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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
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(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
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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
372 375 382
Notes
403
References
407
I n d e x of notation
411
Index
413
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 Air^i 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
I. CONVEX FUNCTIONS OF ONE VARIABLE
—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;)-//(^))
CCl<X<X2
Exercise 1.1.3. Let xi,X2^ • •. be different real numbers and let aj > 0 be chosen so that YlT ^ji^ ^" l^jl) ^ ^^- Show that 00
1
is a convex function and that 00
fri^j)
- fiixj)
= 2aj, Vi;
f'{x) = ^ a j sgn(a; - Xj)
\ix + Xj Vj.
DEFINITIONS AND BASIC FACTS
E x e r c i s e 1.1.4. Show that if fj are non-negative continuous convex functions in a compact interval / and / = YlT fj converges at the end points, then / is continuous and convex in / and /'(a;±0)-^/j(a:±0)
for every x in the interior of / . E x e r c i s e 1.1.5. Show that if fj are convex functions in the interval / and fj{x) ^ / ( x ) G R for every x G / , then / is convex in I and fj -^ f uniformly on every compact interval contained in the interior of / . Show that f\x
- 0) < lim / j ( x - 0) < .ihii fj{x + 0) < fix
+ 0),
j -^ oo,
for every x in the interior of / . Give an example where there is inequality throughout. E x e r c i s e 1.1.6. Let / be an open interval and fj a sequence of convex functions in / having a uniform upper bound on every compact interval J C I. Prove that either fj^^—oo uniformly on every such interval J or else there is a subsequence fj^, converging uniformly on every J to a convex function. E x e r c i s e 1.1.7. Let / be convex in the interval / and bounded above there. Show that / is decreasing (increasing) if / is infinite to the right (left); thus / is constant if / = R. To prove a converse of Theorem 1.1.7 we need a suitable form of the mean value theorem: L e m m a 1.1.8. Let f be a continuous function in a closed interval {x]a < X < h} such that /^(x) exists when a < x < b. If f^(x) > C for all such x then f{b) — f{a) > C{b — a). If instead f'j.{x) < C then f{b)-f{a)
< C then
[a, h]-f{x) - f{a) > C'{x - a)}
is closed because / is continuous, and a E F. The supremum y of F is in F , and y = b since we would otherwise have
f(y+h)-f{a)
= f{y+h)-f{y)+f{y)-f{a)
> C'h+C'{y-a)
=
C'{y+h-a)
6
I. CONVEX FUNCTIONS OF ONE VARIABLE
for sufficiently small /i > 0, contradicting the definition of y. Hence f{b) — f{a) > C'{b — a) for every C < C which proves the lemma. The lemma gives the inequality (1.1.6)
inf / ; < {f{h) - f{a))/{b
-a)<
l^'^)
sup/;, [a,b)
and we obtain: T h e o r e m 1.1.9. If f is a continuous function in the interval I such that f!j. exists at every interior point x of I and increases with x, then f is convex, and J^ f^{t) dt = f{y) — f{x), x,y ^ I. (The same result is true with /^ replaced by f[.) Proof. (1.1.1)" follows at once by (1.1.6) if r^i < x < X2 are interior points of / . Since / is continuous the inequality remains valid if xi or X2 is a boundary point, so the convexity follows in view of Theorem 1.1.5. The second statement follows since the right derivative of J J f!^{t) dt with respect to y is equal to /^(y) by the monotonicity and right continuity. Corollary 1.1.10. Let f be a continuous function in I which is in C^ in the interior of I. Then f is convex if and only if f" > 0 there. If f" > 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,
DEFINITIONS AND BASIC FACTS
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^^-^^)^E^^/(^^)' 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
I. CONVEX FUNCTIONS OF ONE VARIABLE
E x e r c i s e 1.1.11. W^ith 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)<
f{y) + {g' + ^(a + 6 - 2y))h - eh\
ii y + h e J,
which proves (1.1.7). In particular it follows that we can characterize convexity in terms of the second differences also for functions which are not in C^ so that Corollary 1.1.10 is not applicable: C o r o l l a r y 1.1.16. If f is a continuous real-valued function in the interval I then f is convex if for all interior points x G / (1.1.8)
^{f{x
+ /i) + fix -h)-
2f{x))/h^
> 0.
The necessity of (1.1.8) is obvious for the second diff'erence is > 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^'^Ev.'
10
I. CONVEX FUNCTIONS OF ONE VARIABLE
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(;^A,a^)=p-Mog(^A,(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)'
A^o(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)
< mina^A^
,
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:
SOME BASIC INEQUALITIES
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<) . I
l
with equality if and only ifa^/b^
l
is independent
of j .
Exercise 1.2.1. State and prove the analogue of (1.2.4) with means with respect to A i , . . . , A^, and also an analogue for integrals. (Do so also for the later inequalities in this section whenever appropriate.) Using Theorem 1.2.1 instead of Theorem 1.2.2 we shall now give an alternative proof of a more general version of Holder's inequality: T h e o r e m 1.2.5. Ifajk, j = 1 , . . . , iV^ fc = 1 , . . . , n are positive andpj >1, j = 1,...,N, 5 ] f 1/pj = 1, then n
(1.2.5)
N
N
n
En-.^^n(E<0 k=lj=l
j=l
k=l
. y
'
numbers
12
I. CONVEX FUNCTIONS OF ONE VARIABLE
with strict inequality unless the direction of {a^A}'^^-^ is independent
of j .
Proof. Replacing ajk by ajk/Aj where Aj — (Z)fc=i ^jD^^^^ ^ ^ ^ ^ ^ i^ view of the homogeneity of (1.2.5) that it suffices to prove the inequahty when Aj = 1 for jf = 1 , . . . , AT. By (1.2.1) we have then n
N
n
N
n
N
N
E n«.- - E u«i)'^'^ ^ Y:T.pr
fc=ij=i
k=ij=i
j=i
with strict inequahty unless a^^ is independent of j , that is, {a^^j^^i is independent of j . This proves the theorem. The convexity of x i-^ a;^ used to prove Theorem 1.2.2 also yields: T h e o r e m 1.2.6 (Minkowski's inequality). Let aj > 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.
SOME BASIC INEQUALITIES
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 Y^i ^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 ^ok <-J2^'j '^^ ^'j ~ Yl ^k ^^g ^fc' ^f 3=1k=l
j=l
k=l
K Xj = /
J ^Xjk^
Xj^ = / ^ Xjk.
k=l
j=l
(Hint: Note that ^ ^ Xj = 1 and that x^^ = ^^. x'j{xjklx'-)
is a mean value.)
It is possible to improve Theorem 1.2.3 also: T h e o r e m 1.2.7. If aj > 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 = ^
Xjie^ajf
14
I. CONVEX FUNCTIONS OF ONE VARIABLE
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 A^i/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) = P^ogMp{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<^^){a;X) = p
^{^Xjcp(aj))
a well-defined point in / . T h e o r e m 1.2.10. Let p and ip be two strictly monotonic functions defined in the interval I. In order that (1.2.6)
continuous
M^^){a;X)<M(^){a;X)
for all a £ I'^ and X E [0,1]^ with ^ ^ Aj = 1^ it is necessary and sufficient that either (a) if is increasing and po^~^ is convex; or (b) p is decreasing and cp o ^i/j'^ is concave.
SOME BASIC INEQUALITIES
15
Proof. (1.2.6) means that for arbitrary aj G / n
n
1
1
Set bj = tp{aj) and X = ^ ^i^~^- For arbitrary bj in the range of -0, which is the domain of x, we have then n
n
1
1
if (/? is increasing, and the opposite inequaUty if ip is decreasing. Hence the necessity of (a) or (b) follows already when n = 2, and the sufficiency follows by Jensen's inequality. Corollary 1.2.11. Under the hypotheses A4(^)(a; A) = M(^){a] A) if and onlyifcpo^~^ for some constants a and (3.
in Theorem 1.2.10 we have is afhne, that is, cp = aip-}- /3
Proof. If both ^ and ij^ are increasing (decreasing), it follows from Theorem 1.2.10 that (^o%jj~^ and its inverse i/joip~-^ are both convex (concave). Both are increasing, so a glance at a figure suffices to show that they must be affine. If on the other hand (f and ^^ are monotonic in opposite directions, then these inverse functions are decreasing, one is convex and the other concave, which again implies that they are affine. By Corollary 1.2.11 the mean value Mp is equal to A^(y, ) not only with (fp taken as the function x y-^ x^ but also with
ipp{x) = {x^-l)/p=
f
t^-Ut.
The advantage of this choice is that it is applicable also when p = Q and shows right away that the mean value is then the geometric mean. It is also useful in the proof of the following corollary, which explains the special importance of the means M.p. Corollary 1.2.12. Let (p he strictly (0,oo), and assume that (1.2.7)
monotonic
and continuous
on
A^(c^)(ca; A) = cA^(^)(a; A)
for all aj > 0, Aj > 0, j = 1 , . . . , n, with ^ ^ A^ = 1, and all c > 0. Then it follows that M(^^) = Mp for some p, that is, (p = aipp + /3 for some constants a and l3. Proof. By Corollary 1.2.11 we can replace iphy ip—ip{l) without affecting (1.2.7), so we may assume that ip{l) = 0. This reduction is useful because
16
I. CONVEX FUNCTIONS OF ONE VARIABLE
(^p(l) = 0 SO now we must show that ^ = acpp. By (1.2.7) and Corollary 1.2.11 we must have for every c > 0, with uniquely determined coefHcients a(c), P{c), (p{cx) = a{c)cp{x) + /?(c), X > 0. When X = 1 it follows that /?==
(p{cx) = a{c)(p{x) + (p{c) — a{x)(p{c) -f (p{x).
If we give c a fixed value ^ 1, it follows that a{x) == 1 + k(f{x) for some constant k. If we enter this in (1.2.8) and write y instead of c, we obtain the functional equation (p{xy) = k(p{x)(p{y) + (p{x) + (p{y),
x,y > 0.
Assume at first that k = 0. Then we have the functional equation ip{xy) = ip{x) -f (f{y), which means that $ ( t ) = ^(e*) is a continuous function such that $ ( t + 5) = $ ( t ) + $(5),
t.seR.
This implies that $ is a linear function. (By Exercise 1.1.13 $ is both convex and concave, hence afRne, and since $(0) = 0 it follows that $ is linear.) Thus ip{x) = a log a; for some constant a, so the statement follows with p = 0. Assume now that k ^ 0. With the notation f{x) = k(p{x) + 1 we obtain the functional equation f{xy)
= f{x)f{y),
x,y>0.
When X = y we conclude that / > 0, and f{x) > 0 for every x > 0 since f(x) = 0 would imply f(xy) = 0 for every y. Hence l o g / satisfies the same functional equation as in the first case discussed, so l o g / ( x ) = plogx for some p ^ 0. This means that (p{x) = {x^ — l)/fc = aipp{x), a = p/k, which completes the proof. 1.3. Conjugate convex functions (Legendre transforms). By Theorem 1.1.13 a convex function is determined by its affine minorants. We shall now exploit this fact systematically. In doing so it is convenient to work with functions defined on all of R . If / is initially only defined in an interval / , we extend the definition so that / = +oo outside / and change the definition of / at 9 / , if necessary, so that / is equal to its limit from the interior of / there. (This may be finite or +oo.) Then (1.1.1)' is valid for all xi and X2 if we define oo -f oo = oo and 0 • (X) = 0, and / is lower semi-continuous, that is, (1.3.1)
f{x) = lim f{y)
for every x G R.
In what follows we assume that / is defined on R with values in (—oo, 4-oo] and that / is lower semi-continuous. A slight extension of Theorem 1.1.13 gives:
CONJUGATE CONVEX FUNCTIONS
17
L e m m a 1.3.1. For every c < f{x) one can choose A: G R so that f{y) > 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
I. CONVEX FUNCTIONS OF ONE VARIABLE
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=^i,
/'(^) = 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
^ajb, 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
Y^ajbj
< 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
I. CONVEX FUNCTIONS OF ONE VARIABLE
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) + Y^ihij)
-
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
I. CONVEX FUNCTIONS OF ONE VARIABLE
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
I. CONVEX FUNCTIONS OF ONE VARIABLE
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-^ip{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
I. CONVEX FUNCTIONS OF ONE VARIABLE
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 > ^i, ^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
I. CONVEX FUNCTIONS OF ONE VARIABLE
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 C^iY) (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
I. CONVEX FUNCTIONS OF ONE VARIABLE
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.mi
= <Jxo,t^)-<,(xo,t^)V 0 in y . Before proceeding we note that the conditions (i)-(iii) are preserved if we replace u{x^t) by u{x,t) + ^ex^ where e > 0, and U{x) is then replaced by Ue{x) = U{x) -f l^x^. If Ue is convex for every ^ > 0 it will follow that U is convex. It is therefore sufficient to prove the convexity of U when (ii) is strengthened to u'^^{x^t) > e and (iii) is strengthened to u'^x.t)
+ 2u'U{x,t)X + < ( x , t ) A 2 >e,
A G R.
CONVEXITY OF THE MINIMUM OF A FAMILY
31
We shall refer to these conditions as (ii)^ and (iii)^ respectively. Let £^ be a compact subset of X such that U' is non-increasing on E, that is, {U\x)-U\y)){x-y)<0 ifx.yeE. We claim that the measure ofU'{E) — {U'{x)\x G E] must he equal to 0. There are three cases to consider: (a) YP\E is discrete, hence countable, since by the first part of the proof U eC^ and U"{x) >emY, (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))+<,(x,a)(x-y) = "^yiy,«) - < ( ^ , « ) - u'^xi^, a){y -x)
= o{y - x).
The terms on the left-hand side have the same sign and \u'^^{x^ a){y-~x)\ > e\y — x\ by (ii)^, which gives a lower bound for \y — x\. The same argument can be applied to the end point b of / . (c) It remains to prove that if EQ^ then
U'{EQ)
{x e E\ 3t{x) G J{x) \ dl, such that u^^{x,t{x)) has measure 0. lix^y
= 0},
G JBQ then
{u\y)-u'{x))-^u'U^Am^-y) = 4(2/,%))-<(^,^(^))-<'.(^,^W)(y-^)-o(|a;-7/| + |t(:r)-%)|), for u'l^{x^ i{x)) == 0 by the definition of EQ, and this implies u'^^{x^ ^(^)) = 0 by (iii). The terms on the left-hand side have the same sign, and the absolute value of the second is at least e\x — y\. Thus we can choose ^ > 0 so small that e\x-y\
< \e{\x-y\^t{x)-t{y)\),
lix.y
G E^, \x-y\
< 6, \t{x)-t{y)\
< 6.
Hence \x - y\ < \t{x) - t{y)\ and \U'{y) - U'{x)\ = o{\t{x) - t{y)\) then. Let if C X be an interval of length < 6, and let / = / i U • • • U /jv where each Ij has length {b — a)/N. When {b — a)/N < 8 it follows that the set U'{{x G £"0 n K]t{x) G Ij]) is contained in an interval of length o{l/N). Hence N
m{U\EonK))
e EonK-t{x)
elj}))
^0,
32
I. CONVEX FUNCTIONS OF ONE VARIABLE
when AT -^ oo, which proves that m(U'{EQ fl K)) = 0, hence that m{U'{Eo)) = 0. If xo,xi G X and XQ < xi but U'{XQ) > C/'(xi), then £; = {rr G [xo,Xi]]U'{x)
> U'{y)
iix
is closed, U' is non-increasing on E , and [C/'(xi), C/'(a;o)] C U'{E), for if U'{xi) < 7 < U'{xo)^ then x = supf^/ G [xo,xi]; [/'(?/) > 7} is in E and C/'(x) = 7. Since U'{E) has measure 0 it follows that U'{xi) = C/'(rro), so U' is non-decreasing, that is, [/ is convex and [/' is differentiable almost everywhere. To prove that U G C^'^ we may assume that \u'J.^\ < M in X x I. If x,y e X we obtain with s G J{x) and ^ G J(2/) [/(a;) < ^(rr, t) < u{y, t) + u'y{y, t){x - y) ^ \M{x
-
yf
= U{y) -f C7'(2/)(rz; - y) + | M ( x - T/)^ Adding the equation obtained by interchanging (x, 5) and (i/, t) we conclude that {U'{x) - U'{y)){x -y)< M{x - y)\ and since U' is increasing it follows that \U'{x) — U'{y)\ < M\x — y\. This proves the Lipschitz continuity of U'. In the remaining part of the proof we assume again only the hypotheses (i), (ii), (iii). Let a; C X be an open interval where there is a function cp G C^(a;) with a < (p{x) < ft, x G a;, and u'f.{x,t) = 0, u'^'^{x^t) > 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) - <'^(x,t)V 0 except when x belongs to a null set. Next consider the closed set Ea of points x £ X\Y such that a G J{x). We have U'{x) = 'u^(x,a), for every x G Ea^ hence U"{x) = u'l^(x^a) if X e. Ea'i^ not isolated and U' is differentiable at x, so the equality fails only in a null set. We can apply the same argument with a replaced by h. Let E be the set of points x £ X\Y such that U"{x) exists and U"{x) ^ u'J.^{x, t) for some t G J{x) \ dl with u'l^{x^t) = 0, hence u'J.^{x,t) = 0. We must prove that E has measure 0. In doing so we may assume that X is compact
and that ueC^iX
x I).
CONVEXITY OF THE MINIMUM OF A FAMILY
33
It is clear that E = Uf^^^jv where E^ consists of all x e E such that for some t{x) E Jx\dl with u^^{x^ ^(^)) = 0 we have \U'{y) - U'(x) - u'Ux,t{x)){y
- x)| >\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\<
S, \t{x) - t{y)\ < S,
and then we have |rr — y| < o(\t{x) — t{y)\). Let X C X be an interval of length < 6 and let / = / i U • • • U /jv where the intervals Ij have length {h - a)/N. When {b - a)/N < 6 then {x e E^H K\ t{x) G Ij} is contained in an interval of length o{l/N), hence AT
m{{xEENnK})
< Y^m{{xeENnK;t{x)
e Ij})
-^0,
when AT —> oo. This proves that m{EN ft if) = 0, hence that m{EN) = 0 and that m{E) = 0. The proof is complete. The extension to subharmonic functions in Chapter III will require a reformulation of Theorem 1.7.1 where the convexity conditions are suppressed: T h e o r e m 1.7.3. If X C R is an open interval, I ~ [a, b] is a compact interval on K, and u G C'^{X x I), then U{x) == miut^/ u{x^ t) is in C^'^{X) if and only if (i) IfxEX then u'^{x^t) does not depend on t when t G J{x) = {t E I\u{x,t) = U{x)}. (ii) For every compact interval K C X there is a constant AK > 0 such that (1.7.1)'
u'^t{^,tf
< AKu'^^{x,t),
ifx eK.te
J{x)\dl.
Then U £ C'^ and (1.7.3) is valid in the open subset Y of X defined by (1.7.2), and (1.7.4) is valid for almost all x G X. Proof. Ii U e C^'^ we can choose A > 0 so that x i-^ U'{x) + Ax is increasing in a neighborhood of K^ which means that U{x) + \Ax'^ is
34
I. CONVEX FUNCTIONS OF ONE VARIABLE
convex there. If we apply Theorem 1.7.1 to v{x^t) = u{x,t) -f- ^Ax"^ then (i), (ii) follow from (i), (iii) there, for (1.7.1) means that v^^ — u'l^ > 0 and that < ? = <*' < <;<* < AKU':, if AK = A-i- sup;^x/ ^xx- Conversely, assume that (i) and (ii) in Theorem 1.7.3 are fulfilled and set v{x,t) = u{x,t) + ^Ax"^. Then (i) in Theorem 1.7.1 is fulfilled by v, and (ii), (iii) follow inKxl ii AK < A-\-mmKxiu'xxSince mint'u(a:,t) = U{x) + \Ax'^ is in C^'^ in the interior of K, it follows that U E C^'^(X). The statements on the derivatives of U{x) follow at once from the results on the derivatives of U{x) + \Ax'^ given by Theorem 1.7.1. Note that the second part of Theorem 1.7.3 gives a precise bound for the Lipschitz constant of U'\ in particular, we obtain inft u'^xi'^^ t) — AK < U"{x) < sup^'Z/^'^(x,t) for almost every x £ K. Corollary 1.7.4. Let u £ C'^{XxI) and assume that u{x, t) is a convex function of x E X for fixed t £ I and a quasi-convex function of t £ I for fixed x E X. If in addition (1.7.1) is valid when (x^t) E X x I and u[{x^t) = 0, then U{x) = miut^/ u{x,t) is a convex function in X. Proof. The hypotheses contain conditions (ii) and (iii) of Theorem 1.7.1, so we just have to verify condition (i) there. By the quasi-convexity Jx is an interval C / . If it is not reduced to a point then u'f.{x,t) — u'l^{x^t) — 0 when t G J^? and by (1.7.1) this implies that u'^^{x^t) = 0 when t E J{x)^ so u'^{x,t) is constant when t E J{x). Corollary 1.7.5. Let u E (7^(X) where X C R^ is an open set such that Tx = {t; {x,t) E X} for every x E H is an interval, possibly empty. Assume that Tx 3 t \-^ u{x, t) is quasi-convex for hxed x, that u'^^{x^ t) >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) Ai^i + 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.
DEFINITIONS AND BASIC FACTS
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 J^i " /^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)
Aio^i 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 ^i " ^' ^^ ^ ^ •
1
1
When N = 3 this means that Ai^i + 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 = Y^i ^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
DEFINITIONS AND BASIC FACTS
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-^i 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
II. CONVEXITY IN A FINITE-DIMENSIONAL VECTOR SPACE
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 X^i 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:,
5^A,-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^ ^o 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:
DEFINITIONS AND BASIC FACTS
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 X^o ^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;j^iXj. Since AT > 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
II. CONVEXITY IN A FINITE-DIMENSIONAL VECTOR SPACE
Write Xi = Xf - A" where Xf = max(±Ai,0).
Then A = E f \ ^
=
E f A- > 0, and N
N
N
N
^ = J2iK/^)^i = E(VA)^^, E(^^^A) = 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:
DEFINITIONS AND BASIC FACTS
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 <
a convex and relatively open subset of V. If XQ ^ X, then there is an affine but W HX = fb. Thus there is an affine f{x), x ^ X.
Proof. It is convenient to place the origin at XQ SO that an affine hyperplane through xo becomes a hyperplane. (a) First assume that d i m F = 2; the theorem is trivial when d i m F — 1. If d i m X = 0 or d i m X = 1, then X is contained in an affine line and the statement is obviously true then. We may therefore assume that d i m X = 2, which means that X is open in V. Then X = {tx; 0
eX}
=
[jtX t>o
is open and convex, and 0 ^ X. In fact, if xi,X2 G X , ti,t2 > 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.
DEFINITIONS AND BASIC FACTS
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')<\x-y\y2t
if
\x - y - tf\
>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
II. CONVEXITY IN A FINITE-DIMENSIONAL VECTOR SPACE
(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 i^i, 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 = ^ajT 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,
DEFINITIONS AND BASIC FACTS
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
k
^0^'
j,k
Hence D^ is the image by an injective affine map of the set in R^'^"^) defined by djk > 0, i, k
^
ajk < 1, A; < n;
^
ajk < 1, i < n;
(2.1.9)' 2 ^ djk
>n-2.
j,k
It has interior points such as ajk — 74/(n—1), j , k < n^ with (n—2)/(n —1) < yl < 1, so Dn has dimension (n — 1)^. For an extreme point there must be equality in at least (n — 1)^ of these (n — 1)^ + 2n — 1 = n^ inequalities, for otherwise we could find (bjk) 7^ 0 so that bjk = 0 resp. ^.^^ bjk = 0 or ^j^^^ bjk = 0 or Ej,fc 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. CONVEXITY IN A FINITE-DIMENSIONAL VECTOR SPACE
(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
3= 1
1
1
(v) (3 = Pa where P = {Pjk) is a doubly stochastic n x n matrix. Proof. That (ii) implies (i) is trivial. Suppose that (i) is valid and let ^ G R'^. If we set Xj = exp{(jt) and consider the growth of the two sides of (2.1.10) as ^ -^ +00, it follows that (2.1.11)
(/?,0 < m a x ( a / ? , 0 < m a x ( a a , 0 ,
which imphes (iii) by the Hahn-Banach theorem. Thus we can write P = ^X,aa,
A, > 0 , ^ A , = 1.
Hence by the inequality between geometric and arithmetic means (Theorem 1.2.1) (7
a
a
and (ii) follows at once. The equivalence of (iii) and (v) follows from Theorem 2.1.14. To complete the proof we must show that (iv) is equivalent to (2.1.11), which follows from the following elementary L e m m a 2.1.16. Ifx^yE Wx,y)
R^ have decreasing coordinates, < {x,y),
then
a G ©n-
Proof. Let x' = ax. If x' is not decreasing, then we can find j < k such that Xj < x'j^, which implies that ^'jVj + ^'kVk < x'kVj + x'jVk since the difference is {yj — yk){x'j — xj^) < 0. If we interchange x'j^ and x'j we therefore do not decrease the left-hand side, and continuing in this way until x' is decreasing, hence equal to x^ we obtain the stated inequality. End of proof of Theorem 2.1.15. (2.1.11) is valid if and only if it is valid when ^ is decreasing, and then it is by Lemma 2.1.16 equivalent to (2.1.11)' (r,o<(«*,oTaking ^j = 1 for all j or ^^ = — 1 for all j , we obtain the second condition in (iv). If we let (,j = 1 when j < k and ^j = 0 when j > A;, then the first n — 1 conditions follow. Conversely, (2.1.11)' follows from (iv) since every decreasing ^ is a positive linear combination of the cases just considered. The proof is complete.
DEFINITIONS AND BASIC FACTS
49
E x e r c i s e 2.1.7. Prove that (hi) =4> (v) = ^ (iv) in Theorem 2.1.15 without using Theorem 2.1.14. Prove that the conditions there are also equivalent to (^i) Z^r fil^j) — Yl^ fi^j) ^^^ every convex function / on R. (vii) /(/3) < f{a) for every convex function / in R'^ which is symmetric, that is, invariant under permutation of the coordinates. Prove that { P a ; a e A and P is doubly stochastic} is convex if ^ C R^ is convex and ai > a2 > " • > 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
^ajk
k = l,...,m-
^
j=i
ajk < 1, j = 1 , . . . , n.
k=i
If O = {Ojk)'j^k=i is ^^ orthogonal matrix, then O^^^ = (0|j^.)^j^^i is a doubly stochastic matrix. Since the orthogonal group 0{n) has dimension n{n — l ) / 2 < (n — 1)^ if n > 2, all doubly stochastic matrices are not of this form if n > 2. (However, if (
^^
^^ | is doubly stochastic, we can
choose 0 G R so that a n = a22 = cos^ ^, a^ — a2\ = sin^ 0, so the matrix is equal to 0(2) with O =
^ " ' \ ' " " : .) Nevertheless, one can for \ ^ - s m ^ cos^y ^ arbitrary n take P of this special form in condition (v) of Theorem 2.1.15: T h e o r e m 2.1.17 ( H o r n ) . The conditions in Theorem 2.1.15 are also equivalent to (viii) /3 = 0(2)^ ^ijg^g Q ^ Q(^)
Proof. That (viii) ==> (v) in Theorem 2.1.15 is obvious since 0(2) IS doubly stochastic; the problem is to prove that (viii) follows from (iv) in Theorem 2.1.15. First we note that (iv) can be written I'S'I
(ivy
n
n
E^^-^E^.*' s^^r.'. E/^i = E^i' jes
1
1
1
where N ^ = { 1 , 2 , . . . , n} and \S\ is the number of elements in S. If 5 ' = Nn\S then n
jes
1
jes'
50
11. CONVEXITY IN A FINITE-DIMENSIONAL VECTOR SPACE
SO (iv)' is equivalent to
n-|5|+i
jes
1
1
We shall prove the theorem by induction with respect to n, so assume that n > 2 and that the theorem has already been proved in lower dimensions. Let /3 satisfy (iv). We may assume that p is decreasing for the composition of an orthogonal matrix and a permutation matrix is an orthogonal matrix. We know from (iv)' and (iv)'' that
(iv)o
E
^.*^E^^-^E^^ 5CN._2,
n-|5| +l
jes
1 |5| + 1
n
(iv)i
E ^.*-E^^-^^-i^ E^.*-E/'^' '5^N.-2n-\s\
jes
1
jes
If 5, 5" are complementary subsets of Nn-25 then n — |S"| = | 5 | + 2 and n
|5'| + 1
n
n-2
E «; - E z'.- + E«; - E 5/^- = E«; - E ^r n-|5'|
jG5'
1
jes
1
1
If we choose S so that the right-hand side of (iv)i attains its minimum a, it follows that the left-hand side attains its maximum b when S is replaced by S'. Note that b < (3n-i < a, and that if | 5 | = k then 5 = { 1 , . . . ,fc}and 5"i={A: + l , . . . , n — 2} since (3 is decreasing. From (iv)o and the definition of a we obtain for s C S \s\
\s\-\-l
|5| + 1
E/?.<E^^ E/^^+^^E^.*' E/5^+^=Es*' jes 1 jGs 1 jes 1 which is the condition (iv)' for (/?i,...,/J^, a) and ( a ^ , . . . , c^X^i)- Similarly we obtain for s C S'
jGs
n-|s|4-l
i^s
n-\s\
jeS'
k+2
which is the condition (iv)" for (/?fc+i5 • • • ,/3n-2 5^-R^2.1.10) and (0^^4.2? • • • jCHn)By the inductive hypothesis it follows that we can find Oi G 0{k + 1) and O2 G 0{n - k~l) such that
(/3i,...,/?,,a)-Oi')(a*,...,^*,^0,
DEFINITIONS AND BASIC FACTS
51
(If A: = 0 or A: = n — 2 then we take Oi resp. O2 equal to the unit 1 x 1 matrix.) Writing 0=1
J'
^
j G 0 ( n ) with block matrix notation, we
have Rotating the k + 1st and the nth row in O in the plane spanned by them we define a one-parameter family of orthogonal transformations O{0) with 0{6)k-^ij
= Ofc+ij cos 6>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
II. CONVEXITY IN A FINITE-DIMENSIONAL VECTOR SPACE
and write Qj{x,y) = \{E{TjX,y) •^E{x,Tjy)), which is a symmetric bilinear form. Then e{9) = cos^ei + 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<9, we have 2Qj{e{e), e{e)) = g , ( e i , ei) + Q,(e2, 63) + (Qj(ei,ei) - Qj(e2,e2))cos20 + 2Qj(ei,e2)sin20. If the determinant
D =
Qi{ei,ei)
- Qi(e2,e2)
Qi(ei,e2)
Q 2 ( e i , e i ) - (92(62, 62)
Q 2 ( e i , 62)
is equal to 0, then the range i?(ei,e2) of {Qi{e{0), e{6)), Q2{e{9), e{6))) is an interval (or a point), but otherwise i?(ei,e2) is an ellipse run through once in the positive (negative) direction if D > 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.
DEFINITIONS AND BASIC FACTS
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:
DEFINITIONS AND BASIC FACTS
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)
+ y)-f{x),
xeX, ^ 1 , ^ 2 6 ^ , x G X,
xeX,
x +
yeX.
For almost all x £ X we have f{x] —y) = —/(x; y) for all y, that is, f'{x] y) is linear in y and f is diiferentiable at x. If fj is a sequence of convex functions in X such that fj{x) -^ f{x), x E X, then \hR f'jix;y)
yeV,xeX.
If f is differentiable at x, then fj(x; y) -^ f'{x] y) for all y
EV.
Proof. If £• is a finite subset of X and Y — chE, then My = supy / = sup^; f < 00, and if x ±y EY and h G [0,1], we have ifix)
- My)h
< ifix)
- fix - y))h < fix + hy) -
fix)
y)-fix))h
which proves Lipschitz continuity in the interior of S. The existence of the hmit follows from Corollary 1.1.6, and it is obviously a convex function of y, hence continuous, and positively homogeneous. By Theorem 1.1.5 the difference quotient {f{x + hy) — f{x))/h decreases as /i | 0, which implies the uniform convergence on compact sets and (2.1.12). I f e i , . . . , e n is a basis for V and x i , . . . ,Xn corresponding coordinates, we know from Section 1.1 that f'{x]—ej) = —f'{x;ej) except for countably many Xj when Xk is fixed for k ^ j . Hence it follows from the Fubini theorem that / ' ( x ; —Cj) — —f'{x\ Cj), j — 1 , . . . , n, for almost all x £ X. For such x it follows that n
n
1
1
is a convex function vanishing on the coordinate axes. Thus g{y) < 0 for all y G R"", and since 0 = g{0) < \{g{y)-^g{—y)): it follows that g vanishes identically. Thus / is differentiable at almost every x EV . (We could also
56
II. CONVEXITY IN A FINITE-DIMENSIONAL VECTOR SPACE
have appealed to Rademacher's theorem on differentiability of Lipschitz continuous functions, but it is less elementary.) Since fj{x] y) < fj{x + y) - /j(x),
X eX,
x-hy
eX,
we obtain when j -^ +00 if fj -^ f ^
fj{x- y) < fix + y) - fix) = fix-, y) -f oi\y\),
xeX,
x-hyeX.
J—»-00
If we replace y by hy^ divide by h, and let h -^ +0, it follows that
]hiif'jix;y)
> lim -f'^ix\-y)
= -fix;-y)
=
fix\y),
which completes the proof. From Corollary 1.1.10 we conclude at once that if / G (7^(X) where X is open and convex, then / is convex if and only if the second differential fix) is positive semi-definite for every x G X . One calls / strictly convex if f'\x) is positive definite for every x E X. E x e r c i s e 2.1.12. Prove that if u G C^(X x I) where X is an open convex set in R^ and / a compact interval on R, then C/(x) = minii(x,t),
x E X,
is convex if and only if i) li X E X then u'^ix^t) does not depend on t when t G Jix) = {t E
I\uix,t) = Uix)], ii) If a: G X and t E Jix) then n
Y^
XjXkdjdkuix.t)
>0
when A G R^"^"", x E X, tE
Jix),
j,k=0
provided that AQ — 0 if t G dl. Here dj = d/dxj when 1 < j < n and ^o = d/dt. Prove that U E C^^'^iX) then. (This is an extension of Theorem 1.7.1. State and prove similar extensions of Corollaries 1.7.4 and 1.7.5 also.) We have seen that to every convex function there is associated a convex set, the epigraph. It is also possible to associate convex functions with convex sets.
DEFINITIONS AND BASIC FACTS
57
T h e o r e m 2.1.23. Let B bea compact convex set in a finite-dimensional vector space V having the origin as an interior point. Then (2.1.13)
F{x) = mi{t > 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)
x,yeV,
and (2.1.15)
B = {xeV',F{x)
< 1}.
Conversely, ifF is a function satisfying (2.1.14) and B is defined by (2.1.15), then (2.1.13) holds. Proof. It is obvious that (2.1.13) implies the first two parts of (2.1.14), and that (2.1.15) holds. Since B is convex we have F{Xx -h fiy) < 1 if F{x) < 1, F{y) < 1 and X,fi>0,X-\-/i = l. By the homogeneity it follows that F{Xx + fiy) < (A + //) max(F(x), F{y)), if A, /i > 0. li x,y ^ 0 we apply this with x,y replaced by x/F{x),y/F(y) and A,/x replaced by F{x),F{y), which gives the third part of (2.1.14). It is clear that (2.1.14) implies that F is convex, hence that (2.1.15) defines a compact convex set with the origin as interior point, and it is clear that (2.1.13) is then valid. The proof is complete. One calls F the distance function corresponding to B, and we shall sometimes write \X\B for the function defined by (2.1.13). If 5 is symmetric and only then we have \tx\B = \t\\x\B^
V^ G R, X eV,
and \X\B is then called the norm with unit ball B. T h e o r e m 2.1.24. If X is an open convex set in V, X ^ V, B is a compact convex neighborhood of 0 in V, and (2.1.16)
dx,B{x) = inf \x - y\B,
x e X,
58
11. CONVEXITY IN A FINITE-DIMENSIONAL VECTOR SPACE
is the corresponding boundary distance, then dx,B is a concave function in X with boundary values 0. Proof. If X is a half space {x G F ; L{x) > 0}, where L is a linear form 7^ 0 on y , then dx,B{x) = AL(x), x e X^ for some A > 0. In fact, HXQ E X and L{xo) = 1 we can write x = txQ H- z where t — L{x) and L{z) = 0, which gives that dx,B{x) is equal to inf
\txo-\-z - y\B =
Liy)<0
inf
\txo - y\B = t
L{y)<0'
inf
\xo - y\B = L{x)X.
L{y)<0'
Thus dx,B is then a linear function. If we prove that dx,B{x) = inf
rfi/,5(x),
X
eX,
where H denotes an arbitrary half space D X, it follows that dx^B is concave. It is obvious that dx,B ^ dn.B for every H D X^ so we must prove that there is equality for some H. Given x E X we can choose y E dX so that \x - y\B = dx,B{x)' By Theorem 2.1.10 there is a half space H D X with y on the boundary, which implies that dH,B{x) <\x-
y\B =
dx,B{x)
and completes the proof. Exercise 2.1.13. Prove with the notation in Theorem 2.1.24 that — log dx,B is a convex function in X which —> oo at the boundary. It follows from Corollary 1.1.12 that convexity is a local property in the following sense: If / is a function defined in a convex set X and to every point in X there is a convex neighborhood U such that the restriction of / to X n (7 is convex, then / is convex in X. We shall now show that under suitable additional conditions one can infer the convexity of X from the existence of a locally convex function. T h e o r e m 2.1.25. Let X be an open connected subset of a Rnitedimensional vector space V, and let f be a non-constant locally convex function in X, that is, assume that every point in X has a convex neighborhood U C X such that f restricted to U is convex. Let M = s u p ^ / , and assume that Xt = {x £ X; f{x) < t} is compact for every t < M. Then X is convex, and f is a convex function in X. Even iff is just quasi-convex on every line segment contained in X, it follows that X is convex. Proof. Since X is connected we can join any pair of points in X by a polygonal arc contained in X . To prove that X is convex it suffices to show that the number of corners can be reduced, that is, that if [rcoj^i] —
DEFINITIONS AND BASIC FACTS
59
ch{a:o,^i} and [a:i,a;2] = ch{xi,a;2} are contained in X , then [0:0,3:2] C X. Let t be the maximum of / in [xo,a;i] U [a;i,rz;2]; we have t < M since / would otherwise be constant in X. Set A = {A; 0 < A < 1, [(1 - A)rro + XxuX2] C X}. By hypothesis 1 G A and we must prove that 0 G A. Since X is open it is clear that A is open in [0,1]. If A G A t h e n /
is compact for every t < s u p / .
(v) There exists a non-constant function / in X which is quasi-convex on every line segment C X , such that {x G X; f{x) < t}
is compact for every t < s u p / .
Proof, (i) =^ (ii) by Theorem 2.1.24, and (ii) =^ (iii) is trivial. Let \x\ be any norm in V and let R > 0. If (iii) is fulfilled, we can find ^^ > 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
II. CONVEXITY IN A FINITE-DIMENSIONAL VECTOR SPACE
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)<0,
for some t G R"^. Thus {x;g(x) < 0} is contained in X near z and has there a C^ boundary with negative normal curvature in the direction t. Conversely, X is not convex if there exists such a polynomial. Proof. The last statement is trivial, {or z:ket ^ X for small e although z G dX. Let B be the Euclidean unit ball. If X is not convex, then dx,B is not locally concave, so it follows from Theorem 1.1.15 that we can find xo G X and 2/ 7^ 0, c G R and £: > 0, 5 > 0 such that (2.1.17)
dx,B{xQ + hy) > dx,B{xo) + c/i + 6/i^
\h\ < 6.
Choose ZQ G dX so that dx,B{xo) = \xo — zo\- Then (2.1.17)'
\zo - xo\ + c/i -h eh'^ < dx,B{xQ + hy) < \zo -xo - hy\ = \zo -xo\ -h{p,y)
+ 0(|/ip),
where v — {zi^ — xo)/|2:o — xo|. Hence c = —{v^y) and y = —cv -h t where (t,iy)=0.
If \x — X{) — hy\ < \zo — xo\ -}- ch + eh"^ for some h G [—6,8], then x E X by (2.1.17), so a; G X if (2.1.18)
(p{x) = min(|a; — XQ — hy\ — \zo — xo\ — ch — eh?) \h\<.8
is negative. We have ip{zQ + ht) < 1^0 — XQ ^ chv\ — \ZQ — xo\ — ch — eh? = —eh^,
if \h\ < ^,
DEFINITIONS AND BASIC FACTS
61
and (p{zo) = 0 since ko ~
XQ
— hy\ — \zo — xo\ — ch — eh? > 0,
\h\ < 5,
by (2.1.17)'. If we assume that (2.1.17)' is valid with e replaced by 2e, which is legitimate since we can replace our original 5 by 6:/2, then it follows that \zo —
XQ
— hy\ — \zo — xo\ — ch — eh^ > 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|<2
we have q{x) > 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
II. CONVEXITY IN A FINITE-DIMENSIONAL VECTOR SPACE
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\<{\XI\'-Q')/20,
Now consider the subset of balls B{w,r) (2.1.20)
B{w, r) D 5 ( 0 , g),
r2<(|xi|2 + ^2)2/Vsuch that B{w, r)nF
= ili.
By (2.1.19) the corresponding set of (w^r) G R'^"^^ is bounded, and it is closed, hence compact. Choose B{wo^rQ) satisfying (2.1.20) with maximal
DEFINITIONS AND BASIC FACTS
63
VQ. The maximality implies that B{wo,ro) D F contains some point yi, and we want to prove that there is another. Assume the contrary, that B{wo^ro)r\F = {yi}. HO is any vector with (O^Wo — yi) > 0, it follows that B{wo + sO^ro) n F = 0 for small e > 0, for the only possible intersection is then near yi where B{wo -\- eO^ro) is contained in B{wo^rQ). By the maximality B(wo -\-s0,ro) must intersect B{0,g), so there is some point 2/2 ^ 5B(0, g) n dB{wo,ro)^ and it is necessarily unique since ro > g- If we take for 0 the vector 2/2—2/1, we get a contradiction because B{wo + eO, ro)r\ 5 5 ( 0 , g) = ^ for small 5 > 0, since 6 points out of B{wojro) at 2/2. (Draw a figure!) Hence B{wo^ro) fl F contains at least two points so F does not have the unique nearest point property. The proof is complete. Remark, The theorem is valid more generally for the distance function defined by a strictly convex set B with C^ boundary, but it is not true otherwise. We shall end this section with an example from the theory of hyperbolic differential operators. P r o p o s i t i o n 2.1.31. Let p be a homogeneous polynomial of degree m in a (Bnite-dimensional) vector space V, let 6 e V and assume that p is hyperbolic in the sense that p{6) ^ 0 and that the equation p{x -\-t9) = 0 of degree m has only real roots for every x ^V. Then the component T of 0 in {x G V]p{x) / 0} is a convex cone, the zeros oft\-^ p{x -h ty) are real if X £ V and y E T, the polynomial p{x)lp{6) is real, it is positive when X G r , and {p{x)/p{0)y^'^ is concave and homogeneous of degree 1 in T, equal to 0 on the boundary Proof. (See ALPDO, Lemma 8.7.3.) To simplify notation we assume thdit p{9) = 1. Then p{x^t0) = UTi^-'^j) with realty-, sop{x) = UTi-tj) is real. Set T0 = {xe V;p{x -\-t0)^O, t> 0}. Then TQ is open, and 0 eVg since p{0-\-t0) = {l-{-t)'^p{0) only has the zero t = - l . lix e 17, then p{x +10) ^ 0, when t > 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 Ey^e = [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 Ey^e, then p{x + ie0 + isy) 7^ 0 by Hurwitz' theorem if
64
II. CONVEXITY IN A FINITE-DIMENSIONAL VECTOR SPACE
Re5 > 0, and this remains true when R e s = 0 since x + isy is real then. Hence Ey^e is both open and closed, so Ey^e — 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
DEFINITIONS AND BASIC FACTS
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
II. CONVEXITY IN A FINITE-DIMENSIONAL VECTOR SPACE
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)
is a convex set, and f is continuous in the interior of X in ah(X). Proof. It is obvious that X is convex. When proving the other statement we may assume that ah(X) == F , that is, that X has interior points. But then it follows from Theorem 2.1.22 that / is even Gateau differentiable in the interior of X. At the boundary points of X in ah(X) it is not always possible to redefine / so that / becomes continuous with values in (—oo,-foo]: E x a m p l e . f{x,y) = (x^ + y^)/x is convex when 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)
ev^ii-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
11. CONVEXITY IN A FINITE-DIMENSIONAL VECTOR SPACE
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
^E/.-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).
THE LEGENDRE TRANSFORMATION
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 Yl^eA -^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
II. CONVEXITY IN A FINITE-DIMENSIONAL VECTOR SPACE
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(^i) + 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
C^O.
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.
THE LEGENDRE TRANSFORMATION
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;^eF',(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, y^ip{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
11. CONVEXITY IN A FINITE-DIMENSIONAL VECTOR SPACE
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 = {^;^eV',{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,<^), y E V^ which means that ^ G Ki.
Hence K = i^i as
We shall now discuss some applications of the Legendre transformation, beginning with two-person zero sum games. Two players A and B can bet respectively on m alternatives, denoted by A i , . . . , A ^ , and n alternatives denoted by Bi^... ^B^. After both have made their bets they announce them and A pays B the amount ajk (positive or negative) in a given matrix if the bets have been Aj and B^. The problem is to decide with which frequencies A and B should bet on the various alternatives. Suppose that A and B bet on Aj and B^ with probabilities Xj and y^, and write x = ( x i , . . . , Xm)^ y — iv\->" "> 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
= ^^ay^x^j/fc.
THE LEGENDRE TRANSFORMATION
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
The function / i is the supporting function of the convex hull Z of a i , . . . , ttn € R"^, so / i ( 0 = 0 if ^ G Z and / i ( 0 = +00 if ^ ^ Z. Let f2{x) = 0 when x E X and f2{x) — +00 when x ^ X. Then /2 is convex and lower semi-continuous, and f2{0 ~ ^^^^i<j<m^j- We have minmax A{x,y) xex yeY
= min {fi{x) -f f2{x)), xeti'^
and since /2 is finite everywhere, it follows from Corollary 2.2.6 that niinm a x ^ ( : r , ^ ) - - ^eti^ min ( / i ( 0 + / 2 ( - 0 ) = -^}l^,^^J^ xex yeY t,ez i<j<m '^J = max min Ci = m a x m i n ( x , 0 Cez i<j<m -^ iez xex Now $, E Z means that (^ = ^^ akyk for some y G F , so (2.2.15) is proved.
74
II. CONVEXITY IN A FINITE-DIMENSIONAL VECTOR SPACE
Exercise 2.2.5. Show that if in Theorem 2.2.12 x^ is in the relative interior of X, then A{x,y^) — sY^ ^j-^ ^ ^ R-"^- Show when n = m = 2 that 5 is either one of the numbers ajk or else 5 = {aiia22 - aua2i)/{aii
+ a22 - CLU - ^21),
where the denominator is not 0. Next we shall discuss a related problem in linear programming, which we state directly in the mathematical form: T h e o r e m 2.2.13. Let b E R'^ have positive coordinates, tti,... R"^ \ {0} have non-negative coordinates, and c G R"^ . Then (2.2.16)
inf (x, b) = sup{y, c);
where
K = {x e R'^; X > 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, ] ^ A^a^ > 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 {y^a^) < 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 =
^IHKI
+
^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
IL CONVEXITY IN A FINITE-DIMENSIONAL VECTOR SPACE
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 ^iHi + X2H2. I f ^ G F ' \ { 0 } , then (2.3.2) {x e Aiifi + A2if2; {x^O = A i i i i ( 0 + X2H2{0} = Aiifi(e) 4- X2K2{0.
K,{0 =
{^eKj;{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
GEOMETRIC INEQUALITIES
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-^oo. 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<^), defined by the volume function is called the mixed volume of i f i , . . . , KnAs an example, we see that for the case of two convex bodies (2.3.3)
nQ3(5, K , . . . , K ) -
lim {^{K + sB) -
^{K))/e.
The (mixed) volume is translation invariant so we may assume that B contains the origin. The right-hand side is then non-negative, and we shall prove below that this is always true for the mixed volume. When V is Euclidean and B and K both have smooth boundaries with positive Gaussian curvature, it is easy to show that (2.3.4)
n^iB,K,...,K)=
[ JdK
HB{C{x))dS{x)
where dS is the Euclidean surface area and ^{x) is the exterior unit normal n{x) of dK at x, identified with an element in V using the Euclidean metric. In the proof of (2.3.4) we may assume that 0 is an interior point of both B and K , for a translation of B by the vector v just means adding to the right-hand side the integral / {v,n)dS JdK
= 0,
which vanishes by Gauss' theorem since divt; = 0. The Gauss map dK 3 X i-^ (^{x) is a diffeomorphism on the unit sphere since the Gaussian
78
II. CONVEXITY IN A FINITE-DIMENSIONAL VECTOR SPACE
curvature is positive, and since HK{C{X)) = {x,^{x)) it follows that HK restricted to the unit sphere is C^. The same is true of HB- Let (p be the inverse of the Gauss map from dB to the unit sphere. It follows from (2.3.2) that the boundary of K -{- eB is {x + £(p{({x))]x G dK}. Locally we parametrize dK as x = / ( y ) , where y E cu^ 3. compact set in R"^"^, and we set Fit,y) = f{y) + Maf{y)))Then the part of {K + sB) \ K "above" /(a;) has the volume (2.3.5)
r
dt f
Jo
\DF{t,y)/D{t,y)\dy.
Juj
Here DF{t,y)/D{t,y) = det{^iaf)),df/dyi,...,df/dyn-i) when t = 0. Since HB{0 — i^iO^O^ i^ follows that the absolute value of the determinant is equal to HB{C{f)) times the (n — l)-dimensional volume spanned by the tangent vectors df/dyi,... ^df/dyn-iHence (2.3.5) divided by e converges to Jrr^^HB{^{x))dS{x) when e —> 0. Decomposing the boundary dK in suitable coordinate patches we conclude that (2.3.4) holds when both B and K have smooth boundaries with strictly positive curvature. In particular, the mixed volume is the surface area divided by the dimension if B is the unit ball. The condition on B can easily be removed using Lemma 2.3.2, for if B^ I B is chosen according to the lemma so that the boundary of B^ is C^ with strictly positive curvature, then i/^^ -^ HB locally uniformly, and V{XiK-hX2B^) -> Q3(AiK +A2B) for arbitrary A^ > 0. The extension to a general K requires a preliminary discussion of surface area and tangent planes for general convex compact sets (with interior points). Let K E )C have interior points. If we introduce orthonormal coordinates xi,x' = ( ^ 2 , . . . x^) in F , then the projection K' = {a:';(xi,x') E K for some xi} is obviously convex and compact in R^~^, and K = {{xux'y,x'
e K', f_{x')
< xi <
f+{x')}
where =F/± are convex in K\ If A: is a compact subset of the interior of K'^ it follows from Theorem 2.1.22 that f± are Lipschitz continuous in k and differentiable almost everywhere in k. We define the surface measure dS in dK over k so that J cpdS is equal to J cp{U{x'),x')^l
+ \f'4x')\^dx'
+ I
+
UU^'Wdx',
GEOMETRIC INEQUALITIES
when cp e
CQCR
79
X k), and define the exterior unit normal by
n{Mx'),x')
= ±(1, - / 4 ( x ' ) ) / ^ l + |/i(x')|2
almost everywhere on dK with respect to dS. It is classical that these local definitions fit together to a global definition of dS and n when dK is smooth. That this is true in general follows if we show that for every sequence K^ | K with dK^ £ C^ having positive Gaussian curvature the integral j^{f^{x'),x'mnU±{x'),x'))^l
+ \f^{xTdx',
if e C ° ( R x k),
is the limit as /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
IL CONVEXITY IN A FINITE-DIMENSIONAL VECTOR SPACE
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 ^
{^uMK{^,))
where d denotes (n — l)-dimensional volume in a hyperplane, K{0
=
{X£K;{X,0=HK{0},
GEOMETRIC INEQUALITIES
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 oiHKy^i^iv) here is non-negative. If Kn C K'^ then HK^ < HK^, SO it follows that ^{Ki,.. .,Kn-i,Kn) < QJ(i^i,..., 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 r, for r = 1 , . . . , n. Proof, (ii) implies (i), for if K'j = ch{0,ej} then V{Ki,...,Kn) > ^{K[,...,K'J. Since 53(AiK( + --- + A„K;) = A i . . . , A „ | d e t ( e i , . . . , e „ ) | , it follows that I d e t ( e i , . . . , en)\ = nm{K[,
...,K)<
n!QJ(ifi,..., K , ) .
