in a compact set jFf C X and harmonic in the interior of K, such that h > u on dK. Let e > 0. For every 0:0 G dK we have Uj{xo) < h{xo)-\-e for some j , and since Uj — h is upper semi-continuous there is a neighborhood V of XQ such that Uj{x) < h{x) 4-6:,

iix eV

nK.

Here we may replace Uj by Uk for any k > j . By the Borel-Lebesgue lemma we can cover dK with a finite number of such neighborhoods V,

BASIC FACTS ON SUBHARMONIC FUNCTIONS

143

hence Uk(x) < h{x) 4- £ when x G dK^ if k is large enough, and it follows that u < Uk < h-\- e m K. Any decreasing limit of upper semi-continuous functions is upper semi-continuous, which completes the proof. Definition 3.2.1 is often useful as it stands, but it does not indicate for example that the sum of subharmonic functions is subharmonic. We shall therefore give other equivalent properties, analogous to those in Exercise 1.1.12 or Theorem 3.1.12. T h e o r e m 3.2.3. Let u be an upper semi-continuous function in an open set X C R"^ with values in [-oo, oo). Then each of the following conditions is necessary and sufficient for u to be subharmonic in X: (i) Condition (b) in Definition 3.2,1 is fulGUed when K is a closed ball

ex. (ii) If Xs = {x^X\y£Xif\y {x,r)=

— x\< 6], ^ > 0, then the mean value /

(3.2.1)

M^{x,r) = Mix,r)= f u{x + ry) du{y)/cn, x e Xr, is an increasing function ofrE [0,6] ifx G X^. (iii) For every positive measure dfi in the interval [0, ^], ^ > 0, we have (3.2.2) u{x)

/ duj{y)d^{r) 0 and every x E X^ there is a positive measure supported by [0,6] but not by {0} such that (3.2.2) is valid. Note that the integrals are well defined since u is semi-continuous. Proof. It is obvious that subharmonicity implies (i) and that (iii) implies (iv). From (ii) it follows that u{x) < M{x,r) ii r < 6 and x G Xs, hence (ii) implies (iii). To prove that (i) implies (ii) let x £ Xs and set K = {y] \x — y\ < 6}. If (^ is a continuous function on the unit sphere such that u{x + 6y) < (p{y) when \y\ = 1, then condition (i) states that u < h in K if h is the solution of the Dirichlet problem given in K by h{x -\-z)=

P{z/6, y)cp{y) du{y),

\z\ < 6.

I f O < r < 5 t h e n b y (3.1.12) M{x,r)

< / J\z\ = l

h{x-hrz)duj{z)/cn

= h(x) = / J\y\ = l

(p{y) duj{y)/cn.

144


Since u is upper semi-continuous the infimum of the right-hand side over all continuous majorants cp is equal to M{x^6)^ which proves the monotonicity stated in (ii). It remains to prove that (iv) implies that u is subharmonic. We can essentially repeat the argument in the proof of Theorem 3.1.12. Let K be a compact subset of X and let /i be a continuous function on K with h> u on dK, such that h is harmonic in the interior of K. If the supremum V oi V — u — h in K is positive, then it is finite and is attained in a compact subset F of the interior of K, because v is upper semi-continuous. Let XQ E F have minimal distance 2^ > 0 to dK. On every sphere with radius r G (0, 28) and center at x there is some point y at distance 2^ — r to dK, and since v is upper semi-continuous we know that -u < V^ in a neighborhood of y. Hence /

v{x + ry) duj{y) < c^V — Cnv{x),

0 < r < 2^.

^12/1 = 1

If we integrate with respect to a measure d\i with support in [0, S\ with the properties in condition (iv), we get a contradiction with the hypothesis that (3.2.2) holds, for the inequality (3.2.2) must also hold with u replaced \yY V — u — h since h is harmonic. This contradicts the assumption that F > 0 and proves that u the left-hand side by (3.2.2), and the upper limit is < the left-hand side since u is upper semicontinuous. E x e r c i s e 3.2.2. Prove that iiu is upper semi-continuous in X then u is subharmonic if and only if lim^_ô(-^n(^5^) — u{x))/r'^ > 0 for every x G X with u{x) > —oo. (Hint: Prove first that u{x) + £|a:p is subharmonic.) Prove that for every subharmonic function u in X and x E X lim /

\u{x + y) — u{x)\ dy/r'^ = 0, lim /

if u{x) > —oo,

\u{x 4- y)\ dy/r"^ = CXD, if u{x) = —oo.

(Hint: Examine J\y\^^ \u{x -i-y) — c\ dy/r"^ when c > u{x).)


Corollary 3.2.4. u is both subharmonic and superharmonic ifu is harmonic. Corollary 3.2.5. If ui,... subharmonic.

,Uk are subharmonic

145

if and only

then ui -\- - • - -\- Uk is

Corollary 3.2.6. Ifu is a function defined in an open set X such that every point in X has a neighborhood in which the restriction of u is subharmonic, then u is subharmonic in X. Thus subharmonicity is a local property. Corollary 3.2.7. Let Y C X be open sets, and assume that u is a function in X which is harmonic in Y and equal to —oo in X\Y. Then u is subharmonic if and only if u is upper semi-continuous, that is, u{y) -^ —oo ifY3y-^xeX\Y. Proof. Note that (3.2.2) is trivial ii x E X \Y and follows for small 6 from the mean value property of harmonic functions ii x EY. Corollary 3.2.8. Ifu is subharmonic in an open connected set X and not = —oo, then u E Ll^^{X), so u(x) > —oo almost everywhere. Proof. li X £ Xs^ defined as in Theorem 3.2.3, and u{x) > —oo, then u is integrable in {y; \x — y\ < r } if r < (5, for u is bounded above and the integral is > u{x)r'^Cn/n. The subset Y oi X consisting of points such that u is integrable in some neighborhood is open by its definition, and we claim that it is closed in X . In fact, iiy is in the closure and y E Xs^ then we can choose X E X with |a; — ?/| < 6/2 so that u{x) > —oo, and u is integrable in the ball with radius S/2 and center at x, which has y as interior point. Thus Y is open and closed in X , and since X is connected it follows that Y = X or Y = ib, so u E iîoc(^) oi"^ = - o o . Corollary 3.2.7 shows again that positive linear combinations of fundamental solutions E{- — y) are subharmonic, and that log | / | is subharmonic if / is an analytic function. We can prove this more generally: T h e o r e m 3.2.9. Let du be a positive measure in R'^ with support, and set (3.2.3)

u{x) -= j / u{x)

compact

E{x-y)dv{y).

Then u is subharmonic; it is called the potential of the measure dv. The convolution E ^ dv in the sense of distribution theory is defined by u. Proof. li t is a constant, then Et = max(£^, ^) is subharmonic and continuous, by Theorem 3.2.2, and Et I E diS t I —oo. Hence the continuous function Ut{x) = {Et * dv){x) = JEt{x — y)dv{y) decreases to u{x) as

146


t [ —oo, which proves that u is upper semi-continuous. To prove that u is subharmonic it suffices by Theorem 3.2.2 to show that Et * dv is subharmonic, which follows from Theorem 3.2.3: / ut{x + ry) du;{y) J\y\=i = / diy{z) / J

Et{x'\-ry-z)duj{y)

J\y\ = l

> Cn

Et{x-z)dv{z)

= CnUt{x).

J

(This is really a continuous version of Corollary 3.2.5, for the convolution is a superposition of translates.) Since Et - E -^ 0 in L^ we have Ut — u -^ 0 in L^, as t —> — oo, and since convolution is continuous in the distribution topology it follows that u = E ^ dv. The proof is complete. Remark. When n > 2 the fundamental solution is negative everywhere so (3.2.3) is well defined with values in [—00,00) even if dv does not have compact support. Since u is the decreasing limit of the same integral taken only for I2/I < jR when i? -^ 00, it follows that u is subharmonic also in this more general case. However, we may have u = —00 then. Theorem 3.2.9 allows us to give a simple example of a subharmonic function which is not continuous with values in [—00,00). We just take sequences xjt -^ 0 in R"^ \ {0} and a^ > 0 with YlT ^kE{xk) = —1. The sum u{x) = YlT ^kE{x — Xk) is subharmonic, u{xk) = —00 for every fc, but u{0) = —1. lit < -1 then Ut{x) = msix{t,u) is subharmonic and takes values in [t,oo), ut{0) = - 1 but lim^_^Qn^(x) = t < —1 = ut{0). To establish a connection between subharmonicity and Poisson's equation we begin with an elementary result similar to Corollary 1.1.10: P r o p o s i t i o n 3.2.10. Ifu G (7^(X) where X is an open set in R^, and M is defined by (3.2.1), then (3.2.4)

lim(M(x, r) - u{x))/r'^ = Au{x)/2n,

x e X,

r—vO

and u is subharmonic

in X if and only if Au > 0.

Proof. It suffices to prove (3.2.4) when rr = 0. By Taylor's formula n

u{x) =u{0)-^Y^Xjdju{0)i-^ 1

n

^

XjXkdjdku{0) + R{x),

R{x) = o{\x\'^).

j,k=i

Hence /. ,^R{ry)duj{y)/r'^ —> 0 as r ^ 0. The integrals over dBr of the terms in the sums are all zero apart from


147

which proves (3.2.4). Here we have used that /. , ^ 7/? da;(y) = Cn/n since the integral is independent of j . Prom (3.2.4) and Theorem 3.2.3 it follows at once that A?x > 0 if u is subharmonic, and that u is subharmonic if An > 0. If we just assume that Aii > 0 we may conclude that u{x) + e\x\'^ is subharmonic for every 6 > 0, and when ^ | 0 it follows that u is subharmonic. We could also prove the second part of Proposition 3.2.10 using an analogue of Corollary 1.1.16: P r o p o s i t i o n 3.2.10'. Ifu is an upper semi-continuous function in the open set X C R'^ which is not subharmonic, then one can find XQ ^ X and a quadratic polynomial q with Aq < 0 such that q{xQ) = u{xo) and u < q in a neighborhood of XQ. Conversely, u is not subharmonic if there is such a function q. Proof. The last statement is obvious, for x i-> u{x) — q{x) — £\x — rrop would be subharmonic for small ^ > 0, equal to 0 at XQ but < 0 in a punctured neighborhood of XQ. Now assume that u is not subharmonic. We can then choose a closed ball B C X and a function h which is continuous in B and harmonic in the interior of B such that u — h0. Set for 5 > 0 Ve{x) = h{x) — £|a:p. Then u < v^ on dB if e is sufficiently small, but sup^(?z — Ve) > 0. The upper semi-continuous function u — Ve takes its maximum in B at an interior point rro, so u(x)

< Vs(x)

-h u(Xo)

— Ve{xo),

X £

B.

Set q{x) = u{xo) - Ve(xo) + ^

d'^h{xo){x - Xo)'^/a\ - £:|x|^ + ^€\x - Xo|^

|a| 0. It is easy to extend Proposition 3.2.10 to distributions:

148


T h e o r e m 3.2.11. Ifu is a subharmonic function in an open set X and is not = —oo in any component, thus u G L\Q^{X), then Atfc > 0 in the sense of distribution theory. Conversely, ifU G V'{X) and AC/ > 0, then U is defined by a unique subharmonic function u in X. Proof. That Ait > 0 in the sense of distribution theory means by definition that fuAvdx

> 0,

live

C^{X),

v>Q.

If we express Av using (3.2.4) we obtain since (3.2.4) is uniform in x -— / uAvdx 2n J

— hm / u{x){My{x,r) r-ô J

— v{x))/r^

dx

= hm / v{x){Mu{x,r)

— u{x))/r^

dx.

The right-hand side is non-negative by Theorem 3.2.3 iiu is subharmonic, which proves the first statement. To prove the second statement we choose (f G CQ°{BI) such that cp > 0 and (p{x) only depends on \x\, J (p{x) dx = 1. Set ipeix) =:= e-''(p{x/s). Then Ue = U ^ ips e C ^ ( X ^ ) , e > 0, where Xs is defined as in Theorem 3.2.3, and AUe = (AC/) * (^^ > 0, so C/^ is subharmonic by Proposition 3.2.10. Hence it follows from Theorem 3.2.3 that U îêîps, which is defined in X^^s, is a decreasing function of 8 as 5 I 0. Letting e ^^ 0 we conclude that C/ * (^^ is a decreasing function of ^ as 5 I 0, and then it follows from Theorem 3.2.2 that U ^ (ps I u, where u is subharmonic and not identically — oo. Since U ^ (ps -^ U in V^ and U ^ (fs —^ u in LJQ^, it follows that the distribution U is defined by the function u. For any subharmonic function u we have t6 * 0, so the subharmonicity of u is also a consequence of Theorem 3.2.11. If / is an analytic function ^ 0 of a complex variable z, then A log 1/(2:)! is the measure

2TE

rrijdz

where Zj are the zeros and ruj their multiplicities. In fact, in a neighborhood of Zj we have f{z) = [z - Zj)'^^g{z) where g is analytic and ^ 0, so log 1/(2^)1 = 27rmjE{z - Zj) -^ log \g{z)\ where the last term is harmonic. Extending the result on harmonic functions in Exercise 3.1.2 we shall now prove:


149

T h e o r e m 3.2.12. Let Uj be a sequence of subharmonic functions in an open connected set X C R^, which have a uniform upper bound on every compact subset of X. Then either Uj -^ —oo uniformly on every compact subset of X, or else there is a subsequence Uj^ which converges in Ll^^{X). Ifuj ^ —oo for every j and Uj -^ U in V, then U is deRned by a subharmonic function u and Uj -^ u in Ll^^{X). Proof. It suffices to prove this with X replaced by a relatively compact subset, so subtracting a constant we may assume that Uj < 0 for every j . If Uj does not converge to — oo uniformly on every compact set, then we can find jk and Xk such that all Xk belong to a compact subset K oi X and Ujj^{xk) is bounded. We may assume that Xk -^ XQ E X, and to simplify notation we assume that jk = k. By Corollary 3.2.8 we have Uj E L\^^{X) for every j . If B C X is a closed ball with center at a;o, then the sequence Jg Uj is bounded from below. In fact, for large j there is a ball Bj with center at Xj such that B C Bj C X^ and then we have / Uj dx > JB

Uj dx >

m{Bj)uj{xj).

JBi

We can now show as in the proof of Corollary 3.2.8 that if Y is the set of points X e X having a neighborhood N such that the sequence J^ Uj is bounded from below, then Y is both open and closed, hence equal to X. This proves that the sequence Uj is bounded in L J Q ^ ( X ) . We can therefore find a subsequence Ujj^ which converges in the weak topology of measures, hence as a distribution, and the limit is defined by a subharmonic function in view of Theorem 3.2.11. It remains to prove the last statement, so assume now that Uj —» U in D'. Then AU = lim^_,oo A-Uj > 0, so [/ is defined by a subharmonic function u^ by Theorem 3.2.11. With the notation in the proof of Theorem 3.2.11 we have (3.2.5)

% ( ^ ) ^ %• * y^si^) -^ u^ ^six),

X G Xs,

and the convergence here is uniform on compact sets in Xs since the convolutions Uj * ips{x) are equicontinuous there. Choose x ^ 0 in CQ^{XS) and e > 0. Then / {u^^s{x)+e-Uj{x))x{x)

dx -^ / {u^(ps{x)-\-e-u{x))x{x)

dx,

and the integrand on the left is positive for large j . Hence lim / \u — Uj\xdx 3-ôoJ

u in L^^^{X) for p G [ l , n / ( n — 2)), while Uj —> u' in L\^^ for p G [1, n/{n - 1)). For every x e X we have (3.2.6) More generally^ ifK (3.2.7)

iim Uj{x) < u{x),

x e X.

is a compact subset of X and f G C{K),

then

Iim sup{uj - f) < s\ip{u - / ) .

If da is a positive measure with compact support in X such that the potential E ^da in Theorem 3.2.9 is continuous, then there is equality in (3.2.6) and u(x) > —oo for almost every x ^ X with respect to da. Moreover, Ujda —^ uda in the weak topology of measures. Proof. We keep the notation in the proofs of Theorems 3.2.11 and 3.2.12. From (3.2.5) it follows at once that we have a uniform upper bound for Uj on any compact subset of X , and that Iim Uj{x) < u^ ^si^)^

X ^ Xs-

The right-hand side converges to u{x) as (5 -^ 0, which proves (3.2.6). Let M — snpj^{u-f). UK C Xs a n d x G K, thenmax{M,u*(ps{x)-f{x)) | M when ^ I 0. It follows from Dini's theorem that the convergence is uniform on K, which proves (3.2.7). If Et = max(J5, t) for some large negative t, as in the proof of Theorem 3.2.9, then Et is continuous with finite values and E — Et —^ 0 in L^ SiS t -^ - o o for every p G [ l , n / ( n - 2)). Let F ^ X and let 0 < x ^ CîX) be equal to 1 in Y. The positive measures dfij — tlUj converge to d[i = Au in V^, hence in the weak topology of measures. If we set duj = X^/^j ^^^ dp = xcf/x, then Uj = E ^ duj -\- Vj,

u = E ^ du -hv,


151

where A'^j — (1 — x)dlJ^j = 0 in y and Vj —> -y in V\X) as j —> oo, hence Vj ^ ' i; in C^{Y) by Corollary 3.1.4. Hence it suffices to examine the convergence of -B * duj to E ^ du. We have \\{E - Ft) * {duj - du)\\LP < C\\E - EtWLp -^ 0,

when t -^ - o o ,

for the total mass of the measures duj is uniformly bounded. When t is fixed then Et * {dvj — dv) -^ 0 uniformly on every compact set as j -^ oo, because Et is continuous. Hence it follows that E * dvj -^ E ^ dv in L^^^, To prove the corresponding result about the first derivatives we take a smooth approximation to E^ for example E^{x) = E{x){l — xoi^/^))-, where xo ^ CQ^CR^) is equal to 1 in a neighborhood of the origin. Then dE^{x)/dxj-dE{x)/dxj

= -xo{x/6)dE{x)/dxj-E(x){djXQ){x/6)/6

-^ 0

in L^ as ^ -> 0 if p ( l - n) + n > 0. Hence ||(£^^ - E)' * {duj - du)\\LP -> 0 then as 5 -^ 0, uniformly in j , and since E^ * {duj -diy) -^ 0 in ( 7 ^ for fixed ^ as j ^^ oo, it follows that u'j -^ u' in L^^^. Prom Fatou's lemma it follows that

/

( l i m Uj{x))da{x)

> lim /

Uj{x)da{x).

In view of the general inequality (3.2.6) it will follow that (3.2.6) must be an equality almost everywhere with respect to da if we show that (3.2.8)

/ Uj{x)da{x)

-^ / u{x)da{x),

when j -^ oo,

and that the right-hand side is finite. In doing so we may assume that suppdcr C y , and then the statement follows from the fact that (E ^ dvj){x) da{x) = = j{E

* da){y)dv,{y)

E{x — -^ J{E

y)dvj{y)da{x)

* da){y)du{y)

- J{E

*

du){x)da{x),

where the limit is justified by the continuity of £^ * da. To prove that Ujda -^ uda it suffices to show that we may replace da by i/;da in (3.2.8) if 0 < -0 G Co, and that follows if we prove that E * (ipda) is continuous. Now the fact that J5t * dcr | J5 * da as t ^> — oo implies by Dini's theorem that the convergence is locally uniform, that is, {Et — E) ^ da —^ 0 locally uniformly as t -^ - o o . Now 0 < {Et - £") * ipda < {snpip){Et — JS) * dcr, so it follows that also E * ipda is continuous. The proof is complete.

152

III. S U B H A R M O N I C F U N C T I O N S

When E^ da is continuous, it follows that a subsequence of the sequence Uj converges to u almost everywhere with respect to a. However, this is not true for the full sequence. To give an example we choose for z/ = 1, 2 , . . . a finite set Ajy C {y E H^; \y\ < v] such that min^^^^ E{x — y) < —v^ when |a;| < z/. Ordering the functions £^(- — y)lv with 7/ G Aj, as a sequence Uj^ we have Uj —> 0 in Ll^î^^) but lim ^ ^ "^jC^) — ~c>o for every x. The condition on da in Theorem 3.2.13 is obviously fulfilled if da = -0 dx where 0 < I/J G CQ^ and dx is the Lebesgue measure, so lim^-^.oo'^j = '^ almost everywhere with respect to the Lebesgue measure. We may also replace the Lebesgue measure by the area measure dS in any C^ hypersurface, for the area of its intersection with a ball of radius e is 0{e'^~^)^ which suffices to show that {Et — E) t (^dS) —> 0 locally uniformly as ^ -^ —oo. We leave the verification as an exercise but shall give some comments on the case of a hyperplane. P r o p o s i t i o n 3.2.14. Let X be an open set in H^, K a compact set in R'^'^j and / a compact interval on R such that K x I C X. Then there exists a positive constant C such that for all subharmonic functions u in X (3.2.9) /

\u{x',Xn)

-U{x\yn)\dx'

'(X), so if 0 < x ^ CQ^{X) then the mass of du = xA?x has a uniform bound. Choose x so that X = 1 in an open set Y D K x I. As in the proof of Theorem 3.2.13 the function v ~ u — E ^ dv is harmonic in Y and we have uniform bounds for V and its derivatives in K x I. Now (3.2.11)

/

\E{x',Xn)

- E{x',yn)\dx'

- \\\xn\ - bn||,

for it suffices to prove (3.2.11) when 0 < 2/n < ^n? with all absolute value signs removed, and then it follows from the fact that the integral of 2dE{x\xn)ldxn = PH{X.,0) with respect to x' is equal to 1. Hence we obtain / \{E^du){x\xn)

- {E^diy){x',yn)\dx'

< | | x n - yn\ / du,

which proves (3.2.9). Since / JK

\u{x',Xn)\dx'

< / JK

\u{x',yn)\dx'

-\-C\Xn-yn\

\u\dx, JX

yn ^ I,


153

the estimate (3.2.10) follows when we integrate with respect to yn over / . For the restriction of subharmonic functions to hyperplanes we also have a Holder type continuity: P r o p o s i t i o n 3.2.15. Let X be an open set in W^, K a compact set in R ^ - ^ and assume that K x {0} CX. Then (3.2.12) / \vi{x\0)-V2{x',0)\dx' u{Ox) is a subharmonic function (see Exercise 3.2.1), it follows that x y-^ M(0, \x\) is subharmonic. This is equivalent to the convexity stated in the theorem. It is of course also a consequence of (3.2.13), and since dfi can be any positive measure we see from (3.2.13) that one cannot improve on the convexity statement. However, when we study plurisubharmonic functions in Chapter IV, then dfi cannot be arbitrary, and Mu{x,r) will have a stronger convexity property. Exercise 3.2.4. Let X be an open set in R'^, a;o ^ X^ and let IA be a subharmonic function in X \ {XQ}. Show that M ( x o , r ) is bounded above as r —> 0 if and only if for some R> 0 one can write u = v -\- w where v is subharmonic when \x — XQ\ < R and w is harmonic when 0 < |x — xo| < R with M>uj{xojr) = 0 when 0 < r < R. (Hint: Conclude that Mu{xQ^r) is increasing and that the mass of Au near XQ is finite.) Show that u can be extended to a subharmonic function -it in X if and only if u is bounded above in a neighborhood of XQ, and that u{xo) is then the limit of M{xo^r) as r ^> 0. Conclude that in Exercise 3.2.1 one can allow any analytic map (p : X -^ X. Exercise 3.2.5. Prove that ME{x^r) Msix^r) = e(r) if r > \x\.

= E{x) if r < |x| and that

T h e o r e m 3.2.17. Let u be a subharmonic function in R^ such that Mu{0,r) = o(logr) as r —> oo. Then u must be a harmonic function. Proof. Since Mi^(0,r) is an increasing convex function of log r, by Theorems 3.2.3 and 3.2.16, it must be constant if it is o(logr) as r —> oo. Thus Mu{0,r) = u{0) for every r, and it follows from (3.2.13) that Au = c6o for some c > 0. Hence u — cE -\- h where h is harmonic, and since Mu{^^r) — ce{r) + /i(0) we conclude that c = 0. T h e o r e m 3.2.18. If (p is convex and increasing on R, cp{—oo) = limt_*_oo r, then (p{u{x)) < ip{Mu{x,r))

= cp{

u{x-\-ry)duj(y)/cn)

J\y\=i

< / J\y\=i

^{u{x-^ry))du}{y)/cn

=

Mî^^){x,r),


157

where the first inequaUty follows from Theorem 3.2.3 (ii) and the fact that if is increasing, and the second follows from the convexity of (p and Jensen's inequality (see Theorem 1.1.14 and Exercise 1.1.9). This proves the theorem, for it is clear that ^{u) is upper semi-continuous. Remark. It suffices to assume that ^p is defined in an interval containing the range of IA, for ip can then be extended to the whole line with preservation of monotonicity and convexity. Exercise 3.2.6. Let l i i , . . . ,Uk be subharmonic functions in X C R^, let / be a convex function in an open convex set in R^ containing the range of ('Ui,... ,1^^), and assume that / ( s i , . . . ,Sfc) < / ( t i , . . . ,tfc) if both sides are defined and Si < ^ i , . . . , ^ ^ < tk- Prove that f{ui^,..Ûk) is subharmonic. Deduce that log(e^^ + • • • + e^^) is subharmonic and that {u\ -\- •' • -{-vF^Yl'P IS subharmonic if p > 1 and all Uj are non-negative. Corollary 3.2.19. U u is subharmonic when Ri < \x\ < R2 and (p is convex and increasing on R, with (p{—oo) = limt_,-oo ^{t)j then M^p(û){0^ r) is a convex function of e{r) when Ri < r < R2; ifu is subharmonic when \x\ < R then M(p(it)(0,r) is a convex increasing function of e(r) when 0 < r < R with limit ip{u{0)) as r —> 0. In the following results we just discuss subharmonic functions in an annulus and omit the obvious improvement in the case of a ball. T h e o r e m 3.2.20. Ifu is subharmonic log ( /

when Ri < \x\ < R2 then

e 00 it converges to max|^|=i '^{^y)^ which proves the statement. Remark. We can also prove the theorem directly and more easily from the definition of subharmonic functions. If i?i < r i < r < 7*2 < i?2 and u{x) < Mj when \x\ = rj, then u{x) < aE{x) + b when r i < \x\ < r2 provided that ae(rj)-\- b = Mj. Thus max|a.|=^ u{x) < ae{r) -f-b then, which proves the statement. In the same way we also see that im.x^x\=r'î^) is increasing when 0 < r < i? if TX is subharmonic when \x\ < R.

160


Corollary 3.2.23 (Hadamard's three circle t h e o r e m ) . If f is an analytic function in the unit disc in C, then (3.2.14)

(i-|j/(,e'«)|Prf0)'

is a logarithmically convex increasing function of log r G [—00,0), if 0 < p < oo. (We interpret (3.2.14) as max^ |/(ê*^)| ifp — oo.j T h e o r e m 3.2.24. Ifu is a subharmonic function in all of R^ and u{x) < o(log \x\) as X -^ oo, then u is a constant. Proof. By Corollary 3.2.22 we know that M{r) — max|^|=^tA(x) is a convex increasing function of logr with limit ?x(0) as logr —> —oo, and by hypothesis it is o(logr") as logr -^ +oo. Hence it must be constant, so u{x) < u{0) for every x. We can move the origin to any other point so it follows that 16 is a constant. Corollary 3.2.25 (Liouville's t h e o r e m ) . If f is an entire analytic function in C sucii that \f{z)\ = o{\z\) as z ^^ oo, then f is a constant. Proof. By Theorem 3.2.24 we know that log \f{z)\ is a constant. If / ^ 0 we conclude that log f{z) is locally an analytic function with constant real part, hence constant imaginary part, so f{z) is a constant locally, hence globally. Corollary 3.2.23 is obviously false in general when ^ < 0. However, a closely related result is true in an important special case: T h e o r e m 3.2.26. If f is an analytic function in the unit disc in C, /(O) = 0, and / is injective (^^schlicht^^) then (3.2.14) is a logarithmically concave increasing function of logr G (—oo,0) ifp < 0. Proof

By Theorem 3.2.20 we know that f27r

l0g(

/"\p'°8l/(re-)|rfg/27r)

is a convex function of logr when 0 < r < 1, for plog|/(>2:)| is harmonic when 0 < |z| < 1. What must be proved is that it is decreasing, that is, that

I

/o

Jo

is an increasing function of r. Since r—\og\f{re'')\^Re{dlogf{z)/d\ogz)

=

lmid\ogf{re'')/de),


161

we obtain with the notation / = i?e^^ that the derivative is \p\I{r)/r I{r)=

[ ^ Jo

where

R{re'ydip{re^^)/dede.

We must prove that / is positive. The argument (p oi f is uniquely defined for 0 < ^ < 27r when it is chosen for ^ == 0, and we have /^ ^ dcp/dO dO = 27r. Since dip/86 is an analytic function of 6 and not identically zero it has only finitely many zeros for fixed r, corresponding to a finite number of critical values for ip. Let A be an interval C [0, 27r] containing no critical values mod 27r. Then the equation (^(r, 0) = a mod 27r has a fixed number of simple zeros 6i,... ,6^ which are C^ functions oi a E A, corresponding to Ri < R2 < - - < RkThese represent the intersections of the Jordan curve [0,27r] 3 0 i-> f{re^^) with the ray where argi^; = a. The interior Gr — {f{z)\ \z\ < r} contains re*^ when r < Ri but not when i?i < r < i?2, and so on, so it follows that k is odd since the last intersection must lead to the exterior. Since / preserves the orientation the set Gr must lie to the left of the oriented boundary curve, which proves that {—iy~^dip/d6 > 0 at 6j, so I{r) can be written as a sum of integrals / {Ri{af - R2{af -f • • • -I- Rk{af) JA Since Ri{aY - R2{a)P > 0, . . . , Rk-2{ocY - Rk-i{aY is positive, which proves the theorem.

da. > 0, the integrand

Theorem 3.2.26, due to Prawitz [1], has as consequences some important classical results of Koebe and Bieberbach: T h e o r e m 3.2.27. Iff is as in Theorem 3.2.26 then \f"{0)\ the range of f contains {w G C; \w\ < | / ' ( 0 ) | / 4 } , and

r^9i,^

J ^ M _ < l/'WI < J_±l£L

< 4|/'(0)|,

ui/i

Proof. We can assume that /'(O) = 1. By Theorem 3.2.26 w i t h p = - 2 a , a > 0, the integral r27r /•ZTT

J{r)=

/ Jo

\f{re^')\-'^de/27r

is decreasing. To compute J{r) we note that since f{z)/z has no zeros in the simply connected unit disc, we can define g{z) = {f{z)/z)~^ uniquely as an analytic function with ^(0) = 1. Then we have 27r

nZTT

J{r)^

/

id\i2^-2a \g{re"')\'r-"'dd/2'K.

162


li g{z) — XI0^ î-^"^ ^^ ^he Taylor expansion then CQ = 1 and oo

oo

J{r) = ^ r - 2 0 - ) | c , f ,

rJ\T)

0

^Y^r''^^-^^{13

- 2a)|c,f < 0.

0

If a < 1 then all terms with j > 0 are positive and we obtain when r* —> 1 oo

1

S i n c e / ( z ) / z = l + 2 / " ( 0 ) / 2 + O(^2) we have ^(z) = which means that c\ = —a/"(0)/2, so we have

\-azf'{^)j1^0{z^),

o(l-a)|/"(0)/2|2 | / ' ( 0 ) | / 4 . If Id < 1 then

h{z)=^fi{z-0/iCz-l))-f{0 satisfies the hypotheses in the theorem, and since

(z-l we have Hz) = / ' ( O d C P - l){z + z'O + / " ( O d d ' - l ) ' ^ V 2 + and the inequahty \h"{0)\ < 4|/i'(0)| gives

/"(C) /'(O

C77

2|CP 1^ 4|d < i-ld^l-i-KP

Oiz%


163

Prom the proof of Theorem 3.2.26 we know that Re{zf'{z)/f'{z)) rdlog\f'\/dr, so we obtain

=

2r--4 4 + 2r -^ < 9 1 o g | / \/dr < —. 1 — r^ 1 — r^ Integration from r = 0 gives (3.2.15) and completes the proof. Remark. Bieberbach conjectured that \f''^\0)/n\\ < n | / ' ( 0 ) | for every n, which he proved when n = 2. The conjecture was finally proved by De Branges [1]. There and in all results in Theorem 3.2.27 there is equality for the Koebe function f{z) = z/{l + z)^ mapping the unit disc to C slit from I to +00 along R. Already in Exercise 3.2.1 we observed that subharmonicity is invariant under orthogonal transformations. Convexity, on the other hand, is invariant under arbitrary linear transformations, and this is the essential distinction between the two notions: T h e o r e m 3.2.28. Let u be defined in an open set X C R " , and assume that UA{X) = u{Ax) is subharmonic in XA = {x;Ax G X} for every nonsingular linear transformation A. Then u is locally convex. Conversely, if u is a locally convex function in X then UA is locally convex, hence subharmonic. Proof. Let / C X be a closed interval on a line, with distance > ^ > 0 to dX. For the sake of simplicity we assume that / is on the a;i-axis. By hypothesis U£{x) = u{xiêx2^ • • •, sxn) is then subharmonic at distance < 6 from / for every e G (0,1). Hence u{x) — Ue{x)
(iii) we start from a function UQ with the properties listed in Theorem 3.2.30. By adding a constant we can attain that lio < 0 in K. Let M = {x e X \ Y;uo(x) < 1}. This is a compact set. It follows from (ii) that for every x £ M we can find a harmonic function h in X such that h < 0 in K and h{x) > 1. Then we have /i > 1 in a neighborhood of x. By the Borel-Lebesgue lemma we can therefore find finitely many functions / i i , . . . , h^ which are harmonic in X and < 0 in if, such that maxj hj > 1 in M . But then it follows that u{x) — m3.x{uo{x),hi{x)^... ^hN{x)) has the properties required in condition (iii), which completes the proof. In the whole section we have emphasized that l o g | / | is subharmonic if / is an analytic function in an open subset of C. Equivalently, log \u'\ is subharmonic if ix is a harmonic function and \u'\ is the Euclidean norm of u'. We give as an exercise to prove an analogue in R^:


167

Exercise 3.2.7. Let uhe a, harmonic function in an open set X C R"^, n > 3, and set \u'\ = {J2^ \du/dxj\'^)'2. Prove that \U'\'P is subharmonic if P^ (n —2)/(n — 1) but that this is not always true when j9 < (n —2)/(n —1). (Hint: Calculate A(|?z'p + e)^^ when e > 0 and examine the result when the matrix u" is diagonal, with trace 0.) Finally, in analogy with Section 1.7, we shall discuss when the minimum of a family of functions is subharmonic. As a preparation we first prove an analogue of Theorem 1.7.3. T h e o r e m 3.2.32. If X C R^ is an open set, I = [a, 6] is a compact interval on R, and u G (7^(X x / ) , then U{x) = min^^/ u{xj t) is in C^'^{X) if and only if (i) u'^{x^t) does not depend on t when t G J{x) — {t £ I]u{x^t) = U{x)] and x e X is fixed. (ii) For every compact K C X there is a constant AK > 0 such that, with dj = d/dxj, n

(3.2.17)

Y^ \djdtu{x, t)\'^ < AKdû{x, t),

ifx e K,t e J{x) \ dl.

1

Then U E C^ in the open subset YofX Y = {x eX]t^dI For every x eV (3.2.18)

deGned by

and dlu{x,t)^Q

when t e

one can find t e J{x) C I\dl

such that

djdkU{x)

= djdku{x,t)

-

J{x)}.

{djdtu{x,t)){dkdtu{x,t))/dû{x,t),

when j,k = 1 , . . . , n . For almost all x £ X the equation (3.2.18) is valid for every t G J{x) \ dl with dû^x, t) > 0, and (3.2.19)

djdkU{x)

= djdku{x,t),

for every t G J{x) with t e dl or dû{x,t)

j,k =

l,...,n,

= 0.

Proof. That (i) is necessary and sufficient for U to be in C^ follows as in the proof of Lemma 1.7.2. If A is a Lipschitz constant for U' in a convex compact set K C X, then it follows from the proof of Theorem 1.7.3, applied to {s, t) \-^ u{y -f sv^ t) with y £ K and a unit vector v^ that {{da:,v)dtu)^ < AKdlu{x,t), AK = A-{-

sup xeK,tei,\v\=i

lixeK.te

J{x) \ dl,

\{dx,v)'û{x,t)\.

168


Hence (3.2.17) is a necessary condition for U to be in C^'^. Conversely, if (3.2.17) is valid we obtain from Theorem 1.7.3 that s \-^ dsU{y H- sv) for y -}- sv E K has Lipschitz constant A = AK+

sup xeK,tei,\v\=i

\{da:,vyu{x,t)\.

This means that n

-A
0, and that {U"{x)v,v)

=

{u'^,{x,t)v,v),

when t G J{x) and t e dl or u'l^{x^t) = 0. (Verify as an exercise the measurability of the set of points in X where this is not true.) We can determine all partial derivatives using n(n4-1)/2 vectors v chosen appropriately, which proves that (3.2.18) and (3.2.19) hold for the same t when x E X avoids a set of measure 0. As in the proof of Theorem 1.7.1 the statements on U in Y are consequences of the following analogue of Lemma 1.7.2:


169

L e m m a 3.2.33. Let (fi,-. • ,(pN ^ C'^{X)j where X is an open subset ofRP. Then cp = min((pi,... ,(PN) is in C^{X) if and only if for x e X y^j{x) = (pk{oo) = (fix)

= > (p'j{x) = iPkix).

In that case ^ e C'^{X), and for x e X ^'{x) = (f'j{x), {ip"{x)v,v)

when ^j{x) — (p{x),

= mm{{(p'j{x)v,v)](pj{x)

= (p{x)},

v G R^.

For every x e X we have ^"{x) — (Pj{x) for some j with (pj{x) = ^{x). Proof. The first statement follows from Taylor's formula as in Lemma 1.7.2. Hip e C-^{X) we know from Lemma 1.7.2 that for every fixed v G R"^ the function s ^-^ cp(x -{- sv) is in C^ when x -h sv E X. We claim that the second derivative is a continuous function oi x -\- sv. To prove this we may assume that ?; = (1, 0 , . . . , 0). Then diip{x) is a continuous function of xi for fixed x' = {x2^. •., Xn)^ and for x G X d^cpix) = mm{dlipj{x)]ipj{x)

= (p{x)J < N}.

To prove continuity in a neighborhood of XQ G X we may assume that (pj{xo) = (P{XQ) ior j = 1^... ^N and change the labelling so that dl(Pj{xo) = dlcfixo),

j < u\

dfifjixo)

> dl(p{xo),

j > v.

Then we can choose first e > 0 and then d > Q such that iPk{xQ + x) > ipj{xQ + x), dfifkixo-\-x)

> dl(pj{xo-\-x),

if i < ^ < A;, |a;i| = e, \x'\ < 6, if j < z^ < fc, |xi| < e, \x'\ < 6.

When \xi\ < e and \x'\ < 6 it follows from the last statement in Lemma 1.7.2 that (p{xo -i- x) = min-^0,

veR"",

ii(pj{x) = 0,

and for every v there is equality for some such j . Since the zeros of a non-negative quadratic form are contained in a hyperplane if it is not identically zero, we conclude that there is some j with (pj{x) = 0 such that {d,v)'^(Pj{x) = 0 for all v G R"", that is, (p'J{x) = 0 = (p"{x). The proof is complete. We are now ready to prove an extension of Theorem 1.7.1 to subharmonic functions:

170


T h e o r e m 3.2.34. Let X C R'^ be open and let I C H be a compact interval. IfuEC'^{Xx I) then U{x) = min^^/ u{x, t) is subharmonic in X if and only if the following three conditions are fulGUed: (i) If X e X then u'^{x^t) does not depend on t e J{x) I\u{x,t) = U{x)}. (ii) If X G X and t G J{x), then A^^tx > 0 at {x^t). (iii) Ifx^X andte J{x) \ dl then n

(3.2.20)

n

Aû + 2 ^

Then we have U e

= {t G

XjdtdjU + ^

X'^^dû > 0,

AG R'".

C^^\X).