To prove that (i) implies (iii) we shall prove that 9J(iiri,... ,ifn) == 0 if, say, the sets iiTi,..., if^ are contained in a subspace W of dimension r — 1. Let S be a ball in W containing i f i , . . . , X^. Then 5 J ( A i K i + . • • + XnKr,)
< 53((Ai + • • • + Xr)B + Xr-^iKr+i
+ •••+
X^K^),
82
II. CONVEXITY IN A FINITE-DIMENSIONAL VECTOR SPACE
which is 0(Ai -f • • • -|- A^.)'^"^ when A i , . . . , A^ -^ oo for fixed A^_|_i,..., A^. Hence the polynomial ^{XiKi + • • • + )^nl^n) cannot contain any term of degree r with respect to A i , . . . , A^., so 2 J ( X i , . . . , Kj^) = 0. It remains to prove that (iii) implies (ii). Let Wj be the linear hull of Kj. We have to show that if the dimension of every sum Wi^^ -f . . . + Wi^, where 1 < ii < 12 < • • • < V < n, is at least r, then we can find onedimensional subspaces Wj C Wj for which this remains true. The space Wi only enters the conditions when H = 1, and then Wi^ + • • • + Wi^ has dimension at least r — 1. Thus Wl + Wi^ H f- Wi^ has dimension > r* if the line W{ is not contained in a certain hyperplane in Wi, and altogether the conditions are preserved if W[ just avoids a finite number of hyperplanes in Wl. Continuing in this way we can replace all Wj by one-dimensional subspaces Re^ without loss of the hypothesis on dimensions, which proves (ii) and the theorem. The following theorem, due to Fenchel and Alexandrov, will prove that mixed volumes satisfy all the assumptions of Proposition A.2. It will therefore contain numerous inequalities of the form (A.8), (A.9), (A.10). T h e o r e m 2.3.7. IfKi, ....Kn^lC, then (2.3.8) 9J(Kl,K2,i^3,...,i^n)'>23(i^l,i^l,if3,...,i^n)2J(X2,if2,if^ The proof of this theorem and some applications of it will occupy the rest of this section. The case n = 1 is trivial, and the case n == 2 is exceptional and more elementary since by Proposition A.l it does not really go beyond the classical isoperimetric inequality. We shall prove it starting from a beautiful inequality essentially due to Bonnesen: T h e o r e m 2.3.8. Let B he the unit disc in R? and let K G /C(R^) have interior points. Then (2.3.9)
QJ(K -f XB) < 4A93(K, B),
ifr < A < i?,
where r is the largest radius of an inscribed circle and R is the radius of the smallest circumscribed circle. Before the proof we recall that 22J(if, B) = L{dK) so explicitly (2.3.9) states that (2.3.9)'
Q3(K) - XL{dK) + TTA^ < 0,
is the length of dK,
if r < A < i?.
If K is not a disc then r < R and we must have strict inequality for r < A < i?, since the leading coefficient is positive. Note that for the discriminant of the polynomial in (2.3.9)' we have L{dK)^
- 47r2J(X) > n'^{R -
rf
GEOMETRIC INEQUALITIES
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
II. CONVEXITY IN A FINITE-DIMENSIONAL VECTOR SPACE
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
GEOMETRIC INEQUALITIES
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)<^.
if /
z{OD{m{i\...,
Kio, Id) du^io = 0.
Here Z is homogeneous of degree 1 and C^ outside 0. As an orientation we consider first the case where K2^... ,Kn are all equal to the Euclidean unit ball, which means that iJ2(0 = • • • = Hn{0 ~
H{0 = 1^1 and that
If ^ = (1,0,..., 0), for example, then 2'"(^) and iif"(0 have only zeros in the first row and column by the homogeneity, so the coefficient of Ai • • • A^ in D{\iZ"iO + (A2 + • • • + Xn-i)H"iO + Xn Id) is equal to the coefficient of Ai • • • A^-i in the determinant of the other rows and columns in XiZ''{() + (A2 + • • • + A^-i) Id, which is (n - 2)! ^ 3 ^jjiOHence (2.3.11)
DiZ"{0,H"{0,...,H"iO,Id)
= AZ/inin
- 1))
where A is the Laplace operator, for Zii{^) = 0. In particular the righthand side is equal to l/(n|^|) when Z = H. In view of the orthogonal invariance (2.3.11) holds for all ( G S^~^, hence (2.3.12)
n(n-l) /
Z{OD{Z'\O.H'\0.-•
• ,H"{OM) = {A^Z,Z) +
du;{0 {n-l){Z,Z)
where A^; is the Laplace-Beltrami operator in S'^~^ and the scalar products in the right-hand side are taken in L^{S'^~^). (Recall that A = r~'^A^ + {n — l)r~^d/dr-\-d'^/dr'^ in polar coordinates.) The eigenvectors of A^; are the spherical harmonics, of degree /x — 0 , 1 , . . . , with corresponding eigenvalues -/x(/i -f n - 2). (See e.g. ALPDO, p. 111:55.) Thus (2.3.12) is positive when fi = 0, vanishes when fi = 1 and is negative when ji > 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)
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II. CONVEXITY IN A FINITE-DIMENSIONAL VECTOR SPACE
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 + ^2H^iO + ' ' • + 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 ^i = 0 is obtained when Z"{^) is replaced by —6 (8) ^, so it is equal to
(2.3.13)
e^
-Did®e,H'^io,...,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 is positive semidefinite and 7^ 0 for real 6 ^ 0, it follows from Corollary 2.1.32 (applied after removal of the first row and column) that (2.3.13) is negative when 6^0^ which proves the ellipticity. We shall now examine the eigenvalues of the quadratic form in (2.3.10) with respect to a form equivalent to the square of the L^ norm, (2.3.14)
Z^
[
gZ^duiO,
where ^ > 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.
GEOMETRIC INEQUALITIES
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
II. CONVEXITY IN A FINITE-DIMENSIONAL VECTOR SPACE
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)*,
i^i,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
GEOMETRIC INEQUALITIES
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
II. CONVEXITY IN A FINITE-DIMENSIONAL VECTOR SPACE
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 = K^i
+ • • • + 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
When z/ = n — 1 we see again that nW\ = J Ri- • • Rn-i dcu is the area of dK. More generally, introducing du — KAS = {R\ • - • Rn-i)~^ dS^ where dS is the surface measure on dK^ we obtain (2.3.20)'
Wn-r.
= - J ^
/
n[ ^ ) JdK
{/^l, . . . , f^n-l-u}
dS,
V < U,
GEOMETRIC INEQUALITIES
where
KJ j
sire
91
the principal curvatures. In particular,
n{n - 1)
JQK
is proportional to the integral over dK of the mean curvature. When 0 < u < n we also have
Jsn-l
^
V f—1
' ^
—
/n-i\
V n—v
/
H{Ri,...
'
,Rj,-i}du.
In particular,
Wn-i = 11
H{i) du{i) = ^ [
{H{0 + H{-0) duiO^
Here H{$^)-hH{—^) is the distance between the two supporting planes with normals ± ^ , so it is the width oiK in the direction ^. Since J\t\^i du{^)/n = Cn-, it follows that 2Wn-i/Cn is the average width of K. When n = 3 we have two different expressions for W2 and conclude that
JdK
J\i\=^
is 47r times the average width. We have derived all these formulas for K E /Creg. However, since the measures {ACI, . . . ,Kn-i-iy} dS and { i ? i , . . . ,Ri^} du) are positive on dK and S'^~^ respectively and the mixed volumes extend by continuity to general K E IC (with non-empty interior), it is easy to give a general definition of these positive measures. The problem of determining K from these measures has been extensively studied as models of fully nonlinear elliptic equations, but we cannot pursue these developments here. The mixed volumes Wi, with u odd can be defined in terms of the inner geometry of the boundary dK. To verify this we first note that ii K ^ /Creg and v{r) is the polynomial equal to V{K + rB) when r > 0, then
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II. CONVEXITY IN A FINITE-DIMENSIONAL VECTOR SPACE
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
dF/dxj = fj + j^Udjn,
p
n
= /i - E
E
1^=1
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^ = Y^i^i ^ijQ^''Hence the volume of the tube over f{uj) is
(2.3.21)
/ ,/^dx I JLJ
det((5,-,
J\t\
-Ylt,{h),n,))l^^^dt. ^^1
The integrand is a polynomial in t of degree n, so it is clear that the integral will be a linear combination of powers 7*^+^ with a even and 0 < a < n. Now we have for even integers a >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
GEOMETRIC INEQUALITIES
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
^'^"'
n
^ ^ -
Q
where Cp is the volume of the unit ball in KP and
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11. CONVEXITY IN A FINITE-DIMENSIONAL VECTOR SPACE
the summation is taken over all sequences a i , . . . a2a and / ? i , . . . , ^2a of 2a different numbers between 1 and n, and sgn (^) is 0 unless they are permutations of each other and equal to the sign of the permutation then. The measure dvol is the Riemannian volume density on M, so VQ is the volume of M, and R is the Riemann curvature tensor. In the particular case of a hypersurface we have p = 1 and Ci = 2, so half the volume is
Y, r-i'^'"K/(2a-M)!!. 0
li K e /Creg(R'^) and M = dK, of dimension n - 1, this is equal t o
(2.3.24)
H^,,« =
(2mn-l-2.y.^^^ n!
that is, (2.3.25)
W2a+i = 2~''{n-l-2a)\/nl-
When cr = 0 we just get the surface area divided by n, as we already knew. W^hen (7 = 1 and n = 3, the left-hand side is the volume 47r/3 of the unit ball in R^ and the integral in the right-hand side is 4 times the integral of the total curvature, so the equality follows from the Gauss-Bonnet theorem. 2.4. S m o o t h n e s s of convex sets. In Section 2.3 we worked t o a great extent with convex sets having a smooth boundary and strictly positive curvature. The latter assumption is essential to prevent loss of smoothness in the natural operations in convexity theory. We shall discuss this point here in the case of projections, to give a background for later related questions. (See also Section 1.7 and Exercise 2.1.12.) Let TT be the projection R'^"'"^ 3 ( x i , . . . ,3:^+1) •—^ ( a ; i , . . . , x ^ ) £ R ^ along the direction e^+i = ( 0 , . . . , 0 , 1 ) G R ' ' ^ ^ If K G /C(R^+i) it is obvious that TTK G / C ( R ^ ) . If the interior K° oi K is not empty, then 7rK° is equal to the interior of TTK, which is therefore not empty. That it is a subset is obvious. On the other hand, an interior point x G TTK is in the interior of a simplex with vertices in TTK, and these can be approximated arbitrarily closely by points in irK^. Hence x G TVK^. P r o p o s i t i o n 2.4.1. Let K G 1C{W^^^) have non-empty interior K^. Assume that dK G C^ for some fi >2 and that the Gaussian curvature is
SMOOTHNESS OF CONVEX SETS
95
strictly positive at x e dK if {x -h Rcn+i) fl ii:° = 0. Then dnK is in C^ and has strictly positive Gaussian curvature. Proof. liy £ OTTK^ then y = nx for some x G OK for which the hypotheses in the proposition can be appUed. We assume that x = 0 to simpKfy notation. Then K fl Re^+i = {0}, for the intersection must be contained in dK and if it is an interval not reduced to a point, then the normal curvature of dK in the direction Cn^i at 0 would be equal to 0. After suitable relabelling of the coordinates x i , . . . , x^ we may assume that K is defined near 0 by xi > f{x2,...,
Xn+i),
where /(O) = 0,9n+i/(0) = 0.
The points near 0 E R'^"^^ mapped to dnK must also satisfy the equations (2.4.1)
Xi = f{x2,
. . . , X^ + l ) ,
dn-^if{x2,
. . . , Xn-^l) = 0.
These imply that e^-^i is parallel to the supporting plane oi K at re, so they imply that TTX G dirK. Using the implicit function theorem we can solve the second equation (2.4.1) for x^-fi near 0, obtaining Xn-\-i = X{x2,.. • ,Xn) with X G C^~^. Hence OTTK is defined near 0 by (2.4.2) Xi = F{X2, ...,Xn)
= f{x2,
. . . , Xn,X{x2,
. . • , Xn)) = m i u f{x2,
. . • , ^n+l)-
Xn+l
Using (2.4.1) we obtain djF = djf when 2 < j < n, which proves that F eCf^ and that for 2<j,k
= djdkf -
djdn+ifdkdn+if/dl^^f,
where the last equation follows if the equation dn-\-if{x2^..., differentiated with respect to Xk- Thus we have n
n
Yl yjVkdjdkF^ j,k=2
x^, X) = 0 is
n
Y^ yjVkdjdkf j,k=2
iY,yAdn+iff/d^+J 1=2
2.4.3 ^
^
n+1
= min V
yjVkdjdkf.
This is positive by assumption, which completes the proof. The preceding result is of course just a trivial case of Theorem 1.7.1. It was presented as a brief background of the main point in this section which is an example due to Kiselman which shows that even when n — 2 and dK G C^ the boundary of -KK may not be in C^. In particular it shows that C^'^ cannot be replaced by C^ in Theorem 1.7.1 even if IA G C ^ is a convex function of (x,t).
96
II. CONVEXITY IN A FINITE-DIMENSIONAL VECTOR SPACE
E x a m p l e . With the notation in the proof of Proposition 2.4.1 let K be defined near 0 G R^ by (2.4.4)
xi > f{x2, xs) = xlv{x3) -f u{x3),
where u,v e C"^, u{0) = u'{0) = 0 and u'ixs) ^ 0 when xs ^ 0. For / to be convex we need to know that u is convex. To examine if the first term in / is convex we form the second differential '^ylvixs) + ^y2y3X2v'{x3) +
ylxlv'^xs).
It is non-negative if t; > 0, v'' > 0 and 2v''^ < v"v^ which means that {l/vy < 0. As an example we can take v{xs) = 1 + ax^ + 6x3 in a neighborhood of 0 if a^ < b. Taking a < 0 we have -^'(0) < 0. The minimum F{x2) in (2.4.2) is attained when 53/(2:2,^3) = X2v'{xs) •i-u'{xs) = 0. If X2 7^ 0 then the left-hand side is negative when xs = 0, and for fixed small 0:3 > 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')\
< 6}, /(O) = 0,
\x'\<8Ay'\<6.
Then \f'{x')\ < C\x'\ and it follows from Taylor's formula that \f{x')\ < (7|x'p/2. The ball with center (i?, 0) and radius R is defined by 2xiR > | x p , and this implies that xi > 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 r^i-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 ^i > / ( ^ ' ) with / convex, iiU = {x; |:z;| < R}. Since 0 < f{x')
Vi?2 - |rr'|2 = |x'|V(i? + Vi?2 -
\x'\^),
it follows that / is differentiable at 0 with f'{0) = 0. For the same reason / is differentiable at every point with \x'\ < i?/2, say, and since the tangent plane does not intersect the ball we have fix')
+ {fix'),
y') < \x' + y'l Vi?,
if \x'\ < R/2,
\y'\ < R/2.
98
II. CONVEXITY IN A FINITE-DIMENSIONAL VECTOR SPACE
If we take 1^1 < |x'|/2 < i?/4 it follows that \f'{x')\\x'\l2
< 9|xf/4i?,
hence | / ' ( x ' ) - /'(0)1 = If'ix')]
<
9\x'\/2R.
As in the first part of the proof we can deduce a similar estimate for | / ' ( ^ ' ) ~ f'{y')\/W ~ y'\ when \x' — y'\ is sufficiently small by changing to a coordinate system adapted to the point {f{y')^y')- This completes the proof of Lipschitz continuity of / ' . Corollary 2.4.4. If K e /C(R^+^) has C^'^ boundary, then the projection TTK of K in R^ obtained by dropping the last m coordinates has also C^'^ boundary Proof. By Proposition 2.4.3 if is a union of balls of radius R. Hence TTK is a union of balls of radius i?, the projections in R'^, so Proposition 2.4.3 shows that dnK is in C^'^ The convexity assumption here cannot be omitted. We give an example (cf. Trepreau [1]), which can easily be modified to a compact set: E x a m p l e , li K = {x E R^; xi > 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(I^i + 1^2) + dim(I^i fl W2) valid for linear subspaces of V implies that d i m P ( t ^ i ) + dimP(I^2) = dimP(T^i + W2) + dimP(T^i 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
II. CONVEXITY IN A FINITE-DIMENSIONAL VECTOR SPACE
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 a^a:^
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.
PROJECTIVE CONVEXITY
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
II. CONVEXITY IN A FINITE-DIMENSIONAL VECTOR SPACE
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
PROJECTIVE CONVEXITY
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}.
i^i 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 <
XQ}.
If we take xi,xo — x i , X2 as new coordinates and define X2 = 0 as the line at infinity, we obtain the situation already discussed in (b). Hence we have proved T h e o r e m 2.5.7. If Ki, K2 is a projectively convex pair in a twodimensional projective space, and neither is contained in a line, then either one of them is a bounded convex set in the affine space obtained by removing a suitable line, or else Ki consists of one of the components in the complement of two lines, together with an interval on each of these. The two-dimensional case gives us a starting point for the proof of a general separation theorem corresponding to Theorem 2.1.10 and its corollaries. For the inductive proof we also need a lemma: L e m m a 2.5.8. Let K be a projectively convex subset of P{V), let P{W) be a subspace of codimension one, and let XQ G P{V) \{KU P{W)). Then the projection K = {x e P{W);
the line through x and XQ intersects
K}
104
II. CONVEXITY IN A FINITE-DIMENSIONAL VECTOR SPACE
is projectively
convex in P{W).
Proof. Let xi,X2,yi,y2 ^ P{W) be collinear, and let Xi e K, yi ^ K^ i = 1,2. Choose x'^ E K on the Hne through XQ and x^, i = 1,2, and denote the intersection of the Kne through x'^ and X2 with that through XQ and yi by y[. (The Hues intersect since they he in a two-dimensional plane.) Then yl ^ K, and the cross ratios {xi,X2;yi,y2) and {x[,X2;y[jy2) are equal for they are related by a projection from XQ. Since K is projectively convex they are positive. This proves the lemma. T h e o r e m 2.5.9. Let Ki,K2 be a projectively convex pair in P{V), both with interior points. If Li is a maximal subspace C ifx and L2 a maximal subspace C K2, then d i m L i + dimL2 + 1 = d i m P ( y ) ; since Li and L2 are disjoint this means that Li and L2 span
P{V).
Proof. To simplify the notation we assume that Ki is open and that K2 is closed. The statement is trivial when dim.P{V) = 1, and when d i m P ( F ) = 2 it follows from Theorem 2.5.7, so we may assume that n = d i m P ( y ) > 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
PROJECTIVE CONVEXITY
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
11. CONVEXITY IN A FINITE-DIMENSIONAL VECTOR SPACE
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);^i > 0, ...,a:ifc > 0}, and K\ is then the intersection of half spaces of the form k
{a;GR^;^a;,Ci
so K\ is a cylinder in the direction of the X]^j^\^... ,Xn plane. The closure Kl in P^ contains some plane defined by J = 1 , . . . , fc, which has dimension n — k^ while the subspace of P^ defined by a;o +^1H hx^ = 0, Xk-\-i = • -' = Xn = 0 is contained in 1^2 • Thus n2 > k — I, ni > n — k, and since n i + 77,2 = n — 1, it follows that A: = n 2 - f l = n — n i , hence iy = n2 — n2=n—1 — n2=n — k. The plane defined by XQ + xi + • • • -f x^ = 0 is contained in K2 = Ci^i, and it intersects Ki in the subspace where XQ = xi = •" = Xk = 0, of codimension fc + 1. Choosing this plane instead as the plane at infinity and proceeding as before, we obtain in the new coordinates that the intersection of the cylinder Ki with the plane defined by Xk^i = • - • = Xn = 0 is bounded, for Ki intersects the plane at infinity only in the subspace defined hj XQ — " ' = Xk = 0. Every maximal plane IIi C Ki is then of the form 7 —
7
0 5 .y
= 1 , . . . , A:; so it is of codimension A:, and every maximal plane 112 C K2 is defined by an equation ^ Q CJXJ = 0 where Ck-\-i = • • • = c^ = 0, and Co + Yli Cjaj 7^^ 0 if IIi contains an interior point of Ki. Thus Hi D112 is then defined by XQ = • • • = x^ = 0, which is independent of the choice of Hi and 112. The conclusion can be stated invariantly as follows, with the notation in (2.5.3):
PROJECTIVE CONVEXITY
107
T h e o r e m 2.5.10. Suppose that Ki,K2 is a projectively convex pair in P(y) with interior points. If 712 = n2 = n — 1, then Ki is a bounded afRnely convex subset ofP{V) with a suitable subspace of codimension one removed. Ifn2 0 and 77,2 > 0. The case n = 3 is particularly simple, for then it follows that n\ = Til — 712 = n2 = l. Since it is the key to the general case we shall begin by studying it. P r o p o s i t i o n 2.5.11. Suppose that Ki^K2 is a projectively convex pair in P^ with interior points and ni = n2 = I. Then there is a quadratic form Q of signature 2,2 in the homogeneous coordinates such that K° = {{xo -.Xi-.Xi:
x^y, i-iyQ{x)
< 0},
j = 1,2.
To prepare the proof we shall first recall some basic classical facts concerning the geometry of hyperboloids of one sheet. It is convenient to write the equation in the form (2.5.5)
XQXI = X2X3.
For arbitrary (A : //) G P^ we define a line satisfying (2.5.5) by (2.5.6)
XXQ = //X2, A^s = fixi,
and for (A' : fi') G P^ another line satisfying (2.5.5) is given by (2.5.7)
X'xo = M ' ^ 3 5 ^'^2 —
[^x\.
The lines (2.5.6) and (2.5.7) intersect at (/i/x' : AA' : A/x' : /xA') while two lines of the form (2.5.6) (resp. (2.5.7)) are skew. The surface defined by (2.5.5) is the union of the lines of either family. If we have three lines L i , L27 ^3 of the family (2.5.6) then every line intersecting all three belongs to the family (2.5.7), for such a line is uniquely determined by its intersection X with L\ since it must lie in the two-dimensional plane spanned by x and L2 which has a unique intersection with L3.
108
11. CONVEXITY IN A FINITE-DIMENSIONAL VECTOR SPACE
Any three lines L i , ^2,1^3 in a three-dimensional projective space which are pairwise non-intersecting can be brought to the same position by a suitable choice of coordinates. First we choose a plane at infinity which intersects Li^L2'>Ls in three points which are not collinear. We can make them the points at infinity on the coordinate axes, so that apart from the points at infinity Li == {{xi,a2,a3)]Xi
G R},
L2 = {(&i,0:2,63);^2 ^ R } ,
L3 = {{ci,C2,Xs)]X3
e R}.
By a translation we can arrange that a2 = a^ = bi = 0. Then 63 ^ 0, for Li and L2 would otherwise intersect at 0. If C2 = 0 then Li and Ls intersect at (ci,0,0), and if ci = 0 then L2 and L3 intersect at (0,C2,&3). Thus ci ^ 0 and C2 ^ 0, and by changing scales we make ci = C2 — &3 = 1. Thus all such configurations of three lines are equivalent. It follows that the lines intersecting L i , 1/2,^3 generate a surface defined by a quadratic form of signature 2,2 in the homogeneous coordinates. To prove Proposition 2.5.11 it remains to show that F = dKi = 8X2 is generated by lines in the manner just described. Proof of Proposition 2.5.11. If P{W) is a subspace of PiV) of dimension 2, it follows from Lemmas 2.5.4 and 2.5.5, since Uj < 2, that P{W)nKj 7^ 0, j = 1,2, and that the closure is P{W) fl Kj. The proof will require several steps. (a) For every x £ T there exists a line L with x G L C F. Take any two-dimensional plane P{W) C P{V) containing x, and consider the projectively convex pair P{W) fl Kj in P{W). By Theorem 2.5.7 it follows that either P{W) fl F is the union of two lines, or else P{W) fl Kj is for J = 1 or ji = 2 an aSinely convex bounded subset of P{W) with a line removed. In the first case our claim is true, so assume that we have the second case with j = I. Let Li be a supporting line C P{W) of if ° passing through X. Then Li C K2, and Li fl if 1 is a compact interval containing X not equal to Li. In the tangent space of P{V) at x, let T be a two-dimensional space supplementary to the direction of L i , and identify T with R^. For every ^ G R there is a unique two-dimensional subspace PiWe) containing L, with (cos ^, sin^) as tangent vector at x. We have plWe) = P{We^^) for every 6. If the intersection PiWe) fl K^ is an angle for some 0^ then P{We) D F consists of two lines and the claim is true. If this does not happen, then P{We) fl if ° is always an affinely convex open bounded set in P{W0) with some line removed, and x is a boundary point with Li as supporting line. Let s{6) = ±1 if in a neighborhood of x the set PiWe) n if ° lies on the side of Li to which ±(cos 0, sinO) points. Since if ° is open, it is clear that s{9) is continuous, hence constant. On the other hand, s{0 + TT) = —s{6), which is a contradiction proving the claim.
PROJECTIVE CONVEXITY
109
(b) If L is a line C F and P{W) D L is a two-dimensional subspace of P(y), then P{W)nr = LUM where M is another line. This follows from Theorem 2.5.7. (c) Three lines Li^L2,Ls C F cannot lie in the same plane. This follows from (b) above. (d) The intersection of all lines contained in F is empty. For assume that X e L ior every line L C F. This implies that if L is a line with x e L then either L c F o r L f l F ^ {x}, foi if x ^ y E LnT then the hne C F containing y which exists by (a) above must pass through x^ so it is equal to L. Choose a two-dimensional subspace P{W) ^ x. Then P{W) (1 Kj contains a line M for some j , say i = 1. If the line L through x and a point 2/ G M is not contained in F, it follows from (ii) in Lemma 2.5.3 that L\{x} C Kl, for y ^ K2. Hence Ki contains the two-dimensional subspace spanned by x and M , contrary to our assumptions. (e) The intersection of three lines jLi,L2,i>3 C F is always empty. For assume that rz; G Li fl L2 H L3. By (d) we can choose a line Mi C F which does not pass through x, and in a two-dimensional subspace II D Mi not containing x we have by (b) above another line M2 C F. Now L^ fl II = {xj} G F so Lj intersects either Mi or M2. Since we have three lines Lj it follows that two of them, say Li and L25 intersect the same M^, say M i . Then Li^L2^Mi lie in the same plane, which contradicts (c) above. We can now finish the proof. Choose a line L CT and let L i , L2, L3 be the other lines C F in three different planes through L. They have pairwise no point in common, for the intersection would be on L and that would contradict (e). If a; is an arbitrary point G F \ L i , then the intersection of F with the plane II through x and Li consists of Li and a line which intersects both L2 and L3, at the points where these lines meet II. Thus x is on a line meeting L i , L2 and I/3. The same is true if x G F \ L 2 or x G F \ L 3 . Hence F is contained in the hyperboloid of one sheet generated by such lines. If F were not equal to the whole hyperboloid, then the complement would be connected. Thus F must be equal to the whole hyperboloid, and Kj are the two components of the complement. The proof is complete. We are now ready for the final result, where we again use the notation in (2.5.3): T h e o r e m 2.5.12. Suppose that Ki^K2 is a projectively convex pair in P{V) with interior points and that max(ni,n2) < n — 1, that is, min(n°,n2) > 0. Then there is a quadratic form Q in V with signature 71° + 1,712 -}- 1, that is, n° H- 1 positive and 713 -f 1 negative terms in the diagonaUzed form, such that K° is the image in P{V) of the set in V where
i-iyq < 0. Proof. The theorem has already been proved when 7i < 3, so we assume
110
II. CONVEXITY IN A FINITE-DIMENSIONAL VECTOR SPACE
now that n > 4 and that the theorem has been proved for lower dimensions. In the first part of the proof we assume that the defect z^ = 0, that is, that 0 < n^ = Uj, j == 1,2. Without restriction we may assume that rii < n2, and since n i 4-n2 = n — 1 > 3, this imphes that 712 > 2. Choose a subspace Hi C K^ of dimension n i , and introduce homogeneous coordinates so that E l is defined by Xj = 0, ni < j < n. Every subspace P{W) D IIi of P{V) with codimension one is then defined by an equation
(2.5.8)
Y.
""3^3 = 0,
ni + l
where not all aj are equal to 0. Note that n—ni = n24-l > 3 so that we have at least three terms in the sum. Since P{W) must intersect every maximal subspace C i f | in a subspace of dimension > n2 — 1, and the disjoint rii dimensional subspace Hi C Kl is contained in P{W)^ it follows that every maximal subspace oi P{W) contained in K2 has dimension n2 — 1. Hence it follows from the inductive hypothesis that there is a quadratic form Qw in W such that P{W) H i^° is defined by {-IYQW < 0. The form Qw is unique apart from a positive factor, which we can fix by demanding that Qvr(l, 0 , . . . , 0) = 1, for (1 : 0 : • • • : 0) G Hi C K l . We want to prove that there is a quadratic form Q inV such that Q\w = Qw for every W. For j = n i + 1 , . . . , n let Qj be the form defined in the plane where Xj = 0. If / is another such index then Qj = Qj' when Xj = Xji — 0, for also in this plane of codimension two there are points of K\ and of if|. Thus the coefficients of Qj and Q'j agree apart from the terms involving Xj or a;'. Since we have at least three indices to play with, it follows that there is a unique polynomial Q such that Q{x) — Qj{x) when Xj = 0, for every jf =r m + 1 , . . . ,n. Now consider another plane W^ defined by (2.5.8). The quadratic form Qw — Q\w ^^ W must vanish in the intersection of W and every plane Xj = 0, j — ni -f 1 , . . . , n. A quadratic form in a vector space vanishing in two different hyperplanes can be written /1/2 where / i and /2 are linear forms defining the hyperplanes, so it cannot vanish in a third hyperplane. If three different hyperplanes in W are defined by Xj = 0 for some j == n i -f 1 , . . . ,n, it follows that Qw — Q in W, so this is proved except if n2 = 2 and some aj vanishes. But in that case we conclude by taking the limits of planes W with all aj ^ 0 that {-iyQ 0, z = 1,2. Let n ° be a maximal subspace C Kj, and let Uj D 11° be a maximal subspace C Kj. Then II = Hi fl 112 is a subspace of P{V) of
CONVEXITY IN FOURIER ANALYSIS
111
dimension v — 1, by (2.5.4), and we shall prove that it is independent of the choice of Hi and 112. First we shall prove that (2.5.9)
Ii2\ncKl
To do so we consider the space P{W) of dimension n2 + 1 spanned by 112 and a point XQ El\.\. In this space 112 is a maximal subspace C P(VF) 07^2, and it has codimension 1. The space spanned by 11 and XQ is contained in P{W) n Ki and the intersection with 112 is equal to II. Hence it follows from Theorem 2.5.10 that if y G P{W) D K2 then the space spanned by y and n is also in K2 U 11. Now 112 ^^^ n are disjoint subspaces of 112 with dim 112+dim n = 712 + ^ —1 = ^2 — 1, and since 112 ^ ^2 we obtain (2.5.9), for 112 is the union of the lines joining points in II2 to points in 11. From (2.5.9) it follows that a subspace C Ki can only intersect 112 in 11. Hence the intersection Hi fl n2 does not change if we choose another Hi, and thus it is clear that it cannot change if we choose another H2 either. If we take a subspace P{W) disjoint with H and of dimension n — v/ii follows that H U K"^ consists of all lines connecting a point in H with a point in P{W) n K ° . In fact, the dimension of P{W) fl H2 is 712 + n - z/ - n = n^, and since 722 + dimH ==77,2+^^ — l = n 2 — 1, these lines span H2, so we can use (2.5.9) and the fact that there is a plane H2 through every point in K2. This proves the statement for j = 2 and the statement for j = 1 is parallel. By the first part of the proof the intersections P{W) fl K° are defined by a quadratic form in VF. If we extend it to V so that it is constant in the direction of the space corresponding to H, we obtain a quadratic form with the stated properties. The proof is complete. Together Theorems 2.5.10 and 2.5.12 give a complete description of projectively convex pairs apart from the boundaries. In particular, they describe all projectively convex pairs where one of the sets is open and the other is closed. However, in the case studied in Proposition 2.5.11, for example, it is clear that a complete description would also require the study of subsets of r which have convex intersections with each generator. We shall not discuss this problem here. 2.6. C o n v e x i t y in Fourier analysis. Recall that the definition of the Fourier transform in the theory of distributions starts from the Schwartz space 5 ( R ^ ) of all ^ G C ^ ( R ^ ) such that (2.6.1)
SMp\x''D^ip{x)\
< 00,
Va,/?,
X
where a, (3 are arbitrary multiindices. The Fourier transformation if \-^ (p^ (2.6.2)
ifii) = [ e-'^''^^^^{x)dx,
^ e R^,
112
II. CONVEXITY IN A FINITE-DIMENSIONAL VECTOR SPACE
is an isomorphism of <S, with the topology defined by the seminorms (2.6.1), and the inverse is given by (2.6.3)
ip{x) = (27r)-^ Ie'^""^^^^{Qdi,
x E R^.
The space S is translation invariant. However, we shall now discuss more general spaces which are not. Let / be a convex, lower semi-continuous function in R^. To avoid uninteresting complications we assume that / < oo in an open set, and we then define Sf as the set of functions ^ G C ^ ( R ' ^ ) such that (2.6.1)'
snY>e^^''^\x'^D^ip{x)\ < oc,
Va,^.
This implies that e<^'^>|^(a:)| < C ( l + |x|)-^-^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^D^ip, 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^<^'^+^^>^(^ + ir/) d^,
rj e M.
Proof. We have already proved that (p satisfies (i), (ii), (iii) when cp £ Sf. Now assume given a function ip in R'^ + iM satisfying (i), (ii), (iii). By the basic properties of the Fourier transformation in S we know that when T] £ M then cp^ix) = (27r)--y'e^<-'^+^^>(^(^ + i77)de is defined and e^*'^^(^^ G S. In view of (iii) we have (2.6.6) e('^^'i)-fM\x''D^iPr,{x)\
= e-^'W|x"(jD + 277)^e<^'^Vr7(^)l < Ca,p,
TjeM.
If ^1,^2 ^ M then it follows from (ii) that (p{^-\-irji-}-T{rj2 — rji)) is analytic in the strip {r G C; 0 < I m r < 1} and continuous in the closure, and since we have fast decrease at oo by (iii), it follows from the Cauchy integral formula that
^R
has the same value for j = I and j = 2. When we integrate with respect to the other variables it follows that (pjj^ = ip^j^. Thus (prj = (p is independent of 7/. Hence it follows from (2.6.6) and Theorem 2.2.4 that (2.6.1)' holds. The proof is complete. Special cases are given in the following exercises:
114
II. CONVEXITY IN A FINITE-DIMENSIONAL VECTOR SPACE
Exercise 2.6.1. Show that if if is a convex compact set in R^ then the Fourier-Laplace transform of the space C^{K) is the space of all entire analytic functions F in C^ such that for every N
\F{0\ < c^ii + |c|)-^e^^(i-^), c e C". Here HK is the supporting function of K. Exercise 2.6.2. Show that if the interior fi of the closed convex set K is not empty then the Fourier-Laplace transform of 5 / , where f{x) = HK{X), consists of all analytic functions F in R'^ + ifi such that
sup \CD^F{0\
Va,/3.
Exercise 2.6.3. Describe the Fourier-Laplace transform of Sf when f(x) = \x\'P/p, for some p > 1. We shall now give an extension of the preceding results to distributions which are temperate with respect to / in a sense similar to (2.6.1)'. Thus we define 5 r to be the set of distributions u G I>'(R"^) such that for some a and AT (2.6.7)
\u{ip)\
Yl
^^pe-flx'^D^ifl
^ G Co^(R^).
| a | + |^|
If f{r]) < oo, then the fact that {x,rj) < f{x) + /(r/) shows that (2.6.8)
|(ne<-'^>,(^)|
Y^
supe-^|x^Z)^(e<-'^V)l
|a+/3|
< C'ef^'^^l + \ri\)^
Y
s^P l^^^D^^l
if e Co^(R^),
\a-\-/3\
SO e^'^^iA G S'.
The Fourier transform Urf G S' is therefore defined when
f{rj) < oo, and e~^{l -f \rj\)~-^Ur} belongs to a bounded set in S'. To get a clean and simple statement we assume that M = {r/; f{rj) < oo} has interior points, that is, that / is not linear in any direction. Denote the interior of M by M ° as before. Then we claim that there is an analytic function u in R'^ + iM° such that Ur^ is defined by the function R'^ 3 <^ »-> u{^ + 'i'V)- F^^ ^^^ proof let 7/°,..., T/'^ be affinely independent points G M , and let 77 = Yl^ ^jV'^ where Xj > 0 and ^ Q XJ = 1. Then
V,(a:) = e(^'''7^e<^''''>
CONVEXITY IN FOURIER ANALYSIS
115
is in a bounded subset of S when all A^ have a fixed positive lower bound. Since '?ze<'^) = i^rj E o '^e<'^'>, it follows (see Lemma 7.4.1 in ALPDO) that Urj is a C^ function
0
of ^ and 7] with all derivatives 0 ( ( 1 4-1^|)^)- Now -^^e"*^*'^^ is an analytic function of C = ^ + ^^ with values in <S, so it follows that UrjiO ^^ ^^ analytic function of ( when I m ^ G M ° . T h e o r e m 2.6.2. Let f be a convex lower semi-continuous function in R^ which is finite in some open set, and assume that the interior M° of the set M where the Legendre transform f is finite is not empty. Then the Fourier-Laplace transform maps S^ defined by (2.6.7), to the space of analytic functions u in H^ + zM° such that: (i) For every compact subset K of M ° there are constants C and N such that
KOI < 0(1 +ICl)^, ImCGK. (ii) The map M° 3 rj ^-^ Uj^ = u{- -{- irj) G S' can be extended to M so that for every line L C R"^ it is continuous with values in S' when rj e LnM. (iii) e~^^'^\l + \r]\)~-^Urj is bounded in S' when rj G M, for some N. Conversely, given an analytic function U satisfying these conditions, the product V ofe~^'''^'^ and the inverse Fourier transform ofU(- + irj) is independent ofrjE M, and (2.6.9)
e<'^>-^~(^)(l + \rj\)-^v
is bounded in S\
r] e M.
Proof. We have already verified the necessity of these conditions. That the inverse Fourier transform oiU{--\-irj) multiplied by e"^*'^^ is independent of 7] when rj G M ° is a consequence of the Cauchy-Riemann equations, for the derivative with respect to rj is equal to 0. In view of condition (ii) it follows by continuity for all rj G M. Prom (iii) we obtain (2.6.9). Remark. (2.6.9) is in general weaker than (2.6.7); however, we would only have needed that u satisfies (2.6.9) to prove (i)-(iii) (with another N). For suflSciently regular / , the diff'erence is not very great. The results discussed here are quite close to those in Section 7.4 of ALPDO.
CHAPTER III
SUBHARMONIC
FUNCTIONS
S u m m a r y . This chapter is in principle parallel t o Chapter I: subharmonic functions in R*^ are defined as convex functions on R with linear functions replaced by harmonic functions. However, this requires some background knowledge concerning harmonic functions which is provided in Section 3.1. Basic facts concerning subharmonic functions are then given in Section 3.2, mainly by means of the mean value property and the characterization as distributions with positive Laplacian. In Section 3.3 we start with t h e solution of the Dirichlet problem using Perron's method before approaching t h e main topic, t h e Riesz representation formula, which generalizes Section 1.5. A number of applications to one variable analytic function theory are given. Finally, Section 3.4 is devoted to a study of the exceptional sets associated with subharmonic functions. It is proved t h a t t h e polar sets where a subharmonic function can be equal t o — oo are precisely t h e sets where t h e limit of a bounded increasing sequence of subharmonic functions can fail t o be upper semi-continuous. This section will not be needed in t h e following chapters.
3.1. H a r m o n i c functions. The definition of convex functions of one variable in Chapter I was based on a comparison between a function and affine functions, that is, solutions of the differential equation (Pu/dx^ = 0. We defined convex functions of several variables in Chapter II in terms of the restrictions to lines. In this chapter we shall instead discuss functions in R*^ obtained by comparing functions with harmonic functions: Definition 3.1.1. A function u G C'^{X) where X is an open subset of R'^ is called harmonic if u satisfies the Laplace equation n
(3.1.1)
^^ = Y1 d'^^ldx] = 0.
We shall prove below (Theorem 3.1.3) that all distribution solutions of the Laplace equation are in ( 7 ^ , so we could equally well have required u to be in C^ or allowed u to he a. distribution in Definition 3.1.1. After starting from the more classical point of view we shall in fact use distribution theory
HARMONIC FUNCTIONS
117
throughout to simpHfy the arguments even where a more classical approach is not too hard. The inhomogeneous equation corresponding to (3.1.1) (3.1.iy
Au = f
is called Poisson's equation. It is of course well defined also if u and / are distributions, and u \-^ Au is a continuous map in the distribution topology. T h e o r e m 3.1.2. For xeR'^\
{0} let
f (27r)-MogH,
ifn = 2,
V((n-2K),
ifn>2,
where \x\ is the EucUdean norm and Cn is the area of the unit sphere in R"^. Then E is locally integrable in K^, dE/dxj is defined by the locally integrable function X I ^ X j \X I '^/cji, and (in the sense of distribution theory) (3.1.2)
AE = 6Q,
where So is the Dirac measure at 0. This is Theorem 3.3.2 in ALPDO, and we refer to the proof given there on page 1:80. One calls E a fundamental solution^ and (3.1.2) is equivalent to each of the following important identities (ALPDO, p. 1:110): (3.1.2)'
E^Au = u,
(3.1.2)''
A(£; * / ) - / ,
ue S'iR''), /G5'(R-).
That distribution solutions of (3.1.1) (or (3.1.1)' with / G C^) are in C^ is an easy consequence of (3.1.2)': T h e o r e m 3.1.3. If u e V'{X) and Au = f e C ^ ( X ) , then u G C^{X), and u can be extended analytically to a neighborhood of X \ s u p p / in C^. Proof. Choose x ^ CQ^{X) equal to 1 in an open subset Y (E X , and set V = x'^- Then v ^ £' and Av = xf ~^ 9 where g = 2(x',ix') + {Ax)u G £' vanishes in Y. By (3.1.2)' we obtain v = E^{xf)-^E^g, where E * ( x / ) G C^ since xf ^ C'o°, and {E * g){x) = {g,E{x - •)),
x G Csupp^,
is a C^ function of a; G Csupp^. The right-hand side is in fact well defined and analytic in a complex neighborhood. Hence u = v is 3. C^ function in Y. This proves that u G C ^ ( X ) , and if / = 0 it follows that u can be extended analytically to a neighborhood of X.
118
III. SUBHARMONIC FUNCTIONS
Corollary 3.1.4. Ifuj is a sequence of harmonic functions in X converging as distributions to u e V'{X), then u is defined by a (C^) harmonic function, and d^Uj -^ d^u uniformly on every compact subset of X for every multi-index a. Thus the sequential distribution topology and the sequential C^ topology are equivalent when restricted to harmonic functions. In particular, if K is a compact subset of X then SVip\d^u\ < K
Ca,K,X^^p\u\, ' ' X
for all harmonic functions u in X. Proof. That u G C^ is an immediate consequence of Theorem 3.1.3, for A?/ is the Umit of A-Uj in the sense of distribution theory, hence A?x = 0. If we define QJ as g there but with u replaced by Uj, then
Since gj —^ g in S' it follows that Uj{x) —> {g, E{x — •)) = u{x), x eY, and this remains true if we differentiate any number of times with respect to x. All derivatives are uniformly bounded on any compact subset of Y if Uj is just bounded in D', which proves the corollary. We shall now solve the Dirichlet problem in the simplest case, for a ball. Thus we want to find a C^ function defined in the closure of the ball BR = {x eR""; \x\ < R} such that (3.1.3)
Au = f
in BR',
U = cp on OBR,
where / and cp are given in ^ i ^ and OBR respectively. We know from (3.1.2)" that the Poisson equation is satisfied by the convolution E ^ f if f G £'{BR). We shall now modify E so that we obtain a solution with the desired boundary values. (The one-dimensional case of this is Theorem 1.5.1; we shall assume n > 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\
and can then rewrite (3.1.8) in the form (3.1.8)' u{x) = [
GR{X,y)f{y)
dy + [
P{x/R,yMRy)
dcj{y),
x e BR
where du is the Euclidean area measure on the unit sphere. Recall that GR < 0 and note that P > 0; taking u = 1 we obtain (3.1.10)
/ P{x, y) du{y) - 1, J\y\=i
\x\ < 1.
Since P was obtained as a derivative of GR, it follows that x H^ P{x,y) a harmonic function.
is
T h e o r e m 3.1.5. Let u be a continuous function in BR which is harmonic in BRJ and let ip{y) = u{Ry) when \y\ = 1. Then we have (3.1.11)
u{x)=
f
P{x/R,yMy)du{y),
x e BR.
HARMONIC FUNCTIONS
121
Conversely, if cp is a continuous function on S'^~^ then (3.1.11) defines a harmonic function in BR such that u{x) -^ (p{y) when BR 3 X -^ Ry; thus u is continuous in BR with boundary values OBR 3 X \-^ (p{x/R). Proof. If u is harmonic in BR and we take r with \x\ < r < R, it follows from (3.1.8)' that u{x)=
I
P{x/r,y)u{ry)duj{y),
and (3.1.11) follows at once by letting r —^ R ii u is continuous in BR and u{Ry) = (p{y)^ when \y\ = 1. To prove the converse we note that differentiation under the integral sign shows that (3.1.11) defines a function harmonic in BR^ because P{-,y) is a harmonic function. It remains to examine the boundary values. If XQ = Ryo G OBR, then (3.1.10) gives u{x) - ifiyo) = /
P{x/R,
y){(p{y) - (fiyo))
du{y).
J\y\=i
Let e > 0 and choose a neighborhood U of yo such that \
I
P{x/R,y)\(p{y)
-
ip{yo)\du;{y).
Since P{x/R^y) converges to 0 uniformly for y ^ C/ as a; —> XQ, it follows when x is sufficiently close to XQ that \u{x) —
u{0)=
I
u{Ry)dLj{y)lcn,
J\y\=i
valid when u is continuous in S^^ and harmonic in BR. Another important consequence is T h e o r e m 3.1.6 (the m a x i m u m principle). Let X he an open bounded set and let u be a real-valued function which is continuous in X and harmonic in X. Then we have (3.1.13) ^
^
lainu < u(x) < max^A, dX
~
^ ^~
dX
x E X,
122
III. SUBHARMONIC FUNCTIONS
and if one equality holds at some x E X then u is constant in the component of X in X. Proof. If X is a ball the theorem is an immediate consequence of (3.1.10), (3.1.11) and the positivity of P. In the general case it follows that if u assumes its maximum (minimum) at an interior point, then u is constant in every ball C X with this point in its interior. Hence the same value is taken in the component of a: in X , hence also at its boundary points E dX. There is a useful supplement to the maximum principle which concerns the boundary points where the maximum (or minimum) is attained. We state it just for functions in a ball. T h e o r e m 3.1.6' ( H o p f s m a x i m u m principle). Let u be a harmonic function in the ball BR which is continuous in BR. Ifx G OBR and u{x) = raaxdBii u, then the normal derivative Nx = liint^i-Q{u(tx) — u{x))/{t — 1) exists, Nx G [0, oo], and 0 < u{x) - u{z) < 2^-^(1 + |^/it!|)(l - \z/R\Y-''Nx,
z G BR.
Proof. By (3.1.11), (3.1.10) and (3.1.9) we have u{tx) - u{x) = c;^^ / u{tx) — u{x) ^ : ^ = c-\l t - 1
P{tx/R,
y){u{Ry)
- u{x)) du{y),
-^t) f \tx/R - yr{u{x) J\y\ = l
- u{Ry)) du;{y).
If li21t->i-o('^(^^) " '^(^))/(^ - 1) < oo it follows from Fatou's lemma in integration theory that A = 2c-'
f
\x/R - yl-'^Hx)
- u{Ry)) du{y) < oc;
J\y\=i
since \x/R — y\ < \tx/R — y\-\-{l — t) < 2\tx/R — y\^ it follows by dominated convergence that {u(tx) — u{x))/{t — 1) —> A as t -^ 1 — 0. Since u{x)-u{z)
= c-^ I
{l-\zlR\'')\zlR-y\-^{u{x)-u{Ry))dy,
the proof is completed by the obvious estimate \x/R — y\/\z/R 2/(1 - \z\IR).
z E BR, ~ y\ <
The estimate in Theorem 3.1.6' proves that Nx > 0 unless i^ is a constant. It is related to the following result:
HARMONIC FUNCTIONS
123
T h e o r e m 3.1.7 (Harnack's inequality). Ifu is harmonic and nonnegative in BR, then (3.1.14) {R + |:r|)-^^(0) < u{x)R^-''/{R^ (3.1.14)'
- \x\^) <{R-
\u\{))\ <
|x|)-^tx(0),
x G BR,
nu{0)/R.
Proof. Assume at first that u is continuous in BR. Then we have a representation of the form (3.1.11) with <^ > 0, and as in (3.1.12) /
ip{y)duj{y)/cn = u{0).
Hence u{0) miii CnP{x/R,y)
< u{x) <
|2/I=i
u{0)ma.xCnP{x/R,y), liy|=i
which yields (3.1.14). If li is just harmonic and non-negative in BR, then we know that (3.1.14) holds with R replaced by r ii \x\ < r < R, and letting r -> i? we conclude that (3.1.14) is valid. By (3.1.14) \u{x) - u{{))\ < n\x\u{0)/R + 0{\x\'^), which proves (3.1.14)'. Exercise 3.1.1. Let / be an analytic function with 0 < \f{z)\ < 1 when |2;| < 1. Prove that
l/(o)|a+i-i)/a-i-i) < 1/(^)1 < |/(o)|(i-i-i)/a+i-i),
\z\ < i.
Deduce Hurwitz' theorem which states that if fj is a sequence of analytic functions with no zeros in a connected open set X C C such that fj —> / uniformly on every compact subset of X , then / has no zeros in X unless / vanishes identically Exercise 3.1.2. Let Uk be a sequence of positive harmonic functions in a connected open set X C IV^. Prove that unless Uk -^ -foo uniformly on every compact subset of X then there is a subsequence converging uniformly on every compact subset of X to a harmonic function. T h e o r e m 3.1.8 (Riesz-Herglotz). Let u be a harmonic function which is non-negative in BR. Then there exists a unique positive measure da on the unit sphere such that (3.1.15)
(x)= [
u{x) = [
P{x/R,y)da{y),
x E BR,
124
III. SUBHARMONIC FUNCTIONS
and u{ry)du{y)
converges weakly to da as r -^ R.
Proof. By (3.1.12) the positive measures dar = u{ry)duj{y) on the unit sphere have total mass Cnu{0) for every r G (0,i?). Hence we can find a sequence Vk -^ R such that dor^ converges weakly to a limit da, by Kelly's selection theorem or in modern terminology the weak* compactness of the unit ball in the dual of the space of continuous functions on the unit sphere. Since u{xrk/R)=
/
P{x/R,y)u{rky)du;{y)
= /
J\y\ = l
P{x/R,y)dar,,{y),
J\y\ = l
when X G BR, we obtain (3.1.15) when k —> oo. The uniqueness of da and the last statement will follow if we show that (3.1.15) implies that dar{y) = u{ry)dLu{y) converges to da. Let cp e C ( 5 ^ - ^ ) and form for r < i? / ip{z)dar{z) = / J\z\ = l J\z\ =l \z\--
u{rz)(p{z)du{z) P(rz/R,
y)(p{z)duj{z)da{y).
J J\v l\yMz\=l
Now the Poisson kernel has the important symmetry property (3.1.16)
P{tz,y)
= P{ty, z),
for \tz — y^ — i^ -\-\ — 2t{z,y) evident. Hence /
^{z)dar{z)=
J\z\ = l
\y\ - |2;| - 1, 0 < t < 1,
= \z — ty\'^; (3.1.16) is also geometrically
da{y) J\y\ = l
P{ry/R,z)cp{z)d(jj{z) J\z\ = l
j
ip{y)da{y),
for the inner integral converges uniformly to ip{y) by Theorem 3.1.5. But this means that dar -^ da, which completes the proof. Remark. The proof shows that instead of assuming that u is positive, one can assume that J. , ^^ ["^(^y)! du){y) is bounded when r —^ R; the conclusion is the same except that da is not positive unless u is non-negative. E x e r c i s e 3.1.3. Prove that if li is a non-negative harmonic function in BR with the representation (3.1.15), then lim^-^ / ^~*^
\u{x)-a{{y})P{x/R,y)\dx Je<\x-Ry\<2e,\x\
and that this determines the mass a{{y}) at y.
= 0, y E S'
HARMONIC FUNCTIONS
125
E x e r c i s e 3.1.4. Let / be an analytic function in the unit disc in C and assume that Ref{z) > 0 when \z\ < 1. Show that there is a positive measure da on R/27rZ and a real number c, both uniquely determined, such that 1
r
p*^ 4- r
27r Jn/27rz e' - ^ Corollary 3.1.9. Ifu is a positive harmonic function in BR then u can be written as a sum ui -|- U2 of two positive harmonic functions which are not proportional to u unless u is a multiple ofP{/Rj y) for some y G S'^~^. Proof. We can write u and Uj in the form (3.1.15) with positive measures da and daj. The equation u = ui-{-U2 is equivalent to da = dai-\-da2' If the support oida contains two points yi ^ ?/2, then we can write da = dai-\-da2 where daj are positive and yj ^ suppcJo-j. On the other hand, if supp doconsists of a single point ?/, then da = dai -f da2 implies supprfaj C {y} if dai > 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~'^E{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
III. SUBHARMONIC FUNCTIONS
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 = 0 and Aip2 = 0, so 2
A{u o (/?) = I Y^ {8jcpkf{Au) o (p, j,k=i
When n > 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\'^~'^u{x) -^ c as a; —> oo in X . Since \x\'^~'^u{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
III. SUBHARMONIC FUNCTIONS
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 — \^e 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-^e)=
/
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'\^)-^u{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
III. SUBHARMONIC FUNCTIONS
For 0 < £ < 1 it follows that 0 < a^ < C, / ( I + Wiyuiy'.e) dy' < C. We can therefore choose Sj -^ 0 such that u{y'^ej)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
= I
P{x',l,{))P[y',Xn,x')dx'
= P{y\ Xn + 1, 0) < (1 + Xn)P{y\
1, 0).
This proves (3.1.18), which implies ih^
[{l-^\x'\Y^u{x',Xn)dx'
<
Xn^Oj
[{l-{-\x'\^)-2da{x'). J
Combining this with the weak convergence we obtain if x ^ Co(R'^~"^) ]i^
/(I + k f )-t(l -
x{x'))u{x\xn)dx'
Xn-*0 J
<Jil
+
\x'\Y^a-xi^'))da{x').
If 0 < X ^ 1 ^^d X is equal to 1 in a large compact set, then the righthand side is small, and this yields the stronger statement on the weak convergence.
HARMONIC FUNCTIONS
131
To prove the characterization of a it suffices to show that u{t-)/t -^ 0 in LjQ^(iJ) if u is given by (3.1.15)' with a = 0. Let if be a compact subset of F . Then (3.1.19) Ft{y)
= t-'
I PH{tx,y')dx
^ G R ^ - \ ^ > 0.
JK
In fact, since Pnix^y')
is homogeneous of degree 1 — n in {x^y')^ we have
Ft{y') = t"' I
PH{x,y'lt)dx,
JK
and since / P H { X \ X n ^ z ' ) d x ' have / PH{x^z')dx JK
— JPniz'^x^x')
< maxx^i, / PH{x,z')dx ^^^
dx' = 1 for every a;^ > 0 we < {2/cn) sup
\x'—z'\'"^Xnm{K)^
xeK
JK
which gives / ^ PH{X, Z') dx < CK{^ + l^'l)"*^ and proves (3.1.19). Now we obtain when t -^ +oo / JK
u{tx)/tdx
= f
Ft{y')da{y')
JdH
for 1 > (1 -h \y'\Y{t-\-\y'\y
[
(t + b l ) ' " da{y') ^ 0,
JdH
-^ 0 when 1 < t -^ oo. The proof is complete.
E x e r c i s e 3.1.6. Show that if / is an analytic function of z in the half plane C_|- — {z ^C]lmz> 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'^~^u{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'^~^u{tx) -^ PH{X^O) 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 x^i —> 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
III. SUBHARMONIC FUNCTIONS
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'^ix', 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' <{l^Xn)
j{l^\x'\^)-^\da{x')\-\-\cn\a\Xn,
Xn > 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'-^iP{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'^ej)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)
PH{x,y')diy{y'),
where
JdH C=
Ijm-
[
Q-*oo Q JQ
dR f
4.nc-^R-''u^{y)dS{y) JHndBR
< 4nc~^ lim Q~^~^ I g-^oo
u~^{y)dy JHn{B2e\Be)
is finite by (3.1.21). The right-hand side of (3.1.24) is a harmonic function. The theorem follows if we apply Theorem 3.1.8' to the difference between the two sides. Note that the conclusion strengthens the hypotheses (3.1.20) and (3.1.21) which shows that they are quite reasonable. The mean value property (3.1.12) is characteristic for harmonic functions. It can be given in apparently stronger and weaker forms: Theorem 3.1.12. Let u be a continuous function in an open set X C R"^. Then the following properties are equivalent: (i) u is harmonic.
134
III. SUBHARMONIC FUNCTIONS
(ii) IfXs
= {x eX]y
(3.1.25)
eX
if\y-x\<
/ dfi{r) / u{x-\-ry) J J\y\ = l
6}, then duj{y) = u{x) / Cndfi{r), J
for every x E Xs and every measure dfi with support in [0,6]. (iii) For every 6 > 0 and every x G Xs there is a positive measure d/i supported by [0,6] but not by {0} such that (3.1.25) is valid. Proof, (i) implies (ii) by (3.1.12), and it is obvious that (ii) implies (iii). Assume now that (iii) is valid, and let B be an open ball with B C X. Let h be the harmonic function in B which is continuous in B with h = u on dB\ the existence is guaranteed by Theorem 3.1.5. Then v = u — h vanishes on dB, and we shall prove that v = 0 in B. Assume that this is false, for example that M — m^yi^^vix) > 0. Then K = {x e B]v{x) = M} (s B, so the distance 26 from K to dB is positive. Ii x E K has distance 2^ to dB, then every sphere with radius r G (0,2^) and center at x contains some point at distance 2^ — r to dB, hence a point m B\K, so we have / v{x + ry)du){y) < CnM — Cnv{x), J\y\=i
0
<26.
If we integrate with respect to a measure dfi with the properties in condition (iii), we get a contradiction with the hypothesis that (3.1.25) holds, for since h is harmonic the equality (3.1.25) must also hold with u replaced by v. Hence we obtain v = 0 so u — h \s harmonic in B. The proof is complete. We shall end this section by proving two results on harmonic functions which will be used in Section 3.3 to study Green's function of a convex set. The first is closely related to Theorem 3.1.6', and we keep the notation used there. T h e o r e m 3.1.13. Let u be a harmonic function in the ball BR C R"^ which is continuous in BR and has the bound (3.1.26)
\u{y)\<\y-Ren\\
y e OBR.
Here e^ = ( 0 , . . . , 0 , 1 ) . Set I =^ { ( 0 , . . . ,0,2;^); | < Xn/R < 1}. If x G I then n
(3.1.27)
n—1
Y^ \dju{x)\/R + Y.
\dA<^)\ + \dlu{x)\ n-1
+ Y \djdMx)\/\ log(l - xJR)\ < C,
HARMONIC FUNCTIONS
135
where C only depends on n. If e{y) is a continuous function on the unit sphere with e{en) = 0 then there is a continuous function €i on [0,1] with £^(1) = 0 such that if instead of (3.1.26) we assume (3.1.26)'
\u{y) - Q{y)\ < eiy/R)\y
where Q is a quadratic polynomial, (3.1.27)' \djdk{u-Q){x)-8jkdn{u-Q){en)/R\
- iJe^l',
y G SBR,
then < ei{xn/R),
ifj,k
Proof. To shorten notation we assume that R = 1. J^y^^^P{x,y)u{y)du{y) when \x\ < 1, and
CndP{x,y)ldx,
< n, and x e I. Then u{x)
=
- - j ^ 3 ^ - 1^ _ ^|nH-2 ^^3 " %)•
When X G / we have \y — e^l < \y — x\ -{- {1 — Xn) < 2\y — x\, hence \u{y)\ < A\x - y\'^ by (3.1.26). Since /. ^^^P{x,y)dcj{y) = 1 and /i 1=1 \y ~ ^n\'^~^ duj{y) < oo, we obtain the bound (3.1.27) for the firstorder derivatives, and Um^ dju{0,Xn) ^--"1 '
0,
ifjVn,
I -2j\en-y\~''u{y)du{y)/cn,
iij = n.
(Cf. the proof of Theorem 3.1.6'.) Another differentiation gives /i2o/ \/f^ fi Cnd P{x,y)ldx,dXk
o ^Jk , = - 2 ^ 3 ^ +
n(l-kp),
'^n{xj{xk-yk)-^Xk{xj-yj)) \x-y\n+2 ^
. n(n+ 2)(l-|xp),
^^
^
If j^k < n then the second term vanishes and
\d'Pix,y)/dxjdxk\
<—^^^+nin ^n\^ y\
+
3)P{x,y)\x-y\-\
which gives the estimate (3.1.27) of djdkU when j^k < n. The estimate of d'^u follows since u is harmonic. If j < n then there is in Cnd'^P{x^y)/dxjdxn an additional term —2nxnyj\x — y\~'^~'^. Since \x — y\ > I — Xn and \y — en\ < 2\y — x\^ the corresponding contribution to djdnu{x) can be estimated by 4 times du;{y), Cn J\y\=:l k - 2/h
~ Cn i l ^ l ^ i (1^ - e^l + 1 - rZ^n)"""^
136
III. SUBHARMONIC FUNCTIONS
hence by C\ log(l — Xn)\ which completes the proof of (3.1.27). Now assume that j^k < n and that (3.1.26)' holds with Q = 0. By Theorem 3.1.5 P{x^ y)e{y) dy -^ 0,
/
when a; —> e^
so we obtain (3.1.27)' from the proof of (3.1.27) if we note that 1
1
/I \x-yr
\en-yr
\y - en\ duj{y) -^ 0
when / 3 a; —> e^
since the integrand is bounded by the integrable function 2"^|y — e^P '^. For a general Q we write u = v -\- h where h is the harmonic function h{x) = Q{x) + ^{1 -
\x\^)AQ/n,
which is equal to Q on the unit sphere. Then we have proved that \djdkv{x) - 6jkdnv{en)\ < ei{xn),
x e I, j,k < n.
Noting that v{x) - {u{x) - Q{x)) = Q{x) - h{x) = |(|x|2 - l ) A Q / n , we obtain (3.1.27)' for a general Q since djdk\x\'^ = 2Sjk and 5n|xp = 2 when X = en- The proof is complete. The next result concerns convexity properties of harmonic functions. A harmonic function u must of course be affine linear if it is convex in some open set, for if tx is convex in a ball B = {x] \x—Xo\ < r}, then u{x) — L{x) > 0, X G S , for some afRne linear L with u{xo) — L{xo) = 0, and this implies that the harmonic function u — L is identically 0 in B . However, the level surfaces may very well be convex; the fundamental solution (outside the pole) is an example. We shall now prove that the level surfaces must then be strictly convex everywhere or nowhere. In fact, much more is true: T h e o r e m 3.1.14 (Gabriel). Let u be a harmonic function in an open connected set X C R'^, and assume that the level surfaces of u are convex toward increasing values in some neighborhood of a point XQ E X where u'{xo) + 0. If To = {te
R^; u'{xo; t) = 0, u"(xo; t, t) = 0} j^ {0}
then (3.1.28)
u''{xo]y,t)
= a{t)u\xo\y),
y G R^,
HARMONIC FUNCTIONS
137
defines a linear function TQ 3 t \-^ a{t). If A is the hyperplane defined {(a(t), t)] t ETQ} in ttie projective space P ( R © TQ), obtained by adding To a plane at infinity, then u is homogeneous of degree 0 with respect A. This means that for any x G X the function u is locally constant in plane of dimension dimTo spanned by x and TQ.
by to to the
Note that it follows that dimTo is independent of the choice of starting point XQ^ satisfying the hypotheses made. Proof. To simplify notation we assume that XQ = 0. The convexity of the level surface through 0 means that u"{Q]y,y) is positive semi-definite when ii'(0;?/) = 0, so it follows from the Cauchy-Schwarz inequality that u"{id\ y,t) = 0 for every y with u\0; y) = Q iU e TQ. Hence (3.1.28) is valid for some a(t), uniquely determined by t. To prove the theorem it suffices to consider a fixed vector e G TQ \ {0} with (3.1.28/
7x''(0;y,e)-an'(0;y),
y G R".
We must prove that u is locally constant on the lines s ^-^ x -\- s{e — ax)^ that is, that (e — ax^u'{x)) = 0. Since X is connected it sufiices to prove this in a neighborhood of the origin. For the Taylor expansion of u the equation means that oo
- ax,x^)/j\
= 0.
0
(We denote the fcth differential of a function / at x by f^^\x\ti^ • • ",tk) where tj G R"", and write f^^\x] ^^^,..., t^^) if ^ i , . . . , ^-,- occur Ki,... ,KJ times, /^i H \- K,J = k.) Collecting the terms of equal degree we find that the condition is equivalent to (3.1.29)
u^^"^^\0;e,x^)
= aju^^\0;x^),
j > 0,
which is true when j = 0 since u'{0; e) = 0 and follows from (3.1.28)' when j = 1. We shall prove (3.1.29) by induction and start with examining the proof for j == 2 in detail. Choose a smooth vector field T{x) such that (3.1.30)
T(0) = e,
and
{T{x), u'{x)) = 0
in a neighborhood of 0. (We can construct T by projecting the vector field (e, d/dx) orthogonally on the tangent plane of the level surface of u at x.) Differentiation of (3.1.30) in the direction Y gives at the origin u"{0;e,Y)
+
{T'{0)Y,u'm^O,
138
III. SUBHARMONIC FUNCTIONS
that is, if we use (3.1.28)', {r{0)Y-^aY,u\0))=0. This is the only restriction on T'(0), for we can always add to T a sum Yl^~ ^j{^)'^j where y?j(0) = 0 and the vector fields Tj form a basis for the tangent space of the level surface of u at any x close to the origin. This adds E l " ' ^jWTjiO) to T'(0). The convexity condition assumed shows that (3.1.31)
u''{x]T{x),T{x))
>0,
and we have equality when x = 0. Hence the differential vanishes at 0, that is, (3.1.32)
u"'(0; e, e, Y) + 2n"(0; e, T'{0)Y)
= 0,
and the second differential is non-negative, u^^\0; e, e, Y, Y) + 4n'"(0; e, T'(0)y, Y) + 2u"{0]T'(0)y,T'(O)y) + 2^"(0; e,T"(0; F, Y)) > 0. The last term is equal to 2a{T"{0;Y,Y),u'{0)) by (3.1.28)', and differentiating (3.1.30) twice we obtain (3.1.33)
(T"(0; Y, Y),u\0))
+ 2u'\0; T ' ( 0 ) r , Y) + 7/'"(0; e, F, y ) = 0,
so the positivity means that (3.1.34)
^(^)(0; e, e, F, F ) + 'zx'"(0; e, 4 T ' ( 0 ) y - 2aY, Y)
+ 2^"(0;T'(o)r,r'(o)y) - 4a^x"(0;r'(o)y,y) > 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 <2iy (3.1.36) 7x(^+2)(0; e, e, Y^) - 2jau^^^^\0]
e, Y^)-{-j{j - l)a^u^^\0', Y^) = 0.
(This is of course a consequence of (3.1.29), but it will be obtained much faster than (3.1.29) in the course of the induction.) When i/ = 1 we know that (3.1.29) is valid for j = 1,2. The equation (3.1.36) is obvious for j = 0 and follows from (3.1.28) and (3.1.32), with T'{0) = - a i d , when j = 1. When j = 2 it follows from the vanishing of (3.1.35) and (3.1.29) with j = 2. Thus the statement is true when u = 1. Assume that the statement has been proved for some integer u >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
III. SUBHARMONIC FUNCTIONS
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
can simplify the first sum using (3.1.29) to
2^"(0;e,T(^)(0;y^))
i=0
ij-i-
l)\{i +1)!
and the coefficients in the sum become (^i^) when we add the second sum. (This calculation also works when j = i/ 4- 2 if Z = 0, for then we have T O - ^ ) ( 0 ) = 0 when i = j - u - 1.) Prom (3.1.30)' it follows that the j t h differential given by (3.1.37) is equal to 0. The term tA'(0;T(-^)(0;y^)) for i == 0 multiplied by a is equal to ix"(0; e,T(^)(0;y-^)) by (3.1.28)', so subtracting the product of (3.1.37) by 2a we obtain that the j t h differential in the direction Y of u"{x]T{x),T{x)) at 0 is simply (3.1.40)
7x(^+2)(0;e,e,y^) - 2aju^'^^\0;e,Y^)
+ j{j - l)a2^(^)(0;y^)
BASIC FACTS ON SUBHARMONIC FUNCTIONS
141
if j < 2i/ + 1. By the inductive hypothesis this is equal to 0 for j < 2u^ and since u^'{x; T{x)^ T{x)) > 0 it follows that the differential of order j = 2z/+l must also vanish, which proves (3.1.36) for j = 2u -\-l. If Z = 0 then the calculation works also for j = 2u -\- 2, and proves that (3.1.40) is then nonnegative. Since it is a harmonic polynomial vanishing at the origin it must be identically 0, which proves (3.1.36) also when j = 2i/ -h 2. When Z ^ 0 we have the additional term (3.1.39), and we do not have (3.1.29) available when j = u -\-2. The differential of order j = 2u -{-2 is therefore equal to
'^J "^^^V2^(^+3)(0; e, Z(y^+^), y^+i) - 2a{u + 2)^(^+2)(Q. ^(y^-f 1)^ y ^ + i ) ^ ^ / / ( Q ; Z ( y ^ + ^ ) , Z ( y " + ^ ) ) ) . Since this is non-negative, the linear term must vanish, which means that n(^+^)(0; e, Z, y^+^) - a{u + 2)^(^+2)(Q. ^ . y . + i ) ^ Q,
if (Z, u'{0)) = 0.
Thus the harmonic polynomial (3.1.29) with j = i/ -j- 2 is a. function of (x,ii'(0)), which implies that it is equal to 0 since it is of degree > 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.
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III. SUBHARMONIC FUNCTIONS
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 X^and 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\'^~'^u(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) < I I u{x J\y\=i Jre[oM J\y\='^ Jre[o,d]
•^ry)du{y)d^{r),
ifx e Xs. (iv) For every 6 > 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.
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III. SUBHARMONIC FUNCTIONS
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
= lim I I ^^^ Ay\='^ JreR
u{x-\-6ry)duj{y)dii{r).
In fact, the lower limit of the right-hand side is > 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^_^o(-^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).)
BASIC FACTS ON SUBHARMONIC FUNCTIONS
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 i^ioc(^) 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
III. SUBHARMONIC FUNCTIONS
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
BASIC FACTS ON SUBHARMONIC FUNCTIONS
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|<2
Then Aq = —ns and q(xo) = u{xo), u{x) - q{x) < Ve{x) - ^
d'^h{xQ){x - rro)''/^! + e\xf
- \e\x - xo\^
\a\<2
= h{x) - Y^
d''h{xo){x
- x o ) " / a ! - \e\x - XQ\^ < 0
|a|<2
by Taylor's formula, in a neighborhood of XQ. This proves the proposition. li u e C'^ then u < q with equality at XQ implies that d'^u{xo)/dx'j < d'^q(xo)/dx'j for j = 1 , . . . ,n, hence Au{xo) < Aq{xo). In view of Taylor's formula this proves again that u is subharmonic if and only if Au > 0. It is easy to extend Proposition 3.2.10 to distributions:
148
III. SUBHARMONIC FUNCTIONS
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-^o 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 ^i^e^ips, 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:
BASIC FACTS ON SUBHARMONIC FUNCTIONS
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-^ooJ
<2 / \u ^ cps -^ ^ — '^Ix^^5 J
j
oo,
150
III. SUBHARMONIC FUNCTIONS
and the right-hand side —> 0 if ^, ^ -^ 0. This completes the proof that For sequences of subharmonic functions distribution convergence is not only equivalent to local L^ convergence but also to convergence in various stronger topologies: T h e o r e m 3.2.13. Let Uj ^ - o o be a sequence of subharmonic functions in an open connected set X C R"^ converging in V'{X) to the subharmonic function u. Then the sequence is uniformly bounded above on any compact subset of X, and Uj —> 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 ^ C^iX) 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,
BASIC FACTS ON SUBHARMONIC FUNCTIONS
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^^i^^) 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'
I
JK
\u\dx
whcn Xn.yn
^ I\
JX
(3.2.10)
/ \u{x'jXn)\dx'
^C
\u\dx^
JK
when x^ G / .
JX
Proof. It suffices to prove the inequalities when J-^\u\dx = 1. The positive measure Au is then in a bounded subset of X>'(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,
BASIC FACTS ON SUBHARMONIC FUNCTIONS
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'
{\vi\ + \v2\)dx
\v1-V2\dxy
functions Vi, 7;2 in X.
Proof. If e is sufBciently small we can apply (3.2.9) with I = [—^,6:] to vi and to ^2- For Xn ^ I we obtain /
\vi{x',0)-V2ix',0)\dx'
JK
[
{\vi\-^\v2\)dx
JX
-f / \vi{x\Xn) JK
-V2{x',Xn)\dx\
Averaging over all Xn ^ I gives /
\vi{x',{))-V2{x\0)\dx'
JK
< \Cie [ {\vi\-\-\v2\)dx+le-^ JX
[
\v1-v2\dx.
JX
The estimate (3.2.12) is a consequence of (3.2.10) unless the second integral in the right-hand side is much smaller than the first. But then we can choose e so that Ci^^ / (|7;i| 4-|'?;2|)c?rz; = / \v1-V2\dx, Jx JX which gives (3.2.12) with C = \fC\. The proof is complete. E x e r c i s e 3.2.3. Show that (3.2.12) is not valid with a geometric mean in the right-hand side with weight > | on J ^ l-^i — V2\ dx. Summing up, if u^ and u are subharmonic functions and VLJ —> u in J^JQ^.? then Uj —> u in L\Q^ in every hyperplane and limj_vQo'^j(^) — ui^x^ almost everywhere in every hyperplane. We shall return to deeper results of this type in Section 3.4. The monotonicity of the mean values M{x.,r) in (3.2.1) can be supplemented by a convexity property: T h e o r e m 3.2.16. Let u be a subharmonic function in {y G R ^ ; i ? i < \x — y\ < R2}. Then M{x^r) is a convex function of e{r), defined by E{x) — e(\x\)^ that is, elrj
( {2i:)-'\ogr, I _ ^ 2 - „ / ( ( ^ _ 2)c„),
ifn = 2, ifn > 2.
154
III. SUBHARMONIC FUNCTIONS
If dfi = Au then {dv d
i''2—0
/ ri<\x-y\
Ifu is subharmonic (3.2.13)'
in the ball {y G R'^; \x - y\ < R2}, then
/ dii{y) - ——Mu{x,rJ\x-y\
0),
0 < r < R2.
Proof. We may assume that x = 0. li Ri < r < R2 and ^ > 0 is chosen such that g < min(i?2 — r, r — i?i), we have u{y) <
u{y-\- Qz) duj{z)/cn,
\y\ = r.
J\z\ = l Replacing y by ry and integrating over y G S'^~^^ we obtain M(0,r)< / /
u(ry-hQz)duj{y)duj{z)/cl.
Jj\y\ = \z\ = l
The right-hand side is rotation invariant so the inequality can be written M(0, r) <
JJ
M(0, \ry + gz\) du{y) du;{z)/cl
=J
\y\=\z\=i
M(0,
s)dfirA^),
"""^
where dfir^g is a positive measure with / dfir^g = 1,
/ e{s)dfirA^)
= ^(^)^
for the inequality above must hold for arbitrary a, 6 G R i{u{x) = a+bE{x)^ hence M ( 0 , r ) — a-{- be{r). If M(0, •) were not a convex function of e then we could by Theorem 1.1.15 find r G (i?i,i?2), a constant c and ^ > 0, ^ > 0, such that M(0, s) < M(0, r) + c{e{s) - e{r)) - e{e{s) - e(r•))^
\s - r\ < g.
If we integrate with respect to dfir^g^ which is not supported by { r } , it follows that [
M{0,s)dfirA^)<M{0,r),
BASIC FACTS ON SUBHARMONIC FUNCTIONS
155
which is a contradiction proving the asserted convexity. Let if e Co°((i?i,i?2)) and note that A ^ ( b l ) = V^dyl),
if ^{r) = ip"{T) + (n -
l)cp'{r)/r.
This gives J cp{\y\) dfi{y) = J u(ymy\)
dy = c^ J M , ( 0 , r)^{r)r^-'
= Cn f Mu{0,r)d{r''-'ip'{r))
= -c^
dr
f r^-'dMu{0,r)/drip'ir)
dr.
Here we have used that the convexity impUes that Mu{0^r) is Lipschitz continuous and difFerentiable except at countably many exceptional points. Set /
dfi^y) = g{r2) - g{ri),
Ri < ri < r2 < R2-
Jri<\y\
Then g is continuous to the left and uniquely determined up to an additive constant, and we obtain
/*
(gir) - Cnr^-^dMu(0,r)/dr)ip'{r)dr
= 0.
Thus the derivative of g{r) — Cnr'^'~^dMu{Oj r)/dr is 0 in the sense of distribution theory, and this implies that g{r±0) = Cnr'^~^dMu{Ojr±0)/dr-{-C^ which proves (3.2.13) since de/dr = r^~^/cn. liu is subharmonic in {?/; |a: — 2/| < R2} and harmonic in a neighborhood of X, then Mu{x,ri) = u{x) for small r i and (3.2.13)' follows from (3.2.13). If we apply this result to v = u — u^ where Ue{y) = /
E{y -
z)dfi{z),
J\Z-X\<£
it follows that for e: < r < i?2 dr d(X f dr
I
e<\x-y\
dedr\
J
When e —> 0 we have u^ -^ e{\y - x\)fi{{x}), dr d / o<\x-y\
so (3.2.13)' is proved.
\
dedr
which gives
156 Remark.
III. SUBHARMONIC FUNCTIONS A more natural proof is perhaps to note that M(0,|x|) = f
u{Ox)dO,
where O is an orthogonal transformation and dO is the Haar measure on the orthogonal group. Since x »—> 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 <^(^)? ^nd if u is subharmonic in the open set X C R^, then (p{u) is subharmonic in X. Proof. If the distance from a; G X to dX is > 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
=
M^i^^){x,r),
BASIC FACTS ON SUBHARMONIC FUNCTIONS
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^,..^Uk) 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(^u){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<^y^ duj{y))
is a convex function of e{r) when Ri < r < R2. Proof. Since u{x)-\-cE{x) from Corollary 3.2.19 that
is subharmonic when Ri < \x\ < i?2, it follows
J\y\=i is a convex function of e{r) for i?i < r < it2, for every constant c. The statement is therefore a consequence of Lemma 1.2.8. The proof of Theorem 3.2.20 depended on a special trick, but we shall now give a direct proof of a more general result. (It will not be used later on.)
158
III. SUBHARMONIC FUNCTIONS
T h e o r e m 3.2.21. Let cp € C^(R) be an increasing function such that ip" > 0 and define (p{-oo) = limt_,_oo (fit)- Then ^ ^(
^{u{ry))du;{y)/cn) '\y\
is a convex function ofe{r) when Ri < r < R2 for every function u which is subharmonic when Ri < \x\ < R2 provided that (^'^/(/?" is a concave function of(p; this condition is also necessary. Proof. Since u is the decreasing limit of C^ subharmonic functions we may assume that u G C^. Define U by 7/;(r) = /
(p{u{ry))duj{y)/cn
=
^{U(T)).
That t/ is a convex function of e(r) is equivalent to U"(r) -f- (n — \)r-W{r) > 0. Since ip\U)U' = ^'
and ip'{U)U" +
ip"{U)U'^^^",
we have ip'{U){U" + (n -
1)T-^U')
= ^" + (n - l)r'^^'
-
^'''ip"(U)/ip'{Uf.
With the notation UQ{y) — u{ry), ui{y) = u^ri'^v) we have 0
=
where A^; is the Laplace-Beltrami operator in the unit sphere. Hence Cn(V^" + (n - l ) r - V ) = [{^"{uoH^ =
(p\uo)Audu-^
+ ^'{uo){Au
- r-^A^uo))
duj
(p''{uo){ul^r~^\Vu;UQ\^)du,
where V^; is the gradient in the unit sphere. Thus f'iUW
+ (n - l)r-^U')
= j
ip"{uo)uldu/cn
+ / (p'{uo)Auduj/cn
+ r~^ / (p"{uo)\VuUo\^
dui/cn,
BASIC FACTS ON SUBHARMONIC FUNCTIONS
159
where the last two terms are non-negative. Since the other terms are quadratic in ui and by the Cauchy-Kovalevsky theorem UQ and ui can be arbitrary analytic functions when u is harmonic in some neighborhood of the sphere |x| = r, we conclude that the convexity in the theorem is true if and only if (^Jcp'{uo)u,duj/Cny
<
ip'{Uflip"{U)jip"{Uo)uldulCn
for all ui. With / = ^''^ 1^" this means by the converse of the CauchySchwarz inequality that / f{uo)duj/cn
< f{U),
ii(p(U) = /
(p(uo)duj/cn.
With F = f o ip~^ and v = ip{uo) this condition can be written / F{V) du/Cn
du/Cn).
This is Jensen's inequality reversed, so it means that F is concave, which completes the proof. We return now to consequences of Theorem 3.2.20: Corollary 3.2.22. Ifu is subharmonic when Ri < \x\ < R2 then r \-^ mdiX^y^^i u{ry) is a convex function of e{r) when Ri < r < R2. Ifu is subharmonic when \x\ < R, then it is an increasing function when r < R. Proof. As usual it suffices to prove the statement when u E C^. Theorem 3.2.20 we know that
p-'log(
f
By
e^<^yUu{y))
is a convex function of e{r) when Ri < r < R2, for every p > 0. When p —> 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'^i^) is increasing when 0 < r < i? if TX is subharmonic when \x\ < R.
160
III. SUBHARMONIC FUNCTIONS
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^ |/(^e*^)| 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),
BASIC FACTS ON SUBHARMONIC FUNCTIONS
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
III. SUBHARMONIC FUNCTIONS
li g{z) — XI0^ ^i-^"^ ^^ ^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
- fiz)/w)
= f{z) + f{zY/w
+
0{z')
satisfies the conditions in the theorem so it follows that i r ( 0 ) + 2 / ' ( 0 ) V ^ | < 4|/'(0)i,
hence 2|/'(0)V"'I < 8|/'(0)|,
which proves that |w;| > | / ' ( 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%
BASIC FACTS ON SUBHARMONIC FUNCTIONS
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{xi^ex2^ • • •, sxn) is then subharmonic at distance < 6 from / for every e G (0,1). Hence u{x) — Ue{x) <
U£{x -\- ry) du;{y)/cn,
x G / , 0 < r < ^.
'J\y\=i
Since lim£_KO "^ei^ + ^^) ^ u{xi + r y i , 0 , . . . , 0) by the semi-continuity of u, we obtain using Fatou's lemma u{xi,0,...
,0) < /
u{xi-\-ryi,0,...
,0)duj(y)/cn,
xel,0
The right-hand side is a symmetric mean value of u ( - , 0 , . . . ,0) in {xi — r, xi + r ) , so it follows from Theorem 3.2.3 (or Theorem 1.1.15) that u is convex in / . The converse is obvious. Convexity also reappears if a subharmonic function only depends on one variable. More generally, we have:
164
III. SUBHARMONIC FUNCTIONS
T h e o r e m 3.2.29. Let u be a function deGned in an open set X C R^. Then u is subharmonic in X if and only if the function (x^y) H^ U{X) is subharmonic in X x R^. Proof. This is obvious from the fact that (A^; -\- Ay)u = Aa^ti, if one is willing to use compositions of distributions with maps. Otherwise one can just use the characterizations by mean values in Theorem 3.2.3; a rotation invariant mean value in a ball C X x R^ is a rotation invariant mean value in a ball C X. In Corollary 2.1.25 we saw that convex sets can be characterized as sets where there is a locally convex function with compact level sets. We shall now see that existence of such subharmonic functions does not impose any restriction, in contrast to other convexity conditions which we shall encounter in later chapters. T h e o r e m 3.2.30. Let X be an open set in RJ^, not equal to the whole space. Then (3.2.16)
u{x) = \x\ + sup -E{x -y), yedx
x e X,
is a continuous subharmonic function in X such that Xt = {x E X; u{x) < t} is compact for every t. One can also Gnd U G C^{X) having the same properties. Proof. lidx{x) = min^^^x | ^ ~ y | is the boundary distance then (3.2.16) means that u{x) = \x\ — e{dx{x)). Since dx is Lipschitz continuous by the triangle inequality it follows that u is even locally Lipschitz continuous, and since X 3 x \-^ —E{x — y) is harmonic in X for every y G 9 X , it follows that u is subharmonic, because x ^-^ \x\ is convex, li x ^ Xt then —e{dx{x)) < t, which gives a positive lower bound for dx{x)., and if n > 2 we also have \x\ < t, proving the compactness of Xt. If n = 2 we note that if 2/ G dX is fixed then |x| — (27r)~^ log \x — y\ < t, which implies an upper bound for \x\ which again proves the compactness. To construct U we shall use a regularization of u. Take (p G CQ^ rotationally symmetric and non-negative with integral 1, as in the proof of Theorem 3.2.11. Set Uj{x) = /
u{y)(f{{x - y)/ej) dy/e"^ +
£j^/\x\^Ti,
JXj+i
where £j > 0 is chosen so small that the integral is equal to u ^ cp^. > u in Xj_^i^ and Uj < u -{-1 in Xj-2- Choose a convex function x ^ C°°(R) with x{t) = 0 for ^ < 0 and x'(^) > 0 when t > 0. Since Auj > 0 in Xj_^i,
BASIC FACTS ON SUBHARMONIC FUNCTIONS
165
it follows that ^xiV'j -h 1 - i ) > 0 in X^-|_i \ Xj_2/3. We can therefore successively choose aj > 0 so large that k
Uk=uo^rY^
ajxiuj + 1 - i) 1
is> u and strictly subharmonic in Xj^_^i. We have Uk = Ui in Xj if A: > j-\-l and / > j + 1, so [/ = limj_^oo Uj exists and is a strictly subharmonic C^ function in X with U > u. The proof is complete. The following refinement of Theorem 3.2.30 also gives an approximate description of the level sets of subharmonic exhaustion functions: T h e o r e m 3.2.31. Let X C R'^ be an open set and let K be a compact subset. Then the following conditions are equivalent: (i) X \K has no component which is relatively compact in X. (ii) For every x E X \K there is a harmonic function h in X such that supj^ h < h{x). (iii) IfY is an open set with K C Y C X, then there is a continuous subharmonic function u in X such that Xt = {x E X; u{x) < t} is compact for every t, and K C XQ C Xi CY. (iv) There is a function U G C^{X) with AU > 0 satisfying the conditions in (iii). They imply
that
(v) Every harmonic function defined in a neighborhood of K can be uniformly approximated on K by functions which are harmonic in X. Proof. First we prove that (ii) ==^ (i) and that (iii) = > (i). To do so we assume that (i) is not valid. Let a; be a component of X \ if which is relatively compact in X . By the maximum principle we have sup^^ h = sup^ h ii h is harmonic in X , and since duj C if, it follows that (ii) is not valid if a; 7^ 0. Ifu is the function in (iii) then the maximum principle gives 1^ < 0 in a;, so a; must be contained in Y which is not possible for every Y unless a; = 0. Next we prove that (i) => (v). By the Hahn-Banach theorem we must prove that a measure d/x with suppd/x C K which is orthogonal to all functions harmonic in X is also orthogonal to every function harmonic in just a neighborhood of if. Ii x ^ X then X 3 y \-^ d^E{x — y) is a harmonic function in X , so it is orthogonal to dfi. Thus u — E^dii^ which is harmonic in R"^ \ if, vanishes of infinite order at every point in CX, so it follows by analytic continuation that ix = 0 in every component of X \ if which has a boundary point G dX. Since there are no components which
166
III. SUBHARMONIC FUNCTIONS
are relatively compact in X , only the unbounded component u oi X remains to discuss. The convolution
\K
u(x) = / E{x - y) d^{y) JK
vanishes for large |x|, for the Taylor expansion
E{x-y)
=
Y,{d'^E{x)){-yria\
converges uniformly on K^ and the sum of the terms with |a| < iV is a harmonic function oi y E R^ for any A^ since y i-> E{x — y) is harmonic at the origin. Thus u = Q m X \K. Now let /i be a function which is harmonic in a neighborhood Y of K^ and choose x ^ CQ^{Y) equal to 1 in a neighborhood of K. Then x^ = E ^ (A(x/i)), by (3.1.2)', and we obtain hdii=
xhdfi=
/ / E(x - y)dfi{x)A{xh){y)
dy
= I u{y)A{xh){y)dy = 0 since u = 0 in suppA(x/i) CY\K. This proves (v). To prove that (i) = ^ (ii) we can now also use (v). If (i) is fulfilled and x e X \K, then X \{KU {x}) has the same components 3.8 X \K except that {x} has been removed from one of them, so (i) is also satisfied by KU {x}. By (v) we can approximate the harmonic function which is —1 in a neighborhood of K and + 1 in a neighborhood of x by functions harmonic in X with an error < 1, and this proves (ii). It is trivial that (iv) =^ (iii), and the converse follows from the end of the proof of Theorem 3.2.30. To prove (ii) = > (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^:
BASIC FACTS ON SUBHARMONIC FUNCTIONS
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)\'^ < AKd^u{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))/d^u{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 d^u^x, t) > 0, and (3.2.19)
djdkU{x)
= djdku{x,t),
for every t G J{x) with t e dl or d^u{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)'^u{x,t)\.
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III. SUBHARMONIC FUNCTIONS
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<
Y^ VjVkdjdkU < A j,k=i
in the interior of K, in the sense of distribution theory. To prove this we just have to observe that if (^ is a non-negative test function with support in the interior of K^ then I f{{d,vyip)Udx\
= lim\
f {^{x-h ev) ^ip{x -
ev)-2ip{x))U{x)dx/e'^\
= lim I / (f{x){U{x + ev) + U{x - ev) - 2U{x)) dx/e^\
ip{x) dx.
By polarization we conclude that n
-A<
Y^ VjWkdjdkU < A 3,k = l
if V and w are arbitrary unit vectors, which means that {dx^v)U has Lipschitz constant A for every v^ thus U G C^'^ and \U"{x)\ < A almost everywhere. For almost all x E X such that U' is differentiable at a;, it follows from Theorem 1.7.3 and Fubini's theorem for a fixed unit vector v that
{U"{x)v,v) = {u'Ux,t)v,v)
- «„«)V<(x,i),
when t G J{x) \ 9 / , u'^^{x^t) > 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:
BASIC FACTS ON SUBHARMONIC FUNCTIONS
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-^
+ x);ipj{xo + x) = (p{xo -\- x)J
< u],
which proves the continuity at XQ. Thus we have now proved that {d,v)'^(p is a continuous function in the distribution sense, and by polarization it follows that (p" is continuous, thus (p G C'^{X). Replacing ^pj by ipj — (p we may now assume that (p = 0. For every x E X we know then that
{d,vycpj{x)>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
III. SUBHARMONIC FUNCTIONS
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
A^u + 2 ^
Then we have U e
= {t G
XjdtdjU + ^
X'^^d^u > 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<7- /
-\-0{r'^),
hence
min(«(x,t),2/>,K(x,5),2/))da;(2/)+0(7-2)
= -|r /
|«(rr,f)-<(a;,5),7/)|da;(2/)+0(r2),
which proves (i) as r —> 0. If t G J{x) we have by Proposition 3.2.10 Axu{x^ t)/2n — lim r"^ / {u(x + r^/, t) — u{x, t)) duj{y)/cn ^-"^ J\y\=i > lim r - 2 /
{U{x + ry) - C/(x)) duj{y)/cn > 0,
which proves the condition (ii). If t G J{x) \ dl the same argument gives that Au{x, 5 + (rr, A)) > 0 if 5 + {x, A) = t, which proves (iii). Now assume that the conditions (i), (ii), (iii) are fulfilled. The condition (3.2.20) means that d^u > 0 and that n
HARMONIC MAJORANTS AND THE RIESZ REPRESENTATION
171
SO (iii) implies condition (ii) in Theorem 3.2.32, and the condition (i) is the same in Theorem 3.2.32 as in Theorem 3.2.34. Hence U G C^'^(X). With the notation in Theorem 3.2.32 we obtain ii x GY n
AU{x) = AM^, ^)-Yl l^J^M^^ t)\^/dlu{x, t) 1
for some t G J{x) \ dl, hence AU{x) > 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\
J\y\=i
is superharmonic then vi G V.
in X, harmonic when \x - xo\ < R, and
vi
Proof, vi is lower semi-continuous. This is obvious unless |:z; — rcol = R^ and then it follows from the fact that the Poisson integral of a lower semicontinuous function w is the increasing limit of the Poisson integrals of continuous functions Wj | w, which are continuous in the closed ball with boundary values Wj. The definition of superharmonic functions gives that
172
III. SUBHARMONIC FUNCTIONS
vi < V, Now the mean value property (3.2.2) for —vi is obvious for small 6 except when |a:—rcol = i?, and then it follows from the fact that '^1(0;) = v{x) while —vi > —V. Hence vi is super harmonic. If t6 G ZY it follows from the subharmonicity oi u — vi that u < vi^ which completes the proof. Proof of Theorem 3.3.1. If B is an open ball with B C X , it follows from the lemma that in the definition of VQ we may restrict ourselves to functions V which are harmonic, hence continuous, in B. Hence VQ is upper semicontinuous in B , and therefore in all oi X. If it; is a continuous function in X and w > VQ everywhere, then we can find finitely many functions Vi^... ,vjsj in V such that v = min^ Vj < w in a. neighborhood of B. By Lemma 3.3.2 it follows that there is another function v EV with v < w in a neighborhood of B and which is harmonic there. If XQ is the center of B it follows that 'yo(^o) ^'^(^0) = / V dx/m{B) JB
< /
wdx/m{B).
JB
Since w is an arbitrary continuous majorant of VQ., we conclude that -UQ is subharmonic. On the other hand, •^^C^o) > / vdx/m{B)
> / vodx/m{B).,
JB
-z; E V,
JB
so it follows that Vo{xo) = /
vodx/m{B),
JB
which proves first that VQ is continuous, then that VQ is harmonic (Theorem 3.1.12). The proof is complete. Theorem 3.3.1 is so closely related to Perron^s method for solving the Dirichlet problem that we shall digress to discuss it. Let X be a bounded open set in R^, and let / G C{dX). In order to find a solution of the Laplace equation in X with boundary values / we let Uf (resp. V/) be the set of subharmonic (resp. superharmonic) functions in X such that lim
u{x) < f{y),
lim
v{x) > f{y),
y G dX.
We have u 0 then u — v is subharmonic in X and < ^ in a neighborhood of dX, hence < s in X^ so u — v < 0. By an obvious modification of the proof of Theorem 3.3.1 it follows that the functions (3.3.1)
hf = sup u^
hf = inf u
are harmonic, and we have hf < hf. If the Dirichlet problem has a solution it belongs to both Uf and V/, so it must be equal to both hf and hf.
HARMONIC MAJORANTS AND THE RIESZ REPRESENTATION
173
T h e o r e m 3.3.3. If X is an open bounded set in R'^ and f G C{dX), then the harmonic functions hj and hf defined in (3.3.1) are equal. The map C{dX) 3 f ^-^ hf = hj = hf is hnear and is called the Perron generalized solution of the Dirichlet problem. Proof, li f,g e C{dX) hxf = Xhf,
-hf
and 0 < A E R, it is clear that = h-f,
hf -\-hg < hf^g < hf^g < hf -j- hg.
Thus R = {f E C{dX)]hf = hf} is a Hnear space, and f ^-^ hf is hnear in R. If 1/ - ^1 < e and f e R, then hg > hf -{- hg_f > hf — e = hf — e,
hg
-\- hg-f < hf -\- e = hf -\- e.
Hence i? is a closed subspace of C{dX). We shall now prove that if F G C ^ ( R ^ ) then / = F\QX e R. Since F = Fi - F2 where Fi{x) = F{x) + A\x\'^ and F2{x) = A\x\'^ are subharmonic if A is a large positive constant, it suffices to prove this when F is subharmonic and continuous in R'^. Then F eUf, so hf > F^ and since F is continuous in X , it follows that hf G V/, so hf = hf. Thus f e R, and F\dx ^ R a F e C^iW). Such functions are dense in C{dX). For let / G C{dX)^ let {(pu} be a partition of unity and set with summation taken over all u such that we can choose x^ G supp (fj^DdX. Then F G C ^ ( R ^ ) , and ii x e dX then
F{x)-f{x)
=
Y^{f{x,)-f{x))^,{x)
so \F — f\ < e in dX if the partition of unity is so fine that the oscillation of / is < £ in dX fl supp(^iy for every v. The proof is complete. In general the generalized solution of the Dirichlet problem does not assume the desired boundary values everywhere. A trivial example is obtained if X is the unit ball with the origin removed, and / = 0 on the unit sphere. No matter how /(O) is defined, it is clear that sE G Uf when e > 0 and EE G V / if e < 0, so the generalized solution of the Dirichlet problem is 0 no matter what /(O) is. However, at reasonably regular boundary points there is no problem: Definition 3.3.4. A function 6 in X is called a barrier function at y G dX if h is superharmonic in X , h{x) —> 0 as X 3 x -^ y, and for every neighborhood U oi y the restriction of & to X fl CU has a positive lower bound. A simple sufficient condition for the existence of a barrier function is the exterior ball condition that for some 2: ^ X we have |^ — >2^| > \y — z\ if X e X\ {?/}, for then we can take b{x) = E{x - z) - E{y — z).
174
III. SUBHARMONIC FUNCTIONS
T h e o r e m 3.3.5. For the generalized solution hf of the Dirichlet problem in X given in Theorem 3.3.3 when f G C{dX), we have hf{x) —> f{y) ifX3x-^ye dX provided that there is a barrier function b at y. Proof. If e > 0 we can find C^ > 0 so that f{y) -h e + C^ft E V/ and f{y) -e-Cebe Uf. Hence l ^ / ( ^ ) - f{y)\ < ^ + Ceb{x) -> £
2iS X 3 X -^y.
We shall return in Section 3.4 to a more detailed discussion of the boundary behavior of the generalized solution of the Dirichlet problem, but now we return to our main theme and extend Theorem 1.5.1 t o subharmonic functions (Riesz' representation formula): T h e o r e m 3.3.6. Let u be a subharmonic function ^ —oo in the ball = {X E R ' ^ ; | X | < R}, and assume that u has some superharmonic majorant ^ -l-cx). Then u can be written in the form BR
(3.3.2)
u{x) = [/ GR{X, GR{X, y) dpL{y) -f h{x),
x G BR,
JBR
where d/i is a positive measure in BR, the Greenes function GR is defined by (3.1.6), and h is harmonic. Both d/i and h are uniquely determined: h is the smallest harmonic majorant ofu, and d/i — Au. The convergence of (3.3.2) for some x implies that (3.3.3)
/ {R-\y\)dfi{y) < oo.
Conversely the condition (3.3.3) implies that the integral in (3.3.2) converges in BR and defines a subharmonic function ^ —oo with smallest harmonic majorant = 0. Proof. In proving the first statement we may assume that the smallest harmonic majorant is = 0, in particular u < 0. Let d/x be the positive measure Au. If 0 < x ^ 1 ^md x ^ CO{BR) then
I
u^{x) = /
GR{x,y)x{y)dfi{y),
x G BR,
JBR
is subharmonic in BR, AU^ = x^/^5 ^^^ ^x ^^ continuous near OBR with boundary values 0. Hence u — u^ is represented by a subharmonic function i;^ < 0 in BR, and u = u^-\-Vy^ since this is true almost everywhere. Letting X T 1 we conclude that v^^ v where v is harmonic and < 0 in BR, u{x) = /
GR{X, y)dfi{y) 4- v{x),
x G BR.