Proof. Assume that U is subharmonic. Then 0< / Ay\=i

{U{x-\-ry)-U{x))duj{y)

ii X e X and r is small. If 5, t G J{x) then U{x -\-ry) - U{x) < r min((it^(x, t),y), {u'^{x,s),y) 0 0 by condition (iii). For almost all a; G X \ y we obtain AU{x) = Axu(x^t) for some t G J{x), hence AU{x) > 0 by condition (ii). This proves that AU > 0 in the sense of distribution theory, so U is subharmonic. Exercise 3.2.8. State and prove analogues of Corollaries 1.7.4 and 1.7.5 for subharmonic functions. 3.3. Harmonic majorants and t h e Riesz representation formula. In Definition 3.2.1 we considered harmonic functions majorizing a subharmonic function in a compact subset of its domain of definition. We shall now study harmonic majorants in the full domain. T h e o r e m 3.3.1. Let U be a family of subharmonic functions in an open connected set X C R"^, not all = —oo. Let V be the set of superharmonic functions v with u < v for every u ^ U, and assume that not all such functions are = +oc. Then VQ = inf V vev is a harmonic function in X, which is therefore called the smallest harmonic majorant of the family U. For the proof we need a lemma: L e m m a 3.3.2. Let v ^ -f-oo be superharmonic in X, and let XQ ^ X have distance > R to dX. Then vi{x) defined as v{x) if x ^ X and \x — Xo\ > R, and by vi{x) = / /

P{{x-xo)/R,y)v{xo-h

Ry)du{y),

if\x-xo\ hf -{- hg_f > hf — e = hf — e,

hg u, then h{x) > hr{x), x G Br, by Theorem 3.1.5, and since hr > u in Br we have hr{x) > hs{x) ii s < r and X e Bg. Hence the harmonic functions hr < h increase with r, so they converge to a harmonic majorant of u which is < h and must therefore be the smallest harmonic majorant. In particular, Mu{0,r) = hr{0) converges to a finite limit. Conversely, if Mu{0^r) is bounded, then hr increases to a harmonic majorant of u, in view of Harnack's inequality. The smallest one is characterized by h{0) = limr^Rhr{0) = limr-^RMu{0,r), which is precisely the condition (3.3.4). We can choose Vk —> R converging so fast that

yZ /

iH'fky) - u{rky)) du{y)
da{y) and \h{ry)\du;{y) -^ \da{y)\ with weak convergence. Let the Lebesgue decomposition be da{y) = X{y)duj{y) -\-das{y), where A E L^{S'^~^) and das is singular with respect to duj. Then (3.3.7)

lim h{ry) = X{y)

a.e. with respect to duj,

r—^R

and (3.3.8)

lini /

\u{ry) - X{y)\ duj{y) = [

\dasiy)\.

Proof. By hypothesis the total mass of the measures u(ry) duj{y) on S"^'^ is bounded as r —> i?, so we can find a weakly convergent subsequence. If the limit is da it follows from (3.3.5) that (3.3.6) holds. Prom (3.3.6) and the remark after Theorem 3.1.8 it follows that da is the weak limit of h{ry) duj{y)^ hence it is the weak limit o{u{ry) du{y) also in view of (3.3.4). The function \h\ = max(/i, —h) is also subharmonic, and since \h{x)\ < [

P{x/R,y)\da{y)l

x G BR,

we conclude that \h{ry)\ dcj{y) -^ dv < \da\ when r -^ R. The opposite inequality follows for general reasons, since for g G C{S'^~^) we have I / gda\ = Jin^l / g{y)h{ry)

du{y)\ < lim^ J \g{y)\\h{ry)\du{y)

== J \g\du,

which means that \da\ < dv, hence \da\ = du. We shall prove (3.3.7) when ?/ is a Lebesgue point for da. Subtracting a constant we may assume that X{y) = 0, and then the definition of a Lebesgue point is that m{6) = [ J\z\ = l,\z-y\ 0.

HARMONIC MAJORANTS AND THE RIESZ REPRESENTATION

177

For fixed 5 > 0 we have \h{ry)\ < 0{R -r)-hC

f

{R - r){R - r + |z - y\y

\da{z)\

J\z\ = l,\z-y\ 1. Recall from Theorem 3.2.20 that the limit does exist; (3.3.20) is an increasing function of r. If / G H^ then /(e*^) = lim^_,i f{re^^) exists almost everywhere by Theorem 3.3.11. When p > 1, as we assume from now on, then it follows from Theorem 3.3.12 and the remark after the theorem that /(re^^) - f{e'^) ^ 0 in L^ as r -^ 1, so we can identify H^ with a subspace of L'^ on the unit circle. We have a factorization / = Bg where 5 is a Blaschke product and ^, given by (3.3.18), has no zeros in the unit disc. If we write da{0) — X{6) dO + dagiO)^ where dag is singular and < 0 by Theorem 3.3.9, we have by Theorem 3.3.8 |/(e^^)| = | ^ ( e ^ ^ ) | ^ e ^ W . We can write g = SF where S = e x p ( i c + (27r)-^ /

^^^^(7,(0)),

J[0,27r) e'^ - Z

F(.)=exp((27r)-M

-Â(0)d^).

Jo

^

— z

Then l^l < 1, and \S{re'^)\ -^ 1 as r -^ 1 for almost every 0 and

||/||HP

=

r'27r

logF(0)=

[\og\F{e'')\de/{27r), Jo

for 15(0)1 < 1 unless 5 is a constant. One calls F the outer factor and BS the inner factor of / ; we have \BS\ < 1 and the boundary values have absolute value 1 almost everywhere. One calls S the singular factor.


185

T h e o r e m 3.3.15. Let f e H^ and p G [l,oo). Then the closed Unear hull in H^ of the functions z^ f{z), where j = 0 , 1 , . . . , is equal to BSH^ where BS is the inner factor of f. Thus it consists of the functions in H^ with inner factor equal to the product of the inner factor of f and some other inner factor. Proof. Write / = BSF. The linear map T in H^ defined by Tg = BSg is isometric, and T{Fq) = fq ii q e Q^ the space of polynomials. If we prove that FQ is dense in H^ it follows that TH^ is the closure of fQ. It suffices to show that 1 E F Q , for then it follows that Q C FQ^ hence that H^ C FQ. First we note that Fq e FQ, if q is analytic in a neighborhood of the unit disc, for the power series of q is then uniformly convergent to q in the closed unit disc. More generally, Fq G FQ if ^ is a bounded analytic function in the unit disc. In fact, qgF G FQ if qgiz) = q{Qz) and ^ < 1, and /•27r

\\Fq, - Fq\\H. = ( /

\Fie''W\qiQe'')

- qie'')\P dOfr ^ 0,

0-^ 1,

by dominated convergence. Hence we conclude that F,(^)êxp((27r)-i / JeG(0,27r)Md) - o o . Hence J^"" \Ft{e'^) - 1|^rf(9-^ 0 as t -^ - o o , which proves that 1 G F Q and completes the proof. The theorem and the preceding proof are essentially due to Arne Beurling. At least when p = 2 there exist simpler functional analytic proofs with a wider scope, but the point here is to give an example of the power of subharmonic function theory. As in Section 3.1 we shall also discuss the analogue of results for functions in the half space H = {x E Ii^;Xn > tence of a smallest harmonic majorant requires no additional when proving representation formulas we shall at first make condition that 0 is a harmonic majorant.

the preceding 0}. The exiscomment, but the restrictive

T h e o r e m 3.3.16. Let u be subharmonic ^ —oo and < 0 in H. Then there is a measure dji >0 in H, a measure da -cx).

The measures dî and da as well as a are uniquely determined; dji = Au, u{x'^Xn)dx' -^ da{x') weakly as a measure when x^ —> 0, and axn = limt_-+.oo'^(^^)/^ in L\^^{H). Conversely, if dfi > 0 and da < 0 satisfy (3.3.22), and a 0, so u — u^ is subharmonic. For any e > 0 we have u — u^ < —u^ < e outside a compact subset of ^ , so ?x — i^^ < 0. Letting x T 1 we conclude that u{x)=

/ GH{x,y)dii{y)-hv{x),

x e H,

JH

where v is harmonic and < 0. Hence Theorem 3.1.8' appHed to — t' gives the representation (3.3.21). If we choose some x with u{x) > —oo it follows that (3.3.22) holds. Now we claim that (3.3.23)

{l-h\x'\'^)~ûi{x\xn)dx'

-^0,

To prove this we note that (1 -h |3:'p)~ 2^ = ^CnPnix', J PH{x',l,0)GH{x,y)dx'

= J

when oj^ -> 0. 1,0) and that

PH{x\l,0)(E{x-y)-E{x-yn)dx'

= E{y\ \xn - yn\ + 1) - E{y\ x^ + 2/n + 1), which gives / PH{x',l,Q)ui{x\Xn)dx

=

\xn-yn\+l)-E{y\xn-\-yn-^l))dfi{y).


187

When 0 < a;„ < ^ we have Ca;„(l + 2/„)(l + |2/|)-", \E{y',\Xn-yn\

+ l)-E(y',Xn+yn

+ '^)\ 1 and form for a compact set K C H r^

[ ui{tx) dx =^ t^-"" [ F{y/t)dfi{y), J

F{z) = f

JK

^ynf^

GH{x,z)dx,

JK

The function F is continuously differentiable in H and vanishes on dH, and we have |F'(2;)| < (7(1 + l^l)"'^ since {z - x)\z - rc]-^ -{z-

rr:*)|z - x*!"^ =

0{\z\-'')

when X e K and \z\ is large. This proves that \F{z)\ < Czn{l + | ^ | ) ~ ^ , so

t'-^J\F{y/t)\dî{y) < cjyn{t

+ \y\)-''df,iy) ^ 0

by dominated convergence. The theorem follows now from the properties of the harmonic part established in Theorem 3.1.8'. Remark. If we replace the hypothesis u 0 in H, a measure da in dH = R"^"^, and a constant a such that (3.3.21) holds, and (3.3.22)'

[ y„(l + | j / | ) - " dKy) < 00, JH

/" (1 + JdH

\y'\)-^\daiy')

< 00.

188


The measures d/i and da as well as a are uniquely determined; d/i = Au, u{x'^Xn)dx' -^ d(j{x') weakly as a measure when Xn -^ 0, and axn = limt-ôo'^{tx)/t in Ll^^{H). Conversely, if dfi > 0 and dfi, da satisfy (3.3.22)', then (3.3.21) defines a subharmonic function in H such that j{l + \x'\^)-'^\u{x',Xn)\dx' J /

-^ [ {l^\x'\'^)2\da{x% JdH

\u{x)\dx/R^'^^

-^\a\

JRK

Xndx,

as x^-^

0,

i ? - ^ oo,

JK

for every compact set K C H. Proof. The equality (3.1.11)" used in the proof of Theorem 3.1.11 can be replaced by the inequality v{x)
/ij(ImC),

when t -^ 00,

where / is the smallest interval containing the support oiu = uiÛ2. Since u = U1U2 it follows that hi = hj^ + /1/2, which proves the corollary. Hardy spaces in the half plane C^_ = {z G C j i m z < 0} are defined as follows: Definition 3.3.20. The Hardy space H^{C^)^ where 0 0 ^JK

' < 00.

^

Since u{x^y) = \f{x -f iy)\'^ is a subharmonic function in C4. and /

, y)/R^ dx dy < CR/B? u{x/i

-> 0,

i? -^ 00,

it follows from Theorem 3.3.17 that \f{x + iy)\^ dx -> da{x) weakly as 2/ —> 0, where da is a positive measure with finite total mass, and that da{x')

i«--)i'4l(^ In particular, this implies that sup / \f{x + iy)\Pdx< y>oJK

J

da{x) < lim / y-ôJn

\f{xîy)\Pdx.

Hence there is equality throughout, and

(3.3.29)'

WJWHP - hm f / |/(x + iy)^ d x ) ^

If we apply this to a half plane defined by Im 2; > a we conclude that the integral on the right is a decreasing function of y. Applying Theorem 3.3.17 to l o g | / | instead we obtain

(3.3.15)'

E T ^ < » .

HARMONIC MAJORANTS AND THE RIESZ REPRESENTATION if 2^1,2:2,... are the zeros of / in C_|_, because A log | / | = Y^ the Blaschke product

(3.3.16)'

"^T^^ZJ

191 • Hence

S(z) = n

is convergent. (The second factor should be omitted \i Zj — i.) In fact, z — Zj i — Zj _ —2ilviiZj{z — i) z-Zj i - Zj {z - Zj)(i - Zj) can be estimated except for finitely many j by a constant times the terms in (3.3.15)' when ;2: is in a compact subset of C_f.. If we write N

f{z) = gN{z)\[-— ^

it follows that ^jv ^ H^ and that

Z

H^JVUHP

Zj I

=

Zj

II/IIHP,

\z-Zj\ ^ I m z + Im^;. ^ < T^ ;—n" "^ \z — Zj\ |Ini>^j — Im2:| I5

for

._ _ if Im^: -> 0.

Letting iV —> 00 we conclude that / = Bg where g G H^ has no zeros and ll^ll^p = WfWnP. From Theorem 3.1.11 it follows that \og\g{x-\-iy)\

da{x) ^

=-

where da is the limit of log \g{x -h iy)\ dx, so / da'^{x) < 00,

/ da~{x)/{l

+ x^)
-^0

as 2 / ^ 0 ,

for / is the Poisson integral of / and the Poisson integral is a contraction in every L^ space with 1 < ;? < oo, converging strongly to the identity as y —^ 0. Thus we have established analogues of all the properties proved above for Hardy spaces in the disc. There are also representation formulas for subharmonic functions in the whole space, which play an important role in the study of entire analytic functions. Since the most important case n = 2 is somewhat exceptional, we restrict ourselves to that case and leave for the reader to prove an analogue for n > 2. T h e o r e m 3.3.21. Let u be a subharmonic function in C such that u{z) < Ci-{-C2\z\^j 2; E C, for some positive constants Ci, C2 and g G (0,1), and assume that u{0) > —00. Then we have with dfi = Au u{z) = u{0) + f {E{z - C) - E{0) dfîO -^^ = u{0) + (27r)-i / log |1 - z/CI dfiiO, Jc

(3.3.30)

(3.3.31) f J|C|-1 inW.

that

EXCEPTIONAL SETS

207

(iii) There is a positive measure dfi ^ 0 with suppc?// C K such that E ^ dfjL > —oo almost everywhere with respect to dfi. (iv) There is a positive measure d/x 7^ 0 with suppd/i C K such that E ^ dfji is continuous. Proof, (i) =^ (ii). Choose R so large that K C BR^ let U be the set of subharmonic functions u (iii) is trivial, and (iv) = ^ (i) is easy. In fact, if d/x and dv are positive measures of compact support, then (3.4.2)

f(E^diy)dfi=

f{E * d/x) diy,

for this is an immediate consequence of Pubini's theorem if we replace E by the continuous function Et = max(t, J5); and the two sides of (3.4.2) are decreasing limits of the corresponding integrals with E replaced by Et when t I —00. If K is polar we can choose di/ so that E ^ du = —oc in K so the left-hand side of (3.4.2) is equal to —00 while the right-hand side is finite if dji is the measure in condition (iv). What remains is the proof that (iii) = ^ (iv), and that will be postponed until two auxiliary results have been proved. T h e o r e m 3.4.6 ( T h e continuity principle). Let dî he a positive measure of compact support K, set u = E ^ d/x, and let x e K. Ifu{y) —> u{x) as K 3 y ^^ x, it follows that this is also true when R'^ 3 y —> x. Proof. Since u is upper semi-continuous it suffices to show that iiu{x) > -cx) then limj^„^^_^^ u{z) > u{x). Let 2: G R^ be close to x, and let z' e K

208


be the closest point, thus \z' — z\ < \y — z\ when y ^ K. For every y ^ K we have \z' -y\ 0 it follows that uiz) > 2^-2 /

Eiz' - y)dfi{y) - log 2 /

J\x-y\ 2^-2 /

E{x - y) d^{y) - log 2 / + /

dfi{y) E{x-y)dfi{y).

J\x-y\>6

The right-hand side converges to E^d^{x) = u{x) when ^ —> 0. (Note that dji has no mass at x since E * dii{x) > —oo.) The proof is complete. T h e o r e m 3.4.7. Let dfi be a positive measure of compact support such that E * dijL{x) > —oo almost everywhere with respect to dfi. Then there is a sequence of positive measures d/ij with disjoint compact supports Kj such that d/i = ^ ^ dfij and E * d^j is continuous for every j . Proof. Let A be a bounded open set such that ZA is a null set for dfi. By hypothesis the upper semi-continuous, hence dfi measurable function E * dfjL{x) on K is finite almost everywhere with respect to cJ/x. By Lusin's theorem one can then for every s > 0 find a compact subset Ki such that the restriction of £* * dii{x) to Ki is continuous and / ^ dfi < e if Ai = A\Ki. Let d/jii be the product of cf/x and the characteristic function of Ki. Thenrf/x— djii is also a positive measure for which CAI is a null set. Since E ^ d/i = E ^ dfii + £* * {d/i — cf/xi) and the terms are upper semi-continuous, they are both continuous where £" * d/x is continuous, so E * d/ii is continuous on Ki, hence continuous in R"^ by Theorem 3.4.6. Discussing d/x - d/xi in the same say with e replaced by e/2 we obtain a sequence of measures dfij with disjoint compact supports C A, for which E^dfXj is continuous and d/x — Y^l dfij is a positive measure with total mass < e/2^~^. This proves the theorem. End of proof of Theorem 3.4-5. If condition (iii) is fulfilled it follows from Theorem 3.4.7 that there is another positive measure with support in K which has a continuous potential. Hence (iv) is valid, which completes the proof.

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209

C o r o l l a r y 3.4.8. Let Uj ^ —oo be a sequence of subharmonic functions in an open connected set X C R^ converging in V to the subharmonic function u. Then every closed subset of X where (3.4.3)

iim Uj{x) < u{x)

is polar. Proof. If it is not polar, then it contains a compact subset K which is not polar, by Theorem 3.4.2. Hence there is a positive measure dfi with support in K having a continuous potential (Theorem 3.4.5), which contradicts Theorem 3.2.13 which states that (3.4.3) is only true in a null set with respect to dfi. The proof is complete. Corollary 3.4.8 can immediately be strengthened to stating that every countable union of closed sets where (3.4.3) holds is polar. However, much more work is required to prove the full result that the whole set defined by (3.4.3) is always polar (Theorem 3.4.14). The proof begins with a more careful look at the proof that (i) => (ii) in Theorem 3.4.5. Let if be a compact subset of BR for some fixed i?, and recall that we defined there a subharmonic function UK in BR as the upper semi-continuous regularization of the supremum UK of the family UK of subharmonic functions < 0 in BR which are < - 1 in K. Also recall that UK = UK is harmonic in BR \ K, and that C ( | x p - i?^) < UK{X) < 0 for some (7, so the boundary measure of UK is 0, and UK{X)=

/

GR{x,y)dfiK{y),

xeBR,

JK

where dfiK = ^^K has support in K. Thus UK is a C^ function in a neighborhood of BBR^ and by Green's formula the total mass of d^K is given by

MR{K)

= [ dfiK = {ÛK, l)= f duK/duy dS{y). J JdBR

The function u^: is lower semi-continuous. In fact, if n G UK and u{x) > C d x p — jR^), we can continue TX as a subharmonic function in R*^ equal to C(|a;p — i?^) when \x\ > R. li u^ cp^: is a standard regularization, as in the proof of Theorem 3.2.11, then u^ (pe - 8 e UK if ^ > 0 and 0 < e < e{6). This is a continuous function > u — 6 which proves the claim. Let /C denote the set of compact subsets of BR. li Ki^K2 G /C and Ki C ^ 2 , then UKI > î^2 ^^ -^R^ ^^^ since both vanish on OBR, it follows that duKi/driy < duK2/dny, so (3.4.4)

MR{KI)

0 then u{x) 4- s{\x\'^ — B?) < —1 in a neighborhood of K , so ^Kj{^^ > '^(^) + ^(I^P ~ ^) for large j . Hence Urn iZ^. [x) > u{x) + 5(|x|^ - i?^), and if we first let e ^^ 0 and then take the supremum with respect to tt, it follows that UK ^ limuKj > UK- This proves that UKJ —^ UK in V^ which implies that MR{KJ) -> MR{K). The third important property of MR is the strong subadditivity (3.4.6) MR{KiUK2) + MR{KinK2) < MR{KI)^MR{K2), if K i , i f 2 G /C. The first step in the proof is to show that (3.4.7)

UK,UK2(^)

+ UK^nK2(^) > îiTi (^) + '^K2{x),

X G BR.

This is clear if a; G K i for UKy^{x) = UKIUK2{^) — ~ 1 then whereas '^KinK2 ^ '^K2- In the same way we see that (3.4.7) is true in 7^2, so it remains to prove (3.4.7) in BR \ {Ki U K2), where all the terms are harmonic and vanish on OBR. If X G d{KiUK2) then the lower semi-continuity of UK gives lim

{uKiUK2{y) + UK^nK2{y)) > '^KiUK2{x) + UK^nK2{x)

BR\(KiUK2)3y-*x

> UKiix) + UK2ix) > ui{x)-\-U2{x),

ifuj

EUKJ,

j = 1,2.

Hence the maximum principle for the subharmonic function ui + U2 gives ui{x)+U2{x)

< UKûK2ix) -^ UK^nK2{x),

^ ^

BR\{KIUK2),

which completes the proof of (3.4.7). Since the upper semi-continuous regularizations are limits of averages over balls, we obtain (3.4.7) with the terms replaced by their regularizations, and this implies (3.4.6). The strong subadditivity (3.4.6) has an equivalent more useful form: (3.4.6)' MR{BI

U B2) + MR{AI)

+ MR{A2)

< MR{BI)

+ MR{B2)

4- MR{AI

U A2),

if ^1,^2,51,^2 elC, Ai cSi, A2 CB2.

EXCEPTIONAL SETS

211

To prove that (3.4.6)' follows from (3.4.6) we apply (3.4.6) first to Ki = Bi and K2 = AiU B2, and then to Ki = AiU A2 and K2 = B2, which gives MR{BI

U B2) + MR{BI

n (^1 u B2))

< MR{BI)

MR{AI

U B2)

n (^1 u A2))

< MR{AI

+ MR{B2

+ MR{AI U A2)

B2),

U

MR{B2).

+

Addition gives MR{BI)

>

+ MR{B2)^MR{AIUA2)

-f MRiB2 n (Ai U ^2)) >

MR{BiUB2)-^MR{Bin{AiUB2))

MR{BI

U

52) +

On the other hand, if we apply (3.4.6)' to Ai = KinK2, and B2 = K2^ we obtain MR{Kr^K2)

+ MR{Kir}K2)

+ MR{K2)
MR{K) by (3.4.4) and (3.4.5). From (3.4.6) it follows that M*(Ai U A2) + M*{Ai n A2) < M*(Ai) + M*(A2),

if ^1,^2 G M.

If Aj are open this follows at once if (3.4.6) is applied to sequences of compact sets increasing to Aj, and in view of (3.4.9) the general case follows. The passage from (3.4.6) to (3.4.6)" was purely formal so (3.4.6)" extends to Aj, Bj G X if MR is replaced by M*. Theorem 3.4.9. The outer capacity defined by (3.4.8), (3.4.9) is equal to MR on /C, it is increasing, and if Aj G M, j = 1,2,... and V)Âj G M, then

(3.4.10)

M*{[JAi) = 1

\\mM\[JAi). ~^^

1

We have 00

00

(3.4.11)

M*((jA,) 0 and choose for every j an open ^The converse will be proved later.

EXCEPTIONAL SETS

213

set Bj D Aj with M*(Bj) < M*(^^) + e/2^ and U^Bj follows from the strong subadditivity (3.4.6)" that

M%\jB^)<M*{\jAj) 1

+

e M.

Then it

ê/2^,

1

1

SO it follows from the result already proved for open sets that J

M * ( U 5 ^ ) < lim M * ( | J y l , ) + £, J—*00

thus MîjAj)

^^

< lim M * ( | J A , ) ,

^^

J—>CXD

which completes the proof of (3.4.10). Since M*{Ai U A2) < M * ( ^ i ) + M*{A2) we conclude at once that (3.4.11) holds. Assume that A E M and that M * ( ^ ) = 0. We can then find decreasing open set Oi D O2 D • • • containing A with Oi C BR and M*{Oj) < 2~K The set Oj is the union of a sequence Kjk of compact sets with MR{Kjk) < 2~^, each contained in the interior of the following one. Hence UKjk I '^j sts k -^ 00^ where Vj is subharmonic and equal to —1 in Oj D A^ and the total mass of dîj = Avj is < 2~^. Hence dfi = ^ ^ dfij has total mass < 1, and

GR{X,y)dfi{y)

/ '

= Y^Vj

= -00

iix

e A,

1

which completes the proof. Definition 3.4.10. A set A £ M. is called capacitable if (3.4.12)

sup

M ( K ) = M*(A),

ICBKCA

and then one defines M{A) =

M*{A).

We know already that every if G /C is capacitable, but we want to prove that all Borel sets are capacitable. The proof, due to Choquet, requires that one proves more: Definition 3.4.11. A subset A of a compact space K is called analytic if there exists a compact space K, a set A C K, and a continuous map / : K -^ K such that fA = A and A is a Kas set, that is, A = f l ^ i U ^ ^ Kj^k where Kj^k C K is a compact set. The term "analytic set" has of course a completely different meaning in the theory of functions of several complex variables, but confusion should not occur easily.

214


Theorem 3.4.12 (Choquet). Every analytic subset A of a compact subset K of BR is capacitable. In particular, every Borel set C K is capacitable. The second statement follows from the first in view of the following: Proposition 3.4.13. If Aj, j = 1,2,... are analytic subsets of a compact space K, then Uf^^j and HfÂj are analytic. Hence every Borel subset of a compact metric space is analytic. We shall prove these results in order. Proof of Theorem 3.4-12. Let X be a compact space and f : K -^ K a. continuous map such that

A = f{A\

A=f][jK,^,, j=ik=i

where Kj^k is a compact subset of K. It is no restriction to assume that Kj^k increases with k. We have x E A ii and only if there is a sequence /c» such that X G Kj^kj for i "= 1? 2, Thus oo

k. 3 = 1

where the union is taken over all such sequences. Now set oo

k»;ki M*(^) by (3.4.10) as /i -> oo. After fixing a small e > 0 we can therefore choose hi so large that M''{f{Bh^)) > M*{A) - ê. In the same way we can then successively introduce bounds h2^hs^- -. for A:2, fca,... such that with £j = e/2^ we have ior p = 1,2,...

M*{fiBh„...,h,))

> M*{A) -J2ej>

M*{A) - e,

1 OO

A:.;A:i M*(A) - e. If rr G f{Kp) for every p, then the decreasing non-empty compact sets f~^{x) n Kp have in common a point x G Kô — ^^Kp with f{x) = x. This proves that nff{Kp) = f{Koo) C A, for Kô C A, hence sup

M{Ko) > M%A)

fCBKoCA

SO A is capacitable and the theorem is proved. Proof of Proposition 3.4-13. Let us first observe that with the notation in Definition 3.4.11 the graph G — {(rz;,/(a;));x G K^ is a compact subset oi K X K, and that {{x,f{x))]x G A} is a KfjS set there, since K 3 x \~^ {x, f{x)) G G is a homeomorphism. It is mapped to A by the projection in K. Thus A is the projection in if of a Kfjs set m K x K. If Aj is an analytic subset of K we now choose compact spaces Kj and K(JS sets Aj C Kj X K with projection Aj in K. Let if = f j ^ i f j , which is also a compact space by Tychonov's theorem, and let ^3 "= ^ j X Yl Ki C K X K.

It is clear that Aj is also a Kfj^ set, with projection A^ in K. The projection in K of the if^^ set P^Âj is equal to flfÂj, for y G HfÂj means that for every j we can find Xj G ifj so that (xj^y) G Aj, that is, (a:i,a;2, • • • ,y) G n f Aj. Hence n f ' A j is analytic. To prove that UfÂj is analytic we let if = if x N , where N is the natural numbers 1,2,... compactified by a point at infinity. Then A = {{x,j,yyj

G N,(x,7/) G A,} C K X if

projects to UJÂj in if, and A is a if^^^ set. To prove this we write oo

oo

i/=i

11=1

216


where A^^ is a compact subset oi K x K. Then oo

A=f]A\

oo

where I ' ' = {{x,j, y);j € N , {x, y) € \J

1^ = 1

^'^j

/i = l OO

which is a countable union of compact sets. Hence U^Âj is analytic. Every open subset of a metrizable compact space is a countable union of compact metric balls so it is a K(j set. Since the Borel sets form the smallest set of subsets containing open and compact sets which is closed under countable union and intersection, we conclude that all Borel sets are analytic. The proof is complete. Remark. In the following applications of Theorem 3.4.12 it would be sufficient to use it for Kcj^ sets, which is slightly more elementary since one does not have to go out to a larger space in the proof and can dispense with Proposition 3.4.13. However, the beauty and generality of the full result should justify a complete presentation. T h e o r e m 3.4.9'. A set A e M is polar if and only ifM*{A)

= 0.

Proof. By Theorem 3.4.9 it only remains to prove that M * ( J 4 ) = 0 if A is polar. Then there exists a subharmonic function u < 0 in BR which is harmonic outside a compact set, such that CXD

Ac

A = {x e BR]U{X)

= -oo}

=

p j {x G BR;U{X)

0,

so UK = 0 almost everywhere in BR^ hence UK = 0. By (3.4.12) this proves that M*{A) = 0, hence M*(A) - 0. T h e o r e m 3.4.14. Let Uj ^ —oo be a sequence of subharmonic functions in an open connected set X C R'^ converging in V to the subharmonic function u. Then the subset of X where (3.4.3) holds is polar. Proof. It suffices to prove that if BR C X then A = {x E

BR/2]

lim Uj{x) < u{x)}

EXCEPTIONAL SETS

217

is polar. The set A is the union over all rationals a < /3 and integers J of â,/?,J "= {oo e Bji/2]u{x)

> P,Uj{x) < a when j > J } .

By Corollary 3.4.8 we have M{K) = 0 for every compact set K C Aa^^s^jIf we prove that A^^j^^j is capacitable it follows that M'^{Aa^/3^j) = 0, hence M*{A) = 0, so ^ is polar (Theorem 3.4.9). Now {x € Bji/2]Uj{x) < a} is the union of a countable number of compact sets, and so is the intersection with the closed set where u{x) > /3. Hence it follows from Theorem 3.4.12 that Aa,/3,j is capacitable, which completes the proof. Corollary 3.4.15. If K e IC then UK = UK in BR except in a polar set, and UK is continuous at every x E BR where UK{X) = UK{X). IfK° is the interior of K then lim /

^y/

dy = 0,

ifuxix)

>

UK{X).

Proof. The first statement follows from Proposition 3.4.4 and Theorem 3.4.14. Since UK — UK is harmonic in BR \ K we may now assume that x E K. Then UK{X) — —1, so UK{X) — —1 if UK = UK at x. Since UK is upper semi-continuous and UK > — 1, it follows that — 1 < lim uxiy) y—*x

< lini uxiv) y—*-x

< lim UK{y) = UK{X) = —1 y~-^^

so there is equality everywhere. For any x G if we have 0=lim/

{uK{x-^y)-UK{x))dy < (-1

- UK{X))

I

dy

lim / '^'^^Jx-{-yeK°,\y\ 0,

218


for any neighborhood YQ 0 in X HY with w{x) -^ 0 as X HY 3 x —^ y. Then there exists a barrier function at y. Proof. To simplify notation we may assume that y = 0, and it suffices to prove that there is a local barrier. We may also assume that Y = BR for some i?, and we set Xr = X H Br ior 0 < r < R. Denote by b the generalized solution by Perron's method of the Dirichlet problem in XR with boundary data OXR 3 X \-^ \X\. Since | • | is convex, it is a permissible subharmonic minor ant while R is a permissible superharmonic major ant, so we have \x\ < b{x) < R, X e XR. If we prove that b{x) ^^ 0 as XR 3 X -^ 0 it will follow that 6 is a local barrier function. Recall that b is the supremum of the subharmonic functions u in XR with lim u{x) < \zl Mz G BXR, XR3X^Z

SO it suffices to estimate such functions u. To do so we note that at a point z G OXR the upper limit of u{x) — Aw{x) is < \z\ for any ^ > 0. To get a small bound we take r > 0 small and examine u{x) — Aw{x) in Xr instead. This introduces a new boundary d'Xr = X H dXr = {2: G XR\ \Z\ — r}. Since w{x) > 0 in XR the upper limit at such points is also < 0 for large A as long as we stay away from dX. To get bounds near dX we take a positive continuous function x ^ 0 on the unit sphere such that xiv) > 1 if I?/1 = 1 and ry G dX^ and we form the Poisson integral h{x)=

/

P{x/r,y)x{y)du{y),

J\y\ = l,ryeX

which is harmonic in Br and continuous at d'Xr with boundary values x{x/r). When A is larger than some number depending on x we now obtain (3.4.14)

liin Xr3x-^Z

{u{x) - Aw{x) - Rh{x)) < r,

z e dXr-

EXCEPTIONAL SETS

219

This is clear ii z E d'Xr^ when x ( ^ / ^ ) < 1 since Aw is large then, and when x(:r/r) > 1 since the boundary values of h are equal to x(a;/r). On the rest oi dXr^ that is, on dXCldXr-, the boundary values oiu are already < \z\ < r, which proves (3.4.14) since w > Q, h > 0. By the maximum principle for subharmonic functions it follows that u(x) < Aw{x) + Rh{x) + r, b{x) < Aw{x) -h Rh{x) + r, ]hR

x e Xr] x E Xr^

="1

b{x) < Rh{0) -\-r = R I

^-3^-*0

hence and

x{y)My)/(^n

+ r.

J\y\ = l,ryeX J\y\=l,r

When X i 0 in {y]ry e X} the integral goes to 0, and since r > 0 can be chosen small we conclude that b{x) -^ 0 as XR 3 X —> 0, which proves the theorem. T h e o r e m 3.4.17. Let X be a bounded open set in R'^, and let y G dY. Then the following conditions are equivalent: (i) Ifu is the generalized solution by the Perron method of the Dirichlet problem in X with boundary value f G C{dX), then u{x) -^ f{y) as X 3 X —^ y for every f G C{dX). (ii) There is a positive harmonic function u in X such that u{x) -^ 0 as X 3 X —^ y. (iii) There is an open neighborhood Y ofy and a positive superharmonic function w in X r\Y such that w{x) —^QasXnY3x-^y. (iv) There is a local barrier function at y. (v) There is a barrier function at y. The point y is called regular for the Dirichlet problem if these conditions are fulfilled. IfyE dX is irregular, that is, not regular, then y is a point of density of X, that is, (3.4.15)

/ J\x—y\ \x — y\, x E X, so u is positive in X and —> 0 at 7/ by (i). It is trivial that (ii) ==^ (iii), and (iii) = > (iv),(v) by Theorem 3.4.16. The equivalence of (iv) and (v) was proved before that theorem, and (v) = ^ (i) by Theorem 3.3.5. Assume that (3.4.15) is not fulfilled with y = 0 and that X C BR; set Kr = {x e CX; \x\ < r } , where 0 < r < R. Then it follows from Corollary 3.4.15 that Ur = UKr is non-positive harmonic in BR \ Kr, and

220


that Ur{x) -^ —1 a.s Br \ Kr 3 X -^ 0. Hence Ur{x) + 1 is a non-negative harmonic function in BR \ K^ which -^ 0 at 0, and it is positive in the components of BR \ Kr which are not relatively compact in BR. Choose a point Xj in every component Xj of X , and note that (1 + f^|x |) will be positive in Xj. Hence ^ 2~-^(l + U\x.\) is positive harmonic in X and -^ 0 at 0, so 0 is regular. From Proposition 3.4.4 and Theorem 3.4.14 we know that 1 + C/|a..| -^ 0 as Xj 3 a; -^ z where z G dXj and \z\ < \xj\, except when 2: is in a polar set. The union of countably many such polar sets is a polar set E^ such that we have a barrier for all z e dXj \ E with \z\ < \z'\ for some z' E dXj^ for we can use countably many Xj for every Xj. The last condition is always satisfied for some choice of the origin, say one of the vertices of a simplex containing X , which completes the proof. Note in particular that y G dX is regular if y satisfies the cone condition that CX contains a truncated cone with vertex at y. This could also have been proved by an explicit barrier construction. We shall finally discuss a somewhat different approach to the proof of Theorem 3.4.14 which follows the lines of the proofs given by Bedford and Taylor [1] for the analogous results in the case of plurisubharmonic functions. First we observe that the definition of UK and UK in the proof of Theorem 3.4.5 is applicable to any set A G J M , that is, any set A with compact closure contained in BR^ UA{X)

= sup u{x)^

XG

BR^

U£UA

where UA is the set of subharmonic functions < 0 in BR which are < — 1 in A. We have C ( | x p — B?) < UA{X) < 0 for some C, and this is also true for the upper semi-continuous regularization UA\ the functions UA and UA are equal and harmonic in BR \ A. P r o p o s i t i o n 3.4.18. A is polar if and only ifuA = 0. Proof. If-u^ = 0 then UA{X) = 0 for some x G BR. Hence we can choose Uj G UA SO that Uj{x) > —2~^. Then u — Y^ Uj is subharmonic and < 0 in BR, U{X) > —1 and u = —00 in A, so A is polar. Conversely, if A is polar we can choose u subharmonic in R"^, so that u < 0 in BR, U = —00 in A but u ^ —00. Then UA > su for every e > 0, so UA = 0 where u ^ —cx), hence UA = 0So far the argument is very close to Theorem 3.4.5. The following proposition sums up the essential analytical facts which we need on functions such as UA-

EXCEPTIONAL SETS

221

P r o p o s i t i o n 3.4.19. If u is a bounded subharmonic function < 0 in BR such that dfi = Au has compact support in BR and u = 0 on OBR, then u' 6 L^{BR) and (3.4.16)

(U',U')L2

=-

f

udii.

JBR

Ifv is another function satisfying the same conditions (3.4.16)'

=- I

(U\V')L2

then

vdti,

JBR

(3.4.17)

(A^, 1) < (ATX, 1),

ifu < V.

Ifuj satisfy the hypotheses made on u, with supp An^ contained in a fixed compact set, and ifuj I u as j -^ oo, then u'j — u' -^ 0 in L'^. Proof. liAu-= f e C§^{BR), then (3.4.16) follows from Green's formula. To extend it we choose ^ E CQ°{BI) SO that (p > 0, J (pdx = 1, and (p is rotationally symmetric. With (feix) = £~'^(p{x/e) we have u^ = (pe "^ u E C^{BR-S), and Ue = urn BRê\BR-.2e-, if ^ is small enough. If we continue the definition of u^ so that u^ = u in BR \ BR-S-, then u^ G C^{BR) and (3.4.16) holds for u^, Au^ = dîîpe. Thus

\ - 1 in K. Then v - Wy is positive in K but negative in a neighborhood of SBR for large v. The critical values oiv — Wy are of measure 0 (Morse-Sard) so we can find ^ > 0 arbitrarily small so that —5 is not a critical value but v — w^ -{- 6 remains negative in a neighborhood of OBR. NOW /

A{v — Wy) dx =

d{v — Wy)/dn dS^

with the integral taken over a smooth surface C BR where the exterior normal n points toward the set where v — Wy + 6 < 0^ so the integral is negative. Hence / Av dx < Av dx < Av dx JK Jwiyb> -I'm Ka,b- Thus UK^^J\b\ > - 1 in Ka,b, so if rf/i = AUK^^^, it follows from (3.4.19) that / d^l\b\ < dfi = d/i, JKa,b JBR JKa,b which implies that J rf/x = 0 and completes the proof that Ka^b has capacity 0.