HARMONIC MAJORANTS AND THE RIESZ REPRESENTATION
175
Since t; is a major ant of u and we assumed that 0 was the smallest harmonic majorant, it follows that v = 0. Thus (3.3.2) is proved, and (3.3.3) follows if we choose x so that u{x) > —oo. On the other hand, assume that u is of the form (3.3.2) with h harmonic and a positive measure dfi satisfying (3.3.3). Let 0 < r < R. The integral when \y\ > r is then convergent when |a:| < r to a harmonic function while the integral when |2/| < r is the sum of a harmonic function and the potential of d/i restricted to Br- Thus u is a. subharmonic function in Br with Au = dfi, which proves the uniqueness of d/i, hence the uniqueness of h. T h e o r e m 3.3.7. If u ^ —oo is subharmonic in BR, then u has a harmonic majorant if and only if Mu{0,r) is bounded when r —> R. The harmonic function h is then the smallest harmonic majorant ofu in BR if and only ifu
I
{h{ry) - u{ry)) duj{y)/cn -> 0,
when r-^
R.
Thus h{x) = lim hr{x)^
x G BR^
where
r—^R
(3.3.5)
hr{x) =
P{x/r,y)u{ry)duj{y),
x e Br.
One can find a sequence rk -^ R such that u{rky) — h{rky) -^ 0 for almost all y € S'^~^ with respect to du. Proof. If h is harmonic in BR and h > 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) <
oo.
iJ\y\=i
Then Yl^i^i'^kV) — "^{fky)) < oo almost everywhere on 5 ^ ^, which completes the proof.
176
III. SUBHARMONIC FUNCTIONS
T h e o r e m 3.3.8. Ifu is subharmonic in BR and J, , ^ |i^(^2/)| dLu{y) is bounded as r -^ R, then there is a measure da on the unit sphere, called the boundary measure of u, such that u(ry) duj{y) -^ da{y) with weak convergence. The smallest harmonic major ant hofu is the Poisson integral of da (3.3.6)
h{x)=
I P{x/R,y)da{y), 'J\y\=i
x e BR.
We have h{ry)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\<8
\da{z)\ = o((5^-^),
as (5 -> 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\<6
< 0{R - r) + C{R -r)
[ {R-r-\-1)"^
dm{t)
Jo nS
= 0{R-r)-i-C{R-r){m{6){R-r-\-6)-''-^n
/ m ( t ) ( i ? - r + t ) - ^ - ^ dt). Jo
The first term -^ 0 as r -> i?. If m(t) < T ' ^ s when 0 < t < 6, then the second term is < Ce and the third is < Ce{R—r)n J^ {R—r-\-t)~'^ dt < Cen^ which proves (3.3.7). For the Poisson integral / of \du we also have f{ry) —^X{y) as r* -^ i? for almost every T/, and (3.3.9)
/
\firy)
- X{y)\ doj{y) ^ 0 ,
as r - . iJ.
J\y\=i
This is much easier to prove than (3.3.7), for (3.3.9) follows from Theorem 3.1.5 when A is continuous, and we always have J \f{ry)\d(jj{y) < J \X{y)\ duj{y), r < R, so (3.3.9) follows if we approximate A by continuous functions in the L^ norm. Since by Theorem 3.3.7 / \u{'i^y) — h{ry)\ duj{y) -^ 0, J\y\=i
when r -^ R,
we obtain, using (3.3.9) also, that lini /
\u{ry) - X{y)\ duj{y) - Una /
\h{ry) - f{ry)\
duj{y),
which is equal to J \das{y)\ since \das\ is the boundary measure oi \h — f\. The proof is complete. Remark. (3.3.10)
The hypothesis of Theorem 3.3.8 is equivalent to / u-^{ry) duj{y) < C, J\y\=i
r < R,
where u'^ = max(n, 0). In fact, this condition is obviously weaker than boundedness of the integral of \u{ry)\. On the other hand, we have / J\y\ = l
\u{ry)\du{y)=
I J\y\ = l
2u'^ {ry) duj{y) -
I J\y\ = l
u{ry)du{y),
178
III. SUBHARMONIC FUNCTIONS
where the last integral is increasing, hence bounded from below when r > R/2, say, so the integral on the left is bounded if (3.3.10) holds. If we strengthen the conditions on u in Theorem 3.3.8 then we can strengthen the statements about the measure da. In particular we shall prove that suitable upper bounds for u guarantee that the singular part of da is < 0. T h e o r e m 3.3.9. Let u be subharmonic in BR, let (p be a non-negative convex increasing function on R such that (p{t)/t —> +oo as t -^ +oo and (p{t) —»ip(—oo) as t -^ —oo, and assume that (3.3.11)
/ ^{u{ry)) du{y) < C < o o , J\y\=^
0 < r < R.
In the boundary measure da(y) = X{y)duj{y) + das(y) ofu the singular part das{y) is < 0, and the boundary measure of the subharmonic function (p{u) is (p(X{y))duj{y); we have (3.3.12)
/
Mu{ry))
- ip{X{y))\ duj{y) -> 0,
when r ^
R.
Proof. For every ^ > 0 we have by hypothesis t < ecp{t) for large enough t, hence t < e(p{t)-\-Cs' Thus the hypothesis (3.3.11) implies that M^+(0, r) is bounded when r -^ i?, so the hypotheses of Theorem 3.3.8 are fulfilled. I f O < ^ G C ( 5 ' ^ - ^ ) , wehave /
'^{ry)9{y) duj{y) <e
J\y\ = l
^{u{ry))g{y)
duj{y) + C^ /
J\y\ = l
g{y) du{y),
J\y\ = l
SO it follows from (3.3.11) that /
9{y)d(j{y) < eCmaxg
+ C^ /
J\y\ = l
g{y)du{y).
J\y\ = l
If E C S^~^ is an open set and we take the supremum over all g with support in E and 0 < ^ < 1, it follows that / da^
[
du{y),
JE
where da'^ is the positive part of da. Hence the outer da'^ measure of a null set for duj is < Ce for every e > 0, which proves that da^ is absolutely
HARMONIC MAJORANTS AND THE RIESZ REPRESENTATION
179
continuous with respect to du. Let du be the boundary measure of (p{u), which is subharmonic by Theorem 3.2.18, thus /
ip{u{ry))g{y) doj{y) -^ [
J\y\ = l
9{y)du{y\
g G
C{S^-').
J\y\ = l
We take ^ > 0 and restrict r to a sequence Vj -^ R such that u{rjy) — h{rjy) —> 0 almost everywhere (Theorem 3.3.7), hence u{rjy) -^ \{y) almost everywhere by (3.3.7). (Here h is the smallest harmonic majorant of u.) Then we obtain using Fatou's lemma (3.3.13)
/
ip{X{y))g{y)dLj{y)<
J\y\=l
[
g{y)du{y).
J\y\=l
In particular, ip{\) is integrable over 5"^"^. Since das < 0 we have u{x) < [
P{x/R,
y)X{y) du{y),
x G BR,
and in view of (3.1.10) it follows from Jensen's inequality that ^{u{x))
< [
P{x/R,
yMX{y))
du{y),
x e BR,
which implies that diy{y) < (p{X{y))du;{y). Combined with the opposite inequality (3.3.13) this proves that diy{y) = (p{X{y)) duj{y) is absolutely continuous, and then (3.3.12) follows from (3.3.8) applied to (p{u). The proof is complete. We shall now specialize the preceding results to the logarithm of the absolute value of a function which is analytic in the unit disc C C. T h e o r e m 3.3.10. Let f be analytic in the unit disc C C and assume that (3.3.14)
/ Jo
log \f{re'^)\ dO < C,
0 < r < 1.
If zi, Z25 • • • ^re the zeros repeated as many times as the multipUcity, oo
(3.3.15)
Y^{l-\zj\)
This impUes that the Blaschke
product oo
,3.3.16,
^ w = n ^ H
then
180
III. SUBHARMONIC FUNCTIONS
converges to an analytic function with \B{z)\ < 1 and only the zeros Zj when |2;| < 1. The smallest harmonic major ant of log \B\ is = 0. (When Zj — 0 the factor Zj/\zj\ should be omitted.) Thus f{z) = B{z)g(z) where g is analytic and ^ 0. Conversely iff admits a factorization f{z) = fi(z)f2{z) with \fi{z)\ < 1 and f2{z) ^ 0 when \z\ < 1, then (3.3.14) is valid and \h{z)\ < \Biz)l \f2{z)\ > \g{z)\. Proof. By Theorem 3.3.7 the subharmonic function z H-> log|/(2:)| has a harmonic majorant. Since A l o g | / | = 2n^6zj: it follows from (3.3.3) that (3.3.15) is valid, and (3.3.15) implies that the sum defining log|5(2;)| converges. The factor Zj/\zj\ in (3.3.16) makes the argument of all factors with Zj ^ 0 equal to 0 at the origin, and this makes the sum defining the imaginary part modulo 27r also convergent. More precisely, since
1 _ I^^IL
= {\zj\ - I)^^^AN_
ZZj-l\Zj\
^
{zZj-l)\Zj\'
it follows from (3.3.15), which implies \zj\ —> 1, that the product (3.3.16) converges uniformly on any compact subset of the unit disc, if one drops the finitely many factors with zeros there. This proves the convergence and that B has no zeros other than the sequence Zj. Since every factor has absolute value < 1 in the unit disc, we have | S ( z ) | < 1 unless there is no zero at all. Since B has the same zeros as / with the same multiplicities, it follows that / = Bg where g is an analytic function with no zeros in the unit disc, so Theorem 3.3.6 shows that log|p(2;)| is the smallest harmonic majorant of log | / | . If we are given a factorization / = /1/2 with the properties in the statement, then log I/I < log I/2I, so log I/2I > log \g\ since log |^| is the smallest harmonic majorant of log | / | . We shall now restrict the analytic functions further: T h e o r e m 3.3.11. If (3.3.14) is strengthened
to
p2Tr
(3.3.17)
/ Jo
log+ |/(re^^)| d0 < C,
0 < r < 1,
then the function g in Theorem 3.3.10 can be written (3.3.18)
g{z)=exi>{ic+{27r)-'
/ J[0,2TT)
-^da{e)), ^
~
^
where da is the weak limit of log \f{re'^^)\ d9 as r —> 1, and c G R . pointwise limit /(e*^) = limy.-^! f{re^^) exists for almost every 9.
The
HARMONIC MAJORANTS AND THE RIESZ REPRESENTATION
181
Proof. (3.3.18) is an immediate consequence of Theorem 3.3.8. (See Exercise 3.1.4 and the remark after Theorem 3.1.8.) To prove the existence of the pointwise Hmit we first make the stronger assumption /.27r
/
\f{re'^)\de
0
Then the real and imaginary parts of / are harmonic functions with bounded L^ norms on the circles {z\ \z\ — r*}, so we can write
!(ri')^jp{re'\G')dv{Q'\ where dv is a complex valued measure on the unit circle. (See the remark after Theorem 3.1.8.) If dv{e') ^ \{Q')dQ' -f dvs{Q') is the Lebesgue decomposition, it follows by Theorem 3.3.8 that f{re'^^) -^ X{0) for almost all 0. In particular, the special case just studied shows that the Blaschke product (3.3.16) has radial boundary values, and it follows from (3.3.7) that they have absolute value one, for the smallest harmonic majorant of l o g | S | is 0. If in (3.3.18) we split the measure da in its positive and negative parts, we obtain f{z) = B{z)g^{z)/g-{z) where g± is analytic, has no zeros, and | l / p ± | < 1- Hence l/g± has radial limits almost everywhere, by the first part of the proof, and since log |^-b| has boundary values in L^, hence finite almost everywhere, it follows that the boundary values of ^± are finite and non-zero almost everywhere. The proof is complete. T h e o r e m 3.3.12 (F. and M. Riesz). Iff and
is analytic in the unit disc
/•27r
(3.3.19)
lim /
\f{re'^)\de
then /(e*^) = lim^_i f{re^^) exists almost everywhere, and is an L^ function such that '\f{re'')-f{e'')\de^O, / '
r ^ 1.
Proof Since 1/(^^)1 is subharmonic the integral in (3.3.19) is an increasing function so (3.3.19) means that r27r
I
J9\ \f{re'')\de
0
^0
The existence of radial limits was already proved in Theorem 3.3.11. Since / is a harmonic function it follows from (3.3.19) and the remark after
182
III. SUBHARMONIC FUNCTIONS
Theorem 3.1.8 that f{re^^)dO -^ da{6) when r -^ 1, in the weak measure topology. If X is a continuous periodic function then I /
xiO)da{e)\ = ]im\ [
x{0)f{re^')de\
\xm\f{re^')\de.
Now we can apply Theorem 3.3.9 io u = log | / | and ip(t) = e*, which gives that the boundary measure of | / | is |/(e*^)| dO^ hence
l/
x{e)da{e)\< f
\x{e)\\f{e'')\de.
Thus \da\ is absolutely continuous so da{0) — F{0)d0 for some F G L^. But / is the Poisson integral of F{6) dO^ so the radial boundary values are F{0) almost everywhere. This proves that F{0) = /(e*^) almost everywhere and completes the proof of Theorem 3.3.12, for the L^ convergence follows from (3.3.9). Remark. Theorem 3.3.12 is also valid with L^ replaced by L^, if 1 < p < oo. However, that is a much simpler result because the unit ball of L^ is then weakly compact, since L^ is a reflexive space. In the L^ case we could a priori only obtain a measure as a limit of f{re^^) when r -^ 1. This point is underlined in the following result. Corollary 3.3.13. If da is a measure on the unit circle such that [ then da is absolutely
e'^^ da{0) = 0,
3> 0,
continuous.
Proof. The Poisson kernel in the unit disc is in complex notation 1
l-UP
27r \e'^ Since z/{e-'^
-^) /
y[0,27r)
1 / z ~ 2 ^ V "^ e^^ -z
'^ e-i^
-zJ'
= I1T ^^^'^^^ i^ follows that P{z, 0) da{0) = ^
[
-f-
27r y[o,27r) ^'^ '
da{0) = f{z) ^
is analytic in the unit disc. Since /^^ |/(re*^)| rf^ < /rg 27r) l^^(^)l' ^^^ assumptions of Theorem 3.3.12 are fulfilled, so the boundary measure da of the harmonic function / is equal to /(e*^) d^, hence in 1}. There is a local version of the F. and M. Riesz theorem:
HARMONIC MAJORANTS AND THE RIESZ REPRESENTATION
183
T h e o r e m 3.3.12'. Let I be an open interval C (0,27r) and f an analytic function in X == {re*^; 6 e I^l - e < r < 1} for some e > 0, such that II
\f{re'^)\ drde
J Jeel,l-e
Ijm
I \f{re'^)\ dO < oo.
r - ^ 1 Jl
Then /(e*^) = linir—,! /(re*^) exists almost everywhere in I, and f(re^^) — f{e'^) -^ 0 in Ll^{I) when r-^ 1. Proof. Choose x ^ C^W, X = {re'^] 0 G / , 1 - e < r < 1 + e}, so that X =^ 1 in a neighborhood of {re*^; 1 - ^s < r < 1^9 e [a, &]}, where [a^b] is a closed subinterval of I. If x{^) = 1 ^-nd |z| < ^ < 1, then Cauchy's integral formula gives
J
ge
—z
J\C\
^
^
where dL is the Lebesgue measure in C = R^. Letting g —^ 1 through a sequence gj such that f{gje^^) has a weak limit da in / , we obtain f{z) = g(z) — h(z) where
J
e
- z
J\c\
^~ ^
The function /i is analytic outside supp dx- Now consider the function /•f>
G{z) = (z - e - ) ( . - e^^)(27r)-^ /
pie
- ^ W ) .
The integral can be estimated by C/d{z) where d{z) is the distance to J = {e*^; 0 G [a, 6]}, so G is uniformly bounded in {z G C; |z| < 1} outside the sector defined by J . The difference G{z) -{z-
e-)(z -
e"')g{z)
is of the same kind as G but with integral over the complement of [a, 6], so it is uniformly bounded in the sector defined by J . Since g{z) — f{z) = h{z) is analytic in a neighborhood of J , it follows that lini^_^^ J^ ^ |G(re*^)| cf0 < oo. Thus G satisfies the condition in the F. and M. Riesz theorem so 0 1-^ G{re'^^) is L^ convergent as r ^ 1. This proves that 0 H^ f(re'^^) converges in LiQ(,((a,6)) and completes the proof. Remark 1. In the language of microlocal analysis (see e.g. ALPDO I, Section 8.4) the preceding result can be stated as follows: If da is a measure in an open interval / C R, then da is absolutely continuous in [x G / ; (x, 1) i WFA{da)}
U {re G / ; {x, - 1 ) ^
WFA{da)}.
184
III. SUBHARMONIC FUNCTIONS
Remark 2. The hypothesis made in Theorem 3.3.12' is weaker than assuming that lim / / g^lJ J0el,g
\f{Te'')\drdei{l-Q)
< OO.
However, this hypothesis is usually fulfilled in the applications, and it is clearly invariant under conformal mappings. (By a careful choice of x as in ALPDO I, Section 3.1, the hypothesis in Theorem 3.3.12' can be weakened by a factor (1 — r)^ in the double integral.) Definition 3.3.14. The Hardy space H^ in the unit disc C C consists for 0 < p < OO of all analytic functions such that
(3.3.20)
[j^ \f{re'')Yddf
is bounded when r < 1, and one defines ||/||i/p as the limit when r —> 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
-^A(0)d^).
Jo
^
— z
Then l^l < 1, and \S{re'^)\ -^ 1 as r -^ 1 for almost every <9, ||F||ijp, and F is characterized by the fact that F(0) > 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.
HARMONIC MAJORANTS AND THE RIESZ REPRESENTATION
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,(^)^exp((27r)-i / JeG(0,27r)Md)
^:!^x{e)de) - ^
is in FQ for any t G R, for Ft is the product of F by a bounded analytic function. We have \Ft{e'^)\ < 1 when t < 0, and Ft(e'^) -^ 1 as t -> - 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 <0 in R^~^, and a constant
186
III. SUBHARMONIC FUNCTIONS
a < 0 such that (3.3.21) u{x) = / GH{x,y)dfi(y)-\-
/ PH{x,y')da{y')-\-ax^, JdH
JH
x e H,
(3.3.22) / 2/n(l + 12/1)"" dii{y) < oo,
/ (1 + \y'\)-^ JdH
JH
da{y') > -cx).
The measures d^i 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, then (3.3.21) defines a subharmonic function < 0 in H. Proof. Let dfi = Au and write, as in the proof of Theorem 3.3.6, u^{x)=
/
GH{x,y)x{y)dfi{y),
JH
where x ^ Co{H) and 0 < x < 1- Then A{u - u^) == (1 - x)d^i > 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'\'^)~^ui{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).
HARMONIC MAJORANTS AND THE RIESZ REPRESENTATION
187
When 0 < a;„ < ^ we have Ca;„(l + 2/„)(l + |2/|)-", \E{y',\Xn-yn\
+ l)-E(y',Xn+yn
+ '^)\ < I
Cynil + | y | ) - " ,
for Vn ^ ^n 2tnd 0 < y-n ^ ^n respectively, and since x^il + Vn) ^ when Xji < Vn^ this proves (3.3.23) by dominated convergence. Let t > 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)\d^i{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 by u{x) < Ci -\- C2Xn when X G H^ we get the same conclusions except that a < C2 and that da — Ci dx' < 0. An extension of Theorem 3.3.16 analogous to Theorem 3.1.11 is easily obtained: T h e o r e m 3.3.17. Let u be subharmonic ^ —00 in H, and assume that (3.3.24)
lim / u-^{x\ Xn){l + \x'\y Xn-^ + oJ
(3.3.25)
lim R-^+00
/
dx' < 00,
u-^{x)dx/R'''^^
< oc.
JHniB2R\BR)
Then there is a measure d/j. > 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
III. SUBHARMONIC FUNCTIONS
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-^oo'^{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)<
I
P+ix,y)viy)dS{y),
when -y is a subharmonic function in a neighborhood of 5 J . The proof of Theorem 3.1.11 can therefore be used with no essential change to get a harmonic majorant h of u as in (3.1.24), and Theorem 3.3.17 is then a consequence of Theorem 3.3.16 applied to u — h. The proof is complete. We shall now give an important application of Theorem 3.3.17 with n = 2: T h e o r e m 3.3.18. Let u e £'{11), and let u{C) = u{e-'^') Fourier-Laplace transform. Then
(3.3.26)
y,-TTi^^^<-'
be the
I : I T K ; I 2 < - ^
where d , ("2,... are the zeros ofu, with multiple zeros repeated, and (3.3.27)
r 1 log \u{tO\ - /i(ImC)
in
Ll,{C),
where h(y) = a±y when ±y >0 jf[a_, a_|_] is the smallest interval ing suppn. If N{R) is the number of(j with \(j\ < R, then (3.3.28)
N{R)/R
-> (a+ - a_)/7r,
Proof. If supp?x C [a_,a_(_], then
when
R-^00.
contain-
HARMONIC MAJORANTS AND THE RIESZ REPRESENTATION
189
for some constants C and N. We can therefore apply Theorem 3.3.17 to l o g | ^ ( C ) | < l o g C + Ariog(l + | C n + MlmC) in the upper and in the lower half plane. This gives (3.3.26) and proves that (3.3.27) is valid for some function h{y) = a±y when ±y > 0, with tt-f. < a+ and a_ > a_. Since log(l + |^p) = 2 log |^ 4- ^1 when ^ G R, the Poisson integral of ^ i-^ iVlog(l + ^^) is ( H-> 2Ariog |C + i\ when I m ( > 0, so when Im ^ > 0 log\u{0\
< logC+2Ariog|C+i|-fa+ImC, that is, |'a(C)| < C|C+ip^e^+^^^
and similarly when ImC < 0. By the Paley-Wiener-Schwartz theorem this implies that supp?x C [a_, 04-] which proves that d± = a± and that (3.3.27) holds. Hence t-^Y^2n6(^./t
= r^Alog\u(t()\
-> A/i(ImO = (a+-a.)6(ImC),
t-^
00,
in the weak topology of measures, for it agrees with the distribution topology for positive measures, and A is continuous in the distribution topology. Thus r^ V 27r(^(C, A) -^ («+ - « - ) / ^ ( 0 di, t -^ 00, Jn ii (p e Co(C). If (pi,(p2 G CQ{C) and cpi < "0 < <^2 5 it follows that IIE t-' Y^ 27rV'(C,/t) < (a+ - a_) / (^2(0 ^C,
lim t-' Yl 2^V^(0 A) > («+ - «-) / MO dC. If '0 vanishes outside a compact set and is Riemann integrable with respect to the limit measure ^(Im^), we can choose (pi^(p2 so that the right-hand sides are arbitrarily close and conclude that
t-' Y 2^V'(G-A) - («+ - ^-) / i^iO d^, t
00.
JK
(3.3.28) follows when ip is the characteristic function of the unit disc. (3.3.26) is a basic result in the theory of quasianalytic functions, (3.3.28) is due to Titchmarsh, and the stronger result (3.3.27) is due to Beurling. It gives a proof of the following result, Titchmarsh's convolution theorem, which is more natural than the original one based on (3.3.28):
190
III. SUBHARMONIC FUNCTIONS
C o r o l l a r y 3.3.19. If ui,U2 G f ' ( R ) and Ij is the smallest interval containing snpp Uj, then 7i + h is the smallest interval containing the support of the convolution ^ i * 7x2Proof. By (3.3.27) we have t~^ log\uj{t()\
-^ hj.{lm()j
t~^log\u{t()\
—> /ij(ImC),
when t -^ 00,
where / is the smallest interval containing the support oiu = ui^U2. 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 < p < 00, consists of all functions / which are analytic in C-|_ such that (3.3.29)
I I / I I H . - sup ( /
\f{x + iy)]^ dx)
y>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-^oJn
\f{x^iy)\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^) <
00.
We leave for the reader to show by repetition of the proof of Theorem 3.3.9 that da'^ is absolutely continuous. Thus da{x) = das{x) + X{x)dx where / R M^s(^)|/(1 + ^ ^ ) < 00 and das ^ 0. This singular measure corresponds to a singular inner factor
Siz) = exp {ic+l-Jj^^
-
^)daAx)),
which is analytic in C_|_, has modulus < 1 and boundary values of absolute value 1 almost everywhere by the half plane analogue of (3.3.7) since , ^ . .,
liaz
f
das(x)
192
III. SUBHARMONIC FUNCTIONS
Hence we obtain a factorization f = BSh where 5 is a Blaschke product, S a singular inner function, h G H^ has no zeros, ||/i||ifp — ||/||i/p and hiz) = exp ( 1 J ^ ( ^
- ^
)
X{x) dx),
[ \ ^ } ^ < o o , fx^.)d. 0 to a measure dv with Poisson integral / . If ^ > 1 then the reflexivity of L^ shows that there is a weak limit in L^, so dv — f dx with f E L^. If p = 1 this follows from the local version Theorem 3.3.12' of the F. and M. Riesz theorem if we map the half plane conformally to a disc. Hence \\f{-'+iy)-f\\L'>-^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) df^iO -^^ = u{0) + (27r)-i / log |1 - z/CI dfiiO, Jc
(3.3.30)
(3.3.31) f J|C|<1
log(l/K|)dMO
/
z G C,
dfi{0
J\C\
Conversely, (3.3.31) implies that (3.3.30) defines a subharmonic function < Od^l^ log |2:|) as 2 —> oo. Proof. The Riesz representation formula for the disc {z; \z\ < R} gives for \z\ < R (3.3.32) uiz) = i - / log ^ ' ~ ^J dfiiO + r 27r J\^\
P{z/R,
e^'HRe^')
d9.
HARMONIC MAJORANTS AND THE RIESZ REPRESENTATION
193
When 2: = 0 it follows that ^(0) + J - / log ^ rf/x(C) = ^ [^ u{Re^% de
+ C^R'^
rf/^(C) < Ci + C2 - u(0), ^ f dfiiO 0, 27r J\c\
log|l-z/C|dMC)-^ /
log|l-
Jo
For z in a compact set in C, the last term is 0{R^~^) when i? —» 00, for P{z/R, e'^) - P(0, e«*) = 0(|x|/i?), and /»27r
/
/•27r
|^(i?e^^)| de = 2
Jo
/*27r
u^{Re'^) dO Jo
u{Re'^) dO Jo < 47r(Ci + C2R^)
- 27r7z(0).
The preceding term is 0{R^~^) in view of (3.3.31) because log \l — z(/R'^\ = 0{\z\/R) when |C| < R. This completes the proof of (3.3.30). With M{R) = Ju\^jidfi{() and 6: > 0 we have if (3.3.31) is assumed (3.3.33)
/
(1 + I d ) " ' " ' dfi{C) = r
J\C,\>r
{1 +
R)-'-'dM{R)
Jr+0 /•oo
<{Q + e) j
(1+ R)-^-'-'^M{R)
dR
Jr
< C[{1 + r)-^-^ + C'.il + r)-'{0 + e)/e. Since log|l — z/(\ < \z/C\ this implies the convergence of (3.3.30), and we obtain if |2:| == i? is large
/ f
log |i - z/(\ dfiio < \z\ I log |1 - z/Ci d/x(C) < log(2ii) /
«f/.(c)/|CI < c^>l7(i - Q). dfiiO + f
log(l/|C|) dfiiC),
194
III. SUBHARMONIC FUNCTIONS
which proves the last statement. Similar representation formulas hold for subharmonic functions bounded by higher powers of |z|, but they require that one subtracts some terms in the Taylor expansion of log|l - 2:/C| to gain convergence. We shall not develop this theme here but give a classical application of Theorem 3.3.21: Corollary 3.3.22. Let f{z) be an entire analytic function in C such that /(O) > 0 and \f{z)\ < d e ^ ^ k l ^ foj. some positive constants Ci,C2 and g E (0,1). Then the number of zeros Zj of f with \zj\ < R is 0{R^) as R -^ oo, and
(3.3.34)
fiz) = f
{0)11(1-z/zj).
Proof. From Theorem 3.3.21 we know that Xlk |
Yl \'j\''~'
< °o if ^ > 0.
This guarantees the convergence of the infinite product in (3.3.34) and shows that f{z) = g{z)f{0) YI{1 — z/zj) where g is analytic, \g(z)\ — 1 for every z, and ^(0) — 1, which implies g^\. The proof is complete. In many problems one needs to estimate an analytic function in terms of upper bounds for the product by another analytic function for which only an upper bound and a fairly weak lower bound is available. This can often be done using some minimum modulus theorem. We shall now discuss results of that type which can be deduced from the representation theorems in this section. T h e o r e m 3.3.23. Let u ^ —oo be a subharmonic function < M in {z G C; \z\ < R}, and assume that mi\z\=ru{^) ^ ^^ for every r € [0,i?). Then it follows that (3.3.35) 2 R —r u{z) < 6{\z\)m + (1 - S{\z\))M, \z\ < R, where 6{r) = - arcsin — . Proof. It suffices to prove the theorem when i? = 1, M = 0, and m = — 1, thus u{z) < 0, \z\ < 1; inf u{z) < - 1 , 0 < r < 1. \z\=r
By Theorems 3.3.6 and 3.3.8 we have the Riesz representation u{z) = f Giz, OdKO J\i\
+ [ " Piz, e'')d
\z\ < 1,
HARMONIC MAJORANTS AND THE RIESZ REPRESENTATION
195
where d/i >0,da< 0, andG(z,C) = Gi(z,C) = {27r)-Hog{\z-C\/\l-zC\). Harnack's inequality (Theorem 3.1.7) followed from the inequality (3.3.36)
P{-\z\,
1) < P{z, e'') < P{\z\, 1),
\z\ < 1,
and now we need in addition an analogous estimate for G, (3.3.37)
log f ^ < 27rGiz, () < log f ^ , 1 — rg 1 -\- rg
\z\ ^ r, |C| = g.
Since (r - g)/{l - rg) - (r + g)/{l + rg) = 2g{r^ - 1)/(1 - r'^g'^) < 0, it is clear that the left-hand side in (3.3.37) is in fact smaller than that on the right. For fixed ( and fixed r the inequality on the right is valid for all z in a disc with —r(/\(\ on the boundary, r^/|C| in the interior by what we just proved, and the center on the line between them, so the inequality is valid when \z\ = r. Similarly we obtain the left inequality (3.3.37). Let d/i* be the measure d/i rotated to the positive real axis, that is, J ^(r) dfi*(r) = J ipi\z\)dii{z),
^ € Co([0,1)),
let a* = J da < 0, and set (3.3.38)
'v{z)=
f G{z,r)dn*{r) Jlo,i)
+
a*P{z,l).
This defines a subharmonic function < 0 in D = {z £ C;\z\ < 1} since
J{l-r)dfi%r)
=
J{l-\C\)dfi{0
and it follows from (3.3.36), (3.3.37) that (3.3.39)
u{z) < v{~\z\),
\z\ < 1,
v{r) < inf u{z) < - 1 , 0 < r < 1. \z\=r
V is harmonic in Z) \ [0,1), and we shall estimate v by comparison with a function which is harmonic there, equal to —1 on the slit and = 0 on the circular boundary. Let
^(•^) = r T ^ '
^ez)\[0,i),
where we take ( = y/z in the upper half plane. Since ( i-^ (1 - ()/{l H- C) maps the upper half of D on the fourth quadrant, we can define a.rgw(z)
196
III. SUBHARMONIC FUNCTIONS
uniquely G (—7r/2,0). The boundary values are 0 on [0,1) and —7r/2 on dD\{l}. Hence v{z) < - l - ( 2 / 7 r ) a r g 7 i ; ( z ) ,
zeD\[0,l),
for angw{z) = Imlogw{z) is harmonic, and if 0 < ^ < 1 then -^((1 — e)z) < — 1 — {2/7r)w{z) + 6: in a neighborhood of every boundary point. Now , ^
. ^
1 — i\/r 1+ i ^
1— r 1+ r
1i\fr 1 + r'
so it follows from (3.3.39) that 1 -r u{z^ < —\ — (2/7r) arg w{—r) = — 1 + (2/7r) arccos
\ -\r T
— —(2ln) arcsin
\-r
,
\z\ = r,
which completes the proof. Theorem 3.3.23 is due to Beurling (in a more general version) and Nevanlinna, who completed earlier work of Carleman, Schmidt and Milloux. We give some corollaries to suggest why it is useful. Corollary 3.3.24. Ifu ^ —oo is subharmonic in [z G C; \z\ < R], and M{r) = sup|^|<;^tA(z), 0 M{R) r
inf {M{R) -
M{r))/d{r),
0
where 6{r) is defined in (3.3.35). Proof. If the left-hand side is denoted by m, then Theorem 3.3.23 states that M{r) < 6{r)m + (1 - 8{r))M{R), m > M{R) - {M{R) -
that is,
M{r))/6{r).
Corollary 3.3.25. Let u ^ —oo and v be subharmonic functions {z e C; \z\ < R}, and let M(r) = snp^^^^^ u{z), M{R) < oo. T i e n ^(0) < sup {u{z) + v{z)) - M{R) + \z\
inf (M(i?) - M(r))/(!)(r), 0
in
HARMONIC MAJORANTS AND THE RIESZ REPRESENTATION
197
where 6(r) is deGned by (3.3.23). Proof. For every r < i? we have v{0) < sup v{z) < sup {u{z) -\-v{z)) — inf u{z) \z\^r
\z\=r
kl=^
< sup {u{z) -{-v{z)) — inf u{z), \z\
sup \p{z)g{z)l
R>0.
\z\
(Hint: Reflect the zeros of p with \z\ < i? in the circle. See also ALPDO, Lemma 7.3.3.) We shall now prove a minimum modulus theorem for functions which are subharmonic in C by using the representation in Theorem 3.3.21. T h e o r e m 3.3.26 (Wiman-VaHron). Let u be a subharmonic in C with u{0) > —oo which is not constant, and set (3.3.40)
If{)< Q<\ such that (3.3.41)
M(r) = sup u(z), \z\=r and limK-^cx) M{R)/R^
function
m{r) = inf u{z). 1^1=^ = 0, then there IS a sequence Vj -^ oo
^ ( ^ j ) - M(rj) cos(7r^) -^ +oo,
hence m{rj) —> +cx).
Before the proof we shall discuss the important and instructive example u{z) = Re z^, ^ > 0, which is harmonic except on the negative real axis and positive on the positive real axis. (We take arg^: G (—7r,7r].) Thus ATX = 0 except on the negative real axis. Since u and du/dx are continuous when z 7^ 0, we can calculate d'^ujdx^ by pointwise differentiation. However, dujdy
= i{du/dz
- du/dz)
= hoi^^''^
- ^^~^) =
-glmz^-^
198
III. SUBHARMONIC FUNCTIONS
has a j u m p 2^sin7r^|x|^~^ at x < 0, so it follows that in the sense of distribution theory Au =
2gsm7rg\z\^~'^6{y)H{-x)
where H is the Heaviside function; there is no term with support at the origin since Au is homogeneous of degree g—2 > —2. Thus u is subharmonic if 0 < ^ < 1, and Theorem 3.3.21 gives /•OO
(3.3.42)
Rez^ = ^^^^^^^ TT
/
log\l-^ z/t\t'-Ut
Jo
Note that the minimum modulus r^cos(7r^) does not converge to oo unless ^ < | , which motivates this hypothesis in Theorem 3.3.26. Proof of Theorem 3.3.26. By Theorem 3.3.21 we can write u{z)^u{Q)^
/log|l-z/CM/x(C) Jc
where dfi >0 satisfies (3.3.31). As in the proof of Theorem 3.3.23 we rotate the masses so that we obtain a measure d/x* on the positive real axis, that is, /
ip{r)d^i^=
Jr>0
[ cp{\(\)dfi{Cl
^eCo([0,oo)),
Jc
and we introduce /•oo
v{z) = u{0) + / Jo
log |1 +
z/t\dfi*{t),
which is subharmonic in C and harmonic except on the closed negative real axis R _ . Since
ioglli^
M{r) < v{r).
We want to prove that v{—r) — v{r) cos(7r^) —> -1-CXD for a sequence rj -^ oo. If ^ > 0 it follows from the hypothesis M{R) = o{R^) that a{e) = snp{v{—r) — er^ cos{7rQ)) < oo.
HARMONIC MAJORANTS AND THE RIESZ REPRESENTATION
199
The inequality v{z) — eRez^ — a{e) < 0 is then vaUd on R _ , and we claim that it is valid in C. In fact, if ^ < a < | and (5 > 0, then the harmonic function in C \ R_ z^v{z)-eRez^
- a{e) - 6{Rez^ + 1)
is < 0 in a neighborhood of R_ and - ^ —oo at oo since it follows from Theorem 3.3.21 that v{z) < o{\z\^) as 2; —> oo. Hence it is < 0 in C \ R _ , and when ^ -^ 0 it follows that v{z) <£Rez^
-i- a{£),
z e C.
(We have proved a case of the Phragmen-Lindelof principle here.) Thus M{r) < er^ + a(^), so a{e) - ^ oo as ^ —> 0 unless M is bounded, which would imply that n is a constant (Theorem 3.2.24). This case is excluded by hypothesis, so we conclude that the maximum a{e) is assumed at a point r{e) where r{e) -^ 00 as 6: -^ 0. Since m{r{s)) > v{-r{e)) M{ris))
= a{e) + ^r(^)^cos(7r^),
< v{r{e)) < er{£y + a{£),
it follows that m{r{s)) > M{r{s)) cos{7rg) + (1 — cos7r^)a(^), which proves the theorem. There are weaker results available when ^ > | , which guarantee that m{r)/r^ is bounded below for a suitable sequence -^ 00. One may for example apply Theorem 3.3.26 to the subharmonic function U defined by U{z^) = ^^ u{e'^'^'^/^z) where N is chosen so large that g/N < | . We leave this as an exercise for the reader. The main tool in this chapter has been Green's function for a ball (and for a half space). We shall end this section by discussing Green's function also for an arbitrary open bounded convex set X C R'^, n > 2. If y G X we know from Theorem 3.3.5 that there is a harmonic function hy in X which is continuous in X such that hy{x) = E{x—y)^ x G dX. Assuming for the sake of simplicity in statements that the diameter of X is at most 1 when n == 2, that is, that \x — y\ < I for x,^/ G X , we obtain /i < 0 by the maximum principle so Green's function defined by Gj\:(x, y) = E{x—y)—hy{x) satisfies (3.3.43)
Eix-y)
200
III. SUBHARMONIC FUNCTIONS
UfeC^iX)
then i{x) = /
G{x,y)f{y)dy
is a C°° function which satisfies Poisson's equation Au = / and vanishes on the boundary. This follows from Theorem 3.1.3. We shall now examine the behavior of G{x^y) as a; —> dX by a barrier argument. Let x^y ^ X and assume that d{x) < \x — y\/b where d{x) is the distance from x to dX. Choose XQ E dX with \xo - x\ = d(x), let u be an exterior normal of dX at XQ with |z/| = \x — 2/1/4, and set XQ = {z E X\ \z — XQ — Ty\ < 2|z/|}. Then x £ XQ since \x — Xo\ < |z/|, and ii z E XQ then \z — x\ < 2\u\ = \x-y\/2 since {Z-XQ^U) < 0 by the convexity of X . (Draw a figure!) Hence \z — y\ > \x — y\/2 = 2\u\. Prom the maximum principle we now obtain Gx{z, y) > c{E{j^) - E{z - xo - i^)),
z e Xo,
if this is true when z G OXQ. When z G OXQ fl dX this follows if c > 0 since Gx{z,y) — 0 and \z - XQ — v\ > \u\, and when z G OXQ \ dX then Gx{z, y) > E{z -y)> E{2u) and E{z -xo-jy) = E{2u), so the inequality is valid if c{E{u) - E{2u)) = E{2v),
that is, c = E{2iy)/{E{u)
-
E{2v)).
Here c is independent of u when n > 2, and c = log(l/|2z/|)/log 2 if n = 2. Since \E{u) - E{x — XQ - i/)| < Clz/I^'^^lx - xo| we conclude that (3.3.44)
0 < -Gx{x,
y) < Cd{x)\x - y]^'"",
if d{x) <\x-
y\/h.
Here C only depends on the dimension n if n > 2, and (7/ log(2/|a: — y\) is bounded when n = 2. (Note that the proof is just a detailed version of the proof of Theorem 3.3.5.) Using Theorem 3.1.7 we conclude that (3.3.45)
\dGx{x,y)ldx\
ii d{x) <
\x-y\/5.
If 0 < t < 1 t h e n x i-^ Gx{x,y)—t'^~'^Gx{y-\-t{x-y),y) is harmonic in X and positive on dX^ hence in X by the maximum principle. Differentiation with respect to t yields when t = 1 that (3.3.46)
{x - y,d/dx)Gx{x,y)
> -{n - 2)Gx{x,y)
> 0,
if n > 2. When n = 2 the left-hand side is harmonic, > 0 and equal to l/27r when x = ?/ so it is positive also then. Hence the level surfaces {x G X; Gxix^y) = 7 } , 7 < 0, are analytic surfaces which are starshaped with respect to y. Much more is true:
HARMONIC MAJORANTS AND THE RIESZ REPRESENTATION
201
T h e o r e m 3.3.27. Green's function Gx{x^y) of an open bounded convex set X C R'^ is Lipschitz continuous in X x X when \x — y\ has a fixed lower bound, and Gx{x,y) — Gx{y,x). IfYcX is another open convex set, then Gy ^ Gx in Y xY, and ifYj is an increasing sequence of convex open sets with union X, then Gy^ | Gx SLS j —> oo. The level surfaces {x G X]Gx{x,y) = 7} are strictly convex for arbitrary fixed y £ X and 7 < 0. IfdX e C^ and yeX, then n
n
Y^ tjtkd'^Gx{x,y)/dxjdxk, j,fc=i
teW,
Y1^3d^x{x,y)/dxj
=Q,
1
X 3 x ^ y, extends continuously tox ^ X\{y}. IfX = {x G R^; g{x) < 0} where g £ C'^ and g' ^ 0 on dX, then the extension is equal to n
^(^) Z-l ij'^kd^Q{^)/9xjdxk,
where c{x)g'{x) = dGx{x,y)/dx,
x£dX.
Proof. li y E Y C X then the harmonic function Gy{x,y) — Gx(x,y) of X G y is positive on dY, hence in Y, which proves that Gy > Gx- If y EYI and Yj^X then it follows from (3.3.44) that the maximum on dYj of the positive harmonic function Gy.{',y) — Gxi'iy) in Yj tends to 0 as j ^ 0 0 . T h u s G y . [GxNext we shall examine G{x,yo) for fixed yo E X and x close to dX when dX G C^. We can then find R > 0 such that for every z G dX the ball B{z) = {x; \x — u{z)R\ < R} C X, where u{z) is the interior unit normal of dX at z. Let u{x) = —G{x,yo), x E X. If we take R so small that 2R < d(yo,dX) then u is a positive harmonic function in B{z) for every z E dX and u is continuous in B{z). If x G dB{z) then d{x,dX) < |x — z\'^/2R by Pythagoras' theorem applied to the triangle with vertices x,z,z-\- 2i/{z), so we conclude using (3.3.44) and the first part of Theorem 3.1.13 that u'{z + su{z)) has a limit u\z) as s -^ -fO. Since \u''{z + su{z))\ < C\ log{s/R)\/R when 0 < 5 < | i ? , we have \u'(z + siy{z) - u'{z)\ < C{s/R)\ log{s/R)l
0 < s < ^R.
If ( is another point in dX and 6 = \z — (\ is small then \iy{z) — i^{C)\ < C6, hence \u'iz) - u'iOl < \u'iz) - u'iz + 8u{z))\ + \u'{0 - u'iC + SuiO)\ + \u'{z + 6u{z)) -u'iC
+ 6u{0)\ < C6\ log6\
with another constant C. (We have used (3.3.45) to estimate the third term.) Thus u' is Holder continuous at dX of any order < 1.
202
III. SUBHARMONIC FUNCTIONS
To simplify notation we now adapt our coordinates so that the point z G dX considered is the origin and i/(0) = —e^ = ( 0 , . . . ,0, —1). If we solve the equation Q{X) — 0 for Xn we conclude that X is defined near 0 by ^n + Q^{^') < 0 where x' = ( r r i , . . . , X n - i ) , ^o G C^ in a neighborhood of 0, and djdkQoi^) = djdkQ{0)/dnQ{0) for j^k < n. Let q be the quadratic term in the Taylor expansion of ^o ^^ 0. For y G dB{0) close to 0 it follows from the mean value theorem that u{y) - d„uiO){yn+go{y'm+0{\y'\log\y'\))
= a„K(0)(2/„+g(2/'))+o(l2/f )•
With 2/n — ^n + J? 2LS a new variable instead of Xn we can now apply the second part of Theorem 3.1.13, with Q{y) — dnu{0){yn — R-\-q{y'))^ which gives \djdku{x) - dnu{0)djdkq{y')\
< £2{xn),
if j,fc < n and x' = 0.
Here €2 is a continuous function with £2(0) = 0 which is independent of the point z considered. Recall also that \u'{x) — u'{0)\ < Cxn\ loga;^! when x' = 0. lit e BJ" and \t\ < 1, {t,u'{x)) = 0, it follows since {t,u'{0)) = (t,it'(0) — u'{x)) that tn = 0{xn\\ogXn\)' Siucc u''{x) — 0(log|a;n|) we obtain n
^
n —1
tjtkdjdku{x)
3,k = l
- dnu{0) Y2 tj^kdjdkQo{0) = 0{xn\logXn\'^), j,k
i{\t\ < 1.
= l
This proves the last statement in the theorem. When dX G C^ the continuity of dG{x, y)/dx at the boundary allows us to prove that Gx{x,y) — Gx{y,x) by means of the Gauss-Green formula. In fact, let (/?,'0 G CQ^{X) and set u{x) = / Gx{x,y)(p{y)dy,
v{x) = /
Gx{x,y)il^{y)dy.
In view of Theorem 3.1.3 and the continuity oidG{x^ y)/dx at the boundary we have u,v £ C^(X), and both vanish on dX. Hence / {uip — vtp) dx =^ {uAv - vAu) dx = {udv/dn — vdu/dn) dS = 0, Jx Jx Jdx which means that / / G{x, y)ip{y)ij{x) dxdy = / / Gx{x, y)il^{y)^{x) dx dy JJXxX JJxxX
EXCEPTIONAL SETS
203
and proves that Gxi^^y) = Gxiy^^)Finally we shall prove that the level sets Sr^{X) = {x e X] G{x,y) = 7} are strictly convex. This is quite obvious when 7 is large negative. In fact, with the notation E{z) = e{\z\) we have y ~\- ez e Se(£){X) if and only if {E{sz) - hy{y -\- ez))/e{e) = 1, and when e -^ 0 this equation converges to \z\ = 1 iin > 2. If n = 2 the equation converges to (27r)~-^ log \z\ - hy{y) — 0, so we have strict convexity then. Hence it follows from Theorem 3.1.14 that S^ is strictly convex for all 7 < 0 if S^^ is convex for all 7 < 0. It is no restriction to assume that y = 0. At first we also assume that dX G C^ is strictly convex. Choose an open ball B C X with center at 0, and set Xx = XX + (1 - A)JB which increases with A G [0,1]. When A = 0 then Sj{Xo) = Sj{B) is a sphere, hence strictly convex. Let /x be the supremum of all A G [0,1] such that Sn^{Xx) is strictly convex for every 7 < 0. Since Gxx "^ Gx^ when A t // it follows that S^{X^) is also convex, hence strictly convex. Now the boundary of Xx is in C^ and strictly convex, so we have proved already that Sj{Xx) is strictly convex for all A G [0,1] when I7I is sufficiently small. If /x < 1 it would follow for reasons of continuity that Sj{Xx) is strictly convex for all 7 < 0 if A — /x is a sufficiently small positive number, which is a contradiction proving that fi = 1. The convexity of S^{X) is therefore proved when dX G C^ has strictly positive curvature. For arbitrary X we can by Lemma 2.3.2 choose a sequence of open convex sets Yj ] X with C^ strictly convex boundaries, and since GYJ i Gx it follows that Sry{X) is convex, hence strictly convex. The proof is complete. 3.4. Exceptional s e t s . In our discussion of subharmonic functions we have several times encountered exceptional sets, of Lebesgue measure 0: first the sets where a subharmonic function may be equal to —00, then the sets where there may be inequality in (3.2.6), and finally the sets where the Perron solution of the Dirichlet problem does not assume the prescribed boundary values. We shall now discuss the nature of these sets in greater detail. Definition 3.4.1. A set A C R^ is called polar if every point in A has an open connected neighborhood U where there is a subharmonic function u ^ —00 equal to —00 in AOU. A subset of a polar set is polar by the definition, and a polar set is of measure 0 since a subharmonic function ^ —00 is in Lj^^. The preceding definition is local, but one can equally well define polar sets by a global property: T h e o r e m 3.4.2. If A C R"^ is polar then there is a positive
measure
204
III. SUBHARMONIC FUNCTIONS
dfji in BJ^ such that (3.4.1)
[{1 + | x | ) ^ dfi{x) < oo, ViV,
and the subharmonic function E* d/jL equals - o o in A. IfA^j are polar sets, then Uf^Ak is polar.
A; = 1 , 2 , . . . ,
Proof. Let B = {x e li^^lx — XQ\ < R} he 3.n open ball such that there is a subharmonic function u ^ —oo in B with sup^ u < oo and t^ = — oo in BnA. By the Riesz representation formula we know then that the potential of the positive measure which is equal to {R — \x — xo\)Au in B and has no mass outside B is equal to — oo in JB D A. For every ball B with XQ and R rational such that a function u with the preceding properties exists we choose a corresponding measure d/i. Since every open set is the union of such balls, it follows that we obtain a sequence dfij of positive measures with compact support such that A is contained in the union of the sets {x G Ii^;E * dfij{x) — —oo}. Now choose Zj > 0 such that
f{l + \x\ydfij{x) <2Then it follows that d/i = ^ ^ ^jdfij satisfies (3.4.1), and E ^ dpi = —oo in A. To prove the second statement we can use the first result to choose diy^ ^ 0 so that (3.4.1) holds with d/i replaced by duk, and J{l-\-\x\)^duk{x) < 2~^, E^duk = —oo in A^. Then (3.4.1) holds with d/i replaced by diy — ^ ^ dv^.-, and E ^dv — - o o in Uf^Afc. Directly from the definition we also obtain a result on removable singularities: T h e o r e m 3.4.3. Let X be an open set in R'^, A a closed polar subset, and u a subharmonic function in X \A such that u is bounded above in K\A for every compact set K C X. Then u can be uniquely extended to a subharmonic function in X.IfX is connected, then X \A is connected. Proof. Using Theorem 3.4.2 we choose a subharmonic function v in R'^ such that v{x) = —oo a X E A. For ^ > 0 we define (u{x) + £v{x), Ue:{x) — < t -OO,
iixeX\A, iix e A.
Then u is upper semi-continuous in X, for v{y) -^ —oo and u{y) is bounded above ii y ^^^ x e A. The condition (iv) in Theorem 3.2.3 is trivially valid if x G A, and it holds by hypothesis ilx E X\A. Hence Ue is subharmonic
EXCEPTIONAL SETS
205
and u^ —> u in L^Qf^ as E —> 0. By Theorem 3.2.11 it follows that there is a subharmonic function u m X which is equal to u almost everywhere, and it is equal to um X\A. Since A has measure 0 we have u{x) — lim /
u{x + y) dyj I
''-'^J\y\
dy
J\y\
for every x G X , which proves that the subharmonic extension u is unique. If X \ A is not connected, then X\A = Xil}X2 where Xi and X2 are open, non-empty and disjoint. Set u{x) = 1 if x G X i and u{x) = 2 when x G X2, and extend the definition to a subharmonic function uin X using the first part of the proof. Since the mean value of u on any closed ball B C X with center at a point x E X2 cannot be smaller than 2, the ball cannot intersect Xi. Hence X f l X i 0 X 2 is empty. Since A has no interior points it follows that X is the disjoint union of open subsets X (1 Xi and X n X2, so X is not connected. This completes the proof. To discuss the set where there is inequality in (3.2.6) it is useful to observe that lim Uj — lim supix^The limit of a decreasing sequence of subharmonic functions is subharmonic by Theorem 3.2.2, so we are reduced to examining how far the supremum is from being subharmonic. P r o p o s i t i o n 3.4.4. Let Ui^, t E I, be a family of subharmonic functions in the open set X C R'^ which is uniformly bounded above on every compact subset of X, and set u{x) = sup Ui^{x). Then the smallest upper semi-continuous majorant u{x) = liuiy-^x u{y) of u is subharmonic and equal to u{x) almost everywhere. There is a sequence Ui^j such that u{x) is also equal to the smallest upper semi-continuous majorant of v{x) == supjU,^j{x) although v < u. If {?2^}^^/ is compact in Ll^^{X), then u — u. Proof. We begin with the construction of v which has nothing to do with subharmonicity. Label all balls (E X with rational center and rational radius as a sequence J 3 i , 5 2 , . . . so that every such ball occurs infinitely many times. For every j we can choose ij G / so that SMpu,.
> sup 1 6 -
1/j.
206
III. SUBHARMONIC FUNCTIONS
Then it follows that sup^ v > sup^ u > supjg v for every such ball. Hence there is equality and the upper semi-continuous regularizations are identical. Set Vj{x) = max(iA^^,... ,Ui^.), which is a subharmonic function increasing with j and uniformly bounded. If v is the limit then Vj -^ ?; in L J Q ^ ( X ) . By Theorem 3.2.12 t^ is almost everywhere equal to a subharmonic function V, and 1? < y by Theorem 3.2.13. Thus V is upper semi-continuous, so V>v = u>u>v^ and V = v almost everywhere. Now V{x) = lim / V{x-\-y)dy/ / dy ''-^^J\y\
u{x + y) dy
''^^J\y\
\ '
dy < u{x)^
J\y\
since u is upper semi-continuous, so V u{x). If {tt^j^^j is compact we can choose the sequence so that u,^. -^ Ui^ in LJQ^(X) for some t G I. li u^{x) < u{x) then there is a compact neighborhood K oi x such that s\ipy^j^Ut^{y) < u{x). By (3.2.7) this implies that lim sup u,{y)
< u{x),
which is a contradiction proving that u{x) — u{x) with the supremum u(x) attained. The proof is complete. Proposition 3.4.4 shows in particular that the set where n < -u is a subset of the set where v = limvj < v = u^ so it will suffice to discuss the exceptional sets which can occur for increasing sequences of subharmonic functions. Inequality may hold in any polar set, for if li is a subharmonic function in R"", and u'^ = ma.x{u, 0) then the sequence Uj{x) = u{x)/j-^u^(x){l — 1/j) increases to u'^{x) as j -^ oo except in the polar set A = {x G 'R'^]u{x) = —oo}. If one prefers a finite limit one can instead consider max(—1,IAJ(X)) which converges to u'^{x) when x ^ A and to —1 when x G A. Our goal is to prove the converse, that the exceptional sets which occur when one takes the supremum of a family or the upper limit of a sequence are always polar. T h e o r e m 3.4.5. If K C R^ is compact, then the following are equivalent:
conditions
(i) K is not polar. (ii) There is a positive measure d/i ^ 0 with suppd/x C K such E^dii>-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 <0 in BR with u < —1 in K, and set '^Kix) = snpu{x)j ueu
X G BR.
Prom Proposition 3.4.4 we know that the upper semi-continuous regularization UKix) is subharmonic and equal to UK{X) almost everywhere in BR, and it is obvious that C ( | x p — i?^) < UK{X) < 0 for some C. Suppose that UK{X) = 0 for some x G BR. Then we can find Uj subharmonic and < 0 in BR SO that Uj < —1 in K and Uj{x) > —2~^. Hence u = YlT ^j ^^ subharmonic in BR and TX ^ — 00 in BR but u = —00 in K, so K is polar which contradicts condition (i). Thus it follows from (i) that UK{OC) < 0 everywhere so UK{X) is not identically 0. It follows from Lemma 3.3.2 that the subharmonic function UK is harmonic in BR \ K, so d/x = ^UK has support in K. Prom the Riesz representation it follows that E ^ d/i > —C for suitable (7 in a neighborhood of K. Since J? * cf/x is harmonic outside K and -^ 0 at 00 if n > 2, —> cxD at cx) if n = 2, we conclude that E^dfi > —C in R'^, so d^/C has the property required in (ii). (ii) => (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 d^i 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
III. SUBHARMONIC FUNCTIONS
be the closest point, thus \z' — z\ < \y — z\ when y ^ K. For every y ^ K we have \z' -y\<\z' -z\-\-\z-y\<2\z-yl hence E{z' - y) < 2'^-''E{z - y) ii n > 2, E[z' - y) < E{z - y) + log 2 if n = 2. If ^ > 0 it follows that uiz) > 2^-2 /
Eiz' - y)dfi{y) - log 2 /
J\x-y\<6
dfx{y)
J\x-y\<6
+ /
E{z-y)dfji{y).
J\x-y\>6
When z -^ X we have z' -^ x, and by hypothesis the first integral on the right is a continuous function oi z' E K at x. Hence lim
u{z) > 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.
EXCEPTIONAL SETS
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 = {^UK, 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 > ^i^2 ^^ -^R^ ^^^ since both vanish on OBR, it follows that duKi/driy < duK2/dny, so (3.4.4)
MR{KI)
<
MR{K2),
iiKi,K2
E IC, Ki C K2.
210
III. SUBHARMONIC FUNCTIONS
Moreover, we have oo
(3.4.5)
MRif]Kj)
= lim MR{KJ),
H KJ e IC, Ki D K2 D • • • .
In fact, since K = Cif^Kj C Kj, we have UKJ < UK, and ii u E UK and e > 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(^) > ^iiTi (^) + '^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)
< UK^uK2ix) -^ 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^)
+ Mi?(A2).
MR{AI)
A2 = K2, Bi = Ki
+ MR{K2)
+
MR{K2),
which gives (3.4.6). The inequality (3.4.6)' can be generalized to N
(3.4.6)"
MR{[}B^)
N
N
+ Y,MR{A^)
1
1
N
< Y.MR{Bi) +
MR{[JAJ)-,
1
1
liAj.Bj
elC.Aj
cBj,
j =
1,...,N.
In fact, if we apply (3.4.6)' to U^~^Aj^ Ajsf and UJ^~^5^, B^, we obtain N
N-1
MR(\JBj) 1
+ MR{ U Aj) + MR{AN) 1 N-l
<MR{[J
N
BJ) +
MR{BN)^MR{[JAJ).
1
1
If (3.4.6)" is already known with N replaced by iV — 1, we preceding inequality and obtain (3.4.6)" after cancellation of so (3.4.6)" follows by induction. The strength of (3.4.6)" is that it gives excellent control ^^capacity ^^ defined by MR. Let M be the set of all sets with and define M*(A) iov A e M by (3.4.8) M^A) (3.4.9)
- sup{Mi2(if); K3KcA],
M*(A) = inf{M*(0); M3
0DA,
liAeMis O open},
add it to the some terms, of the outer closure C BR^
open, AeM.
212
III. SUBHARMONIC FUNCTIONS
It follows from (3.4.4) that (3.4.8) defines an increasing function on the open sets, so (3.4.9) agrees with (3.4.8) if A is open. We have (3.4.5)'
M%K) = MR{K),
if if
E /C,
for if Kj is a decreasing sequence of compact neighborhoods of K with interior Oj and nf'Kj = K, then M^{K) < M*{Oj) < MR{KJ) -> 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)^Aj G M, then
(3.4.10)
M*{[JAi) = 1
\\mM\[JAi). ~^^
1
We have 00
00
(3.4.11)
M*((jA,)<^M*(A,). 1
IfM''{A)
1
= 0 then A is polar}
Proof. The monotonicity shows that the limit in the right-hand side of (3.4.10) exists and is at most equal to the left-hand side; we must prove the opposite inequality. First assume that all Aj are open. If/C 3 if C VJ^Aj, then K C Uf Aj for some J, by the Borel-Lebesgue lemma, so J
MR{K) = M^{K) < M*(|J A,) < lim 1
J
M*{\\Aj), 1
which proves that the left-hand side of (3.4.10) is < the right-hand side. Now let Aj be arbitrary in M. Fix e > 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
^e/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 M^ijAj)
^^
< 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 d^ij = 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
III. SUBHARMONIC FUNCTIONS
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 Hf^Aj 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
j=l
The sets Bh increase to A, hence f{Bh) increases to A, so M''{f{Bh)) —> 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) - ^e. 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
where
EXCEPTIONAL SETS
215
Set P
1
Since Bhi^...,hp C Kp, we have M''{f{Kp)) > M*{A) - e, and since Kp e IC is decreasing, it follows from (3.4.5) that M{n^f{Kp)) > 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 K^o — ^^Kp with f{x) = x. This proves that nff{Kp) = f{Koo) C A, for K^o 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^^Aj is equal to flf^Aj, for y G Hf^Aj 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 Uf^Aj 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 UJ^Aj in if, and A is a if^^^ set. To prove this we write oo
oo
i/=i
11=1
216
III. SUBHARMONIC FUNCTIONS
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^^Aj 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)
<
-N}.
iV=l
We can take u{x) = J GR{x,y)dji{y) for a suitable positive measure with support in BR. Thus A is the intersection of countably many open sets so A is capacitable by Theorem 3.4.12. li )C 3 K C A then MR{K) = 0 for £u < UK < 0
for every e > 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 ^a,/?,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\
dy/ '
/ J\y\
dy,
for UK = "^K = — 1 in i f ° , and UK is upper semi-continuous. This proves the last statement. We are now prepared to discuss the exceptional sets for the Perron generalized solution of the Dirichlet problem. In Definition 3.3.4 we introduced the notion of barrier function in an open set X C R'^, and we proved in Theorem 3.3.5 that prescribed continuous Dirichlet data are assumed at every point y E dX where there is a barrier function. It would in fact have been sufficient to require the existence of a local harrier function, that is, a strictly positive superharmonic function h defined in X fl F for some neighborhood Y oiy such that (3.4.13)
h{x)-^^
\iX^Y
3x-^y,
inf ^n(y\yo)
& > 0,
218
III. SUBHARMONIC FUNCTIONS
for any neighborhood YQ <^ Y oi y. In fact, if 0 < c < infxn(y\yb) ^5 then b = min(6, c) is superharmonic in X HY and equal to c in X fl ( F \ YQ)? SO we can extend & to a barrier in X by b{x) = c if x E X \Y. We shall now prove the much less obvious fact that the second condition in (3.4.13) can be dropped. T h e o r e m 3.4.16. Let X be an open set in W^, and assume that y G dX has an open neighborhood Y such that there exists a superharmonic function w > 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\
_dx/
dx -^ 1,
as r -^ 0.
J\x\
The set of irregular points is polar. Proof, (i) = > (ii), for if u is the generalized solution of the Dirichlet problem with data f = \ • —y\, then u{x) > \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
III. SUBHARMONIC FUNCTIONS
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 BR^e\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^ = d^i^ipe. Thus
\<\\h — -
{Ue ^ (fe) dfX = —
{u^ife
^ (fe) dfi ^ —
udfl 3.8 S i 0,
for u^ (fs "^ (fe i u- Hence u' e L^^ which proves that u'^ = u' ^ (pe -^ u' in L^{BR), since u'^ - u^ = 0 in a fixed neighborhood of OBR. Hence (3.4.16) is valid for u. By polarization we obtain (3.4.16)', for the right-hand side is symmetric in u and v since it can be written
- / /
GR{x,y)dfiu{x)
dii^{y)
lidjiu = Au and dfiy = Av. The inequality (3.4.17) is obviously valid with u and V replaced by u^ and v^ (see the proof of (3.4.4)), so it follows when To prove the last statement we observe that with dfij = Auj {uj — u'^u'j — U')L2 =2
udjjLj — / Uj d/ij —
udfi < / Ujdjij — I udfi.
Let X be a continuous function > u in BR, and denote by iiT a compact subset of BR containing suppd/x^ for every j . By the monotonicity and upper semi-continuity of Uj we have Uj < x ^^ K for sufficiently large j . Hence / Uj dfij — I udfi <
X dfij -
udfjb -^ / (x — ii) dfi.
We can choose x so that the last integral is arbitrarily small, which completes the proof that u'^ -^ u' in L^.
222
III. SUBHARMONIC FUNCTIONS
P r o p o s i t i o n 3.4.20. For the outer capacity defined by (3.4.8), (3.4.9) we have for every A E M (3.4.18)
M*(A)-(AiZA,l).
Proof. (3.4.18) is valid by the definition of MR{K) if K is compact. Next assume that A is open, and choose an increasing sequence of compact subsets Kjj each contained in the interior of the following one, such that Kj t A. Then UA < UKJ^ and since UKJ = — 1 in the interior of Kj, it follows that UKj I UA = UA' Hence Ait^j -^ A n ^ , so (3.4.18) holds. Now consider a general A. If O G JM is open D A, then UQ
=
M*{0),
and the infimum of the right-hand side is M*(A). To prove the opposite inequality we use that by Proposition 3.4.4 there is an increasing sequence Vj t UA almost everywhere, with Vj = 0 at OBR and Avj supported by a fixed compact subset of BR. Set Oj = {xe
BR', {l-hj-^)vj{x)
< -1},
which is an open set containing A. Since (1 + l/j)vj uoj —»i^A as a distribution, so M*(A) < lim M''{Oj)
= lim {AUQ.A)
< UQJ < I^AJ we have
= (A^A,1),
which completes the proof. The monotonicity and subadditivity of the outer capacity follow from Proposition 3.4.20. We need some additional preparations before we can give another proof of Theorem 3.4.14: P r o p o s i t i o n 3.4.21. Ifu satisfies the hypotheses of Proposition 3.4.19, then one can for every 77 > 0 find an open set O e M with M*{0) < rj such that the restriction ofu to BR \ O is continuous. Proof. Let Ue be the standard regularizations of u used in the proof of Proposition 3.4.19. Then u^ I u 8iS e I 0, and u^ = u for small e outside a fixed compact subset K of BR. li 6 > 0 then Os^e = {x e K]Us{x) > u{x) -\- 8} is open, since u^ is continuous and u is upper semi-continuous. Let Kj be compact sets increasing to 0^,^, each in the interior of the next. Then the support of dfij = ^^KJ is contained in Os^e, and by Proposition 3.4.19 6 / dfij <
{ue- u)diij = - «
-
U',U'K.)
< C | K - u'W L 2 .
EXCEPTIONAL SETS
223
Letting j -^ oo we conclude using Proposition 3.4.19 and the proof of (3.4.18) that S / d/iss < C\\u'^ -
U'\\L2
-> 0,
e ^ 0;
dfis,e = ^uos,e-
Given 7/ > 0 we take 6 = 1/j and conclude that we can choose €j so that M^'iOi/j^ej) < 2"^7/. If O = UOi/j,e, it follows that M'^iO) < rj and that U£. —> u uniformly in the complement. The proof is complete. P r o p o s i t i o n 3.4.22. If K C BR is a compact set and v is a suhharmonic function < 0 in BR with v > —1 in K, then the measure Av satisGes (3.4.19)
/ A^ < {AUK, 1) - M * ( K ) . JK
Proof. Recall that there is a an increasing sequence Wi, of subharmonic functions in C^{BR) with limit UK almost everywhere, hence {Aw^,, 1) -^ {AUKJ1),
zv -> oo.
(See the proof of lower semi-continuity before (3.4.4).) First assume that V e ^^(BR), that T; < 0 in 'BR, and that T; > - 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 Jwiy
Awydx
<
{Awy.l).
Jwiy—6
If we just know that v is subharmonic and > £ — 1 in K, < —e in BR for some e G (0, | ) , we can apply this argument to a regularization vs of v (as in the proof of Theorem 3.2.11). It will only be defined in a somewhat smaller ball Br but it can be taken so large that Wy > —e/2 at the boundary
224
III. SUBHARMONIC FUNCTIONS
for large z/, and this suffices for the argument when the regularizations are < -8/2 there and > ^ - 1 in if. If x G Co(J5^), 0 < x < 1, X = 1 in i^ and w^ < £ — 1 m supp X, it follows that (A7;,x) = lim(Ai;^,x) <
{^'^.A).
6—>0
and when z/ ^^ cx) it follows that for the measure Av we have
L
Av <
{AUKA)-
If V just satisfies the hypotheses in the proposition we can apply the result to (1 — 2e)v — € and conclude when s -^ 0 that (3.4.19) holds. Second proof of Theorem 3.4-14' It suffices to show that if Uj is an increasing sequence of subharmonic functions satisfying the hypotheses made on u in Proposition 3.4.19 and harmonic outside a fixed compact set, then the set where u = limuj is not equal to the upper semi-continuous regularization u has outer capacity 0. By Proposition 3.4.21 we can find an open set O with small outer capacity so that the restriction of Uj to the complement F in Bji/2^ say, is continuous for every j . Let a < 6 < 0 be rational numbers. Then the set Ka,b = {x e F',u{x) < a < b < u{x)} = f]{x
e F; Uj{x) < a} f]{x
G F ; u{x) > b}
is compact; the theorem is proved if we show that the capacity is equal to 0. Dividing by \a\ we reduce to the case where a = —1. Then UK^ ^ > ^' ^5 SO UKa,b >b> -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,... ^Uj^ are plurisubharmonic, then u = m a x ( n i , . . . , 1^^) and ui + • • - + u^, are also plurisubharmonic. Ifu^^i^I, 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\
where dX is the Lebesgue measure in C^, is an increasing function of r E [0, i?) if \z — w\ < R implies w E X and $ is a non-negative continuous function with support in [—1,1], so u is subharmonic. Proof. The first statement follows from Theorem 3.2.3 since u{z + w) is subharmonic in Wj, and the second statement follows since the integral does not change if Wj is replaced by WjC^^^, so it is equal to
/
Mu{z\ r\wi\,...,
r\wn\)^{\w\)
dX{w).
By Theorem 3.2.3 the monotonicity of this integral implies that u is subharmonic. It follows from Theorem 4.1.3 that a plurisubharmonic function ^ —oo in a connected open set can only be equal to —cx) in a polar set, and it is in L\^^ by Corollary 3.2.8. In fact, the pluripolar sets where a plurisubharmonic function can be equal to — cx) are much more restricted since the intersection
BASIC FACTS
227
with any complex line is a polar subset of the line if it has no interior. We shall give a brief discussion of pluripolar sets in Section 4.5. They are much more difficult to handle than polar sets since a precise representation by potentials is not available. It is often convenient to be able to work with smooth plurisubharmonic functions, so we prove: T h e o r e m 4.1.4. Let 0 < cp e Co^iC^) vanish outside the unit ball, and assume that (p{z) only depends on \zi\,... ,\zn\. Ifu G Ljo^ is plurisubharmonic in an open set X C C^ and X^ = {z ^ X]\z — 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'^u/dzjdzk is a measure for j,k — 1 , . . . , n and (4.1.2) is valid in the sense of measure
228
IV. PLURISUBHARMONIC FUNCTIONS
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'^u/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'^u/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'^u{f{z))/dzjdzkWjWk
= ^
j,k=l
u^y^wlw'^,
where
i^,/i=l
n
'^^../.(C) = d'^y^iO/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
IV. PLURISUBHARMONIC FUNCTIONS
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(.))'^'
|CI<1,
it follows by Minkowski's inequality that ioT p > I (4.1.3)
(/
\u{zw d\{z)) ^'^ < c,{\hm + f iog(i/ici) dtiio) = c,\um-
In the case of n complex variables we now consider a plurisubharmonic function which is < 0 in a neighborhood of the polydisc {z G C " ; \zj\ < i?, j = 1 , . . . , n } and finite at the origin. Then we have
(/
HRzWdXiz))'^'
for the integral on the left can be estimated by C^ J \u{0, Rz2,...,
Rzn)f
dX{z2) • • • dX{zn) <•••<
C;^|tx(0)|P.
With the notation in Theorem 3.2.13 we can now conclude that Uj — u is bounded in L'^^^{X) for any p G [l,oo), and by Theorem 3.2.13 we know that Uj — u -^ 0 in L'^^^{X) Up G [1, n/{n — 1)). By Holder's inequality (cf. Corollary 1.2.9) this is therefore true for arbitrary p G [I5O0). The statement about the first derivative follows in the same way if we establish a bound for u' in Lf^^ when p < 2. In the one-dimensional case the Riesz representation formula gives du{z)/dx-idu(z)/dy
= 2du{z)/dz
= 2dh{z)/dz-h
[ -—^-^
-zdfi(().
J (^-C)(i-0 Since z ^-^ (z — C)~^ is in L^ when \z\ < 1, if p < 2, we obtain the desired L^ bound for u' by the arguments which gave such a bound for u above. The repetition of the details is left for the reader. Proposition 3.4.4 is also valid for plurisubharmonic functions, for with the notation there we have Vi^. -^ u in D', hence u is plurisubharmonic by Corollary 4.1.5. Propositions 3.2.14 and 3.2.15 can be strengthened for plurisubharmonic functions:
BASIC FACTS
231
P r o p o s i t i o n 4.1.9. Let X be an open set in C^, K a compact set in H^, and I a ball in R'^ such that K x I C X. Then there is a positive constant C such that for all plurisubharmonic functions u in LIQ^{X) (4.1.4) /
\u(x-\-iy)-u{x-\-iy)\dx
JK
\u{z)\dX{z),
y,y'e
I,
JX
(4.1.5)
/ \u{x 4- iy)\ dx
I \u{z)\ dX{z), Jx
y e I.
Proof. When y and y' only differ in one coordinate, the estimate (4.1.4) is an immediate consequence of (3.2.9), (3.2.10), and the general case follows by the triangle inequality. Assuming that / has interior points we obtain (4.1.5) by integration over y' G / , just as in the proof of (3.2.10). P r o p o s i t i o n 4.1.10. Let X be an open set in C^ and K C R'^ a compact set such that K x {0} C X. Then (4.1.6) ^ ^ / \ui{x) - U2{x)\ dx < C(
ifui
/ {\ui\-{-\u2\)dXj'^^
and U2 are plurisubharmonic
functions in
( / \ui -U2\dXj
\
Ll^^{X).
Proof. The proof is parallel to that of Proposition 3.2.15. Application of Proposition 4.1.9 to Ui and U2 with / = {7/ E R"^; |^| < r} for some small r > 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
{\ui\-^\u2\)dX-\-C'e~'' JX
\u1-u2\dX JX
when 0 < s < r. We choose e so that the two terms are equal, if this is possible when 6 < r, and otherwise we choose e = r, which proves (4.1.6). Exercise 4.1.1. Show that (4.1.6) is not valid with a geometric mean with weight > l / ( n + 1) on J-^ \ui — U2\ dX. Theorem 3.2.16 can also be strengthened:
232
IV. PLURISUBHARMONIC FUNCTIONS
T h e o r e m 4.1.11. Let u be a plurisubharmonic function in {z G C^;i?i < \z — zo\ < R2}, and let n > 2. Then the mean value Mu{zo,r) over the sphere where \z — zo\ = r is a convex increasing function of log r when jRi < r < i?2Note that for n = 1 we cannot assert that M^ is increasing, only that it is convex. Proof. We may assume that ZQ — 0, and by Theorem 4.1.4 it suffices to prove the statement when u G C^. Assume first that u is rotation invariant, that is, u{z) = F{\z\'^), where F G C ^ ( ( i ? ? , i ? | ) ) . Then (4.1.2) means that
>0,
F'{\z\')Y,\w,\'^F^\\zn\j2-^3^^
iiRi < \z\
weC.
Since n > 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
Proof. The first statement follows at once from Corollary 3.2.19 and Theorem 4.1.11; in particular Me^(0,r) is an increasing convex function of logr. As in the remark following Theorem 4.1.11 we have Meu(0,r) = / ^|C|=i
/(r,C)da;(C)/c2n,
/(r,C) -
/ ^^^'^^''^^ (i^/(27r). -'o
Since log/(r, ( ) is a convex function of logr by Theorem 3.2.20, it follows from Lemma 1.2.8 that logMeu(0,r) is also a convex function of logr. The proof of Corollary 3.2.22 now gives the final statement.
234
IV. PLURISUBHARMONIC FUNCTIONS
C o r o l l a r y 4.1.14. Ifu is plurisubharmonic in {z G C^; |z| > 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'^i^) = 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\
^^
where C2n-2 — 7r^~^/(n - 1)! is the volume of the unit ball in C"^"^. Proof. By Theorem 4.1.13 we know that sup^^^^^u{z) creasing function of logr, so i^o < oc. Now
is a convex in-
1
is plurisubharmonic since log|zj| is plurisubharmonic. (See Exercise 3.2.6 and Lemma 1.2.8.) Hence 7/(2:)+^ log |>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\
which is precisely (4.1.7). Note that in the left-hand side of (4.1.7) the mass has been divided by the volume of the ball with radius r i n C ^ - \ s o the mass is compared with a surface area. We shall write (4.1.8)
Uzo,r) =
^ „_, /
<^2n-2^
J\z-zo\
Au/2n
BASIC FACTS
235
when u is plurisubharmonic in{2:;|^ — ^ o | < ' ^ } and extend the definition to r = 0 as the Hmit which exists since (4.1.7)'
t^{zo,r)
= r — Mu{zo,r - 0),
where the right-hand side is an increasing function of r* by Theorem 4.1.11. Note that if Uj —^uin V^ then tuj{zQ^r) -^ tu{zo,r) as a distribution, hence pointwise at every point of continuity of the Umit. (See Exercise 1.1.5.) Definition 4.1.16. If u is plurisubharmonic in a neighborhood of ZQ G C^, then the Umit Uuizo) = tu{zo^ 0) of (4.1.8) as r -^ 0 is called the Lelong number oi u at ZQ. This definition is also valid when n = 1; the Lelong number is then the mass of Au at ZQ divided by 27r. We shall discuss Lelong numbers extensively in Sections 4.3 and 4.4 and content ourselves with some simple observations here. Corollary 4.1.17. Ifu is a plurisubharmonic function in C^ such that e^ is positively homogeneous of degree k, then tu{0,r) = k. Proof. Since u{rz) = u{z)-\-k\ogr, /;:logr, hence rdMu{0,r)/dr = k.
r > 0, we have Mu{0,r)
= Mii(0, l) +
Corollary 4.1.18. If f is an analytic function in X C C " , then the Lelong number of log | / | at ZQ is the order of the zero of f at ZQ. If f is a polynomial, then ^iog|/|(^0 5^) —> m as r ^> oo, where m is the degree of the polynomial. Proof. We may assume that ZQ = 0. Write
/(^) = /,(z) + 0 ( | z r ^ ) ,
z-O,
where fk is a homogeneous polynomial of degree k which is not identically 0. Then Fr{z) = f(rz)/r^ converges locally uniformly to fk{z) as r —> 0, so it follows from Theorem 3.2.12 that log|F^| converges in V to log|/fc| as r -^ 0. Hence ^iog|/|(0,r-) = ^iog|F.|(0,1) -^ ^iog|M(0,1) = k,
r ^ 0,
by Corollary 4.1.17 and the observations before Definition 4.1.16. This proves the first statement. If / is a polynomial of degree exactly m, we have fiz) = fm{z) + 0{\zr-'), Z^OO,
236
IV. PLURISUBHARMONIC FUNCTIONS
where fm is homogeneous of degree m and not identically 0. To prove the second statement we can now argue in the same way as above but letting r -^ oo. In Corollary 2.1.26 we saw how convex open sets are related to locally convex functions as the open sets which have locally convex exhaustion functions. On the other hand we found in Theorem 3.2.30 that subharmonic exhaustion functions always exist. We shall now show that for plurisubharmonic functions the situation is analogous to that for convex functions which will lead to a class of open sets attached to plurisubharmonic functions. Let 5 be a continuous homogeneous distance function in C^ in the sense that (4.1.9)
6{tz) - \t\6{z),
if t G C, ze C^;
6{z) > 0,
iizj^O.
We do not assume that 6 is convex, so 6 may not be a norm. If X C C^ is an open set, we define the boundary distance by (4.1.10)
dx{z)=
mi6{z-C),
z e X,
CGCX
ax
^ C^; if X = C^ we define dx = 0, for example.
T h e o r e m 4.1.19. The following conditions on an open set X C C^ are equivaient: (i) There is a plurisubharmonic component, such that (4.1.11)
{z eX-u{z)
function u in X, not = —oo in any
for every
(ii) IfK is a compact subset of X then (4.1.12) K = {z £ X] u{z) < snpu for all plurisubharmonic
teR.
u in X] (& X.
K
(iii) z \-^ — logdx{z) is a plurisubharmonic function in X for every distance function satisfying (4.1.9). (iv) z H^ — log dx{z) is a plurisubharmonic function in X for some distance function satisfying (4.1.9). Proof. That (i) = ^ (ii) is obvious, for sup^^ u = c < cx) and K C {z E X]u{z) < c} (^ X iiu is the function in (i). It is also trivial that (iii) =^ (iv). We have (iv) = > (i) since u{z) — \zS^ —logdxi^z) is plurisubharmonic if — logdx(^) is plurisubharmonic, and when u(z^ < t we have an upper bound for \z\ and a lower bound for dx{z^^ so (4.1.11) holds.
BASIC FACTS
237
Assuming now that (ii) is fulfilled and that X ^ C^, we shall prove (iii). Let zo e X and w e C^ he chosen so that (4.1.13)
zo-\-TweX,
if r G C, |r| < 1.
Since harmonic polynomials in C, that is, real parts of complex analytic polynomials, are dense in the continuous functions on the unit circle, it follows from Theorem 3.2.3 (i) that to prove (iii) it suffices to show that for every analytic polynomial / ( r ) , r G C, we have
-logdx{zo-\-rw)
< Re/(r),
|r| < 1,
if this is true when |r| = 1. Thus we must show that if (4.1.14)
dxizo + Tw) > |e--^(^)|,
\T\ = 1,
then this remains true when \T\ < 1. The inequality (4.1.14) means that (4.1.14)'
zo + TW^ ae--^M G X,
if a G C^, 6{a) < 1, r G C, |r| = 1.
Fix a and set A = {Xe [0,1]; i^A C X},
where Dx = {ZQ + ri/; + Xae'^^''^; \T\ < 1}.
By (4.1.13) we have 0 G A, and since X is open it is clear that A is open. If we prove that A is closed, it will follow that 1 G A, hence that (4.1.14) is valid when |r| < 1, which means that —logdx is plurisubharmonic. The boundary of the disc Dx is contained in K = {zo-^Tw-^
aAe-^(^); |r| - 1,0 < A < 1},
which is a compact subset of X by (4.1.14). If ix is a plurisubharmonic function in X and u < 0 inK^ then the puUback r ^-^ u{zo-\-Tw-{-Xae~^^'^'^) is subharmonic in a neighborhood of the closed unit disc in C if A G A, so it is < 0 in the unit disc since this is true on the boundary. Hence Dx C K if A G A, and we conclude that Dx C K ii X e A. Hence A is closed, which completes the proof. Definition 4.1.20. An open set X C C^ is called pseudo-convex if the equivalent conditions in Theorem 4.1.19 are fulfilled. By Theorem 3.2.30 this is no restriction when n = 1. Later on it will be important that one can always find a very regular exhaustion function ix, as in condition (i), which essentially defines K in condition (ii):
238
IV. PLURISUBHARMONIC FUNCTIONS
T h e o r e m 4.1.21. Let X C compact subset of X, and define K (^ Y C X, then one can find Theorem 4.1.19 so that the Levi and u <0 inK butu>l inX\
C^ be a pseudo-convex open set, K a K by (4.1.12). IfY is an open set with u G C^{X) satisfying condition (i) in form (4.1.2) is strictly positive definite Y.
Proof. The first step will be to construct a continuous function having the required properties, and then we shall discuss how to regularize it. The proof that (iv) ==> (i) in Theorem 4.1.19 showed that we can find a continuous plurisubharmonic function UQ such that (4.1.11) holds, and by adding a constant we can make TXQ < 0 in K. Let
Kt = {zeX]Uo{x)
which is a compact subset of X. li z E {X \Y) H Ki then z ^ K^ so we can choose a plurisubharmonic function Wz such that Wz < 0 in K but Wz{z) > 1. A regularization Wz of Wz constructed as in Theorem 4.1.4 with a sufficiently small s is then plurisubharmonic in a neighborhood of K2, Wz <0m K, and Wz>lm^ neighborhood of z. Since ( X \ y ) n K i is compact, it follows from the Borel-Lebesgue lemma that we can find finitely many Zj such that W = max W^^ > lm{X\Y)nKi^ and W is continuous and plurisubharmonic in a neighborhood of K2' If VF < C in ^ 2 , it follows that ^ ^ r max(H^(z), Cu^{z)), if ^o(^) < 2, 1 CUQ{Z),
iiuQ{z) > 1
is a continuous plurisubharmonic function in X , for the two definitions agree when 1 < U{){z) <2 by the definition of C. The function U has the required properties apart from smoothness and strict positivity of the Levi form.
Set Xj = {z e X] U{z) < j] and UAz)=
I
U{(:M{z-OlE,)e-'^d\{0^eM\
i-0,1,...,
^Aj +i
where ip is chosen as in Theorem 4.1.4 and EJ > 0 is chosen so small that U < Uj < U -\-l and Uj is strictly plurisubharmonic in a neighborhood of Xj\ Uj < Q m K. Let x ^ C ^ ( R ) be a convex function such that X{t) = 0 when t < 0 and x'(^) > 0 when t > 0. Then x{Uj + 1 - j ) is strictly plurisubharmonic in a neighborhood of Xj \ Xj-i. Hence we can successively choose positive numbers a i , a 2 , . . . so large that 771
Um = Uo + ^
djXiUj + 1 - j )
BASIC FACTS
239
is > U and is strictly plurisubharmonic in a neighborhood of Xm- We have Um = ui in Xj if Z > j^m > j , so u = Um^_,oo'^m exists and is a strictly plurisubharmonic function in C^{X). We have u = UQ < 0 in K, (4.1.11) is fulfilled and u > 1 in X \Y since u >U^ which completes the proof. The original reason for introducing pseudo-convex sets was not the one given here, but it came from the study of domains of holomorphy: Definition 4.1.22. An open set X C C^ is called a domain of holomorphy if there is an analytic function / in X for which the radius of convergence of the power series expansion at every point in X is equal to the boundary distance. The definition given here is equivalent to the conventional one and is more convenient for our purposes. (See CASV, Section 2.5.) By the radius of convergence rf{() of the power series expansion oi f 3.1 ( ^ X,
a
we mean the supremum of all r such that the series is absolutely convergent when 6{z — () < r, where 6{z) = max\zj\. This means that
|Q!|—>oo
so Definition 4.1.22 means that - l o g d x ( C ) = - l o g r / ( C ) = , I M (log|/„(C)|)/|a|, |Q;|—>oo
and this is a plurisubharmonic function. Thus every domain of holomorphy is pseudo-convex. The converse is true but much harder; until the 1950's it was an open question known as the Levi problem. We shall prove it in Section 4.2. The definition of pseudo-convexity may not look very geometrical, but it does give some of the basic properties of convex sets: T h e o r e m 4.1.23. If Xi^ C C^ is a pseudo-convex open set for every L e I, an arbitrary index set, then the interior X of fl^^/X^ is also pseudoconvex. Proof. We have —\ogdx{z) — snp^^j — logdx,{z), which is continuous in X , so the theorem follows from Theorems 4.1.2 and 4.1.19 (iii). Remark. We have not required pseudo-convex sets to be connected, for then Theorem 4.1.23 would not be true. Pseudo-convexity is also a local property:
240
IV. PLURISUBHARMONIC FUNCTIONS
T h e o r e m 4.1.24. If X C C^ is an open set and every point in X has a neighborhood Y such that Y (1 X is pseudo-convex, then X is pseudoconvex. Proof. From (iii) in Theorem 4.1.19 it follows that - log dx{z) is plurisubharmonic in X \ F where F is a closed subset of X . If % is a convex increasing function then u{x) = m a x ( - l o g d x ( ^ ) , x ( k n ) is equal to the plurisubharmonic function x ( k P ) i^ ^ neighborhood of JP, if X grows sufficiently fast. Hence u is a. plurisubharmonic function, and it follows that X is pseudo-convex. We shall now give an analogue of Theorem 2.1.27 which in particular will give a differential geometric condition for pseudo-convexity of X when
dx e C^. T h e o r e m 4.1.25. Let X be an open set in C^ which is not pseudoconvex. Then there is a point ZQ G dX and a quadratic polynomial q with q(zo) = 0, q'{zo) ^ 0, such that q{z) < 0 implies z e X if z is close to ZQ, and n
2_^'Wjdq/dzj
= 0,
Y^S^g/^ZjS^A^'^j'^A; < 0
1
at ZQ for some w 6 C^. Thus {z]q{z) < 0} is contained in X near ZQ and has there a C^ boundary with negative normal mean curvature in the complex direction w. Conversely, X is not pseudo-convex if there exists such a polynomial. Proof. We prove the second statement first. By hypothesis q(zo + Tw) = 2 Re Ar'^ - e:|r|^ for some A G C and e > 0. Choose u G C^ so that {dq{zQ)ldz^v) Then q{zQ -h AT'^V 4- Tw) = -e\T\^ + 0 ( | r | ^ ) <
— —1.
-£\T\'^/2
for small |r|. Hence we can find 6 > 0 such that zo + (Ar^ -f r])u -f rit; G X,
H \r\ < 6, 0< r] < 6]
when | T | = ^ we may also take r] = 0. For a plurisubharmonic function li in X it follows that u{zo -\- {Ar'^ + 7j)iy + rw) < C when | r | = 8 and 0 < T] < 6, and this inequality remains valid when |r| < ^ because of the subharmonicity. When r = 0 we obtain u{zo + rju) < C, which proves that u cannot have the property (i) in Theorem 4.1.19.
BASIC FACTS
241
Now assume that X is not pseudo-convex. If dx is defined by the EucUdean distance function z H-> ( X ^ ^ I ^ J P ) ^ , it follows from Proposition 3.2.10' and condition (iii) in Theorem 4.1.19 that there is some (^ G X , l y E C^ and ^ , 5 G C such that (4.1.15) -\ogdx{(:-^rW)
< - logdx(C)+Re(^r + B r 2 ) - ^ | r p ,
|r| < 6,
where e: > 0, ^ > 0, that is, (4.1.15y
dxiC + rW) >
Choose ZQ G dX so that dx{0
rfx(C)e~^'(^"+^"')^'l"l\
\r\ < 6.
= |C ~ ^o|- Then
<\zo-(-
TW\ = \zo - CI - ReriW, v) + 0 ( | r | 2 ) ,
where v — {ZQ — C)/ko — CI ^^d (•, •) is the Hermitian scalar product in C^. Hence the first-order terms in the Taylor expansions agree, so A\ZQ - C H ( ^ . ^ ) .
that is, W = A{zo - C) + "^^ ('^, ^ 0 * 0 = 0.
If 1^ - C - rW\ < \zo - (|e-i^^(^^+^^')+^l^l' for some r with jrj < 6, then it follows from (4.1.15)' that z e X, hence >2: G X if (4.1.16)
ip{z) = mm{\z - C - rT^j - \zo - (|e-i^^(^-+^^'H^I^I') |T|<<5
is negative. If z{r) = (-\-rW-\-{ZQ
- Qe'^''-^''''
then
- e'\-\") < -e\T\^\zo -
C\e-^'^^'^^''\
for \T\ < 6, and ^'(0) = W - A{ZQ - C) = ^ - By (4.1.15)' we have 1^0 - C - rW\ - \ZQ - (|e-Re(Ar+Br^H.|r|^ > Q^ and if e has been chosen so small that (4.1.15) is valid with e replaced by 2e, we have the lower bound s|rp|2:o - (|e~^^(^^+^'^ ). Hence the minimum in (4.1.16) is a non-degenerate minimum achieved at r = 0 when z = ZQ. By the implicit function theorem it follows for z near ZQ that the minimum is still a non-degenerate minimum taken at a point r which is a C^ function of z. Hence (^ is a C^ function in a neighborhood of ZQ^ and dip{z)ldz = d\z — C\/dz ^ 0 when z = ZQ. Since CP{Z{T)) < - c | r p for some
242
IV. PLURISUBHARMONIC FUNCTIONS
c > 0 and Z{T) is analytic, z'{0) = w^ we conclude that Yl^ Wjdcp/dzj = 0 and that n
|r|2 J2
n
d^if/dzjdzkWjWk^Rer^
j,k=i
J^
d^ip/dzjdzkWjWk
< -c|r|2+0(|r|^),
j,k=i
when z = ZQ and |T| is small. If we add the expression obtained when r is replaced by ir, it follows that when z = ZQ n
y]
WjWkd^if/dzjdzk
< —c.
j,k=i
To complete the proof we just have to replace ip by the second order Taylor expansion at ZQ plus a small constant times \z — ZQI"^, exactly as in the proof of Theorem 2.1.27, and we leave that repetition for the reader. Corollary 4.1.26. If X is not pseudo-convex and ZQ, q are chosen as in Theorem 4.1.25, then for any neighborhood U of ZQ there exists another neighborhood V C U of ZQ such that for every function f which is analytic inU n X there is a function F which is analytic in V such that F = f in a component Vx ofVDX independent of f. IfVDXis connected then F = finVnX. Proof. Introduce local analytic coordinates ( with the origin at ZQ such that n
^o(C) = q{z{0)
= - 2 R e C n + 5 ^ hjkCjCk + O(ICI'),
hu < 0.
j,k=i
Then qo{(i,0,... ,0) < 0 if 0 < |Ci| < ^, say, and by Cauchy's integral formula we obtain / ( ^ ( O ) = ^ [^ 27r Jo
/(^(Ci + re^', C2,..., Cn)) de = F{0
if 0 < r < ^, 0 < ICI < KRe(ri^ and | ( | -h r is small enough, for then the integration takes place over the boundary of a disc contained in X. In fact, if |t| < r and 0 < |C| < KReCn then ^ o ( C i + ^ , C 2 , . . . , C n ) - - 2 R e C n + /Mi|t|' + 0(N|C|) + 0(|Cl') + 0(|t|^) < lhn\t\'
+ 0 ( | C n - 2ReCn < ^hu\t\^
- ReCn < 0.
Now F is an analytic function in {(; |^| < e}, if e is small, li V = {^{O'l ICI < ^} aiid Vx is the component ofVHX containing 2:(0,..., 0,1^), it follows that f{z{C)) = F{() when z{() G Vx- The proof is complete. (Note that if dX E C^ we can always choose V so that y fl X is connected.)
BASIC FACTS
243
T h e o r e m 4.1.27. Let X = {z e C'^giz) < 0} where g e C^ and ^' 7^ 0 when g = 0, be an open set with C^ boundary. Then X is pseudoconvex if and only if (4.1.17) n
n
Y ^ d^g/dzjdzkWjWk
>0
when z G dX and 2_^dg/dzjWj
j,k=-l
= 0.
1
Proof. If (4.1.17) fails at a point ZQ G dX, then the second order Taylor expansion of g at ZQ plus a small constant times l^; —2:oP is a quadratic polynomial q satisfying the conditions in Theorem 4.1.25, so X is not pseudoconvex. On the other hand, assume that X is not pseudo-convex, and choose q and ZQ as in Theorem 4.1.25. Then q{z) < 0 implies g{z) < 0 if z is sufficiently close to ZQ, SO dq/dz and dg/dz differ by a positive factor at ZQ. Changing q by that factor and introducing suitable coordinates with ZQ = 0 we can assume that g{z) = -Rezi
+ ^2(^) + o(|z|^),
q{z) = - R e z i +g2(^),
where ^2 and ^2 are quadratic forms. When R e z i > ^2(0,2:'), where z' = (2:2,..., Zn), it follows that q{e'^zi^ez') < 0 if ^ is small, hence {e'^zi^ez') G X then, so g{e'^zi^ez') < 0, which proves that -Re>2:i H- ^2(05^2:') < 0. Hence it follows that ^2(0,/)-?2(0,/)<0, which implies that n
2_] d^g{0)/dzjdzkWjWk 3,k = l
n
< Y ^ d^q/dzjdzkWjidk, j,k
if wi
=0.
= l
If q has the properties in Theorem 4.1.25, it follows that (4.1.17) cannot be valid, so (4.1.17) implies that X is pseudo-convex. The calculations in the proof of Corollary 4.1.6 show that the condition (4.1.17) is invariant under analytic changes of coordinates, and it is also independent of the choice of defining function g. As we shall see in Proposition 7.1.2 the condition (4.1.7) is actually the only condition depending only on the second-order derivatives which has these invariance properties. One calls dX pseudo-convex when the Levi form is positive semidefinite as in (4.1.17) and strictly pseudo-convex if there is strict inequality in (4.1.17) when w ^ 0. These conditions can in fact be defined under a smoothness condition on dX which is weaker than the hypothesis dX G C^ in Theorem 4.1.27. If dX G C^ we can define X by ^ < 0 where g e C^ and dg ^ 0 when ^ = 0. The conormal bundle AT*^ = {(z,Tdg{z)/dz);z
e dX,T > 0} C T*C^ ^ C^ x C^
244
IV. PLURISUBHARMONIC FUNCTIONS
is then defined and independent of the choice of defining function g. It is a C^ manifold if and only if dX G C^. (See Section 7.3 for a discussion of the geometric properties of NQ-^.) However, as pointed out by Lempert [1] the projective conormal set NQ-^ C C ^ X PQ~^ defined by NQ-^ may be in C^ even if dX is not in C^. Let us examine this condition at a boundary point which we place at the origin so that dg = —2dReZn there. If we solve the equation g{z) = 0 for Re Zn we can then find another defining function g= ~2xn-^
f{z',yn),
z' = {zi,...,Zn-i),
Zj=Xj-^iyj,
where f e C^ near 0 and /(O) = /'(O) = 0. That N^^ is in C^ means then that
1 + pdf/dyn in a neighborhood of 0. Since d/ = 0 at 0 the differentiability at the origin implies that dfiz',0)/dzj = ajiz') + o{\z'\), z' -^ 0, where aj are complex valued linear forms in C^"^ = R^'^"^. Thus the Levi form n—1
J2 j,k
n—1
d^f{0)/dzjdzktjik
= Y^ j,k
= l
daj{z')/dzktjik
= l
is defined on the complex tangent space at 0. It is easily seen that it is determined up to a positive factor. To calculate the Levi form in the direction t at a point {z', ^f{z',yn) + iyn) ^ dX close to the origin where Yl^~ tj^f/^^j ~ ^n(l + ^idf/dyn) = 0, we note that
1 + ^idf/dyn
"
1+
^idf/dyn
is also difFerentiable, with numerator vanishing at the considered point, so the Levi form becomes
(1 + lidf/dyr.) ( X; hd/dz, + t.'^id/dyr?j^ ^ [ ^/'^"/^ . ^1
/ 1+
pdf/dyn
This shows that the Levi form is continuous in the complex tangent bundle {{z,t) edX X C^; {t,dg{z)/dz) = 0} where we have defined it. If (4.1.17) is not fulfilled with the Levi form defined in this way, then X is not pseudoconvex. In fact, if dai/dzi = c < 0 with the notation used above, then f{zi,0,...
,0) = c|zi|2 -^2Re{Azl)
+ o{\zi\^)
BASIC FACTS
245
for some A E C. This implies that one can find r > 0 so that ( z i , 0 , . . . ,0,^j2^i + (5) G X for small ^ > 0 when |>2:i| < r, and also for 6 = 0 when \z\ = r, and this contradicts condition (iii) in Theorem 4.1.19. To prove that conversely X is pseudo-convex when the Levi form is nonnegative we need a lemma. L e m m a 4.1.28. Let g and h be continuous functions in R ^ with h ^ 0 such that g/h G C^. If (p e C ^ ( R ^ ) is even, cp > 0 and J cpdx = 1, (p^ = (p(-/e)/e^, then {g * (Pe)/{h * (fe) —^ g/h in C^ when e -^ 0. Proof. Set G = g/h, so that g = Gh, and let K be a compact set. We have to prove that {Gh) ^ (fe h^ cps
_ (Gh) *(/?£- G{h * ipe) h^ife
and the first derivatives converge to 0 uniformly on ii' as ^ —> 0. numerator in the right-hand side can be written
/
The
{G{x - ey) - G{x))h{x - ey)(p{y) dy.
Since {G{x — ey) — G{x))h{x — ey) = —e{G'{x),y)h{x) + o{e) the numerator is o{e) on K. The first-order derivatives of h^cps are 0{l/e) for we can let the differentiation fall on ip^. What remains is therefore to show that the derivative of the numerator (Gh) * (^^ - G(h * (p^) - G'(h * cp,) converges to 0 uniformly on iiT as ^ -^ 0. The last term converges to —G'h, and the sum of the first two is j e-\G{x
- ey) - G{x))h{x - ey)ip\y)dy
The right-hand side is equal to G'{x)h(x),
-^ -
j{G'{x),y)h{x)ip'{y)dy.
which completes the proof.
T h e o r e m 4.1.27'. If X C C^ has a C^ boundary and the projective conormal set is in C^, then X is pseudo-convex if and only if the Levi form is non-negative in the complex tangent space of dX as defined above. Proof. It just remains to prove that X is pseudo-convex if the Levi form is non-negative. In view of Theorem 4.1.24 it suffices to argue locally, so let X be defined near 0 by - 2 x n + /(^',2/n) < 0 where / G C ^ /(O) = f (0) = 0, and (4.1.18) holds. Now we approximate / by a convolution A = / * v?e as in Lemma 4.1.28 with iV = 2n - 1. Then /^ G C ^ and Lemma 4.1.28
246
IV. PLURISUBHARMONIC FUNCTIONS
implies that (4.1.18) is locally the C^ limit of the same expression with / replaced by f^. For any small 6 > 0 it follows that the Levi form of {z]—2xn + fei^'^Vn) + <5|2;'p} is strictly positive ioT 0 < e < es when 1^:1 < r, where r > 0 does not depend on 6. Hence it follows that {z e C"; 1^1 < r, -2xn + fs{z', Vn) + ^ k f < 0} is pseudo-convex when 0 < e < ES. Letting first e and then 8 tend to 0 we find that this is also true for 6: = ^ = 0, which completes the proof. A somewhat simpler argument also shows that strict pseudo-convexity in the sense defined now is stable under perturbations of X which give a small perturbation of the projective conormal set in the C^ sense. Finally we shall discuss the minimum of a family of plurisubharmonic functions, in analogy with Section 1.7, Exercise 2.1.12 and Theorem 3.2.34. P r o p o s i t i o n 4.1.29. Let u G C^(X x / ) where X is an open set in C^ and I a compact interval on R. Then U{z) == min^^/ u{z, t) is plurisubharmonic in X if and only if the following three conditions are fulfilled: (i) If z E X then du{z,t)/dz does not depend on t when t G J{z) = {tel\u{z,t) = U{z)}. (ii) IfzeX and t G J{z), then n
\ J d^u/dzjdzkWjWk
(iii) IfzeXandte
J{z) \dl,w
n
(4.1.19)
> 0,
eC,
w E C^.
then n
Y2 d'^u/dzjdzkWjWk+2ReY29^^/9zjdtWj+d^u/dt'^ j,k=l
> 0.