CHAPTER IV

PLURISUBHARMONIC

FUNCTIONS

S u m m a r y . In addition to the definition of plurisubharmonic functions Section 4.1 presents t h e basic facts concerning Umits and mean value properties. A new feature compared to t h e parallel Section 3.2 is t h a t there is a class of associated pseudo-convex sets which have the same relation t o plurisubharmonic functions as convex sets have to convex functions. In Section 4.2 existence theorems for t h e Cauchy-Riemann equations in several complex variables are proved in such sets for L^ spaces with respect to weights e""^ where if is plurisubharmonic. This gives t h e tools required in Section 4.3 to study t h e Lelong numbers of plurisubharmonic functions, describing the dominating associated mass distributions. T h e s t u d y of Lelong numbers is extended to closed positive currents in Section 4.4. A brief discussion of exceptional sets is given in Section 4.5; we just show t h a t t h e n a t u r a l exceptional sets are defined by local conditions and closed under countable unions. Instead we pass in Section 4.6 t o t h e study of subclasses of pseudo-convex sets: (weakly) linearly convex sets and C convex sets. These are modelled on the definition of convex sets by supporting planes and intersections with lines, respectively. In section 4.7 we discuss analytic functionals and their Laplace transforms. Besides an analogue of t h e PaleyWiener theorem we prove t h a t more refined support properties related to C convex sets can be detected from properties of t h e Laplace transforms, or rather t h e Fantappie transforms obtained from t h e m by a generalization of the classical Borel transform.

4.1. Basic facts. Convex functions were defined in Chapter II as functions which are convex when restricted to lines. We shall now study an analogue of this in a complex vector space when the lines are taken to be complex. Definition 4.1.1. A function u with values in [—oo, -foo) defined in an open set X C C"^ is called plurisubharmonic if a) u is upper semi-continuous; b) For arbitrary z and w in C^ the function T 1-^ u{z + TW) is subharmonic in the open subset of C where it is defined. If both u and ~u are plurisubharmonic, then u is called pluriharmonic.

226

IV. PLURISUBHARMONIC FUNCTIONS

E x a m p l e . If / is analytic in X then log | / | is plurisubharmonic in X while Re / and Im / are pluriharmonic. (Note that in this context it is useful that we have accepted the function = - o o as a subharmonic function.) It is clear from the definition that plurisubharmonicity is a local property. As an immediate consequence of the definition and Theorem 3.2.2 we have: T h e o r e m 4.1.2. If u is plurisubharmonic and 0 < c E R, then cu is plurisubharmonic. If ui,... Ûj^ are plurisubharmonic, then u = m a x ( n i , . . . , 1^^) and ui + • • - + u^, are also plurisubharmonic. Ifu^îÎ, is a family of plurisubharmonic functions and u{x) = sup^^jiA^(x) is upper semi-continuous with values in [—oo,cx)), then u is plurisubharmonic. If ui,U2^.. • is a decreasing sequence of plurisubharmonic functions, then u = limj_,oo '^j is ^^so plurisubharmonic. From Theorem 3.2.3 we obtain: T h e o r e m 4.1.3. Ifu is plurisubharmonic \zj — Wj\ < Rj, j = 1,... ,n, implies w E X, (4.1.1) M,(^;ri,...,r,)-(27r)-"

JJ

in an open set X C C"^ and then

u{zi-hrie'^\...

,Zn^rne'^-)d0^--

• dO^

0je[o,27v)

is an increasing function particular,

L

of Vj for j = l , . . . , n when 0 < Vj < Rj. u{z + rw)^{\w\)

In

dX{w),

w\ s ifw ^ X}, then the convolution u, :{z) = is a plurisubharmonic

Ju{z-eCMOdX{0

function

in C^{Xe),

and u^ [ u as e I 0 if

Proof. It is well known that the regularization u^ is in C°°; it is proved by introducing z — £( as a new integration variable. That u^ decreases to u follows at once from the monotonicity of the means (4.1.1) and the upper semi-continuity of u. For z ^ X^, w E C^ and sufficiently small r > 0 we have 27r

27r

fue{z

+ rwe'^)d9/27r=

fip{()dX{C)

f u{z+ rwe'^ - sQ de/27T > Ue{z),

0

0

which proves the plurisubharmonicity. Recall that for functions in an open set in C one writes d/dz = ^{d/dx

— id/dy),

Thus A = Ad'^ /dzdz. nate in C^.

d/dz = \{d/dx

-f id/dy)]

z = x -\- iy.

We shall use this notation for each complex coordi-

Corollary 4.1.5. A function u e C'^{X), where X is an open set in C^, is plurisubharmonic if and only if n

(4.1.2)

Y^

d\/dzjdzkWjWk

> 0,

zeX,

WEC.

If u is any plurisubharmonic function in Ll^^{X) then d'û/dzjdzk is a measure for j,k — 1 , . . . , n and (4.1.2) is valid in the sense of measure

228


theory. Conversely^ a distribution for which this is true is defined by a unique plurisubharmonic function. Proof. The statement on C^ functions is an immediate consequence of Proposition 3.2.10. If TX is a plurisubharmonic function it follows with the notation in Theorem 4.1.4 that (4.1.2) is valid in X^ with u replaced by Ue. When e: -^ 0 it follows that the sum in (4.1.2) is a positive measure. For a Hermitian symmetric form H{w) — Y^hjkWjWk the corresponding hermitian symmetric sesquilinear form is given by the polarization identity 4:H{w, v) = H{w -^v)— H{w — v) + iH{w -\- iv) — iH{w — iv)] in particular 4/ijfc is obtained when w and v are the unit vectors along the j t h and fcth coordinate axes. If we apply this to (4.1.2) we conclude that d'û/dzjdzk is a measure for all j and k. Conversely, if u is a distribution for which (4.1.2) holds in the sense of measures, then Au = 4:Y^^ d'û/dzjdzj > 0, so it follows from Theorem 3.2.11 that u is defined by a unique subharmonic function, which we also denote by u. If we take cp in Theorem 4.1.4 as a function of \z\ it follows that Ue I u as e ^^ 0. The inequality (4.1.2) is valid for u^ since it holds for u., so u^ is plurisubharmonic, and by Theorem 4.1.2 it follows that u is plurisubharmonic. The form (4.1.2) is usually referred to as the Levi form of u. Corollary 4.1.6. If X C C and X' C C^' are open sets and f : X -^ X' is analytic^ then f'u = uo f is plurisubharmonic in X for every plurisubharmonic function u in X'. Proof. The statement is trivial when IA = — oo, so we may assume that u G L\^^. With the notation in Theorem 4.1.4 we have f*Ue [ /*ii then, so it suffices in view of Theorem 4.1.2 to prove the statement for plurisubharmonic functions u G C^. We have

^

d'û{f{z))/dzjdzkWjWk

= ^

j,k=l

u^y^wlw'^,

where

i^,/i=l

n

'^^../.(C) = d'^yîO/dCudC^,

C = f{z),

w'^ =

for wl is analytic in z so it is annihilated by d/dzkfollows from Corollary 4.1.5.

Y^Wjdfj,{z)/dzj,

Hence the theorem

Remark. If the differential of / is surjective, we could have made the preceding computation directly on a general plurisubharmonic function u^

BASIC FACTS

229

but the preceding proof works also where this is not true so that puUback is not defined in the sense of distribution theory. Note that the invariance under invertible analytic mappings shows that plurisubharmonicity can be defined for functions on any complex analytic manifold by using local analytic coordinates. The following result, parallel to Theorem 3.2.28, shows that plurisubharmonicity is forced on subharmonic functions by the invariance established in Corollary 4.1.6. T h e o r e m 4.1.7. Let u be defined in an open set X C C^, and assume that UA{Z) = u{Az) is subharmonic in XA — {z]Az G X] for every nonsingular complex linear transformation A. Then u is plurisubharmonic. Proof. Let z ^ X have distance > r to dX. Since u{zi -\- wi^Z2 -\ew2-> • . . , Zn+sWn) is subharmouic in w by hypothesis, we have for 0 < e < 1 u{z) < / J\C\=i

u{zi-\-r(i,Z2-^re(2,"-,Zn-^reCri)du{C)/c2n-i'

Since u is upper semi-continuous and locally bounded above, it follows from Fatou's lemma as e —> 0 that u{z)

< /

u{zi-hr(i,Z2,...,Zn)duj{C)/c2n-l-

J\C\=i The right-hand side is an average of the kind allowed in (3.2.2), so it follows from Theorem 3.2.3 that zi \-^ u{zi,Z2,. •. ,Zn) is subharmonic. The subharmonicity of the restriction to other lines follows from the invariance under complex linear maps. Theorem 3.2.12 is immediately applicable with the word "subharmonic" replaced by "plurisubharmonic". So is Theorem 3.2.13, but we can make an improvement in the L^ class. T h e o r e m 4.1.8. Theorem 3.2.13 is valid for plurisubharmonic functions in X C C"^ with Uj —> u in L^^^{X) for any p G [1, oo) and u'- —> u' mLl^{X){0Ts^nyp£[\,2). Proof. When n — I the statement does not go beyond Theorem 3.2.13, but for the proof when n > 1 we need to make the result uniform. Thus assume that TX is a subharmonic function < 0 in a neighborhood of the unit disc in C, and that tx(0) > —CXD. By the Riesz representation formula we have u{z) = h{z)+

I

log-lp£d;.(C),

230


where h is harmonic < 0 and rf/x is a positive measure,

u{0) = h{Q)+

f

loglCUMO-

By Harnack's inequality 0 > h{z) > 3/i(0) when 1^1 < | , which proves that

(X, Since

(/

(log^p^)'rfA(.))'^' 0 gives when y E I / \ui{x)-U2{x)\dx JK

< C\y\ / {\u1\-\-\u2\) dX-iJX

\ui{x-\-iy)-U2{x-{-iy)\dx. JK

Averaging over all y in a ball with radius s < r with center at 0 we obtain / JK

\ui{x)-U2{x)\dx

0,

F'{\z\')Y,\w,\'^F^\\zn\j2-^3^^

iiRi < \z\ 2 we can for every z find w ^ 0 with ^ ZjWj = 0, so it follows that i^'d'^^P) > O5 and since 0 < \^^jWj\'^ < \z\'^\w\'^j the plurisubharmonicity is equivalent to F\s)

> 0,

F'{s) -f sF'\s)

> 0,

when Rj < s < Rj.

Thus F is increasing and {sF'{s)y > 0, and since sF'{s) is the derivative of F with respect to logs, this proves that F{s) is an increasing convex function of logs. We have M^(0,r) = F{r^)^ so Mi^(0,r) is a convex increasing function of logr^ = 21ogr. liu is not a function of \z\ only we observe that Mu(0, r) does not change if we replace u by the plurisubharmonic function z H^ U{UZ) where [/ is a unitary transformation. Hence Mu{0^r) = My(0,r) if v{z) = / u{Uz) dU, where dU is the Haar measure on the unitary group. Thus v is invariant under the unitary group, so v{z) is a function of \z\ only which completes the proof. Remark.

A shorter proof of the convexity is obtained by noting that M,(0,r) -

/ ^|C|=i

duiO

[

\{re''Od0/{27rc2n)

^0

because the integral with respect to ( is equal to M^(0,r) for every 6. On the other hand, by Theorem 3.2.16 the integral with respect to 0 is a convex function of logr for every ( G 5^^"^, which proves that M-j^(0,r) is a convex function of logr. However, this argument cannot show that Mu{0,r) is increasing, for that is not true when n = 1. We can also strengthen Theorem 3.2.17:

BASIC FACTS

233

T h e o r e m 4.1.12. Let u he plurisubharmonic in [z G C^; \z\ > R}, and assume that M^(0, r) < o(log r) as r -^ oo. T i e n u is pluriharmonic if n >2. Every pluriharmonic function in {z G C^; \z\ > R} can be uniquely extended to a pluriharmonic function in C^ when n > 2. Note that this is false when n = 1. Proof. Prom Theorem 4.1.11 it follows that Mu{0,r) is an increasing convex function of logr, so it is a constant M if it is o(logr). Hence it follows from (3.2.13) that Au = 0, so TX is a harmonic function. But if a plurisubharmonic function is harmonic, then it is pluriharmonic, for it is in C^ and the non-negative hermitian symmetric form (4.1.2) must vanish identically if the trace vanishes. This proves the first statement. If -z; is a pluriharmonic function in {z E C^; \z\ > R}, we choose some r > 2R and introduce /•27r

V{z) = (27r)-^ / Jo

v{zi + re'\ ^2, • • •, Zn) d0,

\z\ < r - R,

Since \{zi + re^^, 2^2,. -. ,^n)| ^ ^ — k | > R, the integral is well defined, and since ?; is a real analytic and pluriharmonic function, so is V. If 1(^2, • • • ^Zn)\ > R then V{z) — v{z), by the mean value property of harmonic functions of one variable, and since the annulus {z G C"^;i? < \z\ < r — R} is connected, it follows that V{z) — v(z) when i? < |2;| < r — jR. Hence V gives a pluriharmonic continuation of 7; to C^. The proof is complete, for harmonic functions are real analytic so the extension is unique. Theorem 3.2.18 is obviously valid with no change for plurisubharmonic functions, and we also obtain an analogue of Corollary 3.2.19 and Theorem 3.2.20: T h e o r e m 4.1.13. If u is plurisubharmonic in [z G C^]Ri < \z\ < R2}, ^ ^ 2, and cp is a convex increasing function on R with (p{—oo) — lim^_,_oo ^(^)? then ip{u) is plurisubharmonic and M^(^)(0,r) is a convex increasing function of logr when Ri < r < R2, and so is logMe^(0,r). Hence r H^ sup|^|^^ u{z) is a convex increasing function of logr when Ri < r R} and u{z) < o(log 1^1) as z —^ oo, then u is a constant ifn > 2. Proof. From Theorem 4.1.12 we know that u is harmonic, and it follows from Theorem 4.1.13 that mdiX\z\=r'î^) = M is independent of r. Hence M — u{z) is a positive harmonic function when \z\ > R which vanishes somewhere on every sphere {z E C^^lz] = r} with r > R. Hence it follows from Harnack's theorem that M — u = 0. (Another proof follows from Theorems 4.1.12 and 3.2.24.) C o r o l l a r y 4.1.15. Ifu is plurisubharmonic in {z G C'^;0 < \z\ < R}, n > 2, then UQ = liiaz^Qu{z) < oo, and u becomes plurisubharmonic in {z G C"^; \z\ < R} if we define u{0) = UQ. The measure dji = Au has no atomic part, and (4.1.7)

] ^2n-2^

f

dKz)-2^rfM^{0,r-0),

J\z\2:| is plurisubharmonic for 0 < |>2;| < i? and -^ —00 as 2; -^ 0, which implies plurisubharmonicity for \z\ < R, so the limit ?x as a distribution when e ^> 0 is defined by a plurisubharmonic function. It is equal to u when 0 < \z\ < i?, hence equal to UQ at 0, which proves the first assertion. To prove the second one we use (3.2.13)', noting that C2n = 2n(72n = 27rC2n-2' Thus we have for 0 < r < jR /

dfi{z) = 2nC2n-2r^''~^dMu{0,r-

0)/dr,

J\z\ +00 at the boundary points of X with t G / . The regularity condition in Theorem 4.1.30 can often be dropped. We give an example which will be useful in Section 4.3. Corollary 4.1.31. Let X be an open set in C^"^^ such that X = {{z,w) eXoX

C',r{z) < \w\ < R{z)}

where XQ is an open set in C^ and 0 < r{z) < R{z), z e XQ. Let u be a plurisubharmonic function in X such that {{z^w) G X]u{z,w) < t,z e K} (i X for every compact subset K of XQ and every t G R, and assume that u{z, e^^w) — u{z, w) when {z, w) e X and 6 eH. Then it follows that U{z)=

ini (z,w)ex

u{z,w)

248


is plurisubharmonic

when z G XQ.

Proof. It is clear that U is upper semi-continuous. Let ZQ G XQ and M > U{ZQ). Then t/ < M in a neighborhood VQ of ZQ. Since {(2:,tt;) G X;ix(2:,ii;) < M,z E VQ} (G X , we can find an open neighborhood V of ZQ and r < R such that if = y x {^(; G C; r < |?x;| < i?} C X and U(z) =

inf

7x(2;,7i;),

z GK

r 0.

At a point w G [r^R] where duy/dw = 0 the last term is the Laplacian in w^ hence the second derivative in the radial direction, and 2d'Ûj^/dzjdw is the radial derivative of dujy{z,w)/dzj. Hence (4.1.19) is fulfilled so Ui, is plurisubharmonic in V which implies that U is plurisubharmonic in V. 4.2. E x i s t e n c e t h e o r e m s in L^ spaces w i t h weights. The purpose of this section is to discuss existence theorems for the Cauchy-Riemann equations in several variables. These are important tools in constructions of analytic functions; preliminary constructions using for example partitions of unity have to be supplemented by such results to achieve analyticity. However, we shall start with the elementary one-dimensional case. Let X be an open set in C and consider the Cauchy-Riemann equation

(4.2.1)

^

= f^ ^^^^^ ^/^^ = \{dldx + idldy).

It is an elliptic equation. If / has compact support it can be solved by convolution with the fundamental solution 5 4 , , ,2 I z 1 — —log z = - — ^ = —. az 47r TT 2: M T^z

EXISTENCE THEOREMS IN L^ SPACES WITH WEIGHTS

249

However, even then one obtains better and more robust L^ estimates of a solution by a duality argument. Suppose that for every / G I/^(X, e~^), the L^ space with respect to e~'^dX where dX is the Lebesgue measure in C = R^, there is a solution u G L^(X, e~^). Here we assume for the time being that ^p and I/J are smooth functions. By Banach's theorem it follows then that there is a constant C such that one can choose u so that (4.2.2)

/|^pe-^rfA 0, noting that with r = |z| A^

> a A l o g ( l + \z\^) = ar-^d/dr{rd/dr)\og{l+r'^)

= 4a/(l -hr^)^

Since 4 e - ^ / A ' 0 < e - ^ ( l + |;zp)2-«/a, it follows that (4.2.1) has a solution with (4.2.8) a f \u{z)\ê-'^^'\l + Izl^)"" dX{z) < f \f{z)\ê-^^'\l-^\z\y-''dX{z), Jx Jx provided that cp is smooth and subharmonic, and the right-hand side is finite. We can get a better estimate by taking instead i^{z) - ip{z) + ax{\z\^),

X{t) = log(l + t) - log(2 + log(l -f t)),

where a > 0. Indeed, Ax{\z\^)/4. X\t)

= x ' ( k P ) + k P x " ( k P ) , and

= (1 + t r \ l - (2 + log(l 4- t ) ) - i ) ,

x\t)

+

tx'\t)

_ (1 + log(l + t)){2 + log(l + t)) + t ^ 1 (1 + t)2(2 + log(l + t))2 - (1 + t)(2 + log(l + t))2 • Hence Aa/A^

< (1 + \z\^){2 + log(l +

\z\''))\

and it follows that we can find a solution of (4.2.1) with (4.2.8)'

a [ \u{z)\ê-'^^'\l Jx

+ \z\^)-''{2 + log(l + 1^^))" dX{z)

< [ \f{z)\'e-^^^\l-^\z\')'-^{2^\og{l + \z\')r^'dX{z\ Jx if the right-hand side is finite. In the following theorem we remove the smoothness assumptions on (p.


251

T h e o r e m 4.2.1. Let X be a connected open set in C and cp a subharmonic function ^ —oo in X, and let a > 0. If f e Lf^îX) and the right-hand side of (4.2.8) or (4.2.8)' is Gnite, then the Cauchy-Riemann equation (4.2.1) has a solution u e L^^^{X) such that (4.2.8) or (4.2.8)' holds. Proof. Let Yj (^ X he open sets increasing to X . By a standard regularization we can find a sequence of subharmonic functions cpj G C^{Yj) with ^j > ^ j + i ill ^j 2iiid limj_,oo ^j = (f in X. If the right-hand side of (4.2.8) is finite it follows that there exists a solution Uj of (4.2.1) in Yj such that (4.2.8)" a f \uj{z)\ê-'^^^'\l JYj

+ \z\^)-^dX{z)
(o,i)(^) we shall now prove an estimate of the form (4.2.11) which will replace (4.2.5). The condition (4.2.12) is satisfied if we set (4.2.13)

ipk = ip-\-{k-

3)^,

A: = 1, 2,3,

provided that (4.2.14)

\dxj? 0 that

^j/j -^

I'e-'^ dX 0, we obtain by Cauchy-Schwarz' inequality and (4.2.16) \{9J)H,\'

< Mjc\f\h-'^dX

< Mi{l+e)\\T*f\\%^ + \\Sf\\jjJ,

when / G VT* H VS, and we claim that (4.2.17)

\{9j)Hj'<M{l+e)\\T*f\\],^,

f

£VT*.

This is clear if Sf — 0. If / is orthogonal to the kernel of 5, which contains the range of T, then T * / = 0, and {g, / ) H 2 = 0 for ^ is in the kernel of S since dg = 0. This completes the proof of (4.2.17). The Hahn-Banach theorem or a projection argument applied to the antilinear map shows in view of (4.2.17) that there is some Ua G L'^{X,e~'^^) (4.2.18)

/ luafe"^'

dX < M ( l + e),

and

such that


257

Hence dua = g> Recalling that î — (p m. Xa, we can choose sequences aj —» cx) and EJ —> 0 such that Uaj converges weakly in L^(Xa) for every a to a limit u. We get du = g in X since d is continuous in V'{X)^ and from (4.2.18) we obtain

L

\u\ê-'^d\<M

for every a. Hence we have proved: P r o p o s i t i o n 4 . 2 . 5 . Let X be a pseudo-convex (p G C^(X) be strictly plurisubharmonic, n

(4.2.19)

c{z) ^

open set in C^ and let

n

\wj\^ < ^ 1

dîp(z)/dzjdzkWjWk,

z e X,

weC",

j,fc=i

where c is a positive continuous function in X. If g E LJ^ ^X locally in X and dg — 0, it follows that one can find u G L'^{X^ e~^) with Bu = g and (4.2.20)

/

Jx

\ufe-^dX
C(N')EKI'' if x'{t) > c{t) and tx''{t) + x'(^) = (^x'(^))' > c{t); when n > 2 these conditions are also necessary but when n — 1 only the second one is. (See the proof of Theorem 4.1.11.) If cp E C^ is just plurisubharmonic and we set ij{z) = ^{z) + alog{l-^\z\^) as in (4.2.7), it follows from our old calculations that 2

Y, j,k=i

n

wjwkdîj{z)/dz,dzk

> a ( l + \z\^)-^

Yl 1

l^^l'-

258


If/ e L?Q^x(X, e~^(H-|-p)^~'') and 9 / = 0, we conclude that the equation du = f has a solution in L^{X, e - ^ ( l + | • p ) " ^ ) satisfying (4.2.8). We can argue similarly with iîz) = ifiz) + a(log(l + \z\^) - log(2 + log(l + |z|2))) to get an extension of (4.2.8)'. Repeating the proof of Theorem 4.2.1 we can remove the smoothness assumptions on (p and obtain a complete extension to higher dimensions: T h e o r e m 4.2.6. Let X be a pseudo-convex open set in C^, cp a plurisubharmonic function in X, and a > 0. If f is in L?Q ^^ locally in X and df = 0, then the equation du = f has a solution u G LYQC{X) such that (4.2.8) (resp. (4.2.8)'j holds, provided that the right-hand side is finite. It is of course no restriction to assume in the proof that X is connected. li if = —oo then / = 0 and the statement is trivial so it suffices to consider the case where cp ^ - o o . Then the proof of Theorem 4.2.1 is applicable with no change. We shall now give some applications of Theorem 4.2.6 which will in particular show that every pseudo-convex open set is a domain of holomorphy; the quite elementary converse was proved in Section 4.1. T h e o r e m 4.2.7. Let (p be a plurisubharmonic function in a pseudoconvex open set X C C^, let B = {z]\z — zo\ < r} be a ball with center ZQ G X, and assume that e~^ ^ L^{B d X). Then one can for every a > 0 find a function U which is analytic in X such that U{zo) = 1 and (4.2.21)

/ \U{z)\ê-^^'\l Jx

+ \z-

< (2 -f a-\l

zolY""""

d\{z)

+ r2)2(n + l ) ^ - ^ ^ - ^ ) / e'^^^) dX{z). JBnx

Proof. We may assume that ZQ — 0. Set fl-{\z\/rr+\ I 0,

ii\z\ r.

Then x(0) = 1 and since % is Lipschitz continuous and f =

dx{z)

~{n + l)i\z\/rr

-{ :

E r z,dz^/{2\z\r),

if \z\ < r, if \z\ > r,


259

we have | / | < | ( n + l)\z\^r~^~^. Now we apply Theorem 4.2.6 with cp replaced by z (—> (p{z) + 2nlog \z\, recalling that z H^ log |2;| is plurisubharmonic since z \-^ log \zj\ is plurisubharmonic (cf. Exercise 3.2.6 or the proof of Corollary 4.1.15). It follows that for every a > 0 we can find another solution of the equation du = f in X such that

< 1 ( 1 + r-2)2(n+ 1 ) ^ - 2 ^ - 2 /

e-^dX{z).

JB

If U{z) — x{^) ~ '^(^) it follows that dU = 0, so C/ is analytic in X. In particular u is continuous, so the convergence of the integral in the lefthand side implies that n(0) = 0, since ip is locally bounded above. Hence {/(O) — 1, and (4.2.21) follows from the triangle inequality. Corollary 4.2.8. Let X be a pseudo-convex open set in C^, let K he a compact subset and let K be the compact subset deGned by (4.1.12). For every ZQ E X \ K one can then hnd U analytic in X such that \U\ < 1 in K but \U{z{))\ > 1, which means that (4.1.12) would not change ifu is required to be the logarithm of the absolute value of a function which is analytic in X. The set X is a domain of holomorphy. Proof. By Theorem 4.1.21 we can choose a plurisubharmonic function (f e CîX) such that (^ < 0 in iiT but (p{zo) > 2. Choose a ball B C X with center at ZQ such that (p > 1 in B, and set (pt{z) — m.ay.{ I'mB. If we apply Theorem 4.2.7 with (^ replaced by (^t, we obtain an analytic function Ut with Ut{zQ) = 1 and

/ mz)\'

< Ce -t

JY

Expressing the harmonic function Ut{z) as a mean value when z ^ K we conclude that sup^^ I^P = 0(e~^), which proves the first part of the corollary. It is now completely elementary to prove that X is a domain of holomorphy, and we refer for details to Theorem 2.5.5 in CASV. One of the most important features of Theorem 4.2.7 is that it is applicable also if if is unbounded below. Then we need a sufficient condition for e~'^ to be locally integrable.


260

P r o p o s i t i o n 4.2.9. There is a constant C such that for every plurisubharmonic function ij; in the unit hall in C^ with -0(0) = 0 and ip(z) < 1 when \z\ < 1 we have (4.2.22)

e-^(^) dx{z) < a

/

Proof. First assume that n = 1. By the Riesz representation formula applied to •^ — 1 we have for \z\ < 1 27r(V(^)-l)= /

log|-^|dM(C)+ /

T^—%MO),

where dfi = Aip > 0 and the boundary measure da is < 0. When z = 0 we obtain /

log^rf/x(C)+ /

J|C| —CXD, then cp(z) — (p{zo) < 1 in a neighborhood of ZQ, SO it follows from Proposition 4.2.8 that e""^ is integrable in a neighborhood of ZQ. Hence G is dense, and the statement follows from Theorem 4.2.7. Remark. It is not be possible to take a = 0 in Corollary 4.2.10. In fact, if (y9 = 0 and X = C^ it would then follow from (4.2.23) that U{z) ^ 0 as z —> oo, if we estimate U{z) by the mean value over the ball with center z and radius \z\/2. But then U = 0 although G = C and C G = 0. In the next section we shall use Theorem 4.2.7 to analyze the properties of the measures associated with the plurisubharmonic function (p. For later reference we shall finally give some additions to Theorem 4.2.7. The first allows one to extend analytic functions from (a neighborhood of) a linear subspace and not only from a point, with precise bounds for the extension. T h e o r e m 4.2.11. Let cp be a plurisubharmonic function in a pseudoconvex open set X C C^, let V bea complex linear subspace of codimension u, and let Xv,r "= {z e X-,dv{z)

< r},

where dy{z) = mm\z — C|.

262


For every analytic function u in Xy^r such that / ^ |?/pe"~'^ dX < oo and every a > 0, one can then End an analytic function U in X such that U — u in V and (4.2.21)'

/ \U{z)\ê-^^'\l

Jx

< (2 + a-\l

+

+ r^f{v

rfy(z)2)-^-^

d\{z)

+ l)V-2--2) /

|7x(;^)|2e-^(^)

d\[z).

J Xv,r

Proof. We may assume that V is defined by z' = ( z i , . . . , z^) = 0, thus dy{z) = \z'\. The proof of Theorem 4.2.7 can now be copied, with n replaced by u^ z replaced by z' in the definition of x, and / = u{z)dx{z')The obvious modifications are left for the reader. Remark. If X = C^ and (p is uniformly Lipschitz continuous, it suffices to assume that / is defined in F , for / can be extended to Xy^r as an analytic function independent of z'. In the following more general extension theorem we shall not give precise bounds for the extension in order to simplify the statement, but the proof does give such estimates. T h e o r e m 4.2.12. Let X be a pseudo-convex open subset ofC^, let G i , . . . , GN be analytic functions in X, and let Y be an open subset of X such that N^{^^Y;GI{Z)

= --- = GN{Z)

= 0}

is a closed subset of X. For every analytic function u in Y one can then find an analytic function U in X such that U{z) = u{z) when z ^ N. Proof. We can choose x ^ C^{X) so that % = 1 in an open set containing N and s u p p x C Y, ior N f] K is hy hypothesis compact for every compact subset K of X. Then / = 9(x?/), where x^ is defined as 0 in X \ y , is in C?^^JX) and vanishes in a neighborhood of N. Set N ,IJ{Z)=^

log {Y,\Gj{z)\%

zeX,

1

which is a plurisubharmonic function equal to — oo in Y. More precisely, if ZQ e N then \z - zo\~'^ = 0(e~^(^)) as 2: —> 2:0, so e~^^ is not integrable in any neighborhood of ZQ. On the other hand, | / p e ~ ^ ^ is a continuous function in X , so it follows from Theorem 4.1.21 that there is a strictly plurisubharmonic function (p G C ^ ( X ) such that l/pg-ni/'-^ ^ L^{X)] if we take cpo G C^{X) strictly pseudo-convex so that X 3 z ^-^ ipo{z) E His proper, then (p = x( iV~^ log|/(2:)|, where f is analytic in X and N is a positive integer, are dense in the topology of îoc(^) ^^ ^^^ ^^^ of all plurisubharmonic functions in X. Proof. First we shall prove that C^ strictly plurisubharmonic functions X are dense. Let v be any plurisubharmonic function in X, let K he a compact subset of X and let Vs{z) be the sum of ^|2:p and a regularization of V defined as in Theorem 4.1.4. If F is chosen as in Theorem 4.1.21, then v^ is strictly plurisubharmonic in Y and 7;e ^^ -y in L^{Y) as e -^ 0. If u is the function given by Theorem 4.1.21 we can take x ^ C ^ ( R ) convex and increasing with x ( 0 = 0 for t < 0 and x"(^) > 0 when t > 0 so that X(IA) > ?; on dY. If we define 14 = max{v^^x{'^)) ^^ ^ 3,nd Ve = x('^) outside y , then V^ is plurisubharmonic and l^ ^^ T; in L'^(K) as £ —> 0. By another regularization we can make Ve smooth and strictly plurisubharmonic also at the set where Ve = x('^)- (See a similar argument in the proof of Theorem 4.4.14 below.) This proves the density claimed. Now assume that cp G C^{X) is strictly plurisubharmonic. Choose a sequence 2:1,2:2,... with Zj ^ Zk when j ^ k, which is dense in X . By Taylor's formula ip{z) > ReAj{z)

-\-ej\z — Zj\^,

\z — Zj\ < r^,

where Sj > 0, Vj > 0, and Aj is an analytic quadratic polynomial with Aj{zj) = (p{zj). Choose x ^ C ^ ( C ^ ) with x(z) = 0 when \z\ > 1 and x{z) = 1 when \z\ < | . Fix some positive integer u and choose (5 > 0 so that 6 < Tj and 6 < ^\zj—Zk\ when j . A: < i/ and j 7^ k. Set e = mini<j |e^'^(^^).

On the other hand, we have for large N

/ ' SO estimating /jv by its mean values gives for small ^ > 0 N-'

log \fN{z)\ < sup (^(z + C) + N-'

log(l + I;. - CP) + N'^

log{C'/g^).

\C\ 1 there seems to be no such straightforward way to concentrate N times the Levi form on analytic hyper surf aces, which is what this theorem does.

LELONG NUMBERS OF PLURISUBHARMONIC FUNCTIONS

265

4.3. Lelong numbers of plurisubharmonic functions. Using Corollary 4.2.10 we shall now give some results on the positive measure A(/p when cp is a. plurisubharmonic function ^ — CXD in a connected open set X C C"^. We recall that by Theorem 4.1.11 the spherical mean values (4.3.1)

Mîz,r)=

[ ip{z + J\C\=i

rOdS{0/c2n

are convex increasing functions of logr when r < d{z^CX) Corollary 4.1.15 (4.3.2)

rdM^{z,r-

0)/dr =

^ ^ ^

/

and that by

d/i(C),

if d/x = A(/?/27r. Here C2n is the area of the unit sphere in C^ and C2n-2 is the volume of the unit ball in C^"^. In Definition 4.1.16 we introduced the Lelong number as the limit (4.3.3)

v^{z) = lim rdM^{z, r -

0)/dr,

r—•O

which exists G [0,oo) in view of the convexity and monotonicity. convexity also gives (4.3.3)'

u^{z) = lim

The

M^{z,r)/logr.

r—•O

We want to study the structure of the set where u^p > 0 and begin with an estimate. L e m m a 4.3.1. Ife~^ is integrable in a neighborhood ofz then it follows that I'lpiz) < 2n. Proof. The convexity of Mcp{z,r) as a function of logr implies that p^{z) < {M^{z,r)

- M^{z,ro))/{logr

-logro),

ifr,ro
CXp f - / cp{z + ru) ^ J\u;\ = l

= exp{-M^{z,r))

dS{u;)/c2n) ^

> r'^^^^ê"^,

and since C'^'^ 3 ( ^-^ $\~'^'^ is not integrable at the origin it follows that e~^ is not integrable in any neighborhood of z ii Uip > 2n. This completes the proof.

266


Prom now on we assume that X C C"^ is a connected pseudo-convex open set, and we shall discuss an operation which modifies the Lelong numbers of the plurisubharmonic function (^ in a controlled way. Set (4.3.4)

X = {{z,w) eX

xC- \w\ < d{z, CX)},

and note that X 3 (z^w) \-^ M^p{z^ \w$ is plurisubharmonic since M^{z, \w\) =

ip{z + wOdS{0/c2n^

{z, w) G X,

J\C\=i and {z^ w) H-> (p{z-\-w() is plurisubharmonic for fixed (. It follows from condition (ii) in Theorem 4.1.19 that X is pseudo-convex, for |2:p —log d{z, ZX) and —l/(logi(; — logd{z^CX)) are plurisubharmonic in X , since t — i > —l/t is convex and increasing on (—oo,0). By Theorem 4.1.21 we can therefore choose a strictly pseudo-convex exhaustion function g G C ^ ( X ) , and replacing g{z, w) by /^ ^ g{z, e^^w) dO we may assume that ^ is a function of z and \w\. L e m m a 4.3.2. Let ^ ^ —oo he a plurisubharmonic function in an open, connected and pseudo-convex set X C C^, and let g be a C^ exhaustion function in X C C^^^ which is invariant under rotation in the last coordinate. If M^p is defined by (4.3.1), then (4.3.5) â{z)=

inf {M^{z,\w\) {z,'w)ex,wô

is a plurisubharmonic (4.3.6)

^ g{z,w) - a log \w\),

a > 0,

function of z e X, and

p^^ = max(zy^ - a, 0), a > 0;

i^îz) < a => (fa{z) > - o o .

Proof. It follows from Corollary 4.1.31 that ipl,{z)=

inf

(M^(^,|^|) + ^(2:,^)+^|^|2+l/log(|ti;|/^)-alog|^|)

{z,w)eX,\w\>e

is plurisubharmonic when e > 0 in the open subset of X where it is defined. (The terms e\w\'^ and 1/ log(|t(;|/£) make sure that \w\ is bounded away from oo and e when the quantity to minimize is bounded.) Since (/P^ | (^Q, when ^ I 0 it follows that ipo, is plurisubharmonic. Iiu^{z) < a then g{z^w)-{-Mfp{z^\w\) —a log \w\ is a decreasing function of \w\ when \w\ is small, for g{z,w) — slog \w\ is decreasing for any e > 0. Hence (pa{z) > —oo, which implies i^^pîz) = 0. Now cpa is obviously a concave function of a which implies by (4.3.3)' that u^^ is a convex

LELONG NUMBERS OF PLURISUBHARMONIC FUNCTIONS

267

function of a, and iîpo{z) ^ ^(p{^) since (ô > ^ + min^. Hence I'lpîz) < max(z/^(2;) — a, 0) since this is true when a = 0 and when a > v^{z). To prove the opposite inequaUty we use that by definition ifociz) < M^{z,r)

+ g{z,r) - a l o g r ,

0< r
ji vanish in a neighborhood of K^^ and modulo C^ we may there replace Xv+i by XM+I ^^ ^^^ others, so 0,

zeX,

weC",

j,k=i

and it follows that n

n

X ^ d'^{(p -\- x{s))dzjdzkWjWk j,k = l

> Y^

6jkWjWk,

j,k = l

SO (/? + x{s) is plurisubharmonic, which completes the proof. Prom Theorem 4.3.5 and (4.3.2) it follows that the Lelong number of 0 can be defined by (4.3.2)'

ve{z) = hm^ /

T

% ( ^ + 0/(^2^.-2^^"-'),

for an arbitrary closed (1,1) form 6 = ^Y^^,k=i^3k^^3 ^ ^^k such that (4.3.9) holds. (The reason for the change of normalization by the factor I will be clear in Section 4.4.) The form O is then said to be positive. When 0 = ^ddip and (p is plurisubharmonic, this agrees with (4.3.2). Now the notion of positive form can be extended to {p^p) forms, and there is a corresponding definition of Lelong number such that Theorem 4.3.3 remains valid. This will be discussed in the next section. Remark. If X is not only pseudo-convex but the deRham cohomology group f f 2 ( x , R ) is trivial, then one can choose (p in Theorem 4.3.5 so that d'^^/dzjdzk — Ojk' It suffices to prove this when 6jk G C^. Then we can first choose a real form g-\-g^ where ^ is a C^ form of type (1,0), such that the differential is equal to 0 = i ^ Ojkdzj A dzk. Thus dg = 0 (the part of type (0,2)). Since X is pseudo-convex we can find u G C^{X) such that du = g, that is, du = g^ which means that 0 = d{g -\- g) = d{du -h du) = dd{u — u) = Hence d"^ {21m. u)/dzjdzk

= 0jk-

2iddliau.