1
Then we have U e C ^ ' H ^ ) Proof. When n = 1 the statement is identical to Theorem 3.2.34, written in complex notation, and since Definition 4.1.1 only refers to the restriction to complex lines, the proposition follows for arbitrary n. The last statement follows from Theorem 3.2.32 since (4.1.19) implies that \d^u/dzjdt\'^ < d'^u/dt'^d'^u/dzjdzj for every j . Remark. (4.1.19)'
The condition (4.1.19) can also be written Tr-7^u(z + ™ , t -f 2 Re r ) > 0, drdr
WEC,
T = 0.
BASIC FACTS
247
This may display the interplay between complex and real variables better. We can also write (4.1.19) in the homogeneous form, with w G C"^"^^, (4.1.19)" n
n
y ^ d'^uldzjdzkWjWk-\-'^^^^^'^d'^uldzjdtWjW
> 0.
As observed in the proof of Corollary 1.7.4, condition (i) is satisfied if u{z,t) is quasi-convex as a function of t, and condition (ii) is obviously satisfied \iu{z,t) is plurisubharmonic for fixed t. Hence the argument used to derive Corollary 1.7.5 from Corollary 1.7.4 gives without any change: T h e o r e m 4.1.30. Let u G C^(X) where X C C x R is an open set such that Tz — {t £ R; (2:, t) G X } is an interval for every z E C, and let I be a compact interval on R. Assume that Tz 3 t \-^ u{z,t) is quasi-convex for fixed z, that u{z^t) is plurisubharmonic in z for fixed t, where it is defined, and that (4.1.19) is valid in X. Set ui(z) -
inf
u(z,t),
when
iHTz^di.
If the interval J^ = {t e I nTz'ju{z^t) — ui{z)} is compact and non-empty when I nTz 7^ 0, then uj is a plurisubharmonic function of z in the open subset ofC^ where it is defined. Remark. If/ is any interval such that the hypotheses in Theorem 4.1.29 are fulfilled with / replaced by any compact subinterval, then it follows that uj is plurisubharmonic in the open set where it is defined. In fact, if Ij are compact subintervals increasing to / , then uj. | uj^ and since uj. are plurisubharmonic it follows that uj is plurisubharmonic. In particular, this conclusion is valid if / is open and u —> +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
IV. PLURISUBHARMONIC FUNCTIONS
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<\w\
Using Theorem 4.1.4 we can choose C^ functions Uj^ which are plurisubharmonic in a neighborhood of K and invariant under rotation in the last coordinate so that u^^ I u in K, hence U^{z)=-
inf
u,,{z,w)
iU{z),
zeV.
r<w
Now Uj,{z^w) is for fixed z e V a. convex function of log \w\, by Theorem 3.2.16 for example, and 2_] d'^Ujy{z,w)/dzjdzktjtk j,k = l
+ JRey^^
d^Ujy{z,
w)/dzjdwtj
j=l
-{-id'^u^{z,w)/dwdw
> 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'^Uj^/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
/ | / p e - ^ dX.
We regard d/dz as an unbounded operator T in L^{X^ e ^), defined in the sense of distributions. Thus Tu = f ii u, f E L^{X, e"^) and f fve-'^dX
= - j ud{ve-'^)ldzd\
= - f udie-'^ d\
v €
C^{X),
where 8 is defined by (4.2.3)
6v = e'^d{ve-'^)ldz
= dv/dz
-
vdip/dz.
Using (4.2.2) we then obtain I I fve-'^ dx\^
f l/l^e-'^ dX f \Sv\'^e-'^ dX,
v e
C^iX),
and since / is arbitrary this is equivalent to (4.2.4)
/ |t;|2e^-2c^ dX < C f \6v\'^e-^ dX,
v €
C^iX).
Conversely, reversing the argument and extending the map L2(X,e-'^) 3 6v^
f fve-'^dX,
v e
C^{X),
by Hahn-Banach's theorem or an orthogonal projection argument, we conclude that (4.2.4) guarantees that (4.2.1) always has a solution satisfying (4.2.2) whenfeL^{X,e-'''). To study the estimate (4.2.4) we integrate by parts: f\6vfe-'^dX
= -
f{{d/dz)6v)ve-'^dX
+ [ d'^(p/dzdz\vfe-'^dX^
= - f 6dv/dzve-'^
f \dv/dz\'^e-'^
dX+I
dX
f \v\'^Aipe'^
dX.
250
IV. PLURISUBHARMONIC FUNCTIONS
Hence (4.2.5)
/ \v\^A(fe-'^ dX < i f \6v\'^e-'^ dX.
This gives the estimate (4.2.4) if e^ = Ce'^Aip/A. Thus we have proved that if (p E C'^ and A(p > 0, then the equation (4.2.1) has a solution u with (4.2.6)
/ | u | 2 e - ^ d A < 4 / Ifl^e'"^ dX/Aip. Jx Jx Here and in what follows it is tacitly understood that u and / are locally square integrable, and the result states that when the right-hand side is also finite then there is a solution u of (4.2.1) satisfying (4.2.6). If if is just subharmonic we can apply the preceding result with cp replaced by (4.2.7)
ij{z) = cp{z)^alog{l
+ \z\^),
where a > 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)\^e-'^^'\l + Izl^)"" dX{z) < f \f{z)\^e-^^'\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)\^e-'^^'\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.
EXISTENCE THEOREMS IN L^ SPACES WITH WEIGHTS
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^^iX) 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)\^e-'^^^'\l JYj
+ \z\^)-^dX{z)<
[ Jx
\f{z)fe-^^'\l^\z\y-''dX{z),
where we have extended the integration and increased the weight in the right-hand side. Hence we can choose a subsequence Uj^ which is weakly convergent in L'^{Yj,e~'^^) for every j . The limit u satisfies (4.2.1) in X , since differential operators are continuous in P ' , and (4.2.8)" holds for every j with Uj replaced by u. Letting j -^ oo we obtain (4.2.8) by monotone convergence. The proof is the same if the right-hand side of (4.2.8)' is finite. In applications discussed later it will be important that the weight e""^ in (4.2.8)' may not be an integrable function. This is clear if A(p contains a point mass of weight > 47r at a point .2:0? for then e~^ grows at least as fast as I2: — 2:o|~^ as z -^ ZQ by the Riesz representation formula. If u is di continuous solution of (4.2.1) this implies that u{zo) = 0. In the case of several variables we shall use the analogue of (4.2.8)' both to construct analytic functions with prescribed zeros and to get information on the measures associated with a plurisubharmonic function. Our next goal is to extend Theorem 4.2.1 to the case of several complex variables ( z i , . . . , 2;^), Zj = Xj + iyj, where the Cauchy-Riemann equation becomes a system (4.2.1)'
du/dzj
= /„
j = l,...,n,
d/dzj
= \{d/dxj
+
id/dyj).
The essential new feature is that this is an overdetermined system of differential equations which cannot be solved unless the compatibility conditions (4.2.9)
dfj/dzk-dfk/dzj=0,
j,k =
l,...,n,
are fulfilled. We can write (4.2.1)' and (4.2.9) in a more compact form by introducing the d operator n
Bu = 2_\
du/dzjdzj
252
IV. PLURISUBHARMONIC
FUNCTIONS
for scalar functions and define df for a differential form / by this action on the coefficients. With / = ^fjdzj the equations (4.2.1)' can then be written du — / , and the compatibility conditions (4.2.9) become df = 0, which reflects the general fact that dd = 0. It is now clear how the CauchyRiemann operator should be extended to forms of higher order, but we shall content ourselves with a discussion of the Cauchy-Riemann operator acting on functions. The first point to reconsider in the proof of Theorem 4.2.1 is the functional analytic argument, where we must now take the compatibility condition into account. L e m m a 4.2.2. Let T be a linear, closed, densely deGned operator from a Hubert space Hi to another H2, and let F be a closed subspace of H2 containing the range ofT. Then the range ofT is equal to F if and only if for some constant C (4.2.10)
\\fU
feFnVT'.
Proof. If the range of T is equal to F , it follows from Banach's theorem that for every g E F one can find u E VT with Tu = g and \\U\\H^ < C\\g\\H2 for some constant C. When f E F f] VT* it follows that \{f,9)H,\ = \{f,Tu)H,\
= \{T*f,u)HA
< ||T7l|i/.C7||ff||H„
which proves (4.2.10). Conversely, assume that (4.2.10) is valid and that F contains the range of T. Given g E F, to find u E Hi with Tu = g means to find u so that {u,T*f)H.={gJ)H,,
fEVr*.
From (4.2.10) it follows that \{9J)HA f E VT* , can be extended to the scalar product with a unique element u in the closure of the range of T* in Hi, that is,
i9J)H, = {u,T*f)H„
/GPT-;
IHIH, < C | M I H „
which completes the proof. In our case T will be defined by the d operator acting on scalar functions, and F will be a space of (0,1) forms annihilated by the d operator. In the
EXISTENCE THEOREMS IN L'^ SPACES WITH WEIGHTS
253
corresponding abstract situation this means that in addition to the Hilbert spaces in Lemma 4.2.2 we have a third Hilbert space H^ and a closed, linear, densely defined operator S : H2 -^ H^ with kernel F containing the range of T. Then the inequality (4.2.10) follows if
(4.2.11)
11/111,^ < C'{\\T*f\\%^ +115/111,3),
/ e PT. n Vs.
This estimate is of course much stronger than (4.2.10), and in fact it also contains information about the range of 5 . However, the inequality (4.2.11) can be discussed with a modification of the method we used in the onedimensional case, and the results obtained are very precise. Let X be an open set in C"^. The Hilbert spaces Hj^ j = 1,2,3, will be L^ spaces with respect to smooth densities e~^^ in X , consisting respectively of scalars, (0,1) forms / = J^^^^i fjdzj, and (0, 2) forms | Yl'j,k=i fjkdzjAdzk where fjk — —fkj- The operators T and S are defined by d for the elements u E Hi and f e H2 such that du G if2 and df E H^. These are closed densely defined operators since d is continuous in the distribution topology. The adjoint of T is the closure of the formal adjoint defined on the set P ( o , i ) ( ^ ) of (0,1) forms with coefficients in CQ^{X), but there is no general reason why it should be sufficient to prove (4.2.11) for such forms. We shall avoid this problem by choosing weights so that P(o,i)(X) is dense in VT* n Vs in the graph norm f ^ | | / | | H , + | | T * / l k i + \\Sf\\Hs- The following lemma shows how this can be done. Note that n
n
1
1
We shall systematically write £'(p,g) for the space of (p, q) forms with coefficients in a space E. L e m m a 4.2.3. Iff G VT* HVS and supp f (s X, then f is in the closure of D(o,i)(X) in the graph norm. Ifx^ Co^{^) and ueVx then xu G VT andTixu) = xiTu)^{dx)u; iff G VT^ then xf e P r * andT'xf-xT^f = _eV^i-^2 (^x, / ) ; iff G Vs then xf G Vs and Sxf - xSf - ^ x A / . Proof li f e VT* H VS and s u p p / (E X, then df G •£'(0,2)(-^) and ^ / = YJldfj/dzj G L 2 ( X ) . If V' G C O ^ ( C ^ ) , /V^rfA = 1, it follows with ipe{z) = £~'^'^ip{z/e) for small e that / * V'e ^ ^{o,i){^) and that when
f^^s^f
inL^o,i); d{f^^,)^df
inL^o,2); Hf^i^s)'^^f
m L \
Since the supports of the regularizations are contained in a fixed compact subset of X for small e, the first statement follows. In the sense of distribution theory we have d{xu) = xdu -\- {dx)u, so the next statement about
254
IV. PLURISUBHARMONIC FUNCTIONS
T is obvious, and so is that on S. If / S VT* and u G 'Dr, then
= if,Txu)H,
- {f,{dx)u)H,
= ixT*f,u)H,
-
{e^'-'Hdx,f),u)H,,
which proves the statement about T*. L e m m a 4.2A. Suppose that there exists a sequence Xj ^ CQ^{X) such that 0 < Xj ^ 1? ^^ every compact subset of X all Xj ^^^ equal to 1 for large j , and (4.2.12)
e-^*+M^Xjf < Ce-'^S
where C is a constant. norm.
k = 1,2, j =
1,2,...,
Then I>(o,i)(-^) is dense in VT* H VS for the graph
Proof. Let / G VT* 02^5, and set fj = Xjf- Then it is clear that fj -^ f in L?Q ^JX,e~'^'^), and similarly for XjSf and XjT*f - We have
\\S{Xif) - X,Sf\\]j^ < I \f\'\dxA'^e-^'dX Jx
which converges to 0 since the integrand is majorized by Ce ^ ^ | / p . Similarly, l|T*(x,/)-X,T*/ll^, < / \f\'e^^-'^^\dxj\'dX, Jx and since the integrand is bounded by Cl/pe"'^^ ^ this converges to 0. Hence Xjf ^^ f ill the graph norm of Py* H P ^ , so elements with compact support are dense. Thus the first part of Lemma 4.2.3 shows that I)(o,i)(X) is dense in the graph norm. With / G X>(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?
i-1,2,....
Other conditions on ip and X will be introduced when they are needed. With the choice (4.2.13) we have n
T 7 - -e^-2^ ^a(e^-^/,)/9z,-,
that is,
1 n
e^T*f
= - ^Sjfj
- {di,, f),
6j = d/dzj
-
dip/dzj.
EXISTENCE THEOREMS IN Z,^ SPACES WITH WEIGHTS
255
Since \\T*f\\jj^ = / |e^r*/Pe-^(ZA, it follows if e > 0 that
^j/j -^
I'e-'^ dX
+ e)\\T*f\\jj^
+ (1 + e-') J l/l^ia^l^e"'^ dX.
1
We have n
j,fc=i
:^ X; \dfj/dzk\'j,k = l
E
dfj/dz,dh/dz,,
j,fc = l
and since / i^jfjhfk
- dfj/dzkdfk/dzj)e~'^
dX
= f{[6j,d/dzk]fj)fke-^dX
=
Jd^ip/dzjdzkfjfke-''dX,
we conclude when / G D(o,i)(X) that /.
n
/ j,fe=i ^
d^/dzjdzkfJke-^dXK
< (1 + e)||TVIIl,, + 115/111,3 + (1 + ^~') / l/l'I^V-^-^ rfA. At this point it is clear that we should require if to be strictly plurisubharmonic, so assume that n
y j d'^ip/dzjdzkWjWk j,k=l
n
> c{z) 2_\ I'^jl^j
z E X, w E C^,
1
where c is a positive continuous function. Then we have proved that
(4.2.15) Jic - (1 + e-^)\di^f)\f\h-'^ dX
256
IV. PLURISUBHARMONIC FUNCTIONS
for every a G R. Choose a fixed a (later on a will -^ +00). If we drop the cutoff functions Xj ^^ (4.2.14) which are not equal to 1 in Xo_|_2, we can choose a function ij) satisfying (4.2.14) which is equal to 0 in Xa^i- We replace ^ hy (p — ^ -\- x{s) where the convex function x > 0 vanishes on (—00, a) but then grows so fast that (p-2^
= ^^- x{s)
-2^>ip,
n
n
j,k=i
1
If we apply (4.2.15) with ipj = (f -{- (j — 3)ip we obtain
(4.2.16)
J c\f\'e-^ dA < (1 + .)||T*/|PH. + WSffn,
when / e 2:)(o,i)(X), hence when / G VT* HDS. (Here T,S,Hi,H^ defined with ^ replaced by (^.) Let g G L L i\{X.) locally, and assume that Bg = 0 and that M = [
are
Igl^e-'^/cdXKoo.
If T/J is chosen so that e~^c is bounded, then g E H2 since ip2 — (p>(p — ip — {(p-2^) = -0, so e-^2 < (e-V'c)(e-^/c). Since 2ip2-(p-(f = (p-2^-cp > 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
EXISTENCE THEOREMS IN L^ SPACES WITH WEIGHTS
257
Hence dua = g> Recalling that ^i — (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\^e-'^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
d^ip(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<
[
Jx
Igl^e'"^/cdX,
provided that the right-hand side is finite. The preceding result is completely parallel to (4.2.6), and we can deduce other versions as in the case of one complex variable. To avoid repetition we note that if % is a C^ function on R, then
f2 w^wkd\{\z\')/dz,dz,=x"{\z\')\J2^jZjf+x'{\zn E M" j,k=l
1
1 n
>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
wjwkd^ij{z)/dz,dzk
> a ( l + \z\^)-^
Yl 1
l^^l'-
258
IV. PLURISUBHARMONIC FUNCTIONS
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 i^iz) = 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)\^e-^^'\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,
EXISTENCE THEOREMS IN L^ SPACES WITH WEIGHTS
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 C^iX) 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.
IV. PLURISUBHARMONIC FUNCTIONS
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|<1
1^1
{-da{e))
= 27r.
J[0,27T)
Both terms are non-negative so it follows that (27r)-i / Jo
(1 - \z\')\z - e''\-'\da{e)\
m=
d/i(C)/27r < l / l o g ( l / i ? ) < 2 ,
[
< 3,
if \z\ < | , HRKe'^.
J\C\
We can choose R so that ^ < R < e 2 ^ for e < 4. Then we have log / \C\>R
z-C 1-zC
dfi{0\
|z|
for some constant C. The inequality between geometric and arithmetic means gives when 1^2:1 < | log
exp
\<.\
JK\
z-C
z-C l-z(
dKO)
dni()/i2nm)
< C
C\-'^d^,{0, J\<\
for the positive measure d/i{()/(27rm) has mass 1 when \(\ < R. The integral of \z — C|~"^ when \z\ < | is bounded since m < 2, which proves (4.2.22) when n = 1. If n > 1 we introduce polar coordinates and write
/
e-^^^UX{z)= f
dSiO [
\w
2n-2
-ip{wO dX{w)/27r,
EXISTENCE THEOREMS IN L^ SPACES WITH WEIGHTS
261
where dS is the area measure on the unit sphere. Hence (4.2.22) follows from the one-dimensional case. When n = 1 the proof shows that e~^ is integrable in a neighborhood of the origin if the mass of At/; at the origin is < in. In fact, then the mass is < in in {z;\z\ < R} if i i is small enough. This is the same as the mass of A'0(i?-) in the unit disc, so m < 2 in the proof above, which proves that e~^ is integrable when \z\ < R/2. On the other hand, if AT/J has a mass > in at the origin, then IJJ{Z) — 21og|2;| is subharmonic, thus bounded above, so e~'^\z\'^ has a positive lower bound in a neighborhood of the origin. Hence e~^ is not integrable there. Thus the set of points such that e""^ is not integrable in any neighborhood is discrete. The following corollary gives an analogue of this fact in higher dimension. Corollary 4.2.10. If (p is a plurisubharmonic function in a connected pseudo-convex set X and cp ^ —oc, then e~^ is locally integrable in a dense open subset G of X containing all points where (p{z) > —oo. For every a > 0 the complement ofG is the set of common zeros of all analytic functions U in X such that (4.2.23)
/
\U{z)\^e-'^^'\l
+ l ^ p ) " ^ " " dX{z) < oo.
Jx Proof. If ZQ is a point where (p{zo) > —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
IV. PLURISUBHARMONIC FUNCTIONS
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)\^e-^^'\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(
EXISTENCE THEOREMS IN L^ SPACES WITH WEIGHTS
263
rapidly increasing. By Theorem 4.2.6 the equation dv = f has a solution in X such that
Jx Thus U = x^ ~ ^ '^^ analytic, hence v E ( 7 ^ , which implies that ?; = 0 on N since e"^'^ is not locally integrable there. The proof is complete. Finally we shall prove that plurisubharmonic functions are not much more general than quotients of log | / | by integers when / is analytic. T h e o r e m 4.2.13. Let X C C^ be a connected pseudo-convex open set. Then the plurisubharmonic functions z i-> iV~^ log|/(2:)|, where f is analytic in X and N is a positive integer, are dense in the topology of ^ioc(^) ^^ ^^^ ^^^ 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<^^-^ and fN{z) = Y^xiiz
- ^,)/^)e^^^(^) -
which is an analytic function in X if V
uj,{z),
264
IV. PLURISUBHARMONIC FUNCTIONS
Since / l ^ i v p e ' ^ ^ ^ dX < C ^ - ^ e ' ^ ^ ^ ' / ^ , it follows from Theorem 4.2.6 that we can find a solution u^ such that
Since u^ is analytic when \z — Zj\ < 6/2^ if j < i/, we can estimate U]s[{zj) by the average over the ball {z] \z — z^| < ^} if ^ < ^ / 2 , which gives \UN{ZJ)\
< C^-^-ie-^^^'/4+^(^(^^H^^).
When Cg < eS'^/A this is o(e^^(^^)) as iV -^ oo, so we have for large N < ie^'^^^^),
\UM{ZJ)\
hence |/;v(^,)| > |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\
If we make the preceding construction for i/ = 1,2,... and choose N = Nj^ sufficiently large, we obtain a sequence of analytic functions Fj, in X and integers iVj, such that N~^ log |Fi^| is uniformly bounded above on every compact subset of X and lim N-^ log \F,{z)\ < ifiz), lim N-'
log \F,{zj)\ = ^{zj),
zeX, j = 1,2,.
iv'—+00
Thus the sequence N~^ log |F^| is compact in L\^^{X)^ and every plurisubharmonic limit is < ^[z) with equality in a dense subset. By the upper semi-continuity all limits are therefore equal to ip^ so the sequence converges to (^. This completes the proof. Remark. When n — 1 one can give a more elementary proof by approximating the measure NA(p/27r with integer point masses, but even then there are some technicahties to cope with particularly if X is not simply connected. When n > 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)
M^iz,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 <
d{z,CX).
Hence M^p{z, r) < y^{z) logr + (7 if 0 < r < ro- By the inequality between geometric and arithmetic means we have j ^-c,(z+ra;) dS{u;)/c2n J\uj\=l
> CXp f - / cp{z + ru) ^ J\u;\ = l
= exp{-M^{z,r))
dS{u;)/c2n) ^
> r'^^^^^e"^,
and since C'^'^ 3 ( ^-^ \(\~'^'^ is not integrable at the origin it follows that e~^ is not integrable in any neighborhood of z ii Uip > 2n. This completes the proof.
266
IV. PLURISUBHARMONIC FUNCTIONS
Prom now on we assume that X C C"^ is a connected pseudo-convex open set, and we shall discuss an operation which modifies the Lelong numbers of the plurisubharmonic function (^ in a controlled way. Set (4.3.4)
X = {{z,w) eX
xC- \w\ < d{z, CX)},
and note that X 3 (z^w) \-^ M^p{z^ \w\) 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) ^a{z)=
inf {M^{z,\w\) {z,'w)ex,w^o
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^^iz) < 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^^p^iz) = 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 i^ipo{z) ^ ^(p{^) since (^o > ^ + min^. Hence I'lp^iz) < 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 <
d{z,CX).
The mean value of M^{z + (,r) when |C| = r' < d{z,ZX) - r IS a mean value of Z i-> ip{z 4- Z) in the ball where \z\ < r -^ r' and invariant under orthogonal transformations of Z. All such mean values can be estimated by Mcp{z^r + r ' ) , so we have M^^ {z, r') < M^{z, r + r') + / Q{Z + r'C, r)dS{0/c2n J\C\=i
- a log r.
If we take r — r'^ divide by logr and let r -^ 0 it follows that
i^^^{z)
for logr < 0 and log(2r*)/logr -^ 1. This completes the proof. We can now prove the main theorem on the level sets of the Lelong numbers: T h e o r e m 4.3.3 (Siu). Let ^p he plurisubhsirmonic and ^ —oo in the connected pseudo-convex open set X C C^, and let J^cp{z) be the Lelong number ofcpatz. Ifa>0 it follows that A{(p,a) = {z e X\v^{z) is the intersection
> a}
of the zero sets of a family of analytic functions in X.
Proof. By Lemma 4.3.1 and Corollary 4.2.10 we have A{ip,2n) C Z{(p) C {z e X]ip{z) = - o o } , where Z{ip) == {z E X ; e ~ ^ ^ Lj^^ dit z}. If we replace ip by the plurisubharmonic function cp^^ constructed in Lemma 4.3.2, we obtain for a > 0 A((^, a + 2n) = A{ipa, 2n) C Z{cpc,) C {z e X] (pa{z) = - o o } C A(v?, a). The discrepancy between the arguments of A on the left and on the right is decreased if we replace cp and a by jcp and ja for some large 7. This gives A{cp, a 4- 2n/7) C Z{{jip)^a)
C A((^, a).
268
IV. PLURISUBHARMONIC FUNCTIONS
Finally, replacing ahy
a — 2n/j
we obtain
j>2n/a
By Corollary 4.2.10 the sets in the right-hand side are intersections of zero sets of analytic functions in X , which proves Theorem 4.3.3. Basic facts on analytic sets show that one only needs a countable intersection in Theorem 4.3.3 and only a finite one on relatively compact subsets. We leave as an exercise to extend Corollary 4.1.18 to the case of an iV-tuple / = ( / i , . . . , JN) of analytic functions: The Lelong number of I l o g ( | / i p -|- . •. -f- l/ivp) at z is then the smallest order of the zeros of fim... ^JN at z. Thus the level sets of v^p can be quite general analytic sets. Theorem 4.3.3 and the preceding example suggest that u^p might be invariant under analytic changes of variables in spite of the fact that the definition using (4.3.1) depends on the metric and vector space structure of C^. This is in fact true, and a proof can be found in CASV, Theorem 4.4.13. It will not be given here since we shall prove a more general result in Theorem 4.4.14 below. The lack of invariance of the definition of Lelong numbers is also seen from the fact that (4.3.2) only involves the Laplacian of ^ and not the other measures 6jk = d^cp/dzjdzk associated with cp. (See Corollary 4.1.5.) However, the reason is simply that A(p determines (/? up to a harmonic function which does not influence the mass density. In order to be able to motivate a general version of Theorem 4.3.3 we shall now restate it in terms of the measures 0jk^ eliminating all direct reference to cp. These measures satisfy the compatibility conditions (4.3.7)
dOjk/dzi =deik/dzj,
dOjk/dzi ^dOji/dzk,
i, A:,Z = 1 , . . . , n .
The second set of equations follows from the first by the Hermitian symmetry 6jk — Okj- These conditions can be stated more succinctly in terms of the (1,1) form 0 = '^Yllk=i^jkd^j A dzk' they mean simply that the real form © satisfies dQ = 0 (or equivalently dQ — 0 or dQ = 0). The equations 6jk =^ d'^ip/dzjdzk mean that 6 = idd(p. Given a real (1,1) form @ = i^Ojkdzj A dzk we shall now discuss the existence of a form ip with iddip = Q. Denote by E the fundamental solution of A / 4 = ^ ^ d'^/dzjdzj in C"^ = R^"^. Since the area of the unit sphere in R^'^ — C^ is 27r'^/(n - 1)! we have £;(z) = - | z | 2 ( i - ^ ) 7 r - ^ ( n - 2 ) ! .
LELONG NUMBERS OF PLURISUBHARMONIC FUNCTIONS
L e m m a 4.3.4. IfOjk G ^ ( C ^ ) andcp = E^Yl7
269
^jj^ ^hen
n
(4.3.8)
d\ldzkdzi
-Oki^Y.
^^I^^o
* {dOkj/dzi -
dOu/dzj)
j=i
-hJ2dE/dzi
* {d0jj/dzk
-
dOkj/dzj),
Proof. Since a differentiation can act on any factor of a convolution, we have n
dif/dzk
n
^Y^E^dOjjIdzk
n
-
j=l
Y^dEldzj^ekj^Y^^^^^nl^^k-dekjIdzj). j=l
j=l
Differentiation of the first term on the right with respect to zi gives n
n
5 ^ d^E/dzjdzj
^Oki + Y^ dE/dzj
3=1
* {dOkj/dzi -
dOki/dzj),
3=1
and since ^ ^ d'^E/dzjdzj
— 6, the lemma follows.
Note that in any open set where the conditions (4.3.7) are fulfilled, the right-hand side of (4.3.8) is a C^ function. This leads to the following result: T h e o r e m 4.3.5. Let X be an open set in C^ and assume that Ojk G V'{X) satisfy (4.3.7), 9kj = Ojk- Then one can find cp e V'{X) real valued such that Ojk — d'^ip/dzjdzk G C ^ ( X ) , j . A: = 1 , . . . , n . If X is pseudoconvex and n
(4.3.9)
Y^ OjkWjWk > 0,
then one can choose (p
we C^,
plurisubharmonic.
Proof. Let Ki,, z/ = 1, 2 , . . . be an increasing sequence of compact subsets of X with union X , each contained in the interior of the next, and choose Xu G CQ^{Kjy^i) equal to 1 in a neighborhood of Kj^, set Xo = ^ and
^ = ^(Xu
- Xi^-i){E * (Xz.+i Yl
^^o))'
270
IV. PLURISUBHARMONIC FUNCTIONS
All terms with v > ji vanish in a neighborhood of K^^ and modulo C^ we may there replace Xv+i by XM+I ^^ ^^^ others, so
- Ojk e
C^{X).
ip is real so Rjk is Hermitian symmetric. When X is pseudo-convex we can by Theorem 4.1.21 choose a strictly plurisubharmonic function s G C^{X) such that {z G X; s{z) < t} is compact for every t. If x is a convex sufBciently rapidly increasing function then n
Y^ {Rjk + d\{s)/dzjdzk)wjWk
> 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^^^i Xj^dzk if Xjk G C. Ife e C^^i){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) <==^ (iv), and by the linear invariance it suffices to prove that (iv) at the origin with Xjk = Oior k
(4.4.2)
te - 0 A ^^-^/(n - p)\,
If 0p_i_i,..., 0 n € C?^^AX) (4.4.1)"
P = J2 ¥^3 ^ ^^3 = l^^l^l^-
are positive,
then
0 A 0p+i A • • • A 0 n > 0.
Proof. The necessity is obvious. To prove the sufficiency we first regularize 0 to a smooth form 0^ in the usual way by convolution with x(*/^)^~^'^ where 0 < x G CQ° and the integral of x is equal to 1. For any open set
CLOSED POSITIVE CURRENTS
273
Y (G X the translation invariant condition (4.4.1) in the assumption is then valid in Y for the smooth form 6^ when e is small enough. By the equivalence of (iv) and (iii) in Proposition 4.4.1 it follows that 0^ is positive, hence the limit © as ^ -^ 0 is positive. Now we can polarize any sesquilinear form (•,•) by 4
k=l
SO it follows that G Ai{Xp^i,dz)
A {flp+i,dz) A---
= 4^-^ J2 i^ ^'® A i(Ap+i + M'fip^udz)
A ( V i + i^^+'flp^i,dz) •. •
where fep+i,..., fc^ run from 1 to 4. Taking Ap4.i,/Xp_|-i,... as the basis vectors we get a representation of the coefficients of 0 which proves that they are measures bounded by a fixed constant times the trace measure. In fact, te = e / 3 " - 7 ( n - p)! = 0 ^
^dz^, A dzj, A • • • A ^dzj^,^
A dz,„_,
where i i < ^2 < • *' • All terms are positive measures < t e , and in view of the unitary invariance it follows that the form (4.4.1) with A^ replaced by (Xj^dz) is bounded by ^elAp+ip • • • |Anp. Hence the total variation of every coefficient is < 2'^~^tQ. If the Hermitian symmetric matrices corresponding to 0p-f i , . . . , On G ^(Ti) ^^^ strictly positive definite, then by a completion of squares we can write n k= l
so (4.4.1)" follows from (4.4.1)' then. Hence (4.4.1)" is always valid with Qj replaced by Qj-\-ie ^ ^ dzjAdzj^ if ^ > 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
IV. PLURISUBHARMONIC FUNCTIONS
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{zd^i,. ..,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
CLOSED POSITIVE CURRENTS
275
Wj = Zj — hj{z')^ d < j < n^ the trace measure is n
te = e A /3^/rf! == 6{wd-^i, ...,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\
is an increasing function
ofr.
Proof. It is sufficient to prove this when 0 G C°°, for if 0-, are smooth regularizations —> 6 for which (4.4.5) is true, and r < r2 < TI < i?, then
/ J\z\
te/r^P < lim / j-*ooJ\z\
tejr^"
< Ikii / ^~^°°J\z\
te^/r'/
< f J]z\
te/rl^,
276
IV. PLURISUBHARMONIC FUNCTIONS
hence l{r)lr'^^ < I{ri)/rl^. DiflFerentiating
Set Q{Z) = \z\^ and note that /? =
I{r) =
^ddg.
jH{r'-Q)eA^P/p\,
where 0 € Cf^^p ^-v) ^^^ ^ ^^ ^^^ Heaviside function, the characteristic function of R^., we obtain r(r-) = 2r f 6{r^
-g)eApP/p\.
We may assume that p > 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) <-
16(r^
- g)gQ A p^/pl = — 16{r^ - g)e A P^/pl =
rl\r)/2p.
This means precisely that (/(r)/r2^)' = {rl\r)
- 2pl{r))lr'^^^^
> 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.
CLOSED POSITIVE CURRENTS
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 z^eWe 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
(4.4.7)
U{z) = - [
\z-
C|-'^/3^ A G - -p\ [
and
\z - Cl'^^^e,
278
IV. PLURISUBHARMONIC FUNCTIONS
where iS = | 5Z ^Cj ^ ^ 0 ^^^ ® ^^^ considered as forms in ( and the integral is a convolution in the sense of distribution theory, then (4.4.8) 2PUjk{z)=p{p^l)
(4.4.9) Here djdzj
[
{Lf^a''-pidL-kAdLjAaP-^)Ae+
Y.U-{z)^p{n-l-p)
f
RjkAdG,
j l^-Cr^^-'^^AG.
and d/dz^ have been denoted by subscripts j and k, L — log \z — (1^,
a = iddL,
where z is regarded as a parameter, Rjk are forms in ( of degree 2p—l with coefficients that are homogeneous functions of z — C,, of degree —2p — 1, and smooth for z ^ (. Proof. In view of the continuity of convolution in distribution theory we may assume that 6 G C^ during the proof. In the main terms of (4.4.8) and (4.4.9) the kernel is homogeneous in z — (^ of degree —2{p + 1) > —2(n — 1) so it is a locally integrable function. This would not be the case when p =^ n — 1, which is the reason why Lemma 4.3.4 is somewhat different. All integrals are thus convergent in a classical sense, and throughout the proof we shall consider z s.s a. parameter and all forms will be forms in (. Let us first compute a explicitly. Since dL = \z- C r ' Yl^'^k - Zk)dCk,
ddL = |z - C|-2 5 ^rfCfcA dCk -dLN BL,
we have a = 2\z — C|~^;S — idL A dL, hence (4.4.10)
\z-Q-^P{2l3f
= {a + ujY = aP + 'pu^OLP-^•, uj =
idLAdL.
Here we have used that w A w = 0 since a; is a product of one forms. Thus we can rewrite (4.4.7) as 2PUiz) = - / ( a + w)P A e = - f{aP + pu A a^"^) A 0 = U\z) + U\z)
= - f aP Ae,
U^z) = - fijAaP-^
pU\z),
AS.
Since a = iddL and a is closed, we can write Q;P A e = d{idL A oP-^ A 6 ) + idL A oP-^ A OS = d{idL A a"-^ A 6 ) + idL A a^'^ A dS,
CLOSED POSITIVE CURRENTS
279
SO Stokes' formula gives U\z)
= -i f dLAa^-^
Ad&.
Here BL A a^~^ is a 2;? - 1 form with coefficients homogeneous of degree l — 2pm(-z,so the second derivatives can be included in the error terms. Next we note that U^iz) = - [{cj-^ A aP-^ -^{p-
l)uj A a^ A a^'^) A 6 .
Here uj-k = i{dL-k AdL-\-dLA ujAaj,=
i^dL
ABLA
BL-^) - 2x01-^ ABL= B{dL ABLA
OBLJ,
iB{LidL)
dL-^) + idL^
-
id{L-^BL), ABlAa,
so we obtain by Stokes' formula [/|(z) rr. - ( p + 1) f idL-kABLAa^-^ + i I
L-^BL A
a^-^ Ade-{p-l)
AG-^i f
f Lj^dL A a^-^ A Be
OLABLA
dL^ A a^-^ A Be.
(Note that a form of bidegree (n - l , n ) (resp. {n,n — 1)) is closed if it is annihilated by d (resp. B).) The derivatives of the last three terms with respect to Zj can be included in the remainder term in (4.4.8), so it remains to examine the derivative of the first term. We have i{dLjj^ ABL -{- dLj. A BLJ) — id{Ljj^ A BL) — Ljj^a + idLj;. A BLJ, and if p > 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
IV. PLURISUBHARMONIC FUNCTIONS
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-^aP -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 >^i^kLfkY~^ is equal to n
The product of (4.4.11) by
n
( X ] ^AkLj^a
- 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)!^o 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©.
CLOSED POSITIVE CURRENTS
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^,-^iQ) = 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-^ip + diieC^ where 0 < p < n — 1 and cJ/i is a measure such that f J\z\
dfi{z)/r^P -^A,
f
A^dA(z)/r-2(^-i) J\z\
5,
as r —^ 0.
282
IV. PLURISUBHARMONIC FUNCTIONS
Then it follows that A = B 4 ^ - ^ - i ( n - p - l)!(n -
l)\/p\.
Proof. Choose an even function x ^ CQ^{IV) and let M{r) = /ui<.^ d/i{z). Then we have for small r j Xi\z\/r)dn(z)
= p
x{tlr)dM{t) /•OO
nOO
= - / Jo
M{t)x'it/r)dt/r
Since M{rt)/r^P = t'^PM(rt)/{rt)^P formly bounded, it follows that
= - / Jo
-^ At^^ as r -^ 0 and is locally uni-
/•OO
/
X{\z\/r)d^l{z)|r^^
^ -A
/ Jo
M{rt)x'(t)dt.
/»00
e^xHt)
dt = 2pA / Jo
t'^-'xit)
dt.
Now ( - A ) " - P - ^ x ( k l A ) = ^(|^|/r-)r2p+2-2n^ where ip € C ^ ( R ) , so J x{\z\/r){-Ar-P-'Aip{z)
d\{z) = r2p+2-2n j ^ ( | ^ | / r ) A ^ ( Z ) dA(z).
Thus the first part of the proof gives j x{\z\lr){~Ar-^ip{z)d\{z)lT''^
=
~r'-'^J^i\z\/r)A^iz)dXiz) /•CO
-^ - ( 2 n - 2 ) 5 /
f2n-3^(^^ ^^^
hence /•OO
(4.4.14)
2]JA /
/•OO
^^^"^xW dt = (2n - 2 ) 5 /
i^n-s^^^) ^^
HereV'(<) = ( - l ) " - P - H 9 V 9 * ^ + ( 2 n - l ) r i a / a t ) " ~ ^ ~ ^ x W , and induction for decreasing p shows that {-l)''-P-\d^/dt'^
- (2n - l ) ( a / 5 « ) < " ^ ) " ' ^ ' ^ i ^ " ~ ^ = r-P-H^P-\n
- p - l)!(n - 2)!/(p - 1)!.
Combined with (4.4.14), this proves the lemma.
CLOSED POSITIVE CURRENTS
Remark. After the first part of the proof has shown that A/B pendent of (^ we could just take ip{z) =\og\z\^ and note that -{l^Y-^ip{z)
= 4^-^(n -p-
283
is inde-
l)!(n - l)\/{p - l)!|z|2^-2^,
which gives A = (74'^"^(n -pl)!(n - l)\/p\ with a constant independent of p; we obtain B = 4(7 by taking p — n - l and the assertion follows. If we now apply Lemma 4.4.11 to the closed positive form 0 and the plurisubharmonic function ip in Theorem 4.4.10, taking the normalizing constants in (4.3.2) and (4.4.6) into account, we obtain 4^-^7r> (n - 1 -p)!i^07r^ = 27r//^(7r^-V(n - 1)!)4^-^-^(n - ; ? - l)!(n - 1)!, that is, (4.4.15)
v^ = 2 7 r V e .
If we apply Theorem 4.3.3 to ip, we obtain an extension to arbitrary closed positive forms: T h e o r e m 4.4.12 (Siu). Let Q be a closed positive current in a pseudoconvex domain X C C^. Ifa>0 it follows that {zeX]iye{z) is the intersection
>a}
of the zero sets of a family of analytic functions in X.
Remark. We would have obtained u^p = VQ if (4.4.7) had been divided by 27r^p. Since 27r^p/p\ — 2n^/{p - 1)! is the area of the unit sphere in R^^~^, this is a very natural normalization. To prepare for a proof of invariance of the Lelong number we shall now give another expression for it. P r o p o s i t i o n 4.4.13. IfQ is a closed positive {n — p,n — p) current in {z G C ; \z\ < R} and 0 < p < n, then (4.4.16) TT-^ /
{iddx{\og \z\)r A e / x ' ( l o g r ) ^ = [
J\z\
te/iC^pr'n.
r < R,
J\z\
for every C^ convex non-decreasing function x on R which is constant in a neighborhood of —oo such that x'(logr) > 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.
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IV. PLURISUBHARMONIC FUNCTIONS
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 ) ^ > ^e(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 z^e(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 < ^e-
Proof. li z E X and r is sufficiently small, then iyeM<
[
teJ{C2pr^n.
r{z) JBJZ)
if Br{z) = {C; 1^: — CI < r}. Since te^ —> t e in C , hence as a measure, it follows when j -^ oo that lim iyQ.{z)<
f
te/(C2pr'n. r{z)
and when r -^ 0 we obtain (4.4.19). 4 . 5 . E x c e p t i o n a l s e t s . Already at the beginning of Section 4.1 we mentioned the exceptional sets which are associated with plurisubharmonic functions: Definition 4.5.1. A set A C C"^ is called pluripolar if every point in A has an open connected neighborhood U where there is a plurisubharmonic function u ^ —oo equal to — oo in A fl C/. The first step in the discussion of polar sets in Section 3.4 was to show that they could also be defined by means of global subharmonic functions. The proof relied on the fact that the singularities of subharmonic functions always come from the Newton potential of a measure. The situation for plurisubharmonic functions is quite different for if ?i is a plurisubharmonic function in C^, then the measures Ujk = d'^u/dzjdzk must satisfy the compatibility conditions (4.5.1)
dujk/dzi
= duik/dzj,
dujk/dzi
=
duji/dzk.
(See (4.3.7).) These relations are destroyed if we multiply by a cutoff function. We shall therefore use another approach where the main step is the following theorem: T h e o r e m 4.5.2. Let u be a plurisubharmonic function < 0 in the polydisc {z G C^] \z\ < 1}, where \z\ = indix\zj\, and assume that u{0) = —1. Then one can for every r G (0,1) and 7 > 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\
(4.5.3)
v{z) < |7^/'^logr,
if\z\ < r and u{z) < - 7 .
The proof is fairly long, so we give an analogue of Theorem 3.4.2 as an application first.
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IV. PLURISUBHARMONIC FUNCTIONS
T h e o r e m 4.5.3 ( J o s e f s o n ) . If A C C^ is a pluripolar set, then one can Gnd a plurisubharmonic function v ^ —oo in C^ such that v = —oo in A. Proof. We shall first prove that if li is a plurisubharmonic function in a neighborhood of the polydisc {z G C^; \z\ < 1} and u{0) > - o o , then one can find a plurisubharmonic function -?; ^ — oo in C"^ such that v{z) = — oo if u{z) — —oo and |2:| < | . Adding a constant to u and multiplying by a positive constant we can assume that u{z) < 0 when |z| < 1 and that u{0) = —1. Then we can apply Theorem 4.5.2 with r = | , 7 = n + 2-^, and conclude that for j = 1,2,... there is a plurisubharmonic function Vj in C^ satisfying (4.5.2) for every j such that, by (4.5.3), Vj{z) < - | 2 ^ > l o g 2
when \z\ < | , and 7i(^) < - n - 2 ^ ' .
From (4.5.2) it follows that Vj is bounded in
L J Q ^ ( C ^ ) , SO
. = ^2-^/^%,
converges in Lj^^ to a plurisubharmonic function v such that v{z) = —00 when \z\ < | and u{z) = - 0 0 and v{z) < 5 ] ^ 2-^/2n ^j^gH-1^^.|^ ^ ^ QU^ Prom this point on the proof runs parallel to that of Theorem 3.4.2. Label as a sequence all rational Zjj G C^ and rational i?^ > 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 = YlT^i'/'^^ ^^ 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 Uf^Aj 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^ -\- ^i 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 M^o < M and MN = |A|. If NQ < ly < N and we
288
IV. PLURISUBHARMONIC FUNCTIONS
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)^/(^~^o\ 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 fj^Aj^Bj and also omit subscripts on the following constructs depending on j . We want to find a polynomial
Q{z)= Yl ^^^^
M
Woo
where |/3|oo = max/?^ and ^ = (^i,..., ^S^) is a multiindex, so that (4.5.6)
Q{z)f{z)=G{z)^R{zy,
G{z)=
Y
g^z^,
\oc\oo
R{z) = Y^
r^z^.
\oc\oo>B
This means that we require that in the power series expansion of Qf the coefiicient of z^ is zero when A < \a\oo < B. In the power series expansion of/, the coefficients fa have absolute value < 1, and the coefficient of z^ in Qf is moo
where /^ should be read as 0 unless all components of 7 are non-negative. The map {Qa)\a\oo
{Pa)\a\oo
EXCEPTIONAL SETS
289
has a triangular matrix M, with diagonal elements /o = /(O) if we order the multiindices a first for increasing |Q!|OO ctnd then in some arbitrary way for fixed |a|oo- Altogether we have N = B'^ variables Qa, and we want to satisfy the NQ = B^ — A^ equations Pa = 0 when A < \a\oo < B. We can now apply Lemma 4.5.6, noting that all minors of M can be estimated by N\ < N^. Hence we obtain a product Qf decomposing as in (4.5.6), such that (4.5.7)
max 1^,1 = 1;
A:^^ > | / ( 0 ) | ^ 7 ( 5 " ^ " ) ,
if A: = max |^,|,
Hoo
for the determinant of a triangular matrix is the product of the diagonal elements. Set g{z) = G{z)/k. Then \g{z)\ < A ^ m a x ( l , |;2|^), so (4.5.8)
A-^ log \g{z)\ < nA'^ log A + log+ \zl
which will lead to the first part of (4.5.2). Since some coefficient of the polynomial g has absolute value 1, we must have max|^|=,i \g{z)\ > 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) < &^a-^-^ + 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
IV. PLURISUBHARMONIC FUNCTIONS
If |2;| < r < 1 and u{z) < —7 it follows that v{z) < ft'^a-^-^ + a - ^ m a x ( - 7 , 6 1 o g r ) = ft^a"^"^ - 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 < b when 7 > 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^^ IV^I ^^ 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
IV. PLURISUBHARMONIC FUNCTIONS
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.
OTHER CONVEXITY CONDITIONS
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 C^iC) such that g <0 in X, Q = 0 and dg ^ 0 on dX. Then it follows that the second differential d? g of g is a positive semidefinite quadratic form in the complex tangent plane Hz at z £ dX. Conversely ifX is open, bounded and connected, and d^g is positive definite in Hz for every z G dX, then X is weakly linearly convex. Proof. If X is (locally) weakly linearly convex, then ^ > 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
IV. PLURISUBHARMONIC FUNCTIONS
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 jj^i 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 jj^i to Jj^i is homotopic to its orthogonal projection to the line Lj through 7^ and Jj^i 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
OTHER CONVEXITY CONDITIONS
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
IV. PLURISUBHARMONIC FUNCTIONS
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(>2^i,... ,^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 — c^i- 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 C^i 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^ '^i((^5Ci) ~ 1)? ^2: G X , that is, wi 7^ W2 and ("^iCi "" '^2C2)/('^i - '^2) ^ X*. Since (1 : 0) and (0 : 1) are in F it follows that (^i 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.
OTHER CONVEXITY CONDITIONS
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 Q^i+^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
IV. PLURISUBHARMONIC FUNCTIONS
complex line parallel to the ^i-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 ^o where Xj — 0 the complement of Se is the triangle Sj — {AG R^; A^ = 0 , Q < Xk < s-,k ^ j , ]^i 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 ^o 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 ^o \ 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:
OTHER CONVEXITY CONDITIONS
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) > 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 ^e^^zM
= C'^ifzie'^w),
0 eRj
w e D,
so (p{wz) = ip^idz{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 | 1. If 2:0 E «5^^~-^, then we
300
IV. PLURISUBHARMONIC FUNCTIONS
can choose a linear form L with L{zo) 7^ 0 such that L{z) = 1 impUes z e CX, for by Proposition 4.6.11 ZQ is not parallel to all affine hyperplanes C CX. Thus the range of cpz does not contain l/L{z) if L(z) ^ 0, which by Theorem 3.2.27 implies that [^'^(O)! < 4/|L(^)|. We can cover the unit sphere by a finite number of sets where some linear form such as L is bounded away from 0 and conclude that
i<\^'M\
zes'2 n - l
In view of (3.2.15) it follows that the functions cpz are in a compact subset of C^[D). We shall prove that they depend continuously on z. Let ZQ G S'2'^-1 and let Zj G S'^^~^ converge to 2:0 as ^' -^ 00. It suffices to prove that if if^^ -^ ^ 'm C ^ ( D ) , then i/j = (Pzo- Since 1 < \i)'{0)\ < C and i^^O) > 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
feA{X).
K
By the Hahn-Banach theorem we can then extend /x to a continuous linear form on C{K), that is, a measure du with support in K such that (4.7.1)
M(/)=
/ f{z)duiz), JK
feA{X),
ANALYTIC FUNCTIONALS
301
but v is far from uniquely determined. For example, if 7 : [0,1] -^ X is a C^ arc in X C C, then (4.7.2)
/x(/) = /
f{l{t))i{t)
dt^
Jo
f f{z) dz,
f e
A{X),
J-f
does not change if 7 is replaced by a homotopic arc 71. D e f i n i t i o n 4 . 7 . 1 . A compact set K C X is called a carrier for fi G ^ ' ( X ) if for every neighborhood u oi K with u C X there is a constant Ceo such that
(4.7.3)
K/)|
f^AiX).
Note that if the measure du in (4.7.1) is replaced by any distribution u with compact support in X , then fi is carried by suppz/, for the topology in A is equivalent to the restriction of the C^ topology. It is clear that the intersection of a decreasing sequence of carriers is also a carrier, so every carrier contains a minimal one. However, there may exist several minimal carriers; that is the reason why the term "support" should not be used. (Some authors use it for minimal carriers.) For example, it is easy to show in the example (4.7.2) that every Jordan arc C X C C with end points 7(0) and 7(1) is a minimal carrier if X is simply connected. If X i C X2 then the adjoint of the restriction map A{X2) —^ ^ ( X i ) is a map A'{Xi) -^ A'{X2). In general it is not injective. For example, M ( / ) — / u i = i •^(^)^'^ ^^ ^^^ equal to 0 as an analytic functional in the annulus { ^ € C ; 0 < | ^ | < 2 } but is 0 as an analytic functional in the disc {z e C; \z\ < 2}. By the Hahn-Banach theorem the map ^ ' ( X i ) -^ A\X2) is injective if and only if the restriction of A{X2) is dense in A{Xi). The range of this map consists of all /z G A'{X2) carried by some compact subset of X i , for such elements /i G A'{X2) can be represented in the form (4.7.1) with suppcJz/ C X i and / E ^ ( X 2 ) , and (4.7.1) with / E ^ ( X i ) defines an element in ^ ' ( X i ) mapped to // in A'{X2)If/x E A'{X) is carried by K, then we can define /x(/) uniquely for every / such that / is the uniform limit of functions in A{X) in some neighborhood uj of if, hence analytic in u. We shall denote this extension by fiK to emphasize that ^IRU) may depend on K. For example, if /x E A'{C) is defined by (4.7.2) where 7 is a C-^ Jordan arc, then JIKU) is defined for K = 7([0,1]) and f{z)=^{z-()~^ii(^K but may change by a multiple of 27vi if 7 is replaced by another arc. (See Lemma 4.7.4 below.) D e f i n i t i o n 4.7.2. If /x E ^'(C"^) then the Laplace transform /i of /x is defined by (4.7.4)
/i(C)=M.(e<^'^>),
CGC".
302
IV. PLURISUBHARMONIC
FUNCTIONS
If /i is written in the form (4.7.1) it is clear that /x is an entire function and that 1/2(01 <exp(supRe(z,C)) [ \du\ This gives the first part of the following theorem: T h e o r e m 4.7.3. If /JL e ^'(C^) is carried by the compact set K, then M{() = pb{() is an entire analytic function, and for every 6 > 0 there is a constant Cs such that (4.7.5)
|M(C)| < Cs exp{HKiO HK{0
=
+ ^|CI), zeK
C ^ C^,
where
^^PM^.O'
Conversely, ifK is a convex compact set in C^ and M is an entire analytic function satisfying (4.7.5) for every 6 > 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 ^i{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<^'<>+<^'^<> dv{x, y),
( ^ C^-
Let
be the Fourier-Laplace transform of du^ considered as a measure in R^'^. Then N must satisfy the condition (4.7.6)
i V « , - C ) = M(C),
CeC",
and since the supporting function h^ of K^ is given by h6{C,v)=
sup x-hiyeKs
{{x,0-^{y^v))=
sup Re(2:,^-ir/) ==ifi^,(^-z77), zeKs
ANALYTIC FUNCTIONALS
when ^,7/ E R'^, we must have (4.7.7) \Ni^,v)\ < Cexp{hs{Im^,Im7]))
303
= Cexp(^K,(Im^-tImr/)),
^,r/ G C " .
Conversely, if we can find an entire function N in C^" satisfying (4.7.6) and (4.7.7)'
| i V ( ^ , r / ) | < C ( H - | ^ | + M)'=exp(/i^(Im^,Im7?)),
^,r/eC",
for some C and k, then it follows from the Paley-Wiener theorem that N is the Fourier-Laplace transform of a distribution with support in Ks defining an analytic functional carried by Ks with Laplace transform /i, which will prove the theorem. The construction of N is an extension problem. By (4.7.6) the function N is prescribed as an analytic function in
^ = {K,-C);CeC"}cc2", and by (4.7.5) its value at (^,r/) where ^ = iC and T] = —( is bounded by C^exp(iZ'x(C) + ^ICD? which is equal to Csexp{hs{Im^^lmr])) since Re( = I m ^ and Im(" = — Imry, which implies Im^ — ilmrj = (. Hence the existence of N follows from Theorem 4.2.11, with ^{^,rj) - 2hs{lm(,Imv)
+ (n + l ) l o g ( l + \^\^ + \rj\'),
^,7/ € C ^
in view of the remark after the proof of that result. This completes the proof. When n = 1 there is a classical proof of Theorem 4.7.3 which gives more. First note that if iiT is a carrier of /x G A'{C) and R > 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'« ./\<:\=R
°^
304
IV. PLURISUBHARMONIC FUNCTIONS
We have fi{z^) = jl^^\0). The Laurent series OO
CO
0
0
which converges for |(| > 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)|
CeC,
we have for arbitrary R> 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'di^iz),
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
/(^) = ^iJJ
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)
When Ce|tt;| < Rew( we can integrate term by term after replacing M{tw)w by the Taylor expansion Y^'o^ Pw^^^M^^\0)/j\ which proves that B{(,w) = BM{0 there. Hence any two functions B{',wi) and B{-^W2) coincide in the angular region where both are defined, for they are equal to
306
IV. PLURISUBHARMONIC FUNCTIONS
BM far away. Thus these functions define together an analytic continuation of BM to {C G C; HK{W) < RewC for some w e C} which is the complement of K. In (4.7.9) we can therefore take s u p p x arbitrarily close to K^ which proves that fi is carried by K. In addition the analytic continuation of BM can give information also about non-convex carriers of /x. We shall now give an analogue of the preceding discussion in the case of several variables. To motivate the definition we make an inversion of the Borel transform of /x G ^ ' ( C ) and note that 5/i(i/C)/C =/^if((i - < ) - ' ) , which is analytic in a neighborhood of the origin in C. If 2;, ^ G C^ then 00
(X)
(1 - (^,0)-^ = E^^'O^" = E^'' E ^"C7^! 0
i=0
\a\=j
converges uniformly when \z\ < Rii \(\ < 1/R. If// G A'iC^) a compact set K and R > 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
ANALYTIC FUNCTIONALS
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 <5a(/) = / ( a ) ,
/€^(C"),
then ^f^(C) = (1 ~ (^5 C))~^- To generalize this example, we need a lemma: L e m m a 4.7.6. If a o , . . . , a^ G C^ and k
(4.7.12)
M / ) = /,
/(' f{Ttjaj)dto...dtk.u
feA{C^),
yk
where S^ is the k simplex { ( t o , . . . , tk); to > 0 , . . . , t^ > 0, ^^Q tj = 1} C R ^ + S then
where a = ( a o , . . . , ak) is a multiindex
with length \a\ = J2Q aj.
Proof. Since
the statement follows from k
I ri(^? /^iO ^^0 ... dtu-i = l/(|a| + A:)!,
^fs5 ^
0
which can be proved by induction with respect to k. In fact, if we set tj = Sj{l - to) ioT 0 < j < k then the integral is equal to
/ (l-to)l^l'+^-4^V^o!rfto /
Jo
r[{s;^/aj\)ds,...dsk-u
75^-1 1
\a\'^Taj. 1
The integral with respect to to is (|a|' -\-k — l)\/{\a\ + fc)!, which completes the inductive proof.
308
IV. PLURISUBHARMONIC FUNCTIONS
With fi defined by (4.7.12) we now set for / G ^ ( C ^ ) k
k
/^ao,...,afc = n ^ - ^ ~ {d^z))fl, 1
t h u s Mao,...,afc ( / ) = / ^ ( J J O ' + 1
where {z, d)f stands for z i-^ {z, d)f{z) = Yl^ ^j^fi^)/^^j' by Euler's homogeneity relations that A; Aao,...,a*(C) = flu 1
{^^d))f),
Then it follows
k + {(,d))m
=
^n(a,-,C>"VI«|!a 0
The Fantappie transform is obtained by dropping the denominator, so k
(4.7.13)
Mfo,...,a.(c) = n(i-(«^'C)r'1
The analytic functional /iao,...,ak is obviously carried by the convex hull of ao5 • • • 5 CLk, but as in the example (4.7.2) (corresponding to A: = 1), there are many other carriers. L e m m a 4.7.7. The analytic functional //ao,.-,afc satisfying (4.7.13) is carried by a compact subset of any open C convex set X containing a o , . . . 5 afc. For any compact set K C X and every k > 0 there is another compact set Ki C X and a constant C such that when ao? • • • 5 ^fc ^ K then |/Xao,...,a.(/)|
feA{C-).
Proof. We shall prove by induction with respect to k that there is a compact set Ki C X and a constant C, both depending on k and a multiindex a = (ofo,..., a^), such that for arbitrary OQ, . . . , a^ ^ K k
I
f{yZtjaj)edto...dtkA
feA{C^).
For a — 0 this proves the lemma, but the more general statement is required for the inductive proof. If we set ^j = ^j^ J
tk-i -^tk = Sfc-i, tk-i = Sfc-iT, tk = Sfc_i(l - r ) ,
then D(to,..., tk-i)/D{so,..., Sk-2, r) = Sk-u so with afc_i(r) = rak-i (1 - T)ak and (3 = (ao, • • •, <^fc-2,1 + «fc-i + «fc) we have
/ = /
f{Sltjaj)edt^...dtk-i s^dso-"dsk-2
/(y;s,a^+5,_iafc_i(r))r"^-(l-r)"^dr.
+
ANALYTIC FUNCTIONALS
309
Since X is C convex we can find a C^ curve [0,1] 3 a \-^ r(cr) G C with r(0) = 0 and r ( l ) = 1 such that T{a)ak-.i + (1 - r{a))ak e X,0 2£ for every ( e X* and some e > 0 but |1 — (a, 0 1 < (S in a neighborhood of (Q. By the Borel-Lebesgue lemma we can therefore find aj E X and 6:^ > 0, j = 1 , . . . , iV, such that X* C {C € a;;|l - (a,-,C)| > 2£,-,i = l , . . . , i V } C wi = { C € a ; ; | l - ( a j , C > | > ej,j = 1,... ,N}
^w.
We may assume that a i , . . . , a„ are linearly independent. If we set $(C) = ( $ i ( C ) , . . . , ^ N { 0 ) e C ^ ,
$,(C) = (1 -
{aj,C))-^
then $(C) is analytic in a neighborhood of cJi", and the range is an analytic manifold since the equations ^j(C) = '^j? i = I5 • • • 5 ^5 can be solved as (j = Z^fc=i ^jk{^ ~ '^k^)' Writing $(C) = w we obtain ioi n < j < N that l/wj is affine linear in 1/wi,..., 1/wn^ so Wjqj{wi,..., w^) = Wi- "Wn for some polynomials QJ of degree n — 1. Set Gj{w) = Wjqj{wi,... ^Wn) — wi- • -Wn for j < n < N. Then the hypotheses of Theorem 4.2.12 are fulfilled with X replaced by the polydisc D = {weC'';\wj\
=
l,...,N},
the functions Gj just defined, and a neighborhood Y oi^{uJi) where u{w) = f{() remains analytic if (j = Yl^=i^3k{^ — '^k^)^ J ~ l ? - - - , ^ - Hence there is an analytic function F in D such that f{() = F($((^)), ( G uJi.
310
IV. PLURISUBHARMONIC FUNCTIONS
If Rj < l/sj^ then Cauchy's integral formula in the polydisc D, that is, repeated application of the elementary Cauchy integral formula in each variable, gives when \wj\ < i?j for jf = 1 , . . . , AT, that F{w) is equal to
(27r)-^ / • • • f F{Rie'^\
... ,RNe'^^)f[{l
- WjR-^e-'^^)-Uei
• -' dON-
liw = $(C) and C G X* then \wj\ < l/2ej. When l/2ej < Rj < l/sj it follows that /(C) in a neighborhood of X* is a superposition of the functions N
(4.7.14) C ^ n (1 - ^7'e'''^(l - («..C))-')~' 1
Now (1 - R-^e-'^^)-^aj
G X , for
|1 - {ajX) - R-'e-'^^\
> 2ej - Rj'
> 0,
if C E X*.
If we expand the last product in (4.7.14) it follows from (4.7.13) and Lemma 4.7.7 that the function (4.7.14) is the Fantappie transform of an analytic functional carried by a fixed compact subset of X and with a fixed bound, so it follows that / is the Fantappie transform of a functional in A'{C'^) carried by a compact subset of X . It just remains to prove that X is a Runge domain, for that implies that wA'(X) —^ A^C^) is injective, hence that the Fantappie transform is injective. For the proof we first observe that since X* is connected by Corollary 4.6.10, the functions (4.7.15)
X3^H^(1-(^,C))"^
are in the closure of A{C^) in ^ ( X ) for every ( e X*. In fact, this is true when C — 0- For a fixed compact subset K oi X the expansion oo
i=o
converges uniformly in a neighborhood of if if |C' — ("I is smaller than some number independent of ( E X*. If (4.7.15) is in the closure of ^(C"^) in C{K) it follows that this is true with ( replaced by ('. Hence it suffices to
ANALYTIC FUNCTIONALS
311
show that polynomials in the functions (4.7.15) and the coordinates Zj are dense in A{X) on K. Choose M so that \zj\ < M for j = 1 , . . . , n if .^ E if. As at the beginning of the proof we can then find points C,j G X* and £j > 0, n < j < N, such that K C {z e C-^Zjl < MJ
= l , . . . , n , |1 - {z,(j)\
> Sj.n < j < N} ^ X.
Another application of Theorem 4.2.12 shows that if / G A{X) is an analytic function F in the polydisc {w G C^]\wj\
< M , l < j < n , \wj\ <e~^,n
then there
< j < N}
such that /(z) = F ( ^ i , . . . , ^ „ , ( l - ( z , C „ + i » - \ . . . , ( l - ( z , C i v ) ) - ' ) when ;$; 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 ^^i-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
fJ^a,b{9f) = lini V
fJ^aMzj - 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
IV. PLURISUBHARMONIC FUNCTIONS
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 f^aM)= 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.
ANALYTIC FUNCTIONALS
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)|
C^C^,
then by the plurisubharmonic extension of Theorem 3.4.4 ]h^t-Hog\M{tC)\ t—^oo
is almost everywhere equal to its upper semi-continuous regularization PM{0^ which is a positively homogeneous plurisubharmonic function, of degree 1, which we mean tacitly in what follows. It is called the indicator function of M. By Theorem 3.2.13 the inequality (4.7.5) is equivalent to (4.7.5)'
PM(C)<^K(C),
CeC",
so Theorem 4.7.3 states that /x is carried by the convex compact set K if (4.7.5)' is valid with M = jl. The indicator function is convex when n = 1. In fact, when ( = ^i+i^2^ Ci > 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'PMi^i
+ ^ 6 ) = (
t2^i)^q"iC2/^i)/ei.
1
Hence Apu = (^1 + ^ i ) ^ " ( 6 / 6 ) / ^ i so q" > 0 since ApM > 0, and this implies that PM is convex in the half plane where ^i > 0, hence in every half plane. As a limiting case we see that pM is also convex (and homogeneous) on lines through the origin, so pM is a convex function in C = R^. In particular, PM{0+PM{-0>O. which is also true for n > 1 since the restriction of PM to a complex line through the origin is then subharmonic and positively homogeneous. Thus, sup
PM(C)
+ ipf
ICKi
so PM is locally bounded below also.
ICKi
PM{0
> 0,
314
IV. PLURISUBHARMONIC FUNCTIONS
T h e o r e m 4.7.11. Under the hypotheses in Theorem 4.7.3 the intersection of all convex carriers of ji is (4.7.18)
{z € C " ; Re{z, () < puiO
VC e C " } ,
where PM is the indicator function of M = fl. It is not a carrier of ji unless PM is convex. Proof. If K is a convex compact carrier then pM < HR by Theorem 4.7.3, so Re{zX) < P M ( C ) , C ^ C^, impHes {z,C) < HK{0, C ^ C^- Hence z G iiT, so the set (4.7.18) is contained in every convex carrier. Suppose now that 0 7^ Re(2;o,Co) > PM{(O) for some Co ^ C'^? ^o ^ C^, hence (o ¥= 0. Choose A so that PM{CO) < A< ReizoXo), and set for some B > 0 (4.7.19)
ff(C)-ARe(zo,C)/Re(zo,Co) + 5(|C-CoRe(C,Co)/|Con + max(0,Re(C,-Co))).
This is a positively homogeneous convex function so H{() = HK{0 for some compact convex set K. We have -H'(Co) = A > PM{(O)^ SO P M ( C ) < H{() in some (conic) neighborhood of (Q. Outside such a neighborhood the second term in (4.7.19) is > cB\(\ for some c > 0. If 5 is sufficiently large it follows that pM < H = HK, SO fi is carried by K. However, HK{CO) = A < Re{zoXo)^ so ZQ ^ K, which means that ZQ is not in the intersection of convex compact carriers. If the compact convex set K defined by (4.7.18) is a carrier then HK{C) ^ PM{0 < HK{0^ SO PM is equal to HK^ which completes the proof. In the one-dimensional case we have a unique convex compact minimal carrier, but that is not true when n > 1: E x a m p l e . If//(/) = / | , | < i , , e R " / ( ^ ) <^^^ f ^ -^(C"), then M(C) - A(C) is a function of (C, C) by the rotational invariance, and
M{t, 0,..., 0) = I
e*«^ (1 - e?)("-^)/2C„_i d^i = 2("-i)/2r((n + l)/2)C7„-ie*r("+^)/2(i +
0{l/t))
as i -^ oo while | argt| < 7r/2 — e for some e > 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 C^iV) is G subharmonic in the sense ofDeRnition 5.1.1, then VQ = {u"{0)] u EV} is a convex conic subset ofS'^iV) 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 C^iy) 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 C^iV) 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'^ok^j^k,
v{^) = Y^Vjk^j^k,
then {u,v) =
"^UjkVjk.
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
V. CONVEXITY WITH RESPECT TO A LINEAR GROUP
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 ^i ==•••== ^^^ = 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&<^i^2 + c^l belongs to V^^ then V^ contains a^l and c^f- If both are zero then 6 = 0, and if just one is zero then 6 = 0 also. Thus the minimality condition (c)' is only satisfied by the non-negative multiples oi^\ or those of ^l? ^^^ they are not positive definite. The corresponding space V consists of the functions which are convex with respect to xi or those which are convex with respect to X2' On the other hand, if we let G consist of the unit matrix only, then we can take V^ generated by ^1+^25 which gives the subharmonic functions in R^. Thus there may exist a positive Vo even if G does not act irreducibly on V. Later on we shall give a criterion for the existence of a positive VQ. P r o p o s i t i o n 5.1.4. UVQ is positive, then one can choose a Euclidean metric in V such that V consists of all u G C^{V) for which nog is subharmonic for every g E G. Proof. Take e ^ VQ positive definite, and let e be the dual form in V. li u E V then 0 < {u^'{x),s) = Au, where A is the Laplace operator corresponding to e, which proves that u is subharmonic, and so is TX o p by condition (ii) if ^ G G. The maximality condition (iv) shows that V consists of all u G C^ such that ^ o ^ is subharmonic for every g G G. Proposition 5.1.4 gives a rather implicit description of V. To clarify the properties of functions in V we shall now develop arguments analogous to the proofs of Theorems 3.2.28 and 4.1.7 by studying limits of the elements in G. Denote by L{V) the set of all linear transformations in V, and define norms with respect to a fixed Euclidean metric form e in y .
SMOOTH FUNCTIONS IN THE WHOLE SPACE
319
L e m m a 5.1.5. Let G be the closure in L{V)\{0} of{ag]0 ^ a eR^g G}. Denote the minimum rank of elements in G by r, and set (5.1.4)
e
WG = { 7 ^ ; 7 e G , r a n k 7 = r}.
Then WQ is a compact subset of the Grassmannian &r{V) of r-dimensional subspaces of V, and there is a constant C independent ofWe WQ such that (5.1.5) IfW
C-%x\\/\\gy\\
eWa
< \\x\\/\\y\\ < C\\gx\\/\\gyl
ifx,y
€W,
geG.
and 7 G G, then jW € WG or jW = {0}.
Proof. First note that the set F of all 7 G G with rank r and norm 1 is compact. Hence there is a constant C such that ||x||/C < ||7x|| < \\x\\, when X is orthogonal to Ker 7 and 7 E F. If ^ G G we obtain \\x\\/C < \\agjx\\ < \\x\\ for the same x if a E R is chosen so that ||a^7|| = 1, for then we have agj E r and the kernel is equal to that of 7. These estimates imply hl^W/hiyW
< C\\x\\/\\y\\ < C'\\jx\\/\\jy\\, Il52:|l/||ffy|| < C2||a:||/l|j/||,
iix,ye iix,yeW
(Ker7)^,
hence
= jV.
This proves (5.1.5). The last statement is clear, for if 7 , 7 ' E G and rank 7 ' = r, then rank 7 7 ' < r so the rank is either 0 or r. For W E W G and ^ E G let a{g,W) be the positive real number such that g maps the Lebesgue measure defined by e in PF to a(^, Wy times that defined by e in gW. If a linear map T in R ^ preserves Lebesgue measure, then the norm of T and the norm of T~^ are > 1, and if the product s u p x , y ( l | r X | | / | | X | | ) ( | | y | | / | | r y | | ) of the norms is < G, then ||T|| < G and ||T-i'|| < G. Hence it follows from (5.1.5) that G-2 < e{a{g, W)gx)/e{x)
< C\
xE W.
If Wi,W2 E WG we shall denote by 6 ( W i , W2) the set of limits of maps Wi 3 X t-^ a(^, Wi)gx E gWi as gWi —^ W2; a priori it might be the empty set if W^i 7^ 1^2.
320
V. CONVEXITY WITH RESPECT TO A LINEAR GROUP
L e m m a 5.1.6. (5{Wi,W2) is for arbitrary Wi, W2 E W a compact set of linear isomorphisms Wi -> 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,W^i), 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 cy^i ^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 ^ T^o 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 a^e 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
V. CONVEXITY WITH RESPECT TO A LINEAR GROUP
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, M^i), 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 <8(VK,W) and qw is invariant under (5{W,W)K So far we have proved the uniqueness of qw, but it is only defined when Wi is in the projection WQ if K in WG- By the minimality condition (c)', VQ is the closed convex hull of the G invariant cone {aq^] g eG,a>0}
C {aqw,; a > 0, P^i G W ^ } -
As at the beginning of the proof it follows that every q E VQ \ {0} can be written as a sum q = 2_] ^jQWj,
where Wj G WQ and Xj > 0.
For any W G WG we can by (2) choose q E VQ with radical W, and since the codimension of the radical of q is minimal it follows that the radical of qw is equal to W when Xj > 0. Thus W is equal to Wj which proves that WQ = WQ. When 1^ == W^ we conclude that every q G Qw is a multiple of qwi so qw was in fact uniquely determined. This completes the proof of (3)-(6). To prove (7) assume that jV C VQ for some 7 G G, where Vb is a G invariant linear subspace of V. Thus gjV C VQ SO Ker7*^* contains VQ for every g. By the minimality condition (c)' the closed convex cone generated by the forms e oj^ o g^ is equal to VQ , so VQ is contained in the radical of every form in VQ, in particular e. Thus VQ = {0}, and VQ = V. The proof is complete. The reason why we included condition (7) is that it suffices to guarantee the existence of a positive VQ: T h e o r e m 5.1.8. Assume that G satisfies condition (7) in Theorem 5.1.7. Then one can find a positive convex cone VQ C S^{V') satisfying (a)-(c). In fact, ifW G WQ then there is precisely one such VQ for every positive definite quadratic form ew in W which is invariant under (5{W, W)
SMOOTH FUNCTIONS IN THE WHOLE SPACE
323
such that VQ contains the dual form of ew, so Vo depends on v ~ 1 parameters if the action of &{W^ W) on W decomposes W into v irreducible subspaces. Proof. Assume that G satisfies condition (7) in Theorem 5.1.7. Let e be a positive definite form in V and let Qe be the closed convex cone generated by eog^ when g E G. As in the proof of Theorem 5.1.7 it follows that all elements q ^ 0 in Q^ are finite sums
^E^°^ where jj E G. The radical of q is then contained in that of e o 7J for j = 1 , . . . , J . Hence the intersection R of the radicals oi qo g^ when g E G is contained in the intersection of the radicals oieojjog^ for any g G G. The annihilator VQ C V oi R is invariant under G and contains the range of 7^, so condition (7) gives that VQ = V. Hence the closed convex cone generated by qo g^ contains positive definite forms. Any minimal G* invariant closed convex cone C Qs is therefore dual to a convex cone Vo C S^{V') satisfying conditions (a)-(c) and which is positive. Let W E WG' Theorem 5.1.7 shows that for every positive convex cone Vo C S^{V') satisfying (a)-(c) the dual cone PQ is generated by the forms Qw o 9* where g E G and qw is the dual of a positive definite form in W invariant under (&{W^ W). Conversely, if we choose such a form ew, then the forms cwi defined in each Wi C WQ by (6) and (5) converge to ew as Wi -^ W in view of the invariance of ew under (3{W, W). Hence the closed convex cone C S'^{V) generated by the corresponding qw contains no forms with radical W° except the multiples of qw- Any smaller invariant cone satisfying (a)'-(c)' must therefore also contain them, in view of Theorem 5.1.7. This proves the minimality of the cone and completes the proof. Corollary 5.1.9. There is a unique positive convex cone Vo C S'^{V') satisfying (a)-(c) if and only if condition (7) in Theorem 5.1.7 is fulfilled, and (3{W, W) acts irreducibly in W when W G W G In the remark given after Definition 5.1.3 we gave an example where condition (7) is not fulfilled and the sets (5(Wi,W2) may be empty. We shall now give an example where 6(Wi,W^2) is never empty although (7) is not valid. To do so we take again V = K^ and let G be the set of upper triangular matrices 9 = [ r. 1 1 • The elements in G of rank 1 are proportional to I
n ) ' ^^ ^^^ range W is the xi-axis. This is a G
invariant space, and 6(VK, VK) is the identity, but (7) is not valid.
The
324
V. CONVEXITY WITH RESPECT TO A LINEAR GROUP
space V must be the space of functions which are convex with respect to Xi.
We shall now list some examples which follow immediately from Theorem 5.1.7. The verification is left as an exercise: 1. liV = R"^ and G = GL{n, R ) , then V is the space of convex functions. 2. li V = R^'^ and G is the linear symplectic group, then V is also the space of convex functions. 3 . li V = H^ and G is the group preserving a non-degenerate indefinite quadratic form Q, then V is the space of functions which are convex in all directions y with Q{y) = 0. If Q = | x p , on the other hand, then V is the space of subharmonic functions. 4. liV = C^ and G = GL{n, C) is the complex linear group, then V is the space of plurisubharmonic functions. The same is true if G is the complex linear symplectic group. 5. If G is the group of complex linear transformations preserving a nondegenerate quadratic form Q, we obtain the space of functions u such that C 3 w y-^ u{wz) is subharmonic if 2: G C^ and Q{z) = 0. The same is true if we replace Q by a non-degenerate indefinite Hermitian form. However, ii Q = \z\'^ then V is the space of subharmonic functions in C^ = R^'^. Note that in all these cases V is uniquely determined because (5(PF, W) acts irreducibly onWe WG5.2. General G subharmonic functions. Let Vo be a convex cone C S^{V') satisfying the conditions (a)-(c) in Proposition 5.1.2, and assume that Vo is positive in the sense of Definition 5.1.3. Then a function u € C^{V) is in the corresponding space V of G subharmonic functions if and only if u''{x) £ Vo for every x £ V. By Theorem 5.1.7 this is true if and only if {u''{x), Q'ly) > 0 for every W G WG- Recall that qw is the dual of a positive definite quadratic form ew in W. The scalar product {u"{x), qw) is the trace of the quadratic form u"{x) restricted to W^ that is, the Laplacian oiW 3 y \-^ u{x + y) at 2/ = 0, taken with respect to the metric form ew in W. Thus V consists of all u G C^iV) such that for every W G WG and
xev (5.2.1)
W 3y^u{x
+ y)
is a subharmonic function oiy with respect to the metric ew- The definition can be applied to functions u G C^{X) when X is an arbitrary open subset of X , and more generally we define: Definition 5.2.1. Let Vo be a convex cone C S'^{V^) satisfying conditions (a)-(c) in Proposition 5.1.2, and assume that Vo is positive in the sense of Definition 5.1.3. Then we define V{X) for any open set X C F as the
GENERAL G SUBHARMONIC FUNCTIONS
325
set of all upper semi-continuous functions in X such that (5.2.1) is subharmonic with respect to the form ew in Theorem 5.1.7 in [y E W] x-\-y E X } , for every x eV and W G WQliu £ C'^{X) it is clear in view of Proposition 3.2.10 that the definition is equivalent to {u"{x)^ qw) > 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^^e V{xJ), x^ = {xe V',x-supp(p c x}, ifo <^e C^{V). (iii) There is a sequence cpj G CQ^{V) such that (pj -^ S in V as j -^ oo, supp (fj is contained in any given neighborhood of 0 for large j , and u^ip,eV{X^,). (iv) Avr^ = ('^''j^iy) ^ 0 ^^ the sense of distribution theory for every W eWG(v) For every q E VQ we have {u",q) > 0 in the sense of distribution theory (vi) With respect to a Euclidean metric in V with dual in VQ the distribution u is defined by a function U such that U o g is subharmonic in g~^X for every g G G; this determines U uniquely Proof, (i) =^ (ii): Let u be defined by a function U G V{X), which is thus in L J Q ^ ( X ) and locally bounded above. If x e V and W G WG then u^ (p e C^ and {U * ^){x -\-w) =
u{x -hw-
y)(p{y) dy,
326
V. CONVEXITY WITH RESPECT TO A LINEAR GROUP
which is a subharmonic function oi w G W since mean values with respect to w can be calculated under the integral sign, (ii) = ^ (iii) is trivial, (iii) ==^ (iv): Since differentiation is continuous in the sense of distribution theory it suffices to show that Awu > 0 if n G C^ fl V{X). As already pointed out, this follows from Proposition 3.2.10. (iv) = > (v) by (4) in Theorem 5.1.7. (v) =^ (vi): Let e be a Euclidean metric form in V which is dual to a positive definite form e E VQ. Then AeU > 0 for the corresponding Laplacian, by (v), and so is (Ae(iA o g)) o g~^ which is the Laplacian corresponding to the form sog^. Hence it follows from Theorem 3.2.11 that u o g is defined by a unique subharmonic function Ug. Since every point is a Lebesgue point for a subharmonic function, it follows that Ug = U o g where U — Uu, so U o g is subharmonic for every g E G. (vi) = > (i): Suppose that [/ is a subharmonic function in X such that U o g is subharmonic in g~^X for every g e G. Let 0 < cp e CQ^ be rotationally symmetric and let J (p{x)dx = 1. Set (fs{x) — ^p{x/8)/6'^ and Us = U^(p6, where ^ > 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's^e 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\
/
\u{wi
+ • • • -h Wk)\^ dWi • • • dWn < OO
•/max \wj\
if Wj G Wj^ Wj G WG and dwj is the Lebesgue measure in Wj^ for j = 1 , . . . , fc. As in the proof of Proposition 5.2.2 we can choose W i , . . . , Wk so
GENERAL G SUBHARMONIC FUNCTIONS
327
that they span V^ and then it follows that u € L\^^ in a neighborhood of the origin. The rest of the theorem is proved in the same way, and we leave the details for the reader. (When r = 1 then u is convex in the directions The estimates in the proof of Theorem 5.2.4 are uniform in u when u has a fixed upper bound and a fixed lower bound at one point. Hence the proof of Theorem 4.1.8 also gives: Theorem 5.2.5. Theorem 3.2.13 is also valid for functions in V{X), with Uj —^ u in L^^^{X) for any p G [l,r/(r — 2)) ifr > 2, uniform convergence ifr = 1; and Uj -^ u' in L\^^{X) for any p G [l,r/(r - 1)). Here 2/(2 - 2) and 1/(1 — 1) should be interpreted as -foo.
CHAPTER VI
CONVEXITY WITH TO DIFFERENTIAL
RESPECT OPERATORS
S u m m a r y . In Section 6.1 we sum up with brief indications of t h e proofs some facts on t h e open sets in R ^ where a differential equation P{D)u = / with constant coefficients can always be solved. Depending on whether / is allowed to be an arbitrary distribution or a C°° function (or a distribution of finite order), we get two classes of admissible open sets depending on P . Those which are admissible for every P are precisely t h e genuinely convex sets. However, more general domains are admissible for individual operators P. In Section 6.2 we prove by methods close t o those used in Section 4.2 t h a t in a pseudo-convex open set in C*^ all equations of t h e form P{d/dzi,... ,d/dzn)u = f can be solved. In fact, we prove more general results for operators in a product space R*^ X C ^ which have this structure with respect to the complex variables. In Section 6.3 we pass to t h e existence of analytic solutions of equations of t h e form P{d/dzi,..., d/dzn)u = / in a pseudo-convex open set C C"^ where / is analytic. We show t h a t it is precisely in the C convex sets t h a t a solution exists for arbitrary P and / .
6 . 1 . P - c o n v e x i t y . Let P{D), where D = {—id/dxi,..., —id/dxn) = —id/dx be a partial differential operator in R"^ with constant coefiicients, and let X be an open set C R'^. A basic question in the study of P(D) is whether the differential equation P{D)u = f in X has a solution u for every / . The answer depends of course on what conditions one puts on u and on / . By a simple explicit construction (see ALPDO Theorem 7.3.10) one can show that there exists a fundamental solution E^ that is, a solution E G VpCR^) of the equation P{D)E = 6. (Here Vp denotes the space of distributions of finite order.) This implies that P{D){E "^ f) = f for every / G 6''(R^). For every / G V'{X) which can be extended to a distribution F E E'^RP') it follows that the equation P{D)u = f is satisfied in X by the restriction of E * F to X , and u G C ^ ( X ) if F G C ^ ( R ' ' ) . Thus obstacles to solvability must come from the behavior of / at dX or at CXD. The geometrical properties of dX are then important. D e f i n i t i o n 6 . 1 . 1 . An open set X C R'^ is called P-convex for supports if to every compact set K C X there is another compact set K C X such
P-CONVEXITY
329
that (6.1.1)
supp^ c k
live
C^{X)
and snpp P{-D)v
C K.
This is Definition 10.6.1 in ALPDO, and in Section 10.6 there we proved: T h e o r e m 6.1.2. The following conditions on the open set X C R'^ and P{D) are equivalent: (i) The equation P{D)u
= f has a solution u G C^{X)
for every
f e C-(X). (ii) The equation P{D)u = f has a solution u G V'{X) for every f G C^{X). (iii) The equation P{D)u = f has a solution u G V'p{X) for every
f e v'p{x). (iv) X is P-convex for
supports.
That (i) = > (ii) and that (iii) = ^ (ii) is trivial, and it is easily proved by functional analysis that (ii) = > (iv) (Theorem 10.6.6 in ALPDO). To prove that (iv) =^ (i) and that (iv) =^ (iii) (Theorem 10.6.7 in ALPDO) one first uses the fundamental solution to construct a sequence Ui, solving the equation near Kj^ for an increasing sequence of compact subsets K^, of X increasing to X . By an approximation theorem which follows from (iv) one can add to u^, a solution of the inhomogeneous equation to make Ui, — Ujy-i so small in J^^_2 that u = lim^_,oo '^i^ exists; it is then a solution of the equation P{D)u = f. There is a condition closely related to Definition 6.1.1 which is relevant when / is allowed to be an arbitrary distribution, not necessarily of finite order: Definition 6.1.3. An open set X C R"^ is called P-convex for singular supports if to every compact set K C X there is another compact set K C X such that (6.1.2)
sing supp^; C K
ii v e S'{X)
and sing supp P(—i?)'y C K.
This is Definition 10.7.1 in ALPDO, and the following theorem is a combination of Theorems 10.7.6 and 10.7.8 in ALPDO: T h e o r e m 6.1.4. An open set X C R"^ is P-convex for singular if and only if P{D) induces a surjective map in V'{X)/C^{X).
supports
Together Theorems 6.1.2 and 6.1.4 show that the equation P{D)u = f has a solution u G V'{X) for every f G V'{X) if and only if X is P-convex
330 VI. CONVEXITY WITH RESPECT TO DIFFERENTIAL OPERATORS
for supports and for singular supports. However, since neither of the two Pconvexity conditions imphes the other, it is useful to discuss them separately rather than aiming directly for existence theorems in V'{X), Remark, liP{D) = Pi{D)P2{D) then X is P-convex for (singular) supports if and only if X is Pi-convex and P2-convex for (singular) supports. The proof is an obvious exercise. We shall not repeat the proofs given in ALPDO of Theorems 6.1.2 and 6.1.4 but shall devote this section to discussing the geometrical meaning of P-convexity for (singular) supports. The term "convexity" will be justified by the following results: P r o p o s i t i o n 6.1.5. Every open convex set X C R'^ is P-convex supports as well as for singular supports; in fact, ch(suppP(-jD)u) = ch(supp'u), ch(sing s\ippP{-D)u)
= ch(sing suppi^),
for
ueS'ilC), u G 5'(R^).
The simple standard proof using Fourier-Laplace transforms can be found in ALPDO, Section 7.3. We shall not repeat it here, for in Section 6.2 we shall prove stronger results in pseudo-convex domains using weighted L^ estimates. A converse of Proposition 6.1.5 will be proved below (Corollary 6.1.11), but first we shall show that P-convexity has the basic property of convex sets, that it is closed under intersection (cf. Proposition 2.1.3). First we need to establish a suitable equivalent form of the definitions: P r o p o s i t i o n 6.1.6. An open set X C R'^ is P-convex for supports if and only if the distance from supp-y to CX is equal to that from snpp P{-D)v to CX for every v G £'{X) (or every v G Cg^{X)). Proof. Assume that X is P-convex for supports. Let v G
CQ^{X)
and
let 6 — imn{dx{x)\x
G supp-u},
where dx{x) — min{|x - y\\y G CX}.
(Here | • | can be any norm.) Choose XQ G supp^; and h G R"^ so that \h\ = 6 3.ndxo-he dX. If 0 < t < 1 then vt = v{- + th) G C^{X), and XQ — th ^ supp-^t, so it follows from (6.1.1) that no compact subset of X can contain suppP{—D)vt for all t G (0,1). Hence {snppP{—D)v) — {h} contains some point in dX, so the distance 6' from suppP(—D)?; to dX cannot be smaller than S] it is trivial that 6' > 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) * (^e) 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
332 VL CONVEXITY WITH RESPECT TO DIFFERENTIAL OPERATORS
6.1.7) that P-convexity for (singular) supports implies that the minimum of dx in / is taken at an end point, which proves the stated quasi-concavity. Suppose now that dx is quasi-concave as stated, let v G CQ°{X)J and let xo G supp'y have minimal distance 6 to dX. If the distance from supp{P{—D)v) to dX is > 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)
for every t e R.
If P = P{d/dz) = P{d/dzi,..., d/dzn), then one can for every set K C X find constants CK stnd TK such that for T > TK, u ^ (6.2.2)
^rl^l
f\P^''\d/dz)u\^e^^^dX
< CK f \P{d/dz)u\^e^^'^
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]
=
2Td^ip/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'^i^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\^e'^^'^ = 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\^e^''^dX
< C j ^ r l ^ l j \P^'"\6)u\^e^^'^
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\'^e^^'^
dX
dX-^-C^T^""^-^
I
\P^'"\8)u\^e^''^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'^ip/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\^e^^^
dX
[ \P{d/dz)u\''e''^'^
dX.
AN EXISTENCE THEOREM IN PSEUDO-CONVEX DOMAINS
335
When T > (2C)2 we obtain
Combining this estimate with (6.2.5) appUed to each P^^\ we obtain the estimate (6.2.2). Next we shall prove a result which implies P-convexity for singular supports. If 16 G £'{^^) and ip G C o ° ( R ^ ) , we shall say that u has ^p derivatives in L^ and write u G H(^^^^ if T^u = t^{x,D)u
G L^
when t^{x,0
= (2 + |^|2)^W/2.
(See ALPDO Section 18.1 for basic facts on pseudo-differential operators, in particular for the definition on p. 111:65 of the standard symbol classes S^.) The calculus of pseudo-differential operators gives t^^p{x^D)tip{x^D) = I d + P ( x , J 9 ) where R G 5"^+^ = f^^y^lS^, Iit^{x,D)u G L^ and s < (f{xo) then t-^p{x^D) G S~^ in a neighborhood of XQ, SO {ldi-\-R{x^D))u G jyJ^? in a neighborhood of XQ^ which proves that u G HY^^ in a neighborhood of XQ. On the other hand, if n G iJ}^? in a neighborhood of XQ and s > (P{XQ) then the fact that t^p{x^D) G i?^ in a neighborhood of a;o shows that t(p{x,D)u G iif/Q? = L^^^ in a neighborhood of XQ. Thus the condition u G H(^ is a natural extension of the conventional definition of the Sobolev spaces i?(s), which are the special cases with cp equal to the constant s. The condition u G Hip is not changed if cp is altered outside a neighborhood of sing suppii, so it is also applicable if ip is just a C°° function defined in a neighborhood of sing suppii. With this notation and N = 2n, R2"^ — C"^, we shall prove: P r o p o s i t i o n 6.2.3. Let cp G CQ^{C'^) be strictly plurisubharmonic the closure of an open neighborhood X of a compact set K C C^. u G £'{K) and P{d/dz)u G -H'((^), it follows that u G ^((^)-
in If
Proof. We shall construct pseudo-differential operators (5^, j — 1 , . . . , n , such that (6.2.6)
-d/dzjT;T^
=
T;T^6J
mod Op 5 " ^
where S~^ = n ^ 5 ^ and Op a denotes the pseudo-differential operator with symbol a. First note that T^T^^p = I - R^ where Rep G Op5~^"^^ = n ^ > _ i O p 5 ^ . We can choose U^ so that U^ - T.^Yl-^-^R^ G O p 5 ^ if /x > -AT - (y9, for AT == 1 , 2 , . . . . Then T^U^ - / G Op 5"°^, and since a left
336 VI. CONVEXITY WITH RESPECT TO DIFFERENTIAL OPERATORS
parametrix can be constructed in the same way it follows that U^T^ — I £ Op 5 - ° ° . (See the proof of Theorem 18.1.9 in ALPDO.) If we define 6j =
-u^u;d/dzjT;T^,
it follows that (6.2.6) holds. Since
Sj = -u^u;T;T^d/dz, - u^u;[d/dzj,T;T^] = -d/dzj
- U^U;[d/dzj,T;T^]
mod O p ^ - ^
and the leading symbol of T*T^ is (2 + iSp)"^, it follows that the symbol of 6j + d/dzj is -dcp/dzj log(2 + | S p ) mod 5~^"^^, where S denotes the variable in the dual of R^^ to avoid confusion with the real part of the complex variable. Assuming at first that u E CQ^{K) we shall apply the identities in Appendix B. With L^ norms and scalar products we have by (6.2.6) \\T^P{d/dz)uf
= {T^P{6)P{d/dz)u,T^u)
+
{Riu,u),
where Ri e OpS~^. The operators 6j may only commute modulo Op S~^ so we shall apply the identities of Appendix B to the algebra of pseudodifferential operators mod Op S~^. The symbol of [Sj^ d/dzk] is equal to d'^ip/dzjdzk log(2 + iSp) mod 5 " ^ + ^ By Proposition B.2 we can write
where Ta,j3{6,d/dz) and d/dzk. Hence (6.2.7) \\T^P{dldz)u\\^
is a sum of products of commutators formed from 8j
= Y,iT^^ap{^.d/dz)P^''\6)u,T^P^^\6)u)
+ {R2U,u),
where i?2 ^ Op 5 " ^ . The symbol of any repeated commutator of [6j^d/dzk] with d/dzi or 8i is the sum of a function in S~^'^^ and log(2 + | S p ) times a function of z. Hence the symbols of the terms in ^a,(3{S-,d/dz) which are products of K commutators can be written as a function of z times (log(2 + |Sp))'^ plus a function in S~^^^. Denote by L the pseudo-differential operator with symbol (log(2 + | S p ) ) 2 , which is a convolution operator. Then it follows that (6.2.8)
\\T^P{d/dz)uf
< C J ] ||T^LI"IP^"\(5)^||2 +
{Rsu,u)
AN EXISTENCE THEOREM IN PSEUDO-CONVEX DOMAINS
337
where R^ G OpS~^. Here we have again used the fact that K < min(|Q;|,|^|) < (|a|-f |/?|)/2. When there is inequaUty then (|a| + | / ? | ) / 2 - ^ > | . We have also used that if A is a pseudo-differential operator of order < 0, then T^A = {T^AU^)T^ mod Op 5 " ^ , and that T^AU^ is a pseudodifferential operator of order < 0, hence L^ continuous. Let Ta,p{6,d/dz) be the sum of the terms in Ta,/3{S,d/dz) which are products of {\a\ -f \/3\)/2 first-order commutators, with the symbol of the true product replaced by the product of the symbols. Then Ta,i3iS^ d/dz) = 0 unless |a| = |^|, and the symbol of fc,,^(^, 9/9^) is 7c,/3(;2)(log(2H-|E;|2))l«l where \a\=ij.
\cx\ = \p\=/i
by Proposition B.4 applied with ^ o = Op 5 + ° / O p 5"^+°. This implies that for arbitrary functions u^ G CQ^{X) ^
/ \ua\'^dX
^
ja,pUaU^dX.
If we take Ua = xTcpL\''^P^'"\6)u, where u G C^{K) and x ^ C^{X) equal to 1 in a neighborhood of K^ it follows that for any 7/ > 0
+ rjY,\\T.,L^"^P^"\s)uf
is
+ {R4,r,u,u),
where i?4,-^ G Op S~^. (Note that L~^ is the sum of an operator in Op S~^ and one in Op 5^ with small norm.) With 7/ = | it follows from (6.2.7) that J2 ||T^i>l"IP^"^((5)7x||2 < 2C\\T^P{d/dz)u\\^ + {R,u,u), where R^ G O p 5 ~ ° ° . Combining this estimate with the estimate (6.2.8) applied to xL'^lii, with P replaced by P^^\d/dz)^ we conclude that (6.2.9) Y, | | T ^ i ' " I P ^ " H 9 / a ^ ) ^ f < C\\T^P{d/dz)uf + {R6U,u), u G C^{K), a
where i?6 G O p 5 - ° ^ . Assume now that u G £'{KQ) where KQ is in the interior of K^ and that P{d/dz)u G i?((p). Let xi ^ C^{K) be equal to 1 in a neighborhood of Ko, and let xo G CQ^{K^) be equal to 1 in a neighborhood of 0. Then
338 VI. CONVEXITY WITH RESPECT TO DIFFERENTIAL OPERATORS
'^T = Xi{^)Xo{D/T)U every N in suppdxi(6.2.10)
is in CQ^(K) and is 0{T~^) If we prove that
T^P{d/dz)ur
=
with all derivatives for
T^P{d/dz)xi{z)xoiD/T)u
is bounded in L^ as r ^ oo, then (6.2.9) applied to u^ shows that r^Ll"lp(^)(a/a^)i6 e L ^ w h e n a is chosen so that P^") is a constant, this is more than claimed in Proposition 6.2.3. To estimate (6.2.10) we choose X2 ^ CQ^ equal to 1 in a neighborhood of KQ SO that x i == 1 in a neighborhood of suppx2- Then we can replace u by X2'^5 and the symbol of [T^Pid/dz),xi{z)]xo{D/T)x2{z) is uniformly bounded in 5 " ^ , so we can move the cutoff function xi to the left and then ignore it. Since P{d/dz) and XO{D/T) commute we can then interchange them. The symbol of [T^P^XO{D/T)] is | i ( 2 + | H n W 2 iog(2 + \E\'){^',
x'oi^M/r)
plus terms of lower order, so the product to the right by U^p is uniformly bounded in O p 5 ^ , for ((/}',Xo(^/'^)/'^) is bounded in S~^. Since R^ = I — U^T^ is of order — oo, and [T^,Xo{DlT)]P{dldz) = {[T^,Xo{DlT)]U^)T^P{dldz)
+
[T^,xo{DlT)]R^P{dld-z),
it follows that T^pP{d/dz)ur is bounded in L^ as r —>- oo, for XO{D/T) is uniformly bounded and T^pP{d/dz)u G L^ by hypothesis. This completes the proof. Proof of Theorem 6.2.1. We proved already after the statement of Proposition 6.2.2 that X is P-convex for supports. Now suppose that u G £'{X) and that if t, so (f < t in sing suppix. If we apply this conclusion to P{~d/dz) instead, the P-convexity for singular supports is proved. As mentioned already, Malgrange has proved an existence theorem for arbitrary systems of the form P{d/dz) in pseudo-convex domains. In that generality pseudo-convexity is also necessary by a theorem of Serre [1] (see also CASV, Theorem 4.2.9). However, it is not clear if that is so when one just considers arbitrary scalar differential operators. There is definitely no analogue of Proposition 6.1.10 which would prove that, for we have:
AN EXISTENCE THEOREM IN PSEUDO-CONVEX DOMAINS
339
P r o p o s i t i o n 6.2.4. If P{D) = {D,t) + c where t G C ^ , c G C and T == {KQwt]w G C } lias dimension two, then an open set X C R ^ is P-convex for (singular) supports if and only if the boundary distance dx satisfies the minimum principle in the intersection of X with any parallel {xo}+T ofT, that is, (6.2.11) ^
^
infdx = m f d x , K
dK
ifKcXn {{xo} + T) is a compact set and dK is the boundary of K as a subset of {XQ} -f T. Proof. Replacing P by e~*^'^*Pe*^'^^ for a suitable 6 we can reduce the proof to the case c = 0. We can also assume that t = ^{i, — 1 , 0 . . . , 0), so that P is the Cauchy-Riemann operator in the variable z = xi -{• 1x2- Let y' — {ys^-' • ^VN) and let M be a bounded open set in R^ with interior points such that M x {y'} C X. If XM is the characteristic function of M, then the support and the singular support oi u = XM ^ ^{x' — y') are equal to M x {y'] but for P(-D)u they are equal to (dM) x {y'}. Hence (6.2.11) follows from Proposition 6.1.6 or 6.1.7, if X is P-convex for (singular) supports. The proof that (6.2.11) implies P-convexity for supports is essentially the same as the proof of Proposition 6.1.10, for a smooth function v such that Pv = 0 in a neighborhood oi u x {y^}, where LJ is an open connected set in R^, must vanish in another neighborhood if it vanishes in a neighborhood of one point in a; x {]/'}. The proof that (6.2.11) implies P-convexity for singular supports requires a similar result on propagation of regularity which is not quite as obvious. A proof can be found in ALPDO Theorem 26.2.3 or rather in Duistermaat-Hormander [1, Lemma 7.1.2], and another will be given below after Proposition 6.2.11. By the remark after Proposition 6.1.4 the proposition remains valid if we take P{D) = q{{D,t)) where g is a non-constant polynomial in one variable. To prove that pseudo-convexity is necessary for the conclusion in Theorem 6.2.1 one would therefore have to work with operators of higher order which do not only act along a complex line. We have no such results and do not rule out the possibility that pseudo-convexity could be replaced in Theorem 6.2.1 by the validity of (6.2.11) for all complex lines T. Remark. There are open sets in C"^ satisfying the condition in Proposition 6.2.4 which are not pseudo-convex. In fact, X = {z E C^; Re 2:1 > 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\^e^^'^dX
< CK I \P{L)u\^e^^'^ 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^^i 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^-i^o-^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 l^iP ^^^ J2^i^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^; (Rej^i,... ,Re2:^,Re2:^_|_i,Im2:7.+i,... .,lmZj,) G X}.