CLOSED POSITIVE CURRENTS

271

4.4. Closed positive currents. Let X be an open subset of C^ and 6 G C^ JX) be a smooth differential form of type {p,p) in X. When p = 1 we defined at the end of Section 4.3 the form to be positive if @ = 2^^,k=i^jk{^)^^j ^ ^^k and (4.3.9) holds. This means that if we restrict © to a complex line with direction A = ( A i , . . . , A^) E C^, that is, pull back © to an open subset of C by a map C 3 w ^^ z^ -{- wX^ then the puUback n

y ^ 9jk{z^-\-wX)XjXk^dw

Adw

j,k=i

is a non-negative multiple of the area form dwi A dw2 = ^dw A dw, where w = wi -{- iw2- Recall that a complex manifold has a natural orientation given by the form i^dzi A dzi A • • • A \dzn A dzn^ ii zi,... ,Zn are complex coordinates; if 2; = F{z) where z are some other complex coordinates, this is equal to | d e t F ' ( z ) p times the same form in the z coordinates. We shall say that a differential form of highest degree is positive if it is a non-negative multiple of the preceding orientation form. It is now natural to define positivity of a smooth (p, p) form by pullback to p-dimensional complex subspaces: P r o p o s i t i o n 4.4.1. Let © E C?^ JX) where X is an open subset of C^. Then the following conditions are equivalent: (i) T i e pullback ofQ to any complex analytic p-dimensional ifold is a positive form of degree 2p. (ii) Condition (i) is fulfilled for complex linear subspaces. (iii) The 2n form (4.4.1)

subman-

© A f Ap+i A Ap+i A . • • A fAn A An

is positive for arbitrary smooth (1,0) forms Aj, j = p + 1 , . . . , n. (iv) Condition (iii) is valid for Xj = X^^î Xj^dzk if Xjk G C. Ife e C^î){X), © = I E ^jkdzj A dzk, then they hold if and only is a positive semideBnite Hermitian symmetric matrix.

if{6jk)

Proof. The last statement follows from (ii) and the motivating discussion above. It is trivial that (i) = ^ (ii) and that (iii) (iv). Since all conditions are local and (i), (iii) are invariant under a change of variables, it suffices to prove that (ii) 0, and letting ^ —> 0 we conclude that (4.4.1)" holds. We shall digress a moment to discuss the positive current © — ^991og | / | when / is an analytic function in X C C"^. It is supported by the zero set of / , for in a neighborhood of a point where / 7^ 0 we can write f = e^ where g is analytic, so l o g | / | = R e ^ = (^ + p)/2, which is annihilated by dd. In a neighborhood C/ of a point ZQ where / = 0 but df/dz ^ 0, we can choose new complex coordinates ( ^ 1 , . . . , (n) such that f = (n- Then e = 1:99 log I/I = ^d^ log Knl/dCndCndCn

A dCn = SiCn^dCn

A dCn-

274


If (/? is an (n — 1, n — 1) form with support in U, then

ioi @ A(p is equal to the unit mass at 0 in the C^j-plane times the terms in cp which do not contain any factor d(ri or d^n- By a partition of unity we conclude that

(4.4.3)

if,

[QA^=[

for all (n — l , n — 1) forms with support in the open subset of X where / =?^ 0 or / ' / 0. Thus 9 is in a natural way the integration current on the regular part of the zero set of / . Now the closed positive current 6 is well defined also where / has zeros of higher order, so (4.4.3) shows that the integration current can be extended from a neighborhood of the regular zero set to all of X , as a closed positive current. If y is an analytic rf-dimensional submanifold of a complex n-dimensional manifold then (4.4.4)

f eAip= JxX

f JY JY

for all compactly supported (rf, d) forms in X defines a closed positive (n — d,n — d) current 0 supported by Y. In fact, in a neighborhood of a point in Y we can choose local coordinates so that Y is defined by Zd+i = • • • = Zn = 0, and then (4.4.4) is valid with n

9 = 6{zdî,. ..,Zn)Y[

\dzj A dzj,

which is obviously a closed form. Since (4.4.4) determines 9 uniquely, this proves the statement. As in the codimension one case there is a natural extension of the definition of the integration current if Y is an analytic variety, with singularities, but it would take us too far to develop the prerequisites for a proof. Referring instead to Lelong [1, 2], Skoda [2], El Mir [1] and Demailly [1], we pass to a discussion of the geometrical meaning of the trace of the integration current for an analytic d-dimensional submanifold of X C C^. In a neighborhood of any point we can then label the coordinates so that Y is defined by Zj = hj{z')^ j = d + l , . . . , n , where z' — {zi^... ^Zd)- With the variables Wj — Zj, j = 1 , . . . , d, and


275

Wj = Zj — hj{z')^ d < j < n^ the trace measure is n

te = e A /3^/rf! == 6{wd-î, ...,Wn)Yl

^dwj A dwj A 6>^/d!,

d-fl d

n

1

d+l

for Zj — Wj + hj{w') when j > d, and dwj^dwj can only occur once. This is a function m times the Euchdean surface measure. At a point where h'j == 0, d < j < n, that is, the tangent plane of Y has the direction of the z i , . . . , ^rf plane, we just obtain m = 1. In view of the unitary invariance this must be true everywhere, so we have proved: P r o p o s i t i o n 4.4.4. If 6 is the integration current on a d-dimensional analytic submanifold Y of X C C^, then the trace measure te is the Euclidean surface measure on Y. Remark. Note that the result means that the Euclidean surface measure is the sum of the surface measures lifted by the projections in the c!-dimensional complex coordinate planes. In fact, ii ijj E C^{X) then

/ '0^9 = / 6 A ipP'^/dl = Y1

^2^^h ^ ^%i A • • • A \dzj^ A dzj^,

where j i < • • • < ja- If the projection Y 3 z y-^ {^h-)" - ^^ja) — ^' ^^ diffeomorphic in y fl suppV', then the term in the sum is the integral of ij) as a function of z' G C^. The example of the integration current shows that it is natural to compare the trace measure of a positive closed (n — p, n — p) current to the volume of a ball of dimension 2p, and this will lead to the general definition of the Lelong number. The following key result is due to Lelong (see Lelong [I])T h e o r e m 4.4.5. 1 / 0 is a closed positive {n — p,n—p)

current in {z G

C-.lzl < R}, and (4.4.5)

/(r) - /

te,

0 < r < R,

J\z\ 0 for the statement is trivial when p = 0. Taking out one of the factors ^ we can then write I{r)=

[H{r^

-g)^ddgAeAf3P-^/p\

= f d{H{r^-g)^dgAQApP-^/p\)+

j 6{r'^ - g)^dgAdgAQ

A^^'^/pi

(Recall the commutativity of even forms and that 0 and fi are closed.) The integral of the first term on the right is 0 by Stokes' formula. Now g(3 - ^dgAdg

= Y2 \zk\'^ ^

^dzj A dzj - ^(^^

"^h^Yl

z^dz^) A ( ^

Zjdzj)

2(^kdzj - Zjdzk) A {zkdzj -

Zjdzk).

Taking the ;?th power and multiplying by O we conclude using (4.4.1)' that e A {g/3P - ppP-^ A {dg A

Bg)^'^

is positive. Hence I{r) 0,

which proves the theorem. Remark.

Since ^ddlogg=

- - -^^dg A dg, g 2g^'

the plurisubharmonicity of log^ explains the positivity oi gP — \dg A dg^ which was the main point in the proof.


277

Definition 4.4.6. If 9 is a closed positive {n - p^n - p) current in an open set X C C^^ then the Lelong number Pe{^) is defined when z e X hy (4.4.6) ve{z) = \imj

te/{C2pr'n.

Br{z) = {( e C^',\C - z\ < r},

where C2p is the volume of the unit ball in C^ and the limit G [0, oo) exists by Theorem 4.4.5. Note that the normalization in (4.4.6) has been made so that the Lelong number of the integration current of a ^^-dimensional analytic submanifold Y is equal to 1 at every point in Y. It is of course 0 outside Y. T h e o r e m 4.4.7. The Lelong number introduced in DeGnition 4.4.6 is an upper semi-continuous function. Proof. Let 0 < x G Co°(R) be an even function decreasing on R^_, and set for £ > 0 T,{z) =

Jxi\--z\/e)te.

This convolution is a C^ function of z for small e, in any relatively compact open subset of X. If /z(^) = / g r^) te then

T,{z) = J x{r/e)dh{r)

= - JUr)x'ir/e)dr/e

= - JUer)x'(r)

dr,

so Ts(z)/e'^P is an increasing function of e with limit as e —> 0 -Mz)C2pJr^''x'{r)dr

= ue{z)C2p J r^''-\2p

- l)x{r)

dr,

which proves the upper semi-continuity of zêWe shall extend the special case of Siu's theorem in Theorem 4.3.3 to arbitrary closed positive currents by means of a reduction to that result, which we did for (1,1) forms already at the end of Section 4.3. The following lemma, closely related to Lemma 4.3.4, is the main technical point in the proof. L e m m a 4.4.8. If 9 G ^ ( ^ . ^ ,,_p)(C'"), where 0 1 {p - l)idL-k ABLA

aj A a^''^ = (p - l)i'^dLj^ ABLA

= {p- l)i^d{dL-k ABLA

OBLJ A a^''^

BLJ A a^-^) -\-{p - \)idL-^ A BLJ A a^'^.

Now another application of Stokes' formula proves (4.4.8). (4.4.9) follows directly by differentiation under the integral sign, for iAI^^r^P = lid^/dr^

+ (2n - l)r-^d/dr)r-^P

= p(p + 1 -

n)r-^P-\

where r = \z\. We shall now examine the sign of the integrand in the main term in (4.4.8) which occurs when one calculates the Levi form of U.

280


L e m m a 4.4.9. IfQ is a positive {n — p^n — p) form and A G C " , then n

(4.4.11)

( Y^

n

XjXkLj-âP -piY^

XjXkdL-^ A BLJ A a^"^) A 6 > 0.

Proof. Let z ^ (. The Levi form

is positive unless A is a multiple oi C, — z. (E",fc=i >î^kLfkY~^ is equal to n

The product of (4.4.11) by

n

( X ] ÂkLjâ

- i ^

XjXkdLi

A ^L^ j

A 9.

The Hermitian form corresponding to the first factor as a form in C is n

n

n

2

which is non-negative since {Ljj^) is semidefinite. Hence (4.4.11) is vaUd when 6 is smooth, and therefore in generaL Before the proof of the next theorem we recall from Section 4.3 that - A | z | 2 - 2 ^ = 47r^/(n - 2)!ô in C^. By induction it follows for 1 < fc < n that (4.4.12)

(-A)^|z|2^-2^ = 4^7r^(fc - l)!/(n - A: - 1)! SQ.

We are now ready to prove an analogue of Theorem 4.3.5: T h e o r e m 4.4.10. Let X be an open pseudo-convex set in C^ and Q G T^'(n-p n-p)(-^) ^^ ^ positive and closed form, 0 < p < n — 1. Then there is a plurisubharmonic function tp in X such that (4.4.13)

(-A)^-^(/? + 4 ^ - P 7 r X n - 1 - p)\iQ E C ^ ( X ) .

Recall that the (n, n) current te defines a positive measure which is here identified with the corresponding distribution t©.


281

Proof. Choose compact sets Ki, C X and cutoff functions Xi^ ^s in the proof of Theorem 4.3.5, and define for z G X «

oo

1

-^

with ^, Xv-\-i^ 0 depending on C,. Then

is in C°° in a neighborhood of if^, and we can calculate d"^/dzjdzk applied to the integral using Lemma 4.4.8. Since d{x^,-îQ) = 0 in a neighborhood of K^^ the error terms Rjj^ give a smooth contribution, and we conclude that in a neighborhood of K^

^fk - 2-^pb + 1) JiLfk Aa^- pidL-^ A BLJ A a^-') A x^+i© ^ C^Thus it follows from (4.4.11) that n j,k=i

where C is a continuous function in X. As in the proof of Theorem 4.3.5 we can now find a function x{s) G (7°°(X) such that cp = ip -^ xi^) ^ C'^{X) is plurisubharmonic. Using (4.4.9) we find that in a neighborhood of K^ we have \Aip{z) -p{n-l-

p)p\ y IC - z\-'P-\^^,te

E C^.

Recalling (4.4.12) with A; = n - p - 1, we obtain (4.4.13). To compare the Lelong numbers of ip and of G, we need a lemma: Lemma 4.4.11. Let ip be an integrable function in a neighborhood of 0 in C^ such that {-^y-îp + diieC^ where 0 < p < n — 1 and cJ/i is a measure such that f J\z\ 0. Proof. By a standard regularization we may assume that G is smooth. (Note that the integral on the right is a continuous function of r.) By Stokes' formula the integral on the left is equal to

i f J\z\=r

ax(iog \z\) A {iddxiiog \z\)y-' A e.

284


Write L = log \z\. When L = logr is constant, then dL -\- dL = 0, hence dx{L) = x'{L)dL, ddx{L) - x'{L)ddL + x"{L)dL AdL = x!{L)ddL. The left-hand side of (4.4.16) is therefore independent of the choice of xSince 0 is smooth we can let x(^) tend to e^*, which gives x(log \A) — k P and iddx{\og \z\) — 2^. We have x'(log^) = 2r^, and (4.4.16) follows from the definition of the trace measure. T h e o r e m 4.4.14. The Lelong numbers are invariant under isomorphisms.

analytic

Proof. Let O be a closed positive (n—p^n—p) current in a neighborhood of the origin in C^, and let z ^-^ f{z) be an analytic isomorphism with /(O) = 0. We claim that for sufiiciently small r we have, with x ^s in Proposition 4.4.13, (4.4.17)

TT-^ /

{iddxilog \f{zW

A e / x ' ( l o g r ) ^ > ê(O).

By Proposition 4.4.13 this will prove that the Lelong number of 0 calculated with the coordinates f{z) is at least equal to zê(O), and if we apply this to the inverse of / also, then it follows that equality holds. To prove (4.4.17) we choose C so that \z\ < e^\f{z)\ in a neighborhood of 0. li 0 < K < 1 and c^ < (1 — K>) logr — KC it follows that (4.4.18)

(p{z) = max(log|/(z)|,/^log|2:| + c^)

is a plurisubharmonic function which is equal to log \f{z)\ in a neighborhood of {z; \f{z)\ = r} and is equal to /^log|2;| + c^ when \z\ is small enough. This is true since |/(2:)|/|2;|'^ -^ 0 as z —> 0 and log 1/(2:)I - /clog \z\ — Cr > logr- - K{C -\-logr) - c^ > 0,

when \f{z)\ = r.

We can choose a plurisubharmonic function (f which keeps these properties and is in C^ for z ^ 0. In fact, if /i is a cutoff function equal to 0 in a neighborhood of {0} U {z; \f{z)\ = r} and equal to 1 in a neighborhood of the set in the complement where (p is not C ^ , and if cp^ is a standard regularization, then (p = hip^ + (1 - h)ip = (p -\- h{ipe - ^) will do for small £, for ip is strictly plurisubharmonic in the support of dh. The integral in (4.4.17) does not change if we replace l o g | / ( z ) | by (p{z)^ and since the integrand is non-negative, it follows that for sufficiently small Q the lefthand side is at least equal to TT"^ /

(z99x(/^log|^| +c^))^ A e / x ' ( l o g r ) ^

J\z\vemKx!{K\ogQ^Cr)lx'{\ogr)Y. We can choose % so that x'(logr) = x'{^^^^Z Q~^^r)• Letting AC ^^ 1 we then conclude that (4.4.17) holds. The proof is complete. Finally we prove a useful semi-continuity statement:

EXCEPTIONAL SETS

285

T h e o r e m 4.4.15. IfQj is a sequence of closed positive currents in X of bidegree {n-p,np), and ifQj -> 0 in T^[n-p,n-p)i^)^ ^^^^ (4.4.19)

lim i/Sj < ê-

Proof. li z E X and r is sufficiently small, then iyeM
t e in C , hence as a measure, it follows when j -^ oo that lim iyQ.{z)
n + 1 find a plurisubharmonic function v in C^ such that (4.5.2)

v{z) < log"^ 1^1,

z G C'';

sup v{z) = 0; \z\ 0 such that there is a plurisubharmonic function Ujy in a neighborhood of {z E C^; \z — Zj^\ < Rjy} with Ui, ^ —00 and Ujy{z) = —00 ii z E A and \z — Zjy\ < Rjy. We can use the first part of the proof to find a plurisubharmonic function v^, in C^ equal to —00 when \z — Zjy\ < | i ? j , and Ujy{z) = —00, hence when z e A, such that Vjy{z) < log(2 + \z\), z E C^, and the L^ norm over the unit ball is < 1. Then v = YlTî'/'^^ ^^ equal to —00 in A, and ?; is a plurisubharmonic function ^ —00. The proof is complete. Corollary 4.5.4. IfAj, pluripolar.

j ' = 1 , 2 , . . . , are pluripolar sets, then UfÂj is

Proof. For every j we can find Uj plurisubharmonic in C^ equal to —00 in Aj. If £j > 0 is sufficiently small then Yl^j'^j converges in Ll^^{C'^) to a plurisubharmonic function which is equal to —00 in UA^. However, it remains to prove Theorem 4.5.2. It is not easy to extend plurisubharmonic functions, but an analytic function in the unit polydisc can be well approximated by a partial sum of the power series expansion if it is followed by a sufficiently wide gap. To use this idea we prove two lemmas:

EXCEPTIONAL SETS

287

L e m m a 4.5.5. Ifu is a plurisubharmonic function < 0 in the polydisc X = {z e C^; \z\ < 1} such that u{0) = -1, then one can find a sequence fj of analytic functions in X such that \fj\ < 1 inX, andj~^ log \fj\ -^ u in L J Q ^ ( X ) , hence limj~^ log |/j(z)| < u{z) for every z, with equaUty almost everywhere. The sequence can be chosen so that j ~ ^ log|/j(0)| -^ —1 as j -^ cx). Proof. If u is plurisubharmonic and < 0 in a pseudo-convex neighborhood U of X , we know from Theorem 4.2.13 that there is a sequence of analytic functions Fk in U and integers Nk such that Nj^^ log \Fk\ -^ u in L\^^{U). For large integers j we write j = Nk^ -\- î where k is the largest integer < j with Nk < y/j and 0 < /^ < Nk, hence i' > \/j — I. Then k and jy tend to CXD with j , and if fj = Fj^ it follows that i " M o g | / ^ | = iy{Nkiy-i-fi)~^log\Fk\

-^ u

m L\^^(U)

as j -> oc.

Hence we can find Zj —> 0 such that j ~ ^ log \fj{zj)\ —> — 1 as j ^^ oo. When \z\ < 1 we have j ~ ^ log \fj{z-\-Zj)\ < 0 for large j , so the sequence fj{z-\-Zj) has the required properties. If u is just plurisubharmonic in X , we can for every r < 1 apply this result to z ^-^ u{rz) and obtain a corresponding sequence frj. If we let Vj -^ 1 sufficiently slowly it follows that fr^j has the required properties. To construct a function with large gaps in the power series expansion we need to be able to control the solution of large systems of linear equations: L e m m a 4.5.6. Let (ajk), j^k = l^...,Nj be a square matrix with determinant A ^ 0, let NQ < N, and assume that all NQ X NQ minors formed from the first NQ rows are < M in absolute value. Then one can find w = (wi,..., wjsf) G C ^ with max \wj\ = 1 such that N

(4.5.4) ^

ajk'f^k = Vj,

j = l,...,N,

where yj = OJ < NQ, and

k=i

(4.5.5)

^max^|%r-^''>|A|/M.

Proof. We shall first prove the lemma when N = NQ -{-1. By Cramer's rule there is a unique solution t^ / 0 for every y 7^ 0, and if yj = 0 for j < No then Awj = yN^jj where Aj is a minor formed from the first NQ rows, so |A| max \wj\ = l^jvl max |Aj|, which proves (4.5.5) with equality in that case. To prove the lemma in general, we denote by M^, the maximum of the absolute values of the u x 1/ minors formed from the first u rows in the matrix (ajk), thus Mô < M and MN = |A|. If NQ < ly < N and we

288


apply the special case above to a (i/ -f 1) x (z/ +1) matrix A formed by the first z/ H- 1 rows with determinant of absolute value M^^-i, it follows that we can choose w with zero components except for the column indices of A, so that (4.5.4) is valid even with yj = 0 when j < v^ and max|y^| > |y^+i| > M^^^/M^. (Note that M^^Q

since A -^ 0.) Since

n' ^

= |A|/M;v„ > A/M,

the largest factor in the product is > (|A|/M)^/(^~ô\ which completes the proof. Proof of Theorem 4-5.2. Using Lemma 4.5.5 we first choose a sequence of analytic functions fj in the polydisc X = {z e C^; \z\ < 1} with the properties listed there. Let a < 6 be positive numbers which will be fixed later, and let Aj^ Bj be the smallest integers > aj, bj. To simplify notation we shall write / , A^ B instead of fjÂj^Bj and also omit subscripts on the following constructs depending on j . We want to find a polynomial

Q{z)= Yl ^^^^

M 1, hence (4.5.9)

max log 1^(^)1 > 0, \z\ = l

which will give the second part of (4.5.2). To prove (4.5.3) we note that kg{z) = f{z)Q{z)

— R{z) where

|Q(^)| ^ n r^y 1^(^)1 ^ ^^"1^1^ n r ,q\z\^< 1. The estimate of R follows by looking separately at terms with a^ > B , 1/ = 1 , . . . , n , for \ra\ < B'^. Hence it follows for \z\ < 1 that (4.5.10)

A-'log\g,iz)\

< A-'{-

log % 4-n log B,- - n l o g ( l - \z\) 4- log(2n) + max(log | / , ( z ) | , Bj log \z\)),

where we have now indicated the dependence on j in the subscripts. When j -^ oo we conclude using (4.5.8) and (4.5.9) that limA"^ log|^j(2:)| is almost everywhere equal to a plurisubharmonic function v in C"^ satisfying (4.5.2), and from (4.5.10) and (4.5.7) we obtain (4.5.11)

v{z) < &â-^-^ + a-^ max(^(z), ftlog |z|),

\z\ < 1,

since - j ~ - ^ log |/j(0)| —> 1 and -logfc,- < ( n l o g B , - log\fjm)B^/A],

l E i j - ' l o g | / , ( z ) | < u(z).

290


If |2;| < r < 1 and u{z) < —7 it follows that v{z) < ft'â-^-^ + a - ^ m a x ( - 7 , 6 1 o g r ) = ftâ"^"^ - a - ^ 7 , where we have chosen b = —7/logr. The right-hand side is minimized when (n -f l)b'^/a'^ = 7, which implies a n 4-1. Then we get v{z) < - 7 n / ( a ( n + l)) = - 7 ^ ' ^ ^ n / ( 6 ( n + l)^'^^) = ( l o g r ) 7 ^ n / ( n + l)^"^^. Since n/{n + 1)^"^ - > 1/4 (the sequence is increasing), the proof of (4.5.3) is complete. 4.6. Other convexity conditions. For open sets in C^ there are some convexity conditions which are stronger than pseudo-convexity but weaker than convexity which are sometimes relevant. We shall devote this section to some of them and give applications in Sections 4.7 and 6.4. First we introduce analogues of the definition of convex subsets of R"^ by means of the existence of supporting hyperplanes: Definition 4.6.1. An open set X C C^ is called linearly convex if for every z G C^ \ X there exists an affine complex hyperplane 11 such that z E U C C^ \ X , and X is called weakly linearly convex if this is true for all z e dX. Since every real hyperplane contains a complex hyperplane, it is clear that every convex open set X C C^ is linearly convex. Note that the complement of a complex hyperplane is connected. This makes linear convexity a much weaker condition than conventional convexity. P r o p o s i t i o n 4.6.2. If X is an open set in C^, then the union F of all affine complex hyperplanes 11 C ZX is a closed set, and X = CF is linearly convex. It is the smallest linearly convex open set containing X. The components of X are weakly linearly convex, and if X is weakly linearly convex, then each component of X is a component of X. Proof. If Xj G F then we can find afline hyperplanes IIj C CX such that Xj G r i j . If Xj -^ X then we can choose a subsequence IIj^ converging to an affine hyperplane 11 C CX, and since x G 11 it follows that F is closed. Since the boundary of X is equal to the boundary of F, the proposition follows, for if X is weakly linearly convex then CX D F D dX so the components of X will be components of X. If y is a complex vector space we shall denote by P{V) the projective space consisting of all complex lines through the origin in V, that is, F \ {0} modulo the multiplicative action of C \ {0}. We identify C^ with an open set in P g = P ( C ^ 0 C) by mapping z G C^ to the class of {z, - 1 ) . A

OTHER CONVEXITY CONDITIONS

291

projective hyperplane in P{V) is the image of a hyperplane in F , so the set of projective hyperplanes in P{V) can be identified with P ( F * ) , where V* is the dual space of V. (See also Section 2.5 for elements of projective geometry.) If X is an open set in P{V), we denote by X* the compact set in P{V*) consisting of hyperplanes not meeting X. When 0 G X C C^ then every projective hyperplane 11 with 11 fl X = 0 has a unique representation of the form H = {2; G C^; (2:, () - 1 — 0}, interpreted as the plane at infinity if C = 0, so we identify X* with {( G C^; (^,C) 7^ 1 V;^ G X}. With this notation (extended to compact sets) we have X — (X*)* and 0 G X* in the preceding proposition. P r o p o s i t i o n 4.6.3. Every weakly linearly convex open set X C C^ is pseudo-convex. Proof. Let if be a compact subset of X , and define K by (4.1.12). We must prove that K is a compact subset of X . Taking for u a real linear function we conclude that K C ch(iC), the convex hull of K. If ZQ G dX we can choose an affine complex linear function L such that ZQ G {z]L{z) = 0} C CX. Since log(l/|L|) is plurisubharmonic in X it follows that |1/L(2:)| < sup^^ IVÎ ^^ z £ K^ so ZQ is not in the closure of iiT, which must therefore be a compact subset of X . Analytic function theory is much more elementary in a (weakly) linearly convex set than in a general pseudo-convex set and was understood much earlier then. In some contexts it is a natural condition. Pseudo-convexity is a local property (Theorem 4.1.24). We shall give an example below which shows that this is not true in general for (weak) linear convexity. However, for sets with a C^ boundary we shall now prove that it is in fact a local property. Note that at a boundary point the tangent plane is then well defined and it contains a unique affine complex hyperplane which is the only possible candidate for the plane II in Definition 4.6.1. We shall call it the complex tangent plane. P r o p o s i t i o n 4.6.4. Let X C C^, n open set with a C^ boundary, and assume convex in the sense that for every z G dX that a; n n^; n X = 0 if 11;^ is the complex X is weakly linearly convex. Moreover, C C " , then LOX is connected and simply transversally atLr]X\LnX.

> 1, be a hounded connected that X is locally weakly linearly there is a neighborhood u such tangent plane of dX at z. Then if L is any affine complex line connected, and L intersects dX

Proof. We shall first prove the transversality statement. Let y be a component of the open subset L fl X of L. If C ^ dX is a boundary point of Y and L is not transversal to dX at (, then L C 11^. By hypothesis

292


this implies that u H L C CX for some neighborhood a; of C, which is a contradiction since Y HOJ ^ ^. Now we shall prove that LH X is connected. Let ZQ,zi be two different points in LH X. Since X is connected we can choose a continuous simple curve [0,1] 9 t i-> z{t) G X such that z{0) = ZQ and z{l) = zi. The intersection of X and the afHne complex line through ZQ and z{t), t G (0,1], is parametrized by the open bounded set (4.6.1)

(jJt = {we

C; zo + w{z{t) - ZQ) G X}.

By the transversality proved above, the boundary of Ut consists of a finite number of C^ curves which vary continuously with t. The set of all t G (0,1] such that 0 and 1 are in the same component of (Jt is therefore open and so is the complementary set of all t such that they are in different components. For small t it is clear that 0 and 1 are in the same component of cjt, so this is also true when t = 1, that is, ZQ and zi are in the same component of

Lnx. We shall now prove that X is weakly linearly convex. Let L be an affine complex line in the tangent plane of X at z G dX; we have to prove that Lnx is empty. Assume that this is not the case. We know then that LnX is an open connected subset of L with C^ boundary, at a positive distance from z by the hypothesis. Let u be the interior normal at z. For sufficiently small £ > 0 the intersection (L 4- eu) 0 X has a component close to LOX, by the transversality of the intersection, and it cannot contain the point z -\- 61^. This contradicts the connectedness of the intersection and proves that X is weakly linearly convex. The transversality proves that LHX has the same topological type for all aflSne complex lines L such that i fl X 7^ 0, for they form a connected set. Hence it suffices to show that there exists one line for which the intersection is simply connected. Choose a ball with minimal radius containing X , and choose complex coordinates Zj — Xj + iyj so that 0 is a point in dX on the boundary of the ball, and the tangent plane is defined by Im^;^ = 0. Thus we may assume that near the origin X is defined by ^/n > ^{x^y')^ X = {xi,...,Xn), y' = (7/i,...,2/n-i), whcrc ip G C^, (f = 0, dcf = 0 at the origin, and cp{x,y') > c(|a:p -f- l^/'P) f^^ some c > 0. Let Ue = {zi] ( z i , 0 , . . . ,0,z^) G X}. This is a subset of the disc where \zi\^ < e/c, it increases with ^, contains a neighborhood of the origin and has C^ boundary varying continuously with e. Hence it must be simply connected, for a compact component of the complement of Ue must be part of a compact component of the complement of Us when 0 < 6 < e so it could only contain the origin which is absurd since 0 G C/^ for every e > 0. This completes the proof.


293

Corollary 4.6.5. Let X be a (locally) weakly linearly convex open subset of C^ with a C^ boundary, and choose a defining function g G CîC) such that g 0 in a neighborhood oi z ^ dX in 11;^, and since g{z) = 0 it follows that d^g is positive semidefinite in n^;. Conversely, if d^^ is positive definite in 11^ then g{() > 0 ii z ^ ( eUz and \C - z\ is sufficiently small, so the statement follows from Proposition 4.6.4. Remarks. 1. Note that the convexity conditions in Corollary 4.6.5 involve the full second differential of g restricted to H^;, whereas the condition (4.1.17) for pseudo-convexity only involves the hermitian part. They are invariant under arbitrary complex projective transformations T, for TII^; is the complex tangent plane of TX at Tz, and g restricted to II^ vanishes of second order at z. When d'^g is positive definite in Hz one can make X strictly convex at 2: by a projective transformation preserving z and II;^. It suffices to prove this at the origin when g{z) = —2ReZn + Q{z) + o(|zp) where the quadratic form Q is positive definite when Zn = 0. With the projective map ^{^{z) — z/{l — az^) where a is a constant we obtain g{i^{z)) = - 2 Re ^n + Q{z) - 2 R e ( a 4 ) +

o{\z^).

Since Q{z) is positive definite when 2:^ = 0 it follows that Q{z) — 1Re{az'^ is positive definite when Re 2;^ = 0 if a is a sufficiently large positive number, which proves the statement. (This should be compared with Proposition 7.1.2, which shows that X can be made strictly convex by an analytic change of coordinates if and only if dX is strictly pseudo-convex.) 2. In Corollary 4.6.5 the hypothesis dX G C^ can be reduced to assuming that dX G C^ and that the projective conormal set defined in Theorem 4.1.27' is in (7^. In fact, this suffices to make d^g defined in the complex tangent space of dX, up to a positive factor, and the extension of Corollary 4.6.5 to this situation can be done as in the proof of Theorem 4.1.27'. 3. From Corollary 4.6.5 it follows that there are weakly linearly convex sets which are not convex. For let X C C^ be a bounded convex set with C^ boundary which is strictly convex except at one point where the Hessian degenerates in one direction which is not in the complex tangent plane. Then all perturbations of X which are sufficiently close to X in the C^ sense remain weakly linearly convex but they are not all convex. In Proposition 4.6.4 we established not only weak linear convexity but also another convexity condition which is analogous to the definition of

294


convex sets as sets with a connected intersection with every real Hne. Before stating a general definition based on it we recall that an open set X C C and its complement in the extended complex plane PQ = CU {00} are both connected if and only if X is connected and simply connected. In fact, if 7 is a closed Jordan curve C X and the complement of X is connected, then it must be contained in the exterior of 7 since it contains the point at infinity, and 7 can then be deformed to a point in the interior of 7. On the other hand, if the complement of X is not connected then it is a union of a compact set K and a disjoint closed set F , and the boundary of a suitable neighborhood of K is then a curve which is not homotopic to a point in X. Definition 4.6.6. An open set X C C^ is called C convex if X f l L is a connected and simply connected open subset of L for every affine complex line L. By Proposition 4.6.4, Corollary 4.6.5 and the remark after its proof there are C convex sets which are not convex. P r o p o s i t i o n 4.6.7. Every C convex open set X C C^ is connected and simply connected. If T is a surjective complex affine map C^ -^ C^ then TX is an open C convex set in C^; if S is a complex affine map C^ -^ C^ then S~^X is an open C convex set in C^. Proof. li zi,Z2 ^ X and L is the line through zi and 2:2, then zi and Z2 can even be connected by a curve in L fl X , so X is connected. To prove that X is simply connected we consider a closed curve 7 C X , which may be assumed piecewise linear with vertices 70,71,72, ••• ^Tiv = To- Such a curve is homotopic to any curve passing in order through these points and which between 7^ and jjî always lies in the intersection of X and the complex line Lj spanned by jj and 7j+i, for Lj fl X is simply connected. The homotopy class is not changed if the points jj are replaced by points 7^ sufficiently close, for if we insert a path from 7^ to 7^ and back to jj we get a homotopic path, and the path from 7^ to 7j to jjî to Jjî is homotopic to its orthogonal projection to the line Lj through 7^ and Jjî if 7^ — Jk is sufficiently small for every k. Thus the homotopy class is independent of ( 7 0 , . . . ,7iv_i) G X ^ , for X^ is connected. Since we can choose all the points in a convex subset of X we conclude that it is equal to 0. A line L C C^ is either mapped to a point by S or else it is mapped bijectively to a line in C^; in both cases it is clear that L f] S~^X is connected and simply connected. If T is surjective then TX is open and obviously connected. If [0,1] 3 t H-> 7(t) G TX is a closed curve, then we can find N so large that for j = 0 , . . . , AT — 1 there is a continuous curve [0,1] 3 t \-^ Iji^) ^ ^ with T^j{t) == 7((j + t)lN), t e [0,1]. Thus the closed curve T consisting of jj followed by a curve from 7j(l) to 7j^.i(0) in the intersection of X and the


295

line joining these two points, j = 0 , . . . , A/" — 1, 7iv = To? is mapped by T to a curve homotopic to 7, for T is constant on the inserted curves. If we apply T to a homotopy from F to a point in X we conclude that 7 is homotopic to a point in TX. If L C C^ is an afRne complex line then X fl T~^L is C convex in T~^L^ hence simply connected. Since T : T~^L —> L is surjective it follows that T{Xr[T~'^L) = {TX)nL is connected and simply connected, which completes the proof. T h e o r e m 4.6.8. IfX C C^ is an open C convex set^ then X is linearly convex. HZQ G C'^\X and L is an afRne linear function with L{z) ^ L(zo), z £ X, then the projection of the cone {t{z — zo);0 ^ t £ C^z E X} in PQ~^ is C convex in the affine space which is the complement of the image of{z]z ^ zo,L{z) = L{zo)}. Proof. We shall start with the case n = 2 which is slightly more elementary. Assuming that 0 ^ X and that every complex line through 0 G C^ intersects X, we must prove that there is a contradiction. We shall first prove that there is a real-valued argument function A G C ^ ( X ) , that is, a continuous function such that (4.6.2) A{wz) = A{z)-\-argw^ if z e X^ w e C^ and |if^-l| < Sz^ \diTgw\ < 7r/2, which implies that A{wz) — A{z)-^diVgw for some determination oihrgw if z, wz G X , hence A{wz) = A{z) implies w > 0. For a fixed z E X it follows from the fact that {w e C] wz e X} is simply connected that we can define such a function in XnCz, and A is uniquely determined there if we prescribe the value at z. lip is the natural projection C^ \ {0} -^ P ^ , it follows that there is a neighborhood LJ oip{z) such that we can define such a continuous function A in X r\p~^{uj). By a partition of unity in P ^ we can construct a function A satisfying (4.6.2) in all of X . Let R = {{p{z),A{z))]z G X}, which is an open subset of P Q X R with convex fibers. Hence we can use a partition of unity in P ^ to construct a C^ map PQ 3 z \-^ (p{z) with graph contained in R. For every z e PQ we can find z e X such that p{z) = z and A{z) = ^ ( i ) , and this determines ^ up to a positive factor. The map z -^ z/\z\ is in C^ as is immediately seen by the implicit function theorem if we locally also prescribe \z\. Hence we obtain a global continuous section without zeros of the canonical line bundle over P ^ , which is a contradiction. Thus we can choose a line L through the origin with L fl X = 0. Let M be a parallel line 7^ L, and let TT be the projection of C^ \ L on M from the origin. The proof of the statement about T in Proposition 4.6.7 gives with obvious changes that irX is connected and simply connected, so it is C convex. Now assume that n > 2, and that the theorem has been proved for lower values of n, but let 2:0 == 0 still. Considering a two dimensional section

296


of X we can find a line L through the origin which does not intersect X. Choose coordinates so that L is the z^-axis. Let TT^ be the projection C^ —> C^'^ defined by 7rn(>2î,... ,^n) = {zi^... ^Zn-i)- By Proposition 4.6.7 the projection vr^X is C convex and it does not contain the origin. Hence there is a hyperplane H' C C^~^ which does not intersect TT^X, SO 7r~-^n does not intersect X. This proves that X is Hnearly convex. Prom the two-dimensional case it follows at once that the projection oiX in PQ~^ with a hyperplane not intersecting X put at infinity is C convex, and this completes the proof. Corollary 4.6.9. If X C C^, n > 1, is an open, bounded, connected set with C^ boundary, then X is (locally weakly) linearly convex if and only if X is C convex. Proof. If X is just locally weakly linearly convex, then X is C convex by Proposition 4.6.4, and if X is C convex, then X is even linearly convex by Theorem 4.6.8. Corollary 4.6.10. If X C C^ is an open C convex set with Q e then X* = {C G C^; {z, Q ^l\Jz e X] is a compact connected set.

X,

Proof. The compactness is obvious for X* is closed and bounded. Let C15C2 ^ ^*5 Ci 7^ C2- Assume first that Ci and (2 are linearly dependent, C2 — cî- The range Y of the projection X 3 z i-^ (2;, (1) G C is C convex by Proposition 4.6.7, so the complement of Y in PQ is connected. Hence even the intersection of X* with the line Cî is connected, and it contains both C,i and ^2- Now assume that C,i and ^2 are linearly independent. Then {((^,Ci)-i,(^,C2>-i);^GX}cC2 is C convex by Proposition 4.6.7 and does not intersect {it; G C'^;wi = 0}. Hence it follows from Theorem 4.6.8 that {((^,C2>-i)/((^,Ci)-i);^€X}cC is C convex, so the complement F in PQ is connected. That {wi : W2) ^ F means that W2{{z,(2) ~ 1) 7^ 'î((^5Ci) ~ 1)? ^2: G X , that is, wi 7^ W2 and ("îCi "" '^2C2)/('î - '^2) ^ X*. Since (1 : 0) and (0 : 1) are in F it follows that (î and (2 are not in different components of X*. The hypothesis that dX G C^ is essential in Corollary 4.6.9. In fact, the following examples show that the classes of open, connected and bounded sets which are respectively locally weakly linearly convex, weakly linearly convex, linearly convex or C convex are all different.


297

E x a m p l e 1. Every open set in C is linearly convex. If X i C C"^^ and X2 C C^^ are open linearly convex sets, then it is obvious that X i x X2 C Qî+^2 is linearly convex. However, if X i C C and X2 C C are open, bounded, connected and simply connected sets with C^ boundaries and Xi X X2 is C convex, then Xi and X2 are convex, hence Xi x X2 is convex. In fact, the pullback of Xi x X2 by a linear map C 3 w ^-^ {aiw + bi,a2W 4- ^2) ^ C^ is {a^^{Xi - 61)) n (a^^(X2 — 62)), which must be connected and simply connected. If X i is not convex we can by Theorem 2.1.27 choose coordinates so that 0 G 9X1 and {x + iy; y < ax^.y'^ -]-x'^ < e^} C Xi if a and e are sufficiently small positive numbers. This implies that X^ = [z G Xi,Im2:i > 0} is not connected, for if we could join (—|^, ^) and {\e^6) in X^ for some positive 6 < m i n ( | ^ , \ae^) by a simple curve in X ^ , we could extend it to a Jordan curve in X i with 0 in its interior, so 9X1 would not be connected. Replacing X2 by a suitable translation we may assume that 0 G 8X2 and that I m z > 0 when z G X2. Then ^ 2 / 7 + il is very close to the half plane [z^lmz > 7} when 7 is small, so (X2/7 4- ^7) n X i — (X2/7 + ^7) n X ^ is not connected. This proves that X i must be convex, and similarly we find that X2 must be convex. If either X i or X2 is not convex, then X i x X2 is linearly convex but not C convex. Using the preceding argument we can also show that the hypothesis in Proposition 4.6.4 that dX G C^ is not superfluous. In fact, we can choose X i , X2 C C and an affine complex line L so that X = Xi x X2 is linearly convex and L fl X is not connected. If Y is one of the components then X = X \ y is locally linearly convex; through the new boundary points in Y the line L is the only affine complex line which remains in CX in a neighborhood. Since it intersects the other components of L fl X , which belong to X , it follows that X is not weakly linearly convex. E x a m p l e 2. To construct a bounded connected weakly linearly convex open set C C^ which is not linearly convex, we shall first construct a compact set K in the real hyperplane H = {(2:1,2:2); 11112:2 = 0} such that the complement of the union of the complex lines L with L f l K — 0 consists of K and another disjoint compact set. To do so we first observe that the complement of H is the union of lines parallel to the zi-axis, and that the intersection of H and a line defined by ao + aizi + a2Z2 == 0 is an arbitrary real line in H with direction not parallel to the zi-axis if ai ^ 0, and is any

298


complex line parallel to the î-axis otherwise. We shall therefore look at real lines in R^ with coordinates xi,2/i,X2. Let 5 be a simplex in R^ = H^ that is, the image of a standard simplex 4

4

So = {Xe R ' ; A , > 0,i - 1 , . . . ,4, ^ A , - 1} c F - {A G R ^ J ] A, - 1}

under an affine map (f. Set S^ = {X E: So] maxj A^ > e}, where 0 < ^ < | ; Se is the union of four simplices C So, each containing a neighborhood of a vertex in So- Since ^ < | the edges of So are subsets of Se, so Se is connected. In the face of ô where Xj — 0 the complement of Se is the triangle Sj — {AG R^; A^ = 0 , Q < Xk < s-,k ^ j , ]î Xj = 1}, which is not empty if e > | , as we will assume from now on. The convex hull Tjk of Sj and s^^ j 7^ ^j is an open subset of 5o, ^12

{A E So; 0 < A3 < £, 0 < A4 < 5,0 < Ai + A2 < e).