342 VI. CONVEXITY WITH RESPECT TO DIFFERENTIAL OPERATORS
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 <0 in K but cp > 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-^id/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 <0 in K2e, ^x'^e
(pe>lmX\
Ke,
> 0 in K,/2, A^/(^e > 0 in ^^72 \ ^ e -
We shall first construct such a function for a fixed x". Let X{x")
= {x';ix',x")
e X},
Ke{x") = {x';{x',x")
G if J ,
and recall that dx is Lipschitz continuous with Lipschitz constant 1. The open set X{x") \ Ke{x") has no component which is relatively compact in X{x"). In fact, if a; is such a component then du C Ke{x")^ so dx{x) > € and \x\ < 1/e on du x {x"}. Hence the same inequalities hold in a; x {x''}, by the minimum principle for d and the convexity of | • |. Prom Theorem 3.2.31 it follows that there is a C^ strictly subharmonic function ip in X ( x " ) such that cp < 0 in if7^/4 but (p > 1 in X{x") \ K^^/4^. Since \dx{x',x")
- dx{x',y")\
< £/4
if \x" - y"\ < e/i,
it follows that (p{x') has the required properties in
{ix',y")eK,/2;W'-y"\<e/4}. By a partition of unity in the x" variables we can find a function with the desired properties in ife/2. We extend it to all of X so that it remains > 1 inX\iC^. Composing (p^ with an increasing convex function we can make ip^ = 0 in K2e without affecting the other properties. A series ^ =
"^^3^2-
344 VI. CONVEXITY WITH RESPECT TO DIFFERENTIAL OPERATORS
will then have the required properties if aj is a positive sequence increasing sufficiently rapidly and (pi is modified to be strictly subharmonic in if 1/2. In fact, if ao = 1 then cpi + ai(pif2 is strictly subharmonic in if 1/4 and is > 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(^i) = {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:^ = ^o- Using Theorem 4.2.12 we can choose / G A{X) so that /(^i,>2^o) = 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 ) = "^{^i^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)
'^ajduldzj
=f
Then every
first-order
346 VI. CONVEXITY WITH RESPECT TO DIFFERENTIAL OPERATORS
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-^Ub^c-^Uc,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 ^a = 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 = {i^e A'{X);
\u{f)\ < sup | / | , / G K
AiX)}
348 VI. CONVEXITY WITH RESPECT TO DIFFERENTIAL OPERATORS
is compact for every compact set if C X . li v E MR
|i>(0| = Ke<-'<>)|<e^-(^),
then
CeC",
where HK{C) = ^^PzeK^^i^X) ^s in Theorem 4.7.3. li u = fi e A'{X), then P(C)/x(C) == />(C), which impUes (6.3.6)
|/x(C)l
P{-d/dz)fi,
CeC^
for some constant C independent of (. This is a very easy consequence of minimum modulus theorems such as Theorem 3.3.23 but follows also from the much more elementary result in Exercise 3.3.1 if the coordinates are chosen so that for example the highest power of Ci has a coefficient independent of the other coordinates. UK is convex it follows from (6.3.6) and Theorem 4.7.3 that /i is carried by K. More precisely, ii K C Ki C X and Ki is a neighborhood of K , then the proof of Theorem 4.7.3 shows that (6.3.7)
|M/)|
feAiX),
where Ci is independent of u. Thus fi belongs to the compact set in defined by (6.3.7) and
\fi{P{d/dz)f\<sup\fl
A\X)
feA\X).
K
It is mapped continuously by P{—d/dz) to another compact set in A\X). This completes the proof if X is convex. When X is only assumed to be C convex we must use the Fantappie transformation, which is much less convenient since the relation between the Fantappie transforms of /x and u = P{—d/dz)fi is not as simple as the relation between the Laplace transforms. We shall therefore interrupt the proof to study the one-dimensional case first. It is very elementary but already confronts this problem. From now on we assume that 0 G X , and at first we require that n = 1. Then there is for every / G A{X) a unique analytic solution of the equation P{d/dz)u = f inX with Cauchy data u{0) = ••- = u^'^-'^\0) = 0, where m is the degree of P. In fact, let E{z) = ^(f ^ d C , 27^^ /|^|=i^ P{Q
zee,
ANALYTIC DIFFERENTIAL EQUATIONS
349
where R is so large that P(C) ^ 0 for |C| > R- Then P{d/dz)E{z)
= ^
(f
27rz Jic\=R
e'^dC = 0,
f\(:\=R ^(C) "
I l/cm,
if i = m - 1,
if P(C) — CjnC^ is of degree m — 1. Hence
«w= r fiOEiz-odc, Jo
with the integral taken over a C^ curve from 0 to 2; in the simply connected and connected set X , is the desired solution, since ^0) Jo ix(^) (z) = / ( z ) E ( — ^ ) ( 0 ) + r
fiOE^^'Hz
- 0
d<:,
Jo which implies P{d/dz)u{z) = f{z) and u^^\0) = 0, j < m. Let us now translate this conclusion to the language with which we started the proof of Theorem 6.3.3. Hue A\C) is carried by a compact set if C X and u = P{—d/dz)fi where /x G v4'(C), then it follows that /x is carried by a compact subset of X. More precisely, let 0 G y ^ X be connected and simply connected with smooth boundary, for example the range of the Riemann mapping of the unit disc to X restricted to a smaller disc. If
k(/)l<sup|/|,
feA{X),
Y
it follows that
K^)|
geAiX),
Y
for ijL{g) — iy{f) if P{d/dz)f
= g^ and we can choose
Jo
E{z-09{0d(
as above. This implies that the Fantappie transform fi-^ can be analytically continued to {( G C; z( ^ I \/z eY}^ so we have proved:
350 VI. CONVEXITY WITH RESPECT TO DIFFERENTIAL OPERATORS
L e m m a 6.3.4. Let X C C be connected and simply connected, and let P be a polynomial ^ 0 . If fi E A'{C) and P{—d/dz)iJ. is carried by a connected and simply connected open set Y (G X, then the Fantappie transform fi^ can be continued analytically to {( G C;z( j^ 1 \/z EY}. To combine one-dimensional information from this lemma we need a classical lemma of Hartogs: L e m m a 6.3.5. Let Z C C"^ be an open set such that 0 G Z and ZQ — {w e C]W0 e Z} is connected and simply connected for every 6 e C^\ {0}. IfF is a function which is analytic in a neighborhood of 0, and ifw i-^ F{w6) has an analytic continuation to ZQ for almost every 9 G C^ \ {0}, then F can be analytically continued to Z. Proof. It suffices to prove analyticity in a neighborhood of any ^o ^ Z \ {0}. Then 1 G ZQ^^ and we can choose an analytic bijection cp of the unit disc D = {w £ C] \w\ < 1} on an open set (p{D) (E ZQ^ containing 1, such that (p{0) — 0. Then ^{D) (s ZQ for all 0 in an open connected neighborhood U oi OQ, for Z is open. When 0 G C/, it follows that D 3 w 1-^ F{(p{w)6) is defined and analytic, so we can form the power series expansion oo
0
for almost all 6. The convergence means that lim j ~ ^ log |aj(0)| < 0, for almost every 6 £ U. The coefficients aj{6) — {j\)~^d^F{ip{w)9)/dw^w=o are analytic functions in C^. If Uo is a neighborhood of 0 where F is analytic and bounded, then we can choose e > 0 such that (p{w)9 G UQ if \w\ < e and 0 e U. Thus \aj{0)\e^
0eU.
Hence it follows from Theorems 3.2.12 and 3.2.13 that for every compact set K C U and every 6 > 0 there is an integer J^^s such that j-Hog\aj{0)\
< 6,
ii0eK,
J>JK,S.
In fact, otherwise this would not be true for some convergent subsequence with plurisubharmonic limit u < 0 almost everywhere in U, hence < 0 everywhere in t/, which contradicts (3.2.7). Thus the power series is uniformly convergent when \w\ < e"^'^ and 0 E K^ which proves that F{(p{w)0) is an
ANALYTIC DIFFERENTIAL EQUATIONS
351
analytic function of {w^ 6) E D x U. This yields an analytic continuation of F to ip{D)U 3 00- Since Zg is simply connected for every 6 we get a unique analytic continuation to Z. End of proof of Theorem 6.3.3. Let us first recall that with the notation (6.3.5) we must show that P{—d/dz)A'{X) HMK is weakly compact. It suffices to prove that there is a fixed compact set Ki C X such that // is carried by K i if /x G ^ ' ( C ^ ) and P{-d/dz)fi G MR- In fact, ii K2 C X is a compact neighborhood of JCi, it follows then that
feA{X).
|M(/)|
The subset B of A'{X) consisting of all /x having such a bound and normed with the best C^ is a Banach space. Furthermore P{-d/dz)A'{C'^)nCMK is a closed subspace Bp of the Banach space CMK^ with the similar definition, for it consists of all u G CMK such that i>{()/P{—() is analytic, so it is closed by the first part of the proof. By Banach's theorem surjectivity of the closed infective map from B to Bp defined by P{—d/dz) will imply that there is a constant C such that {/x G 5 ; ||/X||B < C} is mapped by P{-d/dz) to a set containing P{-d/dz)A'{C'^) fl MK, and the weak compactness follows as in the convex case. Let if be a compact subset of X, let /x G A^C^) and let z/ = P{—d/dz)^ G MK' TO prove that /x is carried by a fixed compact subset of X we choose an arbitrary 0 G C^ \ 0 and let /x^ be the pushforward of /x by the map C"^ 3 2: H-> (z, 0) G C, that is, fi0{f) = f^ifi{;9))),
f€A{Xo),
Xe^{{z,9);zeX}.
By Proposition 4.6.7 Xg is C convex, that is, connected and simply connected. We have fleiw) = H0{e^-) = M(e<'"'^>) - A M ) ,
WEC,
(if{w) = Mil - w)-') = M(l - i^wO))-^) = M^M), when \w\ is small, and similarly for z/. From the fact that i>{() = P{()JJL{() it follows then that i>e{w) = P{w6)fjL0{w), that is, ug = P{—wd/dz)/i0. Now we have \Mf)\
< sup I/I,
Ke - {{z,e)-z
G K } (E X , , / G ^ ( C ) .
Ke
li P{w6) ^ 0, which is true for almost all 0, we conclude using Lemma 6.3.4 that ^I'Q{W) = /ji'^{w0) can be continued analytically to
{w e C]i - WW ^oyweK0}
= {weC]i-
{z,wO) ^^^ o Vz G K},
352 VL CONVEXITY WITH RESPECT TO DIFFERENTIAL OPERATORS
which is a neighborhood of {w;w0 G X * } . Hence Lemma 6.3.5 shows that /i"^ can be continued analytically to a neighborhood of X* independent of u. By Theorem 4.7.8 it follows that // is carried by a fixed compact subset of X , which completes the proof. As in Section 6.1 we could also show by essentially purely functional analytic arguments that a P convexity condition of the form (6.1.1) for carriers of analytic functional is necessary and sufficient in order that P{d/dz)A{X) = A{X). However, little seems to be known about the geometrical interpretation apart from Theorem 6.3.3 and the results to be discussed in Chapter VII, so we shall not develop this theme further here.
CHAPTER VII
CONVEXITY AND CONDITION
(.*)
S u m m a r y . Most of this chapter is devoted to local existence theory for analytic solutions of t h e differential equation du/dzi = f near a b o u n d a r y point of a strictly pseudo-convex domain. This may seem like a very special problem, b u t in fact, via t h e transformation theory of microlocal analysis, it is equivalent t o t h e central question of local solvability for microfunctions. This transformation theory is beyond the scope of these lectures, b u t we shall discuss t h e geometrical aspects in detail. T h e whole chapter is an exposition of t h e thesis of Trepreau [1], b u t the arguments have been rearranged in order t o postpone t h e reference t o microlocal analysis until t h e very end.
7.1. Local analytic solvability for d/dzi. Let X C C^ be an open set and let ZQ E dX. We shall denote by AZQ{X) the germs at ZQ of functions analytic in X , that is, the set of functions / analytic in XnU for some neighborhood U of ZQ, with functions equal in such a neighborhood identified. We want to decide when (T.1.1)
d/dz,:A,,{X)^A,,{X)
is surjective. This is a local version of the question discussed in Lemma 6.3.1. (See also the remark after the proof of Theorem 6.3.2.) However, the problem here is harder than in Theorem 6.3.2 for it refers to a single operator. Finding necessary conditions for local existence is also more difficult than for the global problem, and the conditions obtained are so weak that proving their sufficiency is also more complicated. On the other hand, we shall simplify by assuming that dX G C^ and that dX is strictly pseudo-convex at ZQ. The case where d/dzi is not tangential is elementary: L e m m a 7.1.1. Let X = {z e C-,g{z) < 0)} where Q e C^ and dg/dzi ^0 at ZQ. Then there is a fundamental system of neighborhoods Y of ZQ such that (i) the equation du/dzi = f has a solution u G A{X fl Y) for every f G A{X n y ) , thus (7.1.1) is surjective; (ii) ifv G A{X n Y) and {t,d/dz)v = 0 for some t G C^ then v(z) = v{z') for arbitrary z,z' e X HY with z - z' ^ Ct.
354
VII. CONVEXITY AND CONDITION (^)
Proof. We may assume that ZQ = 0 and write g{z) = - ReL{z) -f o{\z\) where L is complex Unear and dL/dzi / 0. Taking L as a new coordinate instead of zi does not change the operator d/dzi^ so we may assume that L{z) = z\. Then it follows from the implicit function theorem that X is defined in a neighborhood of the origin by x\ > '0(2/i5'2:2, • • • ,^n) where V^ G C^ and '^'(0) = 0. Set y^ = { z G C ^ ; | a : i | < 6,\yi\ < 6,\z2\ <6,...,\zn\
<6},
where Zj = Xj + iyj. For small 6 the intersection X DY^ is connected and simply connected, and remains so if Z2^...',Zn are fixed. Since (|^, Z2J. .. -tZn) E X nYs if \z2\ < 6, ... ^ \zn\ < S, and 6 is small enough, a unique solution u G A{X fl Ys) is given for / G A{X fl YJj) by
'^(^) =
/
f{Cl,Z2,...,Zn)dCi
with the integral taken along a curve in X fl Yi^. This proves (i). For small 6 the set X nV^ is starshaped with respect to 11 = (^, 0 , . . . , 0). In fact, if Xi = '0(2/1,2:2,... ^z^) and
sup IV'/^I + sup Idijj/dyi \ + 2 Y_^ sup \dilj/dzj \ < 1, 2
with the suprema taken for \yi\ < 6^ \z2\ < 6, ... , \zn\ < ^, then Axi + (1 - A)^ - ij{Xyi, \z2,...,
Xzn) > 0,
0 < A < 1,
for the derivative with respect to A is negative and the limit as A —> 1 is equal to 0. Hence z ^ Y^ n X implies Az + (1 — A)n E Y^ C^ X when 0 < A < 1 if (5 is small enough. If z^z' E Ys D X and z — z^ E Ct, then ?;(A^+(1-A)n) = '?;(A^'+(1-A)n) for small A > 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:),... ^i/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^^^i, ^ l ( ^ ) = M^) - \HnnZn " ^HjnZjZn. 3=1 i=i
With ijj[z) — {z\,... , ^ ^ _ i , z ^ + ^i('2^)) ^iid 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)^... ^ipn{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^d^ijJdzidzjd/dC^ i/=i
- dVdCidCj +
d^ilJi/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'^il' + ^(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)-\-Y^aj{zj
-\-CjZn))).
358
VII. CONVEXITY AND CONDITION (^)
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 ~ ^i- 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? + 2^a^-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^~ l^iP ^^^ ^^^ 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 < I xiiig condition will play a major role in what follows, beginning with the following result: P r o p o s i t i o n 7.1.4. If the open strictly pseudo-convex set X C C"^, n > 2, is defined at 0 by g < 0 where g e C^ satisfies (7.1.9), 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 —£
= -2xn - air^ -h r'^ -\- 0{TXn) + O{T'^ + X^) < 0
when (r^Xn) / (O5O) if ^ is small enough. If u is an analytic solution of du/dzi = f defined in {z G X; \z\ < 26:}, we obtain by integration over the interval between (±26:, 0 , . . . , 0, x^) ^ dr ; ^, / .^ 2Xn + aiT^
0 <Xn <£•
360
VII. CONVEXITY AND CONDITION (^)
The integral in the right-hand side tends to +cx) as x^ —> 0 but the left-hand side has a finite limit, which is a contradiction. Hence ai < 1. The proposition is now proved if n = 2. If n > 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
^(C) = - 2 R e C n + Re(aiCi + 2a2CiC2 - PC! + 2a,CiCn) + ^ Z 101' + ^(Kl'). 1
Assuming that al -{- a^ > I > af and that the equation du/d(i = f has a solution which is analytic in X near the origin, we shall prove a contradiction. Since a>2df jdQi — adf /d(2 = 0 we have dv/d(i
=0,
iiv = a2du/d(i
— adu/d(2'
(The change of coordinates was made in order to get a simple diff'erential equation here.) Hence it follows from part (ii) of Lemma 7.1.1 that for sufiiciently small ^ > 0 there is an analytic function V in Y' = {C G C " - i ; 3 C i € C,^(Ci,C') < 0, |(Ci,C')l < <5}, such that •i;(Ci,C') = V{C) if ^(Ci,C') < 0 and |(Ci,C')l < ^- For arbitrary K, A G R the set Y' contains all points in a neighborhood of 0 where 0 > ^ « 2 + AC2, C2,. • •, Cn) = - 2 Re Cn + Re (A(C') + 2a„AC2Cn) 2<j
LOCAL ANALYTIC SOLVABILITY FOR d/dz^
361
where A is an analytic quadratic form. If (7.1.12)
2aiKX + 2a2A + /^^ + A^ + 1 < 0,
then Corollary 4.1.26 gives an analytic continuation of F to a neighborhood of 0. The minimum of the left-hand side of (7.1.12) with respect to K is A^(l — a\) + 2a2A + 1. If a^ = 1 and a2 > 0, this is negative for large negative A, and if af < 1 the minimum with respect to A is +1=
l-al'
I — a\— 02
" o
l~a\
<0
by hypothesis, so we can choose K and A so that (7.1.12) holds. Hence it follows from Corollary 4.1.26 that there is an analytic function F in a neighborhood of 0 in C"^"^ which is equal to F in a neighborhood of an interval 0 < Cn < ^ on the real (^-axis. This implies that 'y(CijC') = ^(CO if ^ ( C I J C ' ) < 0 ^nd (CIJCO is small enough. Since a 7^ 0 we can find an analytic solution of the equation adU((')/d(2 = V{0 in a neighborhood of 0 and conclude that if u{() = u{() + U{('), then u is analytic and aK(C)/aCi = /(C), {a2d/dCi - ad/dC2)u{0 = 0, in X n y for some neighborhood Y of 0. Hence, for arbitrary ^1,^2, {tid/d(:i^t2d/dC2)u{0
=
{ti+t2a2/a)f{0-
If again we choose ti and ^2 so that ti = Kt2 + Xh then Re{aitl
+ 2a2tit2 - fitl) + \ti\'^ + |t2p = (2aiAcA + 2a2A + /^^ + A^ -f-1)|^2|^ +
Re{ctl)
for some c G C, so it follows from (7.1.12) that this is negative for a suitable choice of ^2- Then we obtain ^ ( r t i , r t 2 , 0 , . . . , 0,^n) < 0 if —e < r < ^, 0 < ^n :^ ^ s-nd (r, ^„) 7^ 0, provided that e is sufficiently small. Hence it follows as at the beginning of the proof that u{eti,£t2,0,...,
0, ^n) - u{-eti,
= iti+t2a2/a)
/
-et2,0,...,
0, in)
/(rti,rt2,0,...,0,^n)rfr,
0 < ^^ < ^.
Here the integrand is equal to (2^^ + 7 T ^ ) ~ ^ where 7 is not < 0 since 2^n + 7'7"^ 7^ 0 when —^ < r < ^, 0 < ^n ^ ^ and (r,^^) / 0. Hence the integral is equal to l/y/Cn times /. .^^y /7-dr/(2 + 7r^) -^ 7r/\/27 as in —^ +0, so the integral is not bounded when ^^ —?• 0 if 5 is small enough although the left-hand side has a finite limit. We can choose ti and ^2 so that ti -f a2t2/oi ^ 0, and obtain a contradiction completing the proof. Before proceeding with a discussion of the condition (7.1.10) we shall underline its importance by proving that strict inequality is a sufficient condition. To do so we need a simple and well-known version of the Morse lemma:
362
VII. CONVEXITY AND CONDITION (^)
L e m m a 7.1.5. Let g{s, t) be a C^ function in a neighborhood of (0,0) G R^"^^; assume that dg/ds = 0 and that d'^gjds^ is positive definite at (0,0). Then Q{s,t) has for small \t\ a unique minimum point s = S{t) close to 0, 11-> S{t) is in C^ and t ^ go{t) = g{S{t), t) is in C^ at 0,
^o(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 ^o(t) = g't{S{t),t) e C^ it follows that ^o ^ 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 <0 where g e C^ satisfies (7.1.9), then d/dzi is surjective on AQ{X) if
(7.1.10)'
EKi'
Proof. Since g{zi,0,... ,0) = Re{aizl) + |>2^ip + o(|2:ip), the hypotheses of Lemma 7.1.5 are fulfilled with zi as s and z' — (;22j---5^n) as t. To determine the Taylor expansion oi gQ{z2')... -^Zn) — min;^^ ^(2:1,^25 • • • ? ^n) at 0 we can ignore the error terms and just have to set ^ ^ ajZj + zi = 0 , that is, n
(1 — |ai|^)>2^i = 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 — l^ip, 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 ^o < 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^Y^ijdldzj, 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,d^ip/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, {d
= 0,
364
VII. CONVEXITY AND CONDITION (^)
provided that cp e C^ and dip/df has the direction t in a neighborhood of 0. Before continuing the discussion of surjectivity of (7.1.1) we shall also rephrase Definition 7.1.7 in terms of the boundary distance dx{z) = inf^^x k ~ CI with respect to the norm \z\ = (Xli k i P ) ^ - ^^ i^ ^ ^^ function in Y H X for some neighborhood Y of dX. The result and the proof are close to Theorems 4.1.25 and 4.1.27, and we choose a formulation close to (7.1.13)'. To simplify notation we return to d/dzi convexity. P r o p o s i t i o n 7.1.8. Let z e Y H X, and let ( be the unique point in dX such that |^ — 2:| = dx{z). Then z is on the normal of dX at (. We have ddx{z)/dzi = 0 if and only if(i = zi, which implies that d/dzi is in the complex tangent plane of dX at C,. Then we have (7.1.15)
a^ - log dx{^{r))/drdf
>0
if dX is d/dzi convex at ( as in Dehnition 7.1.7, ip G C^ near r G C, ^25 • • • 5 ^n 3.re analytic functions and ip{T) = z. Conversely, if( E dX and d/dzi is tangent to dX at (, and (7.1.15) holds when (P{T) — z is on the normal of dX at (^ close to C and (p2^- - • ^^n ^^^ complex afRne linear and cpi is real afRne linear, then dX is d/dzi convex at (. Proof. The first statement is obvious, and since ddx{z)/dz='^{z-0/\z-<:\
we have ddx{z)/dzi = 0 if and only if zi = ^i. The complex tangent plane oidX at C is defined by (^ —C? ^C) = 0? so zi = C,\ is also equivalent to d/dzi being tangential at C,. To prove the converse statement we set z = ip{T) and Z •= {w, d/dz)-\-wod/dzi, where w = dip/dr and WQ = dipi/dr. When ddx{z)/dzi = 0 we obtain as in the proof of (7.1.13)' d'^logdx{^{r))/dTdf
= ZZ log dx{z) = {ZZdx{z))/dx{z)
-
\Zdx{z)\^ldx{z)\
No second derivatives of ^p occur so in the condition (7.1.15) it does not matter if one requires (p to be affine. The second term vanishes if also {w^d/dz)dx{z) = 0, that is, {w^z - () = 0, so these conditions mean that d/dzi and {w^d/dz) are in the complex tangent plane of dX at (. When z ^^ ( it follows from (7.1.15) that ZZdx{z) < 0 at C, which means that dX is d/dzi convex at ( since —dx{z) is a defining function for X. The proof of the first part of the proposition is essentially the same as for Theorem 4.1.27 or rather Theorem 4.1.25. Write ei = ( 1 , 0 , . . . ,0) and assume that d^ log dx{z -\-rw -{- fwoei)/dTdf
= c > 0,
ddx{z)/dzi
— 0,
LOCAL ANALYTIC SOLVABILITY FOR d/dzi
365
when T = 0. Then we have by Taylor's formula for some A, 5 G C dx{z
+ ™ + ftOoei) = d^(^)eMAT+BT^)+c|x|^+o(|r|=)_
Since C, is the point in dX which is closest to z^ we have |2: — C| = dx{z)^ z\ = Ci) 3,nd d/dzi is a tangent vector at C, as observed at the beginning of the proof. Set (^(r) = z-\-TW^- fwoei + (C -
z)e'^'"^^''\
Then the triangle inequality gives dxifir))
> d;,(z)|e^-+^^'|(e'^l-l'+<'(l-l') - 1) > d x W c | r | V 2
for small r . Since dxi^piO)) = dx{C) = 0 it follows that d(p{0)/dT also lies in the tangent plane at (, and we obtain d'^dx{^{T))/drdf > 0 which contradicts the d/dzi convexity at ( since d(p(T)/df = WQCI. The proof is complete. In Proposition 7.1.8 the distance dx was taken with respect to the standard Euclidean norm (J2^ k i P ) ^ - We could equally well have used any other positively definite Hermitian form, for it can be reduced to Y^i k j P by an admissible complex linear change of variables. As a limiting case of the distance measured with respect to (7^^12:11^ + ^ 2 k i P ) ^ as if -^ 00 we obtain the distance
4 W = ,^inf
|z-C|
taken for fixed zi. It is also a C^ function oi z in X HYi where Yi is a neighborhood of the part of the boundary where dzi does not vanish in the tangent plane, hence Yi is a neighborhood of the set where d/dzi is in the complex tangent plane. We have d^ > dx, and —d]^ is also a defining function for X at dX fl Yi. In Section 7.2 we shall need the analogue of Proposition 7.1.8 for d]^. Note that if ddx{z)/dzi = 0 and C G dX^ \( — z\ = dx{z), then z is on the normal of dX at ^ and zi = (^i by Proposition 7.1.8. Since dx < d\ it follows that d\ ~ ^x vanishes of second order at 2;, so dd]^{z)/dzi = 0. Conversely, ii dd\-{z)/dzi = 0 it follows that ddx(z)/dzi = 0. In fact, then we have dx{zi -Vwi,Z2,...,Znf
> dx{z)'^
-C\wi\'^
for small wi^ which means that z-hweX
if C K P + K P + • • • + K l ^ <
d]c{zf.
If C G dX^ IC ~ '^^l = d]^{z) and Ci = ^i? i^ follows that d/dzi is in the tangent plane of X at C, so C - 2: is orthogonal to the full tangent plane of dX at ( and not only the tangent plane for fixed zi. Hence dx{z) = d]^{z) if z is sufficiently close to dX, and ddx{z)/dzi = 0 too. Thus the proof of Proposition 7.1.8 gives with no essential change:
366
Proposition in dX such that dx{^) — d}x{z), plane at (. Then (7.1.15)'
VII. CONVEXITY AND CONDITION (^)
7.1.8'. Let z e Yi H X, and let ( be the unique point d — zi and \C - z\ = d]^{z). If dd]^{z)/dzi — 0 then ddx{z)/dzi = 0, and d/dzi is in the complex tangent we have d^ - \0gd\{ip{T))/dTdf
>0
if dX is d/dzi convex at ( as in Definition 7,1.7, ip £ C^ near r £ C, (^2j • • • 5 ^n ^^e 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? • • • ? <^n ^^^ complex affine linear and ipi is real affine linear, then dX is d/dzi convex at (. When z = 0 and g has the form (7.1.9), we have also proved that (7.1.10) is equivalent to d/dzi convexity. Hence Proposition 7.1.4 and Lemma 7.1.1 mean that dX is d/dzi convex at 0 if (7.1.1) is surjective. This conclusion can be strengthened to d/dzi convexity in a full neighborhood: T h e o r e m 7.1.9. IfX C C^, n > 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'(^i,...,^n-i,2/n) =
n—1
Re{aizl-\-2^ajZiZj-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/dU^i - Cif, we obtain by solving for Zn - Cn
eiz) = RedgiO/dU^i = Qd^l
- Ci)^ + Q^z
-()
+ o{\z -
Cf)
- Cl, • • • , ^ n - l - Cn-l) + o(|zi - Cll^ + • • • + \Zn-l - C n - l P )
370
VII. CONVEXITY AND CONDITION (^)
with another quadratic form Q^ depending continuously on (. Since Qoi^i) •' • ')^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
Let e = (C2,..., Cn-i,Vn) e C^-2 X R, assume that E i < j < n 101 + l^n| < ^ and that g{yi,0) is not quasi-convex when \yi\ < 8. By Lemma 7.1.12 we can then choose a compact interval J C {—6^6) such that g{yi^O) is a constant M when yi E J and < M in a neighborhood of J with strict inequality at the end points of open intervals J^, with Ji^^i C Ji, decreasing to J . Let 7 be the arc C dX defined by {(Ci,..-,Cn);(C2,...,Cn-i,ImCn) = ^,ImCiG J,2ReCn = M,/i(C) = 0}; we may assume U so small that the equation h{() = 0 determines R e d when the other coordinates are chosen. Similarly we define the arcs j ^ , by substituting Jj^ for J . We can assume the intervals Jjy so close to J that the arcs are all contained in U HX. Our first goal is to prove that if ^ ^ 7 and /^ is defined as in (7.1.17), then the equation du/dzi = /<^ has no solution analytic in the intersection of X and a neighborhood of 7. To do so we take ^ > 0 small and consider the linear intersection Z^ = {zi; (Z1X2,...,
Cn-i, i ( M + 5) + irjn) G X, Im;2:i G J^}
= {zi'.yi =lmzi
G J^,g{yi,0)
+ h{zi,eY
<M-\-e}.
This is a connected and simply connected neighborhood of the projection of jiy in the zi plane and so close that for large u
z^ = z^ X {(C2,...,Cn-i,U^ + s) + ir,n)}
munx
if £ is sufficiently small. If the equation du/dzi = /^ has a solution in the intersection of X and a neighborhood of Z^ then we obtain by integration that the difference between the values u^_^-^{e) of u{z) at the points in Z^ with h{zi^6) = 0 and yi G dJjy+i is equal to /
/c(^i^ C2,. •., Cn-i, 1{M -\-e)+ irjn) dzi.
'^?+i(^) ^^ bounded as e -^ 0, for it is the value of i6 at a point z with g{z) = -M-e
+ giyuO) < - M + g{yi,e)
< 0.
LOCAL ANALYTIC SOLVABILITY FOR d/dzi
However, since dg{()/dCi M , we have /c(^i,C2,. • • ,Cn-i, liM
371
= —^dg/drji = 0 when C ^ 7? and since 2 Re (^ —
+ e)+ irin) = (dgiO/dCnrHe
- (z, - C i ) ' ) - \
and it follows by residue calculus that the integral is ~ 7ri{dg{()/d(n)~^^~^ which is unbounded as s —> 0. This contradiction proves our claim. Suppose now that there is no neighborhood of the origin where g is quasi-convex as a function of yi when 2^2,..., Zn-i^Vn sire fixed. If we can take J = {0} and 0 — 0 above, then we have proved that d/dzi is not surjective on Ao{X). In any case, since lack of quasi-convexity is a stable property, we can choose a sequence of disjoint arcs Y with the properties of 7 above, converging to {0} so that there is no analytic solution of the equation du/dzi = f(^ in any neighborhood of 7^ when C ^ 7^- The proof of Lemma 7.1.10 can then be applied to construct a function / analytic in [/ n X for a small neighborhood f7 of 0 so that the equation du/dzi does not have an analytic solution in any neighborhood of 0. The proof is complete. Let e, 6 be small positive numbers and set (7.1.19)
U,,s = {ze
C"; I Re^il < s,\Imzi\
< 6,\z2\ < 6,..., | z , | < 6}.
We shall use the notation 6 = (2:25 •. •, ^n-i? 2/n)- li s > C6 for a suitable C, then dh{xi H-^^/i,6)/dxi > 0 in C/^^^, and the equation h{xi -\-iyi^9) = 0 has a solution xi with \xi\ < e for arbitrary (yi^O) with \yi\ < ^, \z2\ < S, . . . , l^n-il < ^5 \yn\ < S. This implies that the fibers C C of Ue^s H X 3 z \-^ z' = {z2->"' ^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, ^i], 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 [^Oj^i] 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^^O
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 and dci^{x) < dx{x). Then dQ,^{x) z=z \x — y\ for some y E X fl dQ.f, thus F{y,t) = 0 and F{y + eFl^iy, t), t) > 0 for small e > 0. This implies y + eF^^iy, 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
VII. CONVEXITY AND CONDITION (^)
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^^e 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 ^i-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 Xs^e] for fixed yi is equal to max^^ d\{xi-\-iyi^z') close to the boundary oiX^^e. 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^^e 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
VII. CONVEXITY AND CONDITION {^)
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^-^ ^is 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 ^il'^^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.
THE SYMPLECTIC POINT OF VIEW
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~^aj: — 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'^a^(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~^aY,{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}^
^iSjk/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
VII. CONVEXITY AND CONDITION {^)
and since all terms vanish when z = 0 we obtain at (0, jdg{0)/dz) d'^A/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\'^-^o{\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 ^ith 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:<0. This symplectic version of p{d/dz) convexity is familiar from the study of (pseudo-)differential operators in a real manifold; it is the first necessary condition on the principal symbol for local solvability. We shall explain the connection in Section 7.4, but first we shall discuss the analogue of the more refined condition (^) known from that case. By Proposition 7.3.2
THE SYMPLECTIC POINT OF VIEW
379
p{dldz) convexity oi dX in an open set U means that {Rep, Impjx; < 0 in S over U when p = 0. li Q e C^ then S is a C'^"^ submanifold of T*C^, and the symplectic form is in (7^~^ in E. Thus p(C) restricts to a C^~^ function in E with Hamilton field in C^~'^. Prom now on we shall assume that A; = 3 so that HJIQP has a uniquely defined orbit through every point with i?Rep ^ 0. Since HuU — [u^ u}^, — 0 for any u^ we have R e p = 0 on the orbit of i^Rep starting at a point where Rep = 0. Thus pi^dldz) convexity means that Im ap does not change sign from — to + at a simple zero on an orbit of the Hamilton field of R e a p where Reap = 0, if a is any complex number, for the p{d/dz) convexity is not changed if p is multiplied by a complex number ^ 0. Definition 7.3.3. A smooth complex valued function p with dp ^ 0 defined in a real symplectic manifold E is said to satisfy the condition ( * ) at a point 0 G E where p = 0 if there is a neighborhood U oiO and a complex number a 7^ 0 such that cfRe(ap) 7^ 0 in [/ and Im(ap) does not change sign from — to + in the positive direction on any orbit of the Hamilton field of Re(ap) in U where Re(ap) = 0. The definition is due to Nirenberg and Treves [1], who also proved that the condition is independent of the choice of a in the sense that if it is fulfilled for one value of a then it follows for any other a with dRe(ap) ^ 0 at 6. (See also ALPDO, Lemma 26.4.10 for a result which is more precise since it takes the size of neighborhoods into account.) The core of the proof is the following lemma (see also ALPDO Lemma 26.4.11): L e m m a 7.3.4 ( B o n y - B r e z i s ) . Let f € C^(Y) where Y is an open set in R ^ , and let v be a Lipschitz continuous vector Geld in Y such that for any integral curve t \-^ y{t) of v we have (7.3.8)
/(7/(0)) < 0 = > 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
whenf = 0,
w is a positive multiple ofv when f = df = Qj
then (7.3.8) is also valid when y is an integral curve ofw. Proof. Let F be the closure of the union of all forward orbits for v starting at a point with f{y) < 0. By (7.3.8) we have / < 0 in F , and the closure of the set where / < 0 is contained in F. Orbits of v which start in F must remain in F by the stability theorems for solutions of ordinary differential equations. Let g{y) = min^^^F \y ~ ^^- By Lemma 2.1.29 we
380
VII. CONVEXITY AND CONDITION (^)
know that g is Gateau differentiable, so on an orbit y —^ y{t) oiw we obtain close to F when y{t) ^ F^ using the differential calculated in Lemma 2.1.29, lim{9{yit^s))-g{y{t))/e = mm{2{w{y{t)),yit)
-z);ze
F, \z - y{t)\^ =
g{y{t))}.
Now {w{z),y(t) -z) 0 = f{z). If d/(;2:) = 0 we have {w{z),y{t) - z) = {cv{z),y{t) - z) < 0 for some c > 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 = Im^i = —^{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.
THE SYMPLECTIC POINT OF VIEW
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-^a^{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](<,dz,)(F,do) 1
where (j = jdg/dzj. we obtain
Since Zj = Xj + iyj, j < n, and Zn = |(/i^ + ^) + Wn,
{V, dzj) = 0, 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
VII. CONVEXITY AND CONDITION (^)
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 i^o^ 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,i^o) to a conic neighborhood of (^o? 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 (^o^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,^o) ^ 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 (0,2^o) G C^ 0 C " . Since x*<^/* == ^ in a neighborhood of (0,^o) in ^^ 0 C"^, by analytic continuation, and X == Xr o -0, V'C^^C) = i^/C/i): we have x ^ = '0*X*^ = '0*icr = cr, which proves the statement. E x a m p l e . The map
{x,iO ^ (^ - m,0'KiO,
X G R",e G R" \ {0},
has a holomorphic extension to a neighborhood of N^n, and it is symplectic since n 1 n
n
1
1
It has the form discussed in Proposition 7.4.1 with X = {z ^ C"^; | Im>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^ =T^iT. 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
VII. CONVEXITY AND CONDITION (^)
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'^
THE MICROLOCAL TRANSFORMATION THEORY
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,i^o) where ^0 ^ R-^ \ {0}, and assume that dp AUJ ^ 0 at (0, i^o)? that is, that dp is not proportional to {dxj ^0) ^^ (0, ^^o)- T i e n there exists a holomorphic homogeneous canonical transformation x defined in a conic neighborhood U of (0, Z(^o) ^nd a strictly pseudo-convex real analytic hypersurface dX C C"^, (O5C0) ^ ^ax^ 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
VII. CONVEXITY AND CONDITION (^)
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
THE MICROLOCAL TRANSFORMATION THEORY
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(^o) 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{^id/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
VII. CONVEXITY AND CONDITION (^)
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 ^i (O5 ^o)j 0 7^ (^0 ^ R^j that is, assume that ifp is the real analytic principal symbol then either p(0,^o) ^ 0 or else dp is not proportional to (dx,^o) ^i (0,^o)- Then P is surjective on C(o,^o) if and only ifp{0,^o) ^ 0 or else p satisfies condition (^) in a neighborhood of (0,(^o) in T*R'^. Proof. In the non-characteristic case p(0,^o) ¥" 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^^^o) 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,^o) 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,^o) 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 ^(^^OJC) ^ 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,i^o) in T*C"^ to a neighborhood of {ZQXO) mapping a neighborhood of (0,z^o) 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,^o)j to
THE MICROLOCAL TRANSFORMATION THEORY
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 , J^i 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
VII. CONVEXITY AND CONDITION (^)
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),mh^iJ2{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) = 5Z^i(^'^)^'' ^ ^ 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 i^i,..., 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 ^i- ^^ 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^jj^i^
• • • 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 l^i, 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
=
^i{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) =
Y^ii^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
for j > 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)^A/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 To^o{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
E
q^'\B)Tj_i,,{A,B){{^dAyB)/^\A)/l\
0
= J2^^'^(B)ik + l)T^,k+i{A,B)p^^\A), k
and this equality follows from the recursion formula (B.2). Remark. Interchanging p and g, A and 5 , and reversing the order of all factors we conclude using the uniqueness that Tj^kiAjB) is obtained from Ffc j ( 5 . A) by reversing the order of the factors in each term. In particular.
398
APPENDIX
Vj^iiA, B) = {3.dAyB/j\ noted.
implies TijiA,
B) = ( - ad jB)^A/jf'! as we already
Next we shall extend the formula to the case where p{A) is a polynomial in several commuting elements Ai^... ^Am G A^ but q(B) is still a polynomial in a single element B. It suffices to examine the case where p{A)
=Pm{Am)'"Pl{Ai).
Repeated use of (B.3) gives (B.4)
p{A)q{B)
= p^{Am)
•
= Pm{Am)---P2iA2)
=
••p2iA2)pi{Ar)q{B) J ] q^''^HB)rj„kMl,
B)P[''\A^)
Y^q^'^'---^'-\B)Tj^,kJA^,B)pii-\Arr^)---r,,,kMuB)p[''\A,).
Here we can commute the operators ^2 This leads to a formula (B.3)'
p{A)q{B)
=
(^2)5... to the right using (B.l).
Y,q^'HB)T^,k{A,B)/'^\A),
valid for all polynomials p in m variables since it is valid for all monomials. Here every T^^k is a linear combination of products of commutators between B,Ai,...,Am containing in all k factors B and aj factors Aj for j = 1 , . . . , m. We shall discuss the properties of r^^^ further later on; right now the important point is that they do not depend at all on the polynomials. We shall now take the final step of letting q{B) be a polynomial in several commuting elements ^ i , . . . , Bn G A. Again it suffices to study the case of a product g(B) = g i ( B i ) . . . g n ( B , ) . By repeated use of the special case dealt with in (B.3)' we get by working p{A) through from left to right (B.5)
p{A)q{B) = ^
= p{A)qi{Bi)
• • • g„(S„)
q[^'\B,)T^^,ls,
{A, B,)qf^\B2)V^.,0,
(A, B^)
where / 3 i , . . . , /?^ are integers > 0 and a ^ , . . . , a*^ are multiindices. In (B.5) we commute the factors ^ 2 ( ^ 2 ) 5 • • • to the left, starting from the left and using a formula equivalent to (B.l),
(B.l)"
Cq{B) = X;g(^)(B)(-ad B)^C/i! 3
when B and C are single elements in A. This gives the following result:
COMMUTATOR IDENTITIES
399
P r o p o s i t i o n B . 2 . Let A i , . . . , A ^ be commuting elements in A, and let Bi,..., Bji be another commuting set of elements in A. Then we have for all polynomials p and q in m and n variables respectively {B.Sy
p{A)q{B)
== ^ g W ( 5 ) r . , ^ ( A , B ) p ( - ) ( A ) ,
where a,/? are multiindices and FQ;,/? are linear combinations of products of commutators formed from A i , . . . , Am, B i , . . . , B ^ , independent of the polynomials p and q. In every term there are altogether aj factors Aj and /3j factors Bj. One can compute Ta,/3 starting from (B.2) when m = n = 1, (B.4) and (B.l) when n = 1, and then (B.5) and (B.l)" in the general case. Tct^f3{A^B) is uniquely determined by (B.3)" The uniqueness follows as when n = m = 1. In our application the commutators formed from A i , . . . , 5 ^ belong to a commutative subalgebra ^ o of A (for example, the set of multiplication operators). The importance of a term in (B.3)" will increase with the number K, of commutator factors which occur in a term in Ta,/3- Each commutator must contain at least one Aj and one B^^ so we have K < min(|a|, |/?|), and min(|a|, |/?|) < (|a| + |^|)/2 with equality only when |a| = \/3\. Writing Tcc,^ for the sum of the terms in FQ,^^ which are products of {\a\ + \l3\)/2 commutators, we have T^^p = 0 when |a| ^ |/3|. We shall compute Ta,(3 explicitly when \a\ — |/3|; the full coefficients FQ;,^ are too complicated to write down. The reason why Fc^,/? is quite simple is that in a term in FQ,^^ which is a product of /^ = \a\ — \^\ commutators, these must all be first-order commutators [Aj,^^]. First in the case n = m = 1 we have by (B.2) with j = fc -f 1 (fc + l)ffc+i,fc+i(A, B) = Vk^k{A, B)[A, B], hence Tk^k{A^B) = [A^B]^/k\. For reasons which will be clear in the general case we sum this up in the generating function obtained by multiplication with (^T]^ where ^,77 are two indeterminates, which gives (B.6)
5 ] ^ S ' r , , , ( A , B ) = exp([eA,7/5])
in the sense that the formal expansion of the right-hand side is equal to the formal series on the left. Next we consider the case where n = \ but m is arbitrary. Then Fc^^^ is calculated from (B.4), where we can ignore commutator terms which appear
400
APPENDIX
when P2 (^2)5 • • • are moved to the right, for they involve commutators of higher order. When k = \a\ we then obtain using (B.6)
m = ra„,a„(^m,5)--.r„„„,(Ai,5) = ]][(t^J'^r7«i!)-
With ^ — (^1, • • • 5^m) a,nd a single 7/ as before, we multiply by ^^7]^ and sum, which gives m
(B.6)'
5 3 r ^ ' r „ , f e ( A , 5 ) = exp([5]Ci^„'?5]), a,fc
1
for Va,k — 0 when \a\ ^ k. We can now discuss the general case in the same way, using (B.5). The commutator terms which appear are moved left do not contribute to TQ,^^, SO we obtain
With ^ = ( ^ 1 , . . . , ^ ^ ) and ry == (771,..., 77^) we multiply by which gives
^"^T]^
and sum,
hence by (B.6)' m
n
Thus we have proved: P r o p o s i t i o n B . 3 . Assume in addition to the hypotheses of Proposition B.2 that all commutators formed from Ai,... ,Bn belong to a commutative subalgebra AQ of A. Then the coefRcients Ta,i3{A,B) in (B.3)" belong to Ao, and Ta,^{A,B) is a Rnite linear combination of products of K < {\a\ + \/3\)/2 commutators formed from Ai,..., Bn. The sum Tot,/3{A, B) of
COMMUTATOR IDENTITIES
401
the terms with K = {\a\ -h \P\)/2 is given by the formal series (B.6)''; the terms are equal to 0 except when \a\ = |/3|. When representing a polynomial p in n variables one has two main choices of notation. One can write
where a = ( a i , . . . ^a^) is an n-tuple of non-negative integers with sum |a| equal to the degree of the monomial ^" = ^^^ "'^n''0^6 calls a a multiindex, which is not really correct; one should call a a multiorder. Alternatively one can consider the polarized form of each homogeneous term in p and write where each index ii runs from 1 to n, fc = 0 , 1 , . . . is the degree of the term, and 'Pi^...Lk is symmetric in the indices. The A:-tuple ^ i . . . ^^ is a true multiindex! The connection between the two notations is that Pa — ViW^'l^^"^ / = 1 . . . 1 2 . . . 2 . . . n . . . n , where a! = ai\ • • • a^!, |Q^|! = (o^iH ho^n)'- We shall momentarily use the true multiindex notation. Let {cjk) be a positive definite ny. n Hermitian symmetric matrix and form in analogy to (B.6)" n
exp ( Y^
CjkijVk) = Y, ^3ik^ ''' ^J^kjj,
• • • ^j.Vk^ • • • Vkjj^l-
j,k=l
If t = {tii...it,)^ 1 ^ ^1 < n , . . . , 1 < Ziy < n, is a symmetric tensor of rank ly we claim that (^•^)
^ ^ Zj^^i^i
"•^3ukjji...jjki...ku
is a positive definite Hermitian symmetric form. The proof is straightforward. We can write (cjk) = $ * $ where $ is an invertible n x n matrix, and then the form (B.7) is equal to
Y^^di^^^k, • "^i.j.^i.kJn...j.hZK
= Yl i^^l-^J^ wh^^^
If ^ is the inverse of $ then
which implies that Yl\'^ji---ju\^ ^ ^YH'^h-^iul'^ ^^^ proves our claim. Switching back to the "multiorder" notation we have proved:
402
APPENDIX
Proposition B.4. Let Cjk, j,k = 1,... ,n, be a positive definite Hermitian symmetric matrix, and write n
(B.7)
exp( 5 ] Cj.CjVk) = E j,k=l
Ca^Cv^.
\a\:=\/3\
For arrays t = {ta}\a\=ti ^he form
(B.8)
t^
Yl Capiat^
is then Hermitian and positive definite. This proposition is essential when analyzing contributions coming from an application of Propositions B.2 and B.3.
NOTES
C h a p t e r I. Most of the results discussed in Sections 1.1, 1.2 and 1.5 can be found in the classical book by Hardy, Littlewood and Polya [1]. Exceptions are Exercises 1.1.10, 1.1.11 and 1.5.1 which are taken from Berezin [1]. Section 1.3 comes essentially from Mandelbrojt [1], and the characterization of the F function in Section 1.4 is due to Bohr-MoUerup [1]. Section 1.6 presents some weaker convexity notions which appear in the study of differential operators of principal type (see ALPDO, Chapter XXVI). They also occur in some of the results on convexity of minima of one-parameter families of functions due to Trepreau [2] and Ancona [1] which are presented in Section 1.7, such as Corollaries 1.7.4 and 1.7.5. C h a p t e r II. The classical text on convex bodies is Bonnesen-Fenchel [1], which in many respects is unsurpassed. Much of the material in Sections 2.1-2.3 can be found there. We have added in Section 2.1 some observations on weak sufficient conditions for convexity motivated by Nirenberg [1]. The proof of Birkhoff's theorem (Theorem 2.1.14) comes from Schaefer [1], and Muirhead's theorem (Theorem 2.1.15) can be found in Hardy-LittlewoodPolya [1]. The related Theorem 2.1.17 is due to Horn [1] where further related results can be found. (See also Kostant [1] and Section 12 in AtiyahBott [1] for a general study of convex functions in Lie algebras invariant under the adjoint representation.) The proof of Motzkin's theorem (Theorem 2.1.29) is mainly taken from Valentin [1]. The results on hyperbolic polynomials at the end of Section 2.1 are essentially due to Carding [1] (see also ALPDO, Section 12.4); special cases are due to Alexandrov [1]. The Legendre transformation was first studied in the generality of Section 2.2 by Fenchel [1, 2]; see also Rockafellar [1]. The basic facts on mixed volumes discussed in Section 2.3 can be found in Bonnesen-Fenchel [1], but not the full Fenchel-Alexandrov inequahty (Theorem 2.3.7), which we prove following Alexandrov [1] using an idea which goes back to Hilbert. The proof of Bonnesen's inequality. Theorem 2.3.8, has been taken from Burago and Zalgaller [1], which contains much more information on geometric inequalities. (The Bonnesen inequality proved here is the original version which was improved in Bonnesen [1] to the optimal L{dK)'^-4:7r^{K) > 47r(i?-r)^, with
404
NOTES
the inner and outer circles concentric. The result was extended by Fuglede [1] to the non-convex case.) The proof of the general Brunn-Minkowski inequality, Theorem 2.3.9, is due to Hadwiger (see Burago and Zalgaller [1] for detailed historical notes). Section 2.4 presents a small subset of the regularity results reviewed in Kiselman [1]. The main part of Section 2.5 is due to Kneser [1], who discussed the case of two or three dimensions. We have extended the result and modified the proofs using a 40-year-old unpublished manuscript. C h a p t e r III. For a more detailed study of harmonic and subharmonic functions than presented in Sections 3.1 and 3.2 we refer to Hayman and Kennedy [1]. Theorem 3.1.13 is somewhat related to a recent paper by Baouendi and Rothschild [1], and Theorem 3.1.14 is due to Gabriel [2]. They are used in Section 3 to prove the theorem of Gabriel [1, 2] that the level sets of Green's function G for a convex set are strictly convex. In the proof we have only used the methods of Gabriel [2] combined with a continuity argument. (A proof based on the methods of Gabriel [1] was given by Rosay and Rudin [1].) The stronger result that {—G{x,y))^^^'^~'^^ is a convex function of (rr, y) when n > 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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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 ! • • • a^i! 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