A line through a point in A G ô which does not intersect Se must exit from 5o through a point in Sj and one in s^ for some j 7!^ A:, so it follows that A G T = Uj^kTjk- li X e R=^ So\{SeUT) then Xj < e, j = 1,..., 4, Y^^ Xj = 1, and Xj -\- Xk > £ ii j "^ k. If we add the inequalities Ai -f A2 > e, Ai + A3 > s, A2 + A3 > s, it follows that Ai -h A2 -h A3 > | £ , hence A4 < 1 — §£ and similarly for all Xj. If 1 — | e < e, that is, e > | , then i? n 5^ = 0 so i? is a compact subset of ô \ Se, and ( | , . • •, | ) G i? since e < | . The inequalities Xj -f A^ > e are equivalent to Aj + Ajt < 1 — e, so it follows that R is the cube defined by ^ < Ai H- A2 < 1 - e, e < Ai + A3 < 1 - e, A2 + A3 < 1 - e. Summing up, when f < ^ < | then K — (pSe C C^ is connected, and the complement of the union of lines not intersecting K consists of K and a parallelepiped Ki — ipR with K r[ Ki = 0. Now take a small open neighborhood X of K in C^, and let F be the closed set which is the union of complex lines disjoint with X. Then X = CF has a bounded weakly linearly convex component Y with Y D K but F fl if 1 = 0, if X is small enough. There is another component Z with Z D Ki and Z fl if = 0, and it is contained in Y, because Y D X so Y D X. Thus Y is not linearly convex. We shall now show that every C convex open set is homeomorphic to a ball. First we give a preliminary reduction:


P r o p o s i t i o n 4.6.11. If X C C^ is linearly convex and ZQ G X, V = {zeC'';zo is a linear subspace of C^ independent intersection of all complex hyperplanes afEne complex hyperplane C CX.

+

299

then

CzCX}

of ZQ, so X -\-V = X. V is the through the origin parallel to an

Proof. li z e V and H is a complex afEne hyperplane C CX, then 11 D {zo + Cz) = ^. Conversely, if 11 fl (2:0 + Cz) = 0 for every such H, it follows that z e V. This proves that V is linear, equal to the intersection of all the planes 11 translated so that they pass through the origin, which is independent of ZQIn particular, Proposition 4.6.11 can be applied when X is C convex. If W is a linear subspace of C^ supplementary to F , then WnX is C convex in W (Proposition 4.6.7), and X = V x {W fl X). There is no affine line contained inWDX. By the following theorem such sets are homeomorphic to a ball, so this is true for every C convex set. T h e o r e m 4.6.12. Let X C C^ be C convex, and assume that no affine complex line is contained in X. Assume that 0 E X , and let B be the unit ball in C^. Then there is a unique homeomorphism cp : B -^ X such that (p{z) e X nCz for every z £ B, and for every z e C^ with \z\ = 1 D 3 w ^ cp{wz) e X nCz, is an analytic bijection with dip{wz)/dw

D = {w eC] \w\ < 1}, G R_j_^ when w = 0.

Proof. Set X, = {we

C;wz E X } ,

ze

C^, \z\ = 1.

Then the condition on (p means that (p{wz) = (pz{w)z where cpz : D -^ Xz is an analytic bijection with (fz{^) = 0 and 0. By the Riemann mapping theorem there is a unique analytic map ipz with these properties, for Xz is connected, simply connected and 7«^ C by hypothesis. We have ê^^zM

= C'îfzie'^w),

0 eRj

w e D,

so (p{wz) = ipîdz{e'"^^w)e'^^z gives a unique definition of (p in JB, independent of 0, and (^ : B —> X is a bijection. It remains to prove that (/? is a homeomorphism. In the proof we may assume that the unit ball is contained in X. Then the inverse of cpz is defined in D with values in D, so the derivative 1/(^^(0) at the origin is bounded by 1, hence | 0, we just have to prove that I/J is injective with the same range as (pz^. That ijj is injective is clear, for if ip{wi) = '^('^2) then ipzj{w{) = ^zj{'^2) for some w{ —> tc'i, by Rouche's theorem, so wl = W2 and Wl — W2' Let y be a connected open set (s Xz^. Then Y C Xz^ for large j , so ip~^{w) is defined iov w ^ Y with values in the unit disc, cpz{^J^w) = w i{w EY. For a subsequence the limit $ of (p~^{w) exists locally uniformly in F , and il){^{w)) = w, ii w e Y. Hence the range of ip contains XZQ • To prove equality assume that WQ E dXz^ and that WQ = '0(ro) for some TQ E D. Choose a linear form L so that woL{zo) = 1 and L{z) = 1 implies z G Cl2. Thus 1/L{zj) ^ Xz^- But for large j we can find TJ G D with TJ -^ TQ SO that ^J{TJ) = \jL{zj)^ which means that 1/L{zj) G Xzj. This contradiction concludes the proof that -^ = (pz^Hence ip is continuous, and since the preceding proof shows that cp is proper, it follows that ip~^ is also continuous. The proof is complete. 4.7. Analytic functionals. If X C C^ is an open set, then an analytic functional in X is a continuous linear form on the space A{X) of analytic functions in X , with the topology of uniform convergence on compact subsets of X. We shall denote the space of analytic functionals in X by A'{X). Thus an element /x G A'{X) is a linear form on A{X) such that for some compact set K C X and some constant C

|M/)l 0, then there exists a unique fi G ^'(C"^) with ft = M, and fi is carried by K. Proof. The necessity of (4.7.5) has already been established. If/i(C) ^ 0 then differentiation of (4.7.4) with respect to C gives for ( = 0 that î{z^) — 0 for every monomial z^. Polynomials are dense in ^(C"^), for the power series expansion of any / G ^ ( C ^ ) converges uniformly on every compact set. Hence it follows that /x(/) = 0 for every / G vA(C^), that is, /x = 0, so /x determines /x. To prove the existence of /x, carried by if, when M is given satisfying (4.7.5), it would suffice to prove that if K^ is the set of points at distance < b from K, then there exists a measure dv with support in K^ having Laplace transform M , that is, M(C) -

/ / e sup^^^^ \z\^ then the Cauchy integral formula

^(^) = ^ /

/(C)(C-^)-'rfC,

feA{C),

is valid for z near K, and gives M(/)

TT^ J\C\=R

Since {( — z) ^ = S ^ l o ^'^^ ^ ^ with uniform convergence for 2: in a neighborhood of the carrier K and |C| == i?, it follows that

1 M(/)

r

2i'« ./\ sup^^^^ \z\, is called the Borel transform of/i, and we have

mfiOdc

M(/)=/ JK\=R

For any M € ^ ( C ) of exponential type, that is, |M(C)| 0

hence |M(^')(0)|

<j\mmCie^^R-^

^j\Cie^{C/jy

< Ci{eCy.

R

This implies that the Borel transform OO

(4.7.8)

BMiO =

Yl'^-'-'M(^\0) 0

is analytic when \(\ > eC. li M = jl and /x is defined by (4.7.1), then MiO=

[

e^^dv{z\

JK

and since the Borel transform of ^ K-> e^^ is OO

0

we conclude that BM{C,)

= JiC - zr'diîz),

for large |^|. If K is any carrier of fi it follows that the Borel transform of //, at first defined in a neighborhood of oo, can be continued analytically

ANALYTIC FUNCTIONALS

305

to the component of oc in P^ \ K. Conversely, if the Borel transform of jl can be continued analytically to a connected neighborhood fl of oo, then // is carried by 9fi, for if x G CQ^{C) is equal to 1 in a neighborhood of CfJ, then

(4.7.9)

Kf) = ^JJdx{0/daiOBf.{Od(AdC,

f e A{C),

where Bp, denotes the analytic continuation of the Borel transform to ft. In fact, if X G CQ^{Q \ {oo}) then the right-hand side is equal to 0, so it suffices to prove (4.7.9) when x(C) = 1 for \(\ < R where R > sup^^;^ \z\. Then we have by Cauchy's integral formula

/(^) = îJJ

dxiO/dUiOiC - z)-'dC A dC, \z\ < R,

which implies (4.7.9) by arguments already given. Since x can be chosen with support in any neighborhood of CO, it follows that // is carried by dft. To get another proof of Theorem 4.7.3 for n = 1 we now assume that we have an entire function M in C satisfying (4.7.5) and define the Borel transform by (4.7.8) for large |C|. With x G CQ^{C) equal to 1 in a sufficiently large disc we define /x by (4.7.9) with Bp, replaced by BM- Then jl = M, for

/i(^)(0) = M(C^) = ^ J J dxiO/dCC'BMiO dCAdC = M«)(0),

smce

which is clear if x is chosen as a function of | ( p . We know already that BM is analytic when \(\ > eC and vanishes at infinity. To continue BM analytically we form for an arbitrary w E C\ {0} /•oo

B{(,w)=

/ Jo

M{tw)e-^'^^wdt,

The integral is absolutely convergent and defines an analytic function of ( m

{CeC;HK{w)

sup^^;^ \z\, it follows that (4.7.10) 00

is carried by

00

Mi^((i - (-,0)-') = E^' E M(^")r/«! = E-?' E ^v(o)r/«! j=0

\a\=j

j=0

\a\=j

is defined and analytic in {( G C^; |C| < l/R}. The germ fi-^ at the origin is independent of the choice of K. The map jl — i > fi-^ consisting in multiplying the terms of degree j in the Taylor expansion of /i by j ! is a bijection of the set of entire functions of exponential type on the germs of analytic functions at the origin, just as in the one-dimensional case. We leave for the reader to repeat the details of the proof. Definition 4.7.4. If /z G ^'(C"^) then the Fantappie transform /x"^ is the germ of analytic function at 0 G C"^ defined by (4.7.10). P r o p o s i t i o n 4.7.5. Ifji G A'iC^) is carried by the compact set K, then the Fantappie transform /x"^ can be continued analytically to the component KQ of the origin in the open set (4.7.11)

K* = { C € C " ; ( z , C ) 7 ^ 1 , V 2 € i f } .

Proof. For any compact neighborhood a; of J^ we can find a measure du with support in OJ representing fi as in (4.7.1). Thus the germ of


307

at the origin is equal to fi-^. For any compact subset K' of KQ we can choose (jj so that {z^ C) 7^ 1 when z G a; and C ^ ^ ' so we have obtained an analytic continuation of ji-^ to a neighborhood of K'^ which proves the statement. Before proceeding we shall discuss some examples. If a G C"^ and 8a is the analytic functional 0 there is another compact set Ki C X and a constant C such that when ao? • • • 5 ^fc ^ K then |/Xao,...,a.(/)| 2ej - Rj'

> 0,

if C E X*.

If we expand the last product in (4.7.14) it follows from (4.7.13) and Lemma 4.7.7 that the function (4.7.14) is the Fantappie transform of an analytic functional carried by a fixed compact subset of X and with a fixed bound, so it follows that / is the Fantappie transform of a functional in A'{C'^) carried by a compact subset of X . It just remains to prove that X is a Runge domain, for that implies that wA'(X) —^ A^C^) is injective, hence that the Fantappie transform is injective. For the proof we first observe that since X* is connected by Corollary 4.6.10, the functions (4.7.15)

X3^H^(1-(^,C))"^

are in the closure of A{C^) in ^ ( X ) for every ( e X*. In fact, this is true when C — 0- For a fixed compact subset K oi X the expansion oo

i=o

converges uniformly in a neighborhood of if if |C' — ("I is smaller than some number independent of ( E X*. If (4.7.15) is in the closure of ^(C"^) in C{K) it follows that this is true with ( replaced by ('. Hence it suffices to


311

show that polynomials in the functions (4.7.15) and the coordinates Zj are dense in A{X) on K. Choose M so that \zj\ < M for j = 1 , . . . , n if .^ E if. As at the beginning of the proof we can then find points C,j G X* and £j > 0, n < j < N, such that K C {z e C-^Zjl < MJ

= l , . . . , n , |1 - {z,(j)\

> Sj.n < j < N} ^ X.

Another application of Theorem 4.2.12 shows that if / G A{X) is an analytic function F in the polydisc {w G C^]\wj\

< M , l < j < n , \wj\ <e~^,n

then there

< j < N}

such that /(z) = F ( ^ i , . . . , ^ „ , ( l - ( z , C „ + i » - \ . . . , ( l - ( z , C i v ) ) - ' ) when ;$; is in a neighborhood of K. Replacing F by a high order partial sum of the power series expansion we obtain the desired approximation of / , which completes the proof. To prepare for the proof of a converse of Theorem 4.7.8 we shall prove a lemma: L e m m a 4.7.9. Let X be a Runge domain and let a ^ b be two points in X. If fia,b is in A'{X), then ljia,b{9f) = 0 for arbitrary f,g E A{X) vanishing on Ln X where L is the complex line through a and b. Proof. By a linear change of coordinates we may assume that b — a has the direction of the ^î-axis. In a moment we shall prove that there are functions fj-,gj G A{X) such that n

(4.7.16)

fiz)

n

= J2i^j

- aj)fj{z),

giz) = J2'^zj -

aj)gj{z).

2

2

Since X is a Runge domain we can find sequences fj^Qj ing to fj^gj in A{X) as i/ —> oo. Hence

G AiC^)

converg-

n

fJâ,b{9f) = lini V

fJâMzj - aj){zk -

ak)g'-{z)fl{z)).

3,k=2

Since /Xa,6 == (1 - (9, z))ii where /z(/i) = j ^ h(ta + (1 - t)b) dt, h G ^ ( C ^ ) , it follows that every term in the sum is equal to 0. To prove the lemma it remains to verify that / (and g) has a decomposition (4.7.16).

312


Assume that this has already been proved in lower dimensions for an arbitrary pseudo-convex X and, to simplify notation, that a2 = • - • = a^ = 0. bmce {z' e C ^ " ^ {z\0) G X} is also pseudo -convex, we can find functions hj{z')^ z' = {zi,... ,Zn-i) which are analytic in {z' G C^"-^; (^',0) G X} such that / ( z ' , 0 ) = Yl^~ ^j^^ji^') there. Using Theorem 4.2.12 again we can find fj G A{X) such that fj{z',0) = hj{z') when (;^',0) G X. Now n-l 2

is analytic in X since the numerator vanishes when z^ — 0, and we have obtained the decomposition (4.7.16). T h e o r e m 4.7.10. ItX is a Runge domain such that iia^b ^ A'i^) arbitrary a^b e X, then X is C convex.

for

Proof. Let a^b^L be as in the proof of Lemma 4.7.9, and assume that the vector a — b has the direction of the 2:i-axis. Then fâM)= pi

=

[ {f^{z.d)ma Jo

+

{l-t)b))dt

n

yZ Cijdjf{tai^{l-t)bi,a2,. Jo

•., a^) dt-i-{aif{a)-bif{b))/{ai

-6i),

2

when / G ^ ( C ^ ) , for / -f zidf/dzi = d{zif)/dzi. where l{z) = Y^^ % ( ^ ~ %) we obtain

Replacing / by Idif

n

(4.7.17)

tia,b{mdif)

= E

\'^3?U{o) - / ( 6 ) ) / ( a i - &i),

/ G ^(C").

2

If iia^h ^ A'{^) then this formula remains valid for all / G A{X). Now we get a contradiction ii Lr\ X is not connected and a, b are in different components, for using Theorem 4.2.12 again we can find / G A{X) equal to 1 in the component of L fl X containing a but equal to 0 in the others. Then the left-hand side of (4.7.17) vanishes by Lemma 4.7.9 but the righthand side is not 0. Making a linear transformation of X if necessary we conclude that LO X \s connected for every complex line L. If some such intersection were not simply connected, then we could choose a C^ Jordan curve in L n X and an analytic function m Lr\X with integral 1 along 7. Another application of Theorem 4.2.12 gives an extension to an analytic function / in X . Since the integral of any / G .A(C^) along 7 is equal to 0, this contradicts the hypothesis that X is a Runge domain, and the proof is complete.


313

An application of Theorem 4.7.8 will be given in Section 6.4. We shall end this section by discussing another formulation of Theorem 4.7.3 which will lead to a result underlining how cautious one must be even in dealing with convex carriers. First note that if M is an entire function of exponential type in C^, that is, |M(C)| 0, we can write PM{(I + «'6) = 6 ^ ( 6 / 6 ) where ^(r) = PM{1 + ir), and a straightforward computation gives in the sense of distribution theory when ^1 > 0 2

iJ2tj9/9Q'PMiî

+ ^ 6 ) = ( 0. Hence it follows that

In the set (4.7.18) we have {z,Q = 0 when (C^C) = O5 a-nd since this quadratic surface is not contained in any hyperplane it follows that the intersection of the carriers consists of the origin only, and it does not carry

CHAPTER V

C O N V E X I T Y W I T H R E S P E C T TO A LINEAR G R O U P S u m m a r y . In t h e preceding chapters we have seen t h a t convex functions, subharmonic functions and plurisubharmonic functions are invariant under respectively t h e full linear group, t h e orthogonal group and t h e full complex linear group. We have also seen t h a t the set of subharmonic functions invariant under the full (complex) linear group consists of convex (plurisubharmonic functions). In this chapter we shall consider t h e spaces of functions attached similarly to other subgroups of t h e linear group. These may be relevant in connection with the convexity with respect to certain differential operators as discussed in Chapter VI.

5.1. S m o o t h functions in t h e whole space. The classes of convex, subharmonic and plurisubharmonic functions have many properties in common. By Theorems 3.2.28, Exercise 3.2.1 and Theorem 4.1.7 they are related respectively to the full linear, the orthogonal and the complex linear groups. The purpose here is to discuss analogous classes of functions associated with any subgroup G of the full linear group GL{V) in a finitedimensional real vector space V. By Ga we denote the group generated by G and the translations in V. We assume that a translation invariant Lebesgue measure dx has been defined in V, for example by means of a basis over R. Definition 5.1,1. A convex conic set V C C ^ ( F ) will be called G subharmonic if (i) V contains every affine function; (ii) V is invariant under Ga, that is, li G P , ^ E Ga implies uo g eV; (iii) the maximum principle is valid in the weak sense that for every compact set K CV (5.1.1)

supii = supn, K

u£V]

dK

(iv) V is maximal with the preceding properties. In this definition we have for the sake of simplicity assumed that the functions in V are smooth and defined in the whole of V but otherwise

316

V. CONVEXITY WITH RESPECT TO A LINEAR GROUP

made the conditions weak. We shall consider other domains and relax the smoothness condition after having analyzed the conditions in the definition. They are fulfilled in all the cases we have encountered so far except the semiconvex and quasi-convex functions in Section 1.6 which will not be covered by the discussion here. If Uj G V and Uj -^ u e C^{V) locally uniformly, then Uj o g -^ u o g locally uniformly, and (5.1.1) is valid for u. In view of the maximality condition (iv) it follows that (v) V is closed in C^{V) gence.

for the topology of locally uniform conver-

The condition (iii) can be replaced by (iii)' li u EV then the quadratic form (5.1.2)

V3y^{u"{0)y,y)

is not negative definite. That (iii) implies (iii)' is clear, for if u E V and v{x) = u{x) — u(0) — {u'{0),x), then T; G P (by condition (i)) and v{0) = 0 but v{x) < 0 by Taylor's formula for all x 7^^ 0 in a neighborhood of 0 if (5.1.2) is negative definite. On the other hand, it follows from (iii)' in view of the translation invariance contained in (ii) that u"{x) is not negative definite at any x eV ii u E V. Hence (iii) follows unless the maximum of n in i f is taken at an interior point where u' = 0 and det-u" = 0, for u'^ must be negative semidefinite at an interior maximum point, so (iii)' implies that detu^' = 0. By the Morse-Sard theorem the set G of critical values of a; H-^ U'{X) is of measure 0 in the dual space V of F , so (iii)' implies that (iii) is valid when u is replaced hy u — {-,6) ii 6 ^ Q. Letting 6—^0inV'\@we conclude that (iii) holds. A quadratic form in V can be identified with an element in the symmetric tensor product S'^{V'). Ii u E C^ we shall denote the quadratic form (5.1.2) also by u"{Q). P r o p o s i t i o n 5.1.2. U the convex cone V C CîV) is G subharmonic in the sense ofDeRnition 5.1.1, then VQ = {u"{0)] u EV} is a convex conic subset ofS'îV) such that (a) Vo is invariant under G; (b) no element in VQ is negative definite; (c) Vo is maximal with the preceding properties. V consists of V n S'^(y). (a)-(c), then subharmonic,

all u E Cîy) such that u"(x) G VQ for every x, hence Vo = Conversely, if a convex cone Q C S'^{V') satisfies conditions the set V of all u G C ~ ( F ) with u"{x) G Q for every x is G and Vo — Q then.

SMOOTH FUNCTIONS IN THE WHOLE SPACE

317

Proof. Let V satisfy the conditions in Definition 5.1.1. Then (a) follows from (ii) and (b) follows from (iii)'. For any convex cone Q C S'^{V') satisfying (a) and (b) the set Q C C ^ ( F ) consisting of all u G CîV) such that u"{x) E Q for every x G F is obviously a convex cone, and (i), (ii), (iii)' hold. By the maximality condition (iv) on P , it follows that V = Vo, and also that VQ has the maximaUty property (c), for Q C Q so Q D V ii Q ^ VQ. Conversely, if Q satisfies (a), (b), (c) then Q will be maximal, for an extension of Q would give rise to an extension of Q. The proof is complete. Note that (a), (b), (c) imply (d) Vo is closed in S'^{V')] (e) Vo contains all positive semidefinite forms. Conversely, (e) implies (b) iiVo is not equal to S'^{V'). Summing up, we have established a one-to-one correspondence between spaces V which satisfy the conditions in Definition 5.1.1 and closed convex cones C S'^{V^) satisfying conditions (a)-(c). There is a natural duality between quadratic forms uinV and quadratic forms 'u in y , for polarization of u defines a linear map V -^ V and similarly v defines a linear map V -^ F , so the trace of the product is well defined (and independent of the order). We shall denote it by {u^ v). If we introduce dual coordinates x mV and ^ in V and write ^(^) = Yl'ôk^j^k,

v{^) = Y^Vjk^j^k,

then {u,v) =

"ÛjkVjk.

Let VQ be the dual cone of VQ in the space S'^{V) of quadratic forms on V , that is, (5.1.3)

V^ = {ve

S\V)-

{u,v) > 0 for every u G VQ}-

Then VQ is defined by VQ in the same way (see Exercise 2.2.3), and the properties (a)~(e) of VQ are equivalent to the following properties of the closed convex cone VQ 7^ {0}: (a)' VQ is invariant under the group G* consisting of adjoints of elements inG; (b)' every element in VQ is positive semidefinite; (c)' VQ is minimal with the preceding properties. The condition VQ 7^ S^{V') is equivalent to VQ ^ {0}, and the condition (b)' is then equivalent to (e) which proves the equivalence of the conditions on Vo and those on VQ . Let W^ be the intersection of the radicals of the elements in VQ. It is invariant under G*, so the annihilator W CV is invariant under G. UVQ^

318


is the space of restrictions of the forms in Vo to W, then the conditions ( a ) (c) are fulfilled by V^ with respect to W, and VQ consists of the quadratic forms in V with restriction to W belonging to V^. Indeed, if we introduce coordinates so that W° is defined by î ==•••== ^^^ = 0, then the forms in VQ are independent of (^^y+i,... so the condition for u to be in the dual cone only involves the coefficients Ujk with j^k < u. If V^ is the space of G subharmonic functions in W corresponding to V^ ^ then V is the space of C^ functions with restriction to W belonging to V^ after an arbitrary translation. It will therefore suflice to study the case where W = V. In that case we can add a finite number of elements in VQ to find a positive definite one. Definition 5.1.3. 'PQ will be called positive if no element 7^ 0 is negative semidefinite or, equivalently, V^ contains a positive definite form. Remark. If G does not act irreducibly on V there may not exist any positive VQ. TO give an example we let 1/ = R^ and G be the group of diagonal invertible matrices. If the semidefinite form a^\ -f 2& W2. IfT G (5{Wi,W2) and S G &{W2, W3), then ST G 6(1^1,1^3) and T'^ G (S(W^2, Wi). In particular, &{WuWi) is a compact subgroup ofGL{Wi), and T-^&{W2,

W2)T =

&{WuWi).

Proof. The first statement has already been proved. To prove the second choose gj G G so that a(gj^W2)gj restricted to W2 converges to 5 , and choose hj G G so that the norm of a{hj,Wi)hj — T on Wi is less than ll{3\H9i,W2)9,\\). Then a{gj,W2)a{hi,Wi)9jhj-ST = a{gj,W2)9Mh3^Wi)hj

-T)

+ {a{gj,W2)9j

-

S)T

converges to 0 on VFi, which proves that ST G ^{Wi,W^). Thus (5(VFi,PFi) is a compact semigroup. But then it is automatically a group since for any T G ©(Wi, VFi) there is some power which is arbitrarily close to the identity, thus a sequence of powers converging to the inverse of T. If T G &{WuW2) and S G (&(W^2,Wî), then ST G &{WuWi) has an inverse R G (d(WuWi). Thus R{ST) is the identity, so T ' ^ = RS e &(W2,Wi). We have T-^&(W2,W2)T

C &{Wi,Wi),

r(5(Wi,Wi)T-^

C (5(W^2,^^2),

which proves that there is equality. We are now ready to discuss the structure of a positive VQ. T h e o r e m 5.1.7. Suppose that Vo satisfies conditions (a)-(c) and that Vo is positive. Let e be a positive definite form G VQ. Then: (1) Every element in VQ is a finite linear combination of compositions £ o J* with 7 G G. For non-zero elements the codimension of the radical is > r, the minimum rank of elements in G. (2) s o 7* G 'pQ ^^^ every 7 G G. In particular, there is an element in VQ with radical W for every W eWc (3) For every W G WQ there is a form qw ^ VQ with radical W^, unique up to a positive factor, and the dual positive definite quadratic form ew in W is invariant under ©(TV, W). (4) Every element in VQ is a finite positive linear combination of the forms qw • (5) &{Wu W2) is not empty ifWi,W2 G WG(6) The forms qw^^w can be normalized so that cyî ^T — ewi if Te&{Wi,W2),Wi,W2 eWc(7) If J G G then 7 F is not contained in any G invariant linear subspace ofV strictly smaller than V.

SMOOTH FUNCTIONS IN THE WHOLE SPACE

321

Proof. By the minimality condition (c)' the closed convex conic hull of the forms e o g^ with g e G is equal to 'PQ- If 9 ^ Tô and {e,q) = 1, it follows from Theorem 2.1.5 that q is the limit of elements of the form J

^

J

aj£ o ^*,

aj > 0, gj e G, ^

1

where J == dimS^{V). ll^jlP < {e,eogj)

A^- = 1, if A^- = aj(e,

eogp^

1

We have aê o ^j = Aj^ o ( ^ j / . / ( e , £ o ^j)), and

< l l ^ j l p d i m F , so g]l^{e,e

o g^^) belongs to a compact

subset M of G . Hence it follows that J

where A^ > 0, Y^^ Xj = 1, and 7J G M . If A^ > 0, then the radical of q is contained in the radical Ker7J = {jjVy of e o 7J, which proves (1). The G* invariance of VQ gives (2), and we pass to the proof of (3)-(6). If VF G W G , then W = jV for some 7 G G of minimal rank r. The radical of e o 7* is the kernel of 7* which is the annihilator W° of W. Thus the set Qw oi all q ^ VQ with radical W° is not empty, it is convex, and Qw U {0} is closed since the radical of an element 7?^ 0 in the closure contains W° and cannot have smaller codimension by (1). Moreover, Qw is invariant under the action of the compact group 6(VF,1^), or rather its adjoint which acts on V/W° where the forms in Qw are defined. By integration over the Haar measure we conclude that there is an element qw in Qw which is invariant under (5(W^, VF)*. We normalize qw so that {e,qw) — 1, and keeip W fixed in what follows. For any g E G the normalized composition q9 = {qwog^)l{e,qw

o g*)

belongs to a compact subset of P Q , and the radical is {g^)~^W° = {gWy, so q^ G Qgw- For any sequence gj G G there is a subsequence, which we also denote by ^ j , such that gjW -^ Wi e WQ and the normalized map a{gj,W)gj converges to a map T G (S(VF, VFi) with adjoint T* mapping ^ 7 ^ 1 ° to V'/W. It follows that q^^ -^ cqw o T"*, where c is a normalizing constant. In fact, if / is a quadratic form in V^ then

(/, q'^ = if o ia{g„W)gj), qw)/{{e, qw ° 9'j)a{gj,W)') ^c{foT,qw)

=

c{f,qwoT'),

322


if we note that (F, qw) only depends on the restriction of F to W. For the same reason 1/c = (e o T,qw) = {e,qw ^ T^)- Hence all elements in the closure K of {{9W,q^);geG}cWGxV'o are of the form {Wi, {qwoT^)/{e, qwoT*)), where T E (5{W, Wi). The form QWi — Qw ^ ^V(^5 Qw o T^) is independent of the choice of T G (S(VF, Mî), for ifS'G 0(1^1, W^) then {qw, o S')/{e, qw, o S') - {qw o (5T)*)/(e, qw o {ST)')

= qw,

since ST G 0 for every x E X and w G WQP r o p o s i t i o n 5.2.2. IfX

is connected and u G V{X)

is not

identically

—CO, then u G Ll^^{X). Proof. Assume that 0 G X and that u{0) > —oo. Shrinking X and adding a large negative constant to u we may assume that u < 0. If Wi G WG it follows that the mean value oiw^-^ W{'^)\ over {w eWi] ewi{w) < r} is < |ti(0)|. For the points with u{w) > —oo we repeat the argument, and after k repetitions we conclude that if Wi^... ,Wk G WG then the mean value of Wix

'•' xWk3

(wi, ...,Wk)^

\u{wi -[-••• + Wk)\

over the product of the balls {wj] ewj i'Wj) < T} is at most equal to \u{0)\ if r is small enough. Now Theorem 5.1.7 shows that we can choose Wi,..., Wk so that W^n-'-nW^ = {0}, that is, Wi,... ,Wk span V. Then it follows that u is integrable in a neighborhood of the origin. As in the proof of Theorem 3.2.11 it follows that u G iJocC-^)T h e o r e m 5.2.3. If X C V is an open set and u G V^{X), following conditions are equivalent: (i) u is defined by a function in V{X),

uniquely determined

then the by u.

(ii) u^ê V{xJ), x^ = {xe V',x-supp(p c x}, ifo 0. Then Us i U diS 6 i 0, Us e C ^ ( X ^ J , and Usog is subharmonic in g~^(X^p^) for every g E G. If e is the dual form of the metric form in F , this means that {Us ^ g^s) = {U'sê o g^) > 0. By (1) in Theorem 5.1.7 it follows that {U's^q) > 0 for every q^V^. With q = qw we conclude that w — i > Us{x-\-w) is subharmonic in {w G W]x-\-w G X } , if W G W G J and letting ^ —> 0 we obtain U G V{X). The proof is complete. In the following theorem r will denote dimW^ when W G WG-, just as in Lemma 5.1.5. T h e o r e m 5.2.4. Ifr = 1 then every u G V{X) is Lipschitz continuous; if r > 1 then every u G V{X) is in L^^^{X) when p G [l,r"/(r — 2)), and u' G L\^^{X) for every p G [ l , r / ( 7 - - l ) ) . Here 2/(2-2) should be interpreted as +00. Proof. The proof is parallel to that of Theorem 4.1.8 so we shall be quite brief. As there we can make Theorem 3.2.13 quantitative and show that if u is di. subharmonic function < 0 in the unit ball in R^, where r > 1, and if p < r/{r — 2), then / \uf dx < Cp\u{0)\P. J\x\ S. We can extend the conclusion to arbitrary v G £'{X)^ for if (^ G CQ^{X) has support in the unit ball, J ip{x)dx = 1, and (feix) = ip{x/£)/e'^^ then the distance from SMpp{{P{—D)v) * (ê) to CX is at least equal to 8' — e. Hence the distance

P-CONVEXITY

331

from supp(7; ^ cp^) to CX is at least 6' — e, and when e —> 0 it follows that 6>6'. To prove the converse we let KS^R = {X G X; |x| < i?, dx{x) > 6}^ where ^ > 0,i? > 0. li V e C^{X) and 8upipP{-D)v C KS,R, then it follows from (6.1.3) that |x| < i? if x G suppi', and dx{x) > S by the condition in the proposition. Hence suppi^ C KR^S-, which proves (6.1.1). P r o p o s i t i o n 6.1.7. An open set X C R'^ is P-convex for singular supports if and only if for every v G £'{Ii^) with sing supp-u (E X the distance from sing supp-u to CX is equal to that from sing supp P{—D)v toCX. Proof. In Definition 6.1.3 we assumed that v G £'{X), but the condition remains true if t; G £ ' ( R ^ ) and sing suppi; (E X. In fact, if x G CQ^{X) is equal to 1 in a neighborhood of sing suppi; then V = x^ ^ ^'i^)-) sing s u p p F — sing supp'u and sing supp(P(—D)y) = sing s\ipp{P{—D)v). With this modification of the definition the proof of Proposition 6.1.6 can be used again with supp replaced by sing supp throughout. We can now easily prove analogues of Proposition 2.1.3: P r o p o s i t i o n 6.1.8. If X^, a e A, is an arbitrary family of open sets in H^ which are P-convex for supports, then the interior X ofHaeAXa is P-convex for supports. Proof. Since dx{x) = miadx^{x), once from Proposition 6.1.6.

iix

E X, the statement follows at

P r o p o s i t i o n 6.1.9. If X^, a £ A, is an arbitrary family of open sets in R^ which are P-convex for singular supports, then the interior X of HaeAXa is P-convex for singular supports. The proof is the same as for Proposition 6.1.8, with reference to Proposition 6.1.7 instead of Proposition 6.1.6. We shall now prove a result which implies the converse of Proposition 6.1.5: P r o p o s i t i o n 6.1.10. If P{D) = {D,t) + c where t e W \ {0} and c G C; then an open set X C R"^ is P-convex for (singular) supports if and only if the boundary distance dx is a quasi-concave function on any line segment contained in X with direction t. Proof. To simplify notation we assume that t = ( 1 , 0 , . . . , 0). Let / = {{xi,y^);a < Xi < b} C X, and set v(x) = w{xi) 0 6{x' — y') where w{xi) = e*^^^ when a < xi 6, then there is a maximal open interval / = {xo -h 5t; a < 5 < 6} with a < 0 < 6 which does not intersect snpp P(—D)v. Since d(e~'^^^^v)/dxi = 0 outside snpp P{—D)v^ it follows that / C supp?;. Hence / is bounded and I C X. The end points are in s\ipp{P{—D)v)^ and by the quasi-concavity of dx we have dx{x) < 6 for at least one of them. Hence the distance from snpp P{—D)v to dX is < 5, so X is P-convex for supports. The proof that X is P-convex for singular supports is essentially the same. We just replace supp by sing supp throughout, and leave for the reader to check that the proof still works then. Note that by the remark after Theorem 6.1.4 we could equally well let P{D) = q{{D,t)) in Proposition 6.1.10 for any non-constant polynomial q in one variable. By Corollary 10.8.10 in ALPDO one part of Proposition 6.1.10 can be stated much more generally, for any P{D) of real principal type. (This means that the principal part p has real coefficients and that p ( 0 = 0 implies p ' ( 0 T^ 0 when (, e W \ {0}.) Then X is P-convex for singular supports if and only if dx is quasi-concave on any line segment contained in X with bicharacteristic direction, that is, direction p'{i) for some ^ € R"^ \ {0} with p(^) = 0; this condition also implies that X is P-convex for supports. A sufficient condition is that there is a continuous function cp in X with {x G X; ip{x) < t} (^ X for every t G R such that (p is convex on all bicharacteristic lines. Corollary 6.1.11. If the open connected set X C R"^ is P-convex for (singular) supports for every P{D) — {t,D) + c with t G R^ and c G C, then X is convex. Proof. This follows from Proposition 6.1.10 and Theorem 2.1.25. Thus we have proved a converse of Proposition 6.1.5. Much stronger results than Proposition 6.1.5 are valid in convex sets: in such sets there is a solution of every (overdetermined) constant coefficient system of differential equations provided that the necessary compatibility conditions are satisfied by the right-hand sides. We cannot prove this theorem here but refer to Chapter VII in CASV or the earlier proofs due to Ehrenpreis and to Malgrange and Palamodov, which are quoted there. See also ALPDO Sections 10.8 and 11.3 for additional results on P-convexity for some specific operators. 6.2. A n existence t h e o r e m in pseudo-convex domains. In Section 6.1 we have seen that it is precisely in convex domains that one can

AN EXISTENCE THEOREM IN PSEUDO-CONVEX DOMAINS

333

solve arbitrary differential equations with constant coefficients. We shall now prove that in a pseudo-convex domain C C^ one can solve a large class of differential equations with constant coefficients related to the complex structure: T h e o r e m 6.2.1 (Malgrange). If X C C^ is a pseudo-convex open set, then X is P-convex for supports and singular supports if P is of the form P{d/dzi,..., d/dzn). In fact, Malgrange even treated arbitrary (overdetermined) systems of operators of this kind. His proof works only in that generality and demands much more background than we can assume here. We shall therefore give another proof for the case of a single operator which is based on weighted L^ estimates in the same spirit as Section 4.2. Appropriate weights to be used in the following result can be obtained from Theorem 4.1.21. P r o p o s i t i o n 6.2.2. Let X be a pseudo-convex open set in C^, and let (f G C^{X) be a strictly plurisubharmonic function such that (6.2.1)

Kt = {z e X ; (p{z) TK, u ^ (6.2.2)

^rl^l

f\P^''\d/dz)u\ê^^^dX

< CK f \P{d/dz)u\ê^^'^

compact C^{K), dX.

Before the proof we note that when r -^ oo it follows from the Carleman estimate (6.2.2) that if suppP{d/dz)u C Kt then supp-u C Kt. In fact, the right-hand side is then 0(e^'^*) as r —> cx). We can choose a with \a\ equal to the degree of the polynomial P such that P^^^ is a constant ^ 0. Using only this term in the left-hand side of (6.2.2) we obtain u = 0 ii (f > t. Replacing P{d/dz) by P{—d/dz) we obtain the part of Theorem 6.2.1 which concerns convexity for supports. Proof of Proposition 6.2.2. With respect to the weighted scalar product

the formal adjoint of d/dzj (6.2.3)

is

6j = e-'^^'^{-dldzj)e'^^'^

= -d/dzj

-

and we have (6.2.4)

[6j,d/dzk]

=

2Tdîp/dzjdzk.

2T{dip/dzj),

334 VI. CONVEXITY WITH RESPECT TO DIFFERENTIAL OPERATORS

Further commutation with djdzi or 8i only means appUcation of d/dzi or —djdzi to d'î^ldzjdzk and does not introduce any new factor r . If we apply Proposition B.2 in Appendix B with Aj = 6j and Bk = d/dzk^ we obtain when u G C^{K) I \P{dldz)u\ê'^^'^ = Yl

dX=

f (PiS)P(d/dz)u)ue^^^

dX

{P^^Hd/dz)TaA^,d/dz)P^''\6)u)ue^^'^dX

a,(3

= Y,

h^^^{8,dldz)P^'"\s)uP^^\8)ue^^'^dX.

Here Ta,/3{8,d/dz) is a sum of products of commutators formed from 6j and d/dzk^ with \a\ resp. |/?| factors of the two kinds in each term. In a term containing the product of /^ commutators there is a factor r'^, and since K < min(|a|, |/3|) < {\a\ + \/3\)/2, it follows from Cauchy-Schwarz' inequality that (6.2.5)

f\P{d/dz)u\ê^''^dX

< C j ^ r l ^ l j \P^'"\6)u\ê^^'^

dX.

To get a lower bound we note that in terms where K < {\a\ + |/9|)/2 the difference must be at least | , so we can estimate them by the right-hand side of (6.2.5) multiplied by r ~ 2 . Thus Y^

ffa^fs{6, d/dz)P^'"^{6)uP^^^{6)ue^^'^ < f \P{d/dz)u\'ê^^'^

dX

dX-^-C^T^""^-^

I

\P^'"\8)u\ê^''^dX.

Here the sum Tcc^^{6^ djdz) of the terms in Vc^^^{b^ djdz) with the maximal exponent K — (|a| + |/3|)/2 for r vanishes if \OL\ ^ |^|, and it is given by n

5 ] ^ V r . , M 5 , a / a ^ ) = exp(2r ^

d^^ldzjdzkijr^k)

j,k=l

according to (6.2.4) and Proposition B.3. We have assumed that cp is strictly plurisubharmonic, which means that the matrix (d'îp/dzjdzk) is uniformly positive definite in K. Hence it follows from Proposition B.4 that with a new constant (1 - CT-i)J2

^'''' / \P^''\s)u\ê^^^

dX R e ^ l } is pseudo-convex, but since the Levi form of dX vanishes identically, the pseudo-convexity can be destroyed by a perturbation of dX near 0 which is arbitrarily small in the C^ sense. Such perturbations only affect the boundary distance X near the normal at 0. Since

dx{zuZ2) =xi-xl+yl-\-

0{\z\^)

340 VL CONVEXITY WITH RESPECT TO DIFFERENTIAL OPERATORS

has negative second derivative with respect to X2 when \z\ is small, it is clear that a sufficiently small perturbation will not affect the condition in Proposition 6.2.4. The proofs of Propositions 6.2.2 and 6.2.3 allow much more general conclusions, for the operators d/dzj can be replaced by arbitrary first-order homogeneous differential operators Lj with constant coefficients. Note that the adjoint L* is equal to —d/dzj if Lj = d/dzj. P r o p o s i t i o n 6.2.5. Let X be an open set in R'^, let Li^...^Ly be first order constant coefficient differential operators, and assume that if G C^{X) satisfies (6.2.1) and that the hermitian symmetric matrix {-L*Lk^)^^k=i is positive definite at every point in X. UP — P{L) = P ( L i , . . . jLiy) then one can for every compact set K C X End constants CK Sind TK such that for r > TK (6.2.2)' ^^l«l

f\p(^){L)u\ê^^'^dX

< CK I \P{L)u\ê^^'^ d\

ifu G

C^{K).

OL

Ifue

£'{K)

and P{L)u G H(^^), it follows that u G H(^^y

There is really nothing to change in the proofs of Propositions 6.2.2 and 6.2.3 apart from obvious changes of notation. Note that L i , . . . ^L^j must be linearly independent over C, for otherwise the matrix {—LjLk(p)^^j^î is singular. As in the discussion after the statement of Proposition 6.2.2 it follows from Proposition 6.2.5 that X is P(L)-convex for supports for any P if there is a function (f satisfying the conditions in the theorem, and as in the proof of Theorem 6.2.1 we find that X is then also P(L)-convex for singular supports. We shall now discuss the geometrical meaning of this condition on X. Let F be a finite-dimensional vector space over R with complexification Vc, and let £ be a complex linear subspace of Vc, of dimension u. We shall identify the elements in C with homogeneous first-order differential operators in V with constant coefiicients. The intersection £ R = CHV is a real linear subspace of V, and there is a natural complex structure in (Re£)/>CR, for the complex vector space £ / ( £ R (8)R C ) is mapped bijectively to ( R e £ ) / £ R when we take the real part. If a basis L i , . . . , L^ for £ R over R is extended to a basis L i , . . . , Lr, Lr+i,..., 1/^^ for £ over C, then Re £ is spanned by the lines R L i , . . . , R i r and the two-dimensional planes Re C L - r + i , . . . , Re CL^^, which are naturally isomorphic to C. If a Euclidean metric is given in V^ then we can first choose L i , . . . , L^ as an orthonormal basis for £ R and then subtract from L r + i ? " - ^Li^ the projection on £ R , SO that R e L j , I m L j become orthogonal to £ R . Then the Euclidean metric form on ^\xjLj + ^^Y^^r^-iô-^j ^^ S i ^ j P^^^ ^

AN EXISTENCE THEOREM IN PSEUDO-CONVEX DOMAINS quadratic form in Zj and in Zj representing the form induced on by the Euclidean metric form. We want it to be hermitian:

341

{ReC)/L^

Definition 6.2.6. If V^ is a finite-dimensional vector space over R and C a complex linear subspace of the complexification Vc, we shall say that a quadratic form in V is hermitian with respect to C if the form induced in ( R e £ ) / ( £ n V) is hermitian with respect to the natural complex structure there. We shall say that a function ip in an open subset X of F is plurisubharmonic with respect to £, if ip is upper semi-continuous and L*Lip < 0 in the sense of distribution theory for every L E C. If the metric form is hermitian with respect to £ , we can choose the basis elements i^r+i, - • ,Ljj so that it is equal to Yl\ ^j + S r + i lîP ^^^ J2î^j^j + I^^X^r+i ^3-^j- ^^ ^^^ moment we are not concerned with the behavior of the metric form in directions outside Re £, for we shall first assume that ReL = V in discussing an analogue of Theorems 4.1.19 and 4.1.21. T h e o r e m 6.2.7. Let V be a real vector space and C a complex linear subspace of the complexiGcation such that ReC = V. The following conditions on an open set X CV are equivalent: (i) There is a function cp in X, not = — oo in any component and plurisubharmonic with respect to C, such that (6.2.1) is valid. (ii) If K is a compact subset of X then the set K of all x ^ X such that (p{x) < s u p ^ (^ for all functions (f which are plurisubharmonic with respect to C in X is (^ X. (iii) a; I—> — \ogdx{x) is plurisubharmonic with respect to C in X if dx is the boundary distance deBned in terms of a Euclidean metric in V which is hermitian with respect to C. Proof. Only the proof that (ii) = ^ (iii) has to be reconsidered. We can assume the coordinates chosen so that £ consists of the vector fields r

L = 2_] h^l^^j 1

V

+ 2 2_. f'j9/dzj,

where Zj = X2j-i-r

+ ^^2j-r?

r+l

thus dimR F = 2i/ - r, and we may assume that the metric form is

N-(E-.'+Ei-/)'1

r+l

Let X — {z ^ C^; (Rejî,... ,Re2:^,Re2:^_|_i,Im2:7.+i,... .,lmZj,) G X}.


This is a tube with base X , so we have a projection X -^ X. If (pis plurisubharmonic with respect to £ in X , and ip is the puUback of (^ to X , then (p is plurisubharmonic since the hfting of L'^Lcp to X is —4 ^ ^ ljlkd'^(p/dzj/dzk' We can add to (p any convex function of (Im ;2;i,..., Im2:^), so it follows from (ii) that X is pseudo-convex. Now the boundary distance in X with respect to the standard hermitian form in C^ is the puUback of dx from X , which proves that dx is plurisubharmonic with respect to C. Definition 6.2.8. If F , £, and X satisfy the hypotheses in Theorem 6.2.7 then we shall say that X is pseudo-convex with respect to C if the equivalent conditions (i)-(iii) there are satisfied. Note that when £ = Vc, then plurisubharmonicity with respect to C means local convexity, so then it is precisely the convex sets which are peudo-convex with respect to C. T h e o r e m 6.2.9. Let V, C and X satisfy the hypotheses and in Theorem 6.2.7, let K be a compact subset of X, and define condition (ii) there. If Y is an open set with K (s Y C X, can find ip E C"^(X) satisfying (6.2.1) so that L*L(p < 0 in X Le /:\ {0}, andcp 1 in X \ Y.

conditions K as in then one for every

Proof. The proof of Theorem 4.1.21 can be repeated with essentially notational changes, so it is left for the reader. By Theorem 6.2.9 we can apply Proposition 6.2.5 to any set which is pseudo-convex with respect to £, so we obtain an extension of Theorem 6.2.1 containing also Proposition 6.1.5: T h e o r e m 6.2.10. Let V be a real vector space and £ a complex linear subspace of the complexification such that Re £ = V. Then it follows that every open set X C V which is pseudo-convex with respect to £ is also P-convex with respect to supports and singular supports if P is any polynomial in Li,... ,Lk G £ . So far we have assumed that ReC = V. The case where ReC ^ V seems much harder to study except when d i m £ = 1. Then £ consists of the multiples of an operator {D,t), and there are two cases depending on whether ReC = T = {Rezt]z E C } has dimension 1 or 2. These were discussed in Propositions 6.1.10 and 6.2.4, but we shall review these results in light of Proposition 6.2.5. P r o p o s i t i o n 6.2.11. Let X C R"^ be an open set, 0 ^ t e C^, and define T = {Rezt;z G C } . Assume that the boundary distance dx{x) from X E X to CX satisfies the minimum principle (6.2.11) when K C

AN EXISTENCE THEOREM IN PSEUDO-CONVEX DOMAINS

343

X n {{xo} + T) is compact and dK is the boundary of K as a subset of {xo} -\-T. Then there is a function cp G C^{X) satisfying (6.2.1) such that {D,t){D,i)cp < 0 inX. Proof, The case where d i m T = 1 is much easier so we shall assume at first that d i m T = 2 and choose coordinates so that t = (1,«), that is, {D,t) = -i{d/dxi-îd/dx2),

-{D,t){D,i)

= A^.,

where x' = (xi,a;2)- We shall also write x" = (0:3,... , x „ ) . Set Ke = {x e X]dx{x)

> e and \x\ < 1/e},

which is a compact subset of X , increasing to X as ^ -^ 0. To construct cp we shall first show that for every ^ > 0 there is a function (p^ G C ^ ( R ' ^ ) such that cpe 2 in X \ if 1/4 if ai is large enough. Having fixed ai we choose a 2 , . . . successively so that ^ Q aj^2-3 is strictly subharmonic in K2-J-1 and > 2^ in X \ K2-J-1. We have ip2-j = 0 in if^ if 2^~^ > 1/s, so the infinite sum ip exists and has the required properties. It remains to discuss the case where T has dimension 1. In that case we just have to replace subharmonicity by convexity in the argument above, and we leave the obvious modifications to the reader. Proposition 6.2.11 combined with Proposition 6.2.5 gives a proof of the P-convexity with respect to singular supports in Propositions 6.1.10 and 6.2.4. However, for a general C with Re £ 7^ T^ it is not clear how to interpret geometrically the condition that there is an exhaustion function ip satisfying the condition in Proposition 6.2.5. It is clear that this implies that the intersection of X and any plane H parallel to Re £ must be pseudoconvex with respect to £ as a subset of H. The boundary distance must also satisfy the minimum principle in H. We profited in the proof of Proposition 6.2.11 from the fact that pseudo-convexity is automatic when d i m i = 1, but that is not true in general, and it is not obvious how to combine it with the minimum condition then. Stronger necessary conditions might be required. 6.3. A n a l y t i c differential equations. We shall now discuss existence theorems for analytic diff'erential equations. The classical CauchyKovalevsky theorem is a very general local existence theorem, but we want to discuss global existence for equations with constant coefficients. The first goal is an analogue of Corollary 6.1.11. L e m m a 6.3.1. Let X C C^ be a pseudo-convex open set. If the equation du/dzi = f has a solution u G A{X) for every f G A{X), then (6.3.1)

X,. =

{zieC;{zuz')eX}

is simply connected for every z' G C^"^. If Zi and wi are in different components of Xz', then they are in different components of X(^' for all (' in a neighborhood of z'. More precisely, ifaeC then the sets X'' =

{{zi,z')eX;{a,z')eX},

X^ — {(;2;i, 2:') G X^\ z\ and a are in the same component of Xz'} are open, and X°' is the union of the components {a} X C ^ - i .

of X^ which

intersect

ANALYTIC DIFFERENTIAL EQUATIONS

345

Proof. Suppose that Xz> is not simply connected. Then it has a component bj which is not simply connected, so we can choose a closed Jordan curve 7 in a; and an analytic function /o in X^', such as /o(î) = {zi—a)~^ with suitable a G 9a;, for which § fo{zi) dzi ^ 0. It follows from Theorem 4.2.12 that we can find / G A{X) such that / ( z i , ZQ) — fo{zi), if zi G Xzi^. Then the equation du/dzi = f cannot have a solution u G A(X), for du{zi,z'Q)/dzi ^ / ( ^ i , 4 ) = fo{zi), zi G a;, implies §^fo{zi)dzi = 0. Suppose now that a and b are in different components of Xz', and let zl G C^"^ be a sequence with lim^_,oo ^2:^ = ô- Using Theorem 4.2.12 we can choose / G A{X) so that /(î,>2ô) = 1 if zi is in the component of a in Xz' but /{ZI^ZQ) = 0 if 2:1 is in the other components. Since df/dzi vanishes in Xz', we can find hk G A{X) such that df{z)/dzi

= Y^{zk -

zok)hk{z).

k=2

(See the proof of Lemma 4.7.9.) By the hypothesis in the lemma there exist functions fk G A{X) such that hk = dfk/dzi^ which means that n

difi^) - J2(^k - zok)fk{z))/dzi = 0. k=2

If a and b are in the same component of Xz'^, it follows that n

f{o^^ 4 ) - / ( ^ 4 ) = "^{î^k - zok){fk{(i, 4 ) -

fk{b,4))-

The right-hand side -^ 0 as z/ -^ 00 while the left-hand side -^ 1, which is a contradiction proving that a and b are in different components of Xz' for all z' close to ^^Q. That X°' and X " are open is obvious; it remains to show that X°' is closed in X^. Assume that {zi^z') G X " \ X " . Thus zi and a are in different components of Xz'. By the parts of the lemma already proved it follows that Ci ^nd a are in different components of X^/ if (C15C') is sufficiently close to {zi^z'), so a neighborhood of {zi^z') is contained in X " \ X " , which is therefore open. The proof is complete. We can now prove an analogue of Corollary 6.1.11: T h e o r e m 6.3.2. Let X C C^ he C convex. differential equation n

(6.3.2)

'âjduldzj

=f

Then every

first-order


with constant ( a i , . . . ,an) 7^ 0 and f G A{X) has a solution u G A{X). Conversely^ ifX C C"^ is a bounded connected pseudo-convex open set such that this existence theorem is true, then X is C convex. Proof. For the first statement it suffices to prove that the equation du/dzi = f has a solution u G A{X) for every / G A{X) if X is C convex. Let IT be the projection of C^ on C^~^ obtained by dropping the first coordinate. Then X' — TTX is C convex by Proposition 4.6.7. For a G C we set X'^ = { / G C ' ' " ^ ; ( a , z ' ) G X}. These open sets cover X\ In X^ = X n Tr~^{X'^) the equation dua/dzi = f has a unique analytic solution with Ua{a^ z') = 0 when z' G X'^\ it is obtained by integration from a to zi in the connected and simply connected fiber of the projection. We have Ua — ui — Ua^h o ^ where Ua^h ^ A{X'^ H X'^) (the intersection may be empty), which means that we have an analytic one cocycle, that is, Ua,b = -y'b,a

inX^nX^,

Ua,b-Ûb^c-Ûc,a = 0,

lu X'^ H X^ D X'^.

We shall recall in a moment how one can find Va G A{X'^) such that Ua,b = Va—^b' This will prove that Ua—Va ore = Ub — vi, on in XaHXb so that we get a function u G A{X) satisfying the differential equation du/dzi = f. To construct Va we first observe that there is a partition of unity cpa G Co°(X^) in X' (thus only finitely many are 7«^ 0 on every compact subset of X' and E a â = 1 i^ ^ ' ) - Then

for the terms are defined as 0 outside suppipb and are therefore in C°°(X^). We have (6.3.3)

Wa-Wb

= ^

^c{Ua,c " Ub^c) = ^

^cUa,b = Ua,b

in X'^ D X^

by the cocycle properties. Differentiation of (6.3.3) gives dwa — dwb = 0 in X^ n X^, which means that there is a form ip G C?^-^^{X') with dip = 0 such that 7/; = dwa in X'^ for every a. Hence it follows from Theorem 4.2.6 that we can find v G C^{X') such that Bv = -0, which implies that Va ='Wa-'^ ^ A{X'^) and that Va - Vb = Wa - Wb = Ua^bSuppose now that X is bounded connected pseudo-convex and open, and that (6.3.2) always has an analytic solution. By Lemma 6.3.1 it follows then that the components of L f l X are simply connected for every complex line L. Let Y be the set of all {z,w) E X x X such that z = w or else z and w are in the same component of Lz,w H X if Lz,w is the complex line containing z and w. This is obviously an open set, for if (z^w) is close to

ANALYTIC DIFFERENTIAL EQUATIONS

347

(z^w) then a curve in Lz^w H X connecting z and w will remain in X if it is projected to Lz^w^ and it can be extended to a curve from z to w. To prove that Y = X x X it remains to prove that Y is closed in X x X . Let {z^w) G {X X X) \Y. It is no restriction to assume that z — w has the direction of the 2:i-axis. Let a = zi and define X^ and X " as in Lemma 6.3.1. Then ^ G X " and it; G X^ \ X " . Now if z and w are sufficiently close to z and w respectively, then L^^^) H X C X " . (It is here that we use the boundedness of X.) Hence z and w cannot be in the same component of L^^w n X , for z and w are in different components of X " . This proves that (X X X ) \ F is open. Thus F = X x X , so the intersection of X with any complex line is both connected and simply connected, that is, X is C convex. The proof is complete. Remark. We do not know if the theorem holds without the hypothesis that X is bounded. Note that in the first part of the proof we only used that Xz' is simply connected and that TTX is pseudo-convex. It would have been enough to assume the necessary condition in Lemma 6.3.1 and that the equation dv = w can be solved when it; is a one form with Bw = 0, in the complex manifold obtained when one identifies points 2:, ^ G X with (' = z' and ziXi in the same component of X;^.'. (See Suzuki [1].) The first part of Theorem 6.3.2 can be extended to an analogue of Proposition 6.1.5 valid for C convex sets: T h e o r e m 6.3.3. If X C C^ is C convex and the differential operator P{d/dz) = P{d/dzi^..., d/dzn) has constant coefficients, not all equal to 0, then the equation (6.3.4)

P(d/dz)u

has a solution u G A{X)

= f

for every f G ^ ( X ) .

Proof. Since ^ ( X ) is a Frechet space, it is well known (see e.g. Bourbaki [1, Chap. IV, §2] or Schaefer [2, Sections 7.3, 6.4]) that the continuous linear map P{d/dz) : A{X) -^ A{X) is surjective if and only if the adjoint P{-d/dz) : A'{X) -^ A'{X) (i) is injective; (ii) has a range which is closed in the weak topology

a{A'{X),A{X)).

If /i G A'{X) and P{-d/dz)iJi = 0, then it follows that P(-C)/i(C) = 0 if jl is the Laplace transform of the image of/x in A^{C'^). (See Section 4.7.) Thus jl{() = 0, which means that /x — 0 (since X is a Runge domain), so (i) is fulfilled. To prove (ii) it suffices to prove that the intersection oi P{—d/dz)A'(X) and (6.3.5)

MK = {iê A'{X);

\u{f)\ (0| = Ke(C), which impUes (6.3.6)

|/x(C)l 0 if (t, d/dz)v = 0, because the interval {X{rz' + (1 - T)Z) + (1 - A)n; 0 < r < 1} is then contained in YsHX and has a direction which is a multiple oft. By analytic continuation to A = 1 we conclude that v{z) = v{z'), which proves (ii). To study the case where d/dzi is tangential we need to reduce the defining function ^ to a simple form by a change of variables and defining function. As a preparation we shall first discuss the case where arbitrary complex coordinates are allowed, which we postponed in Section 4.1. We take zo == 0 to simplify notation.

LOCAL ANALYTIC SOLVABILITY FOR djdzx

355

P r o p o s i t i o n 7.1.2. Let g be a real-valued C^ function in a neighborhood of 0 such that g = 0 and dg ^ 0 at 0. If c E C^ is real valuedj c(0) == 1, and IIJ{Z) = (-01(2:),... î/jn{z)) is analytic at 0 with t/'CO) = 0 and dipj{0)/dzk = 6jk, then (7.1.2)

M(V^(z)) = 2 Re{dg{0)/dz,

z) + Q{z) + o{\z\%

where Q is a quadratic form, and ifw G C^, Y17 dg{0)/dzjWj n 2_] d^Q/dzjdzkWjWk j,k = l

(7.1.3)

n = /J j,k=l

= 0 then

d^g{0)/dzjdzkWjWk.

Conversely, ifQ is a quadratic form satisfying (7.1.3) then one can choose c and i/j as above so that (7.1.2) is valid. Proof. Put ( = ip{z) and note that d'^Q/dzjdzk = when z — 0 and that d/dzj = YZ=i dipj^/dzjd/dCj^,

d^{{cg){tjj{z))/dzjdzk

n d'^/dzjdzk

= ^

d^Jj./dzjd^^/dzkd'^/dCj,d(^,

since -0 is analytic. This proves (7.1.3), for the terms where c is differentiated vanish since ^ = 0 and Y^Wjdg/dzj = 0 at 0. To prove the converse it is convenient to choose the coordinates so that g{z) = —2xn -f 0(|;2;p), hence by Taylor's formula g{z) = -2xn-^2ReA{z)-\-

n ^ HjkZjZk-h j,k=i

o{\z\'^),

where A is an analytic quadratic form and (Hjk) is hermitian symmetric. In Hnn^n^n ctud 2 Rc ^ ^ ~ HjnZjZ-n wc writc z^ = 2xn — Zn and obtain

g{z) = -2xn{l-b)-\-2ReAi{z)-\-

b = Hnn^^Zn

n-l ^

HjkZjZk-\-o{\z\^),

n—1 n—1 + 2 ^ Re iJ^^î, ^ l ( ^ ) = M^) - \HnnZn " ^HjnZjZn. 3=1 i=i

With ijj[z) — {z\,... , ^ ^ _ i , z ^ + î('2^)) îid c = 1 + 6 we have obtained (7.1.2) with the form Q — Yll~k=i-^JkZj^kj which is the hermitian form which by (7.1.3) is invariant under the group of operations allowed. We can

356

VII. CONVEXITY AND CONDITION (^)

apply this result to —2ReZn-\-Q with any Q such that d'^Q/dzjdzk for j , A: = 1 , . . . 5 n — 1, which completes the proof.

= Hjk

The form (7.1.3) is the Levi form of the boundary of {z; Q{Z) < 0} at 0. In view of the obvious invariance under complex linear maps, it is invariantly defined in the complex tangent space, and independent of the choice of defining function g apart from a positive constant factor. This explains why it occurs in Corollary 4.1.27. When dX is strictly pseudo-convex, that is, when the form (7.1.3) is strictly positive definite, we can diagonalize the Hermitian form and put g in the standard form n-l

g{z) = -2ReZn

+ '£\zj\'

+

o{\zn

1

In the problem discussed in this section we must respect the direction d/dzi of the differentiation in (7.1.1), so we can only allow composition with ip(z) = {ipi{z),... jipni^)) when dtpj^/dzi = 0 ioi u ^ 1. We shall then say that '^(2:) is an admissible change of variables. First we prove an analogue of Proposition 7.1.2. P r o p o s i t i o n 7.1.3. Let g be a real-valued C^ function in a neighborhood of 0 such that g = 0, dg/dzi = 0 and dg ^ 0 at 0. If c e C^ is real valued, c(0) = 1, and '0(z) = (ijji{z)^... îpn{z)) is analytic at 0 with d^j,{z)/dzi = 0, 1/ 7^ 1, -0(0) = 0 and di;j{0)/dzk = 6jk, then (7.1.2) is valid with a quadratic form Q such that (7.1.4) n

ZZQ = ZZg{0),

n

ifZ = ^Wjd/dzj

+ wod/dzi,

^Wjdg{0)/dzj

1

= 0;

2

(7.1.5) n

n

d/dzi ReY^Wj{dQ/dzj

- dg)/dzj

= 0, ifz = 0, ReJ2'^jdQ{0)/dzj

1

= 0.

2

Conversely, ifQ is a quadratic form satisfying (7.1.4) and (7.1.5) then one can choose c and ifj so that (7.1.2) is valid. Proof. As in the proof of Proposition 7.1.2 we have at the origin d'^/dzjdzk = d'^/d(jd(k and d^dzidzj

= d^/dCidCj

-{-Y^dîjJdzidzjd/dC^ i/=i

- dVdCidCj +

dîlJi/dzidzjd/dCi

LOCAL ANALYTIC SOLVABILITY FOR d/dzi

357

Since dg{0)/dCi = dg{0)/dzi - 0 we obtain (7.1.4) and (7.1.5) if (7.1.2) holds. To prove the converse we assume as in the proof of Proposition 7.1.2 that Q{Z) = —2xn -\-0{\z\'^). The argument proceeds without change apart from the fact that we cannot eUminate the terms in Ai which depend on zi but obtain n

n —1

(7.1.6) {cg){ip{z)) = -2Re2;n + 2 R e ( z i ^ ' a , - 2 : , ) + ^ 1

HjkZjZk'hoilzl^),

j,k=i

where ^ means that the term with j = 1 shall be multiplied by | . The coefficients here are fixed by (7.1.4) and (7.1.5). In fact, with the notation in (7.1.4) we have n—1

n—1

ZZg{0) = 2_\ HjkWjWk + HIIWQWO + 2Re \ J ajWjWQ j,k=l

1

which determines the coefficients in (7.1.6) apart from a^. When w^ = 0 then (7.1.5) is a consequence of (7.1.4), and when Wn = 2i we obtain d/dzi{id/dzn

— id/dzn)g{0)

= ia^

which determines a^. As before, this completes the proof. We shall now drop the condition that dipj{0)/dzk = Sjk- Assuming strict pseudo-convexity, that is, that the Hermitian form X)j,fc=i HjkZjZk in (7.1.6) is positive definite, we can diagonalize it by first introducing \/HiiZi + ^ 2 ~^ HkiZk/y/Hii as a new variable instead of zi and then diagonalize the remaining form in 2:2,..., Zn-i- Then we have reduced g to the form n

(7.1.6)'

n —1

- 2 R e z n + 2 R e ( z i 5 ] ) ' a , z , ) + Y. I'îl' + ^(1^1')1

1

The basis djdz\^... ,d/dzn-i in the complex tangent space of ^""^(0) at 0 is orthonormal with respect to the Levi form and uniquely determined apart from a multiplication of the first element by a constant of absolute value one and a unitary transformation of the others. Thus we can make ai > 0, aj = 0 ioT 2 < j < n and a2 > 0. However, it is less obvious how one can change a„, for it is affected in several ways if we replace Zj by Zj 4- CjZn where Cn = 0. Then the first quadratic form in (7.1.6)' becomes n

2Re((2;i + c i Z n ) ( | a i ( z i -\-ciZn)-\-Yâj{zj

-\-CjZn))).

358


Here 2Re((aiCi + X^2~ ^j^j + o.n)ziZn) is the term involving ziZnaddition n —1

n —1

n —1

1

j,fc=l

1

In

n —1 1

Introducing Zn — 2xn - Zn, we get a new representation of the form (7.1.6)' with an replaced by a^ + aiCi H- Yl2~ ^j^j ~ î- If ^ coefficient a^ with 1 < j < n is not equal to zero, then we can choose Cj so that this is equal to 0. Otherwise we can choose ci so that aiCi — ci -fan = 0 unless | a i | = 1. In fact, if |ai| 7^ 1 then the map C 3 ci H-> aiCi — ci G C is injective, hence surjective. If ai = 1 then ci — ci takes arbitrary imaginary values, so we can reduce a^, to a real value and make an > 0 by changing the sign of zi, if necessary. Summing up, by multiplication with a function equal to 1 at the origin and an admissible analytic change of coordinates, we have reduced g to one of the forms n-l

(7.1.7)

- 2 Re Zn + Re{aizl

+ 2a2ZiZ2) + J ^ \zj\'^ + o{\z\'^)

or

1 n-l

(7.1.8)

- 2 Re Zn + Re(2:^ + 2anZiZn) + ^

|z^f + o{\z\'^), 1

where a^ > 0, j = 1, 2 and a^ > 0. If we multiply ^ by a constant /€^, and introduce K'^Zn and KZJ as new variables for j < n, we see that ai and a2 are not changed but that a^ is divided by K,. The size of a^ is therefore not invariant, so we could take a^ = 1 in the exceptional case (7.1.8) for a suitable defining function. To show that ai and a2 are invariants in general we consider any reduction of g to the form in (7.1.6)', which includes both (7.1.7) and (7.1.8), (7.1.9) n

g{z) = -2ReZn-\-Q{z)-\-o{\z\'^),

n —1

Q{z) = Re(aiz? + 2â^-2:i2:^) + J ^ |2:^f. 2

1

(We shall not really use the complete reductions (7.1.7), (7.1.8).) If Z is defined as in (7.1.4), with Wn = 0, then n —1

ZZQ ^^\wjf

n —1

+ 2ReY^ ajWjWQ.

0

1

The eigenvalues with respect to the Hermitian form Yl^~ \JY1^~^

I'^jP ^^^ 1 ^

| a j p and 1 (if n > 2). For the restriction to the fields Z with

LOCAL ANALYTIC SOLVABILITY FOR d/dzi

359

W2 = • • • = Wn-1 — 0 the eigenvalues are l d i | a i | . It follows that when Q can be reduced to the form (7.1.9) by multiplication with a positive function and an admissible change of coordinates, then 1 ± \j^Y^~ lîP ^^^ ^^^ maxima and minima of ZZg{0)/\Z\'^ with Z as in (7.1.4) and \Z\'^ defined by the Levi form, with orthogonality of the analytic and the antianalytic vector. For the restriction to Z = wid/dzi -\- wod/dzi the maxima and minima are 1 d= |ai|. In particular, it follows that ai > 0 and a2 > 0 are uniquely determined in (7.1.7), and that ZZg{0) > 0 for every Z if and only if ^ ^ ~ |a^.|2 2, is defined at 0 by g < 0 where g e C^ satisfies (7.1.9), and d/dzi is surjective on Ao{X), then n-l

(7.1.10)

EKI'^1-

Proof. Changing the arguments of ^ i , . . . , Zn-i, starting with zi, we may assume that aj > 0 for j < n, and by an orthogonal transformation of .2:2,..., Zn-i we can make aj = 0 for 2 < j < n. First we shall prove that ai < 1. To do so we note that f(z) = {2zn - aizl - 2a2ZiZ2)~^ is analytic in X when |^| is small enough, for if 2zn — aiz\-\- 2a2ZiZ2 then n-l

^(z) = ^ | z , | 2 ( l + o ( l ) ) > 0 1

ii z ^ 0 and \z\ is small enough. If n = 2 we can replace a2 by 0. Assume that ai > 1. If 0 < x^ < 6: and —£ 3 we shall change the definition of / to f{z) = {2zn - a{zi + a2Z2laf

-

Pz^)-'

for some constants a, /3 which have to be determined so that / is analytic in X near 0. When / is singular we have n-l

Q{Z) = Re((ai - a)zl - {al/a + f3)zl) + ^

\zj\\l

+ o(l)),

1

which is positive for small |z| / 0 if (7.1.11)

|ai - a| < 1,

\al/a + /3| < 1.

We choose first a ^ 0 and then jS so that this is true, and / is then analytic in X near 0. To simplify / we introduce new coordinates (j = Zj when j S{t) is in C^ and t ^ go{t) = g{S{t), t) is in C^ at 0,

ô(o) - ^;(o,o), ^[,'(0) = ^;;(o,o) -^;;(o,o)^';,(o,o)-^^';,(o,o), g{s,t) - go{t) =

'^{g':,{S{t\t){s

- S ^ s - S{t)) + o{\s -

S{t)n

Proof. By the implicit function theorem the equation dg{sj t)/ds = 0 has a unique C^ solution s = S{t) at the origin with 5(0) — 0. Differentiation of the equation g'g{S{t),t)) = 0 gives g'ss'^t + Q'st = 0- Since ô(t) = g't{S{t),t) e C^ it follows that ô ^ C^, and that Qo^Qu

+ Qts^t^Qit-QtsQss

Qst-

The second formula follows from Taylor's formula applied to the difference Q{s,t) - Q{S{t),t). P r o p o s i t i o n 7.1.6. If the open strictly pseudo-convex set X C C"^, n>2, is defined at 0 by g 2î = aiL{z')

— L{z'),

where L{z') =

/^CLJZJ, 2

so ^0(^2^') has the Taylor expansion - 2xn + (1 - | a i p ) - ' Re{ai{aiL{z')

-

L{z'))^)

+ 2(1 - | a i p ) - i Re((aiL(zO -

I(^)L{z')) n-l

+ (1 - \a,\Y'\aiL{z')

- L(z')P + E

\'^\' + ''d^'l')"

LOCAL ANALYTIC SOLVABILITY FOR djdzx

363

If n > 2, then the Levi form at the origin becomes

2

2

and it is positive definite if and only if Y^~ l^jP < 1 — lîp, which is the condition (7.1.10)'. Hence z •-> [z^^..., ^n) projects a neighborhood of the origin in X on a neighborhood of the origin in the strictly pseudo-convex domain in C^~^ defined there by ô < 0. T h e fibers of the projection are topologically circles (approximately ellipses) for small z^-^ • > > -^Zfi^ hence connected and simply connected. Thus the proof of Theorem 6.3.2 gives that djdz\ is surjective on ^ o ( ^ ) Having proved the relevance of condition (7.1.10) we introduce a name for it, phrased in an invariant form: Definition 7.1.7. At a point z on the boundary of an open set X C C^, n > 2, such that dX is defined in a neighborhood by ^ < 0, where g e C^^ g\z) ^ 0, g{z) = 0, and dX is strictly pseudo-convex, we shall say that dX is (t, d/dz) convex if t G C^ \ {0} and either (i) {t,d/dz) is not in the complex tangent plane of dX, that is, {t,dg{z)/dz) 7^ 0, or else (ii) the extended Levi form n

(7.1.13)

n

ZZg{z), Z = Y^Wjd/dzj^w^Yîjdldzj, 1

1

is > 0 for all {WQ, . . . , Wn) € C"""^^ with ^ ^ Wjdg{z)/dzj

= 0.

This definition is independent of the choice of defining function g^ and it is invariant under analytic changes of variables preserving the direction of {t^d/dz)^ for this was proved for djdz\ in Proposition 7.1.3 and the discussion following its proof. The invariance is also easily proved by a direct calculation: If ^{r) — {(PI{T), ... ^cpnir)) is a C^ function defined in a neighborhood of the origin in C and with values in C^, such that dip{0)/dT ^weC" and d(p{0)/df = w^t, then (7.1.14)

d^g{cp{T))/dTdf

= ZZg{z)

-^2Re{dg/dz,dîp/dTdf),

when T = 0 and z = (p{0)^ with Z as in (7.1.13). When dtp/df has the direction t in a. neighborhood of 0 the last term drops out if (^, dg/dz) = 0. The invariance of (7.1.13) follows at a point z G dX where (i) is not fulfilled, that is, {t^dg{z)/dz) = 0, for (7.1.13) is then equivalent to (7.1.13)'

d^g{ip{0))/drdf

> 0, if

(^(0) = z, {t,dg{z)/dz)

= 0, {d0

if dX is d/dzi convex at ( as in Definition 7,1.7, ip £ C^ near r £ C, (^2j • • • 5 ^n ^ê analytic functions and ^{T) = z. Conversely, if( G dX and d/dzi is tangent to dX at (, and (7.1.15)' holds when (p^r) = z is on the normal of dX at ( close to z and ^2? • • • ? 2, is defined by g < 0 where g G C^, g' y^ 0 when g = 0, and dX is strictly pseudo-convex, then dX is d/dzi convex in a neighborhood in dX of every ZQ G dX such that d/dzi is surjective on Azo{X). Conversely, this is also sufficient for surjectivity if g{zo + r e i ) is strictly convex as a function ofr at the origin. For the proof we need a lemma. L e m m a 7.1.10. If the open set X C C^, n > 2, has a C^ strictly pseudo-convex boundary near ZQ G dX, and if d/dzi is surjective on Azo{X), then there is a neighborhood U of ZQ such that d/dzi is surjective on Az{X) for every z eUD dX. Proof. If no such neighborhood exists then we can find a sequence Zj G dX of different points such that Zj -^ ZQ and for every j there is a function fj analytic in Uj (1 X for some open neighborhood Uj of Zj for which the equation du/dzi = fj does not have an analytic solution in the intersection of X with any neighborhood of Zj. We can take the neighborhoods Uj disjoint and then choose Xj ^ C^{Uj) equal to 1 in some smaller open neighborhood Vj of Zj. Then / = YliXjfj — '^ is analytic in X if dw = Yl fj^Xj = 9- The (0,1) form g satisfies dg = 0, and it vanishes in VjHX for every j . Since strict pseudo-convexity is stable under perturbations which are small in the C^ sense, we can choose a pseudo-convex open set Y D X such that g remains C^ and B closed in Y when extended by 0 in y \ X , and Zj G Y for every j . Then it follows from Theorem 4.2.6 that the equation for w has a solution in C^{Y). Since / — fj is extended analytically to a full neighborhood of Zj by —w, the equation du/dzi = f does not have an

LOCAL ANALYTIC SOLVABILITY FOR d/dzx

367

analytic solution in V^ fl X , for the equation dv/dzi — w has an analytic solution in a full neighborhood Wj of Zj such that the equation dv/dzi = fj has no solution in Wj ClVj Ci X. Since every neighborhood of ZQ contains (infinitely many) neighborhoods Vj, the equation du/dzi = f cannot be solved in the intersection of X and a neighborhood of ZQ. The proof is complete. Proof of Theorem 7.1.9. The necessity of d/dzi convexity in a neighborhood of ZQ follows at once from Proposition 7.1.4 and Lemma 7.1.10. To prove the converse statement we assume that ZQ = 0 and that g has the form (7.1.9). The convexity assumption means then that | a i | < 1. By Lemma 7.1.5 it follows that the projection X' of X in C"^~^ along the 2:i-axis has a C^ boundary near 0, and the Levi form of dX' is non-negative at 0 by a calculation made in the proof of Proposition 7.1.6. This remains true for all points in a neighborhood of 0 since d/dzi convexity was assumed there too, so it follows from Corollary 4.1.27 that X' is pseudo-convex near the origin. Hence the proof can be finished as that of Proposition 7.1.6. The only remaining problem is thus the case where with the notation in (7.1.9) we have (7.1.10) but | a i | = 1, hence a2 = • • • = fln-i = 0, so that dX is not strictly convex a.t ZQ = 0 in the direction of the zi-axis. We may assume that ai = 1. Then we still have strict convexity in the xi-direction and can clarify the problem by projecting first in that direction. When doing so we shall use an addition to Lemma 7.1.5: L e m m a 7.1.11. Let f e C^, k >2, in a neighborhood of 0 in R ^ and assume that /(O) = 0, dif{0) = 0, and dffiO) > 0, where di = d/dxi. Then f = g -\- h'^ in a neighborhood of 0, where g ^ C^ and h E C^~^ in a neighborhood of the origin, g is independent ofxi, g{0) = h{0) = 0 and

dihiO) = ^dlf{Q)l2. Proof. The equation dif{x) = 0 has a C^~^ solution xi = X{x'), x' — (x2'>..., Xiv) such that X(0) = 0, and from Lemma 7.1.5 we know that the corresponding minimum value g{x') — f{X{x')^x') G C^. Thus r{x) = f(x) - g{x') is in C^ at 0 and 3?r(0) > 0. By Taylor's formula

r{x) - (xi - X{x'))^2{x) ^j{x)=

- (xi -

X{x'))i^^{x),

[ dif{X{x')-^t{x,-X{x')),x'){l-ty-Ut, Jo

i = l,2.

The positive function '^2 is in C'^"^, and {xi — X{x'))il)2{x) = ipi{x) is in C^-K We have f = g-\-h^ where h{x) = {xi - X{x'))^/^jJ^ is in C^-\ When xi ^ X(x') we have h G C^~^ and since h'^{x) = (xi — X(x'))ipi{x) 2h{x)dh{x)

= Mx)d{xi

- X{x'))

+ (xi - X ( x ' ) ) # i ( x )

= {xi - X{x')){^2{x)d{xi

- X{x'))

+ #i(:r)).

368

VII. C O N V E X I T Y AND C O N D I T I O N ( ^ )

Hence dh{x) = ^M^)~HMx)d{xi

- X{x'))

+ #i(a;)),

xi ^

X{x'),

which proves that the derivatives of h of order k — 1 extend continuously to a neighborhood of the origin, so that h G C^~^. If a defining function g e C^ oi X satisfies (7.1.9) with all aj > 0 then one can find another defining function in C^ of the form —2xn H^{zi,...,Zn-i,yn) where n—1

V'(î,...,^n-i,2/n) =

n—1

Re{aizl-\-2âjZiZj-2anyiyn)-^^\zj\'^-ho{\z\'^); 2

1

here Zj = Xj + iyj. li g E C^ for some k>2 then V G C^. This follows if we use the implicit function theorem to solve the equation g{z) = 0 for XnBy Lemma 7.1.11 applied to ip we can write this new defining function in the form (7.1.16) g{z) = -2Xn n— 1

+ g{yi,Z2,

. . . , Zn-l,yn)

+ K^l,

2

• • •,

Zn-l,ynf

"^

2

2 n-1

/l = \ / l + tti (xi + ^

O'jXjKX + fll)) + ^(kDj 2

where ^" = (^2, • • •, ^ n - i ) , ^ G C^ and ft G C^"^ if dX G C^. The local projection of X in R x C"^"^ along the xi-axis is therefore defined by g{yi^z"^yn) < 2xn^ and the fibers of this projection are intervals. When 7/1 = 0 the Levi form of the projection at 0 is n —1

n—1

2

2

2

which is > I X I 2 ~ k i P since ai > 0 and X^2~ I ^ P — 1- Hence itfte projection {{z2,>..,Zn)

G C''~^^(2/l,2;2,...,2;n_l,2/n)

< 2Xn}

is strictly pseudo-convex near the origin if yi is small enough. For suitable Xn the fiber of the next projection along the yi-axis will not be connected if ^ as a function of 2/1 has an interior maximum. As in Lemma 6.3.1 this will lead to another condition for solvability. To formulate it we recall the notion of quasi-convex function introduced in Definition 1.6.3; they are the functions which on any compact interval in the domain take their maximum on the boundary. The negation of this property is expressed by the following:

LOCAL ANALYTIC SOLVABILITY FOR djdzi

369

L e m m a 7.1.12. If the continuous function u in the open interval / C R is not quasi-convex, then there is a sequence J i D J2 D • • • of compact intervals C / , each in the interior of the preceding one, such that u is constant in J = HjJj and u{t) < u{s),

ift G J i , 5 G J;

u{t) < u{s)

ift G UdJj,

s e J.

Proof. By assumption we can find J i so that max^ji u < maxj^ u = M. Choose a point to G J i where u{tQ) = M , and let J be the maximal interval containing to where u = M. Since J is maximal we can choose Jj D J successively with end points in the interior of J j _ i and at distance < 1/j from J , so that u < M in dJj. The intersection is then equal to J and the lemma is proved. The final necessary condition for surjectivity of (7.1.1) is given in the following: T h e o r e m 7.1.13. If the open strictly pseudo-convex set X C C^, n > 2, is defined at 0 by g < 0 where g e C^ satisfies (7.1.16), and ifd/dzi is surjective on Ao{X), then there is a neighborhood of the origin where g is quasi-convex as a function ofyi when 2^2,..., Zn-i^Vn ^^^ fixed. Proof. By (7.1.16) we know that ^ is a strictly convex function of 2/1 if ai < 1. Since surjectivity of d/dzi implies Yl^~ a | < 1, we may therefore assume in the proof that ai = 1 and that aj = 0 for 1 < j < n, thus n-l

g{z) = -2xn + R^{zl - 2anyiyn) + Yl \^j\^ + ^d^l^)' 1

Then there is a neighborhood U of the origin such that (7.1.17) ^

—1

/c(z) = (2 ^ ( z , - Cj)dgiO/dCj

- dgiO/dUzi

- Cif)

& A{U D X),

1

if C ^ f^ n dX. In fact, by Taylor's formula n

e{z) = 2 R e ^ ( z , - QdQiO/dCj

+ Qd^ - C) + oQz - d ^ ) ,

1

where Q^ is a quadratic form with coefficients depeiiding continuously on (^. When 2 S r C ^ j - 0 ) 9 ^ ( 0 / 9 0 = dgiO/dUî - Cif, we obtain by solving for Zn - Cn

eiz) = RedgiO/dUî = Qd^l

- Ci)^ + Q^z

-()

+ o{\z -

Cf)

- Cl, • • • , ^ n - l - Cn-l) + o(|zi - Cll^ + • • • + \Zn-l - C n - l P )

370


with another quadratic form Q^ depending continuously on (. Since Qoiî) •' • ')^n-i) = ^^~ k j P is positive definite, it follows that Q^ is uniformly positive definite for small |C|, which proves (7.1.17). Choose 5 > 0 so small that (7.1.18)

zeU

a g{z)^0,h{zi,...,zn-i,yn)

= QAyi\<S, Yl

\^j\ ^ Ivnl < ^^

l<j"' ^Zn) are connected and simply connected. In fact, the projection zi H^ yi maps a fiber on {7/1; \yi\ < 8^g{yi^6) < 2xn}^ which is an interval because of the quasi-convexity of ^, and xi varies in an interval for each such yi. If the projection

(7.1.20)

x ; , = {z' e C"-^ 321 e c, {z^z') G u,,s n x}

of Xs^s = Ue,6 n X in C^"^ is pseudo-convex, it follows from the proof of Theorem 6.3.2 that the equation du/dzi = f has a solution u G A{Xe,6) for every / G A{X£^s)- Pseudo-convexity is no restriction when n — 1 = 1, so then it follows that the quasi-convexity condition established in Theorem 7.1.13 implies that (7.1.1) is surjective. When n > 2 we shall prove in Section 7.2 that pseudo-convexity of X'^ ^ follows from d/dzi convexity combined with the quasi-convexity in Theorem 7.1.13, so these conditions together imply that (7.1.1) is surjective.

372

VII. CONVEXITY AND CONDITION {^)

7.2. Generalities on projections and distance functions, and a t h e o r e m of Trepreau. In this section we shall discuss some geometrical facts which will allow us to prove the pseudo-convexity of the projection X'^ ^ left open at the end of Section 7.1. In doing so we shall at first simplify notation by working in a real vector space. Let X be an open subset of R ^ , and let Q be an open subset of X x [-1,1]. We want to study u = nQ where TT is the projection X x [—1,1] —> X. Set Q,^ = {x e X] (x,t) G Ji}, and let dfi^{x) be the distance from x to Cfif The distance is of course Lipschitz continuous in x with Lipschitz constant 1, and it is lower semi-continuous with respect to t since ft is open. Set d{x) = sup dci^(x). te[-i,i]

Since cj = Ufit? it is clear that d{x) < d(^{x) < dx{x) where ^^^(a;) (resp. dx{x)) is the distance from x to Co; (resp. CX). Examples such as n = {{xi,X2,t)]X2

> txi - ^ ^ } ,

u = {(xi,a;2);x2 > -\xi\

- 1},

show that we may have d{x) < d^(x), in this case when X2 > \xi\ — 1. However, we have: P r o p o s i t i o n 7.2.1. Assume that t-axis and that dQ^{x) is a continuous X e Xj unless d{x) = dft^{x) for some point in Cfit with minimal distance to

D. is convex in the direction of the function oft. Then d{x) = d^^{x), t for which there is more than one x.

Proof. Since d{x) < d^{x) < dx{x)j we may assume in the proof that 0 < d{x) < dx{x). The set T = {t e [-l,l];dn^{x) = d{x)} is a closed interval, for dQ^ is a continuous quasi-concave function of t by the convexity of 0 in the t direction. Assume that d{x) < c?^(a:). Then B = {y e R ^ ; \y - x\ < d{x)} (G CJ C X, and B C ^t when t E T. Assuming that B n Cilt consists of a single point y{t) for every t G T we shall prove a contradiction. By the convexity in the t direction we have y{t) G {^/(T); r G dT} for every t e T, and T 3 t \-^ y(t) is continuous since CJl is closed, so y{t) is independent of t G T. Since B C uj this point y belongs to ilfi for some ti G [—1,1]- Let to be the end point of T closest to ti. Then {B \ {y}) nf^ti C r^t for t = to and for t = ti , hence for all t G [to, î], again by the convexity of fi in the t direction. Since K = B Ci Cfiti C ^t^ and K is compact and fi is open, we have K C ilt if \t — to\ is small enough. Hence B \ {y} C fit if ^ G [Ôjî] and \t — to\ is sufficiently small, which contradicts that to is the end point of T closest to ti. This completes the proof.

PROJECTIONS AND A THEOREM OF TREPREAU

373

The hypotheses in Proposition 7.2.1 do not imply any smoothness of duj, for fi could be the cylinder a; x [—1,1]. From now on we shall assume that n = {{x, t) eX (7.2.1)

F e C^X

X [-1,1]; F{x, t) < 0},

X [-1,1]), F^Ô

where

when F = 0.

This implies that (IQ^ {X) is a continuous function. It suffices to prove upper semi-continuity with respect to t at a point where F{x^ t) 0 for small e > 0. This implies y + eF^îy, t) E Cfi^ for |5 — t| < ^£5 and it follows that dQ^{x) < dct^{x) -f €\Fy{t^y)\ then. For the preceding argument we only used that F^{x^t) is defined and continuous in X x [—1,1] but we shall also use the second derivatives with respect to x in the proof of the following proposition, which is related to Corollary 2.4.4 but much less obvious when fi is not a convex set. P r o p o s i t i o n 7.2.2. Assume in addition to the convexity condition in Proposition 7.2.1 that (7.2.1) holds. Then the boundary of u in X has a neighborhood Y such that for every x e Y D u there is a unique closest point p{x) E duj, and d^{x) — sup^^[_i ^j dQ^{x). Proof. Choose a compact set K C X and a compact neighborhood K C X. In view of (7.2.1) we can choose r > 0 smaller than the distance from K to CK so that K nQf for every t E [—1,1] is covered by balls C ftt of radius r. (See Proposition 2.4.3.) Then ^ fl a; is covered by balls C a; of radius r. At a point x e KHUJ with d^{x) < r we have dQ^{x) < r for every t E [-1,1]. If dQ^{x) = d{x), and p E dQt, \x - p\ = da^{x), then p e K, and it follows that x — p is orthogonal to the tangent plane of dilf at p, so the ball B contained in Ot with radius r having p on the boundary has its center a.t p-{- {x — p)r/\x — p\. Since r > \x —p\ it follows that p is the only point in Cfit with |rr: — j?| = dQ^{x). Hence d{x) — d^{x) by Proposition 7.2.1, and p = p{x) is the unique point in Co; closest to x. Exercise 7.2.1. With the notation in Proposition 7.2.2 prove that \p{x) — p{y)\ < 3|x — y\ for (x^y) in the intersection of a; x a; and a neighborhood of the diagonal in y x y . Conclude that the distance function div ^ C^'^{Y n a;) and that \d'^\ < A/d^j almost everywhere in y fla;. The local projection Ct in the Xi-direction of the set X in Theorem 7.1.13 is defined by ^(2/1,2:2,... ,2:n-i,2/n) < 2a;n, so (7.2.1) holds and the convexity in the yi-direction assumed in Proposition 7.2.1 is equivalent to the quasi-convexity in Theorem 7.1.13, with t equal to a constant times 2/1 and X denoting the other variables. However, since we are not able to prove that the boundary of the projection a; of fi in C"^"^ is in C^, we cannot

374


examine if a; is pseudo-convex by studying the Levi form of doj. Instead we shall use Theorem 4.1.30 to show that the boundary distance d^^ satisfies condition (iii) in Theorem 4.1.19, which will give the main result of this section: T h e o r e m 7.2.3. Let X C C"^ he an open pseudo-convex set which has a C^ strictly pseudo-convex boundary in a neighborhood of ZQ G dX. Then d/dzi is surjective on Azo{X) if and only if dX is d/dzi convex in a neighborhood of ZQ in dX and the quasi-convexity condition in Theorem 7,1.13 is fulfilled. Proof. The necessity was proved in Theorems 7.1.9 and 7.1.13. The sufficiency follows from Lemma 7.1.1 unless d/dzi is in the tangent plane of dX at zo, so this will now be assumed. Recall that the condition in Theorem 7.1.13 means that if X^,^ = XnUe^s with 11^,6 defined by (7.1.19), then the fibers of X^^s 3 z \-^ z' = {z2^... ^ z^) G C^"-^ are connected and simply connected \i e > C6 and e is small enough. As observed at the end of Section 7.1, the sufficiency follows from the proof of Theorem 6.3.2 if we prove that the projection X'^ ^ of X^ê in C^~^ along the zi-axis (see (7.1.20)) is pseudo-convex at 0. The projection can be made in two steps, as discussed in connection with (7.1.16). In the first step we project along the xi-axis to a set Xe,8 = { ( ^ 1 , ^ 0 G ( - ^ , ^ ) X D'^~^]g{yi,Z2,...,Zn-\,yn)

< 2Xn},

where D^ = {w ^ C; \w\ < 6]. (The set is independent of e when e > C6.) Lemma 7.1.11 gives that g G C^, and the condition in Theorem 7.1.13 states that Xe^6 is convex in the î-direction. In Proposition 7.1.8' we studied the boundary distance d]^{z) in X for fixed zi. It follows from Proposition 7.2.2 that the boundary distance d{yi^z') in X^^sivi) = i^'] {Vii^') G Xsê] for fixed yi is equal to max^^ d\{xi-\-iyi^z') close to the boundary oiX^ê. Now —d^x{zi^z') is the restriction to X of a C^ defining function for X near 0, so — d^c is strictly convex with respect to x i . By Lemma 7.1.11 the maximum is therefore attained for xi = ip{yi^z') where ip G C^^ and d{yi,z') G C^ in X^ê close to 0. As observed after (7.1.16) the set X^^sivi) is pseudo-convex, so it follows from Theorem 4.1.19 (iii) that u{yuz')

= -\ogd{yi,z')

= -\ogd\{ip{yi,z'),z'),

{yi,z')

G X^,^,

is a plurisubharmonic function of z' for fixed yi. The convexity of X^^s in the yi direction means that u{yi,z') is a quasi-convex function of yi for fixed z'. If du{yi^z^)/dyi = 0 then dd]^{(p{yi^z') + iyi^z')/dyi = 0,

THE SYMPLECTIC POINT OF VIEW

375

and since dd]^{xi 4- iyi,z')/dxi = 0 when xi = (p{yijz') by the definition of (f^ we conclude that dd]^{zijz')/dzi = 0 when zi = (p{yi^z') + i^/i, if du{yi^z')/dyi = 0. By Proposition 7.1.8', we conclude from the d/dzi convexity that (4.1.19)' (with t = yi) is fulfilled for the function u^ which —> H-oo at the boundary points with \yi\ < 6. Now it follows from Theorem 4.1.30 and the remark following it that U{z')=

inf ^

uiyuz')

is plurisubharmonic in X'^ ^ near the origin. Since

U{z')^-\ogdx'Jz') in a neighborhood of the origin, again by Proposition 7.2.2, it follows from Theorem 4.1.19 (iv) that X'^ ^ is pseudo-convex near the origin. The proof is complete. 7.3. T h e s y m p l e c t i c point of view. In this section we shall rephrase Theorem 7.2.3 in terms of symplectic geometry. This sets the stage for application of the transformation theory of microlocal analysis which we shall sketch briefly in Section 7.4. For the basic notions of symplectic geometry we refer to ALPDO Section 6.4 and particularly Chapter XXL If M is a complex manifold of dimension n, then the (1,0) forms on M can be written ^ ^ Cjdzj in terms of local coordinates Zi^... ^Zn. They can be considered as sections of an analytic vector bundle TT^ QNM with the local coordinates (2:1,... ^Zn-,Qi^ - - - -,Cn)- Foi* the sake of brevity we shall denote it by T^M hoping that this will cause no confusion with the cotangent bundle of M as a real manifold. On T^'M there is a natural analytic one form LJ = ^ ^ Q dzj^ independent of the choice of coordinates, with differential n

(7.3.1)

a = diJ = y ^ d(j A dzj.

It is exact and symplectic, that is, a skew symmetric non-degenerate bilinear form on the fibers of r(i,o)(T*M). Since we shall only deal with local questions we assume from now on that M = C^ and use the standard coordinates 2:1,..., 2:^ in C"^. However, the preceding comments will show that our constructions are invariant under local analytic changes of such coordinates. Let X C C^ be an open set defined by ^ < 0 where g e C^ and dg ^ 0 on dX. A real one form R e ^ ^ Cjdzj vanishes on the tangent plane at 2; G dX if R e ^ ^ Cjdzj = 0

376


when R e ^ ^ dg/dzjdzj We shall write (7.3.2)

E = N;^

— 0, that is, when C = jdg{z)/dz

- {{zX);z

edXX

= idQ{z)/dz,j

for some 7 E R.

> 0} C T^M,

where dg = Yl^ dg/dzjdzj, and refer to N^-^ îs the outer conormal bundle. The restriction of a; to S is jdg, so the restriction of a to S is

(7.3.3)

a^ = dj Adg-\- jddg = dj Adg-^j

^

Qjk^Zk A dzj.

Here ^^^ = d'^g/dzjdzk-) and we shall use similar notation in what follows without explanation. The equation 0 = dg = dg + dg valid on S shows that dj/\dg = djA {dg — dg)/2 is purely imaginary on E, and so is the last sum in (7.3.3). This means that E is R-Lagrangian, that is, the restriction of Re (J toY, vanishes^ where n

n

Re cr = y^(cf^j A dxj — drjj A dyj)^

liaa = /^(ci^j A dyj + drjj A dxj).

1

1

Here z = x -{-iy and C — ^ + ^^- To study the restriction of I m a to E it is convenient to use the coordinates given by the following lemma: L e m m a 7.3.1. After a suitable complex linear change of there is a defining function g at 0 E dX of the form

coordinates

n-l

(7.3.4)

g{z) = -2xn-^2ReA{z)

+ X l îl'^^l^ + ^d'^l^)'

where A is an analytic quadratic form and Xj G R. Proof. The first step is to take dzn = —dg at 0, which reduces g to the form —2xn + 0(|/2:p). Replacing Re{azjZn) by 2xn Re{aZj) — Re{aZjZn) in the quadratic terms as in the proof of Proposition 7.1.2, we can write n-l

g{z) = -2xn{l

+ L)^2ReA{z)-\-

Y^

hjkZjZk + o{\z\'^),

j,k = l

where A is an analytic quadratic form and L is a linear form. The defining function g{z)/{l-\-L) has the required property after a linear transformation of the variables z' = {zi,... ^ Zn-i) diagonalizing the hermitian form.


377

With the coordinates and defining function in Lemma 7.3.1, and taking zi^... ^Zn-i^Vnil as coordinates on S, we can now write at the origin (7.3.5) n—1

n—1

i~âj: — dyn Adj -{- i~^j ^

^jdzj A dzj = dyn Adj -\-2j Y ^ ^jdxj A dyj.

1

1

(Recall that dg = {dg — dg)/2 on E and note that dg — dg = —dz^ + dzn = —2idyn at 0.) This is non-degenerate if and only if Xj 7^ 0, j = 1 , . . . , n — 1. Thus E is I-symplectic, that is, the restriction of I m a to E is non-degenerate, if and only if the Levi form of dX is non-degenerate. In particular, E is /-symplectic if dX is strictly pseudo-convex. When \j 7^ 0 for every j we obtain at the origin the standard symplectic form in the variables (7,1/1,... ,yn-i;^n527Airz;i,..., 27An-i^^n-i)- Now recall that when the two form Z~^(J$] is non-degenerate, it identifies tangent and cotangent vectors of E: to a cotangent vector r of E corresponds a tangent vector Hr such that for every tangent vector ^ of E (at the same point) i'â^(t,Hr)

= {t,T).

If u and V are C^ functions in E one writes Hu and H^ for the Hamilton vector fields corresponding to du and dv^ and one defines the Poisson bracket of u and v by {U,V]Y,

= HuV = i~âY,{Hu,Hy)

=

In terms of the coordinates ( r r i , . . . , Xn-i^yi-,..., {0,jdg(0)/dz) from the standard case {u,v}^

= du/dyndv/dj

-

-{v,u}j:-

^n? 7) on E we obtain at

du/djdv/dyn n-l

H- yj(5ii/9xj5i'/92/j —

du/dyjdv/dxj)/2jXj.

1

In particular, still at (0,79^(0)/9z), (7.3.6) {zj,Zk}j: = {zj.Zk}^ = 0, {zj,Zk}j: {Zn^Zj}^ = {Zn.Zj}^

îSjk/jXj, = 0,

j . A: = 1 , . . . , n - 1,

j == l , . . . , n ,

for Zn = iyn + 0(|>2:p) on dX. Let p{() = ^ ^ CjQ be the symbol of a first order differential operator with constant coefficients tangent to dX at 0, which means that Cn = 0. The restriction to E is n—1

E

n—1

Cj7dg{z)/dzj

= J Z (^jlid^/dzj

+ XjZj) + o{\z\),

378


and since all terms vanish when z = 0 we obtain at (0, jdg{0)/dz) d'Â/dzjdzk n —1

n —1 n —1

j=i

k=i

if Ajk =

j=i

We shall interpret this geometrically when dX is strictly pseudo-convex, that is, Xj > 0 for j = 1 , . . . , n - 1. To do so recall from Definition 7.1.7 that dX is called p{d/dz) convex at 0 if in addition (7.3.7) d'^girw -f- fcj/drdf > 0, r = 0, for arbitrary w e C^ with Wn = 0. Now n —1

g{rw-\-fc) =Re

^

n —1

Ajk{rWj-\-fCj){TWk-^fCk)-^^Xj\TWj-\'fCj\'^-ô{\T\'^),

j,k = l

1

so (7.3.7) means that n—1

n—1

2Re J ] AjkWjCk + J ] ] A^(|i(;^f 4- |cjf) > 0,

?/; G C ^ " \

or if we minimize with respect to w j=i

j=i

fc=i

Since (t, d/dz) convexity is invariant under complex linear transformations, we have proved: P r o p o s i t i o n 7.3.2. If X is strictly pseudo-convex with C^ boundary at ZQ and p{d/dz), p{() — Yl^^jCj îth constant Cj, is tangential at ZQ, then dX is p{d/dz) convex at ZQ if and only if i~'^{p(C)jP(C)}s > 0 over ZQ, where S = Ng^. Equivalently

{Rep(C),ImKC)}i: f{y{t))

< 0 for t > 0.

Ifw is another Lipschitz continuous vector field such that (7.3.9) (7.3.10)

{w,df) 0, for otherwise the integral curve of v starting at z would enter B C ZF. The Lipschitz continuity oi w implies {w{y{t)) — w{z),y{t) — z) < Cg{y{t))j so it follows that the right derivative of g{y{t)) is < 2Cg{y{t)), so g{y{t))e~'^^* is a decreasing function (Lemma 1.1.8). Hence the orbit cannot have started in F , so no sign change from — to -|- is possible. Using Lemma 7.3.4 we shall now prove the main result of this section: T h e o r e m 7.3.5. Let dX G C^ be strictly pseudo-convex at ZQ G dX. The restriction p of (i to S satisfies condition (^) in a neighborhood of (^2^0? Co) ^ S, where P{ZQXO) = 0, if and only if dX is d/dzi convex in a neighborhood of ZQ and the quasi-convexity condition in Theorem 7.1.13 is fulfilled. Combination with Theorem 7.2.3 gives: Corollary 7.3.6. IfdX G C^ is strictly pseudo-convex at ZQ G dX and d/dzi is tangent to dX at ZQ, then d/dzi is surjective on Azo{X) if and only if the restriction pof(i to H — N^^ satisfies condition (*) over ZQ. Proof of Theorem 7.3.5. We assume that ZQ = Q and write Q in the form (7.1.16) as in Theorem 7.1.13, thus dh/dxi > 0. Over a neighborhood of 0 the manifold E is parametrized by a;i,yi,^,7, where 9 = {x2,..., Xn-1, y2,. • •, Vn), and since p = jdg/dzi = ^jidg/dxi idg/dyi) we have (7.3.11) Rep = ReCi — jhdh/dxi^ Imp = Imî = —^{hdh/dyi -f ^dg/dyi). Thus dRep ^ 0, so if condition (*) holds then Imp = —\^dg{yi^9)/dyi does not change sign from — to + along the orbits of the Hamilton field H of R e p in the submanifold EQ of E where R e p = 0, that is, /i = 0. It is parametrized by 2/1,^,7. By Proposition 7.3.2 it follows that dX is d/dzi convex in a neighborhood of 0. We want to conclude using Lemma 7.3.4 applied with / replaced by —dg/dyi that —dg{yi^0)/dyi has no such sign change for fixed 9 and increasing yi, which will prove that g is quasi-convex.


381

Let V be the vector field in E defined by

V =

dh d dxi dyi

dh d dyi dxi

in terms of the local coordinates xi^yi,6,j in E. Since Vh = 0 it follows that V restricts to a vector field in EQ, equal to dh/dxid/dyi in terms of the local coordinates yi,0,j there. Since dh/dxi > 0 our task is therefore to prove that —dg{yi^0)/dyi has no sign change from — to + in the positive direction on the orbits of V where h = 0. First we must verify the condition (7.3.9) with / = —dg/dyi^ that is, prove that (7.3.12)

Vdg/dyi

= dh/dxid^g/dyj

>0

when dg/dyi

= 0, h = 0.

At a point where h = 0 and dg/dyi = 0, 2xn = g, the vector d/dzi is tangent to dX, so the d/dzi convexity says in particular that the Hessian of g with respect to x i , y i , as a function in C^, is non-negative. The first derivative of g in the direction dh/dxid/dyi — dh/dyid/dxi is equal to dh/dxidg/dyi, and since dg/dyi = 0, the second derivative is equal to {dh/dxiYd'^gldyl > 0, which proves (7.3.12). The condition (7.3.10) concerns points where not only h = 0 and dgjdyi = 0 but also ddg/dyi = 0. We must then compare the Hamilton field if of R e p = jhdh/dxi with the vector field V. The Hamilton field is defined by (7.3.13)

i-â^{t,H)

= {t,d{jhdh/dxi))

=

jdh/dxi{t,dh),

when t is a tangent to E. We want to compare this with n

(7.3.14)

n

rVs(i,y) - r i Y,{t,d(:j){v,dzj) - r^ J]( ioY h = 0 and dg/dyi

1,

(y,dzi) = idh/dxi

-

dh/dyi,

= 0. We have

Ci = idg/dzi and since ddg/dyi

1

= j{2hdh/dzi

-

-idg/dyi),

= 0 it follows that

(t,dCi) = 2j{t,dh)dh/dzi

=-f{t,dh){dh/dxi

-idh/dyi).

382


For the same reason we obtain {V, dCj) = jV{2hdh/dzj

+

dg/dzj) = jVdg/dzj

= jdh/dxid^g/dzjdyi

= 0.

Hence (7.3.14) gives (7.3.15)

r VE(t, V) = ^{{dh/dxiY

4- {dhidyif){t,

dh).

Comparing (7.3.15) and (7.3.13) we conclude that (7.3.16)

{{dhldxif

+ {dh/dyif)H

=

dh/dxiV,

when /i = 0, dg/dyi = 0 and ddg/dyi = 0. The factors of H and V in (7.3.16) are positive, so it follows from Lemma 7.3.4 that —dg{yi^9)/dyi has no sign change from - to -h for increasing yi and fixed 0^ that is, g is quasi-convex. Conversely, if dX is d/dzi convex, then Proposition 7.3.2 proves that { R e p , I m p } s < 0 when p{() — 0, which implies the analogue of (7.3.12) with V replaced by H. We can then deduce the full condition (^) from the quasi-convexity of ^ by applying Lemma 7.3.4 again. The proof is complete. 7.4. T h e microlocal transformation theory. In Section 7.3 we saw that if X is an open subset of C^ with C^ strictly pseudo-convex boundary, then NQ^ is i?-Lagrangian and /-symplectic. This is also true for N^n. In fact. Re Yl^ (yjdzj vanishes on R^ if and only if Re C = 0, so

Ar5.n = {(a;,iO;^,eeR"} = r*(R"), and the restriction of a to N^n is i 2_\ d^j A dxj, which is i times the standard symplectic form in T*(R'^). There is a local symplectic equivalence between these cases: P r o p o s i t i o n 7.4.1. Let X C C^ he an open set such that dX is real analytic and strictly pseudo-convex in a neighborhood of ZQ G dX, and let iô^ Co) ^ ^dx- ^^^ Co ^ R-^ \ {0}. T i e n there is a holomorphic symplectic map XJ homogeneous of degree one, from a conic neighborhood of{0,iô) to a conic neighborhood of (ô? Co) ^ ^ a x w-iiich maps N^n to NQ-^. Proof. NQ^ is a real conic symplectic manifold with symplectic form all. It has dimension 2n at (ô^Co)? and so has T*(R'^) at (0,Co)- It is

THE MICROLOCAL TRANSFORMATION THEORY

383

classical that there is a homogeneous symplectic transformation Xr of a neighborhood of (0,ô) ^ T*(R'^) on a neighborhood of {zo,Co) ^ ^dxThe standard proofs (see e.g. ALPDO, Theorem 21.1.19) show that Xr can be chosen real analytic when dX is real analytic. Thus x(z, Q = Xr{z, C,/i) is well defined in a neighborhood of (0,2ô) G C^ 0 C " . Since x*2:p — 1 < 0 } , for a / a ^ ( | I m . ^ | 2 - l ) = - i l m ^ . Next we shall prove a simple but important fact concerning i?-Lagrangian and /-symplectic planes which will allow us to recognize conormal bundles of strictly pseudo-convex surfaces. P r o p o s i t i o n 7.4.2. Let T be a real linear subspace ofT^C^ of real dimension 2n, which is R-Lagrangian and I-symplectic. Then T*C"^ = T © iT, and (7.4.1)

X -h 27/ H-> a{x -\-iy^x — iy) — —2ia{x^y)^

is a non-degenerate {n,n).

Hermitian

symmetric

x^y ET^

form in 7*0"^ with

signature

Proof, lit eTniT then 0 = Rea{t/i, s) = lma{t, s) for every s ET since T is i?-Lagrangian, hence t = 0 since T is /-symplectic. This proves that T*C^ =TîT. The quadratic form (7.4.1) is real valued since T is i?-Lagrangian, and it is Hermitian symmetric with polarization X 4- iy, x' + iy^ •-> o{x -h iy., x' — iy')

384


complex linear in x-\-iy and antilinear in x' -\-iy\ for this reduces to (7.4.1) when X — x' and y — y'. If x'^y' G T and a{x + iy^x' - iy') = 0 for all x,y ETJ we obtain with y = 0 that 0 = Rea{x^x'

— iy') = Im.a{x^y'),

0 = Im(j(rr, a;' — iy') =

Iuia{x^x'),

for all x ^ T^ so x' = y' = 0. Hence the form (7.4.1) is non-degenerate. Since it vanishes for x^y in an /-Lagrangian subspace of T of real dimension n, it cannot be positive (negative) definite in a complex subspace of dimension > n, so the signature is (n^n). Thus the signature of the form gives no information unless we restrict the form to a subspace. However, the restriction to the tangent of the fiber of the cotangent bundle can be used to characterize the conormal bundles of strictly pseudo-convex surfaces: P r o p o s i t i o n 7.4.3. IfX = {z e C^; Q(Z) < 0} where g e C^, ^(0) = 0, ^'(0) / 0, and X is strictly pseudo-convex at 0, then S = NQ-^ is in C^ and over a neighborhood of 0 (i) S is R-Lagrangian and I-symplectic; (ii) ifT is a tangent plane ofl^ then T fl (0 © C^) is a reai line; (iii) if the hermitian form (7.4.1) corresponding to T is restricted to 0 ® C^, then it is positive semidehnite with kernel equal to the complex line generated by the line in (ii). Conversely, ifT, C T*(C^) is a conic C^ manifold satisfying these conditions then E = N^-^ with X as above, and dX is real analytic ifT, is real analytic. Proof. If E = NQX sind X is strictly pseudo-convex then (i) was proved in Section 7.3, and (ii) follows from the definition Nex = {{z,ldg{z)/dzy,j

> 0,e{z) = 0}.

To prove (iii) we may assume that g is of the form (7.3.4). A tangent to AT*^ at {0,dg{0)/dz), dg{0)/dz = ( 0 , . . . , 0 , - 1 ) , is defined by a tangent vector t G C"^ to dX and a number 7 G R as {t,jdg{0)/dz + $ ( t ) ) , where n

* i W = J2d^g{0)/dzjdzktk

n

•^^d^g{0)/dzjdzkh,

j = 1,... ,n.

fc=i fc=i

Given ( G C^ we want to find (^,7) and (^',7') so that the sum of the vector corresponding to (t, 7) and i times the vector corresponding to (t', 7') is equal to (0,C)- This requires first of all that t -\- it' = 0, and since Ret^ = R e t ^ = 0, this implies that t^=t'^-= 0. The remaining equations are n-l

0 = 2 ^ d'^g{0)/dzjdzkik 1

= '^>^jij,

j = 1 , . . . , n - 1; (^ == - 7 - ij'^


385

for i-\-iF = 21 Thus t - i t ' = 2t = ( C i / A i , . . . , C n - i / A n - i , 0 ) . Since the tangent vector corresponding to t — if has the base projection t — it'^ the form (7.4.1) becomes

{c,i-zt') = EioiVAi When dX is strictly pseudo-convex, the form is therefore positive semidefinite in 0 0 C^, with kernel generated by dg{0)/dz; in general the number of positive and negative eigenvalues is the same as for the Levi form of dX. Conversely, assume that E satisfies (i), (ii), (iii) in a neighborhood of (0, a ) , a = ( 0 , . . . , 0, - 1 ) . By (ii) the projection T, 3 (zX) ^ z e C gives locally a C^ manifold EQ C C"^ of dimension 2n —1, for it can be obtained by first projecting to {z, Re (n) with bijective differential and then intersecting the resulting conic surface with the plane Re^n = —1- Since E is conic and i?-Lagrangian, it follows that Re{(,dz) = 0 in the tangent plane of E, so if {z,() G E then 2: G EQ and Re{(,dz) = 0 on EQ, thus ( = jdg/dz for some real 7 if ^ G (7^ is a defining function for X with dX = So near 0. We choose g so that dg{0)/dz = a and g = —2xn + ^1 where ^1 is independent of Xn- (See the discussion after Lemma 7.1.1L) Then (n = 7 ( - l — \idg/dyn) so 7 = — Re ^^ and dgjdz — —C/ReCn which is a C^ function, so ^ G C^. If E is real analytic it is clear that g is real analytic. We have proved that E C A^^X' ^^^ since the dimensions are equal we have equality at ( 0 , a ) . By a linear change of variables in C^ we can reduce g to the form (7.3.4). We saw in Section 7.3 that A^ 7^ 0 for j = 1 , . . . , n — 1 since E is /-symplectic, and the first part of the proof showed that all A^ are positive when (iii) is fulfilled. This completes the proof. We can now prove the second geometrical fact that we need. P r o p o s i t i o n 7.4.4. Let p be a holomorphic function homogeneous of degree one in a conic neighborhood of (0,iô) where ^0 ^ R-^ \ {0}, and assume that dp AUJ ^ 0 at (0, iô)? that is, that dp is not proportional to {dxj ^0) ^^ (0, ^ô)- T i e n there exists a holomorphic homogeneous canonical transformation x defined in a conic neighborhood U of (0, Z(ô) ^nd a strictly pseudo-convex real analytic hypersurface dX C C"^, (O5C0) ^ âx^ Co = ( 0 , . . . , 0, —1), such that x maps U fl N^n on a neighborhood of (0, Co) in N^x ^«d x*Ci = V' Proof. Prom the holomorphic analogue of Theorem 21.1.19 in ALPDO it follows that there is a canonical transformation x such that x*Ci — V- Thus x{U n N^n) is a conic i?-Lagrangian and /-symplectic submanifold of C^ of real dimension 2n, but there is no reason why it should be of the form NQ^. However, that can be achieved by another canonical transformation

386


which does not affect the (i coordinate. It is given by the following lemma, which will complete the proof. L e m m a 7.4.5. Let S be a conic C^ R-Lagrangian and I-symplectic submanifold ofT*C^ of real dimension 2n in a neighborhood of z = (0, a), a == ( 0 , . . . , 0, —1). Then there is a canonical transformation of the form

(7.4.2)

Xi^,0 = {z + df iO/dCO,

where /(C)Cn is a quadratic analytic polynomial in (' = {(i,... ,(ri-i), such that x ( S ) is the outer conormal bundle of a strictly pseudo-convex C^ hypersurface dX C C^. If E is real analytic then dX is real analytic. Proof. The maps (7.4.2) form a group preserving i; composition is equivalent to addition of the functions / , and the inverse is obtained by changing the sign of / . By Proposition 7.4.3 it suffices to show that x can be chosen so that x'(;S)T satisfies conditions (i)-(iii) there, if T is the tangent plane of E at z. The condition (i) is automatically fulfilled by the symplectic invariance. We have

x'iz)-'io,e) = {Ae,e), oec^, where A — —9^/(a)/9(^ is an arbitrary complex symmetric matrix with A^n = ^nj = 0, j = l , . . . , n . Thus x'(^)~H{0} ® C"") is an arbitrary complex Lagrangian plane L containing z and transversal to C^ 0 {0}. We have to show that for a suitable choice of A the intersection with the tangent plane T = TgE is equal to Hz and the form (7.4.1) corresponding to T is positive semidefinite in L with kernel Cz. Since T is /-symplectic, we can choose a real symplectic basis e ^ , . . . , e^^, Si^... ^Sn for T with the symplectic form a/i, so that Sn = z. Thus a{ej,ek)

= 0, a{sj,ek)

= 0, a{ej,ek)

= iSjk,

j,k =

l,...,n.

Then the complex vector space L spanned hy Cj — isj, j = 1 , . . . , n — 1, and Sn is Lagrangian, and L fl T = Hsn for if X^^~ ^ji^j ~ ^^j) + ^n^n ^ T^ then 5^^~ {{Im.Cj)ej —(ReCj)£j)-^{Im.Cn)€n — 0 since T f l z T ^ {0}, which implies that Ci = - • - = c^-i = 0 and that Imc^ = 0. We have a{ej -iSj.Ck

-i-isk) = -i{(7{ej,ek)

+ a{£k,ej))

= "^^jk, cr(e^- -iej.En)

= 0,

for j , A: = 1 , . . . , n — 1, so the form (7.4.1) has the desired signature. The only remaining problem is that L may not be transversal to C^SJO}. Now the complex Lagrangian planes in C^ 0 C^ containing z are contained in the hyperplane where Zn — 0, the a orthogonal hyperplane of z. Hence


387

they can be identified with their Lagrangian projections in C"^"^ 0 C^~^ along the ZnCn plane. The Lagrangian planes in C^~^ 0 C^"-'^ which are transversal to C'^~^0{O} are dense in the set of all Lagrangian planes there. (See Corollary 21.2.11 in ALPDO.) Thus we can choose L' Lagrangian with z E L' so close to L that the conditions (ii), (iii) in Proposition 7.4.3 remain valid for L' and at the same time L' is transversal to C"^ 0 {0}. This completes the proof. We shall now use basic microlocal techniques to draw important conclusions from the results of Section 7.3. It is not possible to develop these methods here, so the rest of the section will assume some familiarity with the basic notions of that theory. First recall that the sheaf C over R'^ is a sheaf on the cosphere bundle S'*(R'") = R^ X S ' ^ - \ and that the stalk of C at {x,i) is the quotient of the space of analytic functional ^'(R"^) in C"^, carried by some compact subset of R'^, by those for which {x, ^) is not in the analytic wave front set. (For a definition see ALPDO, Chapter IX, or the original in Sato-KawaiKashiwara [1]. The definition of the analytic wave front set in ALPDO, Theorem 8.4.11, is in the same spirit as Theorem 7.4.6 below.) If X is a pseudo-convex open set in C^ then the stalk at ZQ G dX of the sheaf A^'^ on dX consists of functions analytic in U H X for some neighborhood U of 2:0, modulo those which can be extended analytically to a neighborhood of F fl X for some other neighborhood V of ZQ. This is a modification of the notation used in Sections 7.1-7.3. However, if the equation du/dzi = f with / analytic in U Ci X has a solution in 1^ fl X modulo functions analytic in a neighborhood of .2:0 5 then there is an exact analytic solution in a smaller neighborhood, for the equation dv/dzi = g has an analytic solution in any ball containing ZQ where g is analytic. Hence the results of Sections 7.1-7.3 concerning the operator d/dzi acting in Azo{X) remain valid with our new definition. We shall need the following important result from microlocal analysis: T h e o r e m 7.4.6. Let i^ e W \ {0} and ZQ G C ^ , let X be an open set in C^ with ZQ G dX which has real analytic strictly pseudo-convex boundary at ZQ, and let x be a homogeneous holomorphic symplectic map from a neighborhood of (0,Z(ô) in T*C'^ to a neighborhood of (ZQXO) in T^C^ mapping N^n to NQ-^. Then there exists a sheaf isomorphism T : C -^ x ~ ^ ^ ^ ^ in a neighborhood of (0, ^0) such that if P{z,d/dz) is a holomorphic differential operator at ZQ then T~^x7^P{îd/dz)x^^ is a microdifferential operator at (0, ^0) with principal symbol pox, ifp is the principal symbol of P. Here we have written x^^ for the map from A^'^ to the pullback to 5*(R^), identified with R^ x z S ' " - ^ For a proof of Theorem 7.4.6 we must

388


refer to Kashiwara-Kawai [1, p. 46] (see also Sato-Kawai-Kashiwara [1]). The following theorems show the power of the microlocal transformation theory when combined with the concrete results of Section 7.3. T h e o r e m 7.4.7. Let P be a microdifFerential operator of principal type î (O5 ô)j 0 7^ (^0 ^ R^j that is, assume that ifp is the real analytic principal symbol then either p(0,ô) ^ 0 or else dp is not proportional to (dx,ô) î (0,ô)- Then P is surjective on C(o,ô) if and only ifp{0,ô) ^ 0 or else p satisfies condition (^) in a neighborhood of (0,(ô) in T*R'^. Proof. In the non-characteristic case p(0,ô) ¥" 0? the microdifferential operator P has a microdifferential inverse, so this is a consequence of the basic calculus of microdifferential operators. Assume now that p(0, ^0) = 0. By Proposition 7.4.4 we can find a homogeneous holomorphic canonical transformation x ^^ (O^^ô) mapping a neighborhood in N^n to Ng-^ for some strictly pseudo-convex dX, and such that p is the puUback of ^1 by X- Let Pi be the transform of d/dzi to a microdifferential operator at (0,^0) provided by Theorem 7.4.6. It has the same principal symbol as P. In view of the symplectic invariance of condition (*) it follows from Corollary 7.3.6 and Theorem 7.4.6 that Pi is surjective on C(o,ô) if and only if p satisfies condition (*). By Theorem 2.1.2, Chap. II, p. 359 in Sato-Kawai-Kashiwara [1] we can find invertible microdifferential operators A and B at (0,ô) such that P = APiB, so surjectivity of P is equivalent to surjectivity of P i , which completes the proof. We can also use the transformation theory in the opposite direction: T h e o r e m 7.4.8. Let X — {z] Q{Z) < 0} be an open set with real analytic strictly pseudo-convex boundary near ZQ G dX, and let P{z, d/dz) be a holomorphic differential operator in a neighborhood of ZQ with principal symbol p{zX)- Assume that p{z,() ^ 0 or else that dp{z,Q and {(,dz) are linearly independent at (2^0, Co)? Co = dQ{z{))/dz. Then P{z,d/dz) is surjective on A^^ if and only ifp restricted to NQ^ satisfies condition (^) in a neighborhood of (ZQ? Co) with respect to the symplectic form a/i. Here A^'^ has the modified definition so the statement concerns solvability of the equation P{z^ d/dz)u = f modulo functions analytic in a full neighborhood of ZQ, with u and / analytic in the intersection of X and neighborhoods oi ZQ. However, if ^(^ÔJC) ^ O5 then the Cauchy-Kovalevsky theorem implies that there is also an exact solution then. Proof. Choose ^0 ^ R-'^ \ {0}- By Proposition 7.4.1 we can find a homogeneous holomorphic canonical transformation x from a neighborhood of (0,iô) in T*C"^ to a neighborhood of {ZQXO) mapping a neighborhood of (0,zô) in ^Rn to a neighborhood of (ZQXO) in NQ^- Using Theorem 7.4.6 we transform P{z,d/dz) to a microdifferential operator at (0,ô)j to


389

which we can apply Theorem 7.4.7. This gives the theorem by the same arguments as in the proof of Theorem 7.4.7, and we leave the repetition to the reader. (Note that the Hamilton field of the function p{z, jdg/dz) on iV^x ^^ radial at a zero precisely when the differential oi p{z,dg/dz) is proportional to {(^dz) = dg.) Corollary 7.4.9. Let X he an open set C C^ defined in a neighborhood of ZQ G dX by g(z) < 0 where g is real analytic, g{zo) = 0, dg{zo) ^ 0. Let P{z^ d/dz) be a differential operator with holomorphic coefficients in a neighborhood of ZQ, and denote the principal symbol by p{z,(). Write PO)(^,C) = dp{z,(:)/dzj andp(^\z,0 = 9 ^ ( ^ , 0 / 5 0 - lfp{z,0 = 0 and dp(zX) is not proportional to dg at (zo,(o), where (o = dg{zo)/dZj and if

(7.4.3)

Y^ WjWkd'^g/dzjdzk

+

+ 2Re {WQ ^

Wj{p(^j){z, dg/dz)

J2P^'\z,dg/dz)d'g/dz,dzk)) k= l

+ \wo\'^ J2 P^^\z^dg/dz)p(k){z,dg/dz)d^g/dzjdzk

>0

j,k=l

when z = ZQ, w e C^, WQ G C , Jî Wjdg/dzj P{z^ djdz) is surjective on A^^

= 0 and \w\ -i-\wo\ ^ 0, then

Proof. When WQ = 0 the inequality (7.4.3) means precisely that dX is strictly pseudo-convex. The additional condition is fully expressed by taking WQ = 1. It is easily verified that (7.4.3) is invariant under a linear change of variables and independent of the choice of the defining function g^ so we may assume that ZQ = 0 and that g has the form (7.3.4). Then dg/dz = ( 0 , . . . , 0, —1) so w^ — 0, and since p{z^ dg/dz) = 0 when z = 0 it follows that also p^'^\z, dg/dz) = 0 then. We have d'^g/dzjdzk — ^j^jk for j , A; = 1 , . . . , n, where A^ = 0, so the inequality (7.4.3) with WQ = 1 reduces to n—1

n—1

Y^ \wj\^Xj -f- 2 Re ( ^ 3=1

Wj{p(^j){z, dg/dz)

j=l n—1

+ Y,P^''\^,dQ/dz)d^g/dz^dzk)) fc=i

n—1

+ J2 \P^'\z,dg/dz)fXj j=i

> 0.

390


Taking the minimum with respect to w we find that this is equivalent to n—1

(7.4.4)

n—1

Y. \p^^\z,dg/dz)\'X,

- ^

j=l

\p^,){z,dQ/dz)

3=1 n-1

k=l

when z = 0. The corollary will be proved if we show that (7.4.4) is also equivalent to Re{p,p}E/2i

> 0

at {z, dg/dz)

when z = 0;

S =

N^dX).

The calculation as well as the preceding argument is essentially a repetition of the proof of Proposition 7.3.2. Set f{z) = p{z, dg/dz) and recall that E = {{z, jdg/dz);

z edX,j>0},

thus p{z, () = 7 ^ / ( ^ )

on E,

where m is the order oi p. Since /(O) = 0 it follows that

at 0, and using (7.3.6) we obtain

{f(z),mhîJ2{df/dzjdfWj-df/dzjdfWj)h)^r 3

Introducing df/dzj

= p(^)A„

df/dz,

= p(,) +

Y^/^^d'g/dzjdzk, k

we conclude that

3

3

k

Thus we have identified (7.4.4) with the condition i~^{p^p}E completes the proof.

> 0, which

Remark. Note that the proof of Corollary 7.4.9 has not by far used the full strength of the results in this chapter. It relies only on the suflicient condition in Proposition 7.1.6 and not at all on the subtle arguments in Sections 7.2 and 7.3.

APPENDIX

A . P o l y n o m i a l s and multilinear forms. Let F be a vector space over R which we assume to be finite-dimensional until the end of the section, and let F be an open convex cone in F. A real-valued function / defined in F is called a polynomial of degree < m if for arbitrary x^y GT we have m

(A.i)

fi^-^ty) = 5Zî(^'^)^'' ^ ^ 0'

where the right-hand side is a polynomial in t in the usual elementary sense. A polynomial p{t) = jy^ bjP of degree < m is uniquely determined by its values at 0 , . . . , m, so solving the equations X^JLQ ^J^^ ~ P(^)^ fc = 0 , . . . , m, we obtain bj = ^Cjkp{k),

j = 0,...,m,

k=0

where {cjk) is the inverse of the matrix (j^), i, fc = 0 , . . . , m , with the convention 0^ = 1. We can therefore rewrite (A.I) in the form

fix + ty) = J2Yl^^'^^(^ + ^2/)^^ t > 0. k=Oj=0

If X, 7 / 1 , . . . , y^ G F it follows when iî,..., t^ > 0 that V

m

v—1

/(^ + X^^/^^^) "" Yl Cjkfix -^^t^y^-i 1

j,k=0

and repeating the argument we obtain

1

+ ky^)ti,

392

APPENDIX

with ai < m, ... , ajy < m in the sum. In fact, we have \a\ = ai-\ f-Of^ < m for the non-vanishing terms, for if M is the maximum of |a| when a^ ^ 0, we can choose positive t i , . . . , t^^ so that V

/(a; + s ^ t ^ ^ , ) = ^ a „ ( a ; , 1/1,...,y^)^!"!*^ •••K" 1

is of degree M in 5, so M < m. If we choose T/I, . . . , y^ G F as a basis for V^ which is possible since F is open, it follows that / restricted to {x + txyx H h t^Vv] is the restriction of a polynomial in V^ defined in the usual sense as a linear combination of monomials in the coordinates. Two such polynomials which agree in an open set must coincide, so it follows that in terms of the coordinates x i , . . . , Xi^, with the usual multiindex notation,

!{x) = Y. «"^"' ^ e F,

(A.2)

\oL\<m

where the coefficients a^^ are uniquely determined. Thus we get a unique polynomial extension to V^ which we shall also denote by / . One calls / a homogeneous polynomial (or form) of degree m if in addition to (A.l) we have (A.3)

j{tx) - t ^ / ( x ) ,

t > 0, a; G F.

This means that in (A.2) the summation can be restricted to multiindices with |«| = m. For a form / of degree minV there is a unique symmetric multilinear form / ( x i , . . . , Xm), ^ ^ i , . . . , Xm ^ V, such that (A.4)

/(xi,..., Xm) = f{x)

iixi = '-• = Xm=x eV.

To prove the uniqueness of / we note that it follows from (A.4) and the symmetry and multilinearity of / that fihXi

+ • • • + tmXm)

= fihXi

+ • • • + tmXm,

• • • , hXl + ' " +

= m\ti • • • tmf{xi,

tmXm)

. . . , Xm) + • • •

where the dots denote terms not containing all the factors tj. To prove existence we observe that f{xi,...,Xm)

= {xi,d/dx)'--

{xm,d/dx)f{x)/m\

POLYNOMIALS AND MULTLINEAR FORMS

393

is multilinear and symmetric in a ; i , . . . , Xm G F , independent of x, and that / ( x i , . . . , x i ) = —f{txi)/m\

= f{xi),

xi

eV.

This proves existence and gives another useful expression for the polarized form / . We can also express / by means of the polarization formula (cf. Burago and Zalgaller [1, p. 137]): ^

(A.5)

m

fix,,...,xm) = — Yl (-ir+^"^7(E^^^^)' eG{0,l}"'

1

which is convenient if x i , . . . , x ^ G F since the arguments of / are also in r then. To prove (A.5) we first observe that the sum g{x\^... , x ^ ) in the right-hand side is symmetric in x i , . . . , x ^ and that it vanishes if one of these vectors equals zero. In fact, if Xj = 0 then the terms with EJ = 1 and Ej — 0 are the same apart from the sign so they cancel each other. Thus ^ ( t i X i , . . . ,tmXm) = 0

a ti, . . . ,tm ^ 'R', h ' • ' tm = 0,

and since we have a form of degree m in t i , . . . , t ^ it follows that gytiXij

. . . , tfYiX^Yi) = t i • • • tfYiQyXx^ • • • 5 ^m)^

SO g is of degree 1 in each variable. Now the only term of this degree in fiYlT ^2^2)/^' is / ( x i , . . . , Xm) n r î- ^^ vanishes unless Si = 1 for every z, so g{Xu ...,Xm) = ( - l ) ^ - ^ ^ / ( x i , . . . , Xm), which completes the proof of (A.5). The Brunn-Minkowski inequality is closely related to the functions studied in the following: P r o p o s i t i o n A . l . Let T be an open convex cone in V and let f be a form of degree m >2 which is positive in F. Denote the polarized form by f. Then the following conditions are equivalent: (i) r 3 X H^ f{x)^^'^

is concave. is concave ifx.yeV. (ii) f{y, X , . . . , x)2 > / ( y , y,x,..., x ) / ( x , . . . , x), ifx, y eV. {ay / ( 7 / , x , . . . , x ) 2 > / ( 2 / ^ 7 / , x , . . . , x ) / ( x , . . . , x ) , ifx eV, y eV. (iii) / ( ^ , 7 / , x , . . . , x ) 2 >f{z,z,x,...,x)f{y,y,x,...,x) ifx G T, y,z G V, unless we have f{sy-{-tz^ sy-^tz, x , . . . , x) < 0 when (5, t) ^ (0,0). They imply (iv) f{y,x,

...,xr>

fiy)f{xr~\

ifx,y

6 T.

394

APPENDIX

Proof. It is obvious that (i) implies (i)'. If h(t) — f{x + ty) then h'{t) = mf{y, x-\-ty,...,

x-^ty),

h"{t) =: m ( m - l ) / ( y , y, x-\-ty,...,

x-^ty),

so we obtain with (i = 1/m when ^ = 0 (A.6)

^h{tY

= iih"{t)h{tY-^ + nin - i)h'{tfh{tr~^ = (m - l){fiy,y,x,...,x)f{x)

-

f{y,x,...,xf)f{xY-\

Thus (i)' impKes (ii) and also (iv), since h(tYIt -^ f{yY as t -^ oo. Next we prove that (ii) implies (ii)'. To do so we note that y ^ f{y, x,...,x)^

- f{y, y,x,...,

x)f(x,

...,x)

is a quadratic form in y with polarization equal to 0 at x^ so it only depends on y modulo R x . Since y -\-tx E T ioT large positive t, the condition (ii)' now follows from (ii), and (ii)' implies (i) by (A.6). From (ii)' it follows that f{sy 4- t z , r r , . . . , x ) = 0

=^

f{sy + tz, sy -^tz,x,...

,x) < 0.

Thus the quadratic form (s, t) ^ f{sy -h tz, sy-^tz,x,...,

x)

is either negative definite, singular or indefinite. In the last two cases the discriminant is > 0, so (iii) holds. Taking z = x in (iii) we conclude that (ii)' is valid, which completes the proof. We shall finally discuss a more restrictive condition related to the Alexandrov-Fenchel inequality: P r o p o s i t i o n A . 2 . Let T be an open convex cone in V, and let f be a form of degree m > 3 such that the polarized form f is positive in V^ and satisfies the condition (A.7) f{xi,X2,X3,..., ifxi,..., (A.8)

Xm)'^ > f{xi,xi,X3,...,

Xm G r . Then it follows that f{xu---,xmr>f{xi)---f{x

x^)/(a;2, X2, X 3 , . . . , Xm),

POLYNOMIALS AND MULTLINEAR FORMS

395

Proof. In view of the homogeneity we may assume that f{xj) — l,j = 1 , . . . , m. Let M be the minimum of f{xi^,..., Xi^) when I < ij < m; we claim that M = 1 which will prove (A.8). Suppose that the minimum is attained when zi ^ 12 say. Then M

= / \Xi^,

Xi^ 5 • • • ,

Xi^)

so the two factors on the right must also be equal to M. Thus the number of indices equal to H can be increased while the minimum is still attained, until it becomes equal to m, and then we have M = 1 by assumption. Remark 1. Since / ( x , . . . ,a:,Xn+i,. • • ,x-^) for every n < m is a form of degree n in x, Proposition A.2 contains the following apparently more general inequalities: (A.9) n jyXij

. . . , Xmj

— J[ J[ J \Xi') • ' ' •) Xi^ XTJ,-)-!, • . . , Xjji),

X\^ • • • 5 ^m ^ -^ 5

(A.IO) / [X^ . . . , X 7/, . . . , ?/, Xi^jjî^

• • • 5^ m j

— / \X-) . . . , X , X^-j-j-j-l, . . . , XTTI j / (7/, . . . , y , X^_(_j_^l, . . . , ^772 j .

Remark 2. The condition (A.7) means that when X3, • • •, ^777, F r then the quadratic form q{x) — / ( x , x , a ; 3 , . . . , x ^ ) which is positive in V has signature 1, r where 1 + r is the rank of q. This means that it is of the form ^0 — ^1 — • • • — ^r with suitable coordinates. Thus q is either of rank 1 or else it has Lorentz signature. So far we have assumed that V is finite dimensional and that V is an open convex cone in V. Let us now just assume that y is a vector space over R and that F is a convex cone in V. Denote by V the linear subspace r — r of F generated by F. Defining as before the notion of polynomial (or form) of degree m in F we get a unique extension to F, which is polynomial in the elementary sense in any finite-dimensional subspace. Indeed, the extensions obtained above for two finite-dimensional subspaces lî, V2 must agree in Vi fl V2 since they are both restrictions of the extension obtained in Vi + ^2- (Our earlier hypothesis that F is open can clearly be replaced by the hypothesis that F has interior points.) Polarization works as before. If we just assume that / > 0 in F, then Proposition A.l remains valid. Indeed, in the finite-dimensional case either / vanishes identically then or else / > 0

396

APPENDIX

in the interior r ° of F, and we obtain the equivalence of the inequaUties by continuity from the earher statement applied to r ° . Thus the extension to the infinite-dimensional case is straightforward, but it is important for the application to the set of convex bodies in a finite-dimensional vector space. B . C o m m u t a t o r identities. Let A and B be elements in a noncommutative algebra A. (In our applications A and B will be first-order differential operators with C^ coefficients and A will be the algebra of differential operators with C^ coefficients or else an algebra of pseudodifferential operators.) We shall use the standard notation (adA)jB = [A^ B] for the commutator [A, B] = AB — BA in order to simplify the notation for higher-order commutators. Recall the Jacobi identity a d [ ^ i , ^ 2 ] - [adyli,ad^2]If ^ is a polynomial then (B.l)

piA)B

=

î{adAyB)p(^\A)/j\. 3

Here (ad AyB is a repeated commutator [A, [ A , . . . , [A, B ] . . . ]] with j successive brackets, and the term with j = 0 is Bp{A), so (B.l)'

\p{A),B] =p{A)B

- Bp{A) =

Yîi^dAyB)p^^\A)/jl

To prove (B.l) we write Ai and Ar for left and right multiplication by A in A. These operators commute and Ai = 3.dA -\- Ar, so (B.l) follows from Taylor's formula. The following lemma generalizes (B.l) to arbitrary polynomials in B: L e m m a B . l . Define Tj^k{A, B) for integers j , fc > 0 by ro,o(^5 B) = 1, Tj o(A, B) = 0 when j > 0 and the recursion formulas (B'.2) (k + l)Tj,k+i{A,B)

- [Tj,kiA,B),B]

+ ^

r,_;,fc(A,B)((adA)'5)//!,

o 0, k > 0. Then we have for arbitrary polynomials p and q

(B.3)

p[A)q{B) = Y,

q^'HB)T,,kiA,B)/^\A).

3,k>0

Here Tj^k{^^ B) is a linear combination of products of commutators formed from A and B, with k factors B and j factors A in all, and Tj^k is uniquely determined by (B.3). Proof. From (B.2) we obtain by taking k = 0 that Tji{A,B) = (ad A)^J5/j!, which agrees with (B.l). Taking j = 0 we obtain To,k{A, B) —

COMMUTATOR IDENTITIES

397

0, A; > 0, and then when j = 1 that Ti^kiA^B) = (-3idB)Â/k\, k > 0. For j > l^k > 1 the expUcit form of rj^k{^->B) becomes more compHcated, but it is clear that all Tj^ki^^ B) are linear combinations of products of commutators formed from A and 5 , with k factors B and j factors A in all. The uniqueness of Tj^k follows by induction with respect to j and k. In fact, taking p = q = 1 in (B.3) we obtain Toô{A^B) = 1. Taking q(t) = t^/K\ and ^{t) = t'^/ J\ we obtain in the right-hand side one term F j x ( ^ 5 B) and otherwise terms involving Tj jt(A, B) with j < J and k < K 3.nd {j,k)^ {J, K). It is trivial that (B.3) is true if g is a constant; in fact, (B.l) proves that (B.3) is true also when p is of order 1. Assume that we have already proved (B.3) when ^ is a polynomial of degree < /x. We shall prove that it is also true if q{B) is replaced by qi{B) = q{B)B, which by induction will verify the lemma. By the inductive hypothesis and (B.l)

j,k

l>0

j,k

j,k

j,k

l>0

To verify (B.3) se must prove that this is equal to ^(g(^)(5)5 +

kq(>'-'\B))r,,k{A,B)p(^\A),

that is, after cancellation of some terms, that

J2 2 was proved in Borell [1,2] even for Green's function oi A — V where y ~ 2 is concave. His proofs rely on probabilistic methods which we have not developed here. Theorems 3.2.32 and 3.2.34 are due to Ancona [1], where also more general statements can be found. Theorem 3.2.34 strengthens an earlier result of Trepreau [2]. The presentation of Perron's method in Section 3.3 follows Brelot [1], and the proof of Theorem 3.3.8 has been taken from Garding-Hormander [1]. Theorem 3.3.12 was first proved in Riesz [1]. The full statement of Theorem 3.3.18 is probably due to Beurling, but a major part of it is due to Titchmarsh [1]. Corollary 3.3.22 is a special case of the classical product representations of Hadamard, and Theorem 3.3.23 is a special case of results in Beurling [1], also proved by Nevanlinna [1]. The Wiman-Valiron theorem 3.3.26 in the formulation given here was proved by Kjellberg [1]. The discussion of exceptional sets in Section 3.4 has been taken from Brelot [1], Carleson [1], Doob [1], and Hayman-Kennedy [1]. For the crucial capacitability theorem see Choquet [1]. The concluding remarks are modelled on arguments of Bedford and Taylor [1] for plurisubharmonic functions. C h a p t e r IV. Plurisubharmonic functions were introduced by Lelong and Oka in the 1940's; see the historical notes in Lelong-Gruman [1]. The results in Section 4.1 are mainly direct consequences of their analogues for subharmonic functions. The discussion of pseudo-convex sets has to a large extent been taken from CASV. There is also a large overlap between CASV and the existence theory for the d operator in Section 4.2 and the study of the Lelong number in Section 4.3. However, CASV also studies forms of higher degree while we have here restricted ourselves to existence

NOTES

405

theorems for the Cauchy-Riemann system acting on a scalar function. A weaker form of the main existence theorem in Section 4.2, Theorem 4.2.6, was first proved by Bombieri [1], who strengthened the results in the first edition of CASV. The more precise version given here is due to Skoda [1]. It improves the corresponding statements in the last edition of CASV. We refer to the papers of Bombieri and Skoda for the number theoretical applications. Theorem 4.3.3 is a special case of a theorem of Siu [1]. The simplifications of the proof given here are due to Kiselman [2, 3]; they can also be found in CASV. In Section 4.4 we have presented the basic facts on positive closed currents due to Lelong (see Lelong [1] and LelongGruman [1]) and proved the Siu theorem in full generality, for arbitrary closed positive forms, following the ideas of Skoda [3] and Lelong [3]. The proof of invariance is a specialization of arguments in Demailly [1]. We refer to Thie [1] for the general geometrical meaning of the Lelong number for the integration current of an analytic set. In Section 4.5 we have only given the beginning of a theory of exceptional sets for plurisubharmonic functions, by proving Theorem 4.5.3 due to Josefson [1]. The reader should consult Bedford and Taylor [1] for results analogous to those proved in Section 3.4 (see also Cegrell [1] and Klimek [1]). The case of plurisubharmonic functions is much harder since plurisubharmonicity is a non-linear condition on the Hessian whereas subharmonicity is a linear one. One must therefore develop a kind of non-linear potential theory. Section 4.6 is devoted to stronger versions of pseudo-convexity. Linear convexity was introduced by Behnke and Peschl [1] for the case of two complex variables, with the term planar convexity. The main ideas of the proof of Proposition 4.6.4 and Corollary 4.6.5 were already in that paper, but we have mainly followed Yuzhakov and Krivokolesko [1] here. The interest of weak linear convexity was revived by Martineau [1, 2, 3, 4] and Aizenberg [1, 2] (see also Aizenberg et al. [4]). Theorem 4.6.8 was given in Znamenskij [1] with a short proof which he has elaborated in a personal communication. We give a variant of his proof here. Example 2 after Corollary 4.6.9 is a modification of one given in Aizenberg et al. [4]. Theorem 4.6.12 is due to Yuzhakov and Krivokolesko [1] in the C^ case, and the general proof is essentially the same. The whole section owes much to an unpublished manuscript by Andersson, Passare and Sigurdsson [1]. This is also true for Section 4.7 where we prove the importance of C convexity for the study of the Fantappie transform of analytic functionals. All results proved there were stated by Znamenskij [1]. A proof using integral formulas of Cauchy-Leray type has been given by Andersson [1]. Here we use instead the existence theorems of Section 4.2 based on weighted L^ estimates. We have followed the terminology of Andersson, Passare and Sigurdsson [1]; it varies in the literature.

406

NOTES

C h a p t e r V . This chapter is a presentation of a twenty-year-old unpubhshed manuscript with the aim of putting convexity, subharmonicity and plurisubharmonicity in a systematic framework containing also numerous other similar notions. It remains to be seen to what extent these spaces are useful for general linear groups. C h a p t e r V I . Section 6.1 is just a short summary of basic facts proved in ALPDO, Chapter X. There are very few operators for which one has a complete geometrical understanding of P-convexity with respect to (singular) supports. However, as another case of estimates with weight functions similar to those in Section 4.2, we prove that pseudo-convex domains are always P-convex for supports and singular supports if P is any polynomial in the operators d/dzj in C^. Such results have been proved by Malgrange [1, 2] even for overdetermined systems, but we give a more elementary approach for the case of a single operator. Malgrange [1] also outlined a general program intended to capture natural convexity conditions associated with other subalgebras of the polynomial ring. However, no essential progress seems to have been made, so we shall only make this brief reference to an interesting open problem. In Section 6.3 we discuss analytic differential operators with constant coefficients and prove that it is precisely in C convex sets that they can always be solved analytically. This is an application of the results in Sections 4.5 and 4.6, and as there we have profited from Andersson, Passare and Sigurdsson [1]. Lemma 6.3.1, which is the important step in proving necessary conditions, is due to Suzuki [1] (see also Pincuk [1]); for the full necessity in Theorem 6.3.2, see Znamenskij [2]. Theorem 6.3.3 is due to Martineau [4]. C h a p t e r V I I . The entire chapter is basically an exposition of Trepreau [1] with some improvements suggested by J.-M. Trepreau, which depend on the results of the recent manuscripts Trepreau [2] and Ancona [1]. We have rearranged the arguments so that the microlocal analysis which occurs at the beginning of Trepreau [1] is postponed to the very end, so most of the chapter is accessible without any background in microlocal analysis. However, the importance of the results may not be clear without the consequences based on microlocal analysis outlined at the end of Section 7.4. Corollary 7.4.9 at the end of the chapter was also stated in the preliminaries of Kawai and Takei [1] where it was obtained from results of Kashiwara and Kawai [2] and Sato, Kawai and Kashiwara [1]. The main result of Kawai and Takei is that (7.4.3) can be interpreted geometrically as a combination of bicharacteristic convexity and strict pseudo-convexity of a local projection along the bicharacteristics. This extends the most regular case of the results of Suzuki [1] for first-order differential operators (see the remark after Theorem 6.3.2).

NOTES

407

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C. Kiselman [2], Densite des fonctions plurisousharmoniques, Bull. Soc. M a t h . France 1 0 7 (1979), 295-304. C. Kiselman [3], La teoremo de Siu por abstraktaj nomhroj de Lelong, Aktoj de Internacia Scienca Akademio Comenius 1. Bo Kjellberg [1], On certain integral and harmonic functions^ Thesis Uppsala 1948. M. Klimek [1], Pluripotential theory, Clarendon Press, Oxford, New York, Tokyo, 1991, London M a t h . Soc. Monographs N.S. 6. W . Klingenberg [1], Riemannian geometry, Walter de Gruyter, Berlin, 1982. H. Kneser [1], Eine Erweiterung des Begriffes "konvexer Korper", M a t h . Ann. 8 2 (1921), 287-296. B. Kostant [1], On convexity, the Weyl group and the Iwasawa decom,position, Ann. Sci. Ecole Norm. Sup. ( 4 ) 6 (1973), 413-455. P. Lelong [1], Plurisubharm,onic functions and positive differential forms, Gordon and Breach, New York, London, Paris, 1969. P. Lelong [2], Integration sur un ensemble analytique com,plexe. Bull. Soc. M a t h . Prance 8 5 (1957), 239-262. P. Lelong [3], Sur la structure des courants positifs fermes, Sem. Lelong 1975/76, p p . 135-156, SLN 578, 1977. P. Lelong and L. G r u m a n [1], Entire functions of several complex variables. Springer Verlag, 1986, Grundl. d. m a t h . Wiss. 282. L. Lempert [1], A precise result on the boundary regularity of biholomorphic mappings. M a t h . Z. 1 9 3 (1986), 559-579. B. Malgrange [1], Systemes differentiels a coefficients constants, Seminaire Bourbaki, 1962-63, p p . 246-01-246-11. M. Malgrange [2], Quelques problemes de convexite pour les operateurs differentiels a coefficients constants, Seminaire Leray, 1962-63, p p . 190-223. S. Mandelbrojt [1], Sur les fonctions convexes, Comptes Rendus Acad. Sci. Paris 2 0 9 (1939), 977-979. A. Martineau [1], Sur les fonctionnelles analytiques et la transformation de FourierBorel, J. Anal. M a t h . 1 1 (1963), 1-164. A. Martineau [2], Sur la topologie des espaces de fonctions holomorphes. M a t h . Ann. 1 6 3 (1966), 62-88. A. Martineau [3], Equations differentielles d'ordre infini, Bull. Soc. M a t h . France 9 5 (1967), 109-154. A. Martineau [4], Sur la notion d'ensemble fortement lineellem,ent convexe. An. Acad. Brasil Ci. 4 0 (1968), 427-435. H. El Mir [1], Sur le prolongement des courants positifs fermes, A c t a M a t h . 1 5 3 (1984), 1-45. R. Nevanlinna [1], Uber eine Minimumaufgabe in der Theorie der konform,en Abbildung, Nachr. Ges. Wiss. Gottingen Math.-physik. Kl. 3 7 (1933). L. Nirenberg [1], A characterization of convex bodies, J. Fac. Sci. Univ. Tokyo 1 7 (1970), 397-402. L. Nirenberg and F . Treves [1], On local solvability of linear partial differential equations. I. Necessary conditions, Comm. P u r e Appl. M a t h . 2 3 (1970), 1-38; / / . Sufficient conditions, C o m m . P u r e Appl. Math. 2 4 (1970), 459-509; Correction, C o m m . P u r e Appl. M a t h . 2 4 (1971), 279-288. S. I. Pincuk [1], On the existence of holomorphic primitives, Dokl. Akad. Nauk SSSR 2 0 4 (1972), 292-294; English transl. in Soviet M a t h . Dokl. 13 (1972), 654-657. H. Prawitz [1], Uber Mittelwerte analytischer Funktionen, Ark. Mat. Fys. Astr. 2 0 A ( 6 ) (1927), 1-12.

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I N D E X O F NOTATION

General n o t a t i o n

cHx) V'{X) £'{X) f*9 suppw sing supp u X ^Y

Zx

dX a \a\ a\ a;° ga Section 1.1 /r' fl Section 1.2

functions in X with continuous derivatives of order < A: functions in C^{X) with compact support Schwartz distributions in X Schwartz distributions in X with compact support convolution of / and g support of u singular support of u closure of X is a compact subset of Y complement of X (in some larger set) boundary of X usually a multiindex a = ( a i , . . . , a^) length ai -\- • • • -\- an oi a multifactorial a i ! • • • aî! monomial x^^ ...x^" in R'^ partial derivative, dj — d/dxj right and left derivative of / weighted F mean value

Section 1.3 / Section 2.1 ah ch &n

Section 2.2 / K° Section 2.3 K, QJ(K)

mean value with respect to function cp Legendre transform of / afRne hull convex hull symmetric group Gateau differential of / at x in direction y Legendre transform of / polar set of K (in particular dual cone) set of convex compact subsets of vector space V volume of iiT G /C mixed volume of if i , . . . , K^ G /C

Section 2.6 ^

Fourier-Laplace transform oi (p ^ £'

412

Section 3.1 BR GR

P H GH PH

Gt Pi

open ball in R"^, radius i?, center 0 Green's function of BR Poisson kernel of Bi half space H = {x ^W^\Xn > 0} Green's function of H Poisson kernel of H Green's function of if fl BR Poisson kernel of iif H BR

S e c t i o n 3.2 Mu{x,r) HP

mean value of u{y) when ly — a;| = r Hardy space

Section 4.1 Mu{z]ri,...,rn) d/dzj d/dzj dx{z) Section 4.3 T^cp

mean value of u{Q when \Q — Zj\ = TJ, j = 1,.. ,n \{dldxj - id/dyj), Zj = Xj -\- iyj \{dldxj + id/dyj), Zj = Xj + iyj distance from z e X to CX Lelong number of plurisubharmonic function cp

Section 4.4 te Z/0

trace measure of positive current 0 Lelong number of positive current 0

Section 4.7

AiX) A'iX) A BM

M^ S e c t i o n 5.1

sHV) &r{V)

analytic functions in X analytic functionals in X Laplace transform of /x G A'iC^) Borel transform of entire function M Fantappie transform of /x G A^C^) quadratic forms in vector space V with dual V Grassmannian of r-dimensional subspaces of V

S e c t i o n 7.1

Ao(^)

germs of analytic functions in X 3.i ZQ E dX

Section 7.3 T*M UJ

a {V}E

Hu

cotangent bundle of manifold M canonical one form in T'^M symplectic form Poisson bracket in symplectic manifold E Hamilton vector field of function u inT,

INDEX

Affine 2, 37, 38 Analytic functional 300 Analytic set 213 Arithmetic mean 10 Barrier 173 Berezin's inequality 7 BirkhofF's theorem 46 Blaschke product 179 Bonnesen's inequality . . . . 82 Bony-Brezis lemma 379 Borel transform 304 Brunn-Minkowski's inequality 88 Capacitable set 213 Caratheodory's theorem . . . 41 Carrier 301 Cauchy-Riemann eq. . 248, 251 Choquet's theorem 214 Commutator 396 Concave 1 Condition (^) 379 Cone condition 220 Conjugate function . . . 17, 67 Continuity principle . . . . 207 Convex 1, 36 C convex 294 Convex polyhedron 53 Cross ratio 99 Current 272 Defect 105 Dirichlet problem . . 118, 172 Distance function 57 Domain of holomorphy . . . 239 Doubly stochastic matrix . . 46 Dual cone 71 Epigraph 36 Extreme point 43

Fantappie transform . . . . 306 F. and M. Riesz theorem . . 181 Fenchel-Alexandrov theorem . 82 Fenchel transform 67 Fourier-Laplace transform . . 1 1 2 Fundamental solution . . . . 1 1 7 G subharmonic . . . 315, 324 Gabriel's theorem 136 Gamma function 20 Gateau differential 55 Geometric mean 10 Grassmannian 319 Green's function 119 Hadamard's three circle th. . 160 Hahn-Banach theorem . . . 45 Hamilton vector 377 Hardy space 184, 190 Harmonic function 116 Harmonic majorant 171 Harmonic mean 10 Harnack's inequality . . . . 123 Hausdorff's theorem . . . . 51 Helly's theorem 41 Holder's inequality 11 Hopf's maximum principle . . 1 2 2 Horn's theorem 49 Hyperbolic polynomial . . . 63 Indicator function 313 Inner factor 184 Integration current 274 Inversion 118 I-symplectic 377 Jensen's inequality 7 Josefson's theorem 286 Krein-Milman's theorem . . 44 Laplace equation 116 Laplace transform 301

414

Legendre transform . . . 17, 67 Lelong number . . 235, 270, 277 Levi form 228, 244 Levi problem 239 Linearly convex 290 Liouville's theorem 160 Malgrange's theorem . . . . 333 Maximum principle 121 Mean value pr. 121, 143, 153, 232 Minimal harmonic function . 125 Minimum modulus theorem . 194 Minkowski's inequality . . . 12 Minkowski's theorem . . 43, 80 Mixed volume 77 Motzkin's theorem 62 Muirhead's theorem . . . . 47 Norm 57 Numerical range 52 Outer conormal bundle . . . 376 Outer factor 184 P-convex 328, 329 Permutation matrix . . . . 46 Perron's method 172 Pluriharmonic function . . . 225 Pluripolar 285 Plurisubharmonic function . . 225 Poisson bracket 377 Poisson kernel 120 Poisson's equation 117 Polar 70, 203

Polarization Positive current Positive form . . . . Positive cone Potential Projective convexity . . Pseudo-convex . . 237, Quasi-convex function . Relative interior Relative boundary Riesz-Herglotz' theorem . R-Lagrangian Runge domain Schlicht function Semi-convex function . . Sheaf C Simplex Singular factor Siu's theorem . . . . Starshaped Strictly convex Strictly pseudo-convex . Strong subadditivity . . Subharmonic function . . Superharmonic function . Supporting function . . Titchmarsh's theorem . . Trace measure Weakly linearly convex . Wiman-Valiron's theorem

273, 393 272 270, 271 318 145 . . 99 243, 342 . . 27 42 42 . . 123 376 308 160 . . 26 387 40 184 267, 283 37 6, 56 . . 243 . . 210 . .141 . . 141 . . 69 . . 189 272 . . 290 . . 197

Notions of Convexity

Notions of convexity

Notions of convexity